Phonon-drag magnetoquantum oscillations in graphene

TL;DR

This paper develops a theoretical model for phonon-drag magnetothermopower in graphene under magnetic fields, predicting quantum oscillations and large thermopower values, with comparisons to experimental data.

Contribution

It introduces a novel theory for phonon-drag thermopower in graphene in quantizing magnetic fields, highlighting quantum oscillations and temperature dependence.

Findings

Quantum oscillations of $S_{xx}^g$ as a function of magnetic field and electron density.

Amplitude of oscillations varies with magnetic field and electron density.

Predicted large thermopower values of a few hundred microvolts per Kelvin.

Abstract

A theory of low-temperature phonon-drag magnetothermopower $S_{xx}^g$ is presented in graphene in a quantizing magnetic field. $S_{xx}^g$ is found to exhibit quantum oscillations as a function of magnetic field $B$ and electron concentration $n_e$. Amplitude of the oscillations is found to increase (decrease) with increasing $B$ ($n_e$). The behavior of $S_{xx}^g$ is also investigated as a function of temperature. A large value of $S_{xx}^g $($\sim$ few hundreds of $\mu $V/K) is predicted. Numerical values of $S_{xx}^g $ are compared with the measured magnetothermopower $S_{xx}$ and the diffusion component $S_{xx}^d$ from the modified Girvin-Jonson theory.

Equations

$$\mathbf{U}=A_0^{-1}\sum _{\mathbf{q}}{N}_{\mathbf{q}}^1\hslash {\omega }_{\mathbf{q}}{v}_{\mathbf{q}},$$

$$\displaystyle N_{\bf q}^{1}=\frac{g}{k_{B}T}\sum_{\alpha,\alpha^{\prime}}\tau_{q}\Gamma_{\alpha,\alpha^{\prime}}({\bf q})\Big{[}\frac{f_{\alpha}^{1}}{(\partial f_{\alpha}^{0}/\partial E_{\alpha})}-\frac{f_{\alpha^{\prime}}^{1}}{(\partial f_{\alpha^{\prime}}^{0}/\partial E_{\alpha^{\prime}})}\Big{]},$$

$${\Gamma }_{\alpha ,{\alpha }^{\prime }}(\mathbf{q})=P_{\alpha ,{\alpha }^{\prime }}^{\mathrm{ab}0}(\mathbf{q})f_{\alpha }^0(1-f_{{\alpha }^{\prime }}^0),$$

$$P_{\alpha ,{\alpha }^{\prime }}^{\mathrm{ab}0}(\mathbf{q})=\frac{2\pi }{\hslash }|C_{\alpha ,{\alpha }^{\prime }}(\mathbf{q}){|}^2{N}_{\mathbf{q}}^0\delta (E_{{\alpha }^{\prime }}-E_{\alpha }-\hslash {\omega }_{\mathbf{q}}).$$

$$|C_{\alpha ,{\alpha }^{\prime }}(\mathbf{q}){|}^2=|C(\mathbf{q}){|}^2|J_{n,n^{\prime }}(u){|}^2{\delta }_{k_y^{\prime },k_y+q_y},$$

$$N_q^1=\frac{geE{l}_0^2}{k_{B}T}\sum _{\alpha {\alpha }^{\prime }}{\tau }_q{\Gamma }_{\alpha {\alpha }^{\prime }}(\mathbf{q})q_y.$$

$$M_{xx}=\frac{ge{l}_0^2{v}_s}{A_0{k}_{B}T}\sum _{\mathbf{q}}{\tau }_q\Gamma (\mathbf{q})(q_x{q}_y/q)\hslash {\omega }_{\mathbf{q}}$$

$$M_{yx}=\frac{ge{l}_0^2{v}_s}{A_0{k}_{B}T}\sum _{\mathbf{q}}{\tau }_q\Gamma (\mathbf{q})(q_y^2/q)\hslash {\omega }_{\mathbf{q}},$$

$$\Gamma (\mathbf{q})=\sum _{\alpha {\alpha }^{\prime }}{\Gamma }_{\alpha {\alpha }^{\prime }}(\mathbf{q}).$$

$$\Gamma (\mathbf{q})$$

$$\Gamma (\mathbf{q})=\frac{A_0}{\hslash l_0^2}\sum _n\sum _{n^{\prime }}|C(\mathbf{q}){|}^2|J_{n,n^{\prime }}(u){|}^2{N}_{\mathbf{q}}^0{I}_{n{n}^{\prime }}(\hslash {\omega }_{\mathbf{q}}),$$

$$I_{n{n}^{\prime }}(\hslash {\omega }_{\mathbf{q}})=\int dϵf^0(ϵ)[1-f^0(ϵ+\hslash {\omega }_{\mathbf{q}})]\rho (ϵ-E_n^0)\rho (ϵ+\hslash {\omega }_{\mathbf{q}}-E_{n^{\prime }}^0).$$

$$|J_{nn}(u)|^{2}=\frac{e^{-u}}{4}\Big{[}L_{n}(u)+L_{n-1}(u)\Big{]}^{2},$$

$$\Gamma (\mathbf{q})=\frac{A_0}{\hslash l_0^2}|C(\mathbf{q}){|}^2{N}_{\mathbf{q}}^0\sum _n|J_{nn}(u){|}^2{I}_{nn}(\hslash {\omega }_{\mathbf{q}}).$$

$$M_{yx}$$

$$\sum _{\mathbf{q}}\to \frac{A_0}{(2\pi {)}^2}{\int }_0^{\infty }qdq{\int }_0^{2\pi }d\theta .$$

$$M_{yx}$$

$$S_{xx}^g$$

$${\psi }_{\alpha }(\mathbf{r})=\frac{e^{i{k}_{y}y}}{\sqrt{L_y}}{\chi }_{n,k_y}(x)$$

$$\displaystyle\chi_{n,k_{y}}(x)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-i\phi_{n-1}(x)\\ \phi_{n}(x)\end{array}\right).$$

$$\displaystyle H_{ep}({\bf r})=\sum_{{\bf q}s}\Big{[}V_{{\bf q}s}e^{i{\bf q}\cdot{\bf r}}a_{{\bf q}s}+V^{\dagger}_{{\bf q}s}e^{-i{\bf q}\cdot{\bf r}}a^{\dagger}_{{\bf q}s}\Big{]},$$

$$C_{\alpha ,{\alpha }^{\prime }}(q)=\int {\psi }_{{\alpha }^{\prime }}^{†}(\mathbf{r})V_q{e}^{i\mathbf{q}\cdot \mathbf{r}}{\psi }_{\alpha }(\mathbf{r})d^{2}r.$$

$$C_{k_y^{\prime },k_y}=\frac{V_q}{L_y}{\int }_0^{L_y}{e}^{-i(k_y^{\prime }-k_y-q_y)y}dy=C(q){\delta }_{k_y^{\prime },k_y+q_y}$$

$$J_{n,n^{\prime }}(u)={\int }_{-\infty }^{\infty }{\chi }_{n^{\prime },k_y^{\prime }}^{†}(x)e^{i{q}_{x}x}{\chi }_{n,k_y}(x)dx.$$

$$C_{\alpha ,{\alpha }^{\prime }}(q)=C(q)J_{nn}(u){\delta }_{k_y^{\prime },k_y+q_y}$$

$$\displaystyle J_{nn}(u)=\frac{1}{2}e^{iq_{x}x_{0}}e^{-\frac{u}{2}}\Big{[}L_{n-1}(u)+L_{n}(u)\Big{]}$$

Full text

I Introduction

Graphene, a monolayer of carbon atoms arranged in a honeycomb lattice of hexagons, has a unique band structure. Its electronic states, at the points ${\bf K}$ and ${\bf K}^{\prime}$ of the Brillouin zone, have a linear dispersion relation, described by the Dirac equation. It is an ambipolar material with zero effective mass of the carriers and zero energy gap. Its electrical transport properties have been studied extensively neto ; sarma , since its discovery novo ; novo1 ; kim , with host of intriguing phenomena due to its unusual band structure. In a quantizing magnetic field ${\bf B}$, graphene exhibits the integer quantum Hall effect (QHE) novo1 ; kim , with novel features different from those in conventional two-dimensional electron gas (2DEG), particularly at $n=0$ Landau level (LL).

Thermopower ${\bf S}$, an electric field ${\bf E}$ generated in a sample due to unit temperature gradient i.e. ${\bf S}={\bf E}/(-{\bm{\nabla}}T)$, has been another powerful tool for probing carrier transport. Application of magnetic field ${\bf B}$, in addition to a temperature gradient ${\bm{\nabla}}T$, provides valuable experimental tool to investigate magnetothermoelectric effects. In a 2DEG, in the $xy$ plane, a temperature gradient ${\bm{\nabla}}T\parallel x$-axis and magnetic field ${\bm{B}}\parallel z$-axis generates electric field ${\bf E}$, in the sample, with components $E_{x}=S_{xx}(-{\bm{\nabla}}T)_{x}$ and $E_{y}=S_{yx}(-{\bm{\nabla}}T)_{x}$, where the thermopower $S_{xx}$ and the Nernst-Ettingshausen coefficient $S_{yx}$ are the tensor components of ${\bf S}$. Quantization effects due to magnetic field are reflected in thermopower.

In conventional 2DEG of GaAs heterojunctions (HJs) and Si-MOSFETs, magnetothermopower is investigated in detail, experimentally and theoretically, in the quantum Hall regime but ; flet ; flet-2003 ; tsa . Measured magnetothermopower tensor components $S_{xx}$ and $S_{yx}$ exhibited oscillations as a function of magnetic field, arising due to crossing of Landau level by the Fermi level due to either change in carrier concentration or magnetic field. There are two additive and independent contributions to thermopower ${\bf S}$. In a ${\bm{\nabla}}T$, diffusion component ${\bf S}^{d}$ arises due to diffusion of carriers and the phonon drag component ${\bf S}^{g}$ arises due to the non-equilibrium phonons transferring some of their momentum to the electrons via electron-phonon (el-ph) scattering. The oscillatory behavior of diffusion component is explained by the theory of Jonson and Girvin jonson and Oji oji . It is established that for about $0.1<T<10$ K, the $S^{g}$ contribution dominates $S$ in GaAs HJsbut ; flet ; flet-2003 . $S^{g}$ is important because it gives directly el-ph coupling and is independent of impurity scattering, unlike mobility.

The study of phonon-drag magnetothermopower $S^{g}$, in conventional 2DEG, began with pioneering experimental work of Fletcher et al flet-1986 showing the quantum oscillations as a function of $B$. Its $B$ and $T$ dependence were explained by developing the theory of $S^{g}$ kuba ; lyo ; from , by modifying the Boltzmann theory of phonon-drag in bulk semiconductors ger ; puri , following the $\Pi$-approach due to Herring herring .

In graphene, the experimental and theoretical investigations of thermoelectric effects in zero and quantizing magnetic field are being intensively pursued sank . In monolayer graphene, experimental data of $S$ vs $T$, in zero magnetic field, in temperature regime 10-300 K show largely linear behavior suggesting that the mechanism for thermopower is diffusive zuev . Thermopower measurements in quantum Hall regime are carried out as a function of magnetic field for different gate voltage (i.e. for different carrier concentration) and temperatures zuev ; check ; wei ; wu . Magnetothermopower $S_{xx}$ and Nernst-Ettingshausen coefficient $S_{yx}$ have shown the oscillatory behavior as a function of magnetic field. The behavior of $S_{xx}$ and $S_{yx}$ are in agreement with the generalized Mott relation, extending the theory of Jonson and Girvin jonson to graphene zuev . The peak values of $S_{xx}$ are predicted to be given by $S_{xx}^{\rm peak}=(-k_{B}/e)\ln 2/n$, noting that Jonson-Girvin theory fails for Landau level $n=0$. Similar observations are made in high mobility samples of graphene wu . Zero-field non-linear $T$ dependence is attributed to the screening das-sarma . Measured strong quantum oscillations as a function of $B$ are understood by evolution of the density of states at the Fermi level and $S_{xx}$ becoming zero when the Fermi level lies in the localized states, because of absence of diffusion wu . In all these measurements, it is observed that phonon-drag thermopower component is absent and no evidence of phonon-drag magnetoquantum oscillations, even at low temperatures, attributing to the weak el-ph coupling. We believe that, equally important reason for the absence of phonon-drag component in these samples may be due to their small size ($\sim 300$ nm). It is about $10^{4}$ times smaller than the samples of GaAs HJs ($\sim$ mm in which $S_{xx}$ is large and about $\sim$ mV/K) but . At low temperatures, the smaller dimension of the sample sets the limit for phonon mean free path $\Lambda$, in the boundary scattering regime, as $S^{g}\sim\Lambda$. We expect the phonon-drag to be significant in large samples (few $\mu$m) for e.g. in the samples of Nika et al nika . Moreover, to know the significant contribution of phonon-drag contribution, more data of $S$ is required at low temperature covering sub-Kelvin region in pure samples.

The theory kuba-2009 of zero magnetic field $S^{g}$ has been developed in monolayer graphene, in the boundary scattering regime, as a function of temperature $T(\leq 10$ K) and electron concentration $n_{e}$ for the phonon mean free path $\Lambda\sim 10\mu$m (closer to the samples of Nika et al nika ). At about $10$ K, $S^{g}\sim$ 10 $\mu$V/K. This value is nearly same order of magnitude as that of predicted $S^{d}$ with the peak values few tens of $\mu$V/K, by the modified Jonson-Girvin formula. We have to note that, unlike $S^{g}$, the latter is independent of sample size.

It would be interesting to study the effects of magnetic field quantization on phonon-drag thermopower $S_{xx}^{g}$. In the present work, we theoretically investigate the phonon-drag magnetothermopower as a function of magnetic field $B$, electron concentration $n_{e}$, and temperature $T$. We explore the circumstances and possibilities of its significant contribution to the measured magnetothermopower, by tuning the parameters $B$, $n_{e}$, $T$, and $\Lambda$. For comparison, we also compute diffusion component $S_{xx}^{d}$. The qualitative comparison of our calculations is made with the experimental observations.

This paper is organized as follows. In Sec. II, we provide formalism of phonon-drag thermopower in presence of quantizing magnetic field. In Sec. III, we present our results and discussion. A summary of our work is provided in Sec. IV.

II Formalism of phonon-drag magnetothermopower

In the following we proceed with the calculations by appropriately modifying the theory of Fromhold et al from for the monolayer graphene. We consider an isotropic and homogeneous 2DEG of graphene in the $xy$-plane with the magnetic field ${\bf B}=(0,0,B)$ along the $z$-direction. In presence of an electric field ${\bf E}$ (along $x$-axis) electrons are assumed to be accelerated isothermally (${\bm{\nabla}}T=0$). In the steady state, the non-equilibrium distribution of the electrons in state $\alpha$ is given by $f_{\alpha}=f_{\alpha}^{0}+f_{\alpha}^{1}$, where $f_{\alpha}^{0}=[\exp\{(E_{\alpha}^{0}-\mu_{F})/k_{B}T\}+1]^{-1}$ is the thermal equilibrium distribution function in absence of the electric field for state $\alpha$, $\mu_{F}$ is the chemical potential and $f_{\alpha}^{1}$ is the first-order perturbation due to electric field $E$. These non-equilibrium electrons transfer some of their momentum to the 2D phonons through the el-ph coupling. This causes perturbation in the phonon distribution which is given by $N_{\bf q}=N_{\bf q}^{0}+N_{\bf q}^{1}$, where $N_{\bf q}^{0}=[\exp(\hbar\omega_{\bf q}/k_{B}T)-1]^{-1}$ is the equilibrium distribution of the phonons of energy $\hbar\omega_{\bf q}$ and the wave vector ${\bf q}$. Here, $N_{\bf q}^{1}$ is the perturbation in the phonon distribution, due to electric field, producing the heat current density ${\bf U}$.

We confine our attention to the linear transport regime at liquid helium temperature. Then it is necessary to consider only acoustic phonons, with the 2D character, which interact weakly with the 2D electrons. The phonon heat current density, noting that $N_{\bf q}^{0}$ will not contribute, is given by

[TABLE]

where $A_{0}=L_{x}L_{y}$ is the area of graphene sample and $v_{\bf q}$ is the phonon group velocity.

In the linear response regime ($N_{\bf q}^{1}\propto E$), the heat current density is given by ${\bf U}={\bf ME}={\bm{\Pi}}{\bf J}$, where ${\bf M}$ is the thermoelectric tensor, ${\bm{\Pi}}={\bf M}/{\bm{\sigma}}$ is the Peltier coefficient tensor and ${\bm{\sigma}}$ is the electrical conductivity tensor. From the Onsager relation, the thermopower tensor is defined as ${\bf S}={\bm{\Pi}}/T$. Using the Onsager symmetry relations it has been shown that from $TS_{xx}=\rho_{xx}M_{xx}-\rho_{yx}M_{yx}$ and $TS_{yx}=\rho_{yx}M_{xx}+\rho_{xx}M_{yx}$, where $S_{xx}$ and $S_{yx}$ are, respectively, thermopower and Nernst-Ettinshausen coefficient, $M_{xx}$ and $M_{yx}$ are the components of tensor ${\bf M}$ and $\rho_{xx}$ and $\rho_{yx}$ are the components of electrical resistivity tensor ${\bm{\rho}}$. Hence, the calculation of ${\bf U}$ will facilitate the calculation of ${\bf M}$ and hence $S_{xx}$ and $S_{yx}$.

The solution for $N_{\bf q}^{1}$, in the linear response regime, is found to be

[TABLE]

where $g=g_{s}g_{v}$, $g_{s}(g_{v})$ is the spin (valley) degeneracy, and

[TABLE]

Here, $P_{\alpha,\alpha^{\prime}}^{{\rm ab}0}({\bf q})$ is the transition probability, in equilibrium, for the electron scattering from state $\alpha$ to state $\alpha^{\prime}$ by absorbing a phonon and $|C_{\alpha,\alpha^{\prime}}({\bf q})|^{2}$ is the square of the electron-acoustic phonon interaction matrix element. Equation (2) is the general to the extent that, it is independent of the electronic structure and the type of el-ph coupling.

In a quantizing magnetic field ${\bf B}=(0,0,B)$ with the Landau gauge ${\bf A}=(0,Bx,0)$, the eigenfunctions and energy eigen values are given in Ref. [matu, ]. The energy eigen values are $E_{\alpha}^{0}=E_{n,k_{y}}^{0}=\hbar\omega_{c}\sqrt{2n}$, where $\alpha\equiv(n,k_{y}),n=0,1,2...$ is the Landau level quantum number, $k_{y}$ is the electron wave vector in the $y$-direction, $\omega_{c}=v_{F}/l_{0}$ is the cyclotron frequency, $v_{F}=1\times 10^{6}$ m/s is the Fermi velocity of electron in graphene, and $l_{0}=\sqrt{\hbar/(eB)}$ is the magnetic length. The el-ph matrix element is given by (see Appendix A for details)

[TABLE]

where $|C({\bf q})|^{2}$ is the matrix which describes the el-ph coupling strength and $|J_{n,n^{\prime}}(u)|$ with $u=q^{2}l_{0}^{2}/2$, is the matrix element describing the scattering between Landau levels.

In presence of crossed electric field ${\bf E}=(E,0,0)$ and magnetic field ${\bf B}=(0,0,B)$, the energy spectrum of graphene can be found exactly baskaran ; peres . In the linear response regime, where the applied electric field is low enough, one can obtain energy eigen value for the magnetic state $\alpha$, approximately, as $E_{\alpha}\simeq E_{\alpha}^{0}+eEx_{\alpha}$ by expanding the exact expression given in Refs. [baskaran, ; peres, ] up to first-order in $E/(v_{F}B)$. Here, $x_{\alpha}=k_{y}l_{0}^{2}$. This is nothing but the first-order energy correction due to the week external electric field. Further, assuming that the form of distribution function retains the same with the modified energy, we expand $f_{\alpha}=f_{0}(E_{\alpha})=f_{0}(E_{\alpha}^{0})-eEx_{\alpha}[\partial f(E_{\alpha}^{0})/\partial E_{\alpha}^{0}](=f_{0}(E_{\alpha}^{0})+f_{\alpha}^{1})$, which gives $f_{\alpha}^{1}/[\partial f(E_{\alpha}^{0})/\partial E_{\alpha}^{0}]=-eEx_{\alpha}$. Then, Eq. (2), using the momentum conservation $k_{y}^{\prime}=k_{y}+q_{y}$, for the chosen Landau gauge gives,

[TABLE]

In order to make $N_{q}^{1}$ linear in $E$, we set all terms in $\Gamma_{\alpha\alpha^{\prime}}(q)$ independent of $E$. Inserting Eq. (6) into Eq. (1) we write ${\bf U}={\bf ME}$ and take the phonon group velocity components $v_{q}^{x}=(q_{x}/q)v_{s}$ and $v_{q}^{y}=(q_{y}/q)v_{s}$, $v_{s}$ being the acoustic phonon velocity in graphene. Then, the two components of the thermoelectric tensor ${\bf M}$ are given by

[TABLE]

where

[TABLE]

When we carry out the angular integration, it can be seen that $M_{xx}=0$ because of the $xy$ isotropy. However, this is shown as the limitation of this theory as the experimental results of $M_{xx}$ in conventional 2DEG show its non-zero value from . Later calculations in these systems, taking into account of anisotropy of electrons and phonons, remove this limitation but-1998 . However, in the present work we undertake the evaluation of only $M_{yx}$, as we have considered $xy$ isotropy of the system.

Using Eqs. (3) and (4), Eq. (9) turns out to be

[TABLE]

Summation of over $k_{y}^{\prime}$ is carried out replacing it by $k_{y}+q_{y}$. Since the integrand is independent of $k_{y}$, summation over $k_{y}$ simply gives $A_{0}/2\pi l_{0}^{2}$.

In the presence of disorder, the energy levels (in zero electric field) $E_{n,k_{y}}^{0}$ and $E_{n,k_{y}+q_{y}}^{0}$ are randomized by LL broadening. A simple system average is taken by integrating over $\epsilon=E_{n,k_{y}}^{0}$ and $\epsilon^{\prime}=E_{n,k_{y}+q_{y}}^{0}$ with the weight factor $\rho(\epsilon-E_{n}^{0})\rho(\epsilon^{\prime}-E_{n^{\prime}}^{0})$, where $E_{n}^{0}(E_{n^{\prime}}^{0})$ is the energy of the $n(n^{\prime})$-th LL in absence of disorder and $\rho(x)$ is the LL density of states with convenient line shape function. Now Eq. (10) becomes

[TABLE]

Integration with respect to $\epsilon^{\prime}$, using the Dirac delta function, gives

[TABLE]

where

[TABLE]

Since the phonon-drag thermopower is important at low temperature, the energy of acoustic phonons involved is so small that only intra LL scattering is possible. Thus we set $n=n^{\prime}$ in Eq. (12). Then, with

[TABLE]

Equation (12) gives

[TABLE]

Intra LL transitions are possible as the energy levels are broadened.

Using Eq. (15) in Eq. (8), we get

[TABLE]

The summation over ${\bf q}$ is converted into integration as

[TABLE]

Angular integration coming through $q_{y}=q\sin\theta$ gives $\pi$. Then

[TABLE]

Substituting for $|C(q)|^{2}=D^{2}\hbar\omega_{q}/(2\rho_{m}A_{0}v_{s}^{2})$, where $D$ is the acoustic phonon deformation potential coupling constant and $\rho_{m}$ is the areal mass density of graphene, we obtain

[TABLE]

From Onsager relation, with $M_{xx}=0$, we have $TS_{xx}^{g}=-\rho_{yx}M_{yx}$. Taking $\rho_{yx}=B/(n_{e}e)$tiwari , where $n_{e}$ is the electron density and expressing $B$ in terms of $l_{0}^{2}$, we get

[TABLE]

We note that this equation can also be obtained by following the method of Kubakaddi et al kuba for conventional 2DEG ignoring $\Gamma(q)$ compared to $1/\tau_{q}$.

III Results and Discussion

Since $S_{xx}^{g}\sim\Lambda$ and $D^{2}$, it is essential to choose the reasonable values of these parameters. We numerically evaluate $S_{xx}^{g}$, for $T\leq 20$ K in the boundary scattering regime for which $\tau_{q}=\Lambda/v_{s}$, where $\Lambda$ is the phonon mean free path. Normally, $\Lambda$ is taken to be the smaller dimension of the sample. Thermal conductivity calculations are demonstrated with $\Lambda$ chosen in the range of $3$-$30$ $\mu$m and the choice of $\Lambda=5$ $\mu$m is giving reasonable agreement with the measured thermal conductivity nika ; nika2 . Nika et alnika , to fit the thermal conductivity data, use the effective phonon mean free path $\Lambda_{\rm eff}=\Lambda(1+p)/(1-p)$ by modulating the smallest dimension of the sample using specular parameter $p=0.9$, which enhances $\Lambda$ by a factor of $20$. The value of $0\leq p\leq 1$ is determined by the roughness of the graphene edges. To present our calculations we choose a reasonable value of $\Lambda=10$ $\mu$m.

In the literature there is a range of $D=3$-$30$ eV. We chose $D=20$ eV which is closer to the values of $D$, for unscreened el-ph interaction, used to fit the experimental data of some of the transport propertiesbaker ; huang ; bist ; silva . The line shape function $\rho(x)$ of LL is taken to be Lorentzian with the width $\Gamma=C\sqrt{B}$, where $C=0.5$ meV/$\sqrt{\rm Tesla}$. Other parameter values used are: $\rho_{m}=7.6\times 10^{-7}$ Kg/m2 and $v_{s}=2\times 10^{4}$ m/s.

In Fig. 1, $S_{xx}^{g}$ is shown as a function of $B$, for $T=2,4,6,8,10$ K for $n_{e}=1\times 10^{16}$ m*-2*. We see that $S_{xx}^{g}$ is oscillatory with the height of the peak increasing with the increasing $B$. The position of the peak occurs when the Fermi energy is in the localized state of LL. Interestingly, our calculations show large peak values of the order of few hundreds of $\mu$V/K which is closer to the values observed in GaAs HJs flet-1986 ; tie . We would like to point out that the sample size in GaAs HJs ($\sim$ few mm) is about two orders of magnitude larger than the size of the graphene sample chosen here. The size of the graphene sample ($300$ nm) in the experiment of Zuev et al zuev is about $30$ times smaller than the value of $\Lambda$ used in the present calculation. Scaling the $\Lambda$ down by $30$ times, we get the peak values of $S_{xx}^{g}$ few tens of $\mu$V/K which is comparable to the measured values.

Dependence of $S_{xx}^{g}$ on $n_{e}$ is shown in Fig. 2 for three different magnetic fields, namely $B=2.82,3.88$, and $6.20$ Tesla, at $T=5$ K taking $\Lambda=10$ $\mu$m. Again, the behavior is found to be oscillatory. This is similar to the behavior observed (as function of gate voltage) in the experiment of Zuev et alzuev . The peak value of $S_{xx}^{g}$ is decreasing with the increasing $n_{e}$. This is similar to the $n_{e}$ dependence of zero field $S^{g}$ and $S^{d}$kuba-2009 . Also, it is found that the peak values are smaller for smaller $B$. The number of oscillations contained in $S_{xx}^{g}$ gets reduced with the increase of magnetic field. This is due to the increase of the separation between LLs with increasing magnetic field. Interestingly, we note that the position of the peak values corresponding to three different $B$ are coinciding at $n_{e}=(1,4,7$ and $10)\times 10^{16}$ m*-2*.

In Fig. 3(a), we have shown $S_{xx}^{g}$ as a function of $n_{e}$ for $\Lambda=300$ nm (as taken in Ref. [zuev, ]) at temperatures $T=4.2,10,$ and $20$ K for a magnetic field $B=8.8$ Tesla. For comparison, we have calculated diffusion component $S_{xx}^{d}$ as a function of $n_{e}$, for the same $T$ and $B$, using modified Girvin-Jonson theory (see Ref. [check, ]) and it is shown in Fig. 3(b). Note that $S_{xx}^{\rm d}$ is also found to decrease with increasing $n_{e}$. According to the Girvin-Jonson theory, the peak values due to diffusion component are given by $(k_{B}/e){\rm ln}2/n$. $S_{xx}^{d}$ is found to be much greater than $S_{xx}^{g}$. For example, for $n_{e}\sim 1.5\times 10^{12}$ cm*-2*, at 4.2 K (10 K) $S_{xx}^{d}$ is nearly ten (three) times greater than $S_{xx}^{g}$. The total $S_{xx}=S_{xx}^{d}+S_{xx}^{g}$ is shown as a function of $n_{e}$ in Fig. 3(c) and it is increasing with $T$.

We would like to point out that in graphene, the peak values of diffusion thermopower $S_{xx}^{d}$ are quantized as $(k_{B}/e){\rm ln}2/n$ which differs from the peak value quantization $(k_{B}/e){\rm ln}2/(n+1/2)$ corresponding to the conventional 2DEG. This difference is attributed to the existence of a non-trivial Berry phase $\pi$ in graphenecheck . Unlike diffusion thermopower, it is difficult to establish such peak value quantization for $S_{xx}^{g}$. However, a careful observation of Fig. 3(a) & 3(b) dictates us that $S_{xx}^{g}$ follow $S_{xx}^{d}$ with respect to the locations of peaks. Moreover, the locations of thermopower peaks in 2DEG and graphene are expected to be different due to different Landau level structures as was found in the case of conductivity oscillationsmatu .

In Fig 4, we have shown $S_{xx}^{g}$ as function of $T$ for $B=2.82,3.88$ and $6.20$ Tesla (corresponding to three peak values in Fig. 1). $S_{xx}^{g}$ increases with increasing $T$, more rapidly at lower $T$. At higher $T$, the increase is slower and showing nearly independent behavior for about $T>10$ K. This behavior is similar to the observations in conventional 2DEG flet-1986 ; kuba ; lyo . The faster increase of $S_{xx}^{g}$ with $T$, at low $T$, may be attributed to the increasing number of phonons linearly with $T$. For a given magnetic field, maximum momentum transfer takes place when $\hbar v_{s}q\simeq\Gamma$ setting limit on $q$. As $T$ increases further, the allowed $q$ is limited by the width of LL ($\Gamma\sim B^{1/2}$) and fewer phonons will exchange momentum. In zero magnetic field, such behavior is generally interpreted but ; flet-2003 in terms of the dominant phonon wave vector $q_{D}$ and Fermi wave vector $2k_{F}$. At a given $T$, $S_{xx}^{g}$ is larger for larger $B$. This is consistent with the findings in conventional 2DEGflet-1986 ; lyo .

Inset of Fig. 4, expressing $S_{xx}^{g}\sim T^{\gamma}$, shows the behavior of exponent $\gamma$ as a function of $T$ for different $B$. It is found to decrease and tending to zero with increasing $T$. Moreover, $\gamma$ is found to be larger for larger $B$. We observe that for $T$ closer to $1$ K, $\gamma$ is greater than $2$ which is signature of phonon-drag thermopower. When $S_{xx}^{g}$ is calculated as a function of $T$ (not shown in the figure ) for different $n_{e}$ we expect it to increase with increasing $T$ but to be smaller for larger $n_{e}$. These curves are expected to show again nearly independent behavior at higher $T$, more so for smaller $n_{e}$. Exponent $\gamma$, in this case $n_{e}$ dependent, is again expected to decrease with increasing $T$ .

Important point we notice is that $S_{xx}^{g}$ can be tailored to be as large as few mV/K by reducing $n_{e}$ and working at larger $B$. We suggest that the enhanced phonon drag contribution $S_{xx}^{g}$ can be achieved by polishing the edge of the sample. It is characterized by a specular parameter $p$ with its value $0<p<1$. The perfect reflecting edge gives $p=1$ and very rough edge corresponds to $p=0$ (diffusive scattering). Besides, the larger samples can be grown on piezoelectric substrates. Woszczyna et alwosz have shown that the graphene samples as large as $150\times 30$ $\mu$m2 on GaAs substrate can be prepared.

We would like to point out that the screening of el-ph coupling in magnetic field is ignored, although justification is for zero magnetic field case. In conventional 2DEG screening is found to reduce the phonon drag thermopower significantly both in zero and quantizing magnetic fieldbut ; flet ; flet-2003 ; tsa . However, screening of el-ph interaction in graphene in magnetic field is yet to be established. Low temperature experimental $S_{xx}^{g}$ may throw some light on significance of screening. One can extract the experimental phonon drag $S_{xx}^{g}$ from the experimentally measured $S_{xx}$ values by subtracting diffusion component using generalized Mott formulazuev .

IV summary

In summary, we have studied phonon-drag thermopower $S_{xx}^{g}$ in graphene subjected to a transverse magnetic field. Based on a method, described in Ref.[from, ], a modified theory is developed to calculate $S_{xx}^{g}$ quantitatively. Dependence of $S_{xx}^{g}$ on magnetic field, electron density, and temperature have been studied. With both magnetic field and density, $S_{xx}^{g}$ exhibits oscillatory behavior. Interestingly, we have found an enhanced phonon-drag thermopower with magnitude of the order of few hundreds $\mu$V/K. This value is closer to that obtained in the case of conventional 2DEG at GaAs based semiconductor hetero interface. We attribute this enhanced phonon-drag effect is a consequence of taking the high value of phonon-mean free path, namely, $\Lambda=10$ $\mu$m. We, thus, suggest that phonon-drag effect may have significant contribution in larger samples of graphene. We have also shown the density dependence of $S_{xx}^{g}$ for parameter values which were taken in Ref.[zuev, ]. The diffusion thermopower has also been calculated for the sake of comparison using modified Girvin-Jonson theory. Moreover, the temperature dependence of $S_{xx}^{g}$ is also studied and the exponent of this dependence has been extracted.

Acknowledgement

SSK would like to thank M. Tsaousidou and TKG would like to thank A. Kundu for useful discussions.

Appendix A

A.1 Matrix elements of Electron-phonon coupling in a magnetic field

For a graphene monolayer, lying in $xy$ plane, with a perpendicular magnetic field ${\bf B}=(0,0,B)$, the eigen functions, for Landau gauge ${\bf A}=(0,Bx,0)$, are given bymatu

[TABLE]

with

[TABLE]

Here, $\alpha\equiv(n,k_{y})$, $n=0,1,2,3,..$ is the Landau level index, $k_{y}$ is the $y$-component of electron wave vector, $\phi_{n}(x)=\sqrt{1/(2^{n}n!\sqrt{\pi}l_{0})}e^{-(x+x_{0})^{2}/(2l_{0}^{2})}H_{n}[(x+x_{0})/l_{0}]$ with $x_{0}=l_{0}^{2}k_{y}$ is the harmonic oscillator wave function.

We assume that at low temperature, for the graphene on the substrate, electrons interact with only in-plane acoustic phonons via deformation potential coupling. In suspended graphene, there will be flexural modes, whose contribution is neglected for the graphene on substrateOppen . The deformation potential coupling is assumed to be only due to longitudinal acoustic phonons.

The most general form of electron-phonon interaction Hamiltonian is

[TABLE]

where $a_{{\bf q}s}(a^{\dagger}_{{\bf q}s})$ is the phonon annihilation(creation) operator and $V_{{\bf q}s}$ is the matrix element of a particular phonon mode $({\bf q},s)$. For longitudinal acoustic phonon mode corresponding to deformation potential, the form of $V_{{\bf q}s}$ is given by V_{q}=D\big{[}\hbar\omega_{q}/(2A_{0}\rho_{m}v_{s}^{2})\big{]}^{1/2}, where $A_{0}$ is the area of graphene sample, $D$ is the deformation potential coupling constant, and $\rho_{m}$ is the areal mass density of graphene. The electron-acoustic phonon matrix element, for the scattering between the states $\alpha\equiv(n,k_{y})$ and $\alpha^{\prime}\equiv(n^{\prime},k_{y}^{\prime})$, is given by

[TABLE]

Substituting for $\psi_{\alpha}({\bf r})$ and $V_{\bf q}$, we get Eq. (5) in which the integrals are given by

[TABLE]

and

[TABLE]

At low temperature, the acoustic phonon energy is small and cause only intra-Landau level transitions $(n=n^{\prime})$. Inter-Landau level transitions are expected at higher temperatures and in the studies such as magnetophonon resonance in which optical phonons are involvedFalko . The matrix element corresponding to intra-Landau level transitions is found to be

[TABLE]

where

[TABLE]

with $u=q^{2}l_{0}^{2}/2$. The equation for $|J_{nn}(u)|^{2}$ given in Eq.(14) is similar to the one obtained in Refs.[Nomura, ; Kand, ].