Magnetothermoelectric transport properties in graphene superlattices with one-dimensional periodic potentials

Recently, the thermoelectric transport properties of graphene systems have attracted much experimental [1–4] and theoretical [5–12] attention. The thermopower (the longitudinal thermoelectric response) and the Nernst signal (the transverse response) of graphene in the presence of a strong magnetic field are found experimentally to be large, reaching the order of the quantum limit $k_B/e$, where k_B and e are the Boltzmann constant and the electron charge, respectively [1–3]. Theoretical calculations from the tight-binding model for graphene [11,12] are in agreement with the experimental observation. This makes the graphene-related materials more promising candidates for future thermoelectric applications.

More recently, much attention has been turned toward the transport properties of graphene superlattices with periodic potential [13–21]. In a graphene-based superlattice, a one-dimensional periodic potential may result in a strong anisotropy for the group velocities of the low-energy charge carriers, which are reduced to zero in one direction but are unchanged in another [16]. Interestingly, Brey et al. [17] have shown that such behavior of the anisotropy is a precursor to the formation of further Dirac points in the electronic band structures, and new zero-energy states are controlled by the potential amplitude and the superlattice period. Furthermore, the extra Dirac points and their associated zero-energy modes [17] drastically affect the transport properties [17–20] of the system and also the Landau level (LLs) sequence [21], and hence the plateaus in the quantum Hall conductivity when a magnetic field is applied. Owing to these interesting electronic behavior, graphene superlattices are expected to exhibit novel thermoelectric transport properties. However, theoretical studies of the thermoelectric transport properties of graphene superlattices have not been carried out so far. Such theoretical studies are highly desirable as they can provide theoretical understanding and guidance to the experimental research of the thermoelectric transport in such systems.

In this paper, we perform a numerical study of the thermoelectric transport properties in graphene systems in the presence of a one-dimensional superlattice potential. The effects of disorder and thermal activation on the broadening of the LLs and the corresponding thermoelectric transport properties are investigated. When the superlattice period length is the smallest, the thermoelectric coefficients exhibit unique behaviors at the central LL due to the coexistence of particle and hole LLs. Both the longitudinal and transverse thermoelectric conductivities are universal functions of the temperature and the disorder-induced LL width, and display different asymptotic behaviors in different temperature regimes. The Nernst signal displays a peak with a height of the order of $k_B/e$, and the thermopower changes sign at the central LL. With the increase of the superlattice period length, quite different behaviors near the central LL are observed. Around the Dirac point, the transverse thermoelectric conductivity exhibits a pronounced valley at low temperatures. This is attributed to the splitting of the central LL near zero energy, and hence the graphene superlattices behave as conventional insulators. We further examine the validity of the semiclassical Mott relation in the graphene superlattice systems.

This paper is organized as follows. In the next section, we introduce the model Hamiltonian. In the third section, numerical results based on exact diagonalization and thermoelectric transport calculations are presented for graphene superlattice systems. The final section contains a summary.

We assume that a graphene sheet has L_y zigzag chains in total, with L_x atomic sites on each chain [22]. We consider a one-dimensional periodic Kronig-Penney type of electrostatic potential with alternating potential barriers and wells along the armchair direction of the graphene sheet, as shown in fig. 1, which can be described by

$$\begin{equation} V_i = \left\{ \begin{array}{l@{\qquad}l} V_{0}, &\displaystyle0 \leq x_i \leq \frac{L}{2}, \\ -V_{0}, &\displaystyle\frac{L}{2} < x_i \leq L, \end{array} \right. \end{equation} \tag{ 1 }$$

where $L=2n\sqrt{3}a_0$ is the period of the potential. In the presence of an applied magnetic field perpendicular to the plane of graphene, the lattice model in real space can be written in the tight-binding form

$$\begin{equation} H=-\sum\limits_{\langle{ij}\rangle}te^{ia_{ij}}c_{i}^{\dagger}c_{j} +\sum\limits_{i} V_{i}c_{i}^{\dagger}c_{i}+\text{h.c.}+\sum\limits_{i}w_{i} c_{i}^{\dagger}c_{i}. \end{equation} \tag{ 2 }$$

Here, $c_{i}^{\dagger}\ (c_{i})$ is the fermion creation (annihilation) operator at site i, t is the hopping integral between the nearest-neighbor (next-nearest-neighbor) sites i and j, and V_i is the superlattice potential given by eq. (1). The spin degrees of freedom have been omitted due to degeneracy. Under the applied magnetic field B, the vector potential is introduced into eq. (2) via an additional phase factor a_ij, which is determined by the magnetic flux per hexagon $\varphi =\sum_{{\small \unicode{x2B21}}}a_{ij}=\frac{2\pi}{M}$. M is an integer proportional to the strength of the applied magnetic field B and the lattice constant is taken to be unity. We model charged impurities in substrate, randomly located in a plane at a distance d, either above or below the graphene sheet with a long-range Coulomb scattering potential [23–26]. For charged impurities, $w_i=-\frac{Ze^2}{\epsilon}\sum_{\alpha}1/\sqrt{(\textbf{r}_i- \textbf{R}_{\alpha})^2+d^2}$, where Ze is the charge carried by an impurity, is the effective background lattice dielectric constant, and $\textbf{r}_i$ and $\textbf{R}_{\alpha}$ are the planar positions of site i and impurity α, respectively. All the properties of the substrate can be absorbed into a dimensionless parameter $r_s= Ze^2/(\epsilon\hbar v_F)$, where v_F is the Fermi velocity of the electrons. $\hbar v_F=\frac{3}{2}ta$, where a is lattice constant [24]. For simplicity, in the following calculation, we fix the distance $d=1\ \text{nm}$ and impurity density as 1% of the total sites, and tune r_s to control the impurity scattering strength. The characteristic features of the calculated transport coefficients are insensitive to the details of the impurity scattering and choices of the parameters.

Zoom In Zoom Out Reset image size

In the linear response regime, the charge current in response to an electric field or a temperature gradient can be written as $\textbf{J} = {\hat\sigma}\textbf{E} + {\hat\alpha} (-\nabla T)$, where ${\hat\sigma}$ and ${\hat\alpha}$ are the electrical and thermoelectric conductivity tensors, respectively. The Hall conductivity $\sigma_{xy}$ can be calculated by Kubo formula and the longitudinal conductivity $\sigma_{xx}$ can be obtained based on the calculation of the Thouless number (see appendix) [27]. We exactly diagonalize the model Hamiltonian in the presence of disorder [28], and obtain the transport coefficients by using the energy spectra and wave functions. In practice, we can first calculate the electrical conductivity $\sigma_{ji}(E_F)$ at zero temperature, and then use the relation [29]

$$\begin{eqnarray} &&\begin{array}{l} \displaystyle\sigma_{ji}(E_F, T)=\int\text{d}\epsilon\,\sigma_{ji}(\epsilon)\left(-{\partial f (\epsilon)\over\partial\epsilon}\right),\\ \displaystyle\alpha_{ji}(E_F, T)={-1\over eT}\int\text{d}\epsilon\,\sigma_{ji}(\epsilon) (\epsilon-E_F)\left(-{\partial f(\epsilon)\over\partial\epsilon}\right), \end{array} \end{eqnarray} \tag{ 3 }$$

to obtain the electrical and thermoelectric conductivity at finite temperatures. Here, $f(x)=1/[e^{(x-E_F)/k_B T}+1]$ is the Fermi distribution function. At low temperatures, the second equation can be approximated as

$$\begin{equation} \alpha_{ji}(E_F,T) =-\frac{\pi^2k_B^2T}{3e}\left. \frac{\text{d}\sigma_{ji} (\epsilon,T)}{\text{d}\epsilon}\right|_{\epsilon =E_F}, \end{equation} \tag{ 4 }$$

which is the semiclassical Mott relation [29,30]. The validity of this relation will be examined for the present graphene superlattice system. The thermopower and Nernst signal can be calculated subsequently from [31,32]

$$\begin{eqnarray} &&\begin{array}{l} \displaystyle S_{xx}={E_x\over\nabla_x T}={\rho_{xx}\alpha_{xx}-\rho_{yx}\alpha_{yx}},\\ \displaystyle S_{xy}={E_y\over\nabla_x T}={\rho_{xx}\alpha_{yx}+\rho_{yx}\alpha_{xx}}. \end{array} \end{eqnarray} \tag{ 5 }$$

We first show the Hall conductivity $\sigma_{xy}$ as functions of electron Fermi energy E_F at zero temperature in fig. 2. For comparison, we also show the results of the graphene system in the absence of the superlattice potential. As shown in fig. 2(a), when no superlattice is present, the Hall conductivity exhibits a sequence of plateaus at $\sigma _{xy}=\nu e^2/h$, where $\nu =kg_{s}$ with k a nonzero integer, and $g_{s}=4$ due to the fourfold degeneracy (double-spin and double-valley) [33–36]. The transition from the $\nu =-2$ plateau to $\nu=2$ plateau is continuous without a $\nu=0$ plateau appearing in between, so that a step of height $4e^2/h$ occurs at the neutrality point. Moreover, the central LL shows a peak around the Dirac point, and the width of the peak increases with the increase of the disorder strength. However, when a superlattice potential is applied to graphene sheet, the Hall conductivity exhibits different quantization rules depending on the superlattice period length L. In fig. 2(b), we first show the result for $L=2\sqrt{3}a_0\ (n=1)$. As we can see, there is a splitting of the LL around $E_F=\pm 0.45t$, yielding a new plateau with ${\sigma_{xy}}=\pm 4,\pm 8$, in addition to the original integer plateaus. When the superlattice period length increases to $L=24\sqrt{3}a_0\ (n=12)$, there is a splitting of the central $(n = 0)$ LL, yielding a tiny new plateau with ${\sigma_{xy}}=0$, as shown in fig. 2(c). Due to this large difference in the Hall conductivity and the LLs in the two cases, the thermoelectric transport properties of graphene superlattices are expected to exhibit interesting characteristics.

Zoom In Zoom Out Reset image size

We now study the thermoelectric conductivities at finite temperatures. In fig. 3 and fig. 4, we find that the transverse thermoelectric conductivity $\alpha_{xy}$ shows different behavior depending on the superlattice period length. In fig. 3(a) and (b), we first plot the calculated $\alpha_{xy}$ for $L=2\sqrt{3}a_0\ (n=1)$. As shown in fig. 3(a) and (b), $\alpha_{xy}$ displays a series of peaks, while the longitudinal thermoelectric conductivity ${\alpha_{xx}}$ oscillates and changes sign at the center of each LL. At low temperatures, the peak of $\alpha_{xy}$ at the central LL is higher and narrower than others, which indicates that the impurity scattering has less effect on the central LL. These results are qualitatively similar to those found in monolayer graphene [11], but some obvious differences exist. More oscillations are observed in higher LLs than in the graphene case, consistently with the further lifting of the LL degeneracy in graphene superlattices. ${\alpha_{xx}}$ oscillates and changes sign around the center of each split LL. At low temperatures, $\alpha_{xy}$ splits around $E_F=\pm 0.45t$, which can be understood as due to the presence of $\nu=\pm 4$ Hall plateau by lifting the subband degeneracy. In fig. 3(c), we find that $\alpha_{xy}$ shows different behavior depending on the relative strength of temperature k_BT and the width of the central LL W_L (W_L is determined by the full width at the half-maximum of the $\sigma_{xx}$ peak). When $k_{B}T \ll W_L$ and $E_F \ll W_L$, $\alpha_{xy}$ shows linear temperature dependence, indicating that there is a small energy range where extended states dominate, and the transport falls into the semiclassical Drude-Zener regime. When E_F is shifted away from the Dirac point, the low-energy electron excitation is gapped due to Anderson localization. When k_BT becomes comparable to or greater than W_L, $\alpha_{xy}$ for all LLs saturates to a constant value $2.77 k_B e/h$. This matches exactly the universal value $(\ln 2) k_B e/h$ predicted for the conventional integer quantum Hall effect (IQHE) systems in the case where thermal activation dominates [29,30], with an additional degeneracy factor 4. The saturated value of $\alpha_{xy}$ is in accordance with the fourfold degeneracy at zero energy.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

To examine the validity of the semiclassical Mott relation, we compare the above results with those calculated from eq. (4), as shown in fig. 3(d). The Mott relation is a low-temperature approximation and predicts that the thermoelectric conductivities have linear temperature dependence. This is in agreement with our low-temperature results, which proves that the semiclassical Mott relation is asymptotically valid in the Landau-quantized systems, as suggested in ref. [29].

When the superlattice period length increases to $L=24 \sqrt{3} a_0\ (n=12)$, the calculated $\alpha_{xy}$ and $\alpha_{xx}$ at finite temperatures are shown in fig. 4. As seen from fig. 4(a), $\alpha_{xy}$ displays a pronounced valley at low temperatures, in striking contrast to the $L=2 \sqrt{3}a_0\ (n=1)$ case with a peak at $E_F=0$. This behavior can be understood as due to the split of the valley degeneracy in the central LL. $\alpha_{xx}$ oscillates and changes sign around the center of each split LL. In fig. 4(c), we also compare the above results with those calculated from the semiclassical Mott relation using eq. (4). The Mott relation is found to remain valid at low temperatures.

We further calculate the thermopower S_xx and the Nernst signal S_xy using eq. (5), which can be directly determined in experiments by measuring the responsive electric fields. In fig. 5(a), (b), we first show the calculated S_xx and S_xy for $L=2 \sqrt{3} a_0\ (n=1)$. As we can see, $S_{xy}\ (S_{xx})$ has a peak at the central LL (the other LLs) and changes sign near the other LLs (the central LL). This oscillatory feature is qualitatively similar to those found in monolayer graphene case [11], which has been observed experimentally [3]. At zero energy, both $\rho_{xy}$ and $\alpha_{xx}$ vanish, leading to a vanishing S_xx. Around zero energy, because $\rho_{xx}\alpha_{xx}$ and $\rho_{xy}\alpha_{xy}$ have opposite signs, depending on their relative magnitudes, S_xx could either increases or decreases when E_F is increased passing the Dirac point. In the graphene superlattice case, we find that S_xx is always dominated by $\rho_{xy} \alpha_{xy}$. Consequently, S_xx increases to a positive value as E_F passes zero. This is quite different from the graphene case. At low temperatures, the peak value of S_xx near zero energy is $\pm 1.14\,k_B/e\ (\pm 98.23\ \mu \text{V/K})$ at $k_BT=0.5W_L$. With the increase of temperature, the peak height increases to $\pm 3.97\,k_B/e\ (\pm 342.09\ \mu \text{V/K})$ at $k_BT=1.5W_L$. On the other hand, S_xy has a strong peak structure around zero energy, which is dominated by $\rho_{xx}\alpha_{xy}$. The peak height is $8.22\,k_B/e\ (708.32\,\mu V/K)$ at $k_BT=1.5W_L$.

Zoom In Zoom Out Reset image size

In fig. 5(c), (d), we show the calculated S_xx and S_xy for $L=24 \sqrt{3} a_0\ (n=12)$. As we can see, $S_{xy}\ (S_{xx})$ has a peak around zero energy (the other LLs), and changes sign near the other LLs (zero energy). In our calculation, S_xx is dominated by $\rho_{xy}\alpha_{xy}$. The peak value of S_xx near zero energy is $\pm 1.59\,k_B/e\ (\pm 137.01\,\mu V/K)$ at $k_BT=1.5W_L$. On the other hand, S_xy has a valley structure around zero energy at low temperatures, which is dominated by $\rho_{xx}\alpha_{xy}$. With the increase of temperature, we find S_xy exhibits a peak. The peak height is $3.68\,k_B/e\ (317.11\,\mu V/K)$ at $k_BT=1.5W_L$.

In summary, we have numerically investigated the thermoelectric transport properties of graphene superlattice systems in the presence of both disorder and a strong magnetic field. We find that the thermoelectric coefficients strongly depend on the superlattice period length. When the superlattice period length is taken to be the smallest, the thermoelectric conductivities display different asymptotic behavior depending on the ratio between the temperature and the width of the disorder-broadened LLs. At low temperatures, $\alpha_{xy}$ splits around $E_F=\pm 0.45t$, which can be understood as due to the presence of $\nu=\pm 4$ Hall plateau by lifting the subband degeneracy. In the high-temperature regime, the transverse thermoelectric conductivity $\alpha_{xy}$ saturates to a universal value $2.27\,k_B e/h$ at the center of each LL, and displays a linear temperature dependence at low temperatures. Both thermopower and Nernst signal display large peaks at high temperatures around the central LL. The validity of the semiclassical Mott relation between the thermoelectric and electrical transport coefficients is verified to be satisfied only at very low temperatures. However, when the superlattice period length is increased to $L=24\sqrt{3}a_0\ (n=12)$, the thermoelectric coefficients display quite distinct behaviors. Around the Dirac point, the transverse thermoelectric conductivity $\alpha_{xy}$ and the Nernst signal S_xy exhibit a pronounced valley at low temperatures, in striking contrast to the graphene case, where $\alpha_{xy}$ and S_xy display a peak. These are consistent with the splitting of the central LL in the graphene superlattice case.

This work was supported by the General Financial Grant from the China Postdoctoral Science Foundation (Grant No. 2014M551546), and the NSFC Grant Nos. 11104146 (RM) and 11225420 (LS). This work was also supported by the State Key Program for Basic Researches of China under Grant Nos. 2015CB921202 and 2014CB921103 (LS).

The Hall conductivity $\sigma_{xy}$ at zero temperature can be calculated by using the Kubo formula [37]

$$\begin{equation} \sigma_{xy}= \frac{ie^{2}\hbar}{S}\sum_{{\varepsilon_{\beta}}< {E_F}<{ \varepsilon_{\alpha}}}\frac{\langle\alpha\mid V_x\mid\beta\rangle\langle\beta\mid V_y\mid\alpha\rangle-\text{h.c.}}{(\epsilon_\alpha-\epsilon_\beta)^2}. \end{equation} \tag{ A.1 }$$

Here, $\epsilon_\alpha$, $\epsilon_\beta$ are the eigenenergies corresponding to the eigenstates ∣α〉, ∣β〉 of the system, which can be obtained through exact diagonalization of the Haldane model Hamiltonian. S is the area of the sample, V_x and V_y are the velocity operators. With tuning chemical potential E_F, a series of integer-quantized plateaus of $\sigma_{xy}$ appear, each one corresponding to E_F moving in the gaps between two neighboring Landau Levels (LLs).

The longitudinal conductivity $\sigma_{xx}$ at zero temperature can be obtained based on the calculation of the Thouless number. The Thouless number g is calculated by using the following formula [38]:

$$\begin{eqnarray} &&g= \frac{\Delta E}{\text{d}E/\text{d}N}. \end{eqnarray} \tag{ A.2 }$$

Here, $\Delta E$ is the geometric mean of the shift in the energy levels of the system caused by replacing periodic by antiperiodic boundary conditions, and $\text{d}E/\text{d}N$ is the mean spacing of the energy levels. The Thouless number g is proportional to the longitudinal conductivity $\sigma_{xx}$.