Duan Wenye, Liu Junfeng, Zhang Chao, Ma Zhongshui. The magneto-thermoelectric effect of graphene with intra-valley scattering. Chinese Physics B, 2018, 27(9): 097204
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The magneto-thermoelectric effect of graphene with intra-valley scattering
Duan Wenye1, 2, †, Liu Junfeng3, Zhang Chao2, Ma Zhongshui1, 4
School of Physics, Peking University, Beijing 100871, China
School of Physics, University of Wollongong, New South Wales 2522, Australia
Department of Physics, South University of Science and Technology of China, Shenzhen 518055, China
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
† Corresponding author. E-mail: duanwy@pku.edu.cn
Project supported by the National Natural Science Foundation of China (Grant Nos. 11274013 and 11774006), the National Basic Research Program of China (2012CB921300), and the Australian Research Council Grant (Grant No. DP160101474).
Abstract
We present a qualitative and quantitative study of the magneto-thermoelectric effect of graphene. In the limit of impurity scattering length being much longer than the lattice constant, the intra-valley scattering dominates the charge and thermal transport. The self-energy and the Green’s functions are calculated in the self-consistent Born approximation. It is found that the longitudinal thermal conductivity splits into double peaks at high Landau levels and exhibits oscillations which are out of phase with the electric conductivity. The chemical potential-dependent electrical resistivity, the thermal conductivities, the Seebeck coefficient, and the Nernst coefficient are obtained. The results are in good agreement with the experimental observations.
Graphene, a single-atom-thick honeycomb-like allotrope of carbon, has attracted intense interest since it was discovered in 2004.[1] On the basis of its potential applications in electronic devices, the electromagnetic responses in graphene have been investigated. Its peculiar band structure, described by the effective massless Dirac Hamiltonian at low-energy,[2–5] results in many novel features in the electrical transport under magnetic field.[6–14] It has been shown that graphene has many unusual thermoelectric and thermal transport properties.[15–38] Because of the sensitivity to the ambipolar behavior of graphene, the measurement of thermoelectric transport properties is regarded as an effective tool to deal with the additional characteristics of graphene, which are difficult to obtain from the electric conducting measurements. The Seebeck coefficient and Nernst signal in graphene have been measured in the quantum Hall (QH) regime,[15–21] in which the Seebeck coefficient displays a large peak and the Nernst signal displays a large oscillation at each Landau level (LL). The measurement of the Seebeck coefficient exhibits an abnormal sign change around the zeroth LL in comparison with other LLs.[15–17,19] Later, extensions of the intrinsic thermal properties have been made towards the investigations of the thermoelectric effect in bilayer graphene.[39–41] There have been several theories put forward to explain the Seebeck and Nernst effects of graphene in the QH regime.[42–50] Some of the observed thermoelectric properties are not yet fully understood.
In this work, we perform a systematic study on the thermoelectric effect and thermal transport in the magnetically quantized graphene. We consider the scattering range of impurity to be much smaller than the electron wavelength and much longer than the lattice constant. In this limit, the inter-valley scattering is much weaker than the intra-valley scattering. By employing the self-consistent Born approximation (SCBA), the Green’s function and the self-energy are determined. Based on the Green’s functions, all other coefficients of electrical, thermoelectrical, and thermal transport are calculated within the linear response theory. Although the thermal transport properties may be influenced by the acoustic and the optical phonons through inelastic electron–phonon scattering at high temperature, the electric contribution of graphene dominates over the phonon contribution at low temperatures. The temperature in the experiments has been confirmed below 50 K.[51–53] Therefore, we neglect the effect of phonon on the graphene and concentrate on the influence of the impurity scattering in this work. The calculated results are in good agreement with the experimental findings.
The article is organized as follows. In the next section, we present the Hamiltonian describing the electrons under a strong magnetic field. The intra-valley interaction potential for the impurity scattering is introduced. The Green’s function with a self-energy correction from the intra-valley impurity scattering is calculated within the SCBA. In Section 3, we present the general relations between the thermoelectric transport coefficients. In Section 4, we calculate the thermoelectric response coefficients. The results are compared with the experimental measurements. A summary is given in Section 5.
2. Hamiltonian and Green’s function in SCBA
2.1. Graphene in presence of a magnetic field
Utilizing the k · p method or with the effective-mass approximation,[54–58] the nearest-neighbor tight-binding model of graphene without the impurity scattering leads to an effective Hamiltonian
where vF is the Fermi velocity. In this expression, σx, σy, and σz are the Pauli matrices describing the pseudo spins, τx, τy, and τz are the Pauli matrices describing the valleys K and K′ in the reciprocal space, and σ0 and τ0 are 2 × 2 identity matrices. In the presence of a uniform magnetic field in the direction perpendicular to the graphene sheet, p is replaced by the mechanical momentum π = p + eA. In the Landau gauge, the vector potential takes the form A = (0,Bx,0) (we use the SI units and electron charge is −e (e > 0)). We shall neglect the Zeeman energy since it is much smaller than the LL spacing. The wave function should be written in a basis
where stands for the transpose, ψ(A/B)(K/K′) are the components of the spinors, and A and B represent the two inequivalent carbon atoms in the honeycomb hexagon lattice of graphene. The states can be obtained by solving the eigen equation . The eigenvalues are
and the eigenstates are given by ,
with
where Cn = 1 for n = 0 and for n ≠ 0, is the sign function,
n is an integer number, Hn(x) is the Hermite polynomial, ky is the wave vector along the y direction, , , and η = ± 1 for the K (+) and K′ (−) valleys, respectively.
2.2. Intra-valley impurity scattering
Now we consider the impurity scattering. Utilizing the k · p method or with the effective-mass approximation,[54–60] the effective potential with impurity scatters at (ζ = A,B) is given by
where . If the effective range of the impurity potential is much smaller than the typical electron wavelength, (which is the case near the Dirac point), the r-dependent potentials can be approximated by δ(r − ri). On the other hand, if the effective range of the impurity potential is larger than the lattice constant, the inter-valley scattering between K and K′ valleys can be neglected.[56,58–60] This argument was used to classify the intra-valley potentials as types of long-range potentials by Ando et al.[56,60] The scattering potential is written as
where ri is the impurity position and ui is the intensity of the scattering. The intra-valley scattering forbids the scattering between the states in K and K′ points. Including the scattering potential in Eq. (7), the full Hamiltonian is given by , where .
2.3. Green’s function with the self-energy corrections in SCBA
The Green’s function is defined by . The matrix elements of the unperturbed Green’s function in the representation of the LL basis are diagonal, i.e., , where
specified in a set of quantum numbers α = (η, n, ky).
The electron–impurity scattering is studied through the impurity-averaged Green’s function within the SCBA. The disorder term Eq. (7) is treated perturbatively with respect to the impurity-free Green’s function . Averaging over all possible configurations of random distributions of impurities, the matrix elements of G(ε) are given by Dyson’s equation
where the self-energy Σαα′ (ε) is introduced within the SCBA,
and . The average Green’s function and the self-energy can be derived self-consistently for random impurities ⟨(ui)2⟩ = u2. The concentration of impurities is given by nim. The self-energy can be decomposed into a diagonal part and an off-diagonal part , i.e.,
where ± α = (η, ± n, ky) is a composite quantum number. The off-diagonal elements between the LL indices +n and −n are non vanishing. The self-energy terms in the above equation can be written as
where the dimensionless parameter is a measure of the intensity of scattering within the SCBA.
The impurity-averaged Green’s functions are reduced to the form
where
The scattering matrix elements between different quantum states are given by
where
N = min(|n| − 1, |n′ |−1), M = max(|n| −1, |n′| −1), and is the associated Laguerre polynomial.
3. Linear response theory and transport coefficients
Within the linear response theory, under an applied field and a temperature gradient, the electrical current jC and energy current jE can be written in the general form
where the external force for the temperature gradient is expressed in the form XE = T∇(1/T). The coefficients , , , and are general tensors. They can be expressed in terms of the current–current correlations together with the corrections from the magnetization , ,[61–64] with the subscripts i and j ( = C or E) standing for the charge and the energy currents in the current–current correlations. Utilizing the Kubo–Strěda formalism, the current–current correlation is given by
while the corrections due to the magnetization are ,
where a and b denote the spatial components (x or y), , ( = −eva) and () are the operators of charge current and energy flux, respectively, fF (ε) = 1/(1 + e(ε − μ)/kBT) is the equilibrium Fermi–Dirac distribution function, L2 is the area of the system, and the velocity operators . The physical meanings of the coefficients in Eqs. (18) and (19) are as follows: is the electrical conductivity, is the thermopower, is the thermal conductivity, and the thermoelectric figure of merit . The thermopower and the thermal conductivity are obtained under the condition of zero electric current.
The electrical conductivity is given by and can be written in the form
and can be obtained from the Green’s functions by the following formula:
where
gs is the degeneracy of spin. Applying the relations and dG(ε ± i0)/d ε = −[G(ε ± i0)]2, can be rewritten as
where
The electrical conductivity is expressed as
with . In the calculations of σab (0,μ), we need to use the matrix elements of the velocity operators in the representation of the LL basis
and the vortex correction in the SCBA
Similar with the conductivity, it has been shown that and can be obtained in terms of σab(0,ε) through the following relations:[61,63,65]
4. The magneto-transport coefficients of graphene
The thermoelectric and thermal response coefficients can be obtained by Eqs. (29), (33), and (34). In the following subsection, we will evaluate the conductivity tensor by the Green’s functions within the SCBA.
4.1. Analytical formula for calculating the conductivity
4.1.1. The longitudinal conductivity
The longitudinal conductivity is σxx. Since (rxvx − rxvx) = 0, therefore equation (28) shows . The longitudinal conductivity is , where
with
Taking account of the impurity scattering within SCBA, it is found that
where
and . We define as , where . The self-energy can be obtained by self-consistently solving the equation
Then the Green’s functions are obtained as , so that .
4.1.2. The Hall conductivity
We now calculate the Hall conductivity σxy. is found as
where
We have .
is found as
where we have used the relation δ(ε − H) = (i/2π)[G(ε+0) − G(ε−0)]. can be written as
Under the Landau gauge, we have A = B(0,x,0) for the magnetic field along the z-axis B = (0,0,B). The terms in the trace can be written in the form . becomes . Because the particle number is found as , is expressed in terms of the particle number . Utilizing the density of states and , is found as
Finally, we obtain the Hall conductivity .
4.2. Thermoelectric coefficients
In the following, we present the numerical results of the thermoelectric coefficients as a function of the chemical potential.
4.2.1. Resistivities ρxx and ρxy
Using Eqs. (35), (39), and (42), we can obtain the conductivities σxx and σxy numerically. The longitudinal resistivity ρxx and the Hall resistivity ρxy are obtained by using the equations and .
The longitudinal resistivity ρxx and the Hall resistivity ρxy are depicted in Fig. 1(a) with several values of V. Considering that ρxx is symmetric and ρxy is antisymmetric with respect to μ = 0, only the n ≥ 0 LLs are shown in Fig. 1(a). As expected, ρxx displays the peaks at the LLs and vanishes between each consecutive LL, while ρxy exhibits the typical Hall plateaus. The longitudinal resistivity ρxx and Hall resistivity ρxy are closely related to the localization picture in the 2D integer quantum Hall effect (IQHE), i.e., the extended states are at the center of the broadened LLs and all the other states are localized. The fluctuation of impurity scattering removes the sharpness of the LLs so that the LLs are broadened as shown in Fig. 1(a). The zeroth LL (n = 0), is the charge neutrality point of graphene. When the chemical potential is varied through the zeroth LL, a zero-crossing of Hall resistivity ρxy changes smoothly through zero from its negative quantized value on the hole side to a positive quantized value on the electron side, whereas ρxx moves from a zero on the hole side through a maximum at the charge neutrality point to another zero on the electron side. Our numerical results of the resistivity agree with the observations in Kim’s experiment.[16] These results demonstrate that the system is in the Hall regime. The temperature dependence of the ρxx at the charge neutrality point displays an activated behavior. For the other LLs, the above picture does not change qualitatively. As the temperature increases, the peak values of ρxx decrease and the half widths increase for n > 0 LLs as shown in Fig. 2(a).
Fig. 1. (color online) (a) The longitudinal resistivity ρxx and Hall resistivity ρxy, (b) the Seebeck coefficient Sxx and Nernst coefficient Sxy, (c) the thermal conductivity κxx and thermal Hall conductivity κxy as a function of μ with different V at temperature T = 5 K.
Fig. 2. (color online) (a) The longitudinal resistivity ρxx and Hall resistivity ρxy, (b) the Seebeck coefficient Sxx and Nernst coefficient Sxy, (c) the thermal conductivity κxx and thermal Hall conductivity κxy as a function of μ at different temperatures with V = 0.001.
4.2.2. Seebeck coefficient Sxx and Nernst coefficient Sxy
The Seebeck coefficient Sxx and Nernst coefficient Sxy are presented in Figs. 1(b) and 2(b). Sxx is antisymmetric and Sxy is symmetric with respect to μ = 0. It is found that as a function of chemical potential, the Seebeck coefficient Sxx displays a series of peaks at the LLs for μ < 0 and dips at the LLs for μ > 0. |Sxx| has a maximum while the sign of Sxy is alternated around the LLs (n ≥ 1), corresponding to the extended states at the LLs. The appearance of zero thermoelectric response can be explained by the existence of the localized states in the gap between adjacent LLs, similar to the longitudinal conductivity in the IQHE. S is defined by , where . From this, we have Sxx = ρxxαxx − ρxyαxy and Sxy = ρxyαyy + ρxxαxy. ρaa and αab (a ≠ b) are symmetric functions of μ, while ρab and αaa are antisymmetric functions of μ. As the Fermi level passes through the core of extended states where the longitudinal resistance becomes appreciable and the Hall resistance makes its transition from one plateau step to the next. Their combination leads the oscillatory structure of Sxy in the vicinity of the LLs. As shown in Fig. 1(b), the broadening of the LLs due to the impurity scattering increases the width of the peaks or dips for Seebeck coefficient Sxx and oscillations of Nernst coefficients Sxy. As the temperature increases, Sxx increases and Sxy decreases for n ≥ 1 LLs as shown in Fig. 2(b).
The Seebeck and Nernst coefficients in the vicinity of the zeroth LL have an opposite behavior compared with those in the higher LLs. Instead of Sxx, the Nernst coefficient Sxy displays a large peak while the Seebeck coefficient Sxx now displays an alternate sign. The feature at n = 0 LL reflects the characteristic property of 2D relativistic fermions in the perpendicular magnetic field. Around the zeroth LL (n = 0), the Hall resistivity changes sign because the charges have opposite signs. As in the analysis above, the Seebeck coefficient from the edge states is larger when the chemical potential is below the LL, and the resistivity for the hole-like quasiparticles makes Sxx negative. When the chemical potential lies near the top of the lowest LL, i.e., in the electron side, Sxx behaves similarly as in the higher LLs. Hence, the Seebeck coefficient Sxx displays an alternate sign. Similarly, the Hall effect with the carrier flux driven by the temperature gradient predominates the Nernst coefficient when the chemical potential near the edge of the band lies below the LL. So Sxy is positive. Binding this with that in the electron side leads to a maximum of Sxy in the vicinity of the lowest LL. Experimentally, Sxx changes the sign from a dip to a peak with increasing μ around μ = 0. This discrepancy with our results is related to the lower ρxx value when μ is at the n = 0 LL, which is also observed in other theoretical calculations by different methods.[44,45] Apart from this, the Seebeck coefficent Sxx and the Nernst coefficient Sxy are in good agreement with the experimental results.[15–17]
4.2.3. Thermal conductivities κxx and κxy
Figures 1(c) and 2(c) show the thermal conductivities as a function of chemical potential μ for n ≥ 0 LL. The longitudinal component of thermal conductivity κxx is symmetric while the transverse component κxy is antisymmetric with respect to μ = 0. κxx shows a peak and κxy jumps to a new plateau when the chemical potential equals . These behaviors are similar to those for the conductivities (resistivities). However, the peaks of κxx at those higher LLs split into two peaks which also have been shown in Ref. [47] and these split double peaks smear to one peak with increasing impurity. In addition, the transverse thermal conductivity exhibits a sign reversing behavior when the chemical potential skims over the zeroth LL as shown in Fig. 2(c). The sign reversing in κxy implies the effects of thermally excited electrons and holes for the zeroth LL state. The thermal conductivity contributions arising from electrons and holes are not simply additive, but in combination they give rise to an additional contribution known as the bipolar effect. In order to check the validity of the Wiedemann–Franz law, the thermal conductivity calculated by κxx = L0σxxT (where ) is shown in Fig. 3 for comparison with the numerical results. We see that at the central region of the LL, the Wiedemann–Franz law is violated. As shown in Fig. 3(b), because of the split double-peak structure of κxx and the non-constant space between nearest LLs, the peaks of the nearest LL merge into one peak when n > 6, which is the origin of the out-of-phase oscillations with electrical conductivity in Ref. [43], even though the double peaks are clear when n < 6. At low temperatures, the breadth of the LLs is narrow and the double peak structure is still clear as shown in Fig. 3(a). It is found that the enhancement of impurity scattering results in enhanced κxx and suppressed κxy as shown in Fig. 1(c). This double-peak structure of longitudinal thermal conductivity in high LLs and the anomalous jump at the zeroth LL with a sign reversal in the transverse thermal conductivity have not yet been observed experimentally.
Fig. 3. (color online) The thermal conductivity κxx as a function of μ with V = 0.001 at (a) T = 5 K and (b) T = 10 K. The results calculated from the Wiedemann–Franz law by conductivity σxx are labeled by WFL in the legends.
From the thermal and electrical transport coefficients, the thermoelectric figure of merit ZT is calculated and shown in Fig. 4. ZT exhibits a resonant like structure. The magnetically quantized graphene is a very good thermoelectric material when the chemical potential is pinned to the first LLs. This excellent ZT can be the basis for expanding the thermoelectrical application of graphene.[66,67] For practical applications with sufficient power output, graphene with a high carrier concentration is required. This would require a large magnetic field to locate the chemical potential in the first LL. A trade-off between efficiency ZT and power needs to be considered to determine the optimal condition.
Fig. 4. (color online) The thermoelectric figure of merit as a function of μ (a) at T = 20 K with different V and (b) with V = 0.001 at different temperatures.
5. Summary
We have calculated the charge and thermal transport coefficients in magnetically quantized graphene in the presence of intra-valley impurity scattering. The Green’s function and the self-energy are calculated in SCBA. A detailed investigation of the electrical and thermal properties of graphene in a magnetic field is performed by means of the Kubo–Strěda formula in the presence of impurity scattering and thermal broadening. The results agree with the experimental ones qualitatively. The features of Seebeck coefficient and the transverse thermal conductivity at the zeroth LL are discussed and are associated with the intensity of impurity scattering. The sign reversal in the Seebeck coefficient and the transverse thermal conductivity imply the effects of thermally excited electrons and holes for the zeroth LL state. The impurity scattering may result in a bipolar effect from these thermally excited electrons and holes. Finally, our results of the thermoelectric figure of merit suggest that an external strong magnetic field could be used to effectively enhance the ZT.
