Possible Current-Induced Phenomena and Domain Control in an Organic Dirac Fermion System with Weak Charge Ordering

We show that when the electron and hole densities are unbalanced, observable nonlinear anomalous Hall effect and current-induced orbital magnetization appear at zero magnetic field in the weak charge ordering (CO) state of an organic two-dimensional Dirac fermion system, a-(BEDT-TTF)2I3. These current-induced phenomena are caused by a finite Berry curvature dipole resulting from inversion symmetry breaking and Dirac cone tilting. In the actual system, however, these effects are canceled out between different types of inversion asymmetric CO domains. To avoid the cancellation, we propose a new experimental method to realize the selective formation of a single type of domain using the current-induced magnetization.

curvature, but a sum of the Berry curvature over all occupied states is zero at the equilibrium state, implying that they are topologically trivial conductors. However, when an electric current is applied to these conductors, the electron distribution is shifted in the k space, resulting in a finite sum of Berry curvature over occupied states. Therefore, the phenomena originating from the Berry curvature are induced in their current carrying state.
This study highlights the possibility of observing the current-induced phenomena, namely, nonlinear AHE and current-induced orbital magnetization in the weak CO state of an organic Dirac fermion system -(BEDT-TTF)2I3, when a finite imbalance exists between electron and hole densities. Furthermore, an experimental method is proposed to align the inversion-asymmetric CO domains in order to observe the current-induced phenomena in an actual system consisting of multiple CO domains.
First, we consider a massive Dirac fermion system with no inversion symmetry as a model for a single CO domain system. We employ the following tilted Weyl equation for the k0 valley of -(BEDT-TTF)2I3 [15,16].
Here, (  bands are easily obtained as follows.
Note that the Berry curvature does not depend on the tilting.
For the −k0 valley, the effective Hamiltonian is given by the time reversal of (1): , where k is measured from −k0. Note that ( , , ) x y z    corresponds not to real spin, but sublattice pseudo-spin. The −k0 valley has a dispersion with opposite tilting slope ( 0 v − ) and a Berry curvature with an opposite sign.
The dispersion and the Berry curvature of the k0 and −k0 valleys are schematically represented in Fig. 1 in no orbital magnetization. These results are expected from the time reversal symmetry.
Subsequently, we consider the non-equilibrium current carrying state, wherein the in-plane electric field E is applied to the conducting system and a stationary electric current is flowing. According to the Boltzmann transport theory, the distribution function is shifted Here, is the equilibrium distribution with a chemical potential , and  denotes a constant relaxation time. In the k space, the occupied states move by ( / ) e − E , so that the anomalous Hall current and the orbital magnetization summed up over occupied states may become finite. Suppose that the electric field E is applied along the x-axis, which is the tilting axis of the Dirac cone in the present model.
We also assume a small imbalance between electron and hole densities ( 0 np −), in other words, a positive Fermi energy at zero temperature. As illustrated in Fig Therefore, the contribution from both valleys does not cancel completely under E, resulting in current-induced AHE and magnetization. The former is called nonlinear AHE because it is a second-order response to the electric field. This can be intuitively understood as a breaking of the balance of the average anomalous velocity, as illustrated in Fig. 1(c). The nonlinear AHE can also be interpreted as Hall effect due to the current-induced magnetization.
The magnitude of these current-induced effects is represented by the following 03 3 We evaluated the BCD of a 2D tilted massive Dirac fermion system as the following. The is always zero, because the integrand of (4) is an odd , the k0 and −k0 valleys have equal contributions. Figure 2( where ny is the unit vector in the y-direction, and  denotes the angle between E and the tilting axis (x-axis). Note that the nonlinear Hall current density j (2) is always perpendicular to the tilting axis regardless of the electric field direction. The value of (2) y j depends only on the tilting axis component Ex. In addition, note that the sign of (2) y j never changes even if the electric field E is reversed, because it is a second-order response. The dependence of (2) y j on the orientation of E is schematically illustrated in Fig. 3(a).
where nz is the unit vector in the z direction. The orbital magnetization M is always normal 8 to the 2D plane. The value of M depends only on the tilting axis component Ex, and changes its sign when the electric field is reversed, as depicted schematically in Fig. 3(b). Note that the magnetization is not scaled by ||, but it only depends on the sign of , because the factor || in (6) is canceled by the denominator of the normalizing factor In -(BEDT-TTF)2I3, the band parameters were estimated as . In a previous work, the scattering broadening of the n=0 Landau level was experimentally estimated to be 3 K [20], which corresponds to ~ 2.5 ps if the scattering time does not change under the magnetic field. Assuming the realistic parameters, that is, 16 Fig. 3(c). 9 These values are in the observable range.
So far, we have discussed the single-domain system. In the actual -(BEDT-TTF)2I3 crystal, however, multiple CO domains appear after the CO phase transition. In the multi-domain system, the nonlinear AHE and the current-induced magnetization can hardly be observed because of the cancellation between two types of inversion-asymmetric domains.
To observe the current-induced phenomena in the weak CO state, we have to realize the selective formation of one of the two types of domains. For this purpose, we propose a new experimental method utilizing the current-induced magnetization, which is referred as the current-field-cooling. As schematically depicted in Fig. 4 We might use the current-induced phenomena to investigate the unidentified 10 insulating state of -(BEDT-TTF)2I3. The inset of Fig. 4 displays the temperature dependence of interlayer resistance of -(BEDT-TTF)2I3 at several pressures (after Mori et al. [21]). At P = 1.2 GPa, just below the critical pressure, the system undergoes a transition to the weak CO state at TCO. It is visible that the system exhibits another insulating behavior at low temperatures below the dotted arrow [22,23]. The excitonic instability [24] and the topological insulator state [25] were proposed as the origin of this unidentified insulating state. Note that inversion symmetry remains in both cases. However, if the current-induced phenomena are observed in this unidentified insulating state obtained by the current-fieldcooling, a gapped state with broken inversion symmetry is strongly suggested.
In conclusion, we studied the possible nonlinear AHE and current-induced orbital magnetization at zero magnetic field in the weak CO state of an organic conductor -(BEDT-TTF)2I3, which is a 2D massive Dirac fermion system with Dirac cone tilting and gap opening due to inversion symmetry breaking. This is an ideal system with a finite BCD, and we demonstrated that a single CO domain system exhibits observable nonlinear AHE and current-induced magnetization in the current carrying state. To avoid the cancellation of current-induced effects between different types of domains, we proposed a current-fieldcooling method to enhance the selective formation of a single type of domain. Conceptual representation of the current-field-cooling method. As the system is cooled, the metallic Dirac state undergoes a transition into the weak CO state with multiple inversion asymmetry domains. The electric current induces orbital magnetization in each domain when the current is oriented in the tilting direction. Sample cooling with a finite DC current and an external normal magnetic field is expected to realize the selective formation of one type of domain. Inset: Temperature dependence of interlayer resistance in α-(BEDT-TTF)2I3 after Mori et al. [21]. The solid and dotted arrows indicate transitions from the Dirac state into the weak CO state and the unidentified insulating state, respectively.