Possible Nonlinear Anomalous Thermoelectric Effect in Organic Massive Dirac Fermion System

We propose a novel current-induced thermoelectric phenomenon, the nonlinear anomalous Ettingshausen effect (AEE), at zero magnetic field in inversion-asymmetric conductors. As an example, we discuss the weak charge ordering state in a layered organic conductor alpha-(BEDT-TTF)2I3, which is a two-dimensional massive Dirac fermion system with a pair of tilted Dirac cones. The nonlinear AEE is a thermoelectric analogue of the nonlinear anomalous Hall effect, which is recently observed in alpha-(BEDT-TTF)2I3, and these two effects generally appear simultaneously. The nonlinear AEE generates a transverse heat current, which exhibits rectifying characteristics, namely unidirectionality even under an AC current.


3
CO" state immediately below Pc, where TCO is suppressed significantly. It has been suggested that the weak CO state is a massive DF state, where a small gap opens at the Dirac points [18][19][20]; in fact, this was experimentally confirmed recently [21]. The weak CO state provides an ideal platform for investigating the current-induced Berry curvature effects, although it generally comprises two types of CO domains [9]. In fact, nonlinear AHE has recently been observed in the weak CO state [11].
Herein, we propose a novel current-induced thermoelectric phenomenon, the nonlinear anomalous Ettingshausen effect (AEE), at zero magnetic field in inversionasymmetric conductors. As a candidate system, we discuss the weak CO state of an organic multilayer DF system, -(BEDT-TTF)2I3.
As a simplified model of the weak CO state, we considered a 2D massive DF system comprising a pair of tilted Dirac cones (k0 and −k0 valleys) with a small gap. We employed the following effective Hamiltonian around the svk0 valley ( 1 v s = ) [22,23].
Here, ( , ) xy kk = k is the 2D wave number measured from svk0. The matrices x, y, and z are Pauli matrices, and 0 is a 22  unit matrix. The x-axis is considered to be along the tilting direction of the Dirac cone, and v0 indicates the magnitude of the tilting. In addition, we introduce a mass parameter , which breaks the inversion symmetry and opens the CO gap. Two types of CO domains have a mass parameter with opposite signs. The energy dispersion () E  k and Berry curvature ()  Ωk of the conduction (+) and valence (−) bands around the svk0 valley can be obtained easily as follows: Here, nz is a unit vector along the stacking (z) axis. It is noteworthy that the Berry curvature does not depend on the tilting. We assume a small imbalance between the electron and hole
In the equilibrium state, the total Berry curvature summed up over occupied states is opposite in two valleys, resulting in no Berry curvature effects. However, Berry curvature effects may appear in the current-carrying state with a nonequilibrium distribution, wherein the in-plane electric field E is applied and a stationary electric current is flowing [1].
First, we briefly review the nonlinear AHE in the present model [9,10]. The current-induced nonequilibrium distribution breaks the balance of the anomalous velocity ( / ) ( ) e  E Ωk between two valleys [24], causing a nonlinear AHE. The anomalous Hall current in the current-carrying state is represented as follows: Here, z  Λ is the Berry curvature dipole (BCD). The -component of (2) j is written as (2) j E E where nx and ny are unit vectors in the xand y-directions, respectively.
Next, we discuss a new current-induced thermoelectric effect, the nonlinear AEE, in the similar way as the nonlinear AHE, in the present model. Generally, the Berry curvature correction of the heat current under an electric field is written as follows [25]: Here, is the equilibrium distribution with a chemical potential . The first term of the integral reflects the contribution from the anomalous velocity of wave packets, whereas the second term corresponds to that from the itinerant bulk current such as the edge current. Because jQ is perpendicular to E, a finite jQ corresponds to the AEE at zero magnetic field, which is a thermoelectric analogue of the AHE. The AEE is associated with the anomalous Nernst effect (ANE) by the Onsager relation [25]. Under time reversal symmetry, jQ vanishes because ( ) in the equilibrium state of the present model with time reversal symmetry, the AHE and AEE cannot be expected, resulting from the cancellation between the two valleys.
However, in the current-carrying state, the AHE and AEE are revived as the nonlinear AHE and nonlinear AEE, respectively. According to Boltzmann theory, in the current-carrying state, the distribution function is shifted by Here,  is a constant relaxation time and ( ) ( wave packets. This distribution change causes a nontrivial heat current as follows: Suppose that the electric field E is applied along the x-axis (tilting axis of the Dirac cone).
As illustrated in Fig. 1 Here, z  Θ is a thermoelectric analogue of the BCD and is referred to as the thermoelectric BCD herein. The -component of (2) Q j is written as ( in the present model. The nonlinear AEE originates from the Berry curvature of the system, but not from the group velocity; therefore, it is not directly related to the normal linear thermoelectric nature of the system [26]. Here, we should note that the nonlinear AEE was derived from the linear Boltzmann transport theory, in which the distribution shift () f  k is given by the first order of E and . The second order solution of Boltzmann equation gives a trivial nonlinear (second order) heat current parallel to E or the third order Ettingshausen heat current 7 perpendicular to E. Therefore, the nonlinear AEE derived here gives the lowest order contribution to the perpendicular component of jQ. This is the same approximation as the original derivation of the nonlinear AHE [1].
We evaluated the thermoelectric BCD for a 2D tilted massive Dirac fermion is always zero because the integrand of (4) is an odd function of ky. As for the x-component [] z x  Θ , the two valleys provide equal contributions. Figure 2 Actual -(BEDT-TTF)2I3 crystals are slightly electron-doped, likely due to the partial lack of I3 − ions. They demonstrate a finite Hall effect, suggesting an uncompensated carrier density n − p on the order of 10 15 − 10 16 cm -3 at low temperatures [12].  Fig. 2(b).
The nonlinear AEE in the present system with Dirac cones tilting in the x-direction is represented as In the current-carrying state, conventional linear thermoelectric effects, i.e., the Seebeck and Peltier effects, also occur at zero magnetic field. Moreover, the system must exhibit the nonlinear AHE and nonlinear ANE, which is a counterpart of nonlinear AEE.
To consider the voltage and temperature response to the charge current, we must simultaneously consider the charge and heat current at zero magnetic field.
Here, , , and S denote the electric conductivity, electron thermal conductivity, and Seebeck coefficient, respectively. S is negative for electrons ( > 0). For simplicity, we assumed that these parameters are scalar quantities. The first and third terms of j correspond to contributions from Ohm's law and the Seebeck effect, respectively, whereas the first and third terms of jQ correspond to contributions from the Peltier effect and Fourier's law, respectively. The second term of j represents the nonlinear AHE with . It is noteworthy that  Q is almost negative but  is positive in the case involving  > 0, v0 > 0, and  > 0. The fourth term of j represents the nonlinear ANE, which originates from the breaking of Berry curvature balance between two valleys due to the distribution change under the temperature gradient.
We consider a thermally isolated system in which the charge current j flows along the direction tilted by the angle  from the x-axis (tilting axis of Dirac cones) as shown in Fig. 3(c), in which the components of j and jQ in Eq. (10) are schematically indicated by dashed and solid arrows, respectively. The total heat current is zero ( Q 0 = j ) because of thermal isolation. We define the " "-and " ⊥ "-axes which are rotated by  from the xand y-axes, respectively. Hence, j is parallel to the -axis with j = jn ( n is a unit vector along the -axes). These conditions correspond to the measurements in the shaded rectangular sample in Fig. 3(c). By solving (9) Here, 2 ( / ) ZT S T   is the thermoelectric figure of merit. It is noteworthy that the fourth term of j (the nonlinear ANE) does not contribute to (11)