Electron wavefunctions, energy eigenvalues, density of states and band structures of graphene systems [2, 25, 27]
1. Introduction
Recent years have witnessed considerable interest devoted to the electronic properties of graphene [13]. Graphene, a oneatomthick sheet of carbon atoms arranged in a honeycomb crystal, exhibits unique properties like high thermal conductivity, high electron mobility and optical transparency, and has the potential for use in nanoelectronic and optoelectronic devices. With the size of these devices shrinking through integration, thermal management assumes increasingly high priority, prompting the study of thermoelectric effects in graphene systems.
The thermoelectric (TE) effect refers to phenomena by which either a temperature difference creates an electric potential or an electric potential creates a temperature difference. An interesting transport property, thermoelectric power (TEP) has been a source of information to physicists for over a century [4]. TE devices are used as generators and coolers to convert thermal energy into electrical energy or
The interest in the TEP of a material system stems not only from its relation to ZT but also due to its sensitivity to the composition and structure of the system and to the external fields. The TE effect has been able to shed much light on the interaction of electrons and phonons, impurities and other defects. Further, the three transport parameters
Ever since its discovery, great interest has been evinced in the electronic properties of graphene [13]. Graphene also exhibits interesting TE effects. For instance, compared to elemental semiconductors, it has higher TEP and can be made to change sign by varying the gate bias [1618]. The unique properties, including high mechanical stiffness and strength, coupled with high electrical and thermal conductivity, make graphene an exciting prospect for a host of future applications in nanoelectronics, thermal management and energy storage devices (For reviews on graphene physics, see [2] and[3]). Technical advances have now made possible the realization of tailormade 2D graphene systems, such as singlelayer graphene (SLG), bilayer graphene (BLG), graphene nanoribbon (GNR), graphene dots, graphene superlattices and defected graphene. Most of the experimental and theoretical work has concerned the electrical and thermal conductivity of such systems. (For a review on recent progress in graphene research, see [19]). However, in the recent past, a good amount of literature has accumulated on the TE properties of graphene systems, and a coherent picture is just emerging into understanding TE effect in graphene.
The present work addresses one of the important components of TE transport in graphene, namely, TEP, also referred to, simply, as thermopower. TEP has been a powerful tool for probing carrier transport in metals and semiconductors [812]. Being sensitive to the composition and structure of a system, it is known to provide information complementary to that of resistivity (or conductivity), which alone is inadequate, say, in distinguishing different scattering mechanisms operative in a system.
In this chapter, we review the literature on TEP in graphene systems and present its understanding using the semiclassical Boltzmann transport theory. In Section 1, the electronic structures and phonon dispersion relations for SLG, BLG and GNR systems are described. In the next section, besides a survey of the experimental work, the basic theory of TEP in 2D systems is given, and its relation with other TE transport coefficients is discussed. In Section 3, the diffusion contribution to TEP of graphene systems is discussed. Section 4 deals with the phonondrag contribution to TEP. An analysis of the experimental data, in terms of the diffusion and drag components, is also presented. This is followed by a summary of the chapter.
1.1. Graphene systems
A singlelayer graphene, commonly referred to simply as graphene, is one of the recent nanomaterials. It is a monolayer of graphite with a thickness of 0.34 nm, consisting of carbon atoms in the sp^{2} hybridization state, with the three nearestneighbour carbon atoms in the honeycomb lattice forming
The 2D honeycomb structure of graphene lattice with two equivalent lattice sites, A and B (Figure 1.(a)), can be thought of as a triangular lattice with a basis of two atoms per unit cell, with 2D lattice vectors
Gapless graphene has a charge neutrality point (CNP), that is, the Dirac point, where its character changes from being electron like to being holelike. For pure graphene the Fermi surface is at the Dirac point. The system with no free carriers at
The electronic properties of graphene depend on the number of layers. Generally, the graphene community distinguishes between singlelayer, bilayer and fewlayer graphene, the latter of which refers to graphene with a layer number less than ten. Bilayer graphene (BLG) consists of two graphene monolayers weakly coupled by interlayer carbon hopping, which depends on the manner of stacking of the two layers with respect to each other; typically they are arranged in AB stacking arrangement. The bilayer structure, with the various electronic hopping energy parameters γ
In order to improve applicability, graphene needs to acquire a bandgap. This can be achieved by appropriate patterning of the graphene sheet into nanoribbons. A graphene nanoribbon (GNR) is a quasionedimensional (Q1D) system that confines the graphene electrons in a thin strip of (large) length
One of the strategies adopted to achieve higher mobility in graphene samples is to improve the substrate quality or eliminate the substrate altogether by suspending graphene over a trench. Improved growth techniques have enabled obtaining graphene as a suspended membrane, supported only by a scaffold or bridging micrometerscale gaps schematic of which is shown in Figure 2.(c). Suspended graphene (SG), shows great promise for use in nanoelectronic devices. With most of the impurities limiting electron transport sticking to the graphene sheet and not buried in the substrate, a large reduction in carrier scattering is reported [21] in currentannealed SG samples. However, unlike supported graphene, only a small gate voltage (V_{g} ~ 5 V) can be applied to a SG sample before it could buckle and bind to the bottom of the trench. Despite limited carrier densities, Bolotin
In the following, the thermoelectric property of TEP will be reviewed with regard to the three systems, namely SLG, BLG and AGNR.
1.2. Electronic structures
1.2.1. Single layer graphene
The transport characteristics of a material are intimately related to the energy band structure. The carriers in the graphene lattice are free to move in two dimensions. In the carrier transport of graphene, the carriers — electrons and holes — close to the Dirac points are of importance. Their transport is described by a Diraclike equation for massless particles [2, 3]:
where
The electronic band structure of the energy (
Being interested mostly in understanding electron transport for small energies and relatively small carrier concentrations, only the low
Here,
1.2.2. Bilayer graphene
The effective Hamiltonian for a BLG, in the low energy, longwavelength regime is [3]
where
1.2.3. Graphene nanoribbon
The spectrum of GNRs depends on the nature of their edges. The lowenergy electronic states of GNRs near the two nonequivalent Dirac points (
where γ (=√3
1.3. Phonon dispersion relations
Vibrations in the 2D graphene lattice are characterized by two types of acoustic phonons: those vibrating in the plane of layer with linear longitudinal and transverse acoustic branches (LA and TA), and those vibrating out of the plane of the layer – the socalled flexural phonons (ZA) [13].
The lowenergy inplane phonons have the usual linear dispersion relation
where,
The acoustic flexural phonons (FPs) are described by an approximately quadratic dispersion relation [1, 28]:
with
The existence and possible modification of the ZA modes, as in the case of SG membrane under tension, are known to lead to the unusual thermal transport in graphene [15]. For slowly varying finite inplane stresses, the dispersion relation of the FPs is anisotropic. Assuming uniaxial strain
The quadratic dispersion relation (8) of FPs becomes linear at long wavelengths [28].
2. Thermoelectric power – Basics
Thomas Johann Seebeck observed that a conductor generates a voltage when subjected to a temperature gradient. This phenomenon is called Seebeck effect, and can be expressed as [4, 5, 8, 9, 11]
where
On the other hand, Jean Charles Peltier discovered that when an external voltage is applied, the resulting current flow is associated with a heat flow. The Peltier effect is thus the reverse of the Seebeck effect — it refers to the temperature difference induced by voltage gradient. A third thermoelectric phenomenon, called the Thomson effect after its discoverer, William Thomson, is the reversible evolution (or absorption) of heat in a homogeneous conductor that carries an electric current and in which a temperature gradient is also maintained.
The three effects are related to thermal transport, and the coefficients are interrelated. The TEP is relatively easily measured and most of the available results are about this coefficient. Focusing attention, therefore, on TEP, we give below, in brief, the basic theory of TEP which serves as a basis for description of TEP in graphene systems. Also discussed below is the relation of TEP with other transport coefficients.
2.1. Definition and general relations
The thermoelectric effect is due to the interdependence of potential and temperature gradient in a system where no electric current flows. The absolute TEP,
under opencircuit conditions, where
There are, in general, two contributions to the TEP of the system, namely, the electrondiffusion TEP and the phonondrag TEP. They will be described later in 2.1.2.
2.1.1. Transport coefficients and thermopower
One can write an expression for the thermoelectric power,
where the coefficients, L_{ij}, are, in general, tensors. In order to relate the coefficients to the experimentally measured quantities, such as TEP, it is usual to invert Eq. (11) and write
Here
are the electrical resistivity, the thermopower, the Peltier and the thermal conductivity tensors, respectively;
with the superscript (T) meaning the transpose. The TEP,
This is known as the second Kelvin relation.
There are two approaches to the evaluation of TEP,
This method of computing
2.1.2. Diffusion and Phonondrag thermopower
As mentioned earlier, there are two contributions to the thermopower,
When the assumption of the phonon system being in equilibrium is lifted (which is true, especially, at low temperatures) an additional contribution to
The total heat current density
and, correlatively, the total TEP, S, can be expressed as
The treatment presented here is quite general and is applicable to graphene systems.
One can make a simplistic estimate of the magnitude of the diffusion thermopower [11]. It follows from Eq.(16a) that the Peltier coefficient Π, being the ratio of the rate of heat flow to the electrical current, is just the heat per unit charge. For a nondegenerate electron gas, the thermal energy per carrier will be ~
Eq.(20) suggests a linear temperature dependence, usually observed in degenerate systems at higher temperatures when the phonondrag is unimportant.
2.2. Survey of experimental work
Fundamentally related to the electrical conductivity of a material, the TE transport coefficients are also determined by the band structure and scattering mechanisms operative, and can offer unique information complementary to the electrical transport coefficients. The minimal conductivity at the Dirac point is characteristic of graphene [13]. Away from the Dirac point, the electron concentration dependence of conductivity depends on the nature of the scatterers. At low temperatures, the conductivity of graphene is limited by scattering off impurities and disorder which depend on the sample preparation. In the absence of extrinsic scattering sources, phonons constitute an intrinsic source of scattering [3].
Measurements of the thermoelectric properties of graphene have helped elucidate details of the unique electronic structure of the ambipolar nature of graphene, which cannot be probed by conductivity measurements alone. Table 2 lists the recent experimental investigations made with regard to the thermoelectric properties of graphene. Here, we primarily review the measurements made in the absence of an applied magnetic field. The presence of a magnetic field is expected to reveal some more interesting important features, as in conventional 2DEG [11, 1618, 32].
The TE effect of Dirac electrons has been initially experimentally investigated in graphene samples mechanically exfoliated on~300 nm SiO_{2}/Si substrates [1618]. The number of layers in graphene samples can be identified by optical contrast of the samples cross correlated with scanning probe studies and Raman spectroscopy. A controlled temperature difference ∆
Zuev
Wei






Mechanical exfoliation on 300nm SiO_{2} substrate; 
Gatedependent conductance and TEP measured simultaneously in zero and nonzero magnetic fields, in linear response regime (∆ 
~ 80 μV/K @ RT 
[16] 
Mechanical exfoliation on 300 nm SiO_{2} substrate; 
Gate voltage (V_{g}) and temperature dependent TEP measured in zero & applied magnetic fields; Oscillating dependence of both 
At @ B=8 T, @ CNP 
[17] 
Exfoliation on 300 nm SiO_{2}/Si substrate 

~ 100 μV/K @ RT 
[18] 
Exfoliated & supported on SiO_{2}; W:1.5–3.2; L: 9.5–12.5 μm; G1:3.2 μm parallel to 1.5 μm G2:2.4μm 
Temperature dependence of TEP 
G1, G2: ~ 80 μV/K @ RT 
[34] 
Exfoliated from Kish graphite/HOPG; 
Effect of charged impurities on the TEP near the Dirac point High high Low charged impurities induce high residual 
~ 60 μV/K @ 295 K 
[36] 
Epitaxial on Cface of SiC holedoped: 
Temperature dependence of TEP Sign change observed for 
~ 55 μV/K @ 230 K 
[41] 
Exfoliated on SiO_{2}/Si using ebeam lithography; 
V_{g} dependence of TEP of device for three mobility states Effect of carier mobility on 
~ 50 – 75 μV/K, @ 150 K  [32] 
Fabricated on SiO_{2}/Si with ebeam lithography; 
Low 

[39] 
Suspended CuCVD SLG  The of TEP for 50 < 
9 μV/K @300 K  [45] 
Few atomic layer thick, cm size sample CVD grown on Si/SiO_{2}/Ni substrates 
The 
10 μV/K @300 K  [46] 
Mechanical exfoliation on 300 nm SiO_{2}/Si substrate; 

 @ 300 K 
[51] 
Mechanical exfoliation on 300 nm SiO_{2}/Si substrate; 
Oscillations in 
~ 100 μV/K @ 250 K 
[49] 
Mechanical exfoliation on 300 nm SiO_{2}/Si substrate; 
Electric field tuning of TEP in DualGated BLG demonstrated – originates from bandgap opening; Enhanced TEP; 
@ 250 K 
[52] 


SBLG transistor; Mechanical exfoliation of graphene sheets onto 90 nm SiO_{2}/Si wafer; SLG/BLG identified by optical contrast & Raman 
Optoelectronic response of SBLG interface junction using photocurrent microscopy as function of (Photocurrent is by photoTE effect) 
~ 6 μV/K @ 12 K 
[53] 
SLG  TLG epitaxial on 6HSiC 
TEP (over 300 – 550 K) as function of environment composition  p: 10 μV/K n: 20 μV/K (annealed @ 500K) 
[54] 
SLG–MLG CVD on Cu 
Layerdependence of the graphene Seebeck coefﬁcient is peculiar & unexpected, that exceptionally increases with increasing thickness Gas flow induced voltage in MLG is not proportional to 
~ 30 μV/K (SLG) – 54 μV/K (HLG) @ RT 
[55] 
FLG Pristine: on SiO_{2}/Si substrate, t ~5nm with possible structural defects; Treated: ACN, TPA attachments 
Temperature dependence of TEP Power Factor Enhancement for FewLayered Graphene Films by Molecular Attachments TEP increased ~ 4.5 times Results supported by simulations based on Kubo’s formula 
Pristine: ~ 40 μV/K Treated: 180 μV/K; 300< 
[56] 
FLG On SiO_{2}/Si substrate; SLG & rGO 
Temperature dependence of TEP Enhanced TEP of films with Oxygen Plasma Treatment Treatment generates disorders which open the ππ* gap leading to enhancement of TEP and reduction in 
FLG: Pristine:~80 μV/K Treated:~ 700 μV/K @575 K; SLG: p @low 
[57] 
Checkelsky and Ong [18] have also reported measurements of TEP,
Seol
With a view to investigate the effect of charged impurities on the TEP of graphene near the Dirac point, Wang and Shi [36] have measured both TEP and electrical conductivity of SLG samples with varying degree of disorders as characterized by carrier mobilities ranging from 1.5 – 13.0 x 10^{3} cm^{2}/Vs, and examined the validity of the Mott relation as the lowdensity region near the Dirac point is approached. The fourpoint geometry they employed allowed them to measure the graphene resistivity properly by excluding the contact resistance and ensuring that both
In their recent measurements, shown in Figure (7), Shi and coworkers [32] have investigated the carrier mobilitydependence of TE transport properties of SLG in zero and nonzero magnetic fields. In the absence of magnetic field, they find that, with increase in mobility, the maximum value of
The magnetic field dependence of TEP has also been studied [1618, 32]. In a magnetic field, carriers diffusing under the temperature gradient experience a Lorentz force, resulting in a nonzero transverse voltage. In the quantum Hall regime at a high magnetic field, the curves of
Samples grown by different methods throw light on the different characteristics of TEP in graphene systems. The main graphene production techniques include dry and wet exfoliation, photoexfoliation, growth on SiC, CVD, MBE and chemical synthesis (for a recent review see [40]). Although initially graphene samples have been mechanically exfoliated, with a view to investigate the TE characteristics further, the samples have been produced by other methods as well.
Wu
Kim and coworkers [43] report measurement of TEP on graphene samples deposited on hexaboron nitride substrates where drastic suppression of disorder is achieved. Their results show that at high temperatures where the inelastic scattering rate due to electronelectron (ee) interactions is higher than the elastic scattering rate by disorders, the measured TEP exhibits an enhancement compared to the expected TEP from the Mott relation.
Graphene structures grown epitaxially on metal surfaces could reach sizes up to a micrometer with few defects. They can also be formed on the surface of SiC with the quality and number of layers in the samples depending on the SiC face used for their growth.
The carbonterminated surface can produce few layers with low mobility whereas the siliconterminated surface can give many layers with higher mobility [40]. Chemically exfoliating graphene is another method of preparing good quality and large amount of fewlayer graphene sheets [44].
There exist a few reports of measurements of TEP of CVDgrown graphene [45, 46]. Figure 8 shows the observed temperature dependences. Other investigations have demonstrated the TEP of CVDgrown graphene to be a sensitive probe to the surface charge doping from the environment and the device concept promises use in gas/chemical sensing [47]. An initially degassed ntype graphene sample, upon exposure to gases, was found to become pdoped or further ndoped during exposure depending on the properties of the ambient gases as evidenced by a monotonic change in sign of TEP.
In a graphene sample, the substrate on which the graphene layer is exfoliated, affects the morphology of the graphene specimen and is a source of impurities. In a suspended graphene (SG) sample, on the other hand, the substrate is etched away so that the graphene is suspended over a trench approximately 100 nm deep, with most of the impurities sticking to the graphene sheet [48]. Annealed SG samples showed both ballistic and diffusive carrier transport properties with carrier mobilities more than 2x10^{4} cm^{2}/Vs. The warping of the layers can be avoided with the use of a top gate [21].
In the case of BLG, Nam
There do not seem to be till date any reports on measurements of TEP of graphene nanoribbons.
In the following sections, we discuss, based on the Boltzmann formalism, the present theoretical understanding of the observed phenomena, in terms of the diffusion and phonondrag contributions. An analysis of measured TEP is usually done by separating the two contributions by making use of their characteristic temperature dependences at lower temperatures [11]. Often, in literature, the diffusion component,
3. Diffusion thermopower
Diffusion thermopower,
In this review, we adopt the Boltzmann approach, found to be robust especially for transport in graphene far away from the Dirac point [58]. We give here, in brief, the basic theory of TEP and the expressions used in the present analysis of
3.1. Basic formalism – Boltzmann approach
Low field transport in many of the systems is often described by the Boltzmann transport equation (BTE) [5961]. This semiclassical Boltzmann approach is known to be appropriate for structures in which the potentials vary slowly on both the spatial scale of the electron thermal wavelength and the temporal scale of the scattering processes. The conventional theory of charge carrier transport in 2D semiconductors is based on this formalism, and the TE coefficients are commonly obtained by solving the BTE in the relaxation time approximation [11].
In the regime of large chemical potential, the nature of transport of the massless Dirac fermions through a 2D graphene membrane may be accessed by the Boltzmann formalism [3] and one may write an expression for TEP in graphene systems in terms of the fundamental transport coefficients.
3.1.1. Transport coefficients in graphene systems
We consider a graphene system of length
and
The electric current density, and heat current density, can be evaluated by solving the Boltzmann transport equation in the relaxation time approximation. Assuming the electric field to be weak and the displacement of the distribution function from thermal equilibrium to be small, the electron distribution function
where,
with
In Eqs. (11a) and (11b), the current densities
and
Here,
where the coefficients
In the absence of temperature gradient (
From Eqs. (21), (22), (27) and (28), one obtains expressions for the diffusion contribution to thermopower,
and
respectively. Equation (31) may be expressed as [11]
with
Equations (30) – (33) show that evaluation of the transport coefficients requires a knowledge of the relaxation time(s), τ(
where the sum is over all the relevant scattering mechanisms,
Often in literature, limiting forms of
where,
where the first term reflects the scattering mechanisms. The parameter
Eq. (37) brings out the feature that
A solution of the Boltzmann equation in the relaxation time approximation may be applied exactly when the important collision processes are all elastic [30, 5961]. It is also applicable when the inelastic processes include nonpolar optic and intervalley phonon scattering. If polar optic phonon scattering is also important, the method is applicable only at high temperatures. Solutions of Boltzmann equation when polar optic phonon scattering is dominant may be obtained by applying variational or numerical methods.
3.2. Scattering mechanisms
Central to understanding the TE transport properties of graphene, are the mechanisms causing the scattering of the charge carriers. A better understanding, therefore, of the relative importance of the operative scattering mechanisms, which varies with temperature and carrier concentrations in graphene, enables useful improvements in the transport properties of graphene for various possible TE applications.
Scattering in graphene which could contribute to carrier transport may result from both intrinsic and extrinsic sources. The extrinsic sources may be vacancies, surface roughness arising from rippling of the graphene sheet, disorder, which can create electronhole puddles, and charged impurities, known to be the main scattering mechanism in graphene. Apart from the graphene layers, the substrates may also be a source of impurities. Besides, there are additional scattering sources such as neutral point defects [1, 3]. In principle, the limitation due to the extrinsic scattering mechanisms can be reduced by improved growth/fabrication techniques.
In the absence of extrinsic scattering sources, phonons, which constitute an intrinsic source of scattering in a system, limit carrier mobility at finite temperatures [5961]. Phonon scattering may be due to intravalley acoustic and optical phonons which induce the electronic transitions within a single valley, and intervalley phonon scattering that induces electronic transitions between different valleys [3]. The intravalley acoustic phonon scattering, induced by low energy phonons and considered an elastic process, gives a quantitatively small contribution in graphene even at room temperature due to the high Fermi temperature of graphene. Shishir
In the case of SG, the intrinsic scattering mechanisms limiting electron transport in SG layers are due to inplane and outofplane (flexural) acoustic phonons. Recent investigations of electron and phonon transport in SG indicate that in the free standing case (absence of strain) the major contribution to resistivity and thermal conductance is from acoustic flexural phonons, and this intrinsic limitation can be reduced by the effect of strain [28, 67].
Evaluation of the transport coefficients requires the knowledge of the relaxation times of the scattering mechanisms. In the following, we give the expressions for the momentum relaxation times of the extrinsic and intrinsic scattering mechanisms.
3.2.1. Relaxation times
The expressions for the momentum relaxation times for the various scatterings in graphene systems may be expressed as [3]
where ‘
The overall momentum relaxation time
In eq.(39b),
In the 2D material of graphene at low temperatures, an understanding of electronphonon interaction is important both from basic physics and technology points of view [3]. In typical conductors, electrons are scattered by phonons producing a finite temperaturedependent resistivity
The expressions for the momentum relaxation rates for inplane and flexural acoustic phonon, nonpolar optical phonon, surface polar optical phonon and for roughness, impurity and vacancy scatterings in the graphene systems are given in Table 3.
Suspended graphene (SG) allows for the investigation of the intrinsic properties of the material, unperturbed by the presence of a substrate. It has been realized that mechanical deformations of graphene sheets affect the electronic properties. This is of special relevance for strainengineering aimed at controlling the electronic properties of graphene by suitably engineering the deformations ([67] and references therein). Employing the semiclassical Boltzmann transport formalism, Mariani and Oppen [67] and Ochoa
3.3. Diffusion thermopower in graphene systems
Besides the experimental investigations (see Table 1.), the TEP of graphene has also attracted much theoretical attention. The theoretical investigations made to understand the experimental results have so far been mostly on the basis of the diffusion TEP, ignoring the drag component.
The following features of TEP, first observed in SLG samples by Zuev
In the presence of a quantizing magnetic field, the TEP of graphene exhibits additional interesting effects [1618, 32]; however, these do not form the content of the present review. The theoretical studies of
3.3.1. Diffusion thermopower in SLG
Much of the theoretical investigations of diffusion TEP in graphene has been made on SLG with interest being devoted mostly to the TEP at higher temperatures (10<
Using a phenomenological theory for transport in graphene, close to Dirac point, based on the semiclassical Boltzman approach Peres and coworkers [72, 76] have obtained an expression for diffusion TEP including the scattering mechanism involving midgap states arising from local point defects in the form of vacancies, cracks, boundaries, impurities in the substrate or in corrugated graphene. They find that this mechanism leads to a similar
Lofwander and Fogelstrom [33], have presented calculations for the linear response to electrical and thermal forces in graphene for the case of strong impurity scattering in the selfconsistent tmatrix approximation. At low temperatures, the electronic contribution to TEP is found to be linear in





Acoustic Phonons  [71]  
Optical Phonons  [66]  
Impurities  [72]  
Surface roughness  [66]  
Vacancies  [72]  


Acoustic Phonons (Inplane) 
[28]  
Acoustic Phonons (Flexural) 
[28]  
Acoustic Phonons (Flexural) (strained) 
[28]  


Acoustic phonons  [73]  
Surface polar optical phonons  [74]  
Coulomb impurities  [75]  
Short range disorder  [75]  


Acoustic phonons  [65]  
Optical phonons  [65]  
Edge roughness  [65]  
Impurities  [65] 
Kubakaddi [62] in his study of TEP at low temperatures, has given Mott formula Eq.(36a) for
Motivated by the experiments of [16], [17] and [18], Hwang
Considering scattering of electrons by both impurities and phonons, Bao
Vaidya
The effect of electronphonon scattering processes on TEP of extrinsic graphene has been studied by Munoz [78]. From a variational solution of the Boltzmann equation, he obtained analytical expressions for the transport coefficients and the leading contribution to phononlimited TEP. Figure 13 represents his results of temperature dependence of TEP at different electronic densities. At lower temperatures (
The distinctive features observed in the energy dependence of the relaxation times
In their investigation of the temperature dependence of
The influence that
Mariani and Oppen [67] and Ochoa
Apart from those mentioned above, there have been reports of other related TEP studies. The investigations of Sharapov and Varlamov [83] and Patel and Mukerjee [84], with regard to the effect of opening a gap in the graphene spectrum, find that the TEP is found to be proportional to the band gap. Zhou
3.3.2.Diffusion thermopower in BLG
In the last few years, TEP in BLG has been studied both theoretically [64, 86] and experimentally [49]. Nam
Kubakaddi and Bhargavi [64, 73] give an expression for
In the case of suspended BLG,
3.3.3. Diffusion thermopower in AGNR
The theoretical efforts, to understand the diffusion contribution to TEP of GNRs have been based on different techniques. Divari and Kliros [87] have studied TEP of ballistic wide graphene ribbons with aspect ratio (W/L ≥ 3) using linear response theory and the Landauer formalism. Xing
Recently, in their systematic study of
The above mentioned changed energy dependences of the relaxation times are found to influence the behavior of and lead to distinctive features in
With a proper choice of parameters characterizing the extrinsic scattering mechanisms, and the possibility of modulating the Fermi level with a control on gate and bias voltages [93], the behavior of overall
4. Phonondrag thermopower in graphene systems
As mentioned in section 2, in the presence of temperature gradient
The formal theory of
where
The phonondrag is known to be important at low temperatures [8, 9, 11, 94, 95]. At these temperatures the phonon scattering is dominated by boundary scattering and
4.1. Phonondrag thermopower in SLG
Kubakaddi has studied in detail the lowtemperature behavior of
where γ =
Taking account of boundary scattering as well as phononphonon interaction in the phonon relaxation processes, Bao
In the BG regime, Kubakaddi gives a simple power law for
where
In conventional 2DEG, in the BG regime,
A useful and simple approach to calculate
4.2. Phonondrag thermopower in BLG
The theory of
where
Figure 21 shows the temperature dependence of
Kubakaddi and Bhargavi [64] have studied the influence on
Nam
In BG regime,
With regard to
4.3. Phonondrag thermopower in AGNR
As shown in section 3.3.3, with regard to the diffusion TEP, the geometry and edge roughness can greatly influence the TE properties of GNRs [7, 65, 8890, 109]. A dramatic reduction in phonon transport in ZGNR [7] indicates small value for Λ. However, in an AGNR, the phonon conductance is shown to be at least one order of magnitude higher than the electronic contribution indicating a larger value for Λ in this system [110]. The role of quasionedimensionality, temperature, Fermi energy and ribbon width on S_{g} of a semiconducting ntype AGNR is investigated by Bhargavi and Kubakaddi [100]. The Q1D electrons are assumed to interact,
where, the various quantities are already defined. In the lowT boundary scattering regime,
where,
In the BG regime, the energy integration in Eq.(46) gives
Figure 23 illustrates the temperature dependence of
The dependence of
The above results in AGNR, the TEP measurements and
From Eqs.(42)(44) and (46), it may be seen that
In all the three graphene systems considered above,
5. Summary
In this chapter, we have reviewed the current status of the experimental investigations of the important and interesting transport property, namely TEP in graphene and described a theoretical treatment of the diffusion and phonondrag components of TEP, in graphene systems. The treatment presented, employing the conventional Boltzmann formalism in the relaxation time approximation, gives a basic understanding of TEP in graphene systems, namely, SLG, BLG and AGNR. It gives a description of the dependences of TEP on temperature and gate bias. This understanding is expected to provide a useful guideline for improvement and optimization of performances of graphenebased TE modules.
Measurements of TEP of graphene reveal unique features not observed in metals [9] and conventional 2D semiconductor systems [10]. The graphene systems exhibit a range of TEP values up to 100 μV/K, at room temperature. The TEP changes sign across the CNP as the gate bias is varied. Away from the CNP, the TEP shows a
Future experimental endeavours may aid not only in improving applicability in TE devices but also in understanding better the TE processes in graphene. Graphene may be a suitable system to realize a large range of BG regime. A detailed investigation of lowtemperature (say,
Conventional lowdimensional systems, such as quantum wires and superlattices, are known to provide not only new approaches for achieving higher ZT, but also new applications such as thermal management of integrated circuits [4, 5]. The possibility of increasing ZT through engineering the electron and phonon transport, therefore, makes graphene systems attractive, in future, for applications in efficient thermoelectric devices.
Acknowledgments
This work was supported by UGC (India). The assistance of Mr. A.S. Nissimagoudar in the preparation of the manuscript is acknowledged.
References
 1.
Katsnelson MI. Graphene: Carbon in Two Dimensions. Cambridge: Cambridge University Press; 2012  2.
Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK. The electronic properties of graphene. Rev Mod Phys 2009; 81(1):10962  3.
Das Sarma S, Adam S, Hwang EH, Rossi E. Electronic transport in twodimensional graphene. Rev Mod Phys 2011; 83(2):40770.  4.
Rowe DM., editor. Thermoelectrics Handbook  Macro to Nano. Boca Raton: Taylor and Francis; 2006  5.
Tritt TM., editor. Recent Trends in Thermoelectric Materials Research III. Semiconductors and Semimetals Volume 71. San Diego: Academic Press; 2001  6.
Mahan GD. Good thermoelectrics. In: Ehrenreich H. and Spaefen F. (eds.) Solid State Physics, vol 51, New York; Academic Press, 1998. p 81  7.
Sevincli H, Cuniberti G. Enhanced thermoelectric of merit in edgedisordered zigzag graphene nanoribbons. Phys Rev B 2010; 81(11):11340114..  8.
Barnard RD. Thermoelectricity in metals and alloys. London: Taylor and Francis;1972.  9.
Blatt FJ, Schroeder PA, Foiles CL, Greig D. Thermoelectric power of metals. New York: Plenum Press; (1976).  10.
Ure RW. Thermoelectric effects in III – V Compounds. In: Willardson RK and Beer AC. (eds.) Semiconductors and Semimetals, vol 8, New York; Academic Press, 1972. p 67102  11.
Gallagher BL, Butcher PN. Classical transport and thermoelectric effects in low dimensional and mesoscopic semiconductor structures. In: Landsberg PT. (ed.) Handbook on semiconductors, vol 1, Amsterdam; Elsevier; 1992. p.721816.  12.
Fletcher R. Magnetothermoelectric effects in semiconductor systems. Semicond Sci Tech 1999; 14(4):R1R15  13.
Dresselhaus MS. Quantum wells and quantum wires for potential thermoelectric applications. In: Willardson RK and Weber ER. (eds.) Semiconductors and Semimetals, vol 71, New York; Academic Press, 2001. p 1  14.
Chen G. Phonon transport in lowdimensional structures. In: Willardson RK and Weber ER. (eds.) Semiconductors and Semimetals, vol 71, New York; Academic Press, 2001. p 203  15.
Balandin AA. Thermal properties of graphene and nanostructured carbon materials. Nat Mat 2011;10(8):569–81  16.
Zuev YM, Chang W, Kim P. Thermoelectric and magnetothermoelectric transport measurements of graphene. Phy Rev Lett 2009; 102(9): 09680714.  17.
Wei P, Bao W, Pu Y, Lau CN, Shi J. Anomalous thermoelectric transport of Dirac particles in graphene. Phy Rev Lett 2009; 102(16): 16680814.  18.
Checkelsky JG, Ong NP. Thermopower and Nernst effect in graphene in a magnetic ﬁeld. Phy Rev B 2009; 80(20): 081413(R)14.  19.
Novoselov KS, Falko VI, Colombo L, Gellert PR, Schwab MG, Kim K. A roadmap for graphene. Nature 2012;490(7419):192200  20.
Palacios JJ, FernandezRossier J, Brey L, Fertig HA. Electronic and magnetic structure of graphene nanoribbons. Semicond Sci Technol 2010;25(3):0330031–10.  21.
Bolotin KI, Sikes KJ, Hone J, Stormer HL, Kim P.Temperaturedependent transport in suspended graphene. Phys Rev Lett 2009; 101(9):09680214  22.
Bolotin KI, Sikes KJ, Jiang J, Klima M, Fundenberg G, Hone J, Stormer HL, Kim P. Ultrahigh electron mobility in suspended graphene. Solid State Commun 2011; 146(9): 35155  23.
Levy N, Burke SA, Meaker KL, Panlasigui M, Zettl A, Guinea F, Castro Neto AH, Crommie MF. Straininduced pseudo– magnetic fields greater than 300 tesla in graphene nanobubbles. Science 2010;329 (5991): 54447  24.
Wallace PR. The band theory of graphite. Phys Rev 1947; 71(9);62234.  25.
Min H, Sahu B, Banerjee SK, MacDonald AH. Ab initio theory of gate induced gaps in graphene bilayers. Phys Rev B 2009; 75(15):15511517.  26.
Wakabayashi K, Takane Y, Yamamoto M, Sigrist M. Electronic transport properties of graphene nanoribbons. New J of Phys 2009; 11(9): 095016121.  27.
Fang T, Konar A, Xing H, Jena D. Mobility in semiconducting graphene nanoribbons: phonon, impurity, and edge roughness scattering. Phys Rev B 2008; 78(20):20540318.  28.
Ochoa H, Castro E, Katsnelson MI, Guinea F. Scattering by ﬂexural phonons in suspended graphene under back gate induced strain. Physica E 2012;44 (6):96366..  29.
Ziman JM. Electrons and phonons. Oxford: Claredon Press; 1960.  30.
Ashcroft NW, Mermin ND. Solid state physics. New York: Brooks/Cole; 1976.  31.
Herring C, Geballe TH, Kunzler JE. Phonondrag thermomagnetic effects in n type germanium. I. general survey. Phys Rev 1958; 111(1):3657.  32.
Liu X, Wang D, Wei P, Zhu L, Shi J. Effect of carrier mobility on magnetothermoelectric transport properties of graphene. Phy Rev B 2012;86(15):15541417.  33.
Löfwander T, Fogelström M. Impurity scattering and Mott’s formula in graphene. Phy Rev B 2007; 76(19):19340114  34.
Seol JH, Jo I, Moore AL, Lindsay L, Aitken ZH, Pettes MT, Li X, Yao Z, Huang R, Broido D, Mingo N, Ruoff RS, Shi L. Twodimensional phonon transport in supported graphene. Science 2010; 328(5975):21316  35.
Hwang EH, Rossi E, Das Sarma S. Theory of thermopower in twodimensional graphene. Phy Rev B 2009; 80(23): 23541515.  36.
Wang D, Shi J. Effect of charged impurities on the thermoelectric power of graphene near the Dirac point. Phy Rev B 2011; 83(11): 11340314.  37.
Jonson M, Girvin SM. Thermoelectric effect in a weakly disordered inversion layer subject to a quantizing magnetic field. Phys Rev B 1984; 29(4):193946.  38.
Oji H, Thermomagnetic effects in twodimensional electron systems. J Phys C: Solid State Phys. 1984;17(17):305966  39.
Liu X, Ma Z, Shi J. Derivative relations between electrical and thermoelectric quantum transport coefficients in graphene. Solid State Commun 2012; 152(6):46972.  40.
Bonaccorso F, Lombardo A, Hasan T, Sun Z, Colombo L, Ferrari AC. Production and processing of graphene and 2D crystals. Materials Today 2012; 15(12):56489.  41.
Wu X, Hu Y, Ruan M, Madiomanana NK, Berger C, de Heer WA. Thermoelectric effect in high mobility single layer epitaxial graphene. Appl Phys Lett 2011; 99(13): 133102:13  42.
Bergman DL, Oganesyan V. Theory of dissipationless Nernst effects. Phys Rev Lett 2010;104(6):06660114.  43.
Ghahari F, Zuev Y, Watanabe K, Taniguchi T, Kim P. Effect of electronelectron interactions in thermoelectric power in graphene. Bull.American Phys Soc; 57(1) Mar2012  44.
Puneet P, Podila R, Oleveira L, Tritt T, Rao A. Transport properties of pristine and doped graphene. Bull.American Phys Soc; 57(1)Mar2012  45.
Xu X, Wang Y, Zhang K, Zhao X, Bae S, Heinrich M, Bui CT, Xie R, Thong JTL, Hong BH, Loh KP, Li B, Oezyilmaz B. Phonon transport in suspended single layer graphene. arXiv:1012.2937v1 [condmat.meshall] 14Dec2010.  46.
Babichev AV, Gasumyants VE, Butko VY. Resistivity and thermopower of graphene made by chemical vapor deposition technique. J Appl Phys 2013;113(7): 07610113.  47.
Sidorov AN, Sherehiy A, Jayasinghe R, Stallard R, Benjamin DK, Yu Q, Liu Z. , Wu W, Cao H, Chen YP, Jiang Z, Sumanasekera GU. Thermoelectric power of graphene as surface charge doping indicator. Appl Phys Lett 2011; 99(1): 013115:13.  48.
Fogler MM, Guinea F, Katsnelson MI. Pseudomagnetic fields and ballistic transport in a suspended graphen sheet. Phys Rev Lett 2008; 101(22):22680414.  49.
Nam SG, Ki DK, Lee HJ. Thermoelectric transport of massive Dirac fermions in bilayer graphene. Phys Rev B 2010; 82(24): 24541615.  50.
Hwang EH, Das Sarma S. Acoustic phonon scattering limited carrier mobility in twodimensional extrinsic graphene. Phys B 2008; 77(11):11544916.  51.
Wang CR, Lu WS, Lee WL. Transverse thermoelectric conductivity of bilayer graphene in the quantum Hall regime. Phys Rev B 2010; 82(12): 121406(R)14.  52.
Wang CR, Lu WS, Hao L, Lee WL, Lee TK, Lin F, Cheng IC, Chen JZ. Enhanced thermoelectric power in dualgated bilayer graphene. Phy Rev Lett 2011; 107(18): 18660214.  53.
Xu X, Gabor NM, Alden JS, van der Zande AM, McEuen PL. Photothermoelectric effect at a graphene interface junction. Nano Lett 2010; 10(2):56266.  54.
Sidorov AN, Gaskill K, Nardelli MB, Tedesco JL, MyersWard RL, Eddy CR Jr., Jayasekera T, Kim KW, Jayasingha R, Sherehiy A, Stallard R, and Sumanasekera GU. Charge transfer equilibria in ambientexposed epitaxial graphene on (0001) 6 HSiC. J Appl Phys 2012; 111(11): 113706:16.  55.
Li X, Yin J, Zhou J, Wang Q, Guo W. Exceptional high Seebeck coefficient and gasflowinduced voltage in multilayer graphene. Appl Phys Lett 2012; 100(18): 183108:13  56.
Sim D, Liu D, Dong X, Xiao N, Li S, Zhao Y, Li L, Yan Q, Hng HH. Power factor enhancement for few layered graphene films by molecular attachments. J Phys Chem C 2011; 115(5):17801785  57.
Xiao Ni, Dong X, Song L, Liu D, Tay YY, Wu S, Li L, Zhao Y , Yu T, Zhang H, Huang W, Hng HH, Ajayan PM, Yan Q. Enhanced thermopower of graphene films with oxygen plasma treatment. ACS Nano 2011; 5(4):27492755.  58.
Shafﬁque A, Hwang EH, Galitski VM, Das Sarma S. A selfconsistent theory for graphene transport. PNAS 2007;104(47): 18392–18397.  59.
Nag BR. Electron transport in compound semiconductors. Berlin: SpringerVerlag; 1980.  60.
Ferry DK, Goodnick SM, Bird J. Transport in nanostructures. Cambridge: Cambridge University Press; 2009.  61.
Ridley BK. Quantum processes in semiconductors. 2nd ed. Oxford: Clarendon Press; 1988.  62.
Kubakaddi SS. Interaction of massless Dirac electrons with acoustic phonons in graphene at low temperatures. Phy Rev B 2009;79(7): 07541716.  63.
Vaidya RG, Kamatagi MD, Sankeshwar NS, Mulimani BG. Diffusion thermopower in graphene. Semicond Sci Technol 2010; 25(9): 09200116.  64.
Kubakaddi SS, Bhargavi KS. Enhancement of phonondrag thermopower in bilayer graphene. Phys Rev B 2010;82(15) 15541017  65.
Nissimagoudar AS, Sankeshwar NS. Electronic thermal conductivity and thermopower of armchair graphene nanoribbons. Carbon 2013; 52:20108.  66.
Shishir RS, Chen F, Xia J, Tao NJ, Ferry DK. Room temperature carrier transport in graphene. J Comput Electron 2009; 8(2): 43–50 [54]  67.
Mariani E, Oppen F. Temperaturedependent resistivity of suspended graphene. Phys Rev B 2010;82(19):195403111.  68.
Efetov DK, Kim P. Controlling electronphonon interactions in graphene at ultrahigh carrier densities. Phys Rev Lett 2010; 105(25):25680514.  69.
Min H. Hwang EH, Das sarma S. Chiralitydependent phononlimited resistivity in multiple layers of graphene. Phys Rev B 2012; 83(16):161404(R)14.  70.
Castro EV, Ochoa H, Katsnelson MI, Gorbachev RV, Elias DC, Novoselov KS, Geim AK. Limits on charge carrier mobility in suspended graphene due to flexural phonons. Phys Rev Lett 2010; 105(26):26660114.  71.
Sankeshwar NS, Vaidya RG, Mulimani BG. Behavior of thermopower of graphene in Bloch–Gruneisen regime. Physica Status Solidi B 2013; 250(7): 135662.  72.
Stauber T, Peres NMR, Guinea F. Electronic transport in graphene: A semiclassical approach including midgap states. Phy Rev B 2007;76(20): 205423110.  73.
Bharaghavi KS, Kubakaddi SS. Scattering mechanisms and diffusion thermopower in a bilayer graphene.Physica E 2013; 11621.  74.
Li X, Borysenko KM, Nardelli MB, Kim KW. Electron transport properties of bilayer graphene. Phys Rev B 2011; 84(19)19545315  75.
Das Sarma S, Hwang EH, Rossi E, Theory of carrier transport in bilayer graphene, Phys Rev B 2010; 81(16):16140714.  76.
Peres NMR, Lopes dos Santos JMB, Stauber T. Phenomenological study of the electronic transport coefficients of graphene. Phys. Rev. B 2007; 76(7): 07341214.  77.
Bao WS, Liu SY, Lei XL. Thermoelectric power in graphene. J Phys Condens Matter 2010;22(31):31550217.  78.
Munoz E. Phononlimited transport coefﬁcients in extrinsic graphene. J Phys Condens Matter 2012; 24(19): 19530218.  79.
Sankeshwar NS, Kamatagi MD, Mulimani BG. Behaviour of diffusion thermopower in Bloch–Grüneisen regime in AlGaAs/GaAs and AlGaN/GaN heterostructures. Physica Status Solidi B 2005;242(14):28922901.  80.
Chen W, Clerk AA. Electronphonon mediated heat flow in disordered graphene. Phys Rev B 2012; 86(12): 125443114  81.
Vaidya RG, Sankeshwar NS, Mulimani BG. Diffusion thermopower in suspended graphene: effect of strain. J Appl Phys 2012;112(9):09371116.  82.
Vaidya RG, Sankeshwar NS, Mulimani BG. Diffusion thermopower in suspended graphene. In: Singh V, Katiyar M, Mazharier SS, Das U, Dutta A, Sodhi R, Anantharamakrishna S. (eds.) : SPIE Proceedings Vol. 8549: proceedings of the 16th national Workshop on Physics of Semiconductor Devices, IWPSD2011, 1922 December 2011, IITKanpur, India. Wahington: SPIE; 2012. doi: 10.1117/12.925343  83.
Sharapov SG, Varlamov AA. Anomalous growth of thermoelectric power in gapped graphene. Phs Rev B 2012; 86(3): 03543015.  84.
Patel AA, Mukerjee S. Thermoelectricity in graphene: effects of a gap and magnetic ﬁelds. Phy Rev B 2012;86(7): 0754111 5.  85.
Zhou B, Zhou B. Liu Z, Zhou G. Thermoelectric effect in a graphene sheet connected to ferromagnetic leads. J Appl. Phys 2012; 112(7):07371214.  86.
Hao L, Lee TK, Thermopower of gapped bilayer graphene, Phys Rev B 2010; 81(16): 16544518  87.
Divari PC, Kliros GS. Modeling the thermopower of ballistic graphene ribbons. Physica E 2010; 42(9):24312435.  88.
Xing Y, Sun Q, Wang J. Nernst and Seebeck effects in a graphene nanoribbon. Phys Rev B 2009; 80(23): 23541118.  89.
Ouyang Y, Guo J. A theoretical study on thermoelectric properties of graphene nanoribbons. Appl Phys Lett 2009; 94(26):26310713.  90.
Karamitaheri H, Neophytou N, Pourfath M, Faez R, Kosina H. Engineering enhanced thermoelectric properties in zigzag graphene nanoribbons. J Appl Phys 2012; 111(5):05450119.  91.
Zheng H, Liu HJ, Tan XJ, Lv HY, Pan L, Shi J. Enhanced thermoelectric performance of graphene nanoribbons. Appl Phys Lett 2012; 100(9):09310415.  92.
Mazzamuto F, HungNguyen V, Apertet Y, Caer C, Chassat C, SaintMartin J, Dollfus P. Enhanced thermoelectric properties in graphene nanoribbons by resonant tunneling of electrons. Phys Rev B 2011; 83(23):23542617.  93.
Lin Y, Perebeinos V, Chen Z, Avouris P. Electrical observation of subband formation in graphene nanoribbons. Phys Rev B 2008; 78(16):161409(R)14.  94.
Fletcher R, Zaremba E, Zeitler U. ElectronPhonon interactions in low dimensional structures. In: Challis L. (ed.) Oxford: Clarendon; 2003. p149  95.
Tsaousidou M. Thermopower of low dimensional structures: the effect of electronphonon coupling. In: Narlikar AV and Fu YY. (eds.) Frontiers in Nanoscience and nanotechnology vol 2. Oxford: Oxford University Press; 2010. p477.  96.
Balandin AA, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, Lau CN. Superior thermal conductivity of singlelayer graphene. Nano Lett 2008;8(3): 90207.  97.
Nika DL, Pokatilov EP, Askerov AS, Balandin AA. Phonon thermal conduction in graphene: role of umklapp and edge roughness scattering. Phys Rev B 2009;79(12) 155413112.  98.
Ghosh S, Nika DL, Pokatilov EP, Balandin AA. Heat conduction in graphene: experimental study and theoretical interpretation. New Journal of Physics 2009;11(09) :095012119.  99.
Cantrell DG, Butcher PN. A calculation of the phonondrag contribution to the thermopower of quasi2D electrons coupled to 3D phonons. J Phys C: Solid State Physics 1987; 20(13): 19852003.  100.
Bhargavi KS, Kubakaddi SS. Phonondrag thermopower in an armchair graphene nanoribbon. J Phys Condens Matter 2011;23(27):27530315  101.
Viljas JK, Heikkila TT. Electronphonon heat transfer in monolayer and bilayer graphene.Phys Rev B 2010;81(24):245404 19.  102.
Fletcher R, Pudalov VM, Feng Y, Tsaousidou M, Butcher P N. Thermoelectric and hotelectron properties of a silicon inversion layer. Phys Rev B 1997;56(19) 1242228  103.
Herring C. Theory of thermoelectric power of semiconductors. Phys Rev 1954;96(11): 1163125.  104.
Tsaousidou M, Butcher PN, Triberis GP. Fundamental relationship between the Herring and CantrellButcher formulas for the phonondrag thermopower of twodimensional electron and hole gases. Phys Rev B 2001;64(16): 165304110  105.
Tieke B, Fletcher R, Zeitler U, Henini M, Maan JC. Thermopower measurements of the coupling of phonons to electrons and composite fermions. Phys Rev B 1998;58(4): 201725  106.
Kubakaddi SS. Effect of acousticphonon confinement on the phonondrag thermopower of a twodimensional electron gas in a semiconductor thin film. Phys Rev B 2004;69(03):03531715  107.
Scarola VW, Mahan GD. Phonon drag effect in singlewalled carbon nanotubes. Phys Rev B 2002;66(20):20540517  108.
Kamatagi MD, Sankeshwar NS, Mulimani BG. Wide temperature range thermopower in GaAs/AlGaAs heterojunctions. AIP Conf Proc.2009; 1147:51420.  109.
Chen Y, Jayasekera T, Calzolari A, Kim KW, Nardelli MB. Thermoelectric properties of graphene nanoribbons, junctions and superlattices. J Phys Condens Matter 2010;22(37):37220216  110.
Mazzamuto F, Hung Nguyen V, Nam DV, Caer C, Chassat C, SaintMartin J, Dollfus P. In: proceedingsof 14th International workshop on computational electronics p 14, IWCE, Pisa, IEEE Xplore; 2010.  111.
Tsaousidou M. Theory of phonondrag thermopower of extrinsic semiconducting singlewall carbon nanotubes and comparison with previous experimental data. Phys Rev B 2010;81(23):23542519  112.
Kubakaddi SS. Electronphonon interaction in a quantum wire in the BlochGruneisen regime. Phys Rev B 2007;75(7) 07530917  113.
Vavro J, Llaguno MC, Fischer JE, Ramesh S, Saini RK, Ericson LM, Davis VA, Hauge RH, Pasquali M, Smalley RE. Thermoelectric power of pdoped singlewall carbon nanotubes and the role of phonon drag. Phys Rev Lett 2003;90(06):06550314  114.
Zhou W, Vavro J, Nemes NM, Fischer JE, Borondics F, Kamarás K, Tanner DB. Charge transfer and Fermi level shift in pdoped singlewalled carbon nanotubes. Phy Rev B 2005;71(20): 20542317.  115.
Yu C, Shi L, Yao Z, Li D, Majumdar A. Thermal conductance and thermopower of an individual singlewall carbon nanotube. Nano Lett 2005;5(9):184246  116.
Tsaousidou M. Phonondrag thermopower of ballistic semiconducting singlewall carbon nanotubes and comparison with the semiclassical result. Europhysics Lett 2011;93(4) 4701018  117.
Kubakaddi SS, Butcher PN. A calculation of the phonondrag thermopower of a 1D electron gas. J Phys Condens Matter 1989;1(19):393946  118.
Tsaousidou M, Butcher PN. Phonondrag thermopower of a ballistic quantum wire. Phys Rev B 1997;56(16):R1004410047  119.
Baker AMR, AlexanderWebber JA, Altebaeumer T, McMullan SD, Janssen TJBM, Tzalenchuk A, Lara Avila S, Kubatkin S, Yakimova R, Lin CT, Li LJ, Nicholas RJ. Energy loss rates of hot Dirac fermions in epitaxial, exfoliated and CVD graphene. Phys Rev B 2013;87(04):045401416  120.
Rao A. Private communication 2012.