graphene-based quantum electronics

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Progress in Quantum Electronics 33 (2009) 165–214

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Review

Graphene-based quantum electronics

M. Dragomana,�, D. Dragomanb

aNational Institute for Research and Development in Microtechnology (IMT), P.O. Box 38-160,

023573 Bucharest, RomaniabUniversity of Bucharest, Physics Department, P.O. Box MG-11, 077125 Bucharest, Romania

Abstract

Graphene, which was discovered in 2004, is one of the most recent nanomaterials. Its uncommon

physical properties and its potential applications in the area of quantum electronics have attracted a

lot of attention. Graphene consists of a 0.34-nm-thick monolayer sheet of graphite consisting of

carbon atoms in the sp2 hybridization state, in which each atom is covalently bonded to three others.

Graphene forms the basic structure of other carbon-based materials: when it is stacked it generates

the graphite, when it is wrapped it creates carbon buckyballs, while when it is rolled-up it forms the

carbon nanotube, which is a key material for nanoelectronic devices that working from few hundred

megahertz up to X-rays. Graphene is the strongest material, having a Young modulus of about

2TPa, and is the material with the highest mobility, due to its intrinsic ballistic transport. Graphene

and carbon nanotubes can be easily functionalized by an applied voltage or pressure, or by chemical

absorption of many molecules, such as oxygen or hydrogen. A functionalized graphene or carbon

nanotube represents a bandgap-engineering material, which is the key concept in quantum

electronics. Based on the above properties, many innovative quantum electronic devices can be built

that can enhance research areas such as nanophotonics, nanoelectronics, or terahertz devices.

r 2009 Elsevier Ltd. All rights reserved.

Keywords: Graphene; Carbon nanotubes; Quantum devices; Terahertz; Nanophotonics

Contents

1. Structure and fabrication methods of graphene and carbon nanotubes . . . . . . . . . . . . . 166

1.1. Basic properties and growth methods of graphene . . . . . . . . . . . . . . . . . . . . . . . 166

1.2. Basic properties and growth methods of carbon nanotubes . . . . . . . . . . . . . . . . . 170

see front matter r 2009 Elsevier Ltd. All rights reserved.

.pquantelec.2009.08.001

nding author.

dress: [email protected] (M. Dragoman).

www.elsevier.com/locate/pquantelec

dx.doi.org/10.1016/j.pquantelec.2009.08.001

mailto:[email protected]

ARTICLE IN PRESSM. Dragoman, D. Dragoman / Progress in Quantum Electronics 33 (2009) 165–214166

2. The physics of carbon nanotubes and graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

2.1. The physics of carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

2.2. The physics of graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

3. Carbon nanotube and graphene devices for signal processing up to THz frequencies . . . 194

3.1. Quantum dots, single-electron transistors in carbon nanotubes and graphene . . . . 194

3.2. Field-effect transistors based on carbon nanotube and graphene . . . . . . . . . . . . . 199

4. Nanophotonic devices based on graphene and carbon nanotubes . . . . . . . . . . . . . . . . . 201

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

1. Structure and fabrication methods of graphene and carbon nanotubes

1.1. Basic properties and growth methods of graphene

Graphene is one of the most recent nanomaterials [1] with important applications. It isin fact a monolayer of graphite with a thickness of only 0.34 nm, which consist of carbonatoms in the sp2 hybridization state. Since in this configuration each carbon atom iscovalently bonded to three others, graphene has a honeycomb lattice composed of twointerpenetrating triangular sublattices. It forms the basic structure of other carbon-basedmaterials, in particular, of graphite, which results from stacked graphene, and of carbonnanotubes, obtained by rolling up one or several graphene sheets.Graphene is a native two-dimensional crystal (2D), in which the charged particles form a

2D gas. Graphene can be quite easily manipulated and, in particular, can be deposited on aSiO2 layer grown over a doped silicon substrate. In this configuration, in which the SiO2

layer has a typical thickness t ¼ 300 nm, the Si substrate acts as a gate that induces asurface charge density n. This charge density, related to the applied gate voltage Vg

through [2]

n ¼ �0�dV g=teffi aVg, (1)

shifts the Fermi energy level in graphene. In Eq. (1), e is the electron charge, and e0 and ed

are the dielectric permittivities of air and SiO2, respectively. These gate-induced carriers arean example of electrical doping, process analogous to the chemical doping typical forsemiconductor devices. In contrast to semiconducting devices, where the chemical dopingcannot be changed once completed, the chemical p- or n-doping in graphene can beelectrically tuned by applying negative or positive voltages, respectively. After thegraphene sheet is deposited on the Si/SiO2 structure, electrodes can be patterned over it inorder to produce certain devices. Fig. 1 shows a typical graphene-based FET-like structure,which represents a generic configuration for many graphene devices.The most widespread method of graphene deposition on Si/SiO2 is based on the

mechanical exfoliation of highly-ordered pyrolytic graphite (HOPG) with an adhesive tapeand the subsequent release of the graphene flake on Si/SiO2 after the tape is removed [2].HOPG is a three-dimensional structure formed from vertically stacked graphene sheets.The peeled flakes of HOPG can be engineered to fall directly on the Si/SiO2 surface, onwhich they are immobilized by van der Waals forces. With this quite rudimentary method,graphene flakes with dimensions of 70–100 mm can be successfully obtained.

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Graphene contour

SiO2

n+ Si

Non-coherent light of wavelength λ

Microscope

Fig. 2. Optical setup to make the graphene visible.

Drain Source

Graphene

SiO2

n+ Si – gate

Fig. 1. Typical configuration of graphene deposited on a Si/SiO2 structure.

M. Dragoman, D. Dragoman / Progress in Quantum Electronics 33 (2009) 165–214 167

For a long time, it was believed that graphene, and 2D crystals in general, cannot befound in a free state. Therefore, the above method for graphene production would havebeen disbelieved if not followed by an astounding discovery, namely that graphene canbe seen at an optical microscope when an incident white light passes through a green, blue(or another colour) filter. So, the graphene detection method is quite simple and consists ofthe visual monitoring (with a microscope) of the Si/SiO2 surface illuminated with filteredwhite light (see Fig. 2).

The wavelength l of the investigating component of white light, which reveals thepresence of graphene and of flakes consisting of several layers of graphene, depends on thethickness of the SiO2 layer [3,4]. In the typical case of a 300-nm-thick SiO2 layer, grapheneis optimally observed in green light, while blue light is the best choice when the SiO2

thickness reduces to 200 nm. These wavelengths, which are in agreement with experimentalresults, are estimated by considering that graphene has a thickness of 0.34 nm and an indexof refraction of n ¼ 2.6�1.3i, irrespective of the illuminating wavelength [3]. The refractiveindices of Si and SiO2 vary with the wavelength. The estimation of the investigatingwavelength implies a simple reflection calculation of the air/graphene/SiO2/Si four-layerheterostructure, the associated contrast of which (called optical visibility) depends on l.The optical visibility is defined as C ¼ (R0�R)/R0, where R is the reflection coefficient inthe presence of graphene, and R0 denotes the reflection coefficient of a bare substrate, withno graphene flake on it. The contrast displays a peak at different wavelengths in the visible

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Table 1

Comparison of graphene fabrication methods.

Starting material Brief description of the method Throughput Graphene dimensions

HOPG Repetitive peeling of HOPG Low Large

SiC Reduction of Si atoms at high temperature Low Moderate

GO GO dispersion into hydrazine High Large

M. Dragoman, D. Dragoman / Progress in Quantum Electronics 33 (2009) 165–214168

spectrum depending on the SiO2 thickness, the optimum investigating wavelengthcorresponding to the maximum of C. Once this wavelength is determined, the colour ofthe filter is selected to make the graphene visible. It is astonishing how such a simpleoptical method has produced such an important step forward in nanotechnology. Thisprocedure can be applied irrespective of the method of graphene production.The mechanical exfoliation of graphene is not a controllable process and, in

consequence, has a low yield. There are also more complicated and more systematic waysto grow and handle graphene than the mechanical exfoliation, which include the AFMmicro-cleavage of HOPG pillars, followed by the deposition of the resulting flakes on Si/SiO2 surfaces [5], and epitaxy [6]. The epitaxial growth of graphene uses a SiC substrate,which is heated in ultrahigh vacuum at a very high temperature, typically greater than1000 1C, the result being a sublimation of Si atoms from the surface and the formation of agraphene sheet on top of SiC. This process has a higher yield, with much less defects thanthe exfoliation method, but is still difficult to implement for fabrication of large grapheneareas. Moreover, the integration with any electronic circuit made on Si is not possible dueto the very high temperatures involved in the epitaxial growth.A new method for obtaining large scale graphene with high throughput was proposed

very recently [7]. In this method, the starting material is the graphite oxide (GO), which isdispersed in hydrazine, the result being stable dispersions of hydrazinium graphene (HG).These are further deposited on various substrates and form uniform films of single- or few-layer graphene. In Table 1 we have presented a comparison between the three methods ofgraphene production discussed above.Despite the apparent fragility, graphene is a robust material, irrespective of the method

of production. The graphene can be washed using solvents such as acetone andisopropanol without damage, but the use of ultrasonic agitation or aggressive cleaningagents (e.g. NMP, DMF, alkali etches, the RCA process) is not indicated. Graphene iscompatible with standard resist processing techniques used frequently for semiconductordevices. In principle, graphene is exposed only in a humidity-controlled environment, butno adverse effects on samples left in ambient conditions were observed. So, graphene iscompatible with the standard semiconductor processes, and hence can be used easily forbuilding innovative devices. Moreover, at present graphene is considered the strongestmaterial, with an elastic stiffness of 340N/m and a Young modulus of 1TPa [8].The physical properties of graphene are unique, and some of them are not even fully

investigated or understood. For example, graphene is characterized by an ambipolartransport (see Fig. 3) and the type of charge carriers is selected by tuning the gate voltagein devices with a configuration similar to that represented in Fig. 1.The typical carrier mobility at room temperature is about 15,000 cm2V�1 s�1 [1], but can

reach the astonishing value of 200,000 cm2V�1 s�1 in suspended graphene for a charge

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ρ (kΩ)8

0Vg (V)

-50 50

T = 300 K

T = 30 mK

4

25-25

Fig. 3. Typical dependence of the resistivity in graphene on the gate voltage and temperature.


carrier concentration of n ¼ 4� 109 cm�2 [9]. Thus, the room-temperature mean-free pathfor ballistic transport is about 300�500 nm. The mobility in graphene does not dependsignificantly on doping and temperature, unlike in the case of metals or commonsemiconductors. The resistivity of graphene, especially in low-mobility samples, is notstrongly modified by temperature changes, but depends significantly on the gate voltage.This dependence, which is expected in a 2D gas, is illustrated in Fig. 3. This weakdependence of graphene resistivity on temperature is accounted for by the dominantscattering mechanisms: residual electron–phonon scattering in high-density samples, andscattering from static impurities in low-mobility samples [10].

These unusual properties of graphene, as well as its finite conductivity when the chargecarrier density (and hence the gate voltage) is zero and the (anomalous) room-temperaturequantum-Hall effect, are consequences of the unique transport characteristics of chargecarriers, which are detailed below. The electron and hole states in graphene are correlatedthrough charge–conjugation symmetry (known as chirality) and their transport isdescribed by a Dirac-like equation for massless particles. As a consequence, both electronsand holes in graphene have a linear dispersion relation for small energies E:

E ¼ �jkjvF (2)

where k ¼ ikx+jky denotes the wavenumber of the charge states, vF the Fermi velocity,and the positive (negative) sign is assigned to electrons (holes). The dispersion relation ofcharge carriers in graphene is displayed in Fig. 4. It implies that graphene is asemiconductor with no energy gap, the valence and conduction bands touching each otherin one point, which is termed Dirac point.

The linear dispersion relation in graphene, described by (2), is found in nature in onlyone case: in photons that propagate in vacuum. The meaning of the linearity of thedispersion relation in the two cases, though, is different. In graphene, it signifies that theeffective mass of charge carriers vanishes, and hence the interaction between electrons orholes and the crystalline lattice is very weak. As a result, the charge carriers propagatewithout collisions through graphene, i.e. balistically, with the velocity vF ¼ 106m/sffic/300,where c is the speed of light. As a result, graphene can be modelled as a 2D gas of masslessfermions. On the other hand, in the case of photons, which are bosons, the lineardispersion relation in vacuum E ¼ o ¼ hc=l indicates that photons propagate with the

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ky

kx

Electrons

Holes

E

0 EF

Fig. 4. The dispersion in graphene.


speed of light. In this respect, graphene can be viewed as a slow-wave structure, in whichthe charge carriers propagate with a velocity much slower than c.Despite the vanishing of the effective mass of electrons and holes in graphene, a finite

cyclotronic mass of carriers can be defined as [11]

E ¼ mcv2F . (3)

This cyclotronic mass, which is of the order 0.05m0, with m0 the free electron mass, isobtained from experimental date related to Shubnikov–de Haas oscillations in tunablemagnetic fields. Expression (3) is similar to the Einstein relation E ¼ mc2, but the analogycannot be further extended. Many other physical effects, such as the charge tunnelingeffect through potential barriers or the position of the Landau levels in magnetic fields,have different manifestations in graphene in comparison to any other known material.The possibility of bandgap-engineering is essential for most quantum electronic devices.

Although graphene is gapless, different methods to induce bandgaps in this material basedon spatial confinement of charge carriers exist. For instance, a graphene sheet contactedwith electrodes can be etched into narrow strips, which are called graphene nanoribbons(GNR). The width W of the strip, with a typical length 1–2 nm, determines the requiredenergy gap Eg that opens at the Dirac point. The energy gap is given by the experimentally-determined relation [12]

Eg ¼ a=ðW �W �Þ, (4)

where W* ¼ 16 nm and a ¼ 0.2 eV nm are fitting parameters. Expression (4) shows that wecan produce an energy gap in the GNR, and that band engineering is possible if thegraphene flake is etched at the desired width. For example, if W varies in the 20–90 nmrange, the corresponding band decreases from 100 to 3meV.

1.2. Basic properties and growth methods of carbon nanotubes

Carbon nanotubes (CNTs) are hollow cylinders, which can be regarded as rolled-upsheets of one or several layers of graphene, as shown in Fig. 5.The CNTs were discovered well before the one-atom thick planar graphene sheet [13].

The CNT that consists of a single rolled-up sheet of graphene is termed single-walled CNT

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C

Graphenesheet

� = 0-zigzag axis

Chiral angle

y

x

C

Rolled graphenesheet → CNT

dx

Rolled graphenearound x →semiconductor CNT

Rolled graphene around y →metallic CNT

y

a1a2

Unit vectors of thehoneycomb cell

�

Fig. 5. Carbon nanotube structure.


(SWCNT), whereas the CNT formed from several concentric layers of graphene is calledmulti-walled CNT (MWCNT). The physical properties of SWCNTs and MWCNTs differconsiderably.

Depending on the rolling direction of the graphene layer (or layers), we can obtainCNTs with semiconducting or metallic transport properties. This remarkable opportunity,to adjust the basic properties of the CNT by changing the rolling direction of the graphenesheets, is not encountered in any other material. The manner in which the graphene sheet isrolled is characterized by two parameters: the chiral angle y, and the chiral vector, orchirality, C. The chiral vector links two equivalent crystallographic sites in graphene, andcan be written as

C ¼ na1 þma2 (5)

where n and m are integer numbers, and a1, a2 are unit vectors of the graphene lattice.The set of numbers (n,m) determines entirely the semiconducting or the metallic

character of the CNT. As a general rule, the CNT is semiconducting when n�ma3i, itdisplays a small bandgap (a semi-metallic behavior) if n�m ¼ 3i, with i ¼ 1, 2, 3,y, and ismetallic if n ¼ m [14]. The semiconducting CNTs characterized by (n,0), also called zigzagCNTs, and the (n,n) metallic CNTs, also known as armchair CNTs, are among the mostcommon CNT configurations [15].

The connection between the set (n,m) and the diameter d of the CNT is expressed as

d ¼ aC�C½3ðm2 þmnþ n2Þ�1=2=p ¼ jC j=p (6)

where |C| is the length of the chiral vector, and aC–C ¼ 1.42 A the length of the carbonbond. Formula (6) provides the meaning of the chiral vector: its modulus is identical to theCNT circumference.

The chiral angle is defined as

y ¼ tan�1½ffiffiffi3p

n=ð2mþ nÞ� (7)

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Furnace

Substrate

Carbonsource

Inert gas

Outlet

Fig. 6. CVD for CNT growth.


and takes the typical values of y ¼ 601 for (n,0) zigzag CNTs and y ¼ 301 for (n,n)armchair CNTs. As illustrated in Fig. 5, however, the domain of y values is customarilylimited to the range (01, 301), case in which the reference axes, or y ¼ 01, is chosen as thezigzag axis. The set of integers (n,m), for example (4,2), (9,0), or (10,10), constitutes analternative method for CNT characterization, since both the diameter and the chiral anglecan be expressed in terms of these integers using (6) and (7), respectively.The most common method of growing CNTs is chemical vapor deposition (CVD).

The CVD (for a review, see Ref. [16]) is performed by the catalytic decomposition ofhydrocarbon or carbon monoxide using metallic catalysts at high temperatures. Typicalcatalysts, such as Fe, Co, and Ni, are used in the form of a solid support or as gasnanoparticles. It is also possible to grow CNT using non-metallic catalysts such asmagnetic fluids, or amorphous hydrogenated carbon nanoparticles [16]. The CVD growthis done in a reactor, as represented in Fig. 6. There are many reviews dedicated to othermethods of CNT growth by PECVD [17], to the non-CVD synthesis of CNTs [18], or tothe growth of CNT arrays [19]. Also, CNTs are obtained by laser ablation or arcdischarges.It is worth noting that graphene has just started to be fabricated using CVD [20], more

exactly using a microwave PECVD, which is strongly related to the above CVD methodsfor growing CNTs. In this case, graphene is obtained in a mixture of methane/hydrogenatmosphere. High quality sheets of single-, or double-layer graphene (also called bilayergraphene) were synthesized at 500 1C on stainless steel substrates. High yield in thegraphene production is obtained using a commercial microwave PECVD system.Irrespective of the chosen method to manufacture CNTs, there are two major

drawbacks. The first one is that CNTs are synthesized at high temperatures(700–900 1C) and thus the integration of CNTs with semiconductors processed at muchlower temperatures is problematic. The second drawback is that CNTs cannot befabricated with prescribed properties, especially with well-defined diameters and chiralities,so that the fabricated CNTs are most commonly a mixture of metallic and semiconductingnanotubes. Additional methods to select CNTs with specific properties are thus required[21]. For example, in many applications it is necessary to have only semiconducting CNTs.Then, the metallic CNTs are burned by passing an electrical current through the CNTbundle. Irradiation with a xenon lamp or with a laser has the same effect, according to thereview presented in Ref. [21].Electrophoresis is another method to separate CNTs, based on their mobility in a

solution when biased with a dc electric field, whereas dielectrophoresis sorts the CNTs on

ARTICLE IN PRESSM. Dragoman, D. Dragoman / Progress in Quantum Electronics 33 (2009) 165–214 173

the basis on the electrical permittivity when excited with an ac field. The last methodexploits the fact that the metallic and the semiconducting CNTs have quite differentdielectric constants. Ultracentrifugation and selective growth are other methods able toseparate various CNTs.

In the case of dielectrophoresis, the dielectrophoretic force is given by [22]

FDEP / �Sð�s � �mÞ=ð�s þ 2�mÞrE2rms (8)

where the indices s and m refer, respectively, to the semiconducting and metallic CNTs in amixture, while Erms

2 the rms (root-mean-square) of the ac dielectric field amplitudes. Atfrequencies around 10MHz the dielectrophoretic force is negative for semiconductingCNTs and positive for metallic CNTs, because their dielectric permittivities differ by anorder of magnitude. A very useful and detailed review about CNT synthesis, organizationand sorting is found in Ref. [23].

2. The physics of carbon nanotubes and graphene

2.1. The physics of carbon nanotubes

The physics of CNTs and graphene have many aspects in common since CNTsare rolled-up graphene sheets. The dispersion relation of graphene is deduced assumingthat the complicated energy structure of sp2 bonded carbon atoms can be obtained bymeans of two bands only: the valence band p and the conduction band p*. Thisassumption leads to a simple tight-binding model [24] that uses the nearest-neighborHamiltonian

H ¼E2p �g0gðkÞ

�g0g�ðkÞ E2p

!(9)

related to carrier hoping between the two bands. The overlap of carbon atoms in thegraphene lattice is also accounted for. In the Hamiltonian in (9), E2p is the site energy ofthe 2p atomic orbital, g040 is the carbon–carbon energy, and

gðkÞ ¼ exp½iðkxaÞ=31=2� þ 2 exp½�iðkxa=2Þ=31=2� cos½ðkyaÞ=2� (10)

where k ¼ (kx, ky) a vector in the reciprocal space of the graphene lattice anda ¼ |a1| ¼ |a2| ¼ 3aC–C. The overlap matrix related to the carbon atoms in the graphenehoneycomb lattice is expressed as

So ¼1 sgðkÞ

sg�ðkÞ 1

!(11)

where s is the overlap of the electronic wavefunction over adjacent sites. Then, thegraphene band structure E2D(k) is obtained from the condition

detðH � E2DSoÞ ¼ 0, (12)

which has a two-branch solution,

E�2DðkÞ ¼E2p � g0oðkÞ1� soðkÞ

, (13)

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15

10

5

0

-4

-2

0

2

4-4

-2

0

2

k ya

-2

02

4

kxa

kxa

E+2D

E- 2D

0

-2

-4

-6

-4

4

-4

-2

0

2k y

a

4

Fig. 7. E2D+ (a) and E2D

� (b) dependence on the wave vector.


where

oðkÞ ¼ jgðkÞj1=2 ¼ ½1þ 4 cosð31=2kxa=2Þ cosðkya=2Þ þ 4 cos2ðkya=2Þ�1=2. (14)

The two branches of the energy band structure in (13) are illustrated in Fig. 7a and 7b,respectively for g0 ¼ 3 eV, E2p ¼ 0, and s ¼ 0.13, while Fig. 8 shows the entire bandstructure of graphene for the same parameters, formed by superimposing the two branchesof the dispersion relation, E2D

+ and E2D� .

When E2D+ and E2D

� are superimposed, the distance between these two branches isminimum (the two branches in Fig. 8 actually touch each other) at six points, called Fermipoints. These points, denoted by K and K’ in Fig. 9, form a hexagon that represents thelimit of the Brillouin zone in graphene, and are situated at the Fermi level EF. The Fermienergy level at the K points is considered as a reference, i.e. it corresponds to E ¼ 0.Graphene is thus a zero-bandgap semiconductor, its density of states (DOS) vanishing atthe Fermi level.

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15

10

E2D 5

0

-5

-4-2

0

24

-4

-2

0

4

k ya

2

kxa

Fig. 8. The band structure of graphene.

K’

K

K’

K

K’

K

M

MM

M

M

Γ M

Fig. 9. Brillouin zone of graphene.


The band structure of the CNT follows from that of graphene by specifically taking intoaccount that, by rolling the latter material into a nanotube, the charge carriers becomeconfined in the circumferential direction of the tube. This confinement leads to thequantization rule (see Eq. (6))

kcC ¼ 2pj, (15)

where kc is the circumferential component of k, j ¼ 1, 2, 3,y,N is an integer, and N thenumber of hexagons in a unit cell of the CNT.

The quantization rule (15) suggests that the allowed energy band structure of the CNTcan be understood as being obtained by slicing the 2D graphene band structure into one-dimensional (1D) sections. In particular, in the vicinity of a K point, where ka51 if thewavevector k ¼ |k| is measured from K, o(k) in (14) becomes linearly dependent on k and

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K K

KaKc

Ka

Kc

Metallic CNT Semiconductor CNT

EFEg

Fig. 10. CNT confinement and dispersion relation.


we have

oðkÞ ffi 31=2ka=2þ . (16)

Moreover, if E2p ¼ 0 and s ¼ 0, the energy band structure of graphene in (13) turns into

E2D ffi 31=2g0kaC�C, (17)

expression that leads to the following energy dependence on the wavevector in CNTs [25]:

ECNT;j ¼ E2DðkKa=jKaj þ jKcÞ. (18)

In (18) Ka is the reciprocal lattice vector along the CNT axis and Kc is a unit vector alongthe circumferential direction. The allowed values of the wavenumber k are situated in theinterval (�p/T, p/T), where T is the length of the translational vector T. According torelation (18), the CNT wavevector in the Kc can have only N discrete values, whereas thewavevector component along the Ka direction can vary continuously. As a result, thehexagonal Brillouin zone of graphene is cut by N lines of the form kKa/|Ka|+kKc, whichdefine the behavior of the CNT. More precisely, as shown in Fig. 10, the CNT is metallic ifthe cuts pass through the K point and semiconductor otherwise. In terms of the (n,m) set ofintegers, the CNT is metallic if n�m ¼ 3i, where i is an integer number, and semiconductorotherwise.So, at low energies, around the K points, the band structure of a metallic CNT is formed

from two energy bands, which intersect at the K point and have a linear dispersionrelation, whereas a semiconductor CNTs is characterized by an energy gap Eg that opensbetween the two energy bands. This bandgap is given by

Eg ¼ 4vF=3d, (19)

and has a numerical value of

EgðeVÞ ffi 0:9=dðnmÞ (20)

for a typical value of vF ¼ 8� 107m/s. So, by simply changing the CNT diameter from afraction of a nanometer to several nanometers, the bandgap of the semiconducting CNTscan be engineered to vary in the 20meV–2 eV range. Note that bandgap-engineering is alsoaccomplished in AIII-BV semiconductor heterostructures through modifications in the


chemical composition of the heterostructure, but the resulting variation of the energy gapis smaller than in CNTs.

The bandgap calculated with the tight-binding model described above is given byEg ¼ 2g0aC�C/d, where g0 is the hoping matrix element. This expression, which does notdepend on y, fails to describe in a satisfactory manner semiconducting CNTs with smalldiameters, i.e. CNTs with 4ono9. A more suitable formula for this case is [26]

Eg ¼ 2g0aC�C½1þ ð�1Þp2baC�C cosð3yÞ=d�=d, (21)

where p is an integer that satisfies the condition n�2m ¼ 3i+p and b is a constant. Fortypical values of g0 ¼ 2.53 eV, b ¼ 0.43, and y ¼ 0, we obtain

Eg ¼ 2g0aC�C=d � 4g0ba2C�C=d2. (22)

These results are valid for defect-free graphene and the crystalline CNT that results afterrolling it. However, the linear bandgap dependence on 1/d is a more general outcome,which holds even for disordered graphene [27]. More precisely, the bandgap of theamorphous semiconducting CNT is

Eg ffi 3� ð2Þ�1=2mCo2a3C�C=d, (23)

where mC ¼ 2� 10�23 g is the mass of the carbon atom and o ¼ 1:00 cm�1 the averagephonon energy. The effect of the disorder produced by the defects in the graphene lattice isto induce a faster decrease of the CNT bandgap with its diameter compared to the case ofcrystalline CNTs. The bandgap dependence on the diameter of both crystalline andamorphous CNTs is illustrated schematically in Fig. 11.

The introduction of defects in the graphene lattice is an example of CNTfunctionalization, i.e. of a drastic and controllable change of the physical properties ofCNTs. Ref. [26] is an excellent review of CNT functionalization, which can be achieved inmany ways, including adsorption of individual atoms or molecules (hydrogenation,oxygenation), doping, and application of radial mechanical deformations and electrical ormagnetic fields.

100/d (Å–1)

Eg

(eV

)

1.8Amorphous CNT

Crystalline CNT

0 10

Fig. 11. Diameter dependence of the bandgap of amorphous and crystalline semiconducting CNTs.

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0.6 eV

1 eV

2 eV

Intrinsic CNT

Eg

Ev

Ec

Hydrogen

Fig. 12. Superlattice done by hydrogenation of the CNT.


All parameters of the CNT: the geometry, the band structure, and the binding energies,are modified when hydrogenation is performed, and depend strongly on the hydrogenationpattern. For instance, a superlattice can be obtained by selective hydrogenation of theCNT, as is shown in Fig. 12.Even the shape of zigzag CNTs change when uniformly exohydrogenated at half-

coverage Y ¼ 0.5, where Y measures the different isomers at different hydrogen coverage.For example, the shape of (7,0) CNT changes from circular to rectangular, whereas that of(n,0) CNTs, with n ¼ 8, 9, 10, becomes square. In addition, at half-exohydrogenation, i.e.at Y ¼ 0.5, the semiconducting CNTs have two double-degenerated and dispersionlessvalence and conduction bands that are separated by a bandgap, which increases as theCNT diameter decreases. This bandgap closes for a large n value. Full exohydrogenation,Y ¼ 1, also affects the bandgap of the CNTs: it increases by 2 eV in semiconducting CNTsand takes a maximum value of 5 eV, and opens in metallic CNTs. These transform intosemiconducting CNTs with a bandgap that is somewhat larger than in semiconductingnanotubes with the same diameter.Functionalization by adsorbtion of oxygen molecules transforms low-diameter

semiconducting CNTs, for example the (8,0) CNT, into metallic nanotubes, while anapplied axial strain can increase or decrease the bandgap of a CNT. In the last case theenergy bandgap depends on the strain s according to [28]

dEg=ds ¼ sgnð2pþ 1Þ3g0ð1þ vÞ cos 3 y, (24)

where v is the Poisson ratio, and p ¼ –1, 0, or 1 is related to the integers n and m throughn�m ¼ 3i+p, with i an integer. From (24) it follows that for semiconductor CNTs withp ¼ 1, dEg/ds40, whereas dEg/dso0 for the nanotubes with p ¼ �1. This controlledmodification of the bandgap of semiconductor CNTs under an applied strain can be usedto implement heterostructures with engineered bandgap, as in AIII-BV semiconductors.Another method to tune the electronic properties of individual semiconducting CNTs is

chemical doping, which shifts the Fermi energy level in the valence band for a p-type CNT,or in the conduction band for an n-type CNT. The semiconductor CNT is of p-type innormal atmospheric pressure and at room temperature, its electronic properties beingdetermined by holes. The origin of the non-intrinsic, natural p-type character ofsemiconductor CNTs is not related to any charge transfer from metal contacts patternedover the CNT, chemical contamination, or adsorbtion of molecules (e.g. oxygen orchemical groups) with which the patterned CNT comes in contact. Rather, the p-type

ARTICLE IN PRESS

1

G (

μS)

0

VG (V)0-5 5

n-type

p-type

Fig. 14. The conductance behavior for doped CNTs.

S D

K

CNT

n++SiSiO2

Cover

Fig. 13. CNT MOS device.


conduction is a self-doping phenomenon specific to the nanoscale geometry of the CNT. Itis produced by the curvature-induced rehybridization (charge redistribution among thebonding orbitals), which is a function of the tube diameter [29].

The n-doping of semiconducting CNTs, through which the type of majority carrierschanges from holes to electrons, is essential for the fabrication of electronic and photonicdevices. Potassium (K) is an electron donor for CNTs, and hence enhances considerablytheir conductance. The n-doping is achieved in the configuration illustrated in Fig. 13,which is known as CNT metal-oxide-semiconductor field-effect transistor (MOSFET) [30].In this configuration the CNT is placed between two metallic electrodes (source S anddrain D), on top of a Si/SiO2 structure that acts as an electrical gate. In the process ofn-doping the CNT is introduced in a vacuum vessel that contains K vapors, produced byelectrical heating a potassium source. If the CNT is partially covered with an organicmaterial, for example PMMA, only a part of it is exposed to potassium doping. Typicaldoping levels are nK ¼ 100–100 mm–1, consistent with a few K atoms/1000 C atoms.

Fig. 14 indicates that the type of CNT doping can be easily found by performingconductance measurements of the device. If the CNT is p-doped and the holes are thedominant carriers, the conductance reduces when the gate voltage increases, whichindicates that the Fermi level EF is inside the valence band VB. Similarly, an increase of theconductance with the gate voltage implies that EF is inside the conduction band CB andthat the electrons dominate the electrical conduction.

Intrinsic CNT semiconductors, for which n ¼ p, are encountered in suspended CNT,which are not in direct contact with the gate oxide. As illustrated in Fig. 13, a suspendedCNT can be implemented by micromachining a trench in the central part of the SiO2 layer.

The doping process in either nanotubes or graphene can be easily implemented usingvarious configurations of the splitting gates. This type of doping is called electrostatic or

ARTICLE IN PRESS

CNTD S

SiO2VG1 VG2

doped Si substrate

Fig. 15. The electrical doping of a CNT.

-VG

+VG

CNT

Dielectric (SiO2)Electric field

Fig. 16. The CNT splitting gate configuration to tune the gap of the CNT.


electric doping, and is widespread in CNT or graphene devices because chemical doping isa difficult process in such small structures. In Fig. 15 we present a CNT configuration withtwo buried gates. When VG1o0 and VG240 the device acts as a p�n junction [31], while ina slightly different geometry, with a grounded back gate, it resembles a p–i–n diode if the[32]. The drain–source current of the device represented in Fig. 15 is given by

IDS ¼ I0½expðVDS=nkBTÞ � 1�. (25)

An electrical doping is performed in almost any graphene device, and expression (1),nffiaVg, is an illustration of this doping concept.The band structure of a CNT can be also modulated by applying dc electric and

magnetic fields. The bandgap of a semiconducting CNT is narrowed by the application ofan electric field transverse to the tube axis (a transverse electric field), whereas metallicCNTs are not affected by the applied field. Unlike the CNT MOSFET configuration inFig. 13, in which the Fermi level is shifted in the valence or conduction band beyond athreshold gate voltage, in the split-gate approach the semiconducting CNT is positionedbetween two metallic gates. In this configuration, represented in Fig. 16, the Fermi level iszero.In the split-gate configuration, the uniform field distribution around the CNT

circumference is guaranteed by the SiO2 thickness of the gate, which must be comparablewith the tube diameter [33]; on the contrary, the corresponding dielectric thickness in theCNT MOSFET configuration is large compared to the nanotube diameter. In the split-gate configuration the energy bandgap of a (10,0) nanotube, for instance, changes from1.1 eV when no gate voltage is applied, to 0.4 eV for VG ¼ 9V [33].According to other studies, (n,n) metallic CNTs experience a reversible metallic–semi-

conductor transition beyond a threshold value of the electric field [34]. When subject to


such a transverse electric field, an energy bandgap opens in the metallic CNT and convertsit into a semiconductor, the bandgap of the (n,n) CNT enhancing as the electric fieldincreases until it reaches the maximum value

EgðeVÞ ¼ 6:89=n. (26)

This value corresponds to the gate voltage

VG;maxðVÞ ¼ 12:09=n. (27)

Under the action of a magnetic field B directed along the axis of a CNT, the wavevectorcomponent in the circumferential direction acquires an additional contribution associatedto the Aharonov–Bohm effect. More precisely, its modulus increases from |Kc| ¼ 2(i�p/3)/d, in the absence of the magnetic field to

jK cj ¼ 2ði � p=3þ f=f0Þ=d. (28)

In (28) i is an integer, p ¼ 0 or 71, f the magnetic flux through the CNT and f0 ¼ h/ethe magnetic flux quantum. The wavevector variation as the magnetic flux f (via themagnetic field B) changes can modify the bandgap of a semiconducting CNT or can eveninduce a reversible metal–semiconductor transition. For instance, a metallic CNT at f ¼ 0transforms into a semiconducting nanotube at f ¼ 0.5f0, returns to the metallic state atf ¼ f0, and so on. These metal–semiconductor transitions induced by Aharonov–Bohmphase shifts have been confirmed experimentally in CNTs [35]. Still, high magnetic fields,of 10–50T, are required for a significant change of the bandgap even at low temperatures[35].

From (28) it follows that, when f ¼ 0, the allowed circumferential wave vectors areseparated by 2/d for a fixed p. Because there is a 1D subband for each |Kc| inside theBrillouin zone, a variation of i leads to the generation of a set of subbands. Thus, small-diameter CNTs have only a few subbands, whereas large-diameter CNTs have manysubbands. Close to the Fermi energy, the subbands can be expressed as

EiðkÞ ¼ �ð2vF=dÞ½ði � p=3Þ2 þ ðkd=pÞ2�1=2 ¼ �E0½ði � p=3Þ2 þ ðkd=2Þ2�1=2 (29)

The dispersion relation (29), which is displayed in Fig. 17, describes semiconductingCNTs when p ¼71 and metallic CNTs when p ¼ 0.

E

k

E

DOS

DOS

EE

METALLIC CNT

SEMICONDUCTING CNT

k Eg

Fig. 17. The dispersion relation of CNTs and corresponding DOS.


In the vicinity of the Fermi level the energy band minimum is

Ei ¼ �2vF i=d (30)

for p ¼ 0, which, as represented in Fig. 17, corresponds to two non-crossing levels if ia0and two crossing levels if i ¼ 0. There are no crossing levels for semiconducting CNTs, ascan be inferred from (29). Expression (29) is equivalent to another energy dispersionrelation [36] written as

EiðkÞ ¼ �½ðvF kÞ2 þ ðEig=2Þ

2�1=2 (31)

where vF ¼ 8� 105m/s and Egi are the individual forbidden energy gaps for each subband.

The typical DOS for 1D quantum systems has a series of sharp peaks, denoted as vanHove singularities. In CNTs the DOS is generally described by the formula

rðEÞ ¼X

i

riðEÞ ¼ ð4=pvF ÞX

i

½1� ðEig=2EÞ2��1=2, (32)

the peaks having a mean energy spacing of _vF/d.In semiconducting (n,0) CNTs, the DOS is described by the analytical model [37]

rsðE; nÞ ¼ ðr0=ffiffiffi3pÞjEj

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE2 � E2

1ÞðE22 � E2Þ

q(33)

where r0 ¼ 4a/aC–Cp, a is the zone degeneracy, and E1 and E2 are the van Hovesingularities:

E1 ¼ �jg0ð1þ 2 cosðpp=nÞj (34a)

E2 ¼ �jg0ð1� 2 cosðpp=nÞj (34b)

In (34a) and (34b), p ¼ 1, 2,y is the index of the subband and g0 ¼ 3 eV is the overlapenergy (see Eq. (13)). In the expression of r0, a ¼ 1 if the energy E is located in theBrillouin zone center, and a ¼ 2 otherwise.In the case of (n,n) metallic CNTs, the DOS is given by

rmðE; nÞ ¼ r0jEj

, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

1

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

1

q� A1

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

2

qþ A2

� �s(35)

where

E1 ¼ �jg0 sinðpp=nÞj (35a)

E2 ¼ �g0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5� 4 cosðpp=nÞ

p(35b)

A1 ¼ g0½�2þ cosðpp=nÞ� (35c)

A2 ¼ g0½2þ cosðpp=nÞ� (35d)

In terms of the parameters defined above, the group velocity is CNT is given by

vðE; pÞ ¼ a=prðE; pÞ, (36)

the effective mass of carriers in zigzag CNTs in subband p can be written as

m�ðE; pÞ ¼ 162E3=3a2C�C½ðE1E2Þ

2� E4� (37)


and the effective mass at the conduction band minima for the first subband, with Eg ¼ 2E1 is

m�c ¼ 42Eg=3a2C�Cg0ð2g0 þ EgÞ. (38)

The effective mass in (37) varies dramatically with the energy. For example, m*c/m0 ¼ 0.15 fora (19,0) CNT with a diameter of only 1.5nm and for an energy of 0.1 eV, while for an energy of0.25 eV we have m*c/m0 ¼ 0.45, value that is three times higher.

The carrier density in zigzag CNTs is simply expressed as [37]

n ¼ 2Nc exp½ð2EF � EgÞ=2kBT � (39a)

where

Nc ¼ ð4a=aC�CÞ½ðEg þ kBTÞ=ðEg þ 4g0Þ�. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kBT=3pEg

q. (39b)

From the above relation it is easy to compute the quantum capacitance per unit length,defined as the derivative of the charge per unit voltage as a function of Fermi level. Thisparameter is given by

CqðEF Þ ¼ ð4eNc=kBTÞ expð�Eg=2kBTÞ coshðEF=kBTÞ (40)

In principle, almost all CNT devices are controlled by a gate, which is positioned overthe top of the CNT or is backing it, gate that is characterized by the gate capacitance Cg.So, the total capacitance is

Ctot ¼ CgCq=ðCg þ CqÞ (41)

The conductance of a CNT is given by the Landauer formula

G ¼ ð2e2=hÞTðEÞ (42)

where T(E) is the transmission of the nanotube. In metallic nanotubes in the ballisticregime T ¼ 1, and the conductance at the Fermi level is given by

G ¼ 2� G0 ¼ 2� ð2e2=hÞ ¼ 2� ð12:9 kOÞ�1 ¼ 155ms, (43)

since two subbands are occupied (those crossing at EF). The associated resistance is R ¼ 1/G ¼ 6.5 kO.

The transport is ballistic in a CNT device when the following conditions are fulfilled:

Lfpbd and LfpbL (44)

where Lfp is the mean-free path of the carriers and L is the tube length. The mean-free pathis a few micrometers in metallic CNTs, and of the order of 300–400 mm in semiconductingCNTs at room temperature [38].

The mobility depends on the mean-free path according to

m ¼ ð2G0ÞLfp=Ne, (45)

where N is the carrier density. The carrier density can be estimated by assuming that theconductance of highly doped CNTs is similar to that of metallic CNTs, and that EF ¼ Eg

beyond a certain amount of doping. Then, kF,met ¼ kF,semic, N/2 ¼ 2kF/p and thus

N ¼ 4� 31=2Eg=hvF . (46)

The mobility of CNTs with diameters between 1 and 5 nm varies in the range104�5� 104 cm2V�1 s�1 at room temperature, increasing with the diameter.


Alternatively, when a dc electric field E is applied along the axis of a CNT, the mobilitycan be defined as

m ¼ vd=E. (47)

For example, the simulations in [39] show that the mobility can reach the value of120,000 cm2V�1 s�1 in a (59,0) nanotube placed in an electric field of 10 kV/cm. Themobility can be expressed et/m, where t is the scattering time, and has the form

m ¼ ðng3=40 =4dÞ2½1þ ð1� gcdðnþ 1; 3Þ=n2=3Þ� � 104 cm2=Vs. (48)

where gcd(n+1, 3) is the greatest common divisor between n+1 and 3.The mobility depends on the applied field according to [39]

mE ¼ m=½1þ lðEÞE=vmax� (49)

where

vmax ¼ ð3ng20=8Þ1=3½1þ ð1� gcdðnþ 1; 3Þ=2nÞ� � 107 cm=s (50)

is the peak of the drift velocity and

lðEÞ ¼ HðE � E0Þfm exp½log210ðE=EcrÞ=S� � vmax=Ecrg. (51)

Here, H is the Heaviside function and

Ecr ¼ g0d2f1þ ð8=nÞ2½gcdðnþ 1; 3Þ � 1�g=27n3=2 � 106 V=cm (52)

is the critical dc electric field at which the drift velocity reaches its highest value,S ¼ 1.3+H(E�Ecr)n

1/2/2, and E0 ¼ 350� [1+3gcd(n+1,3)/7]/n2.According to (45) the mobility is a function of the conductance of the nanotube, which

has theoretically only two conducting channels. Nevertheless, new conduction channelscan open during oxygenation or doping (in general, during functionalization), if thechemical potential changes [40]. The dispersion relation of CNTs is formed fromconduction and valence bands linked by six Fermi points (see Fig. 9) but these pointsbecome circles with radii equal to 2|Dmchem|/(6g0aC–C), if the chemical potential shifts witha small amount |Dmchem|. In this case, two new channels open in metallic CNTs at integermultiples of the lateral quantization energy Eq ¼ 6g0aC–C/d for each Fermi point thatbecomes a circle (more exactly, that transforms in the center of the circle), while only onenew channel opens in semiconducting CNTs, at |Dmchem| ¼ (i+p/3)Eq. So, the maximumnumber of open channels for metallic and semiconducting CNTs is given, respectively, by

nmet ¼ 2½1þ 2IntðjDmchemj=EqÞ�, (53)

nsemic ¼ 2fInt½2=3þ jDmchemj=Eq� þ Int½1=3þ jDmchemj=Eq�g. (54)

The conductance dependence of doped CNTs on the chemical potential is presented inFig. 18 where G/G0 is the number of open channels in (53) or (54).An important issue when dealing with any CNT device is the contact resistance. The

total resistance of the CNT can be written as

Rtot ¼ h=4e2 þ ðh=4e2ÞðL=LfpÞ þ Rc (55)

and is formed from three components: a ballistic component, described by the first term inthe right-hand-side of (55), the Drude resistance due to scattering (the middle term), andthe contact resistance Rc [41]. Although in the ideal situation the intrinsic (ballistic)

ARTICLE IN PRESS

G/G

0

2

4

6

8

10

Δ�chem/Eq

1 2 3

Metallic CNT

SemiconductingCNT

Fig. 18. The opening of multiple conduction channels of CNTs via doping.


resistance of the metallic CNT is h/4e2 ¼ 6.5 kO, resistances over 1MO are not uncommonand are due to various parasitic effects CNT. A method to lower this huge resistance up to8–10 kO is to evaporate gold on top of the CNT using a CVD technique, or to work withplanar and annealed gold electrodes. The problem is more complicated in semiconductingCNTs because, depending on the work function difference between the CNT and the metaland on the type of carriers (n or p), the contacts behave as Schottky barriers or are purelyresistive (ohmic). Taking into account that the CNT work function is about 4.5 eV, weobtain ohmic contacts if

WmetðeVÞ44:5þ Eg=2, (56)

where Wmet is the work function of the metal. An example of a good contact electrode forsemiconducting CNTs is palladium (Pd), which satisfies (56) since it has a large workfunction. The conductance of semiconducting CNTs contacted with Pd is higher than0.5� (2G0), which suggests that no parasitic barrier is formed [36,42].

The physical properties of CNTs, and especially the conductance at different positionsalong the nanotube, are commonly investigated experimentally with AFM and STMtechniques. From such conductance measurements at the nanoscale one can extract manyimportant physical properties of CNTs. For instance, the physical nature of the metalliccontacts to semiconducting CNTs can be determined if AFM is used as a movableelectrical nanoprobe that senses locally the electrical properties of the nanotube [43]. Suchmeasurements with a metallic tip that plays the role of a voltage nanoprobe, revealed thatthe contact between Au and a p-type CNT is ohmic. In this case, the conductance wasfound to be about 1.5G0 after annealing at 600 1C in argon environment, which shows thatthe carriers are directly injected in the valence band of the CNT. The Au-p-CNT contacthas a typical resistance of 10–50 kO, which is insensitive to the gate voltage and approachesthe theoretical limit of h/8e2 ¼ 3.2 kO. The mean-free path of the p-type CNT, calculatedfrom

RCNT=L ¼ ðh=4e2Þ=Lfp (57)

was estimated at 300 nm at room temperature for RCNT/L ¼ 20 kO/mm.

ARTICLE IN PRESS

CNT network

SiO2

n+ Si

Fig. 20. The random network of CNTs.

Carrier injection

Au

Ec

EvEF

n-type CNT

Au

Ec

Ev

n-type CNT

Carrier injection

Fig. 19. The contact between the n-type CNT and gold: (a) parasitic p–n junction and (b) Schottky contact.


In deep contrast to the above situation, the contact between gold and an n-type CNThas a high resistances that can be attributed to the Schottky contact or to the parasitic p–njunction that appear near the contact. A parasitic p–n junction, as that represented inFig. 19a, is characterized by a Fermi energy level inside the valence band, while in theSchottky contact the Fermi level is situated in the CNT gap (see Fig. 19b). The contactresistance due to the Schottky contact or the parasitic p–n junction is of the order of MO,but can be reduced to a value close to 2G0 by reducing the barrier. Pd contacts can alleviatethe problems related to the parasitic Schottky barrier.So, in principle, the room-temperature conductance could achieve the ballistic limit of

2G0 in both metallic and semiconducting CNTs if the metal for contacts is suitably chosen.A similar problem is encountered in graphene devices. If we wish to lower further thecontact resistance, several CNTs must be placed in parallel.Thin films of arbitrarily oriented CNTs or CNT networks are also very useful

configurations in some sensors, high-frequency and optical devices [44]. The resistance ofsuch CNT thin films, represented in Fig. 20, is quite low and is given by [44,45]

RS ¼ xðrCNT �NCÞaL

bS (58)

where x is a fitting parameter, rCNT is the density of CNTs, i.e. the number of CNTs perunit area, NC is the percolation threshold, LS denotes the average tube length, and a and bare parameters depending, respectively, on the spatial arrangement of CNTs and on theconductivities of the CNT and of the tube–tube junction. The percolation threshold isdetermined from the relation

LSðpNCÞ1=2¼ 4:236 (59)


Another important tool for CNT analysis is the STM. With its help, the electronicwavefunctions of quantum energy levels in metallic CNTs were computed from theperiodic oscillations in the differential conductance at different positions along thenanotube. This method was applied at low temperatures in short CNTs, which are few tensof nm long [46,47]. In STM measurements, the conductance dI/dV is proportional to thelocal density of states (LDOS) of the structure under consideration. In particular, in shortCNTs the electrons can be viewed as ‘‘particles in a box’’ and have thus quantized energylevels separated by DE ¼ hvF/2LE2meV/L(nm). The STM conductance measured atvarious positions r and bias voltages V is given by

dI=dV /X

jeV�EijodE

jciðrÞj2, (60)

where dE is the energy resolution and ci(r) are the electronic wavefunctions thatcorrespond to the discrete energy levels Ei, i ¼ 1, 2, 3,y. The sum in (60) reduces to asingle term if DEo|Ei+1�Ei|, case in which it is possible to map |ci(r)|

2 by conductancemeasurements at energy Ei, the steps in the I–V characteristic being associated to thedifferent energy states i.

The dispersion relation can be also experimentally obtained with the STM method [47].For metallic CNTs this dispersion relation is linear. The van Hove singularities in theDOS, which depend on energy as E�1/2, correspond to the sharp peaks in the dI/dV

dependence on the bias voltage. The number of occupied subbands is determined from thenumber of van Hove singularities, whereas the distance between two consecutive peaks inthe conductance indicates the width of the bandgap.

In metallic CNTs with a large contact resistance, the low-temperature conductancecurve G(VG) shows a periodic series of spikes, which are due to the Coulomb blockadeeffect. In this case, the CNT can be modelled as a metallic island that is weakly coupled tothe contacts via tunnel barriers and the electrostatic CNT capacitance takes small values,up to 50 aF/mm. This capacitance is expressed as

Cg ¼ 2p�0�r= lnð2z=dÞ, (61)

where z is the gate height. Thus, a charging electrostatic energy of EchE5meV/L(mm) mustbe overcome to inject a single electron in the CNT.

The Luttinger-liquid behavior in metallic CNTs, which originates in the electron–elec-tron interactions, can be studied also with the STM. The Luttinger liquid is a ground statecharacterized by: (1) a low-energy charge, (2) a suppression of the DOS consistent withr(E)p|E�EF|

a, and (3) a spin excitation that propagates with different velocities. TheLuttinger liquid is described by the parameter

g ¼ ð1þ 4Ech=DEÞ�1=2, (62)

which determines the level of electron correlation in CNTs. In (62) DE is the spacingbetween different energy levels. A strong electron correlation, for example, corresponds toa g value of 0.22. For low applied voltages, when eV5kBT, the CNT response is linear andthe conductance depends on temperature according to [48]

GðTÞ / Ta, (63)

where the parameter a is linked to g, while at large biases dI/dVpVa. Therefore one canobtain g from conductance measurements.

ARTICLE IN PRESS

Table 2

CNT physical properties.

Parameter Value and units Observations

Length of the unit vector a ¼ffiffiffi3p

aC�C ¼ 0:49 A aC�C ¼ 1.44 A is the carbon bond length

Current density 4109A/cm2 1000 times larger than the current density in copper

Measured in MWCNTs

Mobility 10,000–50,000 cm2V�1 s�1 Simulations indicate mobilities beyond

100,000 cm2V�1 s�1

Mean-free path (ballistic

transport)

300–700nm in

semiconducting CNT

Measured at room temperature

At least three time larger than the best

semiconducting heterostructures1000–3000nm in metallic

CNT

Conductance in ballistic

transport

G ¼ 4e2/h ¼ 155ms 1/G ¼ 6.5 kO

Luttinger parameter g 0.22 The electrons are strongly correlated in CNTs

Orbital magnetic moment 0.7meVT�1 (d ¼ 2.6 nm) The orbital magnetic moment depends on the tube

diameter1.5meVT�1 (d ¼ 5 nm)

Thermal conductivity 6600W/mK More thermally conductive than most crystals

Young modulus 1TPa Ten times stronger than steel


Other CNT properties, which refer mainly to SWCNTs, are displayed in Table 2. TheMWCNT, which has a Russian-doll structure, each doll being a rolled SWNCT, ischaracterized by a better mechanical stability and a much lower contact resistance than theSWCNT. On the other hand, the electrical behavior of the MWCNT is comparable to thatof the SWCNT because the current is restricted to the outer shell. The differences betweenthe properties of SWCNT and MWCNT can be exploited when investigating fundamentalphenomena such as the Aharonov–Bohm effect or quantum interference. For instance, theAharonov–Bohm effect is observable in MWCNTs in applied magnetic fields with a five-time lower intensity than in the case of SWCNTs. In addition, the MWCNT are moreeasily distinguished with a SEM, the only method that can discern SWCNTs being TEM.

2.2. The physics of graphene

In the first chapter we have discussed briefly some physical properties of graphene. Inthis Section we describe them in more details. We have to distinguish between graphene,considered as an infinite (in practice, very wide) sheet in the (x, y) plane, and the graphenenanoribbon (GNR), which is a strip of graphene with a width of few nanometers.The low-energy dispersion relation of graphene in the first Brillouin zone is given by

Eq. (2) and displayed in Fig. 4. Eq. (2) can be rewritten as

EðkÞ ¼ svF jkj (64)

where s ¼+1 corresponds to the conduction band and s ¼ �1 to the valence band, and

jkj ¼ ½ðkxÞ2þ ðkyÞ

2�1=2 (65)

is the wavenumber of charge carriers in the (x, y) plane. The point for which |k| ¼ 0, andfor which E(|k| ¼ 0) eV, is called Dirac point; the dispersion relation (64) is valid only

ARTICLE IN PRESS

Vg

ρEF

EF

EF

Fig. 21. Fermi level at various gate voltage positions.


around this point. The linear dispersion relation in graphene was experimentally evidencedusing high-resolution angle-resolved photoemission spectroscopy [49]. Infrared measure-ments also confirmed the form of this dispersion relation [50]. The Fermi energy ingraphene is tunable via the gate voltage. For instance, in the resistivity curve in Fig. 4, theFermi energy is placed in the valence band, below its zero value, when Vgo0, and is insidethe conduction band for Vg40 (see Fig. 21)

The behavior of charge carriers in graphene is governed by the Dirac Hamitonian

H ¼ vF

0 kx � iky

kx þ iky 0

!¼ vFrk (66)

where r is the vector composed of 2D Pauli matrices. Graphene is the only known solid-state material in which the carriers behave as particles in quantum electrodynamics (QED).Although some properties of carriers in graphene are analogous to similar phenomenaencountered in QED, such as the Klein effect or paradox, there are others specific forgraphene, e.g. the finite resistivity or conductivity at zero gate voltage. Thus, the physics ofgraphene is unique and its analogies to QED must not be speculated and exaggerated incatchy phrases as ‘‘graphene is a tabletop CERN accelerator’’, etc.

Using the analogy with QED, one can define the chirality, which is a projection of thePauli matrix r on |k|. It is positive for electrons and negative for holes, and indicates thatelectrons and holes are correlated since they stem from the same sublattice of carbon atoms[1].

Using the Hamiltonian in (66), in the neighborhood of the K point in Fig. 9 the Diracequation in an arbitrary potential U is [51]

fvF ½r; p� þmv2Fszgc ¼ ðE �UÞw (67)

where p ¼ (px, py) is the momentum operator, while the term pmvF2 is due to interactions

with the substrate. The solution of (67) for U ¼ U(x) is a 2D spinor

w ¼ ½cA;cB�T . (68)

If cA ¼ fA exp(ikyy) and cB ¼ ifB exp(ikyy), we obtain a single differential equation forthe unknown spinor components:

d2f=dx2 þ ðO2 � b2Þf� u0=O�ð�df=dx� bfÞ ¼ 0, (69)


where the positive (negative) sign corresponds to f ¼ fA (f ¼ fB). In (69) x ¼ x/L,O7 ¼ e�u7D, e ¼ EL/_vF, u ¼ UL/_vF, D ¼ mvFL/_, O ¼ (O+O�)

1/2, b ¼ kyL, andu0 ¼ du/dx. Depending on the shape of the potential U, we have different solutions whichcan be numerically solved.Considering that each k point is two-spin-degenerate, i.e. that gs ¼ 2, and that there are

two valleys in the first Brillouin zone, at K and K’, so that valley degeneracy is gv ¼ 2, theDOS of graphene near the K point [52] is given by

rgrðEÞ ¼ gsgvjEj=2pðvF Þ2. (70)

The intrinsic carrier density follows then as

n ¼

Z 10

dErgrðEÞf ðEÞ (71)

where

f ðEÞ ¼ f1þ exp½ðE � EF Þ=kBT �g�1 (72)

By introducing the dimensionless variable u ¼ E/kBT and Z ¼ EF/kBT, the electrondensity is finally obtained as

n ¼ ð2=pÞðkBT=vF Þ2F1ðZÞ (73)

where Fj(Z ¼ 1/G(j+1)R0Nduuj/[1+exp(u�Z)] is the Fermi integral and G the gamma

function.At thermal equilibrium the Fermi level is located at the Dirac point and corresponds to

EF ¼ 0 eV. The intrinsic carrier concentration in graphene is then

n ¼ p ¼ ni ¼ ðp=6ÞðkBT=vF Þ2 (74)

The above relation tells us that at room temperature the intrinsic carrier concentration is9� 1010 cm�2. In Table 3 we have summarized some important properties of graphene.In the case of GNRs of width W the physical properties change considerably. If the x

axis is taken as the longitudinal axis of the GNR, the wave vector is quantized in thetransverse direction y due to the boundary conditions imposed at the walls, and we have

ky ¼ np=W (75)

Table 3

Graphene properties.

Parameter Value and units Observations

Mobility 40,000 cm2V�1 s�1 At room temperature (intrinsic

mobility 200,000 cm2V�1 s�1)

Mean free path (ballistic transport) 4400nm At room temperature

Fermi velocity c/300 ¼ 1,000,000m/s At room temperature

Electron effective mass 0.06m0 At room temperature

Hole effective mass 0.03m0 At room temperature

Thermal conductivity 5000W/mK Better thermal conductivity than in

most crystals

Young modulus 1.5TPa Ten times greater than in steel


where n ¼71, 72,y The dispersion relation becomes [52]

Eðn; kxÞ ¼ svF ½k2x þ ðnp=W Þ2�1=2 (76)

meaning that both conduction and valence bands are split into a series of 1D subbands,and that a bandgap opens with a width

Eg ¼ Es¼þ1ð1; 0Þ � Es¼�1ð1; 0Þ ¼ 2pvF=W (77)

which depends only on the width of the GNR. It is interesting to compare the aboverelation with the bandgap of semiconducting CNTs, which is inversely proportional to thenanotube diameter.

The DOS of the n-th subband is given by

rGNRðn;EÞ ¼ ð4=pvF ÞðE=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

n

qÞHðE � EnÞ (78)

where H(E�En) is the Heaviside step function, and

En ¼ npvF=W ¼ nEg=2. (79)

The total DOS for the GNR is then

rGNRðEÞ ¼X

n

rGNRðn;EÞ (80)

The DOS is identical for the conduction and valence bands, and displays the same vanHove singularities at energies En.

The electron density is finally given by

n ¼ 4kBT=pvF

XFn40

Sðxn; ZÞ (81)

where Z ¼ EF/kBT, xn ¼ En/kBT and

Sðx; ZÞ ¼Z 1

x

uðu2 � x2Þ�1=2du=½1þ expðu� ZÞ�. (82)

The intrinsic carrier concentration is recovered for Z ¼ EF ¼ 0, with the Fermi levelplaced in the middle of the gap. It is given by

ni ¼ ð4kBT=pvF ÞX

n

Sðxn; 0Þ (83)

When the GNR is narrow and when EgbkBT and S(x, 0)ffixK1(x), with K1(x) the Besselfunction of the first order, we get

ni ffi ð4=W ÞðpkBT=EgÞ1=2 expð�Eg=2kBTÞ (84)

by considering that K1(x)ffi(p/2x)1/2exp(�x). The intrinsic carrier density is quite differentfrom that of graphene if Wo100 nm.

The carrier density in graphene can be changed by an applied gate voltage Vg, the samebehavior being observed in a GNR. In this case the carrier density is given by

nGNR ¼ ð4=pvF ÞX

n

ðE2F � E2

nÞ1=2HðEF � EnÞ, (85)


where eVg ¼ EF. Moreover, as in common semiconductors,

nffi ni expðZÞ; pffi ni expð�ZÞ, (86)

We can further calculate the total charge defined as Q ¼ e(p�n), and obtain thequantum capacitance Cq ¼ qQ/qVg as

Cq ¼ ½2e2kBT=pðvF Þ2� ln½2ð1þ coshðeV g=kBTÞÞ� (87)

which, in the case of eVgbkBT, becomes

Cq ffi e2rGNRðeVgÞ (88)

Thus, from measurements of the quantum capacitance we can find the bandgap as thedistance between two van Hove singularities.Chiral tunneling is a striking and weird feature of the carrier transport in graphene,

which originates from (67) and the gapless character of graphene [53]. More precisely,when carriers have to penetrate a potential barrier of width D with the potential energydescribed by

V ¼V0 0oxoD

0 otherwise

�(89)

their transport is described by (67) with m ¼ 0, i.e. by the Hamiltonian

H ¼ vF

0 kx � iky

kx þ iky 0

!þ V ðxÞ ¼ vFrkþ V ðxÞ (90)

The Dirac spinors C1 and C2 satisfying (90) for an electron wave incident from region 1on a barrier of height V0 and width D and propagating at an angle j1 with respect to the x

axis are given by [54]

C1ðx; yÞ ¼

½expðik1xÞ þ r expð�ik1xÞ� expðikyyÞ; x � 0

½a expðik2xÞ þ b expð�ik2xÞ� expðikyyÞ; 0oxoD

t expðik3xÞ expðikyyÞ; x � D

8><>: , (91)

C2ðx; yÞ ¼

s1½expðik1xþ ij1Þ � r expð�ik1x� ij1Þ� expðikyyÞ; x � 0

s2½a expðik2xþ ij2Þ � b expð�ik2x� ij2Þ� expðikyyÞ; 0oxoD

s3t expðik3xþ ij3Þ expðikyyÞ; x � D

8><>: (92)

In these expressions, when a bias V is applied on the structure, ky ¼ kF sinj1 with kF theFermi wavenumber, the x wavevector components can be approximated by

k1 ¼ kF cos j1, (93a)

k2 ¼ ½ðE � V 0 þ eV=2Þ2=2v2F � k2y�1=2, (93b)

k3 ¼ ½ðE þ eV Þ2=2v2F � k2y�1=2, (93c)

and j2,3 ¼ tan�1(ky/k2,3), s1 ¼ sgnE, s2 ¼ sgn(E�V0+eV/2), and s3 ¼ sgn(E+eV), with E

the electron energy.

ARTICLE IN PRESS

E

eV

V0 -eV/2

x = 0 x = Dx

Fig. 22. Barrier in graphene.


The transmission coefficient through the barrier,

T ¼ s3 cosðj3Þjtj2=s1 cosðj1Þ, (94)

is determined by imposing the requirement of wavefunction continuity at the x ¼ 0 andx ¼ D interfaces.

In the regions where the electron energy E is higher than the potential energy, as is thecase in the first and third region in Fig. 22, charge transport is performed by electrons,whereas in the barrier region, when the electron energy is lower than the potential energy,holes assume the charge transport role.

We have to emphasise that in graphene the term ‘‘barrier’’ does not indicate a region ofevanescent propagation as in common semiconductors, but a region in which chargetransport is undertook by holes instead of electrons. This is the origin of many mistakesfound in many papers that deal with graphene tunneling. Therefore, although it ismathematically possible that the wavevector k2 becomes imaginary for certain combina-tions of the parameters V0, V, vF and j1, this situation does not indicate evanescentpropagation, which is forbidden, but a total reflection of the electron wavefunction at thebarrier boundary. A graphene barrier, irrespective of its thickness, acts as a classicalbarrier for a quantum wavefunction, in the sense that the wavefunction cannot penetrate(not even exponentially) inside the barrier.

If we consider that the bias applied to the barrier is zero, there is an analytic formula forthe reflection coefficient [53]:

r ¼ 2i expðij1Þ sinðk2DÞ½sinj1 � s1s2 sin j2�=fs1s2½expð�ik2DÞ cosðj1 þ j2Þþ

þ expðik2DÞ cosðj1 � j2Þ� � 2i sinðkxDÞg (95)

If |V0|b|E| and in the absence of an applied voltage V, the transmissionT ¼ |t|2 ¼ 1�|r|2 is given by [53]

T ffi cos2 j1=½1� cos2ðk2DÞsin2 j2� (96)

We can observe that there is resonance, i.e. T ¼ 1, when

k2D ¼ pN, (97)

with N an integer, and that, in particular, the barrier is transparent at normal incidence,for j1 ¼ 0. The transmission can be modulated by changing the angle of incidence ofelectrons on the barrier.


The above result regarding T ¼ 1 at normal incidence, is called Klein paradox, or Kleineffect. This effect has many consequences in quantum electronics, most of themundesirable. In quantum electronics, the majority of the devices are composed of asuccesion of various semiconductor layers that play the role of quantum barriers and wells,such an architecture being found today in almost all semiconductor lasers, cascade lasers,modulators, photodetectors, or electronic transistors and resonant tunneling devices.Due to the Klein effect, the band engineering concept applied to semiconductors cannot beapplied to graphene devices. So, the Klein effect is not desired for quantum electronicdevices and can be alleviated by changing the angle of incidence as indicated in (96), bydoping the graphene, or by using magnetic barriers. Many other physical properties, whichare enhanced in graphene or have specific features, such as the Landau levels or the Halleffect, are reviewed in [55].The Klein effect still persists if, in place of a single sheet of graphene, we have two

overlapping layers of graphene; this structure is called bilayer graphene. The bandstructure of the bilayer graphene differs from that of the single-layer graphene. Inprinciple, the dispersion relation of bilayer graphene is given by [55]

EbðkÞ ¼ V2 þ v2F k2þ t2?=2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V2v2F k2

þ t2v2F k2þ t4?=4

q(98)

where V is an applied dc voltage, and t? ¼ 0.4 eV is the nearest-neighbor hopping energybetween two graphene layers. The applied voltage breaks the inversion symmetry of thetwo graphene layers and opens a gap between the two quasi-parabolic conduction andvalence bands at

k2ffi 2V2=v2F . (99)

Thus, the bandgap of bilayer graphene depends on the voltage applied on it, and thereforebilayer graphene can be tuned reversibly from an insulating state to a semiconductingstate, as proven in experiments [56].

3. Carbon nanotube and graphene devices for signal processing up to THz frequencies

3.1. Quantum dots, single-electron transistors in carbon nanotubes and graphene

The relative simplicity of the quantum CNT-based structures contrasts with thesophisticated technological procedures that produce quantum dots in III–V semiconductorheterostructures. For example, a p-type quantum dot formed naturally at the end of ann-type CNT, and hence a p–n junction appears near the contact, which isolates a smallquantum dot between the contacts and the junction. The parameters that characterize aquantum dot, if single-electron charging phenomena are involved, are the charging energy,

Ech ¼ e2=C, (100)

where C is the capacitance of the quantum dot, and the separation between the discreteenergy levels in the dot, given by

DE ¼ hvF=2L. (101)

At low temperatures, a semiconducting CNT placed between two metallic contactsand near a gate electrode, which changes its potential, behaves similar to a quantumdot coupled to two electrodes via tunneling barriers. This analogy is illustrated in

ARTICLE IN PRESS

Gate 1

Gate

S D

CNT

SDot

SiO2/Si substrate

SiO2 nanolayer

Si

SiO2

Gate 2

Gate 3

CNT

I

D

Fig. 23. Various quantum dots in CNTs: (a) a single quantum dot, (b) its equivalent circuit, (c) double quantum

dot, and (d) double quantum dots with local gates.


Fig. 23(a) and (b) [57,58]. Typical parameter values for such a CNT quantum dot areEch ¼ 8.6meV, C ¼ 9.4 aF, and DE ¼ 2.3meV.

Coupling between quantum dots can be realized by introducing additional contacts tothe CNT. For instance, a configuration comprising a CNT with two gate electrodes and sixmetallic contacts is analogous to two quantum dots that are coupled capacitively.Alternatively, a SiO2 nanolayer placed under the CNT and between the source and thedrain contacts, and two gates, as shown in Fig. 23(c), form an equivalent circuit to twocapacitively coupled quantum dots [58].

A three-gate configuration, analogous to two coupled quantum dots, is illustrated in Fig.23(d). It is specially intended for implementing logic operations and quantum computationat the nanoscale [59]. The central gate controls locally the electric field in the nanotube, andhence allows the tuning of the coupling regime between the two dots from weak to strong.

Another way to fabricate quantum dots is to expose a SWCNT, contacted at both endsand laying on a doped Si substrate acting as a gate, to a focused ion beam (FIB) in twoplaces. In this way, two tunnel barriers form at the irradiated damaged regions [60].If these barriers are separated at 50 nm distance, the room-temperature charging energy ofthe quantum dot is about 255 eV, which is ten times larger than the room-temperaturethermal energy. Coulomb oscillations are visible in this case at room temperature becausethe entire structure behaves like a single-electron transistor (SET).

The SET is a transistor formed from two tunneling barriers weakly coupled to an island,which can be metallic or a semiconducting quantum dot. The two, source and drain,barriers are equivalent to a capacitance, labeled with subscripts S and D, respectively, in

ARTICLE IN PRESS

+

-V Island

Tunneling junction

VD

VS

+

+ nS

≈≈C

Rt

nD

Junction

Fig. 24. A quantum dot coupled by two barriers.


parallel with tunneling resistances. The model of an island weakly coupled to two barriers,as shown in Fig. 24, predicts that the energy of the equivalent circuit is

EðnS; nDÞ ¼ ðCSCDV 2 þQ2Þ=2Ctot þ eV ðCSnD þ CDnSÞ=Ctot (102)

where Ctot ¼ CS+CD and Q ¼ QD�QS ¼ �(nD�nS)e ¼ �ne. In (102), nD and nS are,respectively, the number of electrons that tunnel through the drain and source junctions.The shift in the island energy when the electrons tunnel through the drain junction is

DED ¼ EðnS; nDÞ � EðnS; nD � 1Þ ¼ ðe=CtotÞ½�ðe=2Þ � ðen� VCSÞ�, (103)

whereas the corresponding shift when the carriers tunnel through the source junction isgiven by

DES ¼ EðnS; nDÞ � EðnS � 1; nDÞ ¼ ðe=CtotÞ½�ðe=2Þ � ðenþ VCDÞ�. (104)

Then, for a neutral island, for which n ¼ 0, the energy change when a single electronenters or leaves it,

DES;D ¼ �e2=2Ctot � eVCD;S=Ctot40, (105)

is zero only when a threshold bias voltage is applied, which allows electron tunnelingbetween the source, island and drain. This threshold bias voltage is found to be

V th ¼ jV j ¼ e=Ctot, (106)

if CS ¼ CD ¼ C ¼ Ctot/2. The tunneling is suppressed below Vth, and hence the measuredcurrent is I ¼ 0. This regime of electron transport is called Coulomb blockade and isrecognized by the low conductance value around the origin of the I–V curve. In theCoulomb blockade regime, for an applied bias below the threshold voltage, a Coulombenergy gap of e2/Ctot opens in the energy band diagram around the Fermi level, whichblocks the tunneling process between contacts.For larger biases, when V4e/2C, an electron has already tunneled into the island, which

has now n ¼ 1. As a result, the Fermi level increases by e2/Ctot and a new energy gap formsaround it, which prohibits the tunneling of an extra electron from the island to the drainuntil the applied voltage V overcomes the new threshold 3e/2C. Between these twothreshold values no electron flows through the structure unless the electron situated in theisland tunnels into the drain. In this case the island returns to the state with n ¼ 0 and itsFermi level decreases so that another electron can tunnel through the structure. The cycleof tunneling processes is ready to be repeated.

ARTICLE IN PRESS

n=-1 n=0 n=-1

V/(e/Ctot)

Q=CGVG’e/2 3e/2-e/2 -3e/2

1

-1

3

-3

n=0,-1 n=1,-2 n=-1,0 n=-2,-1

n=0,-1 n=1,-2

n=-1,0 n=-2,-1

I DS

(a.u

.)

Q/e=CGVG’/e

0.5 -0.5 -1.5 1.5

Fig. 25. (a) The stability diagram of the SET and (b) the dependence of the drain–source current on the gate

voltage.


To obtain a SET, the single-electron device in Fig. 24, which consists of a weaklycoupled island, must contain an additional gate contact characterized by the capacitanceCG. The total capacitance of the SET is now Ctot ¼ CS+CD+CG and the energy changefor a tunnel event across the source junction becomes

DES ¼ ðe=CtotÞ=f�e=2� ½en�Qp þ ðCG þ CDÞV � CGVG�g, (107)

while the change in energy when an electron tunnels the drain junction is

DED ¼ ðe=CtotÞ=f�e=2� ½en�Qp � CSV � CGVG�g. (108)

Here Q ¼ �ne+Qp, with Qp a parasitic charge, and VG the gate voltage. The gatevoltage can shift the Coulomb blockade regions if na0, the tunneling conditions into andout from the island being, respectively,

�e=2� ½enþ ðCG þ CDÞV � CGV 0G�40, (109)

�e=2� ½en� CSV � CGV 0G�40, (110)

with V0G ¼ VG+Qp/CG.Eqs. (109) and (110) describe a family of intersecting straight lines in the plane (V, VG),

as illustrated in Fig. 25(a). These lines form the so-called stability plot, in which theCoulomb blockade regions are represented as shaded areas. These stable regions are calledCoulomb diamonds and are characterized by a fixed number of electrons. The shape of the


Coulomb diamonds is determined only by the capacitances of the gate and junctions. Incontrast, the number of electrons varies with integer values as the boundary of theCoulomb diamond is crossed. The sequential tunneling through the island becomes thusallowed at a given source–drain voltage. The extent of the blockade region is found byimposing the condition CGV0G ¼ pe, with p an integer, but tunneling occurs if this productis an half-integer multiple of the electron charge. By changing VG at a fixed drain–sourcevoltage, the drain current develops a multipeaked structure, as shown in Fig. 25(b), whichmarks the onset of the Coulomb blockade regime and the occurrence of sequentialtunneling. Current flow takes place only when the electron number in the gate is a halfinteger and the gate voltage difference between consecutive peaks is DVG ¼ e/CG. Thisbehavior is a manifestation of electric charge quantization.Coulomb diamonds are visible at room temperature or at low temperatures in a series of

quantum dots cut into graphene flakes by the reactive ion etching method. These quantumdots act as tunable SETs, the graphene island being electrostatically tuned by one [61] orseveral gates [62]. Theoretical studies demonstrated that quantum dots can be realized alsoin bilayer graphene [63] or Z-shaped GNR [64]. CNT-based SETs are fabricated simply bypositioning SWCNTs on a raised Al/Al2O3 gate and by contacting them with Pdelectrodes. The tunnel barriers appear in this case at the SWCNT bends at the gate edges.Coulomb oscillations up to 125K are detected at charging energies of 12–15mV [65].The quantum dots or the SETs based on CNTs, and possible on graphene, are promisingdetectors for high-frequency signals up to few THz [66].A cryogenic CNT bolometer excited with radiation at 110GHz was fabricated by using

dielectrophoresis to deposit SWCNT bundles over top and bottom metallic electrodes,which are patterned over a doped Si substrate. In this device the SWCNTs are absorber ofradiation, and the temperature response at 4.2K is 0.4mV/K [67].Detection of THz frequencies was also demonstrated in a log-periodic toothed antenna,

shown in Fig. 26, where S and G denote the signal and ground electrodes. In thisconfiguration, the CNTs are deposited in the gap between the two antennas and areordered by dielectrophoresis, a Si lens being placed over the antenna. Measurements of0.7–2.5 THz radiation at various temperatures revealed typical responsivities at 77K of2.5V/W at 1.4 THz and 0.5V/W at 2.5 THz. The responsivity decreases not only with

G G

S

CNTbundle

Fig. 26. The THz detector based on CNT. G and S have the meaning of the ground and signal electrodes.


frequency but also with temperature, the value at 1 THz being ten times lower at 300Kthan at 77K [68].

3.2. Field-effect transistors based on carbon nanotube and graphene

The typical configuration of a CNT field-effect transistor (CNT FET) is represented inFig. 27 and consists of two contacts placed on a single CNT, a bundle of CNTs, or even anarbitrary network of CNTs. The CNT is separated by a thin oxide layer from a dopedsubstrate that plays the role of gate.

The conductance of the CNT FET, defined as dIDS/dVG, changes with five orders ofmagnitudes for a variation of the gate voltage between 0 and 5V. The source–drain currentIDS is roughly constant for a negative gate voltage, which suggests that the contactresistance of the electrodes is much larger than the gate-voltage-dependent CNT resistance.Measurements of the IDS–VG characteristics can be used to extract the hole densityaccording to

p ¼ Q=eL ¼ CVG;th=eL, (111)

where VG,th is the threshold gate voltage, beyond which the holes are depleted,

C=L ¼ 2p�r�0= lnð4h=dÞ (112)

is the capacitance/unit length of the CNT, erE2.5 is the CNT relative permittivity and h

denotes the thickness of the silicon dioxide substrate.The electrical carriers are injected in the CNT via tunneling if the work function of the

contacts is larger than that of the CNT; Pt electrodes, with a work function of 5.7 eV, areexamples of such contacts. The difference in the work function between the CNT and thecontacts determine the height of the barriers, which can be modulated by applying a gatevoltage. For instance, the valence band bends upwards if a negative gate voltage is applied,until a metal-like conductance (i.e. a constant conductance value) is observed, whereas it

Source (S) Drain (D)

Gate (G)

Doped Si-gate

SWCNT

SiO2

Fig. 27. The CNT FET.


bends downwards for positive gate voltages, until the hole transport between the electrodesis suppressed. This behavior is analogous to that of a BARITT diode, formed from twoback-to-back connected Schottky diodes.The traversal time of this CNT FET is as short as 0.1 ps, which corresponds to a

frequency of 10THz. The resulting time constant of the RC circuit is 100GHz for a CNTcapacitance of 1 aF if R is of the order 1–2MO. However, the cutoff frequency can beincreased to 10THz in ballistic CNT FETs with Pd contacts, for which the resistance R isabout 10 kO at room temperature. Many other configurations of CNT transistors havebeen intensively studied, including the top-gate configurations and the Schottky-barrierCNT FETs. Nevertheless, the cutoff frequency of CNT transistors still does not exceed50GHz due to parasitic capacitances, which diminish the performances of CNT transistorsirrespective of the contact type.The top-gate FET-like structure, schematically displayed in Fig. 28, is the typical

configuration of graphene transistors. To implement it, the graphene is first deposited on aSiO2 substrate following mechanical exfoliation, Ti/Au drain and source electrodes arethen patterned over the graphene flake, and, finally, the gate electrode is defined with theelectron-beam lithography technique. The gate is isolated by a 20-nm-thick SiO2 layer,on which the contact, consisting of 10-nm-thick Ti and 100-nm-thick Au layers isdeposited.The graphene transistor is characterized by quite low values of the mobility of charge

carriers, which have an average of 600 cm2/Vs. These low mobilities are due to top-gatedeposition; prior to top-gate deposition their values were about eight times higher, but stilllow for graphene [69]. The explanation resides, probably, in the long distance, of 7mm,between the drain and the source, which prevents ballistic transport at room temperature.The width of this FET configuration was 260 nm. Although well below expectations, themobilities in the graphene FET are still higher than in MOSFETs fabricated on ultrathinbody Si on insulator. Better average carrier mobility, of 5000 cm2/Vs, were found in a FETimplemented on epitaxial graphene grown on SiC, which also showed dc amplification [70].GNRc can be also used to fabricate FET transistors. Metallic GNRs are in fact graphene

stripes with a zigzag edge. These zigzag GNRs, if rolled, generate armchair CNTs. Quite theopposite, GNRs with armchair edges, if rolled, form zigzag CNTs. An armchair GNR can bemetallic or semiconducting depending on the width W of the graphene strip: it is metallic if the

Source

Drain

Gate

Graphene

SiO2

Fig. 28. Graphene FET on a SiO2 substrate.


number of rows of carbon atoms N (number of dimmer lines) satisfies the relation N ¼ 3p+2,where p is an integer, and semiconducting otherwise [71].

The semiconducting armchair GNRs exist in three configurations, depending on thevalue of N, which is determined by the width W. The carbon atoms in the hexagonal latticeare separated by a distance equal to the carbon bond length aC–C. We then divide theGNRs into: a armchairs if N ¼ 8, 11, 14,y; b armchairs if N ¼ 9, 12, 15,y; and garmchairs if N ¼ 10, 13, 16,y. Their respective bandgaps depend on W as [72]

Eg ¼

0:04 eV=W ðnmÞ; aGNR

0:86 eV=W ðnmÞ; bGNR

1:04 eV=W ðnmÞ; gGNR

8><>: (113)

This relation is analogous to that for semiconducting CNTs: the bandgap Eg isproportional to 1/W for GNRs, and proportional to 1/d for CNTs.

Similar to CNTs, the wavenumber of a GNR located along the y axis takes discretevalues in the transverse direction, x, according to

kx ¼ pp=W , (114)

where p is an integer. The corresponding minimum energy value of the conduction band[73] is obtained from

Ep ¼ ð2 eV� nm=W Þjpþ qj, (115)

where q ¼ 13for semiconducting GNRs and q ¼ 0 for metallic GNRs. The first factor on

the right side of (115) corresponds to the gap between subbands in metallic GNRs.The GNR is used in a series of applications, such as interconnections and FET

transistors, where it forms the channel [74]. GNRs have narrower bandgaps thannanotubes, and their mobilities are higher compared to CNTs with the same unit cell, butlower than in CNTs with equal bandgap or carrier density. High-yield GNRs can beobtained by unzipping MWCNTs via plasma etching of nanotubes partially embedded inPMMA [75], or via oxidative processes [76].

Although graphene- or GNR-based transistors could in principle attain cutofffrequencies of THz, the actual stage of technology is not mature enough for achievingsuch performances, mainly because graphene contains defects, cracks, and impuritieswhich lower this frequency up to few GHz. However, the work is in progress and theperformances of graphene transistors are expected to improve when graphene will begrown in a controllable way and without lots of defects.

4. Nanophotonic devices based on graphene and carbon nanotubes

The emission and absorption of light in semiconducting SWCNTs is dictated bydepolarization effects and dipole selection rules [77]. Depolarization refers to the fact thatthe absorbed or emitted light is mostly polarized along the tube axis, light polarizationperpendicular to this direction being insignificant. The dipole selection rules insemiconducting CNTs permit only optical transitions between valence and conductionbands inside the same subband when the light is polarized along the CNT axis. Thedifferent absorption bands in the SWCNT correspond to transitions between pairs of vanHove singularities.

ARTICLE IN PRESS

v2

c2

v1

c1

0

1

-1

Ene

rgy

(a.u

.)

DOS (a.u.)

2 10

2

-2

0

E11-fluorescence E22-absorption

Valence band

Conduction band

4 6 8

Fig. 29. The optical transitions in CNTs.


Around the energy minima and maxima (7E0) of the SWCNT dispersion relation, thevan Hove singularities depend on the energy as (E2

�E02)�1/2. The highest valence band

singularity and the lowest conduction band singularity are separated by [78]

Es11ðdÞ ¼ 2aC�Cg0=d (116a)

Em11ðdÞ ¼ 6aC�Cg0=d (116b)

where g040 is the carbon–carbon energy and the superscript refers to the semiconductingor the metallic character of the nanotube.If we denote by p ¼ 1, 2, 3,y the order of the p valence band and by p0 ¼ 1, 2, 3,ythe

order of the p* conduction band (these bands are positioned symmetrically around theFermi level), the optical transitions for parallel light polarization take place if

dp ¼ p� p0 ¼ 0, (117)

while for perpendicular light polarization they should satisfy the selection rule

dp ¼ p� p0 ¼ �1. (118)

Optical transitions that satisfy this last selection rule are not actually observed becauseof depolarization effects. In SWCNT films made from a mixture of semiconducting andmetallic nanotubes, optical absorption and emission processes occur at the energies

Es11ðdÞ; 2Es

11ðdÞ; Em11ðdÞ; 4Es

11ðdÞ . . .Es22 . . . (119)

Fig. 29 illustrates the first and the second optical van Hove transitions. The first vanHove peaks in the valence and conduction bands, v1 and c1, respectively, determine theemission energy E11 that corresponds to the c1-v1 transition, while the energies of thesecond van Hove peaks, v2 and c2, determine the optical excitation energy E22 associatedto the v2-c2 transition. The energies E11 and E22 are obtained experimentally from thespectrum of fluorescence and absorption, respectively. The optical transition energies arecalculated using relations analogous to (116a) and (116b) and depend on the CNTdiameter, i.e. on the chirality parameters n and m. The link between the interbandtransitions and the (n,m) parameters suggests that the optical absorption spectra ofSWCNTs can be characterized geometrically [79,80]. First principles are commonly used to

ARTICLE IN PRESS

t1

t2>t1

t3>t2

Abs

orpt

ion

(a.u

.)

1

0

2

S1 S2

Photon energy (eV)

1

Fig. 30. The absorption of the CNT films at different thicknesses.


calculate the absorption spectrum, and in general the simulation results agree rather wellwith the experiments [81].

In a more detailed approach, in which the exciton effects in CNTs are taken into accountin a model that includes the propagation of electron–hole pairs on a cylindrical surface, theabsorption spectrum a(o) can be expressed as [82]

aðoÞ /X

n

jcnð0; 0Þj2=fðEn þ EgÞ½ðEn þ Eg � oÞ

2þ ðGÞ2�g, (120)

where cn(x, y) is the wavefunction of the n-th exciton state with energy En, _Gffi0.05 eV is aphenomenological line width, _o is the photon energy of incident light, andEg ¼ Eexc�Ebind is the CNT bandgap, with Eexc the excitation energy and Ebind thebinding energy of the exciton. From relation (120) it follows that the absorption spectrumof the CNT can be written as a sum of Lorentzian curves that depend on the nanotubediameter through Eg.

The optical absorption spectrum of semiconducting SWCNTs with a diameter of 1–2 nmconsists of two main peaks, S1 and S2, which originate from the lowest and the secondelectronic interband transitions. These peaks are displayed in Fig. 30, and appear at theexcitation energies of 0.7 and 1.2 eV, respectively.

The two peaks S1 and S2 are observed in slightly different positions in thin films ofSWCNTs with different thicknesses t. The engineering of absorption peaks is essential forversatile optoelectronic devices. In SWCNT films it can be done by doping, by chemicalexposure, or by applying a high-pressure treatment [83]. For instance, measurements ofabsorption in SWCNT films with t between 0.2 and 2.5 mm and with an average tubediameter of 1.31 nm show that the S1 peak is blue-shifted with 5.5meV/mm [83]. The originof this shift of the absorption peak is the tensile stress supported by each SWCNT whenthe thickness of the film increases. This tensile stress varies between 0.1 and 0.5GPa when t

increases with 1 mm. A thermal treatment, such as heating the film for 1 h at 400 1C, alsoshifts the S1 absorption band from 0.7 to 0.735 eV.

Another method of engineering the absorption spectrum of SWCNT thin films is bymodifying the average diameter of nanotubes in the film. For instance, in a thin filmformed from two layers with dissimilar average diameters of SWCNTs, d1 and d2,respectively, the built-in stress in each layer is different and, as a result, the S1 peak is


shifted. A shift from 0.7 to 0.74 eV can be achieved if the absorption peaks of the twolayers are situated at 0.7 and 0.75 eV, respectively [83]. This shift of the S1 peak can be,however, cancelled by the shift of the Fermi level, because the peak near 0.7 eV is close toEF. It becomes thus possible to implement an optical switch based on a gate potentialvariation that shifts the Fermi level and turns on and off the S1 absorption.Aligned CNT arrays have also optical properties that can be exploited in metamaterial

fabrication and optoelectronic devices. To calculate the optical response of aligned CNTsit is first necessary to compute the effective electrical permittivity eeff as a function of theincident wavelength, and the ratios between the lattice constant and the nanotube externalradius R and between the internal and external radii of the carbon nanotube r. In filmsof aligned CNTs their orientation becomes essential: the films with nanotubes parallelto the surface are called a-aligned, while those with perpendicular CNTs are referred to asb-aligned. The a-aligned films are birefringent because eeff has different values for lightpolarized normal to (p-polarized) or along (s-polarized) the nanotube axis. The differentmethods of computing the effective permittivity and hence the optical absorption ofaligned arrays of CNTs are reviewed in [84].We will further consider a square array of multishell CNTs with an outer radius R and

an inner radius r, which has a lattice constant a ¼ 2qR [85], where q is a constantparameter. We assume that an isotropic and homogeneous insulating media with adielectric constant e0 encloses the CNT array. In this case, at each point inside the CNTand outside the inner core a dielectric permittivity tensor can be defined as

�ðoÞ ¼ �?ðoÞð h

Þ

h

Þ

þ bzbzÞ þ �kðoÞbrbr, (121)

where br; bz and h

Þ

are unit vectors in a cylindrical configuration, eJ(o) the dielectricfunction parallel to the crystallographic c-axis of the nanotube, and e?(o) the dielectricfunction perpendicular to the nanotube c-axis. For an incoming electromagnetic fieldperpendicular to the CNT array (i.e. with the wavevector components ky ¼ kz ¼ 0), theelectric field is parallel to the CNT at any point and hence the radiation is s-polarized. As aresult, the interfaces do not modify the electric field and eeff is a weighted sum of theconstituents. In contrast, the electric field of a p-polarized radiation is strongly perturbedby interfaces, and the effective permittivity in this case is

ð�eff � �0ÞE ¼ f ð�� 0ÞEin (122)

where f is the filling fraction, which is a measure of the quantity of nanotubes in free space,

E ¼ f Ein þ ð1� f ÞEout (123)

is the average electric field of the composite medium, and Ein and Eout are , respectively, theaverage electric fields inside and outside the CNTs.For a single hollow CNT cylinder f-0,

ð�� 0ÞEin ¼ �0aE, (124)

and the effective dielectric constant is given by

�eff ¼ �0ð1þ f aÞ, (125)

where a is the in-plane polarizability per unit volume.

ARTICLE IN PRESS

Ni dot

CNT

1μm

2μm

Si substrate

Light

Fig. 31. A single cell of the CNT PBG.


In the case of a sparse distribution of CNTs f51, the multipolar contribution to theoptical response is small, EffiEout, and we obtain

�eff ¼ �0½1þ af =ð1� af =2Þ�. (126)

The corresponding energy loss function, defined as Im[�eeff�1(o)], equals eeff

�1¼ e0�1[1�af/

(1/af/2)], and the polarizability can be written as

a ¼2

ð1� r2Þð�kD� �0Þð�kDþ �0Þð1� r2DÞ

ð�kDþ �0Þ2� ð�kD� �0Þ

2r2D, (127)

where D ¼ (e?/eJ)1/2 and r ¼ r/R. Expression (127) is a generalized Maxwell-Garnett

effective dielectric function for the aligned array of CNTs.The case of the nonsparse regime is treated by taking into consideration the

electromagnetic interaction between CNTs by means of the Bloch wave method.The above formulae can be used to calculate the effective dielectric permittivity of alignedCNT arrays using e0 ¼ 1, a typical value for the outer radius of R ¼ 5 nm, r valuesbetween 1 and 8, and tabulated values of the real and imaginary parts of eJ(o) and e?(o)for graphite. The simulations are in quite good agreement with experiments in the 0.5–6 eVrange [86].

Aligned arrays of nanotubes can be viewed as photonic bandgap (PBG) structures.Large-area CNT-based PBGs consisting of well-aligned metallic nanotubes on metallic(Ni) dots can be grown by a self-assembly technique [87]. More precisely, bundles ofmetallic CNTs can be grown via PECVD over Ni dots that form a honeycomb structure.Fig. 31 displays a single honeycomb cell of a CNT array formed in this way; this isthe basic cell of the PBG. The PBG consists of many such cells that reflect the incidentlight in a large bandwidth centered on the wavelength of 0.5 mm. Both the bandwidth andthe central frequency can be tuned by appropriately selecting the diameter and height ofthe CNTs.

The nonlinear optical properties of semiconducting CNTs are strong and unusual. Thethird order susceptibility w3(o), for example, has been estimated theoretically in [88] andproven experimentally in [89]. This parameter is generally complex, its real and imaginaryparts characterizing the strength of the quadratic electro-optic effect and the electro-absorption, respectively. The electro-optic coefficient wQEO

3 (o) is related to the change ofthe refractive index in the presence of an applied electric field E0 as

nðoÞ ¼ n0 þ DnCNTðo;E0Þ ¼ n0 þ ð2p=n0Þwð3ÞQEOðoÞE

20, (128)


whereas the electro-absorption is described by wEA3 (o). The magnitude of wQEO

3 (o)increases with the frequency and takes negative values in the long-wavelength region,whereas |wQEO

3 (o)| increases with the diameter of the SWCNT. The sign of susceptibilitywQEO3 (o) changes from negative to positive and back again near the fundamental

absorption edge, and it reaches its maximum value, of 5� 10�4 e.s.u., at a photon energyjust below the bandgap. The electro-absorption coefficient, wQEO

3 (o), is generally positiveand increases with both frequency and SWCNT diameter. However, it decreases rapidlyand changes sign for wavelengths just below the bandgap, becoming negative and showinga resonant peak.The use of SWCNTs as optical switching devices is based on the change in sign of their

susceptibility wQEO3 (o), and the corresponding fast variation of this coefficient from

5� 10�5 to �10�3 e.s.u., for a small modification of the laser wavelength. For instance,ultrafast optical switching, in less than 1 ps, was achieved at 1.55 mm in a 20-mm-thickpolyimide–SWCNT composite film [89]. The switching and nonlinear properties of the thinfilm were tested with a 150 fs laser with a wavelength of 1.55 mm, using the pump–probemethod. Measurements of the exciton decay time, which is in fact the relaxation timecorresponding to direct electron transitions from the conduction to the valence band,found a value of this parameter of 0.8 ps for a SWCNT bandgap of 0.57 eV. Goodagreement was found between the experimentally-determined nonlinear optical suscepti-bility,w3(o), of 10–9–10–10 e.s.u., and the theoretical predictions [88].The absorption peak of CNT thin films can not only be tuned by changing the tube

diameter or the film thickness, but also through a variation of the incident optical power[90,91]. The application of CNT thin films as saturable absorbers in the passive mode-locking technique of lasers is based on this property. Passive mode-locked lasers are amongthe best sources of ultrafast optical pulse, because they produce transform-limited pulseswith duration of ps or even sub-ps. In a mode-locked laser several longitudinal modesoscillate at the same time. If these modes are locked, in the sense that their phases becomesynchronized so that there is a constant phase difference between them, they interfere andproduce uniformly spaced pulses.The saturable absorber is essential for mode-locking, process that not only forms optical

pulses but also cancels the spurious modes of CW lasers. The saturable absorber is anonlinear optical device, the transparency of which changes with the incident opticalpower in a certain bandwidth of light wavelengths. More precisely, the saturable absorberattenuates the incident light for small optical powers, below a certain threshold, but theattenuation drops sharply for higher incident powers. Such a behavior is encountered inultrathin films, less than 1 mm thick, containing SWCNTs with diameters of 1.2 and1.35 nm. The absorption peaks of these films are situated at 1.55 and 1.68 mm, the firstabsorption band, which is of interest in optical communication systems, decreasing with10% for an increase of the light intensity from 0.1 to 104 kW/cm2. The optical bandwidthof CNT-based saturable absorbers can be changed by modifying the diameter or thethickness of the nanotubes.The saturable absorber incorporating nanotubes (SAINT) can be found in two basic

configurations, as displayed in Fig. 32. In the transmission mode (T-SAINT), representedin Fig. 32a, the CNT film is placed between two anti-reflection coatings made from quartzsubstrates. In the reflection mode (R-SAINT), illustrated in Fig. 32b, the CNT thin film ispositioned over a highly reflective mirror and is coated with the same anti-reflectionmaterial as in the T-SAINT design. Passive mode-locking can be also achieved in a

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CNT thin film Mirror

AR coating

Fig. 32. The CNT saturable absorber configurations.

CNT

Al matrixMetal electrode

IR radiation

Transparentelectrode

Fig. 33. IR detector based on a CNT array.


transmission configuration with the help of a CNT film located between two optical fibers,which are covered with ferule. The CNT-based SAINT is presently optimized andconstitutes the focus of many research efforts. For example, the ultrafast broadbandtechnology is available for SAINT [92], since the same CNT saturable absorber mirror isused for doped fiber lasers operating at 1.05, 1.56 and 1.99 mm.

Absorption engineering in SWCNT has significant applications since the absorptionpeaks are situated in the near-IR, and thus can be used for the detection of IR radiation. Inaddition, SWCNT arrays consisting of sub-arrays with different nanotube diameters canwork as tunable IR detectors. Since only the incident optical radiation with energy higherthan the CNT bandgap is absorbed and, as a result, only the conductivity of a certain sub-array increases, one can expect from these detectors high temperature operation, broadwavelength response, and low noise level because of the weaker interaction betweenelectrons and phonons in 1D quantum systems [93]. Therefore, large areas of highly densepacked (1010 cm�2) and aligned CNT arrays are able to span a broad spectrum that rangesfrom 1 to 10 mm. Fig. 33 illustrates such an IR detector based on a CNT array embedded inan Al matrix.

A simple MWCNT-based IR detection method was described in [94]. The nanotubeswere deposited randomly on top of two interdigital copper electrodes, which werepatterned on a Si/SiO2 substrate, as displayed in Fig. 34. Both the separation between thefingers and the width of the copper electrode were 1.5 mm, and the IR photodetectoroccupied an effective area of 60 mm� 60 mm. The current–voltage characteristics of the

ARTICLE IN PRESS

Metal

MWCNT

SiO2

IR radiation

Fig. 34. IR detector based on interdigitated electrodes.


MWCNT photodetector was insensitive to the radiation emitted by an IR lamp, but its IRsensitivity increased significantly if a DC voltage was applied between the electrodes.The photodetectors based on MWCNTs are sensitive in the mid-IR spectral region.

The sensitivity of these detectors can be determined from the modifications in the I–V

characteristic and are experimentally found by positioning an IR lamp at different heightsabove the detector. Typical response times are around 400ms and the sensitivity issignificant only for small lamp–photodetector distances; it increased 103 times when thisdistance is about 25 cm. The photocurrents can be observed at room temperature andunder low voltages, of the order of 1V [93,94].In a similar way, if CNT photodetectors are integrated with a nanosized antenna

working in the IR, the detected signal is boosted one order of magnitude. This result isexperimentally demonstrated in [95].The photocurrents measured in CNT p–n junctions present unusual characteristics that

can be explained by the unique properties of semiconducting CNTs, which, unlike othersemiconductors, have a direct gap in all directions of the Brillouin zone [96]. Thus, onlyband-to-band transitions are encountered in CNT semiconductors. Moreover, theundesired nonradiative transitions are reduced due to the typical low density of defects.Another advantage of the photodetectors based on p–n junctions in CNTs is the expectedlow sensitivity at any changes in temperature.Simulations of the performances of these photodetectors are usually done using a tight-

binding Hamiltonian. For instance, a p–n junction of a (17,0) SWCNT can be simulated bythis method assuming a doping concentration of 5� 10�4 electrons/C atom and modelingthe illuminated region as composed of parallel rings of 17 carbon atoms separated with adistance that varies between 0.07 and 0.14 nm [96]. Estimations of the photoresponse thatincorporate 128 illuminated carbon rings have demonstrated the existence of several sharppeaks in the spectral range extending from IR up to UV. The photoresponse is defined asIph/eF, where F the photon flux and Iph the photocurrent. This broadband response, whichis displayed schematically in Fig. 35, is due to the direct transitions in all SWCNT bands.The photoresponse for energies lower than the bandgap can be explained by photon-assisted tunneling. In addition, the photoresponse is dependent on the CNT lengthand oscillates at various excitation energies of the incoming photons. An integrated

ARTICLE IN PRESS

Si substrate

MWCNTfilm

White light

Polarizer

Reflected light

�

Fig. 36. CNT film optical antenna.

1 2 3 4 5

2

4

6

8

10

Phot

ores

pons

e (a

.u.)

Photon energy (eV)

IR visible UV

Fig. 35. The photoresponse of the SWCNT p–n junction.


optoelectronic device consisting of an emitter (based on the SWCNT electroluminescence)and a photodetector was recently demonstrated in suspended SWCNTs [97] (Fig. 36).

Another important application of CNTs is as an optical antenna. It was recently shownthat a CNT film composed of randomly oriented metallic MWCNTs works as an opticalantenna at optical wavelengths, which emits and receives electromagnetic radiation [98].The antenna effect occurs when the length of the MWCNT is comparable with thewavelength of the exciting electromagnetic waves, denoted with l. As a result, antennas inthe visible part of the electromagnetic spectrum can be fabricated from metallic MWCNTswith a length that varies between 200 and 1000 nm, and a diameter of 50 nm. Due to thestrong anisotropy of the optical response, the antenna effect is reduced when the electricfield of the incident radiation is perpendicularly polarized with respect to the dipole axis,and response is maximum when the length of the antenna is a multiple of l/2.

The experiments demonstrate that, as the observation angle y of the polarizer changes,the reflected light has a maximum intensity when the polarizer direction is taken as y ¼ 0,and is minimum (no reflected light is observed) when this direction corresponds to y ¼ p/2;in the latter case the response is zero. The intensity of the scattered light depends on theobservation direction as

IMWCNTðyÞ ¼ jEincj2cos2 y, (129)


and the optical field scattered from the MWCNT film is maximum if

L ¼ mðl=2Þf ðy; nÞ, (130)

where L is the MWCNT average length, f(y, n) the radiation pattern, n the number ofMWCNTs, and m an integer. In thin films in which the average distance between antennasis much smaller than l,

f ðy; nÞ ¼ ðn2 � sin2 yÞ�1=2. (131)

An optical antenna consisting of a single CNT that acts as a dipole, similar to the case ofradio frequency waves, was recently investigated at l ¼ 0.54 mm [99].CNT films are characterized also by giant optical rectification. This effect, which was

theoretically anticipated [100] and experimentally confirmed [101], can have importantapplications in THz electronics because optical rectification is a common and efficientmethod for generating and detecting THz signals [102]. The phenomenon of opticalrectification relies on the generation or detection of the very small difference in frequencybetween two interacting waves that excite a certain material. Optical rectification isproduced in centrosymmetric crystals and originates in the quadrupole and magneto-dipole contributions to optical nonlinearity, which is enhanced in materials characterizedby a strong spatial dispersion when the excitation wavelength is comparable to thedimensions of the molecules. This situation is encountered in nanocarbon films with athickness between 3 and 4 mm, which are formed from randomly arranged nanocarbonstructures placed on a Si substrate that is transparent for the THz radiation.Measurements performed on such an unbiased film, deposited on a 25mm� 25mmsubstrate, revealed that the ratio between the detected dc voltage and the laser power is0.5 mV/W (or 500mV/MW) if excited by a Nd:YAG laser emitting nanosecond pulses at1064 nm, and 0.65 mV/W (or 650mV/MW) for a wavelength of 532 nm.In quantum computing or optical communication light sources are an important issue.

Therefore, light emission from CNTs is a hot area of research. In particular, an electrically-induced emission of radiation at a wavelength of 1.5 mm was demonstrated from a CNTFET [103,104]. This CNT FET presents ambipolar carrier transport and is fabricated bypositioning a SWCNT over a doped substrate, which acts as a back gate. The source anddrain contacts are defined by electron-beam lithography and a subsequent evaporation of a30-nm-thick layer of palladium. In this CNT FET, two thin Schottky barriers form at thesource and drain contacts, through which thermal-assisted tunneling of electrons and holestakes place. In intrinsic semiconducting SWCNT the electrons and holes are injectedsimultaneously in the CNT FET if the value of the gate voltage is between that of the drainand the source, such that the electric fields induced by the gate have opposite signs at thesetwo contacts. In the ideal case the source is grounded and VDS/2 ¼ VG [104]. As a result ofthe simultaneous injection of electrons and holes, the wavelength of the emitted radiationis in the 1.5–2 mm range for a SWCNT with a length between 50 and 100 mm, and adiameter of 1.5 nm (see Fig. 37).This IR luminescence is generated as a result of the recombination of electrons and holes

injected at opposite (source and drain) contacts. The luminescence is confined in a smallspatial region of the SWCNT, which can be translated along the nanotube by changing thegate voltage. This way, the shape of the emission spot and the intensity of the IR radiationcan be controlled.

ARTICLE IN PRESS

Source Drain

SWCNTSiO2

Gate

IR light

Fig. 37. Light emission from a SWCNT ambipolar transistor.


Simultaneous Raman scattering and fluorescence experiments from SWCNTs [105] haveshown that the intensity of the CNT fluorescence does not fluctuate if the excitationintensity has moderate values, below 70 kW/cm2. This behavior contradicts that of mostmolecules or individual semiconductor quantum dots, in which the fluorescence isintermittent, and displays blinking or bleaching of the emission irrespective of theexcitation intensity. The stability of the SWCNT fluorescence facilitates the fabrication ofphoton sources with very narrow linewidths. Nevertheless, at high values of the excitationintensity the effect of nonradiative transitions becomes significant and the SWCNTfluorescence starts to fluctuate.

The involvement of the CNTs in solar cells is of paramount importance for obtainingclean energy and the reduction of CO2 emission. For example, solar cell heterojunctionswith an open circuit voltage of 0.23V and a conversion efficiency of 0.54% were obtainedby cutting MWCNT to length values of 50–200 nm via plasma fluorination and subsequentdefluorination, and by embedding these cut MWCNTs into poly(3-octylthiophene)/n-Sisolar cells [106]; cutting the MWCNTs diminishes the shorting and shunting effects inphotovoltaic films with thicknesses comparable to the pristine MWCNT length. Also,vertical CNT arrays are used to fabricate enhanced light trapping photovoltaic cells. In thiscase, the CNTs at the same time the back contact of the cell and the scaffold that supportsthe photoactive CdTe/CdS heterojunction. The photons trapped between coated CNTshave a higher probability to be absorbed by the p-type CdTe layer, and so thephotocurrents increase 63 times compared to commercially available Si-based planar solarcells [107].

In another study it was shown that colloidal systems based on CNT, more specificallymixtures of CNTs and nematic liquid crystals, enhance the orientational order of thesystem when no electric field is applied, and could help improve the performances ofoptical displays [108]. Thus, CNTs emerge as key materials in photonics, with manyapplications in a broadband spectral range. The unusual properties of vertically alignedCNT arrays are also reflected in their behavior as black-body [109]. Such sparse andimperfectly aligned SWCNT arrays have an absorptivity higher than 0.98 in a hugespectrum, which extends from UV to far-IR, i.e. between 200 nm and 200 mm.

Graphene has started to show its potentialities in photonics. There are only few papersin this area, but we could mention the gate-variable optical transitions in graphene, whichcould have important applications in photonics [110], and thin films of graphene, which areexcellent transparent electrodes for solar cells [111].


5. Conclusions

The quantum electronic applications of graphene-based materials are of paramountimportance in the development of photonics, THz electronics, and other relating areas.Although the growth of graphene and CNTs, as well as their manipulation, are at thebeginning, the important applications of these nanomaterials will direct the research effortstowards more mature, flexible and reproducible technologies. The range of applications ofCNTs and graphene is expected to extend to other domains, from nanobiology toconstruction and aeronautic industries.

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