modeling and simulation of active plasmonics with the fdtd method by using solid state and...
DESCRIPTION
An approach for the simulation of active plasmonics devices is presented in this paper. In theproposed approach, a multi-level multi-electron quantum model is applied to the solid state partof a structure, where the electron dynamics are governed by the Pauli Exclusion Principle, statefilling and dynamical Fermi-Dirac thermalization, while for the metallic part Lorentz-Drudedispersive model is incorporated into Maxwell’s equations. The finite difference time domain(FDTD) method is applied to the resulting equations. For numerical results the developedmethodology is applied to a metal - semiconductor – metal (MSM) plasmonic waveguide and amicrocavity resonator.TRANSCRIPT
Modeling and Simulation of Active Plasmonics with the
FDTD method by using Solid State and Lorentz-Drude
Dispersive Model
Iftikhar Ahmed, Eng Huat Khoo, Oka Kurniawan, Er Ping Li
Corresponding Authors: [email protected], eplee @ihpc.a-star.edu.sg
Advanced Photonics and Plasmonics Division
Department of Computational Electronics and Photonics
Institute of High Performance Computing, A*STAR, Singapore 138632
An approach for the simulation of active plasmonics devices is presented in this paper. In the
proposed approach, a multi-level multi-electron quantum model is applied to the solid state part
of a structure, where the electron dynamics are governed by the Pauli Exclusion Principle, state
filling and dynamical Fermi-Dirac thermalization, while for the metallic part Lorentz-Drude
dispersive model is incorporated into Maxwell’s equations. The finite difference time domain
(FDTD) method is applied to the resulting equations. For numerical results the developed
methodology is applied to a metal - semiconductor – metal (MSM) plasmonic waveguide and a
microcavity resonator. © 2010 Optical Society of America
OCIS codes: (50.1755) Computational electromagnetic methods; (250.5403) Plasmonics;
(160.6000) Semiconductor materials
1. Introduction
Until a few years ago, the miniaturization of photonics devices was a challenge due to the
diffraction limit, which restricted minimum size of a component equivalent to 2 . However,
recently, a new emerging area called plasmonics made it possible to go below the diffraction
limit. In plasmonics, the wave propagates at the interface of a metal and dielectric and remains
bounded. This characteristic allows the miniaturization of photonics devices below the diffraction
limit. A number of plasmonic structures, which can guide and manipulate electromagnetic
signals, have been presented in literature [1-2]. Some of the promising plasmonic structures are
surface nano-antennas, lenses, resonators, sensors, and waveguides [1-4]. Most of the work in this
area has been done on passive plasmonics [1-3] and active plasmonics is getting much attention
owing to freedom of light wave manipulation [4-7]. At the same time, complementary metal–
oxide–semiconductor (CMOS) technology is also reaching its limits due to size and RC time
delay. It is difficult to abandon the CMOS technology due its numerous applications, cheap
process and mature fabrications technologies; however, the limitations can be overcome by
interfacing plasmonics with electronics. The active plasmonics is believed to be a perfect
candidate for this purpose, because the interface between both technologies (similar
semiconductor materials) is easier to realize than in passive plasmonics.
In our paper, we incorporate the multi-level multi-electron quantum system approach into
Maxwell’s equations for the simulation of the solid state part of a plasmonic structure. The
electron dynamics in the solid state is managed by the Pauli Exclusion Principle, state filling and
dynamical Fermi-Dirac thermalization, and the approach is also applicable to the modeling of
molecular or atomic media [8]. The solid state approach has been presented for the simulation of
active photonics devices such as lasers and optical switches [9-11]. For the simulation of the
metallic part of a plasmonic structure, we incorporate a Lorentz-Drude (LD) model [12-13] into
Maxwell’s equations. The LD model deals with free electrons (intra-band effects) and bounded
electrons (inter-band effects) in metals. In this paper, both a multi-level multi-electron quantum
system and a LD approach are incorporated into Maxwell’s equations, and the resulting equations
are used to simulate active plasmonics devices. For numerical analysis the FDTD method is
applied to the resulting equations. The FDTD method has the potential to model such complex
dynamical media and their physics. A review of some advances in numerical techniques to
couple carrier dynamics with full wave dynamics is presented in [14], for this purpose stochastic
ensemble Monte Carlo (EMC) approach is applied to simulate carrier transport, while FDTD
method is applied to simulate Maxwell equations. Recently, we have presented an approach
which has concept similar to that presented in [14] to simulate nano-devices [15], where coupled
Schrodinger and Maxwell equations are used to simulate coupled carrier and full wave dynamics.
The proposed method has two main advantages compared to the methods presented in [14-15].
First advantage is that the proposed method can model intra-band and inter-band electron
transitions, i.e. transition from one energy level to other, transition from valance band to
conduction band and vice versa, while [14-15] cannot model this. The second advantage is that
the proposed methods can deal with stimulated emission. Due to these reasons the proposed
method is more promising and realistic comparatively. To our knowledge, this is the first time
domain approach which uses a realistic solid state model and LD dispersive model for the
simulation of active plasmonics devices. In subsequent sections, we illustrate the numerical
methodology, numerical results and draw conclusions.
2. Numerical Methodology
In electro-optical systems, electrons and photons are two important components. Photon
absorption causes a transition of the electron from a lower energy state to a higher energy state. In
the inverse process, when an electron moves from the conduction band to the valance band it
emits a photon. Therefore, the interaction between an electron and photon is a key factor in active
photonics devices; the same principle is adopted in this paper for the modeling and simulation of
active plasmonics devices. The LD model is used to deal with intra-band (Drude model) and
inter-band (Lorentz model) effects of metal, while the solid state media model deals with
transient intra-band and inter-band electron dynamics, the carrier thermal equilibrium process for
direct band gap semiconductors. The solid state model is formulated in such a way that it
automatically incorporates band filling and nonlinear optical effects associated with carrier
dynamics. The formulation adopted for solid state media is applicable only to direct band
structures. Although the formulation developed to simulate active plasmonics devices is
complicated, it covers all the necessary physical effects efficiently.
The proposed methodology consists of two parts, one for metals (LD model), and the other for
semiconductors. The time domain Maxwell equations are written as
t
DH
(1)
t
BE
(2)
where, ED r0 and HB r0 .
Model for metals
At near infrared and optical frequencies, the relative permittivity “ r ” of numerous metals
becomes frequency dependent “ )( r ” and can be described by a Lorentz-Drude (LD) model.
The Lorentz-Drude model for relative permittivity is written as
222
2
22
2
LL
pLL
D
pDr
jjjj
)( (3)
where pD is the plasma frequency and D is the damping coefficient for the Drude model.
L is a weighting factor, PL is the resonance frequency, and L is the spectral width for the
Lorentz model. For the Lorentz model different number of oscillators can be considered; for
simplicity we considered only one oscillator and L is its oscillation frequency.
After putting Lorentz-Drude model into equation (1), it becomes
Ejjj
Ejjj
EjH
LL
pLL
D
pD
222
2
22
2
0
(3a)
After using auxiliary differential equation (ADE) approach and some simplifications of equation
(3a), we get
t
QJ
t
EH L
D
00
(4)
DDD
pD Jt
JE
0
2 (5)
LLL
LL
pLL Qt
Q
t
QE 2
2
22
(6)
where, equation (4) is Maxwell equation with both Drude “ DJ ”and Lorentz”t
QL
0 ” terms,
equation (5) is for Drude model and equation (6) is for Lorentz model and are obtained from
equation (3a). Maxwell’s equations in scalar form after incorporation of Drude and Lorentz terms
(4) and (2) are written as
t
QJ
z
H
y
H
t
E LxDx
yzx
111
00
(7)
t
QJ
x
H
z
H
t
E Ly
Dyzxy
111
00
(8)
t
QJ
y
H
x
H
t
E LzDz
xyz
111
00
(9)
y
E
z
E
t
H zy
r
x
0
1 (10)
z
E
x
E
t
Hxz
r
y
0
1 (11)
x
E
y
E
t
H yx
r
z
0
1 (12)
As an example, after simplifying equations (5) and (6) and then by inserting them into equation
(7) for xE field component, we get
n
Lx
n
Lxx
n
Lxx
n
xx
x
n
Dx
n
xx
n
Dxx
x
n
y
n
z
x
n
x
x
n
x
QQQE
JEJt
z
H
y
HtEE
1
0
2
1
2
1
0
1
1
2
1
(13)
Equations (5) and (6) are simplified for x-direction in the form of equations (14) and (15)
respectively
n
x
n
xx
n
Dxx
n
Dx EEJJ 11 (14)
111 n
Lxx
n
Lxx
n
x
n
xx
n
Lx QQEEQ (15)
Where,
21
21
D
D
xt
t
,
21
2
02
D
pD
xt
t
22
22
21
2
LL
pLL
xt
t
t
,
22
22
21
22
LL
LL
xt
t
tt
22
21
1
LL
xt
t
,
021 xx
x
t
Similarly equations for yE and zE field components can be obtained.
Model for solid state materials, equation (1) can be written as
t
P
t
EnH
2
0 (16)
whereas equation (2) remain unchanged. In equation (16), P is the macroscopic polarization
density and represents total dipole moment per unit volume and can be expressed as
)()(),( rNtUtrP hdipm (17)
Where, )( tU m is atomic dipole moment, )( rN hdip is dipole volume density for level h within
energy width ( E ) and is specified by the number of dipoles ( 0N ) divided by unit volume ( V ),
r represents x, y and z directions. In semiconductor medium, the electron dynamics is modeled by
discretizing the conduction and valance band into different energy levels as shown in Fig. 1. In
this figure the subscript v stand for valance band and c stands for conduction band. Lines drawn
between different levels of conduction and valance band (red) show inter-band transitions, while
the lines (green) drawn in between different energy levels of conduction band represent intra-
band transitions, similarly for the case of valance band. The valance band levels are labeled as
h_v, while the conduction band levels are labeled as h_c. The occupation probability with respect
to time for each level in the conduction and valance band, and the effect of carrier densities for
different level pairs has been studied in [8]. It was observed that the electrons relaxation time to
equilibrium is much slower in conduction band as compared to the valance band. In this paper ten
energy levels are considered for both conduction and valance bands. From these levels we can
calculate intra-band and inter-band transition times, these levels are also helpful to find Fermi-
Dirac-Thermalization dynamics in the semiconductor and are discussed latter in the section.
For inter-band transition and spontaneous decay from energy level h_c to h_v, the expression is
given as
h
kjihv
n
kjihvn
kjihC
n
kjihz
n
kjiz
n
kjihy
n
kjiy
n
kjihx
n
kjix
ahn
kjih
N
N
N
PAPAPAN
,,)(
,,)(
,,)(
,,,,,,,,,,,,,,...
2
1
2
1
2
1
2
1
2
12
1
2
1
2
1
2
1
2
1
1
(18)
Where, hN is number of electrons per unit volume transferred from conduction band level h_c
to valance band level h_v, ah is inter-band transition frequency , h is inter-band transition time.
Similarly the relations for intra-band transition for conduction and valance bands are also derived
and are given below.
Intra-band transition for conduction band from energy level h_c to h_c-1 is given by
),(
,,)(
,,)(
,,)(
),(
,,)(
,,)(
,,)(
,,),(hhC
kjihC
n
kjihCn
kjihC
hhC
kjihC
n
kjihCn
kjihC
n
kjihhC
N
N
NN
N
N
N1
2
1
2
1
2
11
1
2
11
2
11
2
1
2
11
11
(19)
Where, ),( 1 hhCN is number of electrons per unit volume, transferred from level h_c to h_ c-1,
C is intra-band transition time in conduction band.
Intra-band transition for valance band from energy level h_v to h_v-1 is given as
),(
,,)(
,,)(
,,)(
),(
,,)(
,,)(
,,)(
,,),(hhV
kjihV
n
kjihVn
kjihV
hhV
kjihV
n
kjihVn
kjihV
n
kjihhV
N
N
NN
N
N
N1
2
1
2
1
2
11
1
2
11
2
11
2
1
2
11
11
(20)
Where, )1,( hhVN is number of electrons per unit volume transferred from level h_v to h_ v-1,
V is intra-band transition time in valance band.
In equations (18-20) 0
VhN and 0
ChN are initial values of volume densities of the energy states at
level h in valance and conduction band respectively, while Vh
N and Ch
N are latter values.
After some mathematical derivations, equation (17) for polarization can be written as
),(),()(
)(),(
)(
)(2
),(),(2),(),(
00
2
22
2
2
2
2
trEtrNrN
rNtrN
rN
rNU
h
trPtrAUhdt
trdP
dt
trPd
kCh
Ch
hdip
Vh
Vh
hdip
khah
hkkkhah
ahhk
hhk
(21)
where k = x, y and z, h represent the de-phrasing rate at hth energy level after excitation, Ak is
the vector potential, hah
ckhU
3
3032 , here is Planck’s constant divided by 2 , c
is speed of light in free space, 0 is permittivity of free space.
Equation (21) is further simplified and discretized as
n
kjix
n
kjiVh
n
kjiChkh
h
ahn
kjihx
h
h
n
kjihx
h
n
kjixh
ahah
n
kjihx
ENNUt
tP
t
t
Pt
AUt
P
,,2
1,,
2
1,,
2
12
21
,,2
1
,,2
1
,,2
122
2
222
1
,,2
1
).2(
4
.2
2.
.2
424
(22)
Whereas the rate equations for conduction and valance bands are written as
pump
n
kjihh
n
kjihh
n
kjihn
kjiCh
n
kjiCh WNNNtNN
,,),(,,),(,,)(,,,, 2
11
2
11
2
11
2
1
1
2
12 (23)
pump
n
kjihh
n
kjihh
n
kjihn
kjiVh
n
kjiVh WNNNtNN
,,),(,,),(,,)(,,,, 2
11
2
11
2
11
2
1
1
2
12 (24)
In equations (23) and (24), the term pumpW is used for electrical pumping. The Fermi-Dirac-
thermalization dynamics in the semiconductor is obtained by taking the ratio between upward and
downward intra-band transitions for two neighboring energy levels. The relation between two
energy levels can be obtained by the intra-band transition rate equations, as an example, by using
the rate equation (20) for valance band we get
TBK
hvEhvE
hv
hv
hhv
hhve
rN
rN))()((
)(
)(
),(
),(
)(
)(1
1
1
1
(25)
Where ),( 1hhv is downward transition rate, while the ),( hhv 1 is upward transition rate between
levels vh _ and 1vh _ in the valance band, BK is Boltzmann Constant, T represents
temperature, )(hEvand )1( hEv
are energies at levels vh _ and 1vh _ respectively . Similarly
relation for conduction band can also be obtained. The approach adopted for intra-band
thermalization can be applied to inter-band transition also but the contribution is negligible due to
a large energy gap. Therefore, in this paper thermalization effect is only considered for intra-band
transition.
After incorporation of above effects into the polarization equation and then resulting equation
into Maxwell’s equation, the discretized electric field equations for FDTD approach are written as
M
h
n
kjixh
n
kjixh
n
kjiz
n
kjiz
n
kjiz
n
kjiz
n
kjix
n
kjix
PP
HHz
tHH
y
tEE
1 2
11
2
1
2
1
2
1
2
12
1
2
1
2
12
1
2
1
2
12
1
2
1
2
1
2
1
1
2
1
1
,,,,,,
,,,,,,,,,,,,
(26)
M
h
n
kjiyh
n
kjiyh
n
kjiz
n
kjiz
n
kjix
n
kjix
n
kjiy
n
kjiy
PP
HHx
tHH
z
tEE
1 2
11
2
1
2
1
2
1
2
12
1
2
1
2
12
1
2
1
2
12
1
2
1
2
12
11
2
1
1
,,,,,,
,,,,,,,,,,,,
(27)
M
h
n
kjizh
n
kjizh
n
kjix
n
kjix
n
kjiy
n
kjiy
n
kjiz
n
kjiz
PP
HHy
tHH
x
tEE
1 2
11
2
1
2
1
2
1
2
12
1
2
1
2
12
1
2
1
2
12
1
2
1
2
1
2
1
1
2
1
1
,,,,,,
,,,,,,,,,,,,
(28)
For further illustration, the difference in relative permittivity of both semiconductor and metal
cases, and as a result their effect on electric field density is discussed below.
In general electric field density can be written as
ED r0 (29)
ED )( 10
EED 00
Ep 0
pED 0
Where p is polarization density and is linked to susceptibility of the material. Here for solid state
material we add a termP , which is dependent on dipole volume density, pumping density and
atomic dipole moment.
P EED 00 (30)
P ED r0 (31)
After putting equation (31) into equation (1), we get equation (16)
For LD model equation (29) becomes frequency dependent and is written as
ED r )(0 , where, Ldr )(
ED Ld )( 0 . Let 1 f to make it similar to that for semiconductor, here
is constant value and therefore f is also constant, whereas d and L are complex values.
Where, d is frequency dependent permittivity from Drude model and L is frequency
dependent permittivity from Lorentz model.
ED Ldf )( 10
ED )( 10 , where Ldf
EED 00
EP 0 (32)
PED 0 (33)
It can be observed that permittivity dependent electric field density equations (31) and (33) both
for semiconductors and metal have similar relations and both are dependent on polarization. In
the case of semiconductor, it is obtained by electron dynamics between conduction and valance
bands, depending on Pauli exclusions principle, state filling effect and Fermi- Dirac
thermalization. In the case of metal, the relative permittivity is dependent on resonance frequency
and relaxation time of free and bounded electrons (Drude and Lorentz model). The formulation
solve the similar problem but with two different ways because in metals conduction and valance
bands are overlapped while in the case of semiconductors they are separated due to band gap
energy in between them. Therefore, there is difference in simulation approaches. The
semiconductor part of the proposed method although contains essential carrier dynamics,
however, more features can be added such as noise due to spontaneous emission, carrier
diffusion, carriers heating and cooling effects. For LD part of the algorithm more number of
oscillators can be included depending on the range and accuracy requirements in the optical
spectrum. For simplicity, we used one oscillator which is enough for the applications under study.
3. Numerical Results
For numerical results two examples are considered. In the first example an MSM waveguide,
while in the second example a micro-cavity is simulated. The structure for first example is shown
in Fig. 2. It consists of a gallium arsenide (GaAs) slab which is sandwiched between two parallel
gold (Au) plates. Because of the materials arrangement of waveguide we can call it a metal -
semiconductor – metal (MSM) waveguide. In the initial set up, length of the semiconductor slab
is 4 m; thickness is 50 nm, while the width is 100 nm. The thickness of each gold plat is 100
nm, whereas, the length and the width are same as of the semiconductor slab. For semiconductor,
the effective mass of electrons and holes in the conduction and valence band is 0.047 em and
0.36 em respectively, where em is the mass of free electron. The carrier density is 3 x 10 22 3m .
For simulation, 10 energy levels are used for both the conduction and the valance band. The
refractive index is 3.54. Transition time parameters are taken from [16]. The parameters for the
Lorentz-Drude model are
1210 411903π2 .pD rad/sec, 1210 8112π2 .D rad/sec, 1210 2758π2 .L rad/sec,
1 , 024. L and are taken from [12]. A Gaussian profile with wavelength of 800 nm is
injected as a source at the center of the semiconductor slab. The field propagates equally towards
both ends. In our previous works, we have simulated and validated both solid state [9-10] and LD
dispersive [13] models individually. In [9] the solid state part of the algorithm is applied to
enhance the light energy extraction from elliptical microcavity using external magnetic field for
switching applications and was found very accurate. In [10] the solid state model was used to
analyze light energy extraction from the minor arc of an electrically pumped elliptical
microcavity laser. Whereas, in [13] LD part of the algorithm is applied to observe the interactions
between magnetic and non-magnetic materials for plasmonics applications, and results were
accurate. Fig. 3 shows snapshots of the side, top and front views of the structure with the
propagating field inside the MSM waveguide. The dimensions of the waveguide allow only one
mode to propagate, however, by changing the height or width of the slab, more than one mode
can be allowed to propagate.
Fig. 4 shows the electric field intensity with respect to time at different pumping densities. This
figure shows, as expected, that with the increase in pumping density the field intensity also
increases, and vice versa. Due to the plasmonic effect, the field at the interface of metal and
semiconductor enhances at optical frequencies and it creates more electrons and holes in the
semiconductor, which can allow the field to propagate longer distances. An approach to enhance
the propagation distance by incorporating quantum dots in the dielectric medium is studied in [5]
for splitters and interferometers.
Fig. 5 illustrates the electric field intensity with respect to wavelength at different pumping
densities. It is observed that the cutoff wavelength is around 800nm. This shows that at higher
wavelengths, depending on the pumping field density, there is no more change in the field
intensity and curves almost become horizontal; in other words, the semiconductor medium
saturates and allows smooth and maximum field transmission. Fig. 5 also shows that at higher
pumping densities there is a shift in the cutoff wavelength. The reason is that at higher pumping
density the carrier level inside the semiconductor increases due to band filling effect and as a
result, there is a change in the refractive index causing a shift in the wavelength.
For further analysis the width and height of the semiconductor slab are varied. Fig. 6 illustrates
the effect of different widths and heights of the semiconductor slab on the propagating field,
which is observed at a distance of 1 m away from the source. The pumping density for this
analysis is 12w/m .
Fig. 6(a) shows that the total electric field intensity increases with the increase in slab width,
while the height is kept constant (100 nm). For example, when the slab width is 200nm, the loss
in the field intensity at a distance of 1 m from the source is 3% (the actual signal is 0.5, because
the source is equally transmitting in both directions, the field intensity at the observation point is
.47). It is observed that as the width increases, the rate of increase in field intensity becomes
smaller than at smaller widths. Further increase in width may cause other modes to propagate,
and may affect the field intensity at the observation point and, as a result, the propagation
distance. Fig. 6(b) illustrates the total electric field intensity with respect to different heights of
the slab, while the width is kept constant (100 nm). The total electric field intensity increases with
the increase in height. For example, when the slab height is 200 nm, the signal loss is 3.91%, after
travelling the same distance as in Fig. 6(a). Fig. 6(b) shows that after 100 nm, the electric field
intensity is not significantly even with further increase in height. The reason is the thickness of
the metal plates which reach at a level beyond which any variation in height does not show any
impact on the free electrons at the interface of metals. Usually at optical frequencies the field
absorption in metals is few tens of nanometers. These results help us to obtain the optimized
height of the structure.
In the second example, a micro-cavity with radius “R” = 700 nm and thickness “t” = 196 nm is
simulated. The layout of the cavity is shown in Fig. 7 (a). Fig. 7 (b) shows the field distribution in
conventional semiconductor resonator, and depicts a symmetrical pattern for electric field
distribution. In Fig. 7 (c) two cylindrical shaped gold nanoparticles are embedded into the micro-
cavity, each nanopartical has a radius of 140 nm and a thickness of 224 nm. It shows that field
around nanoparticles is confined, and this affects the symmetry of field distribution. Fig. 7 (d)
shows the field distribution with four cylindrically shaped nanoparticles embedded inside the
micro-cavity, it shows field oscillation along two nanoparticles only and suppresses the field in
other directions. This arrangement demonstrates a dipole antenna-like behavior of the micro-
cavity. Fig. 8 shows that by embedding the gold nanoparticles in the micro-cavity, the resonance
frequency can be shifted. The resonance wavelength with two nanoparticles is 640.2 nm, while is
661.20 nm with four nanoparticles, and without nanoparticles is 642.8 nm. The results show that
by using the different combinations of gold nanoparticles inside the semiconductor micro-cavity,
it is possible to shift the resonance frequency to the desired value.
4. Conclusions
In this paper, we presented an approach for the simulation of active plasmonics devices. The
method includes a multi-level multi-electron quantum approach for the simulation of the
semiconductor part and a Lorentz-Drude model to simulate the metallic part of active plasmonics
devices. The methodology is applied to a MSM waveguide and a micro-cavity resonator;
however, it can be applied to other active nano-photonics and plasmonics structures. We hope the
proposed approach will pave the way for new applications in the field of active plasmonics which
is still in its development stage.
Acknowledgements
The authors wish to express their sincerest gratitude to Prof Wolfgang J. R. Hoefer principle
scientist at institute of high performance computing for his advice and useful discussion.
References
1. S. A. Maier, Plasmonics: Fundamentals and Applications, Springer-Verlag, Berlin,
(2007).
2. M. L. Brongersma, and P. G. Kik “Surface Plasmon Nanophtonics, Springer, Dordrecht
Netherlands, (2007).
3. I. Ahmed, C. E. Png, E. P. Li, and R. Vahldieck, “Electromagnetic propagation in a novel
Ag nanoparticle based plasmonic structure”, Opt. Express. 17, 337 – 345, (2009).
4. K. F. MacDonald, Z. L. Samson, M. I. Stockman and N. I. Zheludev, “Ultrafast active
plasmonics” Nature Photonics, 3, 55-58, (2009).
5. A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonics
integration with dielectric loaded SPP waveguides”, Physics Review B. 78, 045425-8
(2008).
6. J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, "Plas-MOStor: A
metal-oxide-Si field effect plasmonic modulator," Nano Letters, 9, 897-902,
(2009).
7. Hill, M. T. et al. Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides.
Opt. Express 17, 11107-11112, (2009).
8. Y. Huang, and S. T. Ho, “Computational model of solid state, molecular, or atomic media
for FDTD simulation based on a multi-level multi-electron system governed by Pauli
exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,”
Opt. Express 14 , 3569-3587, (2006).
9. E. H. Khoo, I. Ahmed and E. P. Li, “Enhancement of light energy extraction from
elliptical microcavity using external magnetic field for switching applications” Appl.
Phys. Lett. 95, 121104-121106, (2009).
10. E. H. Khoo, S. T. Ho, I. Ahmed, E. P. Li, and Y. Huang, “Light energy extraction from
the minor surface arc of an electrically pumped elliptical microcavity laser”, IEEE J. of
Quantum Electronics 46 , 128-136, (2010).
11. Y. Huang, S. T. Ho, “Simulation of electrically-pumped nanophotonic lasers
using dynamical semiconductor medium FDTD method” INEC 2008. 2nd IEEE
International, 202-205, (2008).
12. D. Rakic, A. B. Djurisic, J. M. Elazar and M. L. Majewski “Optical properties of Metallic
films for vertical-cavity optoelectronic devices” Apl. Optics. 37, 5271-5283, (1998).
13. I. Ahmed, E. P. Li and E. H. Khoo, “Interactions between magnetic and non-
magnetic materials for plasmonics”, Inter. Conf. on Materials and Advanced
Tech., Singapore. (2009).
14. K.J. Willis, J. S. Ayubi-Moak, S. C. Hagness and I. Knezevic, “Global modeling of
carrier-field dynamics in semiconductor using EMC-FDTD”, J. Comput. Electron. 8,
153-171, (2009).
15. I. Ahmed, E. H. Khoo, E. P. Li and R. Mittra, “A hybrid approach for solving coupled
Maxwell and Schrodinger equations arising in the simulation of nano-devices, “ IEEE
Antennas and Wireless Propaga. Letters, 9, 914-917, (2010).
16. S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, “Capture of
photoexcited carriers in a single quantum well with different confinement structures,”
IEEE J. Quantum Electron. 27, 1669-1675, (1991).
Figure Caption List:
Fig. 1. The discretization of conduction and valence bands for a multi-level multi-electron model
of a direct band gap semiconductor for simulation with the FDTD method.
Fig. 2. A semiconductor slab sandwiched between two parallel gold plates.
Fig. 3. Snapshots of the electric field intensity for top, side and front views of the MSM structure.
Fig. 4. Electric field intensity with respect to time at different pumping densities.
Fig. 5. Field Intensity with respect to wavelength at different pumping intensities.
Fig. 6. Total electric field intensity for MSM waveguide measured at 1 micrometer away from the
source (a) electric field with respect to the width of the slab, (b) electric field intensity with
respect to the height of the slab.
Fig. 7. Structure and results for a micro-cavity (a) 3D view, (b) electric field distribution in the
conventional micro-cavity, (c) electric field distribution in a micro-cavity with two cylindrically
shaped gold nanoparticles, (d) electric field distribution with four cylindrical shaped gold
nanoparticles embedded inside the micro-cavity.
Fig. 8. Resonance wavelengths for a semiconductor micro-cavity with and without different
combinations of gold nanoparticles.
(a)
Fig. 1.
Fig. 2.
Semiconductor
Metal
Fig. 3.
Top View
Side View
Front View
Y
Z
X
Y
X
Z
Fig. 4.
Fig. 5.
Fig. 6.
(a)
(b)
Fig. 7.
(a) (b)
(c)
(d)
Fig. 8.