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 [1-3]. Graphene, a one-atom-thick 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 nano-electronic 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 [1-3]. 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 [16-18]. 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 tailor-made 2D graphene systems, such as single-layer 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 [8-12]. 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 semi-classical 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 phonon-drag 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 single-layer 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 sp2 hybridization state, with the three nearest-neighbour 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 hole-like. 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 single-layer, bilayer and few-layer 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 A-B 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 quasi-one-dimensional (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 micrometer-scale 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 current-annealed SG samples. However, unlike supported graphene, only a small gate voltage (Vg ~ 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 Dirac-like 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, long-wavelength regime is [3]
where
1.2.3. Graphene nanoribbon
The spectrum of GNRs depends on the nature of their edges. The low-energy electronic states of GNRs near the two non-equivalent 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 so-called flexural phonons (ZA) [1-3].
The low-energy in-plane 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 in-plane 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 open-circuit conditions, where
There are, in general, two contributions to the TEP of the system, namely, the electron-diffusion TEP and the phonon-drag 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, Lij, 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 Phonon-drag 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 non-degenerate 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 phonon-drag 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 [1-3]. 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, 16-18, 32].
The TE effect of Dirac electrons has been initially experimentally investigated in graphene samples mechanically exfoliated on~300 nm SiO2/Si substrates [16-18]. 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
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Mechanical exfoliation on 300nm SiO2 substrate; |
Gate-dependent conductance and TEP measured simultaneously in zero and non-zero magnetic fields, in linear response regime (∆ |
~ 80 μV/K @ RT |
[16] |
Mechanical exfoliation on 300 nm SiO2 substrate; |
Gate voltage (Vg) and temperature dependent TEP measured in zero & applied magnetic fields; Oscillating dependence of both |
At @ B=8 T, @ CNP |
[17] |
Exfoliation on 300 nm SiO2/Si substrate |
|
~ 100 μV/K @ RT |
[18] |
Exfoliated & supported on SiO2; 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 C-face of SiC hole-doped: |
Temperature dependence of TEP Sign change observed for |
~ 55 μV/K @ 230 K |
[41] |
Exfoliated on SiO2/Si using e-beam lithography; |
Vg dependence of TEP of device for three mobility states Effect of carier mobility on |
~ 50 – 75 μV/K, @ 150 K | [32] |
Fabricated on SiO2/Si with e-beam lithography; |
Low- |
|
[39] |
Suspended Cu-CVD SLG | The of TEP for 50 < |
9 μV/K @300 K | [45] |
Few atomic layer thick, cm size sample CVD grown on Si/SiO2/Ni substrates |
The |
10 μV/K @300 K | [46] |
Mechanical exfoliation on 300 nm SiO2/Si substrate; |
|
| @ 300 K |
[51] |
Mechanical exfoliation on 300 nm SiO2/Si substrate; |
Oscillations in |
~ 100 μV/K @ 250 K |
[49] |
Mechanical exfoliation on 300 nm SiO2/Si substrate; |
Electric field tuning of TEP in Dual-Gated BLG demonstrated – originates from band-gap opening; Enhanced TEP; |
@ 250 K |
[52] |
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S-BLG transistor; Mechanical exfoliation of graphene sheets onto 90 nm SiO2/Si wafer; SLG/BLG identified by optical contrast & Raman |
Optoelectronic response of S-BLG interface junction using photocurrent microscopy as function of (Photocurrent is by photo-TE effect) |
~ 6 μV/K @ 12 K |
[53] |
SLG - TLG epitaxial on 6H-SiC |
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 |
Layer-dependence 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 SiO2/Si substrate, t ~5nm with possible structural defects; Treated: ACN, TPA attachments |
Temperature dependence of TEP Power Factor Enhancement for Few-Layered 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 SiO2/Si substrate; SLG & rGO |
Temperature dependence of TEP Enhanced TEP of films with Oxygen Plasma Treatment Treatment generates disorders which open the π-π* 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 103 cm2/Vs, and examined the validity of the Mott relation as the low-density region near the Dirac point is approached. The four-point 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 co-workers [32] have investigated the carrier mobility-dependence of TE transport properties of SLG in zero and non-zero 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 [16-18, 32]. In a magnetic field, carriers diffusing under the temperature gradient experience a Lorentz force, resulting in a non-zero 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, photo-exfoliation, 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 co-workers [43] report measurement of TEP on graphene samples deposited on hexa-boron nitride substrates where drastic suppression of disorder is achieved. Their results show that at high temperatures where the inelastic scattering rate due to electron-electron (e-e) 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 carbon-terminated surface can produce few layers with low mobility whereas the silicon-terminated surface can give many layers with higher mobility [40]. Chemically exfoliating graphene is another method of preparing good quality and large amount of few-layer graphene sheets [44].
There exist a few reports of measurements of TEP of CVD-grown graphene [45, 46]. Figure 8 shows the observed temperature dependences. Other investigations have demonstrated the TEP of CVD-grown 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 n-type graphene sample, upon exposure to gases, was found to become p-doped or further n-doped 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 2x104 cm2/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 phonon-drag 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) [59-61]. This semi-classical 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, 59-61]. It is also applicable when the inelastic processes include non-polar 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 electron-hole 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 [59-61]. 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 in-plane and out-of-plane (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 electron-phonon interaction is important both from basic physics and technology points of view [3]. In typical conductors, electrons are scattered by phonons producing a finite temperature-dependent resistivity
The expressions for the momentum relaxation rates for in-plane and flexural acoustic phonon, non-polar 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 strain-engineering aimed at controlling the electronic properties of graphene by suitably engineering the deformations ([67] and references therein). Employing the semi-classical 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 [16-18, 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 self-consistent t-matrix approximation. At low temperatures, the electronic contribution to TEP is found to be linear in
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Acoustic Phonons | [71] | |
Optical Phonons | [66] | |
Impurities | [72] | |
Surface roughness | [66] | |
Vacancies | [72] | |
|
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Acoustic Phonons (In-plane) |
[28] | |
Acoustic Phonons (Flexural) |
[28] | |
Acoustic Phonons (Flexural) (strained) |
[28] | |
|
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Acoustic phonons | [73] | |
Surface polar optical phonons | [74] | |
Coulomb impurities | [75] | |
Short range disorder | [75] | |
|
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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 electron-phonon 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 phonon-limited 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. Phonon-drag thermopower in graphene systems
As mentioned in section 2, in the presence of temperature gradient
The formal theory of
where
The phonon-drag 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. Phonon-drag thermopower in SLG
Kubakaddi has studied in detail the low-temperature behavior of
where γ =
Taking account of boundary scattering as well as phonon-phonon 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. Phonon-drag 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. Phonon-drag 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, 88-90, 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 quasi-one-dimensionality, temperature, Fermi energy and ribbon width on Sg of a semiconducting n-type AGNR is investigated by Bhargavi and Kubakaddi [100]. The Q1D electrons are assumed to interact,
where, the various quantities are already defined. In the low-T 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 phonon-drag 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 graphene-based 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 low-temperature (say,
Conventional low-dimensional 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.
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