plasmonic nanofocusing and guiding structures for nano … · guiding structures for nano-optical...

PLASMONIC NANOFOCUSING ANDGUIDING STRUCTURES FOR NANO-

OPTICAL SENSOR TECHNOLOGY

Martin Lyndon Kurth

BAppSc (QUT), MAppSc (QUT)

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Chemistry, Physics and Mechanical Engineering

Science and Engineering Faculty

Queensland University of Technology

2018

i

Keywords

diffusion, film plasmons, local field enhancement, localised surface plasmons,

nanofluidics, nanofocusing, nano-optics, plasmonic waveguides, plasmonics, surface

enhanced Raman spectroscopy, surface plasmons

ii

Abstract Surface-enhanced Raman spectroscopy (SERS) is a powerful spectroscopic

technique that allows for the label-free identification of unknown substances with

high degrees of sensitivity and specificity. These qualities make it highly suited to

applications related to national security and environmental monitoring, such as the

detection and identification of trace amounts of chemicals, drugs, and explosives in

the air. As the SERS enhancement is primarily due to the enhancement of both the

incident and Raman scattered fields by the localised surface plasmon (LSP)

resonance of a metallic nanostructure, SERS sensitivity can be increased by

increasing the local electric field.

A process known as nanofocusing, which is often achieved using tapered

metallic structures, allows for the strong localisation and enhancement of

electromagnetic energy into regions far beyond the diffraction limit of light. As a

consequence, nanofocusing offers the potential for the development of new nano-

optical sensors, high-resolution near-field imaging techniques, the delivery of

electromagnetic energy to quantum dots, and the trapping and precise manipulation

of molecules/atoms and objects at the nanoscale. Strong localisation of

electromagnetic waves can also be achieved in plasmonic waveguides, which are

used to guide light in optical circuits, but can also be used in various sensing

applications.

One of the challenges in SERS-based sensing is the delivery of molecules of

interest to the SERS hotspots. This has prompted researchers to investigate micro-

and nanofluidic systems as a method of sample delivery. By combining metallic

structures with micro/nanofluidic delivery to the field hotspots, vapour molecules of

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interest can be guided to the regions of strong field enhancement, at which point they

can be detected by spectroscopic means.

In order to lower the detection limit of a SERS-based sensor, the electric field

enhancement/localisation and/or the concentration/delivery rate of target molecules

to the sensing region needs to be increased. This need to increase the electric field

strength and/or the concentration/delivery rate of target molecules to the field

hotspots of a plasmonic nanostructure so as to lower the detection limit of a SERS-

based sensor provides the motivation for this thesis.

The thesis focuses on two main themes related to the development of new

nano-sensors which utilise strongly localised fields in plasmonic nanostructures.

The first theme is the investigation of the delivery rates of air samples that may

contain explosives, drugs, or other undesirable substances to the electromagnetic

field hotspots produced by metallic nanostructures and the methods in which the

delivery rate of the molecules can be increased to lower the detection limit. Analysis

is also undertaken into the mechanical stability of the sensing structures. The second

theme is the investigation and characterisation of two plasmonic nanostructures for

the sub-wavelength confinement of electromagnetic energy and the evaluation of

their applicability to sensing applications.

In order to address the first theme described above, an investigation into the

optimal design of a nano-optical sensor combining an array of nanoholes and

pressure-driven sample delivery is reported. The parameters for which either fluidics

or molecular diffusion is the dominant transport mechanism are determined for three

different molecules (NH3, SO2, and TNT) and the effects of increases in the pressure

gradient and chamber pressure on the delivery rates are also investigated.

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Secondly, an analysis of the increased functional capabilities resulting from the

combination of tapered metal rod nanofocusing structures with nanofluidic flows to

achieve enhancement of the detection sensitivity through the nanofluidic delivery of

the tested molecules to the plasmonic hotspots is reported. The associated drag

forces and mechanical stress in a tapered rod in air and water are also investigated

for different structural parameters, including the breakdown conditions as the

limiting acceptable flow parameters. The detection limit for a flow-through sensor

using surface-enhanced Raman spectroscopy is evaluated.

In order to address the second theme highlighted above, two different

plasmonic structures are considered. The first structure consists of a high-

permittivity dielectric wedge on a metal substrate. The geometry differs from

nanofocusing in conventional tapered metal structures, as the plasmon propagates

towards the tip in the direction of increasing dielectric wedge thickness. It is

demonstrated that the structure supports the nanofocusing of surface plasmon

polaritons with negative group velocity (negative refraction), displays the formation

of a caustic, which corresponds to the point of mutual transformation of surface

plasmon polariton modes at a critical dielectric thickness, and also displays

enhancement of the local electric field at the metal-dielectric and dielectric-air

boundaries in both the adiabatic and strongly non-adiabatic regimes.

The second structure considered in this work is an L-shaped gap surface

plasmon waveguide (L-GSPW), which consists of a dielectric strip sandwiched

between two metal films. The fabrication process is free from some of the difficulties

encountered when fabricating trench and gap plasmon waveguides. The investigation

identifies a number of desirable characteristics of the L-GSPW as an integrated

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optical component, with significant propagation distance for the guided modes, high

transmission through 90 degree bends, and low cross-talk between neighbouring

waveguides being reported, as well as characteristics that will enable it to be used in

SERS-based sensing, as well as in other sensing applications.

The findings reported in this thesis have important implications for the

development of new sensors for which strong local field enhancement is highly

desirable. Targeted delivery of samples to the sensing regions will lead to the

development of a new range of sensing platforms, or may boost the capabilities of

those that already exist. The findings related to the investigated waveguide will also

lead to the development of components for integrated optical circuits.

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Table of Contents

Keywords .................................................................................................................................. i

Abstract .................................................................................................................................... ii

Table of Contents ...................................................................................................... vi List of Figures ....................................................................................................................... viii

List of Tables ..........................................................................................................................xv

List of Abbreviations ............................................................................................................ xvi

Statement of Original Authorship ....................................................................................... xviii

List of Publications and Manuscripts .................................................................................... xix

Acknowledgements ............................................................................................................... xxi

Chapter 1: Introduction .......................................................................................1

1.1 Background .....................................................................................................................1

1.2 Research problem ...........................................................................................................4

1.3 Thesis structure and aims ................................................................................................7

Chapter 2: Literature Review ............................................................................11

2.1 Detection of drugs and explosives ................................................................................12

2.2 Raman and surface–enhanced Raman spectroscopy ....................................................17

2.3 Plasmonics ....................................................................................................................21

2.4 Structures for observing SERS .....................................................................................38

2.5 Nanoholes .....................................................................................................................44

2.6 Waveguides ..................................................................................................................49

2.7 Nanofocusing ................................................................................................................55

2.8 Fluidics .........................................................................................................................62

2.9 Molecular diffusion ......................................................................................................70

2.10 Finite Element Analysis ................................................................................................72

Chapter 3: Nanofluidic Delivery of Molecules: Integrated Plasmonic Sensing with Nanoholes ..........................................................................................................75

3.1 Introduction ..................................................................................................................75

3.2 Methodology and considered structure .........................................................................77

3.3 Results and discussion ..................................................................................................81

3.4 Conclusions ..................................................................................................................92

Chapter 4: Nanofluidics in Nanofocusing Tapered Rod Structures ..............95

4.1 Introduction ..................................................................................................................95

4.2 Maximum stress on a tapered metal rod in a viscous fluid flow ..................................97

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4.3 Nanofluidic delivery of vapour molecules to the field hotspot of a tapered nanorod .108

4.4 Conclusions ................................................................................................................111

Chapter 5: Plasmon Nanofocusing with Negative Refraction in a High-Index Dielectric Wedge..................................................................................................... 115

5.1 Introduction ................................................................................................................115

5.2 The structure and adiabatic approximation .................................................................117

5.3 Position of the plasmonic caustic ...............................................................................122

5.4 Numerical results and discussions ..............................................................................125

5.5 Conclusions ................................................................................................................135

Chapter 6: Gap Surface Plasmon Waveguides with Enhanced Integration and Functionality ................................................................................................... 137

6.1 Introduction ................................................................................................................137

6.2 Methods ......................................................................................................................139

6.3 Results and analysis ....................................................................................................142

6.4 Conclusions ................................................................................................................153

Chapter 7: Conclusion ..................................................................................... 155

References ............................................................................................................... 161

Appendices .............................................................................................................. 185

Appendix A ...........................................................................................................................185

Appendix B ...........................................................................................................................186

Appendix C ...........................................................................................................................187

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List of Figures

Figure 1.1. Some of the various applications that utilise surface plasmons. ................2

Figure 2.1. Examples of some surface and vapour explosives detection methods (adapted from [85]). .......................................................................14

Figure 2.2. The scattering process for a photon incident onto a molecule. The energy of a Rayleigh-scattered photon is equal to the energy of the incident photon, while the energies of Stokes and anti-Stokes scattered photons are lower and higher, respectively, than the energy of the incident photon. These energy differences are ± the difference between the excited and ground vibrational states. ......................................18

Figure 2.3. A semi-infinite metal-dielectric interface along which plasmons can propagate. ...............................................................................................25

Figure 2.4. A surface plasmon travelling along an interface (x,y) and decaying evanescently into both media, where αd

-1 and αm-1 are the penetration

depths into the dielectric and metal, respectively. ........................................27

Figure 2.5. Dispersion relation of SPs on a flat Ag surface, with the permittivity of Ag modelled by the Drude-Lorentz Model. c/ωp is the plasmon wavelength and εd = 2.25 is the permittivity of the dielectric material. Descriptions of the curves are provided in the text below. ...........29

Figure 2.6. Otto configuration - a laser beam is coupled into a prism at an angle θi (greater than the critical angle). Total internal reflection generates an evanescent field in the air gap, which then excites SPs at the metal-air interface. ..................................................................................30

Figure 2.7. Kretschmann configuration - a laser beam is coupled into a prism at an angle θi (greater than the critical angle). Total internal reflection generates an evanescent field which penetrates the gold film and excites SPs at the metal-air interface. ...........................................................31

Figure 2.8. Using a grating to excite surface plasmons. .............................................32

Figure 2.9. (a) a thin metal film, surrounded by two identical dielectric half spaces, that supports both a symmetric and antisymmetric plasmon mode; (b) symmetric charge distribution; (c) antisymmetric charge distribution. ...................................................................................................34

Figure 2.10. Wavenumbers of the symmetric mode and antisymmetric mode in a gold metal film surrounded by air (symmetric structure) at a wavelength of 632.8 nm as a function of film thickness. The solid curve represents the symmetric mode, while the dashed curve represents the antisymmetric mode. .............................................................35

Figure 2.11. (a) an asymmetric structure consisting of a metal film of thickness H surrounded by dielectrics with permittivities εd1 and εd2. The zero

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plane is a distance h1/2 from the upper interface and h2/2 from the lower interface; (b) two symmetric structures, for which the dispersion relations can be easily solved and then combined (adapted from [76]). .................................................................................................... 36

Figure 2.12. Wavenumbers of the quasi-symmetric mode and quasi-antisymmetric mode in a gold metal film surrounded by dielectrics with permittivities of ε1 = 1 and ε2 = 1.1 (an asymmetric structure) at a wavelength of 632.8 nm, as a function of film thickness. The solid curve represents the quasi-symmetric mode, while the dashed curve represents the quasi-antisymmetric mode. ................................................... 37

Figure 2.13. An illustration of how the polarisation of the incident field affects the charge distribution and, hence, changes the coupling strength between the two nanoparticles [177]. .......................................................... 40

Figure 2.14. Face-to-face nanojunction and edge-to-edge nanojunction, as described in [181]. ....................................................................................... 41

Figure 2.15. A nanohole array milled on a gold film deposited on a glass substrate. Here, the hole period is a0 and the hole diameter is d. ................ 46

Figure 2.16. Various waveguide structures - (a) stripe, (b) gap plasmon waveguide, (c) trench, (d) V-groove, (e) wire, and (f) wedge. .................... 50

Figure 2.17. The fundamental mode in a silver-vacuum gap plasmon waveguide structure, using the parameters from [246] (gap width = height = 100 nm and vacuum wavelength of 632.8 nm).............................. 51

Figure 2.18. Various nanofocusing structures – (a) metal wedge on a dielectric substrate, (b) metal wedge surrounded by uniform dielectric, (c) tapered metal strip with decreasing width and constant thickness, (d) tapered metal rod, (e) ‘wizard hat’, (f) tapered metal film on a dielectric hemisphere, and (g) dielectric wedge on a metal surface. ........... 58

Figure 2.19. The different regimes of fluid flow, according to the Knudsen number. ........................................................................................................ 63

Figure 2.20. The reflection of molecules from a smooth wall (specular reflection) and a rough wall (diffuse reflection). ......................................... 64

Figure 2.21. The flow of air in the x-direction between two parallel plates, separated by 1 µm, with partial-slip boundary conditions ........................... 65

Figure 2.22. The velocity profiles of air flowing between two parallel plates 1 µm apart. The dashed (blue) curve represents no-slip boundary conditions, while the solid (red) curve represents partial slip boundary conditions. The slip length is given as Ls. .................................................. 66

Figure 2.23. (a) A finite element analysis mesh and (b) a finite element analysis solution. .......................................................................................... 73

Figure 3.1. The scheme of the considered structure consisting of a nanohole of diameter d in a metal membrane surrounded by air (a) and on a dielectric substrate (b) with a concentric hole of diameter D. The

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difference between the inlet Pin and outlet Pout pressures of the air in the chamber result in a nanofluidic flow through the hole (a). .....................77

Figure 3.2. Typical distribution of the y-component of the air velocity near a nanohole with the diameter d = 500 nm in a membrane of thickness h = 50 nm, the inlet pressure Pin = 1.15 atm, and the outlet pressure Pout = 1 atm. .........................................................................................................81

Figure 3.3. (a) The air mass flow rate M through a hole as a function of the hole area, and (b) the air mass flow rate Mm per unit area of the hole, as a function of hole area at two different membrane thicknesses h = 50 nm (thick curves) and h = 250 nm (thin curves). The inlet and outlet pressures are 1.15 atm and 1 atm (dash-and-dot curves), 3.15 atm and 3 atm (dashed curves), and 5.15 atm and 5 atm (solid curves). ......82

Figure 3.4. The maximum (critical) pressure difference ∆Pmax across a SiN membrane of radius r = 5 µm and K = 0.8 as a function of its thickness hd. ..................................................................................................85

Figure 3.5. The dependences of the ratios of the mass airflow rates M through a nanohole at the inlet pressures of 5.15 atm (solid curves), 4.15 atm (dotted curves), 3.15 atm (dashed curves), and 2.15 atm (dash-and-dot curves) to the mass airflow rate M1.15 at the inlet pressure of 1.15 atm on hole area at two different membrane thicknesses of (a) 50 nm, and (b) 250 nm. The pressure variation across the membrane for all the curves is 0.15 atm. ........................................................................................86

Figure 3.6. The nanofluidic molecule delivery rates for ammonia, sulphur dioxide and TNT (curves 2, 4 and 6, respectively) compared to the diffusive delivery rates for the same molecules (curves 1, 3 and 5, respectively) for the inlet pressures of (a) 1.15 atm, (b) 3.15 atm, and (c) 5.15 atm. The concentrations of the considered molecules in the air at normal atmospheric conditions correspond to the required benchmark monitoring sensor sensitivity of 1 ng/L [364]. The pressure drop across the membrane is 0.15 atm, and the membrane thickness h = 50 nm. The circles (i), (ii), and (iii) indicate the critical hole diameters dc for ammonia, sulphur dioxide, and TNT, respectively. ..................................................................................................90

Figure 3.7. The ratios of the nanofluidic and diffusive delivery rates as functions of pressure difference ∆P across the membrane of h = 70 nm for the two different outlet pressures: (a-c) 1 atm and (d-f) 5 atm, three different hole diameters: (a,d) d = 250 nm, (b,e) d = 500 nm and (c,f) d = 1000 nm, and for the three considered types of residual gas molecules: NH3 (dotted curves), SO2 (dashed curves), and TNT (dash-and-dot curves). The straight horizontal lines show the unit ratio of the delivery rates, thus identifying the pressure differences for which the nanofluidic delivery rate is smaller or larger than the diffusive delivery rate. .................................................................................................92

Figure 4.1. (a) A nanofocusing tapered metal rod with a hotspot at the rounded tip of radius r and the taper angle q, in a fluid flow with the uniform initial velocity of U. (b) A schematic of a sensor combining

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nanofluidic delivery of a fluid sample into the gap of width W with an array of the nanofocusing ‘wizard hat’ structures [78]. ............................... 98

Figure 4.2. (a) Force distribution on a rod with a taper angle of 36 degrees, for air velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (b) Cumulative force distribution on a rod with a taper angle of 36 degrees, for air velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (c) Normalised cumulative drag force on rods with a taper angle of 5 degrees (solid curve) and 36 degrees (dashed curve) for an air velocity of 100 m/s. All rods in (a)-(c) have a tip radius of 2 nm. ........................................................................................... 103

Figure 4.3. (a) Stress distribution on a rod with a taper angle of 36 degrees and tip radius of 2 nm for air velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), in increments of 20 m/s. (b) Maximum stress on the rod at air velocity of 100 m/s as a function of taper angle, for a tip radius of 2 nm (solid curve) and 5 nm (dashed curve). The horizontal line represents the yield strength of gold. (c) Position of maximum stress on the rod as a function of taper angle, for an inlet velocity of 100 m/s and a tip radius of 2 nm (solid curve) and 5 nm (dashed curve). ........................................................................................... 104

Figure 4.4. (a) Force distribution on a rod with a taper angle of 36 degrees, for water velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (b) Cumulative force distribution on a rod with a taper angle of 36 degrees, for water velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (c) Normalised cumulative drag force on rods with a taper angle of 5 degrees (solid curve) and 36 degrees (dashed curve) for an air velocity of 100 m/s. All rods in (a)-(c) have a tip radius of 2 nm. ..................................................................... 107

Figure 4.5. Velocity of water producing a maximum stress that exceeds the yield strength of gold as a function of taper angle, for a tip radius of 2 nm (solid curve) and 5 nm (dashed curve). ................................................ 108

Figure 4.6. The sensing region at the tip of a tapered rod with a taper angle of 36 degrees and a tip radius of 5 nm. .......................................................... 109

Figure 4.7 Air velocity around the tip of a tapered rod with a taper angle of 36 degrees. ...................................................................................................... 110

Figure 5.1. (a) High-index tapered dielectric layer with the permittivity εl and taper angle α between a metal half-space with the permittivity εm, and cladding with the permittivity εc; an SPP mode with the wave vector q propagates in the direction of increasing thickness of the layer. (b) The dependences of the SPP wave number q on thickness h of the dielectric layer: (1) λ0 = 459.2 nm, εl = 9, εm = – 6.5 (e2 = 0); (2) λ0 = 632.8 nm, εl = 9.4, εm = – 9.3 (e2 = 0); (3) λ0 = 632.8 nm, εl = 9.29, εm = – 9.3 (e2 = 0); λ0 is the vacuum wavelength; the real parts of the metal permittivities correspond to silver and gold, respectively [379],

xii

and εc = 1 (vacuum). (c) A zoomed-in version of the dependences in (b). ...............................................................................................................118

Figure 5.2. (a) The dependences of the SPP wave number q on thickness h of the dielectric layer for εm = – 9.3 (e2 = 0), λ0 = 632.8 nm, εc = 1 and three different permittivities of the layer: (1) εl = 7; (2) εl = 8.5; (3) εl = 9.29 (the same as curve 3 in Figure 5.1); λ0 is the vacuum wavelength, and the real part of the metal permittivity corresponds to gold [379]. (b) A zoomed-in version of the dependences in (a). ................119

Figure 5.3. (a) The typical dependences of the magnitude of the electric field amplitude in the focused plasmon on local thickness h of the tapered layer in the adiabatic approximation for the low-q modes (curves 1a and 2a) and high-q modes (curves 1b and 2b) for the two structures with λ0 = 459.2 nm, εl = 9, e1 = – 6.5, e2 = 0, εc = 1 (curves 1a and 1b), and λ0 = 632.8nm, εl = 9.4, εm = – 9.3, e2 = 0, εc = 1 (curves 2a and 2b). (b) The dependences of the group velocity of the plasmon on thickness h of the high-index dielectric layer: (1) λ0 = 459.2 nm, εl = 9, e1 = – 6.5, e2 = 0, εc = 1; (2) λ0 = 632.8nm, εl = 9.4, εm = – 9.3, e2 = 0, εc = 1. ......................................................................................................120

Figure 5.4. The distributions of the magnitude of the electric field |E| in the structure for the different taper angles and imaginary parts of the metal permittivity: (a) α = 5o, e2 = 0.1; (b) α = 5o, e2 = 1.12 (the imaginary part of the gold permittivitty); (c) α = 40o, e2 = 1.12. The other structural parameters are: εl = 9.4; λ0 = 632.8 nm, e1 = – 9.3 (gold at the considered wavelength), εc = 1; the colour scales for |E| are in the same arbitrary units, and subplots (b) and (c) correspond to the same incident SPP amplitudes at the tip of the wedge. (d) The profile of the magnitude of the electric field along the plane at 0.1 nm above the metal-dielectric interface for α = 5o, e2 = 0.1 (corresponding to subplot (a)) ..............................................................................................126

Figure 5.5. The distributions of the z-component of the electric field Ez in the MII nanofocusing structure with the taper angle α = 20o, εl = 9.4; λ0 = 632.8 nm, e1 = – 9.3 (gold at the considered wavelength), εc = 1, and two different imaginary parts of the metal permittivity: (a) e2 = 0.1; (b) e2 = 1.12 (the imaginary part of the gold permittivitty). (c) The profile of the magnitude of the electric field along the plane at 0.1 nm above the metal-dielectric interface for α = 20o, e2 = 1.12 (corresponding to subplot (b)). ...................................................................126

Figure 5.6. (a,b) The dependences of the maximum local field enhancement for the field intensity near the caustic at the distance of 0.1 nm above: (a) the metal-wedge interface, and (b) the wedge-cladding interface for the gold-wedge-air MII structure with e1 = – 9.3, e2 = 1.12, εc = 1, λ0 = 632.8 nm, and for the three different relative permittivities of the wedge εl: 9.4 (curves 1), 8.5 (curves 2), and 7 (curves 3). (c) The dependences of the distance between the tip of the wedge and the maximum of the SPP field near the caustic on taper angle α for the

xiii

same structural structure and the same wedge permittivities: 9.4 (curve 1), 8.5 (curve 2), and 7 (curve 3). ................................................... 133

Figure 6.1. The considered L-GSPWs of width W formed by partial enclosure of a thin SiO2 film of thickness D between two gold films of thicknesses H1 (overlay) and H2 (underlay). .............................................. 138

Figure 6.2. Experimental realisation of L-GSPWs with the parameters W = 600 nm, H1 = H2 = 100 nm, D = 170 nm, and different waveguide lengths L = 7, 10, 15, 20 μm (only L-GSPWs with the three larger lengths are shown in the presented microscopic image). ............................................. 140

Figure 6.3. The fabricated L-GSPW with approximately the same parameters as those in Figure 6.2a−c but with a sharp 90 degree bend and the lengths of the bend arms La = 7 μm. .......................................................... 141

Figure 6.4. The experimentally observed output radiation from the output gratings (shown by the arrows) for the four L-GSPWs with the indicated different lengths L and the following input powers Pin: (a) 0.68, (b) 1, (c) 1.75, and (d) 4.89 mW. ...................................................... 142

Figure 6.5. The experimental (solid curve) and theoretical (dashed curve for the fundamental L-GSPW mode) dependences of the normalised power output from the output grating on distance Lp that the generated guided plasmon travels along L-GSPW at the vacuum wavelength λvac = 775 nm; the structural parameters are the same as for Figure 6.2a-c, and the coupling efficiency for the output grating is assumed to be 100%, while the theoretical and experimental coupling efficiencies for the input grating are ~4.5 and ~3.3%, respectively. The grey band shows the 90% confidence interval for the obtained experimental dependence. ................................................................................................ 142

Figure 6.6. Typical calculated distributions of the magnitude of the electric field (a,c) and the y-component of the electric field (b,d) for the fundamental (a,b) and second (c,d) L-GSPW modes. The structural parameters are the same as for Figure 6.2a−c. ........................................... 145

Figure 6.7. The theoretical dependences of the real (solid curves) and imaginary (dashed curves) parts of the effective index for the fundamental (curves 1 and 2) and second (curves 3 and 4) L-GSPW modes on waveguide width W; the other parameters being the same as for Figure 1d−f. The horizontal solid and dashed lines correspond to the real and imaginary parts, respectively, of the effective refractive index of the gap plasmon in a uniform gap (in the absence of the overlay termination). .................................................................................. 146

Figure 6.8. The image of the obtained output from the grating at the end of the second arm of the L-GSPW with a sharp bend (Figure 6.3), shown by the arrow. ................................................................................................... 148

Figure 6.9. Two closely spaced L-GSPWs in the back-to-back (a) and back-to-front (b) configurations in an integrated circuit with the width of the metal screening partition S separating them. ............................................. 149

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Figure 6.10. Distributions of the magnitude of the electric field of the fundamental L-GSPW mode for D = 170 nm, H = 100 nm, and W = (a) 300 nm, (b) 600 nm, (c) 900 nm, and (d) 1200 nm. ..............................151

Figure 6.11. The magnitude of the electric field of the fundamental L-GSPW mode in air at the SiO2-air boundary for D = 170 nm, H = 100 nm, and W = 600 nm. .........................................................................................152

Figure 6.12. Effect of reducing overlay thickness H on magnitude of the electric field of the fundamental L-GSPW mode in air for D = 170 nm, W = 600 nm, and H = (a) 100 nm, (b) 30 nm, (c) 10 nm, and (d) 5 nm. ..............................................................................................................152

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List of Tables

Table 2.1. A summary of some of the nanofocusing structures reported in the literature, as well as their configurations (IMI = insulator-metal-insulator, MIM = metal-insulator-metal, and MII = metal-insulator-insulator) [278]............................................................................................. 57

Table 2.2. Diffusion coefficients for different molecules in air at atmospheric pressure and a temperature of 293 K. .......................................................... 71

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List of Abbreviations

ARROW Anti-resonant reflecting optical waveguide

ATR Attenuated Total Reflectance

EFGF Electric Field Gradient Focusing

EIM Effective Index Method

EM Electromagnetic

EOT Extraordinary Optical Transmission

FEA Finite Element Analysis

FIB Focused Ion Beam

GOA Geometric Optics Approximation

IMI

IMS

IR

Insulator-Metal-Insulator

Ion Mobility Spectrometry

Infrared

L-GSPW L-Shaped Gap Surface Plasmon Waveguide

LIBS Laser Induced Breakdown Spectroscopy

LRSPP Long-Range Surface Plasmon Polariton

LSP Localised Surface Plasmon

MCWG Metal Clad Waveguide

MII Metal-Insulator-Insulator

MIM Metal-Insulator-Metal

MS Mass Spectrometry

NSL Nanosphere Lithography

NSOM/SNOM Near-field Scanning Optical Microscope

RM Resonant Mirror

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SAW Surface Acoustic Wave

SEIRAS Surface Enhanced Infrared Absorption Spectroscopy

SERS Surface Enhanced Raman Spectroscopy

SP Surface Plasmon

SPP Surface Plasmon Polariton

SPR Surface Plasmon Resonance

TE Transverse Electric

TEM Transmission Electron Microscope

TERS Tip Enhanced Raman Spectroscopy

TM Transverse Magnetic

TNPR Tunable Nanoplasmonic Resonator

WCRS Waveguide Confined Raman Spectroscopy

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QUT Verified Signature

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List of Publications and Manuscripts

Refereed publications

M. L. Kurth and D. K. Gramotnev, ‘Nanofluidic delivery of molecules: Integrated

plasmonic sensing with nanoholes’, Microfluidics and Nanofluidics, 14, 743-751

(2013).

D. K. Gramotnev, S. J. Tan, and M. L. Kurth, ‘Plasmon nanofocusing with negative

refraction in a high-index dielectric wedge’, Plasmonics, 9, 175-184 (2014).

D. K. Gramotnev, M. G. Nielsen, S. J. Tan, M. L. Kurth, S. I. Bozhevolnyi, ‘Gap

surface plasmon waveguides with enhanced integration and functionality’, Nano

Letters, 12, 359-363 (2012).

Conference publications and posters

M. L. Kurth, D. K. Gramotnev, ‘Increasing the sensitivity of plasmonic sensors by

means of nanofluidics’, Australasian Conference on Optics, Lasers and Spectroscopy

2009, December, Adelaide, Australia, pp. 550-551, Presentation 61.

K. Yamaguchi, M. Fujii, M. L. Kurth, S. J. Goodman, D. K. Gramotnev, P.

Fredericks, M. Fukuda, ‘The plasmonic Raman sensor using periodic nanofocusing

arrays’, SPIE Photonics West, January 2010, San Francisco, USA, Paper 7577-45.

M. L. Kurth, D. K. Gramotnev, ‘Effect of Inlet Pressure on the Delivery of Vapour

Molecules to a Nanoscale Aperture’, 19th Biannual Congress of the Australian

Institute of Physics, 5 - 9 December 2010, Melbourne, Australia.

M. Larkins, W. Olds, M. Kurth, S. Goodman, E. Tan, E. Jaatinen, D. K. Gramotnev,

‘Coupling of light into arrays of nanoholes via the Kretschmann geometry’, 19th

xx

Biannual Congress of the Australian Institute of Physics, 5 - 9 December 2010,

Melbourne, Australia.

M. L. Kurth, D. K. Gramotnev, ‘Nanofluidics versus Diffusion: Molecule Delivery

to Nanofocusing Tapered Rod’, 19th Biannual Congress of the Australian Institute of

Physics, 5 - 9 December 2010, Melbourne, Australia.

xxi

Acknowledgements

I’d like to thank my former principal supervisor, Dr Dmitri Gramotnev, for

giving me the opportunity to work on such an interesting research project and for his

support throughout my candidature. His enthusiasm for physics is inspirational.

I’d also like to thank my principal supervisor, A/Prof. Esa Jaatinen, for his

efforts in helping me to submit my thesis, my associate supervisors, A/Prof. Peter

Fredericks and Dr Thor Bostrom, for their assistance, and my former associate

supervisor, Dr Steven Goodman, for his countless words of wisdom.

I’m also extremely grateful to Dr David Pile for getting me up to speed on the

field of plasmonics and Dr Daniel Mason for his support and advice whenever I

needed a sounding board. Thanks must also go to Dr Gillian Isoardi and Dr Kristy

Vernon for their great advice and constant encouragement, Dr Konstantin Momot for

providing assistance when needed, and Dr Kai Knoerzer for his valuable modelling

advice.

I’m grateful for the help provided by Mark Barry and Ashley Wright from the

HPC group, as I wouldn’t have been able to complete this research without their

assistance.

To my past and present colleagues, of whom there are too many to name, thank

you for your friendship and support throughout my time at QUT. I know that dealing

with me can be difficult, so I appreciate the effort that you made to get me to

socialise. Special mention must go to Eugene Tan for his role in our productive

working partnership.

Finally, and most importantly, I’d like to thank my parents and sister for their

patience and unwavering support. Amid the chaos of my PhD candidature, I could

always count on them to be there for me, and for that I am eternally grateful.

1

Chapter 1: Introduction

This chapter provides a background to the research problem (Section 1.1), puts

the research problem in context (Section 1.2), and gives an overview of the reported

research and states the specific aims of the thesis (Section 1.3).

1.1 BACKGROUND

Developments in fabrication and characterisation techniques have enabled

researchers to conduct investigations at scales that were previously inaccessible. In

the 1960s, for example, lithographic techniques were only capable of minimum

feature sizes of the order of tens of micrometres [1]. However, the desire for

increased computing power has resulted in the continued miniaturisation of

electronic components. In 1965, Moore predicted a doubling in the number of

transistors on an integrated circuit every two years [2]. This trend has continued

since 1970, in part as a result of developments in fabrication techniques

Through the use of instruments and techniques such as electron beam

lithography, focused ion beam milling [3-6], and nanoimprint lithography [7-9],

researchers are now able to fabricate and observe structures at the nanoscale level.

Furthermore, the ability to use computers to model proposed structures and devices

and run simulations has allowed structural parameters to be optimised prior to

fabrication. These developments have led to major advances in a variety of research

fields, including physics, engineering, biology and chemistry.

One of the recent aims in nanophotonics research has been the replacement of

electronic-based components used in signal processing with faster, optical-based

2

analogues. This has resulted in the development of optical communications systems

based on optical fibres and photonic circuits [10]. However, as a result of the

diffraction limit, photonic devices are much larger than their electronic analogues,

which limits the miniaturisation of circuits. A possible solution to this problem is to

guide and concentrate light beyond the diffraction limit by harnessing surface

plasmons (SPs), which are surface electromagnetic (EM) waves that exist on the

interface between a metal and a dielectric. They are strongly localised at the interface

because the real parts of the permittivities of the two materials have opposite signs.

Figure 1.1 highlights some of various applications that utilise surface plasmons.

Figure 1.1. Some of the various applications that utilise surface plasmons.

Surface plasmon research, which is known as plasmonics, investigates the

propagation, localisation, and guidance of light in metallic nanostructures [11].

Research into plasmonic circuits has led to developments in areas such as sub-

wavelength waveguiding [12, 13], plasmonic switches [14], interconnects [15], and

splitters [16, 17].

Another reason for the increased interest in plasmonics research in recent years

is that the field has become more interdisciplinary and turned more towards potential

3

applications, especially for use in biological investigations [18, 19]. For instance,

gold nanoparticles have been used in labelling, delivering, heating, and sensing

applications [20]. Owing to the strong localisation at the metal-dielectric interface,

SPs are very sensitive to refractive index changes at the surface, which is the basis

for surface plasmon resonance (SPR) sensing [21-23]. This allows for label-free, real

time biosensing [24-30]. The enhanced electric fields and strong localisation

produced by SPs can also be used in spectroscopic applications such as surface-

enhanced Raman spectroscopy [31, 32] and tip-enhanced Raman spectroscopy [33,

34]. SPs also play an important role in high-resolution near-field microscopy [35-38]

and photovoltaic cells [39-42]. All of these developments have been possible, in no

small part, as a result of improvements in fabrication techniques.

Advances in fabrication techniques and microfluidics have also led to the

development of “lab-on-a-chip” devices, which have grown in popularity over the

past 20 years [43]. These devices miniaturise and integrate laboratory functions onto

a chip. Chip components may include syringes or micro-pumps to produce pressure-

driven flows, channels for fluid flow, chambers for mixing and reaction, and heaters

for reaction control [44]. There have also been efforts to integrate optical systems

into chips to allow for the trapping or manipulation of molecules [45], as well as

sensing [26].

Lab-on-a-chip devices have a number of advantages over benchtop techniques.

They give the ability to perform fast, sensitive analysis with small sample volumes,

reduce waste generation, and allow for the possibility of measurements to be made

outside of the lab environment [46, 47]. Perhaps the most common example of such a

device is the lateral flow test (e.g., home pregnancy test) [48]. Another example is a

4

point-of-care assay that has the ability to simultaneously diagnose HIV and syphilis

in remote locations [49].

These advances in nanophotonics and microfluidics will allow for the design of

new, highly sensitive optical sensing devices for the detection, identification, and

investigation of chemicals and explosives in the air.

1.2 RESEARCH PROBLEM

Combating the threat of terrorism, the desire to prevent the transportation of

illicit drugs across borders, and the need to monitor for the presence of dangerous

substances in the air are all driving forces behind the development of new sensing

techniques. Attempts to destroy aircraft in flight with explosives concealed in

clothing and footwear [50] have reinforced the vulnerability of airlines and led to

heightened security measures in airports. As a result, airline passengers have been

subjected to invasive searches and lengthy delays in an effort to minimise risk.

Therefore, there is a need for the development of sensors that can be used in airports

and combine a high degree of specificity, selectivity, and speed. Such sensors could

also be used to detect illicit drugs in airports, as well as for indoor or outdoor air

quality monitoring.

Current drug and explosive detection techniques include ion mobility

spectrometry [51-55], mass spectrometry [56-58], direct terahertz probing, and

Raman spectroscopy (see Section 2.1 for a more detailed background on detection

methods). Some of these techniques have already been implemented in detectors

specifically designed for national security purposes. Ion mobility spectrometry, for

example, is the most widely-used technique for the detection of trace levels of nitro-

organic explosives on carry-on luggage in US airports and is used widely in Australia

5

[54]. In its most common form, surfaces of interest are swabbed, the swab is inserted

into a machine in which the target molecules are ionised and the time taken for the

ions to pass through a drift chamber is used to determine the substances present.

Another type of screening device that may be used in an airport is an explosives

trace-detection portal [59]. The portal is also known as a “puffer machine” because it

uses puffs of air to dislodge potential substances of interest from the bodies and

clothing of passengers, before sampling them for analysis. Similar devices that rely

on the principles of mass spectrometry also exist. However, puffer machines have

encountered a number of problems since being deployed and many have been

withdrawn from service or remain in inventory [60]. A summary of commercially

available devices can be found in a review paper by Caygill et al. [61].

A technique that has not been explored as extensively as the aforementioned

techniques is Raman spectroscopy and its associated off-shoots. Raman spectroscopy

has been successfully used for the standoff detection of explosives [62-64], but lacks

the sensitivity required for trace detection monitoring. Surface-enhanced Raman

spectroscopy (SERS), on the other hand, offers the required sensitivity and has been

shown to have potential national security applications, with examples of the use of

SERS for the detection of explosives [43, 65-67], half-mustard agent [68], and an

anthrax biomarker [69] reported. Some of the chemical warfare agents and their

stimulants and breakdown products that have been detected using SERS, as well as

the sensitivities of different SERS substrates for these measurements, can be found in

the review paper by Hakonen et al. [70].

SERS enhancement is primarily a result of the magnification of both the

incident and Raman scattered fields by the localised plasmon resonance of metallic

6

nanostructures. These regions of high electric field intensity are known as “hotspots”

and can exist, for example, on roughened metal surfaces [31, 71], in the junctions

between closely-spaced metallic nanoparticles [72-74], or at tips of tapered metallic

focusing structures [75-78].

One of the challenges in SERS-based sensing is the delivery of the molecules

of interest to the SERS hotspots. This has prompted researchers to use microfluidic

systems as a method of sample delivery [43, 79, 80]. SERS-based fluidic devices can

be split into two broad groups – colloidal-based systems, in which suspended metal

nanoparticles in solution are manipulated, and those which incorporate metallic

nanostructures into the fluidic channels [47]. One of the reasons that fabricated

metallic nanostructures will be investigated in this work is because they can be used

without the need to implement complex flow geometries that may be required when

using colloids.

By combining metallic structures with micro/nanofluidic delivery to the field

hotspots, vapour molecules of interest can be guided to the regions of strong field

enhancement, at which point they can be detected by spectroscopic means. However,

explosives can be difficult to detect because of their low vapour pressures. For

instance, the detection of TNT in vapour phase at room temperature is difficult

because it has a low vapour pressure of approximately 2 × 10−4 torr at 25°C [81]. As

a result, detection techniques that sample air need to either sample large volumes, or

have low detection limits [82]. Therefore, the electric field enhancement and/or the

concentration/delivery rate of target molecules to the sensing region/s needs to be

increased in order to lower the detection limit of SERS-based sensors.

7

The need to investigate new platforms to achieve the field localisation required

for SERS-based sensing, in addition to evaluating existing plasmonic structures for

their suitability as optofluidic sensing platforms, provides the motivation for this

thesis.

1.3 THESIS STRUCTURE AND AIMS

The thesis consists of four separate investigations that consider various aspects

of plasmonic nano-sensor design.

Chapter 3 investigates the pressure-driven delivery of three different

residual/gas molecules of interest in air to the plasmonic field hotspots created by an

array of circular nanoscale apertures. Using finite element analysis and partial-slip

boundary conditions, first-order approximations of the nanofluidic delivery rates are

obtained and compared with delivery rates due to diffusion.

The specific aims of Chapter 3 include:

1. the calculation of the nanofluidic delivery rates of three molecules of interest:

ammonia, dioxide, and TNT, to nanoholes as a function of hole diameter,

chamber pressure, and pressure gradient.

2. the investigation of the mechanical stress on the structure and determination

of the pressure gradient that can be sustained under experimental conditions.

3. the comparison of the delivery rates of the molecules of interest to nanoholes

as a result of both fluidics and diffusion.

Chapter 4 contains the theoretical and numerical analyses of the functional

capabilities resulting from combining tapered rod nanofocusing structures with

nanofluidic flows to achieve enhancement of the detection sensitivity through the

8

nanofluidic delivery of molecules for testing to the plasmonic hotspot at the tip of the

focusing structure. The mechanical stability of the tapered rod structures and their

tolerance to the nanofluidic flows are determined for two distinctly different cases –

air and water flows. The associated drag forces and mechanical stress are determined

and investigated for different structural parameters, including the breakdown

conditions as the limiting acceptable flow parameters. The detection limit for such a

flow-through sensor is also evaluated.


1. to investigate the dependence of parameters such as taper angle, tip radius,

and fluid velocity on the mechanical stability of the metallic nanofocusing

structures and the induced mechanical stress and drag forces in both air and

water.

2. to investigate the rates of delivery of molecules in air to the hotspots created

by nanofocusing tapered rods.

Chapter 5 contains the analysis of nanofocusing by a dielectric wedge on a

metal substrate in both adiabatic and non-adiabatic regimes, for the cases where the

dielectric permittivity of the wedge is smaller or larger than the magnitude of the real

part of the metal permittivity. This analysis is based on the geometric optics

approximation (GOA), but also employs the finite element method (COMSOL

MultiphysicsTM), since the GOA is not applicable at the caustic, nor for larger taper

angles. Initially, the case of zero or very small dissipation in the metal is considered,

followed by the derivation of analytical expressions for the position of the plasmonic

caustic and the corresponding plasmon wavenumber under some general

assumptions, which are then compared with numerically determined values. Next,

9

SPP nanofocusing is considered in the case of non-zero metal permittivity in both the

adiabatic and the non-adiabatic regime and, finally, field enhancements are

calculated in order to evaluate the dielectric wedge for use in SERS-based sensing

applications.


1. to investigate and characterise SPP nanofocusing in a high-index dielectric

wedge, particularly for the case of εd > |εm|.

2. to numerically determine the effect of parameters such as material

permittivities and taper angle of the wedge on the position of the caustic and

derive an analytical expression for the position under some general

assumptions.

3. to determine the maximum electric field enhancement and its position at both

the Si-metal and Si-air interfaces as a function of taper angle in order to

evaluate the structure’s possible applicability to SERS-based sensing.

Chapter 6 proposes and analyses numerically and experimentally an L-shaped

gap surface plasmon waveguide (L-GSPW) consisting of a dielectric strip

sandwiched between two metal films. The structure resembles a trench waveguide

[83] because of the high aspect ratio. As the gap width is determined by the thickness

of the deposited dielectric layer, it avoids the problems associated with the FIB

milling or lithographic fabrication of high aspect ratio rectangular slots. Finite-

element frequency domain calculations were conducted using COMSOL

MultiphysicsTM. The experimental investigations detailed in this chapter, which

include the fabrication of straight and sharply-bent waveguides, as well as the

determination of propagation distances and quantification of energy transmission

10

through a sharp bend, were conducted by the group led by Professor Sergey

Bozhevolnyi at the University of Southern Denmark.


1. the theoretical investigation of the proposed L-shaped gap surface plasmon

waveguide and the comparison of the findings with experimental

observations.

2. the determination of the cut-off structural parameters, if any, of the guided

modes of the waveguide.

3. the evaluation of the suitability of the L-GSPW for use in sensing

applications.

11

Chapter 2: Literature Review

In Section 2.1 of the thesis, some of the methods currently employed in the

detection of drugs, explosives, and other chemicals of regulatory interest are

considered. The relative pros and cons of the various detection methods in a transport

hub, such as an airport, are discussed. The importance of factors such as speed,

specificity, and selectivity are highlighted.

After concluding that SERS is a viable candidate for a new generation of

highly-sensitive sensing devices, Section 2.2 recounts the development of SERS and

explains the mechanisms behind the SERS enhancement factor.

In Section 2.3, key concepts in plasmonics are highlighted. The dispersion

relation for a plasmon on a metal-dielectric interface is derived from Maxwell’s

equations and the techniques used to generate plasmons are discussed.

Various SERS substrates, their enhancement factors, and fabrication methods

are described in Section 2.4. The benefits of using ordered structures for SERS

investigations are highlighted, as control over the physical parameters during the

fabrication process results in a greater probability of reproducible results.

Section 2.5 contains a comprehensive review of the uses of nanohole arrays in

the field of optics. The review starts with Bethe’s theory of diffraction by small

holes, continuing on to Ebbesen’s observation of the extraordinary transmission of

light through subwavelength holes, and finishes with the current research. The

conditions required for the generation of plasmons in nanohole arrays are described

and the effect of parameters such as film thickness, hole diameter, choice of metal,

and probe chemical are listed.

12

In Section 2.6, different types of waveguides are highlighted and their

applicability in sensing applications is discussed.

The concept of nanofocusing is introduced in Section 2.7 and the process by

which light can be focused beyond its diffraction limit is described. Various

nanofocusing structures are also discussed.

After highlighting the problems associated with delivering molecules into the

hotspots produced by nanofocusing structures, Section 2.8 introduces the mechanism

of fluidics as a possible method by which the delivery of target molecules to electric

field hotspots can be controlled. The Navier-Stokes equations are introduced and the

different flow regimes are highlighted. The partial-slip boundary conditions are

explained and their applicability is discussed.

In Section 2.9, the transport phenomenon of diffusion is introduced and the

basic equations that characterise diffusive motion are given.

The final section of the literature review, Section 2.10, provides an overview of

the numerical method used to solve the differential equations encountered throughout

the project. The benefits of the finite element method over other numerical methods

are highlighted.

2.1 DETECTION OF DRUGS AND EXPLOSIVES

The threat of terrorism, the desire to prevent the transportation of illicit drugs

across borders, and the need to detect the presence of potentially dangerous

chemicals in the air continues to drive the development of techniques for detecting

these undesirable substances. There are a number of techniques currently available

for the detection of such substances, each with their own pros and cons. Golightly et

13

al. [84] identified a number of factors that make the detection of analytes a challenge

in homeland security applications, including:

1. substances with a broad range of molecular weights need to be detected,

2. not all of the tested substances may be in a database,

3. there needs to be a very low error rate, since errors could have deadly

consequences, and

4. there may be a need for continuous monitoring.

According to Moore [82], these detection methods can be subdivided into three

groups:

1. those that detect vapours or particles emitted from the materials;

2. those that detect dissolved or suspended solids in solutions; and

3. those that probe solid materials.

Methods that detect particles or vapours are the most appropriate for

monitoring in an airport setting where factors such as sensitivity, specificity, and

speed are all of vital importance. Sensitivity is required to detect trace amounts of

explosives, which often have low vapour pressures. Specificity is necessary in order

to reduce the occurrence of false-positives, which disrupt the screening process. Fast

screening times are needed in order to screen sufficient numbers of passengers in a

timely manner, so as to prevent long queues. Figure 2.1 shows some of the various

techniques that can be used to detect explosives on surfaces and in the air.

14

Figure 2.1. Examples of some surface and vapour explosives detection methods (adapted from [85]).

Sniffer dogs have been shown to be extremely proficient in the detection of

trace amounts of explosives [86, 87]. Dogs are fast and offer real-time detection, but

their biggest drawback is that they can only work for a limited number of hours each

day and need regular breaks to recover. Large numbers of dogs and handlers are also

expensive to train and maintain.

Ion mobility spectrometry (IMS) is the most commonly used technology for

the detection of trace amounts of nitro-organic explosives on passengers and their

hand luggage in US airports [54], but can also be used for the detection of narcotics

in airports [52], as well as chemical warfare agents [55]. A surface of interest is first

swabbed and the sample is vapourised through heating. The vapours are ionised at

atmospheric pressure and then introduced into a drift tube which contains a buffer

gas that opposes the motion of the ions while an electric field is applied. The drift

time, which is the time taken for ions to travel the length of the drift tube, is a

function of charge, mass, and the size of the ion, with the drift time being

characteristic of different ions [61]. However, two downsides of IMS are its low

resolving power and limited selectivity [88].

15

Mass spectrometry (MS) is a technique that uses the mass-to-charge (m/e) ratio

of charged particles to separate and analyse substances such as explosives and drugs

[89]. The sample first undergoes vapourisation and is then ionised. Electric or

magnetic fields within the analyser then separate the ions based on their mass-to-

charge ratio. Time of flight analysers are based on the principle that ions having

different m/e ratios have different flight times and are therefore collected one after

the other. Other methods, such as the quadrupole mass analyser and ion trap, employ

the geometric separation method, where ions having different m/e ratios are

separated according to their position at the collection spot [90]. MS has been shown

to be a sensitive and selective tool for the detection of various explosives [58].

Surface acoustic wave (SAW) devices are thin film polymer-coated

piezoelectric crystals with characteristic acoustic resonant frequencies. When target

molecules from a sample are adsorbed on the film, the resonant frequency changes,

which indicates their presence. SAW sensors have been shown to detect trace

amounts of explosives [91-94] and for the real-time vapour phase detection of

cocaine molecules through the use of surface acoustic wave (SAW) quartz resonator

devices, which are designed for vapour phase detection [95].

Direct terahertz (THz) probing is another technique that can be used for the

detection of illicit drugs and explosives. THz radiation lies in the far IR region (from

0.1 to 10 THz), with waves in this range able to penetrate many non-polar dielectric

materials such as clothing materials, cardboard, and plastics. Most explosives and

explosive related compounds have spectral fingerprints in this range [61], which

makes THz spectroscopy well suited to the detection of explosives. In this region of

the spectrum, the signatures are low-frequency motions between molecules in the

16

crystalline structure and also within the large molecules themselves [96]. As a

detector of illicit drugs, THz spectroscopy has been shown to be able to identify

substances in their real-world ‘street’ forms (which may include ‘cutting agents’ and

other impurities), and not just as pure crystalline materials [97]. This technique has

been shown to detect drugs such as cocaine, MDMA (ecstasy), and diamorphine

(heroin) [97]. In addition, integrating an optoelectronic THz micro source into a

glass-substrated microchip within the near-field distance enables the non-invasive

and sensitive detection of biomolecules [98].

Laser-induced breakdown spectroscopy (LIBS) uses a high-intensity laser to

break down the sample into plasma, with this plasma then emitting light with

characteristic frequencies that can then be detected with a spectrometer. LIBS is able

to provide quick, multi-element analysis of bulk samples (solids, liquids, gases, and

aerosols) in the parts-per-million range [99]. One of the drawbacks of this technique

is that the laser pulse could potentially initiate bulk amounts of sensitive explosives

[63].

The final detection technique to be considered is surface-enhanced Raman

spectroscopy. SERS is highly applicable to national security applications for a

number of reasons. SERS enables chemical fingerprinting on the molecular level

[100], as well as multiplexing without spectral overlap, which allows multiple

species to be detected simultaneously [84]. This is possible because SERS provides

spectra with narrow bandwidth [101]. SERS has been used extensively for the

detection of explosives [102-109] and the development of handheld SERS-based

detectors for virus and explosives detection [110, 111] means that SERS can meet

17

point-of-use requirements [84]. A table summarising detection limits of SERS for a

range of explosives can be found in a review paper by Hakonen et al. [70].

The follow section addresses Raman scattering and the way in which surface-

enhanced Raman spectroscopy greatly enhances Raman signals.

2.2 RAMAN AND SURFACE–ENHANCED RAMAN SPECTROSCOPY

Raman scattering is the inelastic scattering of visible, near-infrared, or near-

ultraviolet photons on the vibrational and/or rotational modes of molecules. When a

photon hits a molecule, energy from its vibrational and/or rotational states may be

exchanged and a photon of either higher or lower energy emitted. The energy

difference is equal to the vibrational energy Evib of the molecule and/or the rotational

energy Erot of the molecule. If monochromatic light is used for the excitation, the

energy difference can be recorded as a vibrational, rotational, or rotational–

vibrational Raman spectrum [112].

Raman spectroscopy can be explained using the classical EM field description.

When a material interacts with an oscillating EM field, the dipole moment that is

induced can be expressed as:

Ep α= (2.2.1)

where α is the polarisability of the material and E is the oscillating electric field

that is interacting with the material. If the vibrational/rotational motion of the

molecule is sinusoidal in nature, then the dipole moment can be expressed as:

( ) ( ) ( )( )ttQ

QEtEp ininin ω−ω+ω+ω

∂

α∂+ωα= coscos

2cos maxmax

max0 (2.2.2)

where α0 is the polarisability of the molecule at its equilibrium position, Emax

is the maximum electric field amplitude, ωin is the angular frequency of the incident

18

photon, Qmax is the maximum vibrational amplitude of the molecule, ω is the angular

frequency of the scattered photon, and ∂α/∂Q is the change of polarisability with the

change in position.

The first term corresponds to Rayleigh scattering, since the scattered photon is

not frequency shifted. The first cosine in the second term relates to the scattering that

occurs when photons interact with molecules that are in their vibrational (or

rotational) excited state. In this case, the photons gain energy and will be blue

shifted, in what is known as anti-Stokes scattering. The second cosine in the second

term relates to the scattering that occurs when photons interact with molecules in

their vibrational or rotational ground state. In this case, the photons lose energy and

are red shifted, which is known as Stokes scattering.

It can be seen from Eq. (2.2.2) that in order for Raman scattering to occur,

there must be a change in polarisability during the molecular vibration. That is:

0≠∂

α∂Q

(2.2.3)

If this condition is not met, the second term in Eq. (2.2.2) disappears and Rayleigh

scattering occurs. The three types of scattering are illustrated in Figure 2.2.

Figure 2.2. The scattering process for a photon incident onto a molecule. The energy of a Rayleigh-scattered photon is equal to the energy of the incident photon, while the energies of Stokes and anti-Stokes scattered photons are lower and higher, respectively, than the energy of the incident photon. These energy differences are ± the difference between the excited and ground vibrational states.

19

However, a major drawback of Raman is that it is an extremely weak process,

with molecular cross-sections in the order of 10-30 – 10-25 cm2/molecule [32]. This

precludes the use of Raman in the detection of trace amounts of substances. A good

fluorescence cross-section, by comparison, is of the order of 10-17 - 10-16 cm2 [113].

Due to the non-elastic electron relaxation to the lower edge of the excited level, a

fluorescence spectrum is usually quite broad. On the other hand, detailed analysis of

the molecule under investigation is possible with Raman spectroscopy as a result of

the Raman transitions being much sharper [114]. There was a need to increase these

cross-sections and thus the technique known as surface-enhanced Raman

spectroscopy was developed.

In 1974, Fleischmann et al. [115] observed an extremely strong Raman signal

from pyridine adsorbed on an electrochemically roughened silver electrode.

However, they incorrectly attributed this significant signal strength to a large

increase in the electrode’s surface area, with its resultant increase in the number of

absorbed molecules sampled. In 1977, separate research groups [116, 117] concluded

that the increase in the observed Raman signal was far in excess of the increase in the

number of molecules interrogated as a result of the surface roughness. In 1978, it was

proposed that the excitation of surface plasmons was responsible for the dramatic

increase in the Raman cross-section [71, 118]. This proposal sparked an interest in

this phenomenon, which became known as surface-enhanced Raman scattering. After

a further 20 years of research, a second wave of interest occurred in 1997 after

Kneipp et al. [100] and Nie et al. [119] reported enhancements that were sufficiently

large to obtain the Raman spectrum of single molecules.

20

The general consensus is that two mechanisms are responsible for producing

enhanced Raman from molecules on a metal surface: electromagnetic enhancement

[31] and chemical enhancement [120]. Electromagnetic enhancement, which is the

dominant contributor towards the overall enhancement, involves the magnification of

both the incident and Raman scattered fields by the localised surface plasmon (LSP)

resonance of the metallic nanostructure. The chemical enhancement is a product of

the shifting or broadening of the electronic states of the analyte molecule because of

either their interaction with the metal or the origin of new electronic states [121]. The

polarisability of the molecule is enhanced as a result of charge transfer between the

metal and adsorbate [122].

The overall SERS enhancement is the product of the incident electric field

intensity enhancement and scattered electric field intensity enhancement by:

( ) ( )( ) ( )

2

000

0

s

slocallocalSERS EE

EEIωω

ωω∝ (2.2.4)

where Elocal(ω0) is the magnitude of the electric field at the excitation

frequency at the site of Raman enhancement, Elocal(ωs) is the magnitude of the local

electric field of the Raman shifted frequency, E0(ω0) is the magnitude of the incident

electric field at the excitation frequency, and E0(ωs) is the magnitude of the incident

electric field at the Raman shifted frequency [31, 123].

If the Stokes shift is small, there is minimal difference between the incident

and scattered wavelengths and the enhancement factor scales as [(E(ω0)]4 [124]. This

relationship demonstrates the need to realise SERS substrates that maximise the local

electric field enhancement.

21

SERS spectra have been obtained from adsorbates deposited on a variety of

metals. At first, only noble metals such as Ag, Au, and Cu (known as Group 11

metals) were used [125]. However, researchers have since turned their attention to

non-Group 11 metals as well, with nanostructures also fabricated using transition

metals (including Pt [126], Fe [127], and Co [128]) and metal oxides [129-131] now

reported in the literature. SERS signals from non-Group 11 nanostructures are lower

than those from noble metals, but such materials also offer some potential

advantages, including the ease of control of important parameters such as band gap,

phonons coupling strength, and surface morphologies [132].

The substrates responsible for the sensitivity achievable through surface-

enhanced Raman spectroscopy will be examined in Section 2.4.

As has been discussed, surface plasmons are largely responsible for the

enhancement observed in SERS experiments, as a result of the excitation of localised

surface plasmons. Therefore, this review will now focus on the field of plasmonics.

2.3 PLASMONICS

2.3.1 Background

Surface plasmons (SPs), also known as surface plasmon polaritons (SPPs)

[133], are charge density oscillations coupled to EM waves that propagate along a

metal-dielectric interface. The field amplitude of an SP is a maximum at the

interface, with the field decaying exponentially into both media. This strong

localisation of the field at the interface has resulted in SPs finding many applications,

as was highlighted in the introduction.

Wood was the first to observe SP phenomena in 1902 [134]. He illuminated a

ruled metal grating with an incandescent lamp and observed sharp and narrow bright

22

and dark bands in the reflected light. This behaviour was only observed when the

incident light was polarised with the magnetic field component parallel to the grating

grooves. Since he had great difficulty in explaining the reason for the large change in

intensity for only a small change in wavelength, he described them as anomalies.

Lord Rayleigh carried on the work from Wood [135, 136], yet there were

discrepancies between his predictions and Wood’s experimental data. It was Fano

who produced the first theoretical breakthrough in 1941 by identifying two types of

anomalies – a sharp anomaly (edge of intensity), which agreed with Rayleigh’s

calculations, and a diffuse anomaly associated with the excitation of surface waves

(surface plasmons), which agreed with Wood’s observations [137]. Pines and Bohm

authored a series of papers [138-140] on the effects of electrons passing through a

metal film and proposed that energy losses were a result of the excitation of

conducting electrons creating plasma oscillations. Ritchie extended the work to thin

metallic films and predicted the existence of self-sustained collective oscillations at

metal surfaces (surface plasmons) that could be excited by fast-moving charged

particles [141]. Powell and Swan then verified the existence of surface plasmons

through a series of electron energy loss experiments [142, 143]. In the late 1960s, the

optical excitation of surface plasmons was demonstrated [144, 145] using techniques

which will be described in Section 2.3.6.

2.3.2 Dielectric function of a free electron gas

The free electrons that are abundant in metals provide the negative permittivity

required for surface plasmons to exist [146]. The dielectric response of a metal can

be described using the Drude-Lorentz Model, with the dielectric function of the

electron gas given as:

23

( )γω−ω

ω−=ωε

ip

2

2

1 (2.3.1)

where ωp is the bulk plasma frequency (the frequency of bulk longitudinal electron

excitations) and γ is the damping frequency. In the case of weak damping (where γ

→ 0), this equation simplifies to:

( ) 2

2

1ωω

−=ωε p (2.3.2)

Likewise, the relative permeability µr describes the magnetic response of a

material to an applied magnetic field. However, since this work is limited to linear,

isotropic and non-magnetic materials, µr reduces to 1 [114].

The choice of metal used in plasmonic applications depends on the nature of

the application, the incident wavelength, and the available budget. Silver displays the

lowest losses for both visible and near infrared wavelengths, but it can suffer from

corrosion if exposed to the environment [146, 147]. Gold suffers from greater losses

than silver and is vastly more expensive, but it does have superior chemical stability,

and is thus less prone to environmental degradation. Copper has been investigated as

a cheaper alternative to gold but, like silver, it degrades when exposed to the

environment [146].

2.3.3 Maxwell’s equations and boundary conditions

Maxwell’s equations can provide a starting point for the analysis of the

properties of SPs. These equations can be expressed in the following form:

ρ=∇ D. (2.3.3)

0. =∇ B (2.3.4)

24

t∂∂

−=×∇BE (2.3.5)

t∂∂

+=×∇DJH (2.3.6)

where E is the electric field, D is the electric displacement, H is the magnetic

field, B is the magnetic flux density, ρ is the density of free charge, and J is the

density of the free currents.

Furthermore,

ED rεε0= (2.3.7)

HB rµµ0= (2.3.8)

where εr and µr are the relative permittivity and relative permeability.

The equations at the metal-dielectric interface must satisfy the following

boundary conditions:

|||| dm EE = (2.3.9)

⊥⊥ = ddmm EE εε (2.3.10)

|||| dm BB = (2.3.11)

⊥⊥ = dm BB (2.3.12)

where the subscripts ⊥ and || denote the field components normal to the

interface and tangential to the interface, respectively.

25

2.3.4 Surface plasmons on a planar dielectric-metal interface

Let us consider the semi-infinite metal-dielectric interface shown in Figure 2.3.

Figure 2.3. A semi-infinite metal-dielectric interface along which plasmons can propagate.

Metal occupies the region z < 0 and the region z > 0 is dielectric. We first

consider a TM wave propagating along the interface in the x-direction. For this

polarisation, the magnetic field is perpendicular to the plane of incidence and the

electric field is parallel to the incident wave plane. Therefore, a TM wave has a

magnetic field component Hy and electric field components of Ex and Ez.

Eq. (2.3.10) reveals that the normal component of E (Ez) must be discontinuous

across the boundary in order to satisfy the boundary condition. This discontinuity

results in polarisation changes at the interface, which creates time-dependent

polarisation charge at the interface [148]. TM waves fulfil all of the boundary

conditions and can therefore propagate along the interface.

TE waves, on the other hand, have the electric field parallel to the plane of

incidence, which means that they have B components in the x and z direction and an

E component in the y direction. Since the presence of a normal component of the E

field is required to produce surface charges, surface modes do not exist for TE

polarisation. Therefore, we neglect any further consideration of TE waves and will

focus solely of waves with TM polarisation.

Next, we consider the Helmholtz equation, which is given by:

26

02

22 =+∇ EE εω

c (2.3.13)

The solutions to Eq. (2.3.13) are sought in the form of a wave that propagates

along the interface in the x-direction, while exponentially decaying into the ± z-

directions:

( ) { }ztiiqxEEE dzdxdd α−ω−= exp,0, ,, for z > 0 (2.3.14)

( ) { }ztiiqxEEE mzmxmm α+ω−= exp,0, ,, for z < 0 (2.3.15)

where Ex and Ez are the x- and z-components of the electric field, respectively, c is

the speed of light, ω is the angular frequency, and αd and αm are the reciprocal

penetration depths into the dielectric and metal, respectively.

The reciprocal penetration depths, which are the distances from the interface at

which the fields drop to 1/e [149], can be found by substituting Eq. (2.3.14) and

(2.3.15) into Eq. (2.3.13) to give:

dd cq εωα 2

22 −= (2.3.16)

mm cq εωα 2

22 −= (2.3.17)

The penetration depths of a surface plasmon into a dielectric and metal are

shown in Figure 2.4.

27

Figure 2.4. A surface plasmon travelling along an interface (x,y) and decaying evanescently into both media, where αd

-1 and αm-1 are the penetration depths into the dielectric and metal, respectively.

In the absence of dissipation, the SP will propagate an infinitely long distance

on a smooth film. However, as a result of free-electron and interband damping, SPs

have a decay length, known as the propagation constant, which is determined by the

imaginary part of the SP wavenumber and given by [114, 150]:

( )qLsp Im2

1= (2.3.18)

This length, which is the propagation distance before the intensity of the

plasmon drops by 1/e, is typically between 10 and 100 µm for optical frequencies

[114]. If the surface isn’t smooth, scattering can cause the field amplitude to

decrease, which shortens the predicted propagation length. SPs can be scattered into

other SPs (in-plane scattering) or into light (out-of-plane scattering) [133]. The losses

associated with this scattering can be of the same order as the ohmic losses in the

case of as-deposited rough Ag films, for example [151].

2.3.5 The dispersion relation of surface plasmons

The dispersion relation of the SP can be derived by applying Eq. (2.3.3) to Eq.

(2.3.14) and (2.3.15) and taking ρ = 0, as a consequence of the absence of charge in

either half-space. This results in:

28

0,, =− zddxd EiqE α for z > 0 (2.3.19)

0,, =+ zmmxm EiqE α for z < 0 (2.3.20)

The application of the boundary conditions 2.3.9 and 2.3.10 necessitates that at

z = 0:

xmxd EE ,, = (2.3.21)

zmmzdd EE ,, εε = (2.3.22)

Inserting Eq. (2.3.19) and (2.3.20) into Eq. (2.3.21) and (2.3.22) gives:

d

d

m

m

εα

εα

=− (2.3.23)

This relationship gives us the first requirement for the existence of a plasmon -

εm and εd must have opposite signs. Since the dielectric can only take positive values,

it means that the metal must have a negative permittivity. Secondly, q needs to be

positive for the mode to propagate, which means that |εm| must be > εd.

Inserting Eq. (2.3.16) and (2.3.17) into Eq. (2.3.23) gives:

md

md

cq

ε+εεεω

= (2.3.24)

Since εm is complex, q must also be complex.

Looking at Figure 2.5, it can be seen that the SP dispersion relation (curve 1)

lies to the right of the light line (line 2), meaning that the SPs have a longer wave

vector than light of the same energy [133].

29

Figure 2.5. Dispersion relation of SPs on a flat Ag surface, with the permittivity of Ag modelled by the Drude-Lorentz Model. c/ωp is the plasmon wavelength and εd = 2.25 is the permittivity of the

dielectric material. Descriptions of the curves are provided in the text below.

Curve 1 in Figure 2.5 shows the dispersion relation of SPs on a metal-

dielectric interface, as was described earlier. The metal (silver), with a bulk plasma

frequency of 11.9989 x 1015 s-1, has the frequency-dependent permittivity described

by the Drude-Lorentz model, while the dielectric has a permittivity of 2.25. Note

how the curve approaches, but doesn’t cross, curve 2 as the wavenumber decreases.

Curve 2 is the dispersion curve of the bulk wave in the dielectric medium (also

known as the dielectric light line). At higher frequencies, curve 1 approaches curve

3, which represents the SP condition (ωp/(εd+1)1/2).

When the frequency of an external EM field is larger than the bulk plasma

frequency, the metal becomes transparent, which means that the EM field will be

able to propagate inside the metal. This is the case indicated by curve 4.

At low frequencies, where the dispersion curve is close to the dielectric light

line, the transverse component is dominant in the electric field of SPs, while the

30

longitudinal component only becomes comparable with the transverse component at

very large wavenumbers [150].

As a consequence of the SP line lying to the right of the light line, an SP cannot

simply be excited by shining light onto a metal surface because the wave vector of

the incident bulk radiation is too short to satisfy Snell’s law. This also means that

SPs cannot directly radiate into light. Therefore, special techniques must be used to

increase the wave vector of the bulk radiation, thereby enabling phase-matching

[133].

2.3.6 Prism coupling

One way to excite SPs with photons is through the use of a prism and a

mechanism known as attenuated total reflectance (ATR). The Otto configuration and

Kretschmann configuration, which are two geometries used for prism coupling, are

shown in Figure 2.6 and Figure 2.7, respectively.

Figure 2.6. Otto configuration - a laser beam is coupled into a prism at an angle θi (greater than the critical angle). Total internal reflection generates an evanescent field in the air gap, which then excites

SPs at the metal-air interface.

In the Otto configuration, a bulk wave that is incident onto the prism at an

angle greater than the critical angle is reflected from the prism-air interface. This

total internal reflection generates an evanescent field in the air gap. The air gap, if

sufficiently small, will act as a tunnel barrier and allow the excitation of SPs on the

metal-dielectric interface [148]. By adjusting the angle of incidence until a dip in a

31

plot of reflected wave intensity versus angle of incidence is observed, the

wavenumber of the incident wave parallel to the interface can be matched to the

wavenumber of the SPs. However, there are some impracticalities associated with

this geometry, which has limited its widespread use. Both the prism and metal need

to be clean and smooth to achieve efficient coupling and there needs to be careful

control over the size of the air gap and the orientation of the prism with respect to the

metal surface [148].

Figure 2.7. Kretschmann configuration - a laser beam is coupled into a prism at an angle θi (greater than the critical angle). Total internal reflection generates an evanescent field which penetrates the

gold film and excites SPs at the metal-air interface.

The Kretschmann configuration, which is also based on ATR, doesn’t suffer

from the issues described above. It consists of a thin metal film deposited on the base

of the prism and, as is the case with the Otto configuration, an evanescent wave is

produced by total internal reflection. SPs cannot be generated at the prism-metal

interface because the wave vector of SPs at this interface is larger than the photon

wave vector in the prism, for all angles of incidence [150]. However, if the metal is

sufficiently thin, it can act as an evanescent tunnel barrier and SPs can be generated

on the outer surface of the metal. This requirement that the film should be thinner

than the penetration depth of the evanescent wave places some limitations on the

suitability of the Kretschmann configuration. Therefore, other methods of SP

excitation need to be considered for optically thick films and non-planar surfaces.

32

2.3.7 Grating coupling

Gratings can be milled or etched into the surface of the metal, or deposited on

the metal. The translational invariance of the interface is broken by the grooves in the

grating, which can change qx [148]. In this configuration, the EM wave is incident

onto a grating with a period a0 at an angle qi.

Figure 2.8. Using a grating to excite surface plasmons.

Phase matching will occur when the following condition is met:

00

2sina

mkq ixπq ±= (2.3.25)

where a0 is the grating period and m is some integer value.

The critical amplitude of the grating can be determined by monitoring the

intensity of the reflected wave. The strongest coupling occurs when the intensity is

observed to drop to zero [133]. This concept can also be extended to 2D metallic

structures, such as arrays of nanoholes (as will be discussed in Section 2.5).

Having control over the grating period and angle of incidence makes this

method of SP excitation much more flexible than prism coupling, as gratings can

provide a greater range of wavenumbers. Furthermore, there is no need to consider

the film thickness or air gap spacing.

33

Through the use of circular gratings, surface plasmons can be made to

converge to a point, by what is known as a plasmonic lens [152, 153]. SPs can be

excited using either linearly [152-154] or radially [155] polarised light. Using an

annular ring milled in a 150 nm thick Ag layer on glass, it was demonstrated that

radially polarised light can produce a much sharper spot at the focal point than is

achievable with light that is linearly polarised [155].

2.3.8 Surface plasmons in multilayer planar structures

Having considered SPs propagating along a flat, semi-infinite metal-dielectric

interface, consideration should also be given to SP propagation in multi-layer

structures. The two main geometries for three-layered structures are the metal-

insulator-metal (MIM) geometry and the insulator-metal-insulator (IMI) geometry.

The characteristics of a three-layered IMI geometry will be discussed here.

A symmetric IMI geometry consists of a thin metal film sandwiched between

two insulators that have the same permittivity, while an asymmetric structure

consists of a thin metal film sandwiched between two dielectrics with different

permittivities. When the metal film is sufficiently thin, the SPs on the two interfaces

are able to couple and form new modes [156].

The terminology used to describe these two modes varies from author to

author, so care must be taken when interpreting results. Authors may define the

modes as symmetric/antisymmetric with respect to charge distribution [76, 157],

symmetric/antisymmetric with respect to field distribution [77], long/short-range

SPPs [156, 158] and even/odd modes [159].

Symmetric (with respect to charge distribution) plasmons are characterised by

an antisymmetric magnetic field profile across the film, meaning that the magnetic

34

field is zero in the middle of the film. Conversely, antisymmetric (with respect to

charge distribution) plasmons have a symmetric field distribution which means that

there is a non-zero magnetic field in the middle of the film [160]. These two modes

are illustrated in Figure 2.9.

For the purpose of this analysis, it will be assumed that the structure is infinite

in the x- and z- direction and that the plasmon propagates in the x-direction, as was

the case in the example in Section 2.3.4. In this scenario, the magnetic field profiles

and charge distributions are as follows:

Figure 2.9. (a) a thin metal film, surrounded by two identical dielectric half spaces, that supports both a symmetric and antisymmetric plasmon mode; (b) symmetric charge distribution; (c) antisymmetric

charge distribution.

The dependences of the wavenumbers of the symmetric and antisymmetric

mode on the thickness of the metal film are shown in Figure 2.10. The wavenumber

of the antisymmetric mode decreases with decreasing film thickness and approaches

that of bulk waves in the surrounding dielectric. Since a small fraction of the energy

is confined in the metal, this mode can have a long propagation length, which is why

they are often referred to as long-range SPPs. Due to the low degree of field

localisation, this mode is of less interest in SERS-based sensing applications. On the

35

other hand, the wavenumber of the symmetric mode increases rapidly with

decreasing film thickness, which means that it becomes strongly localised and,

conversely, its propagation length decreases. Symmetric film plasmons can

experience nanofocusing (see Section 2.7) and are therefore highly relevant to SERS-

based sensing applications. Symmetric modes have higher frequencies than the

respective frequencies for single-interface SPs, while antisymmetric modes have

lower frequencies [114].

Figure 2.10. Wavenumbers of the symmetric mode and antisymmetric mode in a gold metal film surrounded by air (symmetric structure) at a wavelength of 632.8 nm as a function of film thickness. The solid curve represents the symmetric mode, while the dashed curve represents the antisymmetric

mode.

The dispersion relations for the symmetric and antisymmetric plasmon modes,

respectively, can be expressed as:

( )md

dmmh

εαεα

−=αtanh (2.3.26)

( )dm

mdmh

εαεα

−=αtanh (2.3.27)

When an IMI structure is asymmetric (the metal is sandwiched between two

different dielectrics), purely symmetric or antisymmetric plasmons no longer exist

36

because the wavenumbers of the plasmons at the two interfaces are no longer equal.

A technique known as the zero-plane method [161] can be used to derive the

dispersion relations for the two plasmon modes in an asymmetric IMI structure. For

an asymmetric structure, and unlike a symmetric structure, the magnetic field of

symmetric film plasmon is non-zero in the middle of the metal film. However, there

still exists a plane at which the magnetic field is zero and this plane is termed the

‘zero plane’. The asymmetric structure, with a film thickness given as H, can be split

into two symmetric structures, each consisting of a metal film (of thickness h1 and

h2) surrounded by the two dielectric half-spaces.

Figure 2.11. (a) an asymmetric structure consisting of a metal film of thickness H surrounded by dielectrics with permittivities εd1 and εd2. The zero plane is a distance h1/2 from the upper interface

and h2/2 from the lower interface; (b) two symmetric structures, for which the dispersion relations can be easily solved and then combined (adapted from [76]).

If ( ) 2/21 hhH += , the dispersion relations for the symmetric and

antisymmetric plasmon modes, respectively, are given as:

( )mdi

dimimh

εαεα

−=αtanh (2.3.28)

( )dim

mdiimh

εαεα

−=αtanh (2.3.29)

37

where (i = 1,2), αm and αdi are the reciprocal penetration depths in the metal

and dielectrics, respectively, εm and εdi are the permittivities of the metal and

dielectrics, respectively, and hi is the thickness of the metal films.

Values for h1 and h2 are determined so that the above equations give the same

values of q. The dispersion relations for the quasi-symmetric and quasi-

antisymmetric plasmon modes, respectively, can therefore be expressed as:

−+

−= −−

md

dm

md

dm

m

hεα

εαεαεα

α 2

21

1

11 tanhtanh1(2.3.30)

−+

−= −−

2

21

1

11 tanhtanh1

dm

md

dm

md

m

hεαεα

εαεα

α(2.3.31)

The dependences of the wavenumbers of the quasi-symmetric and quasi-

antisymmetric mode on the thickness of the metal film are shown in Figure 2.12.

Figure 2.12. Wavenumbers of the quasi-symmetric mode and quasi-antisymmetric mode in a gold metal film surrounded by dielectrics with permittivities of ε1 = 1 and ε2 = 1.1 (an asymmetric

structure) at a wavelength of 632.8 nm, as a function of film thickness. The solid curve represents the quasi-symmetric mode, while the dashed curve represents the quasi-antisymmetric mode.

38

The quasi-antisymmetric mode displays a cut-off thickness of hc ~ 50 nm in the

considered structure. It will be a bound mode for h > hc and a leaky mode for h < hc,

since the wavenumber of the plasmon at hc is equal to that of a bulk wave in ε1 [156].

Having covered the basics of plasmonics, attention is now turned to metallic

substrates that can be used for surface-enhanced Raman spectroscopy.

2.4 STRUCTURES FOR OBSERVING SERS

A number of different strategies for the generation of plasmonic hotspots

(regions of high electric field) for surface-enhanced Raman spectroscopy have been

considered. They include roughened surfaces [115, 127, 128], metal colloids in

suspension or aggregation [121, 162, 163], and various tailored structures with nano-

scale features [164-166]. The fabrication of nanostructures can be separated into two

broad categories: ‘top-down’ and ‘bottom-up’ approaches. Top-down fabrication

methods involve starting with bulk material and using techniques such as focused ion

beam (FIB) milling or electron beam (e-beam) lithography, as well as various other

lithographic and etching methods, to remove material and thereby produce the

desired nano-scale structures [3-6, 167]. Such techniques allow for precise control

over the position and size of the nano-scale features, but can be time- and labour-

intensive processes and may require expensive equipment [168]. Bottom-up

fabrication methods involve adding atoms or molecules to build the desired

nanostructure from the base up. Such methods include chemical aggregation and

self-assembly [162, 163, 169, 170]. However, these techniques can suffer from a lack

of control over the positioning of features. The review article by Jahn et al. [171]

provides an excellent summary of the different methods by which plasmonic

nanostructures for SERS applications can be fabricated.

39

One of the most desirable attributes for SERS-active substrates is the ability to

give reproducible and reliable results. Factors such as non-uniform surface roughness

and variations in colloid size and distribution on the surface make quantitative

analysis with SERS a challenge [112]. There are obviously many factors to be

considered when selecting an appropriate SERS substrate. Below are descriptions of

some of the many substrates used in surface-enhanced Raman spectroscopy. It

should be noted that different research groups calculate enhancement factors in

different ways, so care should be taken when comparing results reported by different

groups [172, 173].

Spherical gold and silver nanoparticles have been studied extensively over a

number of years. Although Raman enhancement is possible from individual metal

nanoparticles, significantly higher signals are achievable from the junctions between

pairs, clusters, and even aggregate films of nanoparticles [174]. At these hotspots, the

electric field is strongly enhanced and highly localised. The gap between two

closely-spaced nanoparticles can produce a hotspot as a result of coupling between

plasmon modes [175]. It has been concluded that nearest-neighbour coupling, rather

than radiative or long-range coupling, is responsible for this field localisation [73].

Closely-spaced, coupled metallic nanoparticle pairs are known as plasmonic

‘dimers’. They exhibit strong coupling and are red-shifted when illuminated by light

polarised along the interparticle axis, while minimal coupling and blue-shifting is

observed for light polarised orthogonal to this axis [176]. This is illustrated in Figure

2.13.

40

Figure 2.13. An illustration of how the polarisation of the incident field affects the charge distribution and, hence, changes the coupling strength between the two nanoparticles [177].

Nanocubes have also been shown to enhance Raman signals as a result of

highly localised fields made possible by their equidistant sharp edges [178]. They

can be synthesised by seed-mediated growth, with this process capable of tunable

and highly-controllable edge lengths [74]. The Raman signal is enhanced when the

incident field is polarised along one of the edges of the nanocube. A single Ag

nanocube is capable of single molecule detection if its sharp corners are coupled with

the layer of Ag or Au on which it is deposited [179]. Rabin and Lee [180]

experimented with various configurations of Ag nanocube clusters on a silicon

substrate and concluded that closely-packed, linear, face-to-face clusters of

nanocubes should be avoided in order to achieve better sensitivity. Gao et al. [181]

showed that polymer-grafted metal nanocubes can be self-assembled to form

nanojunctions and that the edge-to-edge geometry produces the strongest field

localisation in the interparticle gap.

41

Figure 2.14. Face-to-face nanojunction and edge-to-edge nanojunction, as described in [181].

Bharadwaj et al. define the optical antenna as ‘a device designed to efficiently

convert free-propagating optical radiation to localised energy, and vice versa’ [182].

The interaction of an antenna with incoming EM waves results in a highly localised

field which can be used for spectroscopy. As is the case with nanocubes, the

localised optical fields are located at the sharp corners of the structure, but the

fabrication process can be quite different. Schuck et al. used electron beam

lithography to fabricate gold bowties antennas, which are essentially two metallic

nanotriangles orientated tip to tip and separated by a small gap [183]. An |E|2

enhancement of > 103 was observed for a gap width of 16 nm. By fabricating gold

bowties on top of silicon posts, Hatab et al. were able to achieve maximum SERS

enhancements of 2 × 1011 from isolated bowties and 7 × 1011 from low density

bowtie arrays when the maximum separation was ~8 nm [184]. Observations made

by Fromm et al. [185] implied SERS chemical enhancement of > 107 from p-

mercaptoaniline molecules on a single gold bowtie, which was attributed to charge

transfer between the molecule and the Au film.

However, it can be difficult to bring two nanoparticles together to form a gap

with a controlled distance of a few nanometres. Therefore, there are some limitations

surrounding the use of coupled nanoparticles for reproducible SERS measurements.

42

Nanorods are another structure capable of SERS enhancement. Nanorods are

capable of supporting two plasmon modes, which depend on the polarisation of the

incident field. Light polarised along the long axis results in a long wavelength

plasmon (longitudinal mode), while light polarised orthogonal to the long axis results

in a short wavelength plasmon (transverse mode) [186]. When two nanorods are

placed in close proximity on a substrate, they can couple, with the coupling strength

dependent on both the polarisation and the orientation of the two rods with respect to

each other [187]. Shi et al. [188] investigated gold nanorods with different aspect

ratios and found that the enhancement factor decreases with increasing aspect ratio.

De Angelis et al. [189] investigated SERS with the super-hydrophobic delivery

of molecules to four different plasmonic structures – a micropillar array, a

micropillar array topped with Ag nanoparticles, a micropillar array in which one of

the pillars was replaced with a nanocone, and a nanocone and circular grating. The

electric field enhancement from the cone with the grating was calculated to be

approximately 35, while the Raman enhancement was 1.5 × 106.

Banaee and Crozier [164] compared gold nanorings, nanoring dimers, and

nanodisk dimers, with observed SERS enhancement of 4.2 × 106, 3.9 × 106, and 5.8 ×

105, respectively. Their simulations predicted that ring dimers would produce a

greater field enhancement than rings, which led them to suggest that chemical

enhancement or surface roughness may have contributed to the SERS enhancement

observed from the rings.

Film over nanosphere substrates [190-192] consist of metal films deposited

over nanospheres of a material such as SiO2 [191] or polystyrene [193]. Lin et al.

[194] measured the maximum enhancement factor for Ag deposited on polystyrene

43

nanospheres with diameters of approximately 1000 nm to be 4.3 × 106 at an

excitation wavelength of 532 nm, while the LSPR was excited at 550 nm. Modelling

showed that the electric field between two nearby spherical nanoparticles is

amplified when the incident wavelength approaches half of the nanosphere diameter.

Linn et al. [165] achieved SERS enhancements of up to 108 from an array of

polymer nanopyramids with sharp tips covered with a gold film with a thickness of

30 nm. Changing the film thickness changed the sharpness of the tips, which resulted

in a change in the enhancement. Jeon et al. [195] were able to achieve an

enhancement factor of 4.2 × 107 from dual length-scale Ag nanotip arrays.

Increasing the SF6 etching time from 3 min to 4 min decreased the enhancement

factor because the structures changed from triangular pyramids to a conical shape.

Gold-coated polymer positive and inverted pyramid array structures have also been

used as SERS substrates [196]. Using rhodamine 6G as the target analyte,

enhancement factors over the entire surface were measured to be 7.2 × 104 for

positive pyramids and 1.6 × 106 for negative pyramids.

Using multilayered particles consisting of alternating layers of silver and

dielectric disks, tunable nanoplasmonic resonators (TNPRs) (Ag/SiO2/Ag/SiO2)

stacked vertically can produce a SERS EF of (6.1 ± 0.3) × 1010 from a single TNPR

[197].

Jackson and Halas [198] investigated Ag and Au nanoshell films and were able

to achieve SERS enhancements of up to 1010 from Ag nanoshells with silica cores.

Simulations suggest that roughening the nanoshell contributes less than an order of

magnitude to the total SERS enhancement.

44

Chou et al. [199] fabricated a pulsed laser-nanostructured sapphire surface with

an Au coating for use as a SERS substrate for the detection of nitroaromatic vapours.

The field producing the Raman signal is a combination of the incident field, the field

reflected from the air-gold interface, and the field reflected from the substrate-gold

interface, and is therefore a function of film thickness. By comparing signals from

structured and unstructured parts of the sapphire surface, it was shown that 2,4-DNT

was only able to be identified because of the increase in enhancement produced by

the structured surface.

It can be seen that a wide variety of structures have been fabricated and used to

obtain SERS measurements, with a range of fabrication techniques being employed.

Possibly the least-complex periodic structure to fabricate is an array of holes in a

metallic film, which will now be considered.

2.5 NANOHOLES

Nanohole arrays are of interest in sensing applications because of the enhanced

EM fields generated around the edges of the holes, their sensitivity to surface

dielectric changes, and their ease of fabrication. The setup is easier to align than

reflective mode SPR, since experiments can be conducted at normal incidence.

Nanohole array sensors can have a much smaller active area than reflective mode

SPR sensors, which allows for greater miniaturisation and integration into lab-on-a-

chip devices [200]. By adjusting the parameters such as hole shape and period,

interesting optical effects can be observed, which means that arrays can be easily

tuned for specific purposes [201, 202].

Nanohole arrays reported in the literature have commonly been fabricated

using techniques such as e-beam lithography [167, 203] and FIB milling [27, 204].

45

However, these processes are highly impractical for the production of large arrays

because of cost and time factors, so techniques such as nanoimprint lithography may

be more preferable for such tasks [205]. A summary of fabrication techniques for

nanohole arrays can be found elsewhere [206].

For the purpose of this review, only cylindrical apertures will be considered.

However, it should be noted that investigations have also been conducted into a

variety of apertures, including rectangular [207-209] and cross-shaped apertures

[210-212]. The review article by Gordon et al. [213] provides a good overview of

these different aperture shapes.

The properties of nanohole arrays can be understood by first considering the

transmission of light through a single sub-wavelength circular hole. Bethe’s solution

for the transmission of normally incident light through a single sub-wavelength

circular hole in a conducting screen [214] is given by the equation:

( )2

4

2764

πkrT = (2.5.1)

where r is the radius of the hole and k = 2π/λ is the wave vector of the incident

light of wavelength λ.

Since T ∝ (r/λ)4, the transmission would be expected to drop rapidly as the

wavelength increases. Additionally, if the real depth of the hole is taken into account,

the transmission efficiency is further reduced [215].

However, by patterning a gold film with a regular, periodic array of sub-

wavelength holes, the transmission of light can be increased by orders of magnitude.

Therefore, peaks with transmission efficiencies greater than one can be observed,

even when the individual holes are of such a size that they do not allow the

46

propagation of light [207]. This phenomenon, known as extraordinary optical

transmission (EOT), was first observed by Ebbesen and colleagues in 1998 [216] and

has resulted in significant interest in the physics behind this phenomenon and its

potential applications. Nanohole arrays have since found applications in areas such

as polarisation control [217], filtering [207], second harmonic generation [218], and

surface plasmon resonance sensing [27, 219, 220]. In addition, nanohole arrays have

been shown to be viable SERS substrates, with SERS signals observed from various

substances deposited on nanohole arrays in Au [202, 221-227], Ag [228-231], Cu

[232], and Al [233].

Figure 2.15. A nanohole array milled on a gold film deposited on a glass substrate. Here, the hole period is a0 and the hole diameter is d.

The EOT process can be split into three basic steps – first, light is coupled to

SPs on the incident surface, it is then transmitted through the holes to the second

surface, and finally, re-emitted from the second surface. SP modes can be supported

on both sides of the film, with the difference in εd of the materials in direct contact

with the metal determining the offset of the modes [207].

EOT through nanohole arrays provides an ideal platform for surface plasmon

resonance sensing. Brolo et al. [27] observed a red shift in the wavelength of

47

maximum transmission when they modified the surface of a gold nanohole array

with an ethanoic solution of mercaptoundecanoic acid and a further wavelength shift

when the surface was modified by the protein BSA. This transmission peak shift is

similar to the absorption-induced shift in the angle of minimum reflectivity observed

in SPR sensing experiments with gold-coated prisms in the Kretschmann

configuration [234]. However, as was previously mentioned, one of the downsides of

the Kretschmann configuration is that it is bulky, which inhibits miniaturisation of

the sensor. Sensors based on nanohole arrays, on the other hand, do not suffer from

this problem because the periodicity of the array satisfies the phase-matching

condition for normally incident light. This is evident from Eq. ((2.3.25)).

At normal incidence, combining Eq. (2.3.24) and (2.3.25) gives [235]:

( )2

1

22

0max ,

ε+ε

εε

+=λ

dm

dm

ji

aji (2.5.2)

where a0 is the grating period, i and j are integers, εm is the permittivity of the

metal, and εd is the permittivity of the surrounding medium.

The minima occur at a shorter wavelength than the maxima:

( ) dji

aji ε

+=λ

22

0min , (2.5.3)

Lee et al. [236] fabricated a large (30 µm × 30 µm) array of 300 nm diameter

holes in silver and obtained SERS spectra using three different excitation

wavelengths (532 nm, 633 nm, and 752 nm). A maximum enhancement factor of 8 ×

105 was achieved when the plasmon resonance was nearly centred between the

excitation wavelength of 633 nm and the Stokes scattered wavelength. Therefore,

even though the Raman scattering should have been more efficient with the 532 nm

48

excitation source because it was a shorter wavelength, the enhancement factors were

lower because it was off resonance.

For a given wavelength, Eq. ((2.5.2)) will give a value for the periodicity that is

slightly larger than the experimentally obtained value because of coupling between

propagating surface plasmons and localised surface plasmon resonances [28]. This is

also the reason why the position and intensity of the transmission maxima varies with

hole diameter [221].

In order to achieve significant resonances, the film thickness should be at least

comparable to the finite penetration depth of the EM field into the metal [237].

Degiron et al. found two hole depth-dependent transmission regimes for silver sub-

wavelength cylindrical hole arrays. The SPs on the two surfaces are uncoupled for

holes in thick films, with the transmission rising exponentially as hole depth

decreases. The maximal transmission intensity levels off due to SP coupling, which

results in a broadening of the peak [238].

Since it is evident that parameter changes can have a significant effect on the

transmission of light through sub-wavelength holes, there has been a great deal of

research into the effects of parameter changes such as hole diameter, period, depth,

and shape on transmission spectra, as well as the effect on the electric field and

SERS enhancement.

Brolo et al. [202] obtained SERS signals from oxazine 720 molecules absorbed

on arrays of nanoholes of different periodicities in a gold film, with the overall

intensity of the signal from the absorbed molecule dependent on the period of the

array. The array that produced the largest transmission factor at the excitation

wavelength also produced the largest enhancement.

49

For a fixed array period, the SERS intensity was found to increase with

increasing hole diameter, while for a fixed diameter, the intensity of the SERS

spectra increased as the period of the array was decreased [223]. The decrease in

SERS intensity with increasing array period is due to the red shifting of the

transmission spectra, since Eq. (2.5.2) doesn’t take into account the perturbation of

the SP modes, as has been previously highlighted. A maximum enhancement factor

of 4.2 × 105 was obtained from an array with the diameter of 370 nm, a period of 500

nm, and a hole depth of 550 nm.

Investigations were carried out on nanohole arrays in a silver film to evaluate

the absolute SERS enhancement factors as a function of lattice spacing [230]. Using

514.5 nm light as an excitation source, the enhancement was found to be (6 ± 3) ×

107. However, it was determined that a majority of the enhancement (a factor of

about 105) was due to the roughness of the film, while localised plasmons near the

edges of the apertures had a much less significant contribution (a factor of about 6 ×

102) on the enhancement.

While it is clear that nanohole arrays provide an ideal platform for SERS-based

sensing, simulations have shown only moderate field enhancement from nanohole

arrays [230]. Therefore, attention now turns to other structures which may enable

high localisation of the electric field.

2.6 WAVEGUIDES

2.6.1 Introduction to waveguides

As was discussed in the introduction, one of the driving factors behind

plasmonics research has been a desire to miniaturise components for use in optical

circuits. To this end, there is a need to investigate structures that can be used for the

50

guiding of SP modes. Early studies attempted to guide waves using metallic

nanoparticle chains, but propagation lengths were limited to less than 1 µm because

of strong attenuation [239]. Since then, a variety of waveguiding nanostructures have

been analysed both numerically and experimentally. These include stripes, slots and

trenches, V-grooves, and wedges. Examples of some of these structures are shown in

Figure 2.16, although the characteristics of each structure will not be discussed.

Figure 2.16. Various waveguide structures - (a) stripe, (b) gap plasmon waveguide, (c) trench, (d) V-groove, (e) wire, and (f) wedge.

A simple waveguide to fabricate is a metal stripe surrounded by an infinite

homogeneous dielectric, with stripe waveguides having been investigated both

numerically and experimentally [240-242]. Since stripe SP modes are very close to

the light line, they are prone to radiation losses [243]. Furthermore, the lateral

confinement in stripe waveguides is diffraction limited, which limits their application

potential [244, 245].

51

If a second metal stripe is added in parallel, a gap plasmon waveguide (also

known as a slot waveguide) is formed. The gap may either be partially or fully filled

with a dielectric. In such a structure, SPs may couple across the gap and result in

what is known as a gap plasmon [246, 247]. The field localisation in the gap takes

the form of two coupled fields near the metal edges. The electric field distribution for

the fundamental mode in a gap plasmon waveguide is shown in Figure 2.17.

Figure 2.17. The fundamental mode in a silver-vacuum gap plasmon waveguide structure, using the parameters from [246] (gap width = height = 100 nm and vacuum wavelength of 632.8 nm).

Slot waveguides are capable of achieving good field confinement and long

propagation distances, while supporting a single mode [245]. Pile et al. found that,

rather than being related to gap plasmons, the fundamental GPW mode is actually

related to the four corner wedge plasmons that propagate along the edges of the gap.

The GPW may support an additional mode if the width is reduced and the height

increased. In addition, increasing both the width and the height results in a decrease

in the localisation, which results in an increase in the propagation length [246].

A similar waveguiding structure is a trench waveguide, which can be formed

by increasing the thickness of the parallel strips. This increase in thickness results in

a significant change in the modal characteristics because of the presence of the

vertical edges. For a 400 nm thick gold layer at an excitation wavelength of 1.55 µm,

52

the TM polarised SPs detach from the upper and lower surface of the metal. Instead,

two coupled SPs supported by the sidewalls are produced and are the source of the

main field lobe [248].

However, waveguides are not only used in optical circuits, with a range of

waveguide-based sensors reported in the literature. Note that the following section

does not consider waveguides that are tapered in nature, as they will be considered in

Section 2.7.

2.6.2 Waveguides for use in sensing

The fields of a guided mode in the cladding layer of a waveguide are a typical

evanescent wave that can be utilised for sensing purposes. Specific biomolecular

interactions occur when the target analyte is immobilised at the surface of a

waveguide, which results in a change in the refractive index of the cladding. The

guided mode will feel this change, which can then be measured by using either

interference or resonance [249].

A number of waveguide-based interferometric biosensors have been developed

that show impressive detection capabilities. Such interferometers include the Mach-

Zehnder interferometer [250, 251], Young’s interferometer [252-255], and the

Hartman interferometer [256]. The intensity output of a Mach-Zehnder

interferometer, for example, changes when a refractive index change of a sample in

one of the arms alters the phase of the evanescent field of a guided wave, which

results in a change in the output interference pattern from the two arms. A long

interaction length between the sample and guided wave may be required to ensure

sensitivity of such sensors [21].

53

Conventional slab waveguide sensors consist of a high-index core sandwiched

between a substrate and the analyte which is to be analysed. A series of total internal

reflections (TIRs) at the film–substrate and film–cladding interfaces guide light

through the core. At each point of total internal reflection along the propagation

direction, an evanescent wave is formed. This wave decays exponentially from the

interface and acts as a probe that is responsible for sensing changes in the refractive

index of the analyte that is under investigation [257].

The resonant mirror (RM) waveguide design utilises the leaky mode at the

waveguide-substrate boundary to probe refractive index changes at the sensing

surface. The configuration is similar to that of the Kretschmann geometry (see

Section 2.3.6), but a high-index dielectric replaces the metal film, and it is separated

from the prism by a low-index dielectric [258]. The structure forms a resonant cavity,

which enhances the evanescent field. As a result, RM sensing is more sensitive than

SPR sensing.

Metal clad waveguides (MCWG) [259-262] may consist of a silica sol-gel

[259] or low refractive index gel (e.g. agarose) [262] deposited on a titanium-coated

glass slide. The deposition of the buffer layer on top of the metal extends the

evanescent field into the sensing region, which gives the MCWG greater penetration

than SPR and RM sensors [259].

The reverse symmetry waveguide is characterised by a nanoporous silica layer

(n = 1.19) that is introduced as the lower cladding layer, as opposed to the use of a

material with a refractive index that is greater than that of the solution which is being

interrogated. Consequently, more light will be concentrated towards the sensing

surface, which will increase the sensitivity of the sensor [263, 264].

54

Anti-resonant reflecting optical waveguides (ARROW) consist of a low-index

core embedded in a high-index ARROW layer [265]. This differs from TIR

waveguides, which feature a high-index core material surrounded by a low-index

cladding [266]. As liquid in a microchannel can be used as a low-index core, such

waveguides can be used as sensors in optofluidic systems [267-269].

Gu et al. [270] combined an optical waveguide with metallic nanoparticles to

concentrate and enhance the local EM field for SERS excitation. This waveguide-

assisted nanoparticle substrate produced an average SERS EF for 4-MBA of 1.5 ×

108. Fu et al. [271] used Ag nanoparticles in the evanescent field produced by a

nanoporous dielectric waveguide for SERS. Wang et al. [272] investigated, both

numerically and experimentally, the coupling of the leaky mode resonance of a

polyimide waveguide with the plasmon resonance of Ag nanoparticles to achieve

highly-sensitive evanescent field-excited SERS.

Tang et al. [273] theoretically analysed a SERS sensor that combined a gold

slot waveguide and a Si3N4 strip waveguide, which resulted in a design that produces

SERS signals and extracts these scattering signals on a chip. This resulted in an EF

of 102 – 103.

Although waveguides can provide a flexible sensing platform, there is usually a

trade-off between propagation distance and electric field localisation. In order to

maximise the field localisation, nanofocusing structures will be considered, as they

are designed to focus light into small volumes which can then be used as SERS

hotspots.

55

2.7 NANOFOCUSING

The diffraction process hinders the localisation of light beyond a scale of

~λ/2n, where λ is the wavelength of the light and n is the refractive index of the

surrounding medium [114]. This means that the intensity of an electromagnetic wave

at a given power can only be increased to a certain level by conventional means.

However, a process known as nanofocusing (also known as superfocusing) allows

light to be focused down to nanoscale regions as small as a few nanometres [11].

Nerkarayan showed theoretically that as a surface polariton propagates through a

slender wedge, its wavelength infinitely deceases as it approaches the tip of the

structure. As a result, there is a dramatic increase in the strength of the electric field

[274].

It is important to draw a distinction between nanofocusing and the localisation

of optical energy and enhancement of optical fields within nanoscale spatial regions

that can be achieved using nano-optical antennas [275], as was seen when discussing

nanocubes in Section 2.4, for example, or as a result of the lightning-rod effect [150,

276, 277]. When a nano-optical antenna is illuminated with bulk electromagnetic

radiation at a resonant frequency, strong resonant enhancement of the localised near

field of the supported plasmonic mode can be achieved. The lightning rod effect, on

the other hand, is a non-resonant effect that arises from the high local charge

densities present near the sharp metal corners or tips of structures such as a metallic

SNOM tip [277]. It is characterised by the strong enhancement of the local near field

of incident bulk radiation in the vicinity of such structural elements [278].

Nanofocusing differs from these two mechanisms of field localisation because

it does not directly involve the interaction of bulk radiation with metallic

56

nanostructures [278]. Rather, gradually confining structures are used to transform

surface plasmons into localised surface plasmons. As the plasmons approach the tip

of the structure, their phase and group velocities approach zero asymptotically,

resulting in them slowing down, adiabatically stopping, and accumulating at the tip

[75]. If this process occurs sufficiently quickly so as to prevent losses in the metal

and dissipation is weak, the optical energy may be nanofocused.

Nanofocusing of plasmons can take place in both the adiabatic and the non-

adiabatic regime. In the adiabatic regime, the geometrical optics approximation

(GOA) (also known as the adiabatic approximation [76, 77, 279, 280] or WKB

approximation [281, 282]) can be used to analyse nanofocusing structures.

For the GOA to be applicable, the taper angle of the nanofocusing structure

should be sufficiently small so that the wave vector of the propagating SP mode does

not change significantly within one wavelength in the structure [11, 75, 279]. This

condition is given by the equation:

11

<<−

dxdqx (2.7.1)

where qx is the x-component of the plasmon wave vector and x is the direction

of plasmon propagation. When Eq. (2.7.1) is satisfied, the plasmon will not

experience significant reflections from the taper as it travels towards the tip [77].

However, Gramotnev et al. [283] reported that this applicability condition may be

too restrictive for certain structures. For example, the rate of changing wavenumber

for a tapered rod is smaller than for a tapered groove, which means that Eq. (2.7.1)

should be given as:

57

dxdqx

1−

≲1 (2.7.2)

Structures with small taper angles, while satisfying the conditions for the

adiabatic regime, may suffer from increased dissipative losses because the SPPs must

travel a longer distance to reach the tip of the structure. However, increasing the

taper angle may lead to reflective losses if the conditions for the adiabatic regime are

breached. Therefore, an optimal taper angle at which these two loss mechanisms

balance should exist. [75, 279, 284].

Nanofocusing has been considered in a range of structures, some of which are

shown in Table 2.1 below. A more comprehensive list of nanofocusing structures, as

well as a description of their SPP modes, can be found in a review article by

Gramotnev and Bozhevolnyi [278]. It should be noted that the vast majority of

nanofocusing structures have insulator-metal-insulator or metal-insulator-metal

configurations.

Nanofocusing structure Configuration Reference

Metal wedge on a dielectric substrate IMI [76]

Metal wedge surrounded by uniform dielectric IMI [77, 274]

Tapered metal rod IMI [75, 283, 285-288]

Tapered gap MIM [284, 289-291]

Tapered metallic grooves MIM [279, 292-295]

Dielectric wedge on a metal surface MII [296]

Table 2.1. A summary of some of the nanofocusing structures reported in the literature, as well as

their configurations (IMI = insulator-metal-insulator, MIM = metal-insulator-metal, and MII = metal-

insulator-insulator) [278].

58

A number of these nanofocusing structures are shown in Figure 2.18.

Figure 2.18. Various nanofocusing structures – (a) metal wedge on a dielectric substrate, (b) metal wedge surrounded by uniform dielectric, (c) tapered metal strip with decreasing width and constant thickness, (d) tapered metal rod, (e) ‘wizard hat’, (f) tapered metal film on a dielectric hemisphere,

and (g) dielectric wedge on a metal surface.

Analysis of adiabatic nanofocusing within the GOA has been carried out for a

number of structures including a metal on a dielectric substrate (Figure 2.18a) [76], a

metal wedge surrounded by vacuum (Figure 2.18b) [77], and sharp metallic grooves

[279].

59

Verhagen et al. investigated nanofocusing in laterally tapered Au waveguides

(Figure 2.18c) [297] and found that the symmetric bound mode (with respect to

charge) on the substrate side of the metal film was able to be nanofocused and that

this mode was similar to the mode responsible for focusing in a conical rod.

In many typical nanofocusing structures, the SPP experiences increased

localisation as a result of the decreasing film thickness in the direction of

propagation. However, if SPPs are focused in two dimensions that are normal to the

direction of propagation, greater localisation can be achieved.

The tapered metal rod (Figure 2.18d) is an example of one such structure that

has been studied extensively. Babadjanyan et al. [285] investigated nanofocusing in

pointed cones and demonstrated that the wavelength of a surface plasmon decreases

to zero as it moves towards the tip of a cone. Stockman [75] introduced the concept

of the GOA (as described above) for analysis of tapered rods and predicted field

enhancements of three orders of magnitude at the tip.

Gramotnev et al. [283] numerically analysed non-adiabatic nanofocusing in

tapered nanorods and confirmed the existence of an optimal taper angle and

demonstrated the existence of an optimal length in order to achieve maximum local

field enhancement at the tip of the structure. They also demonstrated that local field

enhancements of up to three orders of magnitude are possible using the optimised

parameters.

Vogel and Gramotnev [280] analysed tapered rods with parabolic perturbations

(convex and concave) of the conical shape of the taper and demonstrated that the

local field enhancement at the tip of a convex tapered gold rod can be stronger than

that achievable from a concave tapered rod, or even an ideal cone, given the same

60

taper angle. It was also demonstrated that the local field enhancement at the tip of a

rod with a convex taper is relatively insensitive to shape imperfections, while rods

with concave tapers suffer from a reduction in the field enhancement as a result of

shape imperfections.

Gramotnev and Vogel proposed a ‘wizard hat’ structure (Figure 2.18e), which

is a conical metal rod formed out of a metal film deposited on a dielectric substrate

or optical fibre [78]. Plasmons can be generated from either the air or substrate side

through the illumination of a spiral grating by circularly polarised light, or a circular

grating by radially polarised light. If the internal radius of the grating is at the

optimal radius (the distance from the centre of the structure at which losses due to

dissipation in the metal are equal to the increase in amplitude due to the annular

focusing of the plasmon), the grating is optimised for producing the maximum

plasmon field enhancement in the structure. For an optimal taper angle of 36 degrees

[283] and a tip radius of 5 nm, field enhancements of up to ~ 3500 were calculated,

while field enhancements of up to ~ 10000 were calculated for a tip radius of 2 nm.

These large field enhancements suggest that the structure could be used for single

molecule detection. This structure will be the focus of investigations in Chapter 4 of

this thesis.

The dielectric hemisphere covered in a tapered metal film [38] (as seen in

(Figure 2.18f) combines the benefits of both the annular focusing of plasmons, as

well as nanofocusing as a result of the tapered film. The field of the inner plasmon

generated on the inner surface (quasi-symmetric with respect to charge) only

becomes significant near the tip of the structure, where the tapered film thickness

becomes less than the skin depth. The structure has an achievable focal size of ~ 20

61

nm and an electric field intensity enhancement of up to ~ 150 times. As a result, the

inner plasmon can be used for highly sensitive detection near the tip.

Vedantam et al. [298] fabricated and characterised a silver-coated metal-

insulator-metal plasmonic dimple lens structure that tapered from an initial SiO2

thickness of 330 nm down to 1 nm. A semicircular grating was used to couple in 633

nm light, which was then focused by the tapered dimple lens to a focal point at the

centre of the structure.

Choo et al. [290] experimentally investigated an Au-SiO2-Au gap plasmon

waveguide with a three-dimensional linear taper. This structure, which they termed a

3D nanoplasmonic photon compressor, was first optimised using FDTD. These

simulations predicted a maximum intensity enhancement of ~ 3 × 104 from 830 nm

focused to an area of 2 x 5 nm2. The structure was then fabricated using electron

beam-induced deposition and showed highly-localised field confinement at the tip.

Although extensive research has been conducted into nanofocusing in

structures with MIM and IMI configurations, there has only been one significant

investigation into nanofocusing in an MII structure - a high-permittivity dielectric

(Si) wedge on an Ag substrate (Figure 2.18g) [296]. Unlike the previously discussed

structures, the incident SP propagates in the direction of increasing dielectric

thickness and a gradual transformation of the mode leads to the SPs slowing down

and becoming fully confined within the dielectric layer, which results in a build up of

optical energy. Although this enhancement is maximal at the metal-dielectric

interface, there was sufficient energy at the dielectric-air interface that could be

useful in sensing and coupling applications. This configuration will be investigated

in Chapter 5 of this thesis.

62

These are just some of the examples of structures that can be used to focus EM

energy into extremely small volumes, which makes them highly suitable for

integration into SERS-based sensors. However, one of the majors challenges faced in

the development of SERS-based sensors is delivering molecules for testing into the

region of enhanced electric field. For that reason, there has been significant interest

in the integration of SERS substrates into micro/nanofluidic systems in order to

produce optofluidic platforms for enhanced sensing.

2.8 FLUIDICS

2.8.1 Background to micro/nanofluidics

Microfluidic systems have been developed for a multitude of applications in

chemical and biological sciences. When the dimensions of analytical systems are

reduced to micrometre or sub-micrometre scales, their detection volumes become

smaller, and thus high-sensitivity detection techniques are required to perform

functions such as the detection of analytes and the continuous monitoring of

reactions [47]. However, fluid flow at the micro/nanoscale level behaves in a

different way to fluid at larger scales. The Navier–Stokes equations assume fluid to

be a continuum, which means that each elementary volume of the fluid should

contain enough molecules for statistical properties such as velocity and pressure to

be defined [299]. The Navier-Stokes equation in its most general form (applicable to

compressible fluid flow) is given as [300]:

( ) fvvvv+∇∇

++∇+−∇=

∇+

∂ .31. 2 ξηηρ vp

dt(2.8.1)

where ρ is the fluid density, v is the fluid velocity, p is pressure, η is the shear

viscosity, ζ is the bulk viscosity, and f represents other body forces (e.g., gravity).

63

In the case of an incompressible fluid, ∇(∇.v) = 0, which means that the

equation can be reduced to:

fvvv+∇η+−∇=

∇+

∂ρ vp

dt2. (2.8.2)

The Navier-Stokes equations assume fluid to be a continuum, so their

applicability depends on the evaluation of the ratio of the mean free path of the gas

molecules to the characteristic dimension of the flow domain. This ratio is known as

the Knudsen number, Kn. If the condition for the applicability of the Navier-Stokes

equations is not met (see Figure 2.19), the fluid will no longer be in local

thermodynamic equilibrium, which means that rarefaction effects may be observed.

In the past, rarefied flows were only encountered in areas such as high altitude

research and vacuum science [301, 302]. However, technological advancements have

enabled the investigation of the flow of gas through and around micro- and nanoscale

geometries, which means that fluid flow needs to be analysed in a different way.

Figure 2.19 illustrates the different flow regimes, as defined by the Knudsen number.

Figure 2.19. The different regimes of fluid flow, according to the Knudsen number.

As explained above, the Navier–Stokes equations are used to describe flow that

is a continuum (Kn < 10-3). For 10-3 < Kn < 10-1, the Navier-Stokes equations can be

used if the boundary conditions are modified [303, 304]. Between Kn = 10-3 and Kn

= 10, the Navier-Stokes equations begin to break down because the stress-strain

relationship for the fluid become non-linear within the Knudsen layer [305]. In the

64

free molecular regime (Kn > 10), the collisions of molecules with solid boundaries

will strongly influence the flow field, since intermolecular collisions are negligible.

However, it should be noted that the Knudsen numbers related to the different flow

regimes often depend upon the geometry under consideration and should therefore

only be used as a guide [305].

The boundary conditions used in the slip regime can be understood by

considering the boundary interactions. In 1879, Maxwell proposed two models

related to the interaction of gas molecules with a solid surface [306]. The specular

reflection model treats the interaction as an elastic collision, meaning that the net

shear stress and total energy exchange with the surface are both zero. In the diffuse

reflection model, the molecules colliding with the surface are re-emitted with a

temperature equal to the wall temperature and velocity distributions being half-range

Maxwellian at the wall temperature [307]. These two types of reflection are

illustrated in Figure 2.20.

Figure 2.20. The reflection of molecules from a smooth wall (specular reflection) and a rough wall (diffuse reflection).

The Maxwellian boundary condition, which is also applicable to a curved

surface, is given as [306, 308]:

65

0

2

=

∂∂

+∂∂−

=y

slip xv

yuu λ

ss

(2.8.3)

where uslip is the tangential fluid velocity at the wall, (u,v) are the components

of the velocity tangential and normal to the wall, respectively, x is the distance from

the inlet and y is the distance normal to the wall, and σ is the tangential momentum

accommodation coefficient (TMAC). The TMAC represents the proportion of

molecules that are reflected diffusively. As such, it can take values from zero to one,

which corresponds to specular and diffuse reflection, respectively [304]. In the case

of diffuse reflection, molecules are reflected with zero average tangential velocity

[309]. Typical TMAC values for air flowing past aluminium, iron, and bronze range

from 0.87-0.97, with surface roughness accounting for some of the variation for the

same surface material [309].

Eq. (2.8.3) was used to modify the boundary conditions for the flow of air

between two plates that were separated by a distance of 1 µm, as illustrated in Figure

2.21. Note how the velocity at the walls is non-zero.

Figure 2.21. The flow of air in the x-direction between two parallel plates, separated by 1 µm, with partial-slip boundary conditions

The velocity reaches a minimum at the walls and a maximum midway between

the plates. Figure 2.22 shows the velocity profile between the plates midway

between the inlet and outlet for both no-slip and partial- slip boundary conditions.

66

Figure 2.22. The velocity profiles of air flowing between two parallel plates 1 µm apart. The dashed (blue) curve represents no-slip boundary conditions, while the solid (red) curve represents partial slip

boundary conditions. The slip length is given as Ls.

Note that the blue curve shows a velocity of 0 m/s at the boundary, while the

red curve has a non-zero velocity at the boundary. If the hydrodynamic velocity field

is extrapolated beyond the solid-fluid interface, the quantity known as the slip length

can be defined. The slip length represents the imaginary distance beyond the

interface at which the tangential component of the fluid velocity vanishes [310] and

is given by the equation:

λs

s−=

2sL (2.8.4)

For 10-1 ≤ Kn ≤ 10, the flow enters what is known as the transition regime and

the first-order slip conditions begin to break down. Numerical methods based on

kinetic theory (e.g., direct simulation Monte Carlo (DSMC)) can be used to analyse

flow in this regime, but these techniques are computationally intensive [311].

Therefore, researchers have attempted to extend the Navier-Stokes equations into the

67

transition regime through the use of higher-order slip flow boundary conditions [312-

314]. However, these attempts are usually biased towards calculating an accurate

mass flow rate [314], or an accurate velocity profile [303], so the ongoing challenge

is to develop a model that will give accurate results for the streamwise pressure

distribution, mass flow rate, and velocity profile [315].

2.8.2 Sensing with micro/nanofluidic delivery

In recent years, a number of research groups have shown that SERS sensing

and micro/nanofluidic delivery can be combined to enhance sensing capabilities. A

variety of different approaches have been used to drive the flow and deliver samples

to regions in which the SERS spectra can be recorded.

Piorek et al. [43] devised a SERS vapour sensor based on the concept of free-

surface microfluidics. In this setup, the microchannel is open to the environment, and

an electrical heating element forms a transpirational pump, with evaporation-driven

water loss from the distal end maintaining flow through the channel. The device was

able to detect vapour from solid-phase 2,4-DNT at 1 ppb.

Lee et al. [316] combined gold-pattern microarray wells and hollow gold

nanospheres with a microfluidic gradient generator in order to produce a SERS-based

immunoassay platform for cancer biomarkers. Fan et al. [317] used optrode fibre

sensors as SERS sensing elements in microfluidic chips. They were able to

demonstrate the ability of the system to multiplex by simultaneously monitoring the

ratios of two analytes (NBA and oxa) concentrations in a concentration gradient

generating fluidic chip.

White et al. [318] used an optofluidic ring resonator with a liquid core for

SERS detection. The core acts as both a ring resonator and a sample delivery

68

mechanism, with a SERS signal from the analyte adsorbed on Ag nanoclusters

resulting in a detection limit as low as 400 pM.

Using embedded fibres on-chip, Ashok et al. [319] developed an alignment-

free Raman spectroscopic detection scheme on a microfluidic platform, which they

termed Waveguide Confined Raman Spectroscopy (WCRS). They stated that the

applicability of WCRS could be extended by implementing SERS.

Huh et al. [320] developed a device featuring embedded electrokinetically

active microwells to obtain a SERS signal from nucleic acid sequences associated

with Dengue virus serotype 2 (DENV-2). When the microwells are electrically

actuated, species from the interrogated solution can be attracted or repelled as a

result of electrokinetic effects [321].

For those optofluidic platforms in which metallic nanoparticle colloids are

manipulated within liquids, it is important to ensure that the solutions are

homogeniously mixed in order to allow for quantitative analysis by producing

identical SERS sites [47].

Quang et al. [322] used a micropillar array in a microfluidic channel to ensure

that the analytes to be detected (dipicolinic acid (DPA) and malachite Green (MG))

and metal nanocolloids were mixed homogenously for SERS sensing. Using a

portable Raman spectrometer, the limits of detection for the two analytes were

determined to be 200 ppb and 250 ppb, respectively.

Wang et al. [323] developed a device with a step junction between a

microfluidic and nanofluidic channel that traps nanoparticles and molecules, at which

point capillary forces act to produce nanoparticle-molecule SERS clusters. Using this

69

device, the SERS enhancement factor was determined to be 108, compared with 106

using the conventional method.

An alternative optofluidic platform is one in which metallic nanostructures

with identical SERS-active hotspots are integrated into a chip. One of the most

popular nanostructures for optofluidic sensing is a periodic array of nanoholes. While

most such nanostructures have been successfully integrated into optofluidic devices

for use in surface plasmon resonance-based biosensors, the same principles can be

applied to SERS-based sensors.

Escobedo et al. [324] presented a flow-through nanohole array sensor which

locally concentrates the analyte prior to sensing through the application of an electric

field, in a technique known as electric field gradient focusing (EFGF). EFGF, in

addition to the pressure-driven flow, causes charged analytes to concentrate and

move to the active sensing regions.

De Leebeeck et al. [201] integrated a 20 µm × 20 µm nanohole array into a

microfluidic chip platform to develop a sensor for chemical and biological

applications. It was demonstrated that the sensor had a sensitivity of 333 nm/RUI,

was able to spatially resolve a cross-stream concentration gradient in a microfluidic

flow, and could monitor protein binding.

Eftekhari et al. [325] combined nanohole arrays featuring different periodicities

and holes of different diameters with fluidics to achieve the real-time detection of the

sequential assembly of a monolayer and biomolecules. By using a flow-through,

rather than flow-over geometry, a 6-fold increase in adsorption kinetics was

observed.

70

Yanik et al. [326] investigated a biosensing arrangement that combined an

array of gold nanoholes and microfluidics. They compared flow-over and flow-

through geometries, both experimentally and through simulations, and demonstrated

that the flow-through geometry produced a 14-fold improvement in the mass

transport rate constants because it guided the analyte to the sensing region, rather

than relying on diffusive transport. Their simulations also showed that the turbulence

of the flow as it passes through the holes results in stirring of the solution, which

further enhances the mass transport rate.

Yanik et al. [327] developed an optofluidic sensor for the direct detection of

intact viruses by functionalising the gold nanohole array described in [326]. When

biomolecules/pathogens either bind to or are immobilised on the nanohole array, a

red shift in the plasmon resonance is observed as a result of the refractive index

increase.

As has just been demonstrated, molecules of interest can be transported to the

sensing region/s when a fluid is subjected to a pressure gradient. However, they may

also be transported as a result of thermal motion or a concentration gradient.

2.9 MOLECULAR DIFFUSION

Diffusion is a fundamental process by which matter may be transported or

mixed and arises as a result of the random thermal motion of molecules. Robert

Brown is credited with the first observation of this phenomenon in 1827, when he

noted the “random walks” of pollen suspended in fluid [328]. In 1905, a physical

explanation for these observations was presented by Albert Einstein [329]. He

derived an expression for the diffusion coefficient of a molecule D in terms of the

71

viscosity of the medium η, the temperature of the system T, and the radius of the

diffusing molecule r. This expression is given as:

rTkD B

πη6= (2.9.1)

Einstein also recognised that distance of the particle, r, from its original

position after time, t, could be expressed as:

Dtr 62 = (2.9.2) where r2 = x2 + y2 + z2

The diffusion coefficients for different molecules in air at atmospheric pressure

and a temperature of 293 K are shown in Table 2.2.

Molecule Diffusion coefficient

(cm2/s) Reference

ammonia 0.216 [330]

sulphur dioxide 0.132 [331]

TNT 0.064 [332]

Table 2.2. Diffusion coefficients for different molecules in air at atmospheric pressure and a

temperature of 293 K.

These diffusion coefficients are, to a first approximation, inversely

proportional to pressure [333]. Knowledge of the diffusion coefficient and

concentration gradient allows the diffusive flux to be calculated using Fick’s first law

[334]:

xCDJ

∂∂

−= (2.9.3)

72

where J is the flux of molecules, D is the diffusion coefficient, and dC/dx is

the concentration gradient. The negative sign indicates that diffusion is occurring

from high to low concentration.

By considering conservation of mass, Fick’s second law can be obtained. This

describes the change in concentration, due to diffusion, over time.

2

2

xCD

tC

∂∂

=∂∂ (2.9.4)

These are the basic equations that describe the diffusive motion of molecules

and can be used to characterise diffusive transport processes.

Having covered the research necessary to provide a detailed background to the

project, the final section of the literature review will cover the primary method of

analysis used in this thesis.

2.10 FINITE ELEMENT ANALYSIS

The numerical technique used throughout this project is the finite element

method (FEM). It can be used to solve problems in areas such as electromagnetics

[335], structural mechanics [336], and fluid dynamics [337].

FEM is used to obtain a set of linear algebraic equations from partial

differential equations, which enables approximate solutions to boundary-value

problems to be found [338]. The geometry is first plotted and the material parameters

and boundary conditions are defined. Next, the geometry is partitioned into small

shapes known as mesh elements. A meshed geometry is shown in Figure 2.23a. In

two dimensions, elements are typically triangles or quadrilaterals. Unlike the finite

difference time-domain (FDTD) method, FEM can easily handle complicated

geometries because irregular meshes can be implemented. Simple functions are used

73

to approximate the unknown solution over each mesh element [338]. A solution is

found using either an iterative or direct solver. Iterative solvers store only nonzero

matrix elements, so they are primarily used for systems with large numbers of

unknowns. Direct solvers, on the other hand, require the storage of the entire matrix,

so they are more suited to smaller systems [339]. The mesh needs to be suitably

refined so as to ensure that the obtained solution is sufficiently close to the exact

solution for the model under consideration. This can be achieved by checking for

changes in the solution after the model has been refined and run again. Figure 2.23

illustrates a both a typically meshed geometry and the resultant solution.

Figure 2.23. (a) A finite element analysis mesh and (b) a finite element analysis solution.

This concludes the background literature review. As was detailed in Section

1.3, the following four chapters contain investigations into the pressure-driven

delivery of three different residual/gas molecules of interest in air to the plasmonic

field hotspots created by an array of circular nanoscale apertures (Chapter 3), the

functional capabilities resulting from combining tapered rod nanofocusing structures

with nanofluidic flows to achieve enhancement of the detection sensitivity through

the nanofluidic delivery of molecules for testing to the plasmonic hotspot at the tip of

the focusing structure (Chapter 4), nanofocusing by a dielectric wedge on a metal

74

substrate in both adiabatic and non-adiabatic regimes, for the cases where the

dielectric permittivity of the wedge is smaller or larger than the magnitude of the real

part of the metal permittivity (Chapter 5) and an L-shaped gap surface plasmon

waveguide (L-GSPW) consisting of a dielectric strip sandwiched between two metal

films (Chapter 6). The findings of these investigations will have important

implications for the design of new, highly sensitive optical sensing devices that could

potentially be used for the detection, identification, and investigation of chemicals,

drugs, and explosives.

75

Chapter 3: Nanofluidic Delivery of Molecules: Integrated Plasmonic Sensing with Nanoholes

3.1 INTRODUCTION

Nanohole arrays in metal films have been shown to display enhanced

extraordinary optical transmission effects at resonant wavelengths, due to surface

plasmon resonances [27, 211, 216, 340-344]. These effects have led to developments

in areas including sensing [202, 222, 232, 345-347] and imaging [348]. Recent

efforts have succeeded in combining fluidics with nanohole arrays, which has

resulted in an increase in the sensitivity of nanohole array-based sensors [325, 326,

349-352]. Through the use of optofluidic elements, molecules in the fluid can

essentially be guided to the plasmonic hotspots within the apertures, where they can

then be detected optically. In addition, nanohole array sensors may have further

advantages and useful features that may make them more competitive compared with

the other options of nano-optical sensor designs. These advantages may include the

relative simplicity of fabrication, potentially large sensing area, simplicity of

pumping electromagnetic energy into the sensor and subsequent retrieval of the

optical signal, etc.

One of the major goals in sensor design for the detection and identification of

trace amounts of molecules in the air or liquid by spectroscopic means is related to

maximising the spectroscopic signal strength to ensure lower detection limits and

thus lower detectable concentrations of the molecules. This can be achieved by

increasing the local field enhancement in the nanoscale structure (which increases the

76

signal from the target molecules in the sensing region), or by increasing the rate of

delivery of the target molecules to the sensing region. Increase and optimisation of

the local field enhancement in nano-optical structures is typically achieved by using

the so-called plasmon nanofocusing structures [11, 78, 283, 294, 353] or plasmonic

nano-optical resonators and nanoantennas [182, 354-361]. Unfortunately, local field

enhancement in nanohole arrays [362] is typically significantly smaller than in the

optimised nanofocusing structures [78, 283, 294]. Therefore, the best way to increase

sensitivity of detection in sensors with nanohole arrays is to increase the rate of

delivery of the target molecules to the electromagnetic hotspots in the nanoholes.

One of the ways to achieve this is to use nanohole arrays in a thin metal membrane

and apply a pressure gradient across the membrane to drive the air or liquid through

the nanoholes and thus deliver the trace molecules in the fluid to the detection

hotspots [325, 326, 349, 351]. However, the detailed theoretical analysis and

optimisation of the fluid flow through such nanoholes are needed to enable

optimisation of the sensor design and understanding of its ultimate capabilities and

functionalities.

Therefore, the aim of this chapter is to conduct the theoretical and numerical

investigation of the airflow through circular nanoholes in metal membranes. The

delivery rates of three different target molecules – ammonia (NH3), sulphur dioxide

(SO2), and TNT – to nanoholes of different diameters are determined and

investigated as functions of chamber pressure and pressure difference across the

membrane. Partial slip boundary conditions are used to ensure the correct application

of macroscopic hydrodynamics to determine the nanofluidic flows in the nanoholes.

The determined nanofluidic delivery rates of the target molecules to the nanoholes

are compared with the molecular diffusion delivery rates in the absence of a pressure

77

difference across the porous metal membrane. Critical nanofluidic regimes are

determined where the molecular diffusion and nanofluidic delivery rates are equal.

Application possibilities for the design and optimisation of nano-optical air

monitoring sensors are also discussed.

3.2 METHODOLOGY AND CONSIDERED STRUCTURE

For simplicity, the analysis will be conducted for a cylindrical nanohole with

the diameter d in a freestanding metal membrane of thickness h surrounded by air

with trace molecules (residual vapours of substances) to be detected and/or identified

– Figure 3.1. The obtained results can be easily generalised to nanohole arrays if the

distances between the nanoholes in the array are sufficiently large, so that the airflow

through each of the holes can be analysed separately.

Figure 3.1. The scheme of the considered structure consisting of a nanohole of diameter d in a metal membrane surrounded by air (a) and on a dielectric substrate (b) with a concentric hole of diameter D.

The difference between the inlet Pin and outlet Pout pressures of the air in the chamber result in a nanofluidic flow through the hole (a).

The numerical finite element analysis of the airflow through a nanohole was

conducted using the weakly compressible Navier-Stokes equation in the COMSOL

MultiphysicsTM 3.5a software package (COMSOL, Inc). Due to its cylindrical

geometry, the numerical analysis was conducted in two dimensions in the cylindrical

coordinates. The computational window was selected sufficiently wide (~ 4 µm in

78

diameter) to minimise the effect of its boundaries on the calculated velocity fields.

The air pressure at the top boundary of the computational window (inlet pressure –

Figure 3.1) was assumed to be larger than at the bottom boundary (outlet pressure) –

to ensure the airflow through the hole. The distances between the inlet and the porous

metal membrane, and between the membrane and the outlet were taken as 3 µm and

12 µm, respectively. The significant difference between these distances was assumed

because of the larger impact of the outlet boundary on the resultant air velocity field.

The obtained results were checked for numerical convergence (within ~ 1%)

including the effect of the boundaries of the computational window. The temperature

throughout the structure was assumed to be constant and maintained at 293 K. This

assumption implies isothermal expansion of the air upon passing through the

nanohole. Though this is certainly an approximation, it is not expected to cause

significant errors in the calculated air flow rates, because nanoscale structural

dimensions result in rapid heat exchange effects (particularly at higher ambient

pressures), and some possible reduction in the air temperature upon its passing

through the nanohole is not expected to cause any significant changes in the air

viscosity, or pressure, or density.

In practice, the considered structure can be fabricated by depositing a thin

metal film onto a dielectric substrate (wafer), etching a circular hole in the film with

the diameter d, and then etching another concentric hole through the dielectric

substrate with the diameter D (Figure 3.1b). It is reasonable to assume that D > d,

because holes with larger (microscopic rather than nanoscopic) diameters are easier

to etch in relatively thick substrates (without using expensive nanofabrication

techniques), and because larger holes in the substrate would not cause additional

significant air dragging effect that could potentially reduce the air flow rates through

79

the nanohole in the thin metal film. A detailed analysis of possible relationships

between D and d is not presented in this chapter, but this is an option for a practical

realisation of the proposed sensor where the dielectric substrate with a bigger hole

works just as a support for the metal film. In this case, the diameter of the

computational window can be assumed equal to the diameter of the hole in the

dielectric substrate (with the respective boundary conditions). Alternatively, a thin

metal film can be deposited onto a preformed dielectric membrane (e.g., a SiN

membrane of ~ 50 nm thickness typically used for transmission electron microscope

imaging) and then a nanohole can be milled using focused ion beam lithography

through the metal film and the underlying dielectric (e.g., SiN) membrane.

The Knudsen number Kn = λL/L is the ratio of the mean free path of the

molecules λL in the fluid to the characteristic length of the device L [305]. The

Knudsen number can be used to classify fluid flows into one of four regimes:

continuum regime (Kn ≤ 10-3), slip regime (10-3 < Kn ≤ 10-1), transition regime (10-1

< Kn < 10), and free molecular flow regime (Kn >10) [363]. The no-slip boundary

conditions are only applicable in the continuum regime of a viscous fluid flow. This

is the standard boundary condition for the macroscopic hydrodynamics with viscous

fluid sticking to a solid interface so that the velocities of the solid and fluid are the

same. However, when the mean free path for the fluid molecules becomes

sufficiently close to the typical dimensions of the structure (e.g., the diameter of a

nanohole), the fluid can no longer be considered as a continuum and ‘partial slip’

boundary conditions are introduced instead to allow for molecular interaction,

collisions and momentum transfer between the boundary and the fluid molecules

[312]. In this case, the tangential fluid velocity is no longer assumed to equal zero at

the boundary, but a slip length is introduced as an imaginary distance into the solid

80

beyond the solid-fluid interface, at which the tangential component of the fluid

velocity vanishes when extrapolating the hydrodynamic velocity field [310].

The first-order slip length in the air near the walls of a nanoscale structure

can be defined by the equation [309]:

( ) ssλ /2 −= LsL (3.1)

where σ is the tangential momentum accommodation factor (TMAC). Eq. (3.1) is

typically used for Kn ≤ 0.1. For higher values of Kn, higher-order slip boundary

conditions can be used [314], although there is currently no general consensus as to

which type of such conditions gives the most accurate results [315]. The typical

pressures and hole diameters investigated in this chapter lead to Knudsen numbers

that correspond to a wide range of nanofluidics regimes from the slip regime to the

transition regime. Therefore, a decision was made to employ the first-order slip

boundary conditions in order to obtain a first-order approximation of the nanofluidic

delivery rates. This choice of the boundary conditions means that the delivery rates

calculated with lower chamber pressures might be somewhat understated for smaller

hole diameters.

The slip length (Eq. (3.1)) is a function of the fluid composition, temperature

and density. In the case of a compressible fluid like air, increasing the chamber

pressure results in a decrease in the slip length, because the slip length is directly

proportional to the mean free path for the molecules in the fluid, which is inversely

proportional to pressure. For example, for the inlet air pressures between 1.15 atm

and 5.15 atm, the mean free path for the air molecules varies between 57 nm to 13

nm, respectively.

81

3.3 RESULTS AND DISCUSSION

3.3.1 Nanofluidic flow rates

The typical velocity field for the airflow through a nanohole with d = 500 nm,

membrane thickness h = 50 nm, inlet pressure Pin = 1.15 atm, and outlet pressure Pout

= 1 atm is shown in Figure 3.2. It can be seen that the velocity field of the air is

predominantly localised near the hole and its maximum, as expected, is achieved in

the middle of the hole with the typical velocities reaching significant values of ~ 50

m/s.

Figure 3.2. Typical distribution of the y-component of the air velocity near a nanohole with the diameter d = 500 nm in a membrane of thickness h = 50 nm, the inlet pressure Pin = 1.15 atm, and the

outlet pressure Pout = 1 atm.

Using the obtained velocity field distributions (Figure 3.2), the rates of

airflow through the nanoholes with different diameters d between 200 nm and 1500

nm were calculated. The density of the air and the y-component of its velocity were

evaluated at every point over the surfaces y = ± h/2 of the hole openings. The product

of these two variables was then integrated over the hole openings in order to

determine the mass flow rates of air through the hole as a function of hole area

(Figure 3.3a). The typical required sensitivity of an air monitoring sensor (e.g., for

residual vapours of explosives) at the normal atmospheric conditions should be ~ 1

ng/L [364]. Assuming that this concentration of the trace molecules of ammonia,

82

sulphur dioxide, and TNT is present in the air (at the normal atmospheric conditions),

and using the calculated airflow through the hole, the rate of the nanofluidic delivery

of the trace molecules to the nanohole (i.e. the rate at which these molecules pass

through the nanohole) is determined. The efficiency of detection of residual

vapours/gases in the air is assumed to be proportional to the number of vapour/gas

molecules that reach the nanohole with the enhanced plasmonic field, where these

molecules can then be detected (e.g., by means of local spectroscopy, etc.).

Figure 3.3. (a) The air mass flow rate M through a hole as a function of the hole area, and (b) the air mass flow rate Mm per unit area of the hole, as a function of hole area at two different membrane

thicknesses h = 50 nm (thick curves) and h = 250 nm (thin curves). The inlet and outlet pressures are 1.15 atm and 1 atm (dash-and-dot curves), 3.15 atm and 3 atm (dashed curves), and 5.15 atm and 5

atm (solid curves).

As expected, the mass flow rate increases (nonlinearly) with increasing hole

area and/or ambient pressure. In particular, this clearly shows that increasing the inlet

pressure, even while maintaining a constant pressure difference across the

membrane, can increase the nanofluidic delivery of air and airborne residual vapour

molecules to nanoholes. Figure 3.3b is a plot of the mass flow rate of air per unit

83

area of the hole, as a function of hole area. It can be seen that the slope of the curves

in Figure 3.3b decreases with increasing hole diameter (hole area) and/or decreasing

ambient pressure. This is because increasing hole diameter and/or reducing ambient

pressure reduces the overall impact of the hole walls (and the applied partial slip

boundary conditions) on the airflow inside the hole.

Also, as expected, increasing membrane thickness results in a reduction of the

airflow through a hole at a fixed pressure difference across the membrane (compare

the thick and thin curves in Figure 3.3). This is due to increased friction and

viscosity effects inside a longer hole, which result in a reduction of the typical air

speed. However, this flow reduction does not seem to be very strong and is limited in

Figure 3.3 to typically less than ~ 30% for a 5 times increase in the membrane

thickness (compare the thick and thin curves in Figure 3.3a). This finding may be of

significant practical importance.

As discussed in the introduction to this chapter, the goal of this investigation

is to analyse the capabilities of an optical sensor with nanofluidic delivery of tested

molecules to the hotspots in nanoholes in a metal membrane. Increasing sensitivity of

such a sensor is related to increasing nanofluidic delivery rate of the tested molecules

to the nanoholes. This rate can be increased through an increase of the airflow either

by increasing the diameter of the nanohole (though large holes may lead to a

reduction of local field intensity and, thus, sensitivity of the detection), or by

increasing pressure difference across the porous membrane with the holes. However,

increasing pressure difference across the membrane subjects the membrane to

increased mechanical stress that may lead to the mechanical breakdown. To avoid

this, membrane thickness should be increased. Since increasing thickness of the

membrane results in only relatively small reduction of the airflow (Figure 3.3),

84

simultaneous increase of the pressure difference across the membrane and its

thickness may be a viable option for the overall increase of the airflow and thus

sensitivity of the sensor.

To investigate this possibility, note that the maximum (critical) pressure

difference that can be applied to a membrane without breaking it can be evaluated

using the following equation [365, 366]:

ErhKP yield

yieldm

ss

≈∆ 29.0max [Pa] (3.2)

where rm is the radius of the membrane, syield is the yield stress, E is Young’s

modulus of the membrane material, and K = (A – Ah)/A, where A is some selected

area of the membrane and Ah is the area of all holes within this selected membrane

area.

The pressure difference ∆Pmax determined by Eq. (3.2) corresponds to the

maximum achievable airflow through the membrane. For example, take a membrane

formed by a thin gold film deposited onto a preformed SiN membrane with a radius

rm = 5 µm and thickness hd (the overall membrane thickness will then be h = hd + hm,

where hm is the thickness of the deposited metal film). Under the assumption that the

mechanical strength of this SiN–Au membrane is largely determined by the

mechanical strength of the preformed SiN membrane (with syield = 4×109 Pa and E =

385×109 Pa) [366]. Therefore, the metal film can be ignored when estimating the

mechanical strength of the considered membrane, and it is assumed that h = hd in Eq.

(3.2). This gives a reasonable lower estimate of the maximum pressure difference

that could be applied across the membrane without breaking it. For the hole arrays

with square lattice, hole diameters 250 nm, 500 nm, 1000 nm, and lattice periods

equal to twice the respective hole diameters, K = 0.8 and Eq. (3.2) determines the

85

respective maximum (critical) pressure differences ∆Pmax as functions of hd (Figure

3.4).

Using the obtained values of ∆Pmax, the maximum possible mass airflow

through the nanoholes for given structural parameters for the membrane can be

estimated. In particular, it can be seen that the considered pressure difference of 0.15

atm in Figure 3.2 and Figure 3.3 is easily achievable in practice and is significantly

below the typical pressure differences (Figure 3.4) that may result in a destruction

(mechanical breakdown) of the membrane.

Figure 3.4. The maximum (critical) pressure difference ∆Pmax across a SiN membrane of radius r = 5 µm and K = 0.8 as a function of its thickness hd.

An interesting result can be obtained when considering the ratio of the mass

flow rates through a nanohole at different ambient pressures as a function of hole

area A (Figure 3.5). It can be seen that there is an optimal hole area (hole diameter)

at which the ratio of the airflow is maximal (Figure 3.5). In other words, whereas it

is quite obvious that increasing ambient pressure results in increasing mass airflow

rates through a hole (Figure 3.3a), there is an optimal hole area (diameter) at which

this increase is the strongest (Figure 3.5). This optimal hole diameter increases with

increasing thickness of the membrane (compare Figure 3.5a and Figure 3.5b). The

effect appears to be more pronounced in the event of larger ambient pressure in the

86

chamber (compare, for example, the solid and dash-and-dot curves in Figure 3.5a

and Figure 3.5b). This will be important for further optimisation of the nanohole

array sensors to ensure their maximal possible sensitivity and operational

capabilities.

Another useful result that follows from Figure 3.5 is that, for example,

increasing ambient pressure in the chamber ~ 5 times results in almost the same (~

4.5 times) increase in the mass airflow rate through a hole of optimal diameter. This

is approximately correct for both the considered membrane thicknesses (compare the

solid curves in Figure 3.5a and Figure 3.5b). Thus the impact of membrane thickness

on the optimal flow rate is rather weak. Further, while for small hole diameters (less

than ~ 1 µm) increasing h results in decreasing the ratio M/M1.15, for larger hole

diameters (larger than ~ 1 µm) increasing h results in the opposite tendency, i.e.

increasing ratio M/M1.15 (compare the solid curves in Figure 3.5a and Figure 3.5 b).

Figure 3.5. The dependences of the ratios of the mass airflow rates M through a nanohole at the inlet pressures of 5.15 atm (solid curves), 4.15 atm (dotted curves), 3.15 atm (dashed curves), and 2.15 atm

(dash-and-dot curves) to the mass airflow rate M1.15 at the inlet pressure of 1.15 atm on hole area at two different membrane thicknesses of (a) 50 nm, and (b) 250 nm. The pressure variation across the

membrane for all the curves is 0.15 atm.

87

3.3.2 Nanofluidic and molecular diffusion flow rates

The conclusion in the previous section that increasing membrane thickness h

causes only relatively weak reduction in the mass airflow through a nanohole is an

important outcome as it allows increasing membrane thickness to enhance its

mechanical strength and thus allow larger pressure differences across the membrane

which result in a significant overall increase of the airflow and thus the molecule

delivery rates to the nanoholes. In order to demonstrate quantitatively higher

efficiency of the nanofluidic delivery of residual molecules in the air to nanoholes,

these delivery rates to the delivery rates due to molecular diffusion in the air must be

compared. Indeed, even in the absence of pressure difference across the membrane

with a nanohole, residual molecules in the air will still have an opportunity to reach

the plasmonic hotspot near the nanohole by means of their diffusion in the air.

If the plasmonic field hotspot near a nanohole in the metal film is

approximated by a hemisphere, the rate of delivery of residual molecules into this

hotspot by means of molecular diffusion can be evaluated by Fick’s first law of

diffusion [367]:

( )rnnrDJ −= 02π (3.3)

where J is the diffusive flux (i.e. the number of residual molecules that enter the

hotspot per second), r is the radius of the hemisphere approximating the hotspot (i.e.

the radius of the sensing region created by the nanohole), D is the diffusion

coefficient of the trace molecules in air, n0 is the concentration of the molecules at an

infinite distance from the hotspot, and nr is the concentration of molecules at the

boundary of the hotspot (hemisphere of radius r).

Here, it is also assumed that the diameter of the hotspot (approximated by a

hemisphere) is noticeably larger than the mean free path of the molecules of interest

88

in the air. If this is not the case, the approximation of molecular diffusion given by

Eq. (3.3) is not valid. The analysis given below also makes the assumption that nr =

0, which means that the molecules entering the hotspot are either quickly destroyed

by the enhanced electromagnetic field, or transformed as a result of interaction with

the field into a different state. Physically, this situation may occur when the

molecules to be detected enter the hotspot region, emit, for example, the Raman

signal, and then get destroyed by the thermal effects. In this case:

02 rDnJ π= (3.4)

In what follows below, it is assumed that the radius of the hemisphere

representing a plasmonic hotspot in a nanohole is equal to the radius of the nanohole

itself d = 2r. As an example, the diffusion delivery of residual molecules of NH3,

SO2 and TNT in the air are considered. The values of the diffusion coefficient D for

these molecules were taken to be 0.216 cm2/s for ammonia [330], 0.132 cm2/s for

sulphur dioxide [331], and 0.064 cm2/s for TNT [332] at atmospheric pressure and a

temperature of 293 K. These coefficients are inversely proportional to pressure [333].

The typical required sensitivity of an air monitoring sensor (e.g., for residual vapours

of explosives) at the normal atmospheric conditions should be ~ 1 ng/L [364]. Using

this value of the residual vapour density, the concentrations of the considered

molecules n0 at the considered inlet pressures in the chamber are determined.

Subsequently, using Eq. (3.4), the diffusive delivery rates for the residual molecules

of ammonia, sulphur dioxide, and TNT in the air were calculated and compared to

the determined nanofluidic delivery rates for the holes with diameters within the

range 200 – 1500 nm at the inlet pressures of 1.15, 3.15, and 5.15 atm, and the

pressure difference across the membrane of 0.15 atm – Figure 3.6.

89

The circles are used to indicate the critical hole diameters dc at which the

diffusive and nanofluidic delivery rates are equal (Figure 3.6). If d < dc, then

molecular diffusion appears to be more effective than nanofluidics in terms of

delivery the residual molecules to the nanoholes. If d > dc, the situation is reversed,

and nanofluidics becomes a more efficient way of delivery the tested molecules to

the hotspots in the nanoholes (Figure 3.6).

The nanofluidic delivery rates appear dependent upon type of molecules used

(Figure 3.6). This is a result of using a fixed residual vapour density of 1 ng/L at

normal atmospheric conditions, which means that lighter molecules will have higher

molecule concentrations and, therefore, correspond to larger nanofluidic (and

diffusive) delivery rates (Figure 3.6). As expected, lighter molecules (e.g., NH3)

correspond to larger critical hole diameters, because they are characterised by larger

values of the diffusion coefficient D and thus larger diffusive delivery rates.

Increasing inlet (ambient) pressure in the chamber results in increasing nanofluidic

delivery rates (compare Figure 3.6a-c). This is predominantly related to the

increasing nanofluidic delivery rates due to increased mass airflow through the holes

(see Figure 3.3 and Figure 3.5). On the contrary, the diffusive delivery rates appear

almost independent of the ambient pressure in the chamber. This is because a

reduction of the diffusion coefficient D with increasing pressure in the chamber is

almost exactly compensated by an increase in the molecule concentration n0 – see

Eq. (3.4).

90

Figure 3.6. The nanofluidic molecule delivery rates for ammonia, sulphur dioxide and TNT (curves 2, 4 and 6, respectively) compared to the diffusive delivery rates for the same molecules (curves 1, 3 and

5, respectively) for the inlet pressures of (a) 1.15 atm, (b) 3.15 atm, and (c) 5.15 atm. The concentrations of the considered molecules in the air at normal atmospheric conditions correspond to the required benchmark monitoring sensor sensitivity of 1 ng/L [364]. The pressure drop across the membrane is 0.15 atm, and the membrane thickness h = 50 nm. The circles (i), (ii), and (iii) indicate

the critical hole diameters dc for ammonia, sulphur dioxide, and TNT, respectively.

91

The obtained critical hole diameters appear rather large in Figure 3.6, which

is an indication that at the considered pressure differences across the membrane,

nanofluidics may noticeably increase sensor sensitivity only at rather large hole

diameters, though increasing ambient pressure in the chamber and/or increasing

molecular weight of the tested molecules results in a significant enhancement of the

efficiency of nanofluidic delivery mechanism (Figure 3.6c).

Another way to increase the nanofluidics delivery rate is to increase the

pressure difference across the membrane. As shown in Figure 3.4, the critical

pressure differences that may lead to the mechanical breakdown of the membrane are

significantly larger than those used for Figure 3.6. This leaves significant

opportunities for further enhancement of the operational capabilities of the described

nanofluidic sensor. For example, Figure 3.7 shows the results of the comparative

analysis of the nanofluidic and diffusive delivery rates for the same types of residual

molecules as functions of the pressure difference ∆P across the membrane.

In particular, it can be seen that, for the considered structural parameters and

pressures, the nanofluidic delivery rates could be as much as ~ 20 times larger than

those due to molecular diffusion, especially for such relatively heavy molecules as

TNT (which is of particular practical importance). This clearly illustrates the

significant capabilities of the nanofluidics delivery mechanism to enhance the

sensitivity and operational capabilities of the described nano-optical sensors. The

considered pressure differences across the membrane are well below the typical

critical pressure differences (Figure 3.4) that could result in the mechanical

breakdown of the membranes.

92

Figure 3.7. The ratios of the nanofluidic and diffusive delivery rates as functions of pressure difference ∆P across the membrane of h = 70 nm for the two different outlet pressures: (a-c) 1 atm and (d-f) 5 atm, three different hole diameters: (a,d) d = 250 nm, (b,e) d = 500 nm and (c,f) d = 1000 nm,

and for the three considered types of residual gas molecules: NH3 (dotted curves), SO2 (dashed curves), and TNT (dash-and-dot curves). The straight horizontal lines show the unit ratio of the delivery rates, thus identifying the pressure differences for which the nanofluidic delivery rate is

smaller or larger than the diffusive delivery rate.

3.4 CONCLUSIONS

This chapter has presented a detailed numerical comparative analysis of the

nanofluidic and molecular diffusion rates of delivery of residual vapour/gas

molecules in the air to plasmonic hotspots in nanoholes in a thin metal or composite

membrane. In particular, structural and material parameters for which either of these

delivery mechanisms of residual molecules is dominant were determined and

93

analysed. The larger the molecular mass of the residual tested molecules in the air,

the greater the advantages offered by the nanofluidic delivery mechanism compared

to molecular diffusion. For example, under practically achievable conditions and for

the typical residual concentrations of explosive (TNT) vapour in the air, nanofluidics

may be up to ~ 20 times more efficient than molecular diffusion in terms of

delivering the residual molecules to submicron holes in a membrane. The most

appropriate ways for optimisation and enhancement of the considered structures from

the view-point of design of efficient nano-optical air monitoring sensors and

measurement techniques were identified and investigated. In particular, it was shown

that the two major approaches to sensitivity enhancement in such sensors are related

to increasing pressure difference across the porous membrane and/or increasing

ambient air pressure in the sensing chamber. It was also shown that increasing

ambient pressure in the chamber results in the most rapid increase in nanofluidic

molecule delivery rates for nanoholes of the determined optimal diameter that

increases with increasing thickness of the membrane.

The obtained simulation outcomes will be important for the optimal design of

nano-optical sensors and measurement techniques using nanohole arrays and porous

membranes with nanofluidic flows. These include air monitoring sensors for the

detection of residual vapours of explosives by means of surface-enhanced Raman

spectroscopy [202, 362] or surface-enhanced infrared absorption spectroscopy [347].

This investigation was published in the article ‘Nanofluidic delivery of

molecules: Integrated plasmonic sensing with nanoholes’. Refer to Appendix A of

this thesis for further details.

95

Chapter 4: Nanofluidics in Nanofocusing Tapered Rod Structures

4.1 INTRODUCTION

In the previous chapter, the delivery of molecules of interest to the sensing

region produced by a nanohole was investigated. However, the local field

enhancement in nanohole arrays is typically significantly smaller than in the

optimised nanofocusing structures. As was described in the literature review,

nanofocusing typically results in strong localisation and enhancement of the local

electric field near the tip of a tapered metal structures, including tapered metal rods

[75, 280, 283, 285], metal wedges [77, 274, 368], tapered metallic grooves, tapered

sections of thin metal films on a dielectric surface [76, 297], etc.. Conical metal rods

have the potential to be one of the best types of highly effective plasmon

nanofocusing structures with achievable field enhancements at the tip of ~ 1600 for a

gold-vacuum nanofocusing tip of optimal length and taper angle [283]. The

preliminary pre-focusing of the plasmon energy due to its annular propagation on a

flat metal surface, combined with the subsequent nanofocusing by a tapered metal

rod, has led to the design of the so-called ‘wizard hat’ nanofocusing structure with

the achievable local field enhancements at the tip up to ~ 3500 – ~ 10000 for a tip

radius between ~ 5 nm and ~ 2 nm, respectively [78]. Such local field enhancements

are associated with up to ~ 1016 Raman signal enhancements, which are ~ 2 orders of

magnitude higher than those typically required for single-molecule detection [119].

However, in order to use a tapered rod in spectroscopic applications, its tip and

the sample (or a molecule) to be analysed must be situated in close proximity (within

96

a distance of about a few nanometres), as is the case with techniques such as tip-

enhanced Raman spectroscopy (TERS) [34]. This is a potential cause for reduced

sensitivity of nanofocusing structures (including tapered rod structures) with regards

to their capabilities to detect residual amounts of substances in the environment (such

as air or water). Indeed, if the concentration of the molecules to be detected is so

small that none of them are situated within the hotspot created by a nanofocusing

structure, then no Raman signal can be generated, irrespective of the achievable local

field enhancement. Thus, the molecules should be delivered to the created hotspot by

means of some delivery mechanism, such as molecular diffusion or nanofluidics.

Nanofluidics has already been demonstrated to have a significant potential for

increasing the efficiency of molecule delivery to plasmonic hotspots, for example, in

a porous membrane with nanoholes [213, 325, 326]. However, investigations into the

use of a pressure gradient to deliver molecules for testing to the plasmonic hotspot at

the tip of a nanofocusing tapered metal rod are yet to be conducted.

Therefore, the aim of this chapter is to present a detailed theoretical and

numerical analysis of the functional capabilities resulting from combining tapered

rod nanofocusing structures with nanofluidic flows to achieve enhancement of the

detection sensitivity through the nanofluidic delivery of molecules for testing to the

plasmonic hotspot at the tip of a focusing structure. The determination and analysis

of the nanofluidics regimes critical for the mechanical stability of the metallic

nanofocusing structure and the induced mechanical stress and drag forces will be

carried out, with this analysis conducted for both air and water. The detection limit

for the flow-through sensor is also evaluated, thus enabling optimisation of

nanofocusing sensor technologies for the detection and monitoring of residual

substances in these most common environmental media.

97

4.2 MAXIMUM STRESS ON A TAPERED METAL ROD IN A VISCOUS FLUID FLOW

The considered rod structure with nanofluidics is shown in Figure 4.1a. The

nanofluidic flow is expected to be perpendicular to the axis of the tapered rod. A

fluid sample with the residual molecules to be tested (for example, by means of

surface-enhanced Raman spectroscopy) can be delivered by means of microfluidics

into a microscopic gap of width W between two solid interfaces (Figure 4.1b).

Tapered rod nanofocusing structures can be fabricated on one of these surfaces, such

that their rounded tips are positioned close to the middle of the gap. For example,

these could be ‘wizard hat’ nanofocusing structures [78], where the plasmons are

generated by an incident circularly polarised laser beam (incident wave in Figure

4.1b) in spiral gratings of circular shape (with the phase shift of 2π within one

revolution) coaxial with the conical section of the nanofocusing metal rod (Figure

4.1b). Such structures combine the advantages of the efficient sub-wavelength

nanofocusing by means of a section of a tapered metal (e.g., gold) nanorod with the

initial diffraction-limited pre-focusing of the plasmon energy due to its annular

propagation (convergence) on a flat surface [78]. One of the important questions of

this design is related to the mechanical stability of the fabricated nanofocusing

structures and the critical fluidic regimes leading to the breakdown of the conical

sections of the wizard hats.

98

Figure 4.1. (a) A nanofocusing tapered metal rod with a hotspot at the rounded tip of radius r and the taper angle q, in a fluid flow with the uniform initial velocity of U. (b) A schematic of a sensor combining nanofluidic delivery of a fluid sample into the gap of width W with an array of the

nanofocusing ‘wizard hat’ structures [78].

The analysis of such critical regimes and mechanical stress in the tapered

sections of the wizard hat structures will be conducted in this chapter in the

approximation of an infinitely long tapered rod in a uniform fluid flow with the

velocity U (Figure 4.1a) under normal atmospheric conditions. It is appreciated,

however, that the flow of a viscous incompressible fluid in a uniform microscopic

gap (Figure 4.1b) cannot be uniform across the gap, but is rather given by the simple

analytical equation which reduces to reveal that the maximum velocity occurs at a

distance of W/2 from either wall, and is given by [307]:

−−−−= Kn

dzdPWU

v

v

ss

µ22

41

2

2

max (4.1)

99

where ρ is the fluid density, µ is dynamic viscosity, dP is the pressure difference in

the fluid along the direction of its motion (i.e. along the gap – Figure 4.1b), Kn is the

Knudsen number, and σv is the tangential momentum accommodation coefficient.

The Knudsen number, Kn, is the ratio of the mean-free path of the molecules to the

gap width, W. This equation assumes that partial slip boundary conditions are

employed on the walls of the gap, since the tangential component of the fluid

velocity will be non-zero for Kn > 10-3 [312]. The non-uniformity of a viscous flow

(and the impact of other rods on the fluid velocity field in the gap) will result in a

reduction of the drag force exerted on the conical section of each of the wizard hat

structures. Therefore, the consideration of the uniform fluid flow (Figure 4.1a) with

the velocity U that is equal to the maximal velocity in the middle of a microscopic

gap (Figure 4.1b) will give somewhat overestimated values of stress in the rods and

thus a lower estimate of the critical fluidic regimes resulting in the mechanical

breakdown of the nanofocusing structures. This will be sufficient for the

understanding of approximate design limitations for the proposed nanofocusing

detection technique combined with nanofluidics.

Drag is the net force that results from the translational motion of a body in a

viscous fluid (or the motion of the fluid with respect to the body), exerted in the

direction opposite to the direction of the relative flow velocity. Both pressure drag

and friction drag contribute to the total drag on a cylinder in a viscous fluid flow

[369]. A uniform flow of a fluid with the density ρ, dynamic viscosity μ, and velocity

U around a cylinder with the radius R, can be characterised by a dimensionless

parameter known as the Reynolds number [369]:

100

µρ

=UR2Re (4.2)

If the Reynolds number is sufficiently small, the fluid flow is laminar and the

drag is predominantly due to friction caused by the fluid viscosity. In this case, the

drag force per unit length on a cylinder in a creeping flow is given by [370]:

)]UR/(log[3.70U4=FD ν

πµ(4.3)

where ν is the kinematic viscosity. However, this equation is only valid up to

Re ≈ 1 [371]. If Re > 1, empirical approaches are typically used to fit a dependence

to the experimental data [372, 373]. For example, it was found that the drag force on

a section of a cylinder of unit length in a fluid flow normal to the axis of the cylinder

is given as [371]:

D2

D RCUF ρ≈ (4.4) where

32

D Re101C−

+= (4.5)

is the drag coefficient for the cylinder. The drag coefficient is a

dimensionless quantity that is used to quantify the drag on an object in a fluid. Eq.

(4.4) and (4.5) are approximately valid within the range of the Reynolds numbers 1.0

< Re < 2×105 [371].

Before determining the mechanical stress distribution in tapered metal rods,

the structure must first be characterised. A tapered nanofocusing rod can be

approximated to be a cylinder with a diameter that decreases constantly with

increasing height. As such, the tip of the rod is assumed to be flat, rather than

rounded for this theoretical analysis. The radius of the rod at a distance x from the

base of the rod is:

101

( ) ( )2/tanxLR q−= (4.6) For the purpose of this analysis, the minimum radius of the rod is defined as

the radius at which the tapered section of the rod would transition into the rounded

tip. Therefore, by combining Eq. (4.4), (4.5), and (4.6), the drag force on an element

of a tapered rod in a viscous fluid flow can be expressed as:

( )( )

µ

−

q

ρ+−

q

ρ=

−32

2D

xL2

tanU2101xL

2tanUF (4.7)

Using Eq. (4.6), the stress at any cross-section along the length of the rod can

therefore be determined using the equation:

( )3

xL2

tan2

M32

−

q

π

=s(4.8)

where M is the bending moment [374].

Plotting stress against distance from the tip allows the maximum stress on the

tapered rod to be determined for a given taper angle and fluid velocity. If the

maximum stress exceeds the yield strength of the material from which the rod is

fabricated, the rod will fail. The yield strength of gold can depend on the history and

heat treatment of the material. Although the yield strength of gold has been shown to

be diameter-dependent when in the form of a nanowire [375], it will be taken to be

independent of diameter for the purpose of this analysis.

However, there is an additional limiting case to this problem when considering

the flow of air around a tapered rod. Air can be considered incompressible up to a

velocity of ≈ 0.3 Ma (≈ 100 m/s), where Ma is the Mach number, which is defined

as the ratio of the initial flow velocity to the speed of sound [369]. This region,

which ends at 0.4 Ma, is known as the shockless regime, and is strongly dependent

102

on Re [376]. Above this velocity, shock waves will intermittently alternate from one

side of a cylinder to the other, with a further velocity increase leading to permanent

shock waves being produced simultaneously on both sides of the cylinder. Thus, due

to the unknown impact of such shock waves on the structural integrity of the tapered

rod, and to simplify the analysis by neglecting the investigation of compressible

fluid, this investigation will focus only on air velocities of 0.3 Ma and below. For

this reason, the maximum air velocity considered will be 100 m/s.

As this investigation presents only a first-order approximation of the drag force

on a tapered rod, the effects of boundary slip on the drag force are neglected. By

using the finite element analysis package COMSOL MultiphysicsTM to model the

drag force around cylinders of different diameters with both no-slip and partial slip

boundary conditions for air with a velocity of 100 m/s, it was determined that no-slip

boundaries result in a drag force that is ~ 6.8% to 12.5% larger than partial slip

boundaries. Consequently, the results presented here are an upper limit on the drag

force on the structure.

The structure to be analysed, as seen in Figure 4.1, is a tapered rod with a

length of 1 μm, which has been considered elsewhere [78]. It has been found that the

taper angle that results in maximal local field enhancement at the rounded tip of a

gold rod in air and for λvac = 632.8 nm is ~ 36 degrees [283], so this optimal angle

will be the taper angle that is highlighted in these results. As was discussed in the

introduction to this chapter, the field enhancement for a tip radius of 2 nm has been

calculated to be approximately three times larger than for a tip radius of 5 nm, but the

authors focused on rods with a tip radius of 5 nm because their fabrication is thought

to be more practically feasible. As a result, the analysis reported here will consider

103

tapered rods with a tip radius of 2 nm (as a limiting case) and 5 nm for the stress

analysis, while analysis of delivery rates of molecules to the sensing region will only

consider rods with a tip radius of 5 nm.

The drag force on the rod will act in the direction parallel to the velocity of

the fluid, which is defined as the z-direction in Figure 4.1. Using Eq. (4.7), Figure

4.2 was generated.

Figure 4.2. (a) Force distribution on a rod with a taper angle of 36 degrees, for air velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (b)

Cumulative force distribution on a rod with a taper angle of 36 degrees, for air velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (c) Normalised

cumulative drag force on rods with a taper angle of 5 degrees (solid curve) and 36 degrees (dashed curve) for an air velocity of 100 m/s. All rods in (a)-(c) have a tip radius of 2 nm.

Figure 4.2a is a plot of Eq. (4.7) for a taper angle of 36 degrees and air

velocities ranging from 20 m/s to 100 m/s. It is evident that the majority of the drag

force is generated towards the bottom of the structure, which is as expected, since the

surface area of the structure in the fluid stream increases with distance from the tip.

104

Figure 4.2b shows the cumulative force distribution as a function of velocity.

The cumulative force at a given value of x is the sum of the forces on the rod from

the tip of the rod to that point.

Figure 4.2c shows the normalised drag force on the rod for taper angles of 5

degrees (solid curve) and 36 degrees (dashed curve), at an air velocity of 100 m/s, in

order to illustrates how the force distribution on the tapered rod changes with taper

angle. It can be seen that the percentage of drag force concentrated towards the tip of

the rod decreases as taper angle increases and this will affect the magnitude and

position of maximum stress on the tapered rod.

Figure 4.3. (a) Stress distribution on a rod with a taper angle of 36 degrees and tip radius of 2 nm for air velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), in increments of 20 m/s. (b) Maximum stress on the rod at air velocity of 100 m/s as a function of taper angle, for a tip radius of 2 nm (solid curve) and 5 nm (dashed curve). The horizontal line represents the yield strength of gold.

(c) Position of maximum stress on the rod as a function of taper angle, for an inlet velocity of 100 m/s and a tip radius of 2 nm (solid curve) and 5 nm (dashed curve).

105

Figure 4.3 illustrates the stress on a rod with a taper angle of 36 degrees as a

function of air velocity, the maximum stress on a tapered rod as a function of taper

angle and tip radius, and the position of maximum stress as a function of taper angle

and tip radius.

Figure 4.3a was generated using Eq. (4.8) and shows the stress on a rod with a

taper angle of 36 degrees rod as a result of the drag force. It shows that stress

increases as the distance to the tip decreases, rising to a maximum value at a distance

of 20.3 nm from the tip, before quickly dropping away towards 0 Pa. It can be seen

that the magnitude of the stress on the rod is dependent on fluid velocity, with this

relationship being linearly proportional for a given taper angle over the velocity

range considered. Conversely, the position of maximum stress is highly insensitive to

velocity over the velocity range considered.

Figure 4.3b illustrates the maximum stress on a tapered rod as a function of

taper angle for air velocities of 100 m/s and tip radii of 2 nm (solid curve) and 5 nm

(dashed curve). The horizontal solid line indicates the yield strength of gold, which is

taken to be a conservative value of 55 MPa [377]. It can be seen that for a tip radius

of 5 nm, the maximum stress does not exceed the yield strength for an air velocity of

100 m/s over the considered range of taper angles. However, for a tip radius of 2 nm,

the maximum stress will exceed the yield strength for taper angles below 6.1 degrees

when the air velocity is 100 m/s. A rod with a taper angle of 5 degrees and a tip

radius of 2 nm can withstand a maximum air velocity of 75 m/s, while a rod with a

taper angle of 6 degrees and a tip radius of 2 nm can withstand a maximum air

velocity of 98 m/s.

106

Figure 4.3c shows the position of maximum stress for a tapered rod with a tip

radius of 2 nm (solid curve) and 5 nm (dashed curve) for an air velocity of 100 m/s,

as a function of taper angle. It can be seen that, for the considered tip radii, the

position of maximum stress rapidly moves towards the tip as the taper angle is

increased from 5 to 15 degrees, then continues moving towards the tip at a slower

rate with increasing taper angle. Increasing the tip radius from 2 nm to 5 nm for a

given taper angle has a significant effect on the position of maximum stress for taper

angles below 15 degrees, with the effect becoming less significant above this taper

angle. A rod with a length of 1 μm and a taper angle of 5 degrees has a radius of less

than 100 nm at the base, so even a small change in the tip radius can significantly

affect the observed stress distribution. This finding may have important practical

implications if the goal is to use rods with small taper angles in flow-through

applications, but can be largely disregarded for taper angles in the range of the

optimal taper angle for the gold wizard hat in air.

The same analysis of force and stress on the structure can be conducted using

water, rather than air. Figure 4.4 was generated using Eq. (4.7), with the density and

viscosity of water replacing the corresponding values for air.

The force distributions in Figure 4.4a-b bear a striking resemblance to those in

Figure 4.2, although the magnitude of the drag force is larger because of the greater

density and viscosity of water. Figure 4.4c shows the normalised drag force on the

rod for taper angles of 5 degrees (solid curve) and 36 degrees (dashed curve) at a

water velocity of 100 m/s, in order to illustrate how the force distribution on the

tapered rod changes with taper angle. This figure is similar in appearance to Figure

4.2c, although the percentage of drag force concentrated towards the tip of the rod

107

decreases more dramatically with increasing taper angle than was seen for the

corresponding increase in taper angle in air.

Figure 4.4. (a) Force distribution on a rod with a taper angle of 36 degrees, for water velocities ranging from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (b) Cumulative force distribution on a rod with a taper angle of 36 degrees, for water velocities ranging

from 20 m/s (bottom curve) to 100 m/s (top curve), increasing in increments of 20 m/s. (c) Normalised cumulative drag force on rods with a taper angle of 5 degrees (solid curve) and 36 degrees

(dashed curve) for an air velocity of 100 m/s. All rods in (a)-(c) have a tip radius of 2 nm.

Due to the significantly higher density and viscosity of water, the drag force on

a tapered rod due to water flowing at a given velocity is much greater than the drag

force produced by air on a rod with the same taper angle. For this reason, the analysis

108

is limited by the yield strength of gold. As such, it is necessary to determine the

velocity of water at which rods will fail, as a function of taper angle. Eq. (4.8) was

used to determine the minimum velocity at which the maximum stress on the rod

exceeded the yield strength of gold. Figure 4.5 illustrates the breaking velocity of

tapered rods with a tip radius of 2 nm (solid curve) and a tip radius of 5 nm (dashed

curve), as a function of taper angle.

Figure 4.5. Velocity of water producing a maximum stress that exceeds the yield strength of gold as a function of taper angle, for a tip radius of 2 nm (solid curve) and 5 nm (dashed curve).

This indicates that the maximum velocity of water that a tapered rod can

sustain increases with increasing taper angle, with this increase being non-linear in

nature. It also demonstrates that, as was seen in the analysis for air, the tip radius

strongly influences the maximum stress on the rod, although this difference is

noticeable across the entire range of taper angles that were considered. In fact,

increasing the tip radius from 2 nm to 5 nm increases the breaking velocity by a

factor of ~1.5 times for a given taper angle.

4.3 NANOFLUIDIC DELIVERY OF VAPOUR MOLECULES TO THE FIELD HOTSPOT OF A TAPERED NANOROD

It has been shown that the yield strength of gold will be the limiting factor for

taper angles below 6.1 degrees subjected to an air velocity of 100 m/s, while the

compressibility of air will be the limiting factor to this analysis above this angle. In

109

the case of water, the yield strength will be the limiting factor for a range of the taper

angles considered. Having established the fluid velocity required to damage the

tapered rod as a function of taper angle, these values can then be used to determine

the flow rate of air or water to the field hotspot produced at the tip of a tapered

nanorod.

As the nature of the field enhancement and size of the sensing region for

tapered rods in water have not been reported in the literature, the analysis of delivery

rates will be confined to air. In addition, for the purpose of this analysis of delivery

rates, only tapered rods with a tip radius of 5 nm will be considered, as this was the

tip radius of the wizard hat previous considered in the literature [78].

The sensing region for a tapered rod with a tip radius of 5 nm in air is defined

as a circular virtual surface in the x-y plane with a radius of 10 nm and origin at (0,L)

[78], as shown in Figure 4.6. The mass flow rate of fluid through this surface is

defined as:

∫ dAvz ρ. (4.9)

where vz is the component of the fluid velocity normal to the sensing region and dA

is an element of the virtual surface.

Figure 4.6. The sensing region at the tip of a tapered rod with a taper angle of 36 degrees and a tip radius of 5 nm.

110

COMSOL MultiphysicsTM was used to model the airflow around a tapered rod

with a tip radius of 5 nm in a chamber with height = 1.5 µm and length = 20 µm, as a

function of taper angle. Eq. (4.1) can be used to determine the appropriate inlet

pressure to achieve a desired air velocity for a given set of microchannel dimensions.

For a microchannel with the dimensions listed above, an inlet pressure of 2.05 atm

(given an outlet pressure of 1 atm) will be sufficient to produce an air velocity of 100

m/s in the middle of the channel. Figure 4.7 shows that there is a non-zero velocity at

the surface of the tapered rod when partial slip boundary conditions are employed.

Figure 4.7 Air velocity around the tip of a tapered rod with a taper angle of 36 degrees.

The mass flow rate through the sensing region varied by less than 5% over the

range of taper angles considered, so it can be considered to be essentially invariant

with taper angle. The mass flow rate of air to the sensing region at the tip of a rod

with a taper angle of 36 degrees was calculated to be 1.28x10-14 kg/s. The

concentrations of various molecules in air required to allow one molecule per second

to flow through the sensing region, for an air velocity of 100 m/s, can then be

determined.

111

It was reported that a 1 s acquisition time was sufficient to obtain a SERS

spectrum from 200 pg of TNT deposited on a nanoscale textured substrate with an

enhancement factor of 104, as a result of the presence of NO2 scissoring and

stretching modes in the spectrum [378]. Assuming a field enhancement of ~ 3500 at

the tip of a tapered rod with a tip radius of 5 nm, a SERS enhancement factor of ~

1014 may be possible, meaning that a far lower detection limit could be achievable.

A concentration of 36 pg/L of trinitrotoluene (TNT) molecules, 10 pg/L of

sulphur dioxide molecules, or 2.7 pg/L of ammonia molecules will be sufficient to

deliver one molecule per second to the sensing region and, given the possible

enhancement factor, these concentrations could be sufficient to produce a SERS

signal.

These results compare extremely favourably with the typical required

sensitivity of an air monitoring sensor of 1 ng/L that was used as the basis of the

calculations in the previous chapter. It can therefore be concluded that combining

tapered rod nanofocusing structures with nanofluidic flows will enable the

development of a new type of SERS-based sensor with a low detection limit.

4.4 CONCLUSIONS

In this chapter, the force and stress distributions on a tapered metal rod placed

in a viscous fluid flow were analysed and the delivery rate of air to the tip of the rod

was evaluated. An equation was derived to allow the calculation of the drag force on

each element of a tapered rod, from which both the magnitude and position of stress

on a rod as a function of taper angle and fluid velocity were able to be determined. It

was shown that the maximum stress on a rod is highly dependent on fluid velocity,

112

taper angle, and tip radius, while the position of maximum stress is a function of

taper angle tip radius, but highly-insensitive to velocity over the range investigated.

It has been demonstrated that gold rods with a tip radius of 2 nm can withstand

an air velocity of 100 m/s if they have a taper angle greater than 6.1 degrees, while

rods with a tip radius of 5 nm can withstand an air velocity of 100 m/s over the range

of taper angles considered. The breaking velocity of gold rods in water is highly-

dependent on taper angle and tip radius, with breaking velocity being the limiting

factor for the parameters considered.

Finally, the delivery rates of molecules to the sensing region at the tip of the

tapered rod were calculated, with 36 pg/L of trinitrotoluene (TNT) molecules, 10

pg/L of sulphur dioxide molecules, or 2.7 pg/L of ammonia molecules being the

minimum concentrations required to deliver one molecule per second to the sensing

region of a rod with a taper angle of 36 degrees.

Thus, it can be concluded that tapered gold rods are highly-suited to use in

flow-through spectroscopic devices because they combine the benefits of strong field

localisation at the tip with the strength to withstand a significant air velocity, which

will allow large numbers of new molecules to be pumped into the sensing region in a

given time interval. This will allow for the detection, identification, imaging and

analysis of single molecules, which will be important in the development of new

nano-optical sensors for the detection of trace amounts of substance such as

explosives and drugs.

The analysis described in this chapter can be applied to tapered rods fabricated

from other metals (e.g., silver or aluminium) by comparing the maximum stress

generated with the yield strength of the respective metal. In order to expand this

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research to allow for the calculation of the delivery rates of different molecules to the

sensing region in water, the field localisation at the tip of a tapered rod in water must

first be characterised, as has been done for tapered rods in air [283]. This will allow

the size of the sensing region to be determined, from which the delivery rates can be

calculated.

As the sensing region in air is miniscule when compared with the size of the

sensing chamber, and since the flow is not directed to the sensing region as in the

previous chapter, the delivery rate of molecules to the sensing region of a tapered rod

could be increased by implementing flow geometries that direct more air towards the

sensing region. This could be the subject of future investigations.

115

Chapter 5: Plasmon Nanofocusing with Negative Refraction in a High-Index Dielectric Wedge

5.1 INTRODUCTION

As has been discussed in previous chapters, a range of metallic nanostructures

have been considered for SPP nanofocusing, including sharply tapered metal rods

[4,5,7-9], tapered metallic gaps and grooves [3,6,10-12,22-24,29,38,39], sharp metal

wedges and tapered metal films on dielectric substrates [3,13-17]. In a typical

nanofocusing structure, the focused SPP mode experiences increased localisation and

enhancement of its local field along the direction of the taper, i.e. in the direction of

progressively reducing spatial dimensions of the guiding metallic structure.

There is only one recent paper that considered a structure in which SPP

nanofocusing occurred in the direction opposite to the direction of the structural taper

[296]. In this case, nanofocusing was achieved using a high-refractive index

dielectric wedge on a metal substrate – that is, a metal-insulator-insulator (MII)

structure. Increasing the thickness of the dielectric layer results in increasing the SPP

effective refractive index and, thus, its localisation and local field enhancement

[296]. The analysis was largely limited to the adiabatic regime with dielectric layers

(wedges) having small taper angles and permittivities that were smaller than the real

part of the metal permittivity. Therefore, the smallest achievable localisation region

for SPP in these structures could not be smaller than the localisation region for SPP

on an interface between the metal and the semi-infinite dielectric.

116

In addition, interesting physical effects associated with the possible formation of

plasmonic caustics in the wedge have so far been overlooked. The authors noted that,

despite the electric field enhancement being maximal at the metal-wedge interface,

there was still significant field enhancement at the wedge-cladding interface which

may be utilised for spectroscopic applications. Therefore, it is also of interest to

observe how the magnitude and location of the field maximum varies with taper

angle in order to optimise the dielectric wedge for use in SERS-based sensing

applications.

The aim of this chapter is to demonstrate and analyse SPP nanofocusing by a

dielectric wedge on a metal substrate (MII nanofocusing structure) in adiabatic and

non-adiabatic regimes, for the cases where the dielectric permittivity of the wedge is

smaller or larger than the magnitude of the real part of the metal permittivity. This

analysis is based on the geometric optics approximation (GOA), but also employs the

finite element method (COMSOL MultiphysicsTM), since the GOA is not applicable

at the caustic (the derivative of the SPP wavenumber with respect to the x-coordinate

goes to infinity, which is unrealistic), nor for larger taper angles.

Initially, the case of zero or very small dissipation in the metal is considered in

order to understand the underlying physics behind the SPP modes under

investigation. It will be shown that this MII structure leads to a range of important

physical phenomena associated with nanofocusing, including formation of plasmonic

caustics within a nanoscale distance from the tip of the wedge, nanofocusing of SPPs

with negative refraction, mutual transformation of focused SPP modes, and

significant local field enhancements in the adiabatic and strongly non-adiabatic

regimes. Simple analytical equations determining the position of the plasmonic

117

caustic corresponding to mutual transformation of SPPs with positive and negative

refraction are derived in the adiabatic approximation. Typical field distributions and

local field enhancements are determined, interpreted and discussed and the structure

is evaluated for use in SERS-based sensing applications.

5.2 THE STRUCTURE AND ADIABATIC APPROXIMATION

The considered MII structure consists of a tapered dielectric layer with the taper

angle α and large dielectric permittivity εl (e.g., silicon or chalcogenide glass)

between a metal half-space with the permittivity εm = e1 + ie2 and cladding (e.g., air)

with the permittivity εc < |e1| (Figure 5.1a). The tip of the dielectric wedge is

assumed to be infinitely sharp because, as will be seen below, this does not affect the

nanofocusing conditions and obtained local field enhancements (unlike in the

previously considered nanofocusing configurations, where radius or size of the tip

was a crucial parameter determining local field enhancement). An SPP mode is

excited on the metal-cladding interface (i.e. some distance from the tapered dielectric

layer). The generated SPP propagates in the direction of increasing thickness of the

layer (i.e. opposite to the direction of the taper), as indicated by the wave vector q in

Figure 5.1a. The vacuum wavelengths of 459.2 nm and 632.8 nm were chosen

because of their practical availability and the need to have relatively small metal

permittivities, such that εl ≥ |e1|

The adiabatic approximation is applicable if the taper angle is sufficiently small,

such that the variations of the SPP wave number q = q1 + iq2 within one SPP

wavelength are negligible [75, 279]:

1)( 11 <<− dxqd (5.1)

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In this approximation, the SPP does not “feel” the taper, and its propagation

parameters are determined by the local thickness of the high-index layer.

Figure 5.1. (a) High-index tapered dielectric layer with the permittivity εl and taper angle α between a metal half-space with the permittivity εm, and cladding with the permittivity εc; an SPP mode with the wave vector q propagates in the direction of increasing thickness of the layer. (b) The dependences of the SPP wave number q on thickness h of the dielectric layer: (1) λ0 = 459.2 nm, εl = 9, εm = – 6.5 (e2 = 0); (2) λ0 = 632.8 nm, εl = 9.4, εm = – 9.3 (e2 = 0); (3) λ0 = 632.8 nm, εl = 9.29, εm = – 9.3 (e2 = 0); λ0 is the vacuum wavelength; the real parts of the metal permittivities correspond to silver and gold,

respectively [379], and εc = 1 (vacuum). (c) A zoomed-in version of the dependences in (b).

For the sake of simplicity, and to reveal the essential physics behind the

considered SPP modes, the case of zero or very small dissipation (i.e. e2 ≈ 0) is

considered initially. It can be seen that the number of existing SPP modes and their

dispersion in the considered MII structure depend upon the permittivity of the layer

(dielectric wedge) and its thickness (Figure 5.1b,c and Figure 5.2a,b). For example,

as was shown at [380], if the layer permittivity εl > |e1| and the layer thickness h is

smaller than a certain cut-off (critical) thickness hcr1, two different SPP modes exist

119

in the layer, which correspond to the two different (upper and lower) branches of

curves 1 and 2 in Figure 5.1b,c. The mode corresponding to the lower branches of

curves 1 and 2 in Figure 5.1b,c will be called the low-q mode, and the mode

corresponding to the upper branches will be called the high-q mode. The low-q mode

is the one that is asymptotically transformed into the SPP mode on the metal-

cladding interface if the thickness of the layer tends to zero. Both of the modes

merge at the cut-off thickness hcr1 of the layer (Figure 5.1b,c). No SPP modes exist if

the layer thickness h > hcr1 (curves 1 and 2 in Figure 5.1b,c).

Figure 5.2. (a) The dependences of the SPP wave number q on thickness h of the dielectric layer for εm = – 9.3 (e2 = 0), λ0 = 632.8 nm, εc = 1 and three different permittivities of the layer: (1) εl = 7; (2) εl = 8.5; (3) εl = 9.29 (the same as curve 3 in Figure 5.1); λ0 is the vacuum wavelength, and the real part of the metal permittivity corresponds to gold [379]. (b) A zoomed-in version of the dependences in

(a).

In the adiabatic approximation, as the generated SPP mode propagates from the

left along (and opposite to the direction of) the taper (Figure 5.1a), its group velocity

decreases to zero at h = hcr1 (curves 1 and 2 in Figure 5.1b,c and Figure 5.3b). This

results in an accumulation of the SPP energy near the turning point, which thus

represents a plasmonic caustic with an infinite increase of the SPP amplitude (Figure

120

5.3a). The physical nature of this caustic is similar to that of the SPP total external

reflection from a dielectric layer with a thickness exceeding the critical thickness hcr1

[380]. However, there are two major differences between the considered effects in

the tapered MII structure and the SPP total external reflection [380]. Firstly, the MII

nanofocusing structure is associated with significant (in the adiabatic approximation,

infinite – Figure 5.3a) enhancement of the local field near the caustic. Secondly, the

low-q SPP mode with positive refraction (the lower branches of curves 1 and 2 in

Figure 5.1b,c) is not just reflected from the caustic at h = hcr1 (as in the case of total

external reflection [380]), but is rather transformed into a different, high-q, SPP

mode with negative refraction (the upper branches of curves 1 and 2 in Figure

5.1b,c) – see below for more detail.

Figure 5.3. (a) The typical dependences of the magnitude of the electric field amplitude in the focused plasmon on local thickness h of the tapered layer in the adiabatic approximation for the low-q modes (curves 1a and 2a) and high-q modes (curves 1b and 2b) for the two structures with λ0 = 459.2 nm, εl = 9, e1 = – 6.5, e2 = 0, εc = 1 (curves 1a and 1b), and λ0 = 632.8nm, εl = 9.4, εm = – 9.3, e2 = 0, εc = 1 (curves 2a and 2b). (b) The dependences of the group velocity of the plasmon on thickness h of the high-index dielectric layer: (1) λ0 = 459.2 nm, εl = 9, e1 = – 6.5, e2 = 0, εc = 1; (2) λ0 = 632.8nm, εl =

9.4, εm = – 9.3, e2 = 0, εc = 1.

121

To demonstrate negative refraction of the high-q SPP mode, Figure 5.3b shows

the dependences of the SPP group velocity on the thickness of the dielectric layer. It

can be seen that the group velocity changes sign at the plasmonic caustic (the upper

branches of the group velocity curves in Figure 5.3b, correspond to the lower

branches of curves 1 and 2 in Figure 5.1b,c. Therefore, despite the fact that both the

branches of curves 1 and 2 in Figure 5.1b,c correspond to positive vectors q pointing

in the positive x-direction, the direction of the energy flow in the high-q mode (the

upper branches) is in the negative x-direction. The energy flow at the caustic is zero,

which reflects the infinite (in the adiabatic approximation) accumulation of energy at

this point (Figure 5.3a).

If the layer permittivity εl < |e1|, SPP modes exist in the structure at any thickness

of the layer (Figure 5.2). When the layer permittivity is reduced just below |e1|, three

different SPP modes can exist in the MII structure at a given thickness of the layer

within the range: hcr1 > h > hcr2 (curves 3 in Figure 5.1 and Figure 5.2 with hcr1 ≈ 20

nm and hcr2 ≈ 5 nm). Two of these modes, corresponding to the lower and upper

branches, are characterised by positive refraction, while the intermediate mode has

negative refraction. Both critical values of the layer thickness hcr1,2 correspond to the

infinite derivative of the wave number q with respect to h (curves 3 in Figure 5.1b,c

and Figure 5.2a,b and zero group velocity of the SPP mode (Figure 5.3b). Thus, there

are two different plasmonic caustics in the MII wedge structure (Figure 5.1a) at h =

hcr1 and h = hcr2, which correspond to the points of mutual transformation of the

respective modes. When the layer permittivity is reduced to a critical value εlc, the

two caustics reduce to just one (i.e. hcr1 = hcr2 – see curves 2 in Figure 5.2a,b

corresponding to εl ≈ εlc ≈ 8.5). At lower permittivities of the layer (εl < εlc), the

plasmonic caustics disappear altogether from the SPP dispersion curve (curves 1 in

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Figure 5.2a,b). If h is increased, the upper branches of curve 3 in Figure 5.1b and

curves 1 to 3 in Figure 5.2a tend to plateaus that correspond to the respective SPP

wave numbers at the interface between the metal and a dielectric halfspace with the

permittivity εl.

5.3 POSITION OF THE PLASMONIC CAUSTIC

The position of the plasmonic caustic and the corresponding plasmon wave

number can be determined analytically under some rather general (for this problem)

conditions. The general dispersion relation for the plasmons in an MII structure with

the layer thickness h and permittivity εl, placed between a metal substrate (complex

permittivity εm) and cladding (permittivity εc), is given as:

tan(klzh) =klzεl (αcεm + αmεc )klz

2εcεm − αcαmεl2 (5.2)

where

2

22

cq cc

ωε−=α , αm = q2 − εm

ω2

c 2 , klz = εlω2

c 2 − q2 (5.3)

ω is the angular frequency, and c is the speed of light.

For simplicity, in the derivation below, dissipation in the metal is assumed to be

zero (i.e. e2 = 0), in which case klz is either real or imaginary. The first caustic

(between the low-q and high-q SPP modes) typically occurs at a small layer

thickness (Figure 5.1b,c and Figure 5.2a,b and relatively small values of klz, so that:

klz h <<1 (5.4)

In this case, Eq. (5.2) can be reduced to:

123

h =εl (αcεm + αmεc )klz

2εcεm − αcαmεl2 (5.5)

which means that h is a function of the wave number q, and this function has a

maximum (or inflection – curve 2 in Figure 5.2b) at the critical value qcr1 of the

wave number corresponding to the first (lower) caustic (Figure 5.1b,c and Figure

5.2a,b).

If it is also assumed that:

εc klz2εm << αcαmεl

2 (5.6)

(the validity of this condition near the caustic will be demonstrated below), then the

first term with klz2 in the denominator in Eq. (5.5) can be neglected. In this case,

differentiating Eq. (5.5) with respect to q and equating the derivative to zero, gives

the following equation for q = qcr1:

011

22 =+== crcr qq

ccm

qq

mmc dq

ddq

d αεα

αεα (5.7)

Here, substituting Eqs.(5.3), gives:

qcr12 = εc

ω2

c 21+ c1/ 3

1− c−2 / 3 ; c ≡ |εm|/εc (5.8)

In order to demonstrate that the assumption Eq. (5.6) is valid, we need to make

sure that the obtained values of qcr1 satisfy condition (5.6). To do this, Eq. (5.8) is

substituted into inequality (5.6) and the following assumptions are made:

mc εε << , lc εε << (5.9)

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which are the natural assumptions for the considered structure with a high-index

dielectric layer whose permittivity is assumed to be larger than, or close to, the real

part of the metal permittivity. As a result, inequality (5.6) can be reduced as:

3/13/2 ~

11

cε−c

+cε<<ε mmc ,

which is always satisfied under the general conditions (5.9).

The substitution of Eq. (5.8) into Eq. (5.5) gives the cut-off layer thickness hcr1

that corresponds to the first (lower) caustic (Figure 5.1b,c and Figure 5.2a,b). The

calculated values of hcr1 are in good agreement (within ~ 10%) with those from the

exact numerical solution of the dispersion relation (Figure 5.1b,c and Figure 5.2a,b.

However, the analytical values of qcr1 (Eq. (5.8)) noticeably differ from the

respective values of qcr1 obtained using the numerical analysis for the lower caustic

(Figure 5.1b,c and Figure 5.2a,b). This is mainly because of the rapid variation of q

near the caustic (Figure 5.1b,c and Figure 5.2a,b). In order to improve the agreement

for both hcr1 and particularly qcr1, an additional correcting factor A = 2.5 is introduced

in Eq. (5.8):

qcr12 = Aεc

ω2

c 21+ c1/ 3

1− c−2 / 3 (5.10)

In this case, under the considered conditions (Eq. (5.9)), the analytically

calculated hcr1 and qcr1 agree with their numerical values (Figure 5.1b,c and Figure

5.2a,b) within ≾ 4% and ≾ 30%, respectively (for larger wedge and metal

permittivities of ~ 9 – see condition (5.9) – this agreement is within < 2.5% for both

hcr1 and qcr1).

Note again that the derived equations (5.5) and (5.10) are applicable only for

the determination of the position of the first (lower) caustic (Figure 5.1c and Figure

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5.2b). The position of the second (upper) caustic on curves 3 in Figure 5.1b and

Figure 5.2a cannot be accurately predicted by the derived equations (Eqs. (5.5) and

(5.10)), because the corresponding wave numbers (and their z-components in the

layer klz) appear to be sufficiently large to breach condition (5.4). In addition, if e2 is

not close to zero, then Eq. (5.7) is no longer valid. No simple analytical solution for

the caustic has been found in this case.

5.4 NUMERICAL RESULTS AND DISCUSSIONS

The considered adiabatic approximation (and its applicability condition (5.1)) is

not valid near the caustic, even for small taper angles, because the derivative of the

SPP wave number with respect to the x-coordinate turns to infinity (Figure 5.1b,c

and Figure 5.2a,b). This is why the approximation gives an unrealistic, infinite

increase of the SPP amplitude at the caustic (Figure 5.3a). Therefore, although the

analysis based on the adiabatic approximation and zero dissipation offers an

important physical insight into the optical effects in the MII nanofocusing structures,

this approach cannot give reliable quantitative evaluations of, for example, local field

enhancements near the plasmonic caustic and real field distributions in the structure,

even at small taper angles. As a consequence, numerical finite-element analysis using

the COMSOL MultiphysicsTM software package was used for small and large taper

angles. The resultant field distributions are shown in Figure 5.4a-c and Figure

5.5a,b.

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Figure 5.4. The distributions of the magnitude of the electric field |E| in the structure for the different taper angles and imaginary parts of the metal permittivity: (a) α = 5o, e2 = 0.1; (b) α = 5o, e2 = 1.12 (the imaginary part of the gold permittivitty); (c) α = 40o, e2 = 1.12. The other structural parameters are: εl = 9.4; λ0 = 632.8 nm, e1 = – 9.3 (gold at the considered wavelength), εc = 1; the colour scales for |E| are in the same arbitrary units, and subplots (b) and (c) correspond to the same incident SPP amplitudes at the tip of the wedge. (d) The profile of the magnitude of the electric field along the

plane at 0.1 nm above the metal-dielectric interface for α = 5o, e2 = 0.1 (corresponding to subplot (a))

.

Figure 5.5. The distributions of the z-component of the electric field Ez in the MII nanofocusing structure with the taper angle α = 20o, εl = 9.4; λ0 = 632.8 nm, e1 = – 9.3 (gold at the considered

wavelength), εc = 1, and two different imaginary parts of the metal permittivity: (a) e2 = 0.1; (b) e2 = 1.12 (the imaginary part of the gold permittivitty). (c) The profile of the magnitude of the electric field along the plane at 0.1 nm above the metal-dielectric interface for α = 20o, e2 = 1.12 (corresponding to

subplot (b)).

127

Figure 5.4 and Figure 5.5 further demonstrate the pattern of SPP nanofocusing in

the considered MII structures. Initially, the incident SPP propagates in the direction

of increasing thickness of the dielectric layer. The propagating SPP mode is

predominantly localised at the metal-wedge interface (Figure 5.4 and Figure 5.5) and

its wave number increases with increasing thickness of the wedge (Figure 5.1b,c and

Figure 5.2a,b). When the incident SPP reaches the caustic, it experiences significant

field enhancement (caused by the reduction of its group velocity – Figure 5.3b) and

scattering. If εl > |e1| (curves 1 and 2 in Figure 5.1), the SPP mode does not exist

beyond the plasmonic caustic, which means that all of its energy must be

transformed into reflected or scattered waves (apart from the fraction of the energy

that dissipates in the metal). The incident SPP mode can be scattered into a range of

other modes including modes guided by the dielectric (tapered) slab, bulk scattered

waves, and back-reflected SPP modes. The reflected SPP modes can include the low-

q reflected mode and high-q reflected mode. The reflection into the low-q mode can

be seen in Figure 5.5c, where the standing wave pattern on the left of the wedge tip

(positioned at x = 0) can be seen formed by the incident and reflected SPP modes at

the metal-cladding interface. However, the small amplitude of this standing wave

pattern suggests low efficiency of reflection of the incident low-q SPP into the low-q

reflected mode at the caustic. Decreasing the taper angle results in further reduction

of the already weak standing wave pattern on the left of the tip (Figure 5.4d), which

is an indication of significantly decreased reflections into the low-q SPP mode.

Some scattering can occur at the caustic (and/or in the taper itself – due to non-

adiabaticity at larger taper angles) into forward-propagating modes in the dielectric

wedge. The interference of these forward-scattered guided (by the wedge) modes

128

and/or bulk waves is demonstrated by the periodic field pattern beyond the caustic

(Figure 5.5c). That these forward-scattered waves are not types of SPP modes is

highlighted by the following arguments. Firstly, SPP modes cannot exist in the

considered structure (with εl > |e1|) at h > hcr1. Secondly, the period of the field

pattern beyond the caustic (Figure 5.5c) is significantly larger than the wavelength of

the SPP mode, which is a confirmation that this pattern is not formed by SPPs. Thus,

the periodic field pattern beyond the caustic (Figure 5.5c) is formed by bulk and/or

guided waves in the dielectric wedge. The weak decay of the amplitude of this

interference pattern with increasing x in Figure 5.5c is primarily caused by the

spread of the scattered energy within the gradually thicker wedge (and, possibly,

losses to bulk radiation). This periodic pattern of non-zero field beyond the caustic

disappears for small taper angles (Figure 5.4d), which means the absence (or

significant reduction) of this type of forward scattering in the adiabatic regime of

nanofocusing (even though the adiabatic approximation is always breached at the

caustic).

As a consequence, in both the adiabatic and nonadiabatic regimes, the

dominant scattering process for the incident low-q SPP mode at the caustic is its

back-reflection (transformation) into the high-q SPP mode with negative refraction

(Figure 5.4a,d and Figure 5.5a,b). The observation of this back-reflection pattern as

a result of the conducted numerical analysis corroborates the conclusions made in the

previous sections on the basis of the adiabatic approximation.

It is also important to note that this transformation of the incident low-q SPP

mode into the back-propagating high-q SPP mode with negative refraction is a

significantly different physical process (though it has similar physical origins)

compared to the SPP total external reflection from a high-permittivity dielectric layer

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[380], where no mode transformation occurred. SPP total external reflection occurs

from the physical termination (structural discontinuity) of the dielectric layer of a

thickness greater than hcr1 [380], which causes the inversion of the direction of the

SPP wave vector upon reflection. In the MII nanofocusing (tapered) structure, there

is no structural discontinuity at h = hcr1, which means that the wave vector of the

incident SPP does not have anything to be inversed by. Therefore, the direction of

the wave vector should remain the same upon reflection from the plasmonic caustic,

but the energy flow (SPP group velocity) should be inversed to ensure energy

conservation, and this can only happen if transformation occurs into the high-q SPP

mode with negative refraction. This is also the reason for the discussed lack of

efficient reflection of the low-q incident SPP mode into the low-q reflected SPP

mode at the caustic.

The reflected high-q SPP mode experiences further nanofocusing as it propagates

from the caustic towards the tip of the wedge. In this process, the phase velocity of

the high-q SPP mode tends to zero near the tip of the wedge (though the direction of

the phase velocity remains in the direction opposite to the taper due to SPP’s

negative refraction). The reflected high-q and incident low-q SPP modes interfere

producing an interference pattern with the period reducing towards the tip of the

wedge (Figure 5.4a,d and Figure 5.5a,b). This reduction in period demonstrates (is

caused by) further nanofocusing of the high-q SPP mode with negative refraction.

Importantly, this interference pattern, and thus nanofocusing of the high-q SPP mode

with negative refraction, can be seen in Figure 5.5b corresponding to the structure

with the realistic dissipation level in the gold substrate. Thus, nanofocusing of SPP

modes with negative refraction is not just a theoretical exercise, but is rather an

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observable effect in realistic structures (though noticeably dampened by dissipation

in the metal).

As can be seen from the comparison of Figure 5.4a,b and Figure 5.5a,b,

introduction of realistically increased levels of dissipation in the metal (Figure 5.4b

and Figure 5.5b) results in a noticeable shift of the field pattern corresponding to the

plasmonic caustic to the right, towards larger thicknesses of the dielectric wedge.

The position of the plasmonic caustic is determined by the critical layer thickness

hcr1. Therefore, it is argued that this effect has the same physical origins as the

increase of the critical thickness hcr1 of the dielectric layer (required for SPP total

external reflection) with increasing dissipation in the metal [380].

Another important physical outcome (which was neither analysed nor discussed

in [296]) is the significant nanofocusing effect by dielectric wedges at large taper

angles - far beyond the adiabatic approximation (Figure 5.4c and Figure 5.5a-c). For

example, strong local field enhancement occurs for large taper angles ~ 40o, and the

maximum of the SPP field is spatially separated from the wedge tip, occurring under

the wedge within about 15 nm beyond the tip (Figure 5.4c). This demonstrates that,

contrary to other nanofocusing configurations, field localisation and enhancement in

the considered MII structure is not directly related to the sharp tip of the wedge, but

is rather a phenomenon occurring inside the structure within some considerable

(though nanoscale) distance from the tip. Such rapid concentration of the SPP energy

within the distances as small as ~ 15 nm and into a spatial region of approximately

the same dimensions is a unique feature of the considered MII structure in the

strongly non-adiabatic regime of nanofocusing (Figure 5.4c). In addition, because

the tip of the dielectric wedge does not play a significant role in the process of

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nanofocusing, its fabrication requirements could be comparatively less precise,

which should simplify fabrication of the MII structures.

The significant and rather abrupt decrease of the SPP field in the immediate

proximity to the tip (Figure 5.4d and Figure 5.5c) is related to the boundary

conditions near the tip of the high-permittivity dielectric wedge. This strong field

reduction near the tip of the wedge can also be seen in Figure 5.4b,c. A boundary

condition for the component of the electric field normal to the wedge-cladding

interface can be written as: Ec⊥εc = El⊥εl , where Ec⊥ and El⊥ are the normal

components of the electric field in the cladding and dielectric wedge (layer),

respectively. Since εl >> εc, it follows that Ec⊥ >> El⊥ , which is the physical reason

for the significant step-wise variation of the SPP field across the cladding-wedge

interface (Figure 5.4a-d and Figure 5.5c). Since three different interfaces meet at one

point near the tip of the wedge, this causes the need to satisfy the three different

boundary conditions, which creates a complex local field pattern with a strong and

sharp local minimum of the field near the tip (Figure 5.4c and Figure 5.5c). The

spatial dimensions of the region with the reduced field near the tip could be ~ 10 nm,

which is much smaller than the wavelength of the incident SPP. In other words, the

boundary conditions near the tip of the wedge produce a local disturbance of the SPP

field in the form of a small region with significantly reduced field inside and outside

the wedge.

This local reduction of the field of the incident SPP near the tip of the wedge

(particularly inside the wedge) causes a technical difficulty with the definition and

evaluation of local field enhancement near the caustic. It would be reasonable to

define the local field enhancement near the caustic as the ratio of the field maximum

at the caustic to the amplitude of the incident SPP mode inside the wedge near the

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tip. However, because of the local reduction of the SPP field near the tip, the

amplitude of this field cannot be taken as the amplitude of the incident SPP mode in

the structure. Taking this field as a reference field in the definition of the local field

enhancement at the caustic would result in artificially large enhancements (up to ~

2000 times for the field intensity). On the other hand, taking the amplitude of the

incident SPP mode outside of the wedge (on the left of the tip of the wedge) as the

reference field could understate the local field enhancement inside the wedge

because the field in the wedge is reduced compared to the incident plasmon outside

the wedge by a factor up to ~ εl/εc >> 1. Nevertheless, keeping this in mind, the local

field enhancement at the caustic will be defined relative to the amplitude of the

incident SPP mode outside the wedge (on the left of the tip – Figure 5.1a). This is

because of the discussed significant uncertainties with choosing an appropriate

reference value for the SPP amplitude inside the wedge near the tip. It is therefore

necessary to keep in mind that the obtained local field intensity enhancements in the

wedge at the caustic (Figure 5.6a) could be regarded (at least for some applications)

as effectively underestimated by a factor up to ~ (εl/εc)2 >> 1 due to the high

dielectric permittivity of the wedge.

It is interesting that, if εl > |e1|, the local field enhancement at the plasmonic

caustic significantly increases with increasing taper angle beyond the adiabatic

regime (curve 1 in Figure 5.6a). For smaller values of the wedge permittivity this

increase is significantly less pronounced (if it exists at all – curves 2 and 3 in Figure

5.6a). As a consequence, for large wedge permittivities, strongly non-adiabatic

regime is preferable for achieving large local field enhancements at the caustic,

despite the increased scattering in the taper and/or at the caustic (evidenced by the

interference patterns outside the wedge and beyond the caustic – Figure 5.5c). It is

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argued that one of the major mechanisms for achieving larger local field

enhancements at large taper angles is related to reduced dissipation of the SPP modes

due to smaller distances between the tip of the wedge and the caustic (Figure 5.6c).

The fact that these distances are typically less than ~ 200 nm and could be as small as

~ 10 nm (Figure 5.6c) demonstrate the clearly nanoscale nature of the considered

plasmonic effect. Further, the typical localisation of the plasmonic field within the

hotspot at the caustic can vary between ~ 100 nm for small taper angles (Figure 5.4b)

down to ~ 20 nm for large taper angles (Figure 5.4c).

Figure 5.6. (a,b) The dependences of the maximum local field enhancement for the field intensity near the caustic at the distance of 0.1 nm above: (a) the metal-wedge interface, and (b) the wedge-cladding interface for the gold-wedge-air MII structure with e1 = – 9.3, e2 = 1.12, εc = 1, λ0 = 632.8 nm, and for the three different relative permittivities of the wedge εl: 9.4 (curves 1), 8.5 (curves 2), and 7 (curves 3). (c) The dependences of the distance between the tip of the wedge and the maximum of the SPP

field near the caustic on taper angle α for the same structural structure and the same wedge permittivities: 9.4 (curve 1), 8.5 (curve 2), and 7 (curve 3).

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The intensity enhancements above the wedge near its interface with the cladding

appear (expectedly) somewhat smaller (Figure 5.6b). This is because the dominant

localisation of the SPP modes occurs at the metal-wedge interface (Figure 5.4a-c and

Figure 5.5a,b. Nevertheless, because of the discussed mechanisms associated with

the boundary conditions at the wedge-cladding interface (with εl >> εc), there is a

significant jump in the local field in the cladding above the caustic (Figure 5.4a-c).

Once again, larger local field enhancements in the cladding are predicted for larger

wedge permittivities and large taper angles far beyond the adiabatic regime of

nanofocusing (curve 1 in Figure 5.6b). This field enhancement in the cladding above

the wedge could be utilised for the development of nano-optical sensors, such as

SERS-based sensors, and new detection techniques.

As discussed above, if the wedge permittivity is smaller than the magnitude of

the real part of the metal permittivity (εl < |e1|), there is a possibility of two different

caustics in the structure (curves 3 in Figure 5.1 and Figure 5.2). However, the

second (higher) caustic that corresponds to transformation of the SPP mode with

negative refraction into the SPP mode with positive refraction but even larger wave

number (the top branch of curves 3 in Figure 5.1 and Figure 5.2) is unlikely to be

realised in practice due to excessive SPP dissipation at the corresponding large

values of qcr2. An exception could be a situation where the second caustic merges or

nearly merges with the first caustic (curves 2 in Figure 5.2a,b). If, for example, εl ≈

εlc ≈ 8.5, the reflected high-q SPP mode does not exist (because there is no branch

corresponding to a negative refraction SPP – curves 2 in Figure 5.2), but there is a

forward-propagating SPP mode beyond the caustic. As a result, there is no

interference pattern between the wedge tip and caustic, but there is a forward-

propagating SPP mode (with relatively large q – curves 2 in Figure 5.2a,b.

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Therefore, in this case, the dominant process at the caustic is forward transformation

of the incident SPP mode into the forward-propagating high-q SPP mode beyond the

caustic. The physical reasons for this process to be dominant at the plasmonic caustic

are again related to the absence of a structural discontinuity at the caustic. As a

result, the incident SPP mode cannot be effectively reflected back (because this

would have required inversion of its wave vector in a continuous structure), which

leaves only one efficient option conserving the energy – transformation into the

forward-propagating high-q SPP mode. The respective local field enhancements in

this case are presented by curves 2 in Figure 5.6 a-c.

5.5 CONCLUSIONS

This chapter has described and analysed theoretically a unique MII nanofocusing

structure formed by a high-permittivity dielectric wedge on a metal substrate. This

structure is unique, in the fact that it supports SPP nanofocusing in the direction

opposite to the taper of the wedge, producing a range of nanoplasmonic effects

including nanofocusing of SPPs with negative refraction, formation of a strong

plasmonic caustic within a nanoscale distance from the tip, mutual transformation of

SPP modes in the process of their nanofocusing, and significant local field

enhancements at the metal-wedge and wedge-cladding interfaces in adiabatic and

strongly non-adiabatic regimes. Simple analytical equations were derived from rather

general structural assumptions to predict the position of the caustic in the dielectric

wedge in the adiabatic approximation. Localisation of the SPP field was shown to

occur within spatial regions as small as a few tens of nanometres near the caustic at

the metal-wedge interface. One of the significant and rather unexpected outcomes

was that SPP nanofocusing appeared to be efficient in the strongly non-adiabatic

regime where taper angles of the dielectric wedge are as large as ~ 40o. This aspect is

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likely to make fabrication of the considered structures more reasonable because they

do not require precise technologies to control and maintain small taper angles to

produce significant localisation and field enhancements. In addition, the considered

structure offers a possibility of efficient SPP nanofocusing within propagation

distances as small as a few tens of nanometres (at large taper angles), which is a

significant distinction of this MII structure from the previously considered

nanofocusing configurations. This may allow for the further miniaturisation of

nanofocusing-based sensors for use in SERS applications, which can utilise the

enhanced field at the wedge-cladding interface.

Unlike in other nanofocusing configurations, SPP nanofocusing in the MII

structures is neither directly related to, nor determined by, the sharp tip of the wedge,

but is rather caused by the physical phenomena associated with the SPP propagation

along the metal-wedge interface at some considerable (though nanoscale) distance

from the tip. It is only for taper angles > 40o that an additional sharp field maximum

starts to appear at the sharp tip of the wedge – not considered in this analysis because

it seems to be neither practically achievable, nor physically justifiable (at least for the

considered structural parameters) due to its typical width of less than 1 nm.

The potential applications of the considered nanoscale plasmonic effects may

include nano-optical sensors, with a particular emphasis on SERS-based sensors,

new detection techniques and photovoltaic devices.

This investigation was published in the article ‘Plasmon nanofocusing with

negative refraction in a high-index dielectric wedge’. Refer to Appendix B of this

thesis for further details.

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Chapter 6: Gap Surface Plasmon Waveguides with Enhanced Integration and Functionality

6.1 INTRODUCTION

Metallic nanostructures have been intensively investigated both theoretically

and experimentally for their unique properties that offer the possibility for the

development of a new generation of efficient waveguides and interconnects with

strong subwavelength localisation of optical signals in highly integrated nano-optical

devices, circuits, and sensors [11, 149, 159, 381, 382]. These structures are also

expected, through plasmon nanofocusing, to provide efficient coupling of

electromagnetic radiation and optical communication systems to nanoscale objects,

single molecules, and nanoelectronic devices and components [11]. As was

highlighted in Section 2.6.2, waveguides have been used in refractive index and

SERS-based sensing, as well as being the basis of interferometric biosensors and

optofluidic sensors.

A range of different metallic nanostructures that are capable of guiding optical

signals have been proposed and described. These include metal nanorods [383] and

nanostrips [384], V-shaped metallic grooves [385-393], triangular metal wedges

[392-395], and slot plasmonic waveguides [246, 247, 396-400]. Plasmonic slot and

groove waveguides appear to be one of the best options for the design of efficient

subwavelength interconnects and optical components with a high degree of

integration [11, 401], as well as being suited to a variety of sensing applications [273,

402-405]. However, fabrication of slot and groove plasmonic waveguides typically

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presents a challenge, as it requires precise nanoscale lithography [399] or focused

ion-beam milling to fabricate a nanoslot or a groove with precise nanometre

dimensions and high aspect ratio in a metal film or membrane. Further technical

problems arise if a rectangular profile of the slot is required or desirable. In addition,

such waveguides may display significant cross-talk when placed close together [406,

407], which may reduce the degree of integration of such plasmonic slot and groove

waveguides. Additional screening structures proposed in [407] to reduce the

crosstalk between two closely spaced slot waveguides require further precise

fabrication with nanoscale resolution.

The proposed L-shaped gap plasmon waveguides (L-GSPWs) resemble, in

their geometry, the trench plasmonic waveguides [83] and are formed by a thin

dielectric strip partially enclosed between two metal films (Figure 6.1).

Figure 6.1. The considered L-GSPWs of width W formed by partial enclosure of a thin SiO2 film of thickness D between two gold films of thicknesses H1 (overlay) and H2 (underlay).

In this structure, the width of the gap is determined by the thickness of the

deposited dielectric film, rather than by the resolution of a lithographic technique.

This opens a possibility for a much more precise, repeatable, and broad variation of

the geometrical parameters of the gap/slot, including fabrication of gaps/ trenches

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with very high aspect ratio filled with dielectric. Such structures can easily be

fabricated using a standard two-step lithography combined with film deposition (see

Figure 6.2a-c).

The aim of this chapter is to theoretically and experimentally analyse an L-

GSPW configuration that is free from the indicated deficiencies of the previously

considered slot waveguides and, once characterised, to evaluate the L-GSPW for use

in sensing applications. The experimental work reported in this chapter was

conducted by Michael G. Nielsen at the University of Southern Denmark.

6.2 METHODS

The numerical analysis of the L-GSPW was conducted by means of finite

element frequency domain analysis using the COMSOL MultiphysicsTM software

package. A model resembling Figure 6.1 was created, with H1 = H2 = 100 nm, D =

170 nm, λvac = 775 nm, εd ≈ 2.11, and εm = −23.6 + 1.7i (ref [408])]. The thickness

of the gold underlay (H2 ≈ 100 nm) was significantly larger than the skin depth (~30

nm). Therefore, for all purposes, including the experimental investigation of these

waveguides, the effect of the Si substrate on the L-GSPW modes can be neglected,

and the gold underlay can be assumed infinitely thick as H2 → +∞. A parametric

sweep of W was conducted and the real and imaginary parts of the effective

refractive index were calculated over the range W = 50 nm to 1400 nm. Additionally,

H1 was reduced over the range 100 nm to 5 nm in order to determine the effect of the

thickness of the gold overlay on the magnitude of the electric field in the air

surrounding the waveguide.

L-GSPWs were also fabricated on a Si substrate covered in thick gold

underlay, as shown in Figure 6.2.

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Figure 6.2. Experimental realisation of L-GSPWs with the parameters W = 600 nm, H1 = H2 = 100 nm, D = 170 nm, and different waveguide lengths L = 7, 10, 15, 20 μm (only L-GSPWs with the three

larger lengths are shown in the presented microscopic image).

The thickness of the gold underlay (H2 ≈ 100 nm) was significantly larger than

the skin depth, so the effect of the Si substrate on the L-GSPW modes can be

neglected, as was explained above. The SiO2 strips on the thick gold underlay

(Figure 6.2a-c) were fabricated using the standard lithographic procedure with the

subsequent lift off applied to the 170 nm thick SiO2 film with a ~3 nm thick titanium

adhesion layer deposited by means of RF-sputtering. The second lithography, with

the lift off applied to the gold overlay (with H1 ≈ 100 nm and ~3 nm titanium

adhesion layer), was then used to form the guiding structure with an overlap of width

W ≈ 600 nm of the SiO2 strip and the gold overlay (Figure 6.1). L-GSPWs were

fabricated with four different lengths (L ≈ 7, 10, 15, and 20 μm), to enable the

determination of the propagation distance of the L-GSPW mode(s). Each of the

fabricated waveguides had input and output diffraction gratings with the period Λ ≈

0.512 μm at the ends of the guiding structure to ensure efficient coupling of the

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waveguide mode and bulk radiation. The grating fringes were formed during the

second lithography with the lift off applied to the 100 nm thick gold overlay. The

thickness D ≈ 170 nm of the SiO2 layer was measured using ellipsometry (TFProbe

ellipsometer, Angstrom Sun Technologies), which also gave the refractive index of

the SiO2 layer εd ≈ 2.11 at the wavelength λvac = 775 nm. The metal permittivity at

the same wavelength was εm ≈ −23.6 + 1.7i [408].

The input grating was illuminated by a tightly focused laser beam of

wavelength λvac = 775 nm, power equal to Pin, and polarisation of the electric field

along the z-axis. The output signals (with the power equal to Pout) were measured by

means of a CCD camera from the output gratings. The power of the incident

generating beam Pin at the input grating was increased with increasing length of the

waveguide L to compensate for dissipative loss of the guided mode.

Using the same techniques described above, a sharply bent L-GSPW was also

fabricated (Figure 6.3) and tested in order to measure the transmission of the

generated signal through the bend.

Figure 6.3. The fabricated L-GSPW with approximately the same parameters as those in Figure 6.2a−c but with a sharp 90 degree bend and the lengths of the bend arms La = 7 μm.

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6.3 RESULTS AND ANALYSIS

6.3.1 Investigation of coupling efficiency and propagation length

Figure 6.4 shows the output signals (with the power Pout) that were measured

from the output gratings by means of a CCD camera.

Figure 6.4. The experimentally observed output radiation from the output gratings (shown by the arrows) for the four L-GSPWs with the indicated different lengths L and the following input powers

Pin: (a) 0.68, (b) 1, (c) 1.75, and (d) 4.89 mW.

Figure 6.5. The experimental (solid curve) and theoretical (dashed curve for the fundamental L-GSPW mode) dependences of the normalised power output from the output grating on distance Lp that

the generated guided plasmon travels along L-GSPW at the vacuum wavelength λvac = 775 nm; the structural parameters are the same as for Figure 6.2a-c, and the coupling efficiency for the output

grating is assumed to be 100%, while the theoretical and experimental coupling efficiencies for the input grating are ~4.5 and ~3.3%, respectively. The grey band shows the 90% confidence interval for

the obtained experimental dependence.

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Figure 6.5 shows the experimentally measured normalised signal η = Pout/Pin

from the output grating as a function of distance Lp = L − Lgr that the excited L-

GSPW mode travels along the waveguide (here, Lgr = 3Λ ≈ 1.536 μm is the length of

the input grating). The solid curve represents the statistical exponential fit:

−

=GSPL

zexp0ηη (6.1)

to the experimental points (Figure 6.5), where the efficiency of conversion of the

guided L-GSPW mode into the bulk radiation in the output grating is assumed to be

100%, η0 = η(z = 0) ≈ 0.033 ± 0.007 (with its standard error), and LGSP ≈ (6.7 ± 0.7)

μm is the power propagation length of the L-GSPW mode (with its standard error) in

the considered waveguide. The obtained value of η0 means that the efficiency of

conversion of the incident bulk radiation into the L-GSPW mode was ~3.3 ± 0.7%.

The grey band in Figure 6.5 shows the 90% confidence interval for this fit. This

means that the actual dependence of the output power on distance Lp must lie within

the grey band with the probability of 90%. The error ranges for η0 and LGSP,

corresponding to the 90% error band for the obtained experimental dependence, are

±0.015 and ±1.7 μm, respectively.

The numerically calculated propagation length for the fundamental mode with

the considered L-GSPW parameters [W = 600 nm, H1 = H2 = 100 nm, D = 170 nm,

λvac = 775 nm, εd ≈ 2.11, and εm = −23.6 + 1.7i [408] gave LGSP ≈ 5.4 μm. Assuming

also that the efficiencies of coupling between the fundamental L-GSPW mode and

bulk radiation in the input and output gratings were 4.5 and 100%, respectively, a

theoretical dependence of the normalised power output on the waveguide width W

(the dashed curve in Figure 6.5) was obtained. Though the theoretical value of LGSP

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≈ 5.4 μm is smaller than the obtained experimental value LGSP ≈ 6.72 μm, it appears

to be within the experimental error for the 90% confidence interval. Similarly, the

theoretical distance dependence (the dashed curve) in Figure 6.5 also lies within the

error band for the 90% confidence interval for the experimental curve.

The differences between the theoretical and experimental Lp-dependences of

the output signal (Figure 6.5) could be due to fabrication imperfections, especially

near the edge of the SiO2 film (Figure 6.2a-c and Figure 6.3). The larger

experimental value LGSP ≈ 6.72 μm of the propagation length compared to its

theoretical value LGSP ≈ 5.4 μm for the fundamental mode could also be due to

excitation of the second L-GSPW mode, whose propagation length can be larger near

the cutoff width Wc2 (Figure 6.7), and whose excitation efficiency could be increased

by the fabrication imperfections (such as, for example, changing thickness of the

SiO2 layer near its edge formed by the lift off).

L-GSPWs are subject to the conventional trade-off for plasmonic waveguides

between plasmon localisation (ensuring a larger degree of integration) and

propagation distance of the guided modes. The experimentally observed and

theoretically predicted propagation distances LGSP for the fundamental GSPW mode

appear large enough to ensure propagation of the guided signal over more than a

dozen of plasmonic wavelengths (or even longer when operating at telecom

wavelengths), which is typical for other types of gap surface plasmon waveguides

and is deemed sufficient for the design of nanoscale plasmonic interconnects and

devices. In some cases, L-GSPWs offer further advantages in this regard, for

example, compared to slot waveguides with silicon core [399]. As a result of the

small dielectric permittivity of the SiO2 (compared to silicon), the considered L-

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GSPWs can be used in a wide range of frequencies (including optical and near-

infrared frequencies, as opposed to just telecom wavelengths [399]) and typically

have larger propagation distances.

6.3.2 Numerical analysis of the L-GSPW

The numerical analysis of the considered L-GSPWs (Figure 6.2a-c) shows that

the waveguide supports two modes with distinct field distributions (Figure 6.6). In

particular, it can be seen that for both the L-GSPW modes, the dominant portion of

the field is concentrated in the gap under the gold overlay.

Figure 6.6. Typical calculated distributions of the magnitude of the electric field (a,c) and the y-component of the electric field (b,d) for the fundamental (a,b) and second (c,d) L-GSPW modes. The structural parameters are the same as for Figure 6.2a−c.

Note that, for the second mode, Ey changes its sign between the terminations of

the SiO2 layer and the gold overlay (Figure 6.6d). This means that the charge at the

interface between the gold overlay and the SiO2 layer also changes its sign when x is

varied from 0 to 600 nm (Figure 6.6d). Generation of a mode with such symmetry of

the charge and field distributions in L-GSPW is inefficient when using the incident

146

laser beam with the polarisation along the z-axis. Therefore, although the structure

supports two different modes at the considered L-GSPW parameters, only the

fundamental mode can be efficiently generated in the experiment. Therefore, the

explanation of the obtained experimental results should mainly involve the

fundamental L-GSPW mode (Figure 6.6a,b).

Both real and imaginary parts of the effective refractive index for the

fundamental and second L-GSPW modes show nontrivial dependences on the

waveguide width W (Figure 6.7).

Figure 6.7. The theoretical dependences of the real (solid curves) and imaginary (dashed curves) parts of the effective index for the fundamental (curves 1 and 2) and second (curves 3 and 4) L-GSPW

modes on waveguide width W; the other parameters being the same as for Figure 1d−f. The horizontal solid and dashed lines correspond to the real and imaginary parts, respectively, of the effective

refractive index of the gap plasmon in a uniform gap (in the absence of the overlay termination).

In particular, for both the modes, there exist cutoff waveguide widths Wc1 ≈ 100 nm

and Wc2 ≈ 515 nm. If W < Wc1 (or W < Wc2), then the real part of the effective

refractive index of the fundamental (or second) mode becomes smaller than the real

part of the refractive index (~1.405) of the surface plasmon in the gold-SiO2(170

nm)-air structure. Thus, the fundamental and second modes exist as localised guided

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L-GSPW modes only if W > Wc1 and W > Wc2, respectively. Otherwise, they leak

into the surface plasmon in the gold-SiO2(170 nm)-air structure.

Though increasing the L-GSPW width W causes the real parts of the effective

refractive indices of both modes Re(neff) to monotonically increase (curves 1 and 3 in

Figure 6.7), they do not tend to or reach the real part of the effective refractive index

for the gap surface plasmon in the uniform gap with no termination of the overlay

(the horizontal solid line in Figure 6.7). Therefore, the L-GSPW modes can always

be represented by a propagating gap plasmon experiencing successive reflections

from the termination of the gold overlay and the termination of the SiO2 layer. This

conclusion is confirmed by the distributions of the electric field in the L-GSPW

modes (Figure 6.6), showing that the dominant portion of the field (mode energy) is

concentrated in the gap under the gold overlay.

The imaginary part of the effective refractive index of both L-GSPW modes

Im(neff) (curves 2 and 4 in Figure 6.7) display significant maxima (corresponding to

minima of the propagation length) at around W ~170 nm (for the fundamental mode)

and W ~600 nm (for the second mode). Further increase of W results in a monotonic

decrease of dissipation of both the L-GSPW modes. However, similar to the real

parts of the effective refractive indices, their imaginary parts neither tend to, nor

reach, the imaginary part of the refractive index for the gap surface plasmon in the

uniform infinite gap with no termination of the overlay (the horizontal dashed line in

Figure 6.7). Therefore, plasmon dissipation in the L-GSPW is always larger than in

the uniform gap filled with the same dielectric and having the same width D.

148

6.3.3 Experimental investigation of a sharply bent L-GSPW

As was mentioned in the Methods section, a sharply bent L-GSPW was also

fabricated (Figure 6.3) and tested in order to measure the transmission of the

generated signal through the bend.

Figure 6.8. The image of the obtained output from the grating at the end of the second arm of the L-GSPW with a sharp bend (Figure 6.3), shown by the arrow.

Figure 6.8 indicates significant transmission of the generated signal through

the bend in the waveguide. Notably, no scattering was observed coming from the

waveguide bend, with the overall image quality being very similar to that of images

obtained with straight waveguides (compare Figure 6.4a−d and Figure 6.8). It was,

however, found problematic to make quantitative estimation of the bend

transmission. The main problem was that, in this experimental setup, a polarisation–

dependent beam splitter was used, which results in different energy distributions (and

thus coupling efficiencies) in the focused laser beam for straight and bent

waveguides (compare Figure 6.4a-d and Figure 6.8). In addition, the straight and

bent waveguides and the corresponding coupling gratings appeared to come out

differently (compare Figure 6.2a-c and Figure 6.3), making it questionable to

directly use the values of coupling efficiency and mode propagation loss determined

149

with the straight waveguides for the analysis of the bend waveguide configuration.

However, the important point is that significant and detectable energy transmission

through the sharp 90 degree bend in L-GSPW has been confirmed experimentally.

Highly efficient energy transmission through a sharp 90 degree bend with

minimal radiation losses should be expected because in the considered geometry of

the bend (Figure 6.3), the overlay metal film effectively screens the mode, thus

preventing it from leaking it into bulk radiation at the bend. Further optimisation of

the bend to suppress the plasmon reflected back into the first arm of the bend and

thus achieve the maximal possible power transmission can be achieved by

introducing a corner defect into the bend, as was done for the V-groove [409] and

slot waveguides [410].

6.3.4 L-GSPWs in different configurations

The same screening effect of the metal overlay is also a reason for the reduced

cross-talk between two closely spaced and positioned back-to-back L-GSPWs

(Figure 6.9a).

Figure 6.9. Two closely spaced L-GSPWs in the back-to-back (a) and back-to-front (b) configurations in an integrated circuit with the width of the metal screening partition S separating them.

150

In addition, for not very large thicknesses D of the SiO2 layer, the mode field

near the termination of this layer is close to zero (Figure 6.6a,b), and this is another

reason for low cross-talk in the two-waveguide geometry shown in Figure 6.9a. For

example, for the considered structural parameters of the two L-GSPWs separated by

the partition of the width S = 20 nm (Figure 6.9a), the coupling length Lc (i.e. the

length within which the energy from one L-GSPW is coupled into the neighbouring

L-GSPW across the partition) is ~103 and ~16 μm for the fundamental and second

mode, respectively. Reducing the width of the partition to S = 10 nm results in Lc ~51

and ~8.5 μm for the considered modes, respectively. The significantly smaller

coupling lengths for the second mode are explained by larger field in this mode near

the partition (Figure 6.6), which results in more efficient coupling. Considering the

fundamental L-GSPW mode, even for a 10 nm wide partition, the coupling length

significantly exceeds the mode propagation length, thereby eliminating the issue of

cross-talk in practical design considerations.

Effective reduction of the cross-talk by metal screening occurs not only in the

back-to-back configuration of two L-GSPWs (Figure 6.9a) but also in the back-to-

front arrangements (Figure 6.9b). In this case, it is sufficient to have a nanoscale

separating metal partition of a thickness larger than the skin depth (~30 nm) to ensure

effective cross-talk suppression between the neighbouring L-GSPWs.

6.3.5 Evaluation of the L-GSPW for use in sensing applications

Having characterised the L-GSPW, attention finally turns to the potential for

the use of this waveguide in sensing applications. In keeping with the focus of this

thesis, this analysis will be limited in scope to potential for this structure to be used

in SERS-based sensing. It has been noted that the dominant portion of the field of

both modes is concentrated in the gap under the gold overlay, as is illustrated in

151

Figure 6.6. However, this figure also indicates that there is still a region of intense

electric field in air at the edge of the gold layer and this intense electric field can be

utilised to generate a SERS signal. Figure 6.10 shows the effect of varying W on the

field distribution in the waveguide and in the air.

Figure 6.10. Distributions of the magnitude of the electric field of the fundamental L-GSPW mode for D = 170 nm, H = 100 nm, and W = (a) 300 nm, (b) 600 nm, (c) 900 nm, and (d) 1200 nm.

This confirms that, over a wide range of waveguide widths, there is still a

region of intense electric field in the air around the termination of the gold overlay.

Figure 6.11 shows the amplitude of this field in the air at the SiO2-air boundary for D

= 170 nm, H = 100 nm, and W = 600 nm (the parameters of the fabricated straight

waveguides).

152

Figure 6.11. The magnitude of the electric field of the fundamental L-GSPW mode in air at the SiO2-air boundary for D = 170 nm, H = 100 nm, and W = 600 nm.

The field distributions for the other values of W considered in Figure 6.10 are

virtually indistinguishable from the distribution for W = 600 nm, and therefore have

not been displayed. The amplitude of this field in the vicinity of the edge of the gold

layer is of the same order of magnitude as the maximum field amplitude below the

gold overlay and can be utilised for SERS-based sensing if molecules of interest are

located in this region. Figure 6.12 shows the effect of reducing the thickness of the

gold overlay, H, for D = 170 nm and W = 600 nm.

Figure 6.12. Effect of reducing overlay thickness H on magnitude of the electric field of the fundamental L-GSPW mode in air for D = 170 nm, W = 600 nm, and H = (a) 100 nm, (b) 30 nm, (c)

10 nm, and (d) 5 nm.

153

This has the effect of increasing the size of the region of intense electric field

and its maximum amplitude, especially when the gold overlay thickness, H, becomes

smaller than the skin depth. These findings confirm that the structure can be used for

SERS-based sensing.

However, as was discussed in Section 2.6.2, there are sensing techniques other

than SERS that utilise waveguides. For example, it may be possible to use the L-

GSPW to detect refractive index changes at the SiO2-air interface by introducing

resonant structures, such as mirrors or gratings, and monitoring the changing guiding

characteristics when the refractive index outside of the waveguide is changed, which

happens when molecules bind to the surface.

In addition, as a result of the significant propagation distances and high

transmission through sharp 90 degree bends that are achievable, the L-GSPW could

prove useful in the development of nano-optical devices (e.g., Mach-Zehnder

interferometers) for sensing applications. As S-bends with large bend radii are not

needed to prevent bend losses [411, 412], more compact geometries can be

implemented, which may provide an easier solution than using evanescent coupling

to replace the S-bends [250].

6.4 CONCLUSIONS

In conclusion, this chapter has reported on experimentally observed and

theoretically determined findings related to a new type of gap surface plasmon

waveguide that combines simple and reliable fabrication, possibility of high mode

localisation and integration, significant propagation distances for the guided modes,

and high transmission through sharp 90 degree bends with low radiation bend losses.

Further analysis of the waveguides with sharp bends, directional couplers, and other

154

integrated optics components using L-GSPWs will be necessary to optimise their

performance and better understand the underlining physical principles. Nevertheless,

the obtained results have already demonstrated significantly improved functionality

and fabrication of the proposed L-GSPWs, as well as their significant potential for

the design of integrated optics components.

Furthermore, it has been demonstrated that this waveguide geometry displays

characteristics which could enable it to be an effective platform for SERS-based

sensing, as well as other forms of sensing, although further investigations would

need to be conducted in order to optimise the L-GSPW for specific sensing

applications.

Elements of this investigation were published in the article ‘Gap surface

plasmon waveguides with enhanced integration and functionality’. Refer to

Appendix C of this thesis for further details.

155

Chapter 7: Conclusion

This thesis has focused on two main themes related to the development of new

nano-sensor technology. The first theme was the investigation of the delivery rates of

air samples that may contain explosives, drugs, or other undesirable substances to the

electromagnetic field hotspots produced by metallic nanostructures and the methods

in which the delivery rate of the molecules can be increased to lower the detection

limit of the sensor. The second theme was the investigation and characterisation of

plasmonic nanostructures for the sub-wavelength confinement of electromagnetic

energy and the evaluation of their applicability to sensing applications.

In Chapter 3, the optimal design of a nano-optical sensor combining nanohole

arrays and pressure-driven sample delivery was investigated. The delivery rates of

different residual vapour/gas molecules in the air to the plasmonic hotspots in a

nanohole array by means of a pressure and a concentration gradient were compared

and the most appropriate methods for the optimisation of the system from the

viewpoint of the design of efficient nano-optical air monitoring sensors were

identified and investigated. The parameters for which either fluidics or molecular

diffusion was the dominant transport mechanism were then determined for three

different molecules (NH3, SO2, and TNT). The effects of increases in the pressure

gradient and chamber pressure on the delivery rates were also investigated. The

findings reported in this chapter have important implications for the development of

optimised nanohole array-based optofluidic sensors used for the detection of residual

vapours of explosives and other undesirable substances by means of surface-

156

enhanced Raman spectroscopy or surface-enhanced infrared absorption

spectroscopy.

In Chapter 4, the nanofluidic delivery of molecules for testing to the field

hotspot created by a tapered rod nanofocusing structure in a viscous fluid flow was

investigated. The force and stress distributions on the rod in air and water were

analysed and the mass flow rate of air to the tip of the rod was evaluated in order to

determine the delivery rates for three different molecules (NH3, SO2, and TNT) for

testing to the sensing region of the structure. An equation was derived to allow the

calculation of the drag force on each element of a tapered rod and was used to

determine both the magnitude and position of maximum stress on a rod as a function

of tip radius, taper angle, and fluid velocity. The findings reported in this chapter will

be important in the development of new nano-optical sensors for the detection of

trace amounts of substances such as explosives and drugs and also for use in

environmental monitoring.

In Chapter 5, a metal-insulator-insulator structure that consisted of a high-

permittivity dielectric wedge on a metal substrate, with the permittivity of the

dielectric layer being greater than the magnitude of the permittivity of the metal (εd >

|εm|), was described and analysed. The geometry differed from nanofocusing in

conventional tapered metal structures, as the plasmon propagated towards the tip in

the direction of increasing dielectric wedge thickness. The structure was shown to

support the nanofocusing of SPPs with negative group velocity (negative refraction),

the formation of a caustic which corresponds to the point of mutual transformation of

SPP modes at a critical dielectric thickness, as well as the enhancement of the local

electric field at the metal-dielectric and dielectric-air boundaries in both the adiabatic

157

and strongly non-adiabatic regimes. Simple analytical equations were also derived to

predict the position of the lower caustic in the dielectric wedge in the adiabatic

approximation. Based on the findings reported in this chapter, the potential

applications of the considered structure may include nano-optical sensing, including

surface-enhanced Raman spectroscopy, as well as the development of new detection

techniques and photovoltaic devices.

In Chapter 6, an L-shaped gap surface plasmon waveguide (L-GSPW)

consisting of a dielectric strip sandwiched between two metal films was proposed

and analysed numerically and experimentally. The ease with which the waveguide

was fabricated through standard lithography and thin film deposition is one of the

attractive features of the design. The investigation identified a number of desirable

characteristics of the L-GSPW as an integrated optical component, with significant

propagation distance for the guided modes, high transmission through 90 degree

bends, and low cross-talk between neighbouring waveguides being reported. The L-

GSPW was also shown to have characteristics that will enable it to be used in SERS-

based sensing, as well as in other sensing applications.

In conclusion, this thesis has presented four investigations that are highly

relevant to various aspects of nano-optical sensor technology, with findings that

should be of interest to researchers in a variety of fields.

The main findings of the thesis can be summarised as follows:

1. It was shown that the sensitivity of nanohole array sensors can be enhanced

by increasing the pressure difference across the porous membrane and/or

increasing the ambient air pressure in the sensing chamber. The most rapid

increase in molecule delivery rates by fluidics, for nanoholes of the

158

determined optimal diameter, can be achieved by increasing the ambient

pressure in the chamber.

2. Increasing the membrane thickness results in a relatively small reduction in

the mass airflow through a nanohole. This is an important outcome because

increasing the membrane thickness to enhance its mechanical strength, and

thus allowing larger pressure differences across the membrane, results in a

significant overall increase in the molecule delivery rates to the nanoholes.

3. The larger the molecular mass of the residual tested molecules in the air, the

greater the advantages offered by fluidics as a delivery mechanism, compared

to molecular diffusion. Under experimentally achievable conditions, and for

the considered concentration of explosive (TNT) vapour in the air, fluidics

may be as high as ~ 20 times more efficient than molecular diffusion in terms

of delivering the molecules of interest to submicron holes in a membrane.

4. The maximum stress on a tapered nanofocusing rod in a viscous fluid flow

was found to be strongly dependent on fluid velocity, tip radius, and taper

angle, while the position of maximum stress is a function of taper angle, but

highly-insensitive to velocity over the range investigated.

5. It was demonstrated that gold rods can withstand an air velocity of 100 m/s if

they have a taper angle greater than 6.1 degrees, while the breaking velocity

of gold rods in water is highly-dependent on taper angle, as a result of the

significantly higher density and viscosity of water.

6. A concentration of 36 pg/L of TNT molecules, 10 pg/L of SO2 molecules, or

2.7 pg/L of NH3 molecules in air will be sufficient to deliver one molecule

159

per second to the hotspot at the tip of a tapered nanofocusing rod with a tip

radius of 5 nm.

7. The major scattering process of the low-q incident mode at the caustic in a

dielectric wedge was determined to be its back-reflection (transformation)

into a high-q mode, which displays negative refraction. This process is quite

different to SPP total external reflection from a high-index dielectric layer

[380], since that process does not display mode transformation.

8. SPP nanofocusing appears to be efficient in the strongly non-adiabatic

regime, even when the taper angle of the dielectric wedge is as large as ~ 40

degrees. Furthermore, the fact that the SPP field maximum is spatially

separated from the tip demonstrates that, unlike in other nanofocusing

configurations, the localisation and enhancement of the field is not directly

related to the sharp tip of the wedge, but is rather a phenomenon occurring

inside the structure, within some nanoscale distance from the tip.

9. Although intensity enhancement at the wedge-air interface appears somewhat

smaller than at the metal-wedge interface, this hotspot could still be utilised

in sensing applications, such as SERS. The observation that increased local

field enhancements are expected for larger wedge permittivities and taper

angles beyond the adiabatic regime is significant.

10. Since it is not necessary to use wedges with small taper angles in order to

achieve significant localisation and field enhancements, as well as the fact

that the tip of the dielectric wedge does not play a significant role in the

process of nanofocusing, the fabrication process may require less precision

160

than previously considered nanofocusing structures, which is an important

practical implication.

11. Both real and imaginary parts of the effective refractive index for both

fundamental and second L-GSPW modes were found to be dependent on the

waveguide width W, with cutoff widths Wc1 ≈ 100 nm and Wc2 ≈ 515 nm for

the two modes. The fundamental and second modes exist as localised guided

L-GSPW modes only if W > Wc1 and W > Wc2, respectively. If not, they leak

into the surface plasmon in the gold-SiO2 (170 nm)-air structure.

12. Significant energy transmission through a 90 degree bend in an L-GSPW was

achieved experimentally. Since the mode is effectively screened by the metal

film overlay, it is prevented from leaking into bulk radiation at the bend. The

overlay also reduces cross-talk between adjacent waveguides that are closely-

spaced and placed in either back-to-back or back-to-front configurations. In

the case of the back-to-front geometry, a metal partition of thickness greater

than the skin depth (~ 30 nm) is sufficient to suppress cross-talk.

13. By utilising the localised electric field at the edge of the gold overlay, the L-

GSPW can be used for SERS-based sensing. It may also be possible to use

the waveguide to detect refractive index changes at the SiO2-air interface by

introducing resonant structures and monitoring the changing guiding

characteristics when the refractive index outside of the waveguide is changed.

The L-GSPW could also prove useful in the development of nano-optical

components (e.g., Mach-Zehnder interferometers) for sensing applications as

a result of the significant propagation distances achievable and high

transmission through sharp 90 degree bends.

161

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Appendices

Appendix A

The article resulting from the investigations reported in Chapter 3:

M. L. Kurth and D. K. Gramotnev, ‘Nanofluidic delivery of molecules: Integrated

plasmonic sensing with nanoholes’, Microfluidics and Nanofluidics, 14, 743-751

(2013).

DOI: https://doi.org/10.1007/s10404-012-1093-5

Statement of contributions:

Martin L. Kurth – Conducted the analysis, data analysis, discussion and paper

writing.

Dmitri K. Gramotnev – Originally suggested the idea, conceptual design, data

analysis, discussion and paper writing.

186

Appendix B


D. K. Gramotnev, S. J. Tan, and M. L. Kurth, ‘Plasmon nanofocusing with negative

refraction in a high-index dielectric wedge’, Plasmonics, 9, 175-184 (2014).

DOI: https://doi.org/10.1007/s11468-013-9610-2


Martin L. Kurth – Conducted the analysis, data analysis and discussion.



Shiaw Juen Tan – Conducted the analysis, data analysis, discussion and paper

writing.

187

Appendix C


D. K. Gramotnev, M. G. Nielsen, S. J. Tan, M. L. Kurth, S. I. Bozhevolnyi, ‘Gap

surface plasmon waveguides with enhanced integration and functionality’, Nano

Letters, 12, 359-363 (2012).

DOI: https://doi.org/10.1021/nl203629m


Martin L. Kurth – Conducted the analysis, data analysis and discussion.



Michael G. Nielsen - Designed the fabricated structures, fabricated the investigated

samples, characterised the samples with SEM and optical microscopy, made a

substantial contribution to the intellectual content, and finalised the versions to be

published.

Shiaw Juen Tan – Conducted the analysis, data analysis and discussion.

Sergey I. Bozhevolnyi - Conceptual design, data analysis, discussion and paper

writing.

plasmonic nanofocusing and guiding structures for nano … · guiding structures for nano-optical...

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