Giant Nonlinear Hall Effect in Twisted Bilayer WTe2

In a system with inversion symmetry broken, a second-order nonlinear Hall effect can survive even in the presence of time-reversal symmetry. In this work, we show that a giant nonlinear Hall effect can exist in twisted bilayer WTe2 system. The Berry curvature dipole of twisted bilayer WTe2 ({\theta} = 29.4{\deg}) can reach up to ~1400 {\AA}, which is much larger than that in previously reported nonlinear Hall systems. In twisted bilayer WTe2 system, there exists abundant band anticrossings and band inversions around the Fermi level, which brings a complicated distribution of Berry curvature, and leading to the nonlinear Hall signals exhibit dramatically oscillating behavior in this system. Its large amplitude and high tunability indicate that the twisted bilayer WTe2 can be an excellent platform for studying the nonlinear Hall effect.


Ⅰ. INTRODUCTION
Over the last few decades, the study of various Hall effects has become an important topic in condensed-matter physics. For Hall effect or anomalous Hall effect, it is well known that an external magnetic field or magnetic dopants are necessary to break the time-reversal symmetry [1]. However, a recent theoretical study [2] predicted that a second-order nonlinear Hall effect can survive in the presence of time-reversal symmetry but with inversion symmetry broken. Subsequently, the experimental study on bilayer and multilayer WTe2 [3][4] has validated this prediction in terms of the observed transverse nonlinear Hall-like current with a quadratic current-voltage characteristic. The nonlinear Hall effect arises from the non-vanishing dipole moment of Berry curvature in momentum space, i.e. Berry curvature dipole [2].
The experimental observations of the nonlinear Hall effect are * Correspondence author: hmweng@iphy.ac.cn mainly limited to the two-dimensional systems [3][4][14][15]. It is well known that the twist angle between adjacent layers can be used as a new degree of freedom to modulate the electronic structures of two-dimensional system, which has attracted great attention since the recent discovery of the unconventional superconductivity and correlated state in the twisted bilayer graphene [16][17] and twisted bilayer of transition metal dichalcogenides [18]. Very recently, it was shown that the strained twisted bilayer graphene [12] and WSe2 [13][14] present a large nonlinear Hall response. The experimental results demonstrate that there exist a significant nonlinear Hall effect in bilayer WTe2 [3], it is interesting to know how the twist regulates the nonlinear Hall effect in this system.
In this work, we study the nonlinear Hall effect in twisted bilayer WTe2 by using first-principle calculations combined with the semiclassical approach. Multiple twisted bilayer WTe2 structures are constructed, and the twist angles range from 12° to 73°. It is found that the twisted bilayer WTe2 has a more complicated band structure compared to the prefect bilayer system. We chose the twisted bilayer WTe2 with twist angle θ = 29.4° as a typical system, and show that the Berry curvature dipole of twisted bilayer WTe2 can be strongly enhanced.

Ⅱ. METHODS
The nonlinear Hall effect originates from the dipole moment of Berry curvature over the occupied states. In a system with time-reversal symmetry but broken inversion symmetry, when applied an oscillating electric field (1) Here, bd D is the Berry curvature dipole, adc  is the Levi-Civita symbol and  is the relaxation time. The Berry curvature dipole can be written as [2] , where nk f is the Fermi-Dirac distribution function, and is nonzero, which is given by [19] , 2 2 ( ) where n E and n are eigenvalues and eigenwave functions, respectively.
The first-principle calculations are performed by using the Vienna ab initio simulation package (VASP) [20] with the projector-augmented wave potential method [21][22][23]. The exchange-correlation potential is described using the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form [24]. Spin-orbit coupling (SOC) is taken into account self-consistently. The energy cutoff of the plane-wave basis set is 350 eV. A vacuum region larger than 15 Å is applied to ensure no interaction between the slab and its image. In our optimization, all structures are fully relaxed until the force on each atom is less than 0.02 eV/Å. The Van der Waals interactions between the adjacent layers are taken into account by using zero damping DFT-D3 method of Grimme [25]. The maximally localized Wannier functions [26][27][28] for the d orbitals of W and p orbitals of Te are generated to compute the Berry curvature and Berry curvature dipole [29].

Ⅲ. RESULTS AND DISCUSSIONS
The bilayer WTe2 exhibits an orthorhombic lattice, the optimized lattice constants are calculated to be a1=3.447 Å and a2=6.284 Å . We construct a series of twisted bilayer WTe2 structures based on the method described in Ref. [30]. For simplicity, we start with the normal stacking of perfect bilayer WTe2. As shown in Fig. 1 It is important to note that the twist angle will change slightly after structural relaxation.  Besides, the relatively smaller formation energy means that the twisted bilayer WTe2 may be formed in an efficient way.   and Dyz can switch their signs dramatically within a very narrow energy region. On the other hand, we find that the magnitude of Berry curvature dipole in our considered twisted bilayer WTe2 is much larger than that in previous reports [3][4][11][12][13][14][15]. For example, the peak of Dyz locates near the Fermi level, which is calculated to be ~1400 Å. As a comparison, the Berry curvature dipole in monolayer or bilayer WTe2 [3,11] is estimated to be in the order of 10 Å, and ~200 Å in strained twisted bilayer graphene [12].
To have a clearly understanding of the features of Berry curvature dipole in the twisted bilayer WTe2, we analyze the band structure and the distribution of Berry curvature. In general, the large Berry curvature appears at the band edge, as displayed in Fig. 3(b), where shows the band structure and Berry curvature , nk z  along the Γ − Y line. As can be seen from Fig. 3(b), the entanglement of multiple bands around the Fermi level causes a complicated distribution of Berry curvature, while the large Berry curvature dipole arises from the drastic change of Berry curvature in momentum space, as indicated by Eq. (2). We find that the peaks of Berry curvature dipole show in Fig. 3(a) mainly come from the contribution of the Berry curvature near the band edge shown in Fig. 3(b), for instance, the peaks of Berry curvature dipole originate from the band edges at corresponding energy   Fig. 3 . S2).

Ⅳ. CONCLUSIONS
In conclusion, we have predicted that a giant nonlinear Hall effect can exist in twisted bilayer WTe2 system. We show that the twist can greatly change the band structure of bilayer WTe2.