Abstract
Recently, it has been pointed out that the twisting of bilayer WSe_{2} would generate topologically nontrivial flat bands near the Fermi energy. In this work, we show that twisted bilayer WSe_{2} (tWSe_{2}) with uniaxial strain exhibits a large nonlinear Hall (NLH) response due to the nontrivial Berry curvatures of the flat bands. Moreover, the NLH effect is greatly enhanced near the topological phase transition point which can be tuned by a vertical displacement field. Importantly, the nonlinear Hall signal changes sign across the topological phase transition point and provides a way to identify the topological phase transition and probe the topological properties of the flat bands. The strong enhancement and high tunability of the NLH effect near the topological phase transition point renders tWSe_{2} and related moire materials available platforms for rectification and second harmonic generations.
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Introduction
The study of longperiod moiré superlattices formed in van der Waals heterostructures has emerged as a central topic in condensed matter physics^{1}. After the observation of correlated insulator and superconductivity in twisted bilayer graphene(TBG) with flat bands^{2,3,4,5,6,7,8,9,10,11}, it was proposed that there are moirémediated flat bands in twisted transition metal dichalcogenide heterobilayers and homobilayers^{12,13}. Recently, correlated insulating phases and possible superconductivity signatures were discovered in twisted bilayer WSe_{2} at twist angles between 4^{∘} to 5^{∘}^{14} and twisted doublebilayer WSe_{2}^{15}.
Besides the correlated phases, it was shown that the flat bands of the moire superlattices also exhibit nontrivial topological properties. For example, at 3/4 filling in hBNaligned TBG, the degeneracy of the bands with nontrivial Chern numbers is lifted by electronelectron interactions and results in quantum anomalous Hall states which were observed recently^{16,17,18}. These observations clearly demonstrate that the topological properties of the flat bands also have important consequences on the nature of the correlated phases.
Similar to TBG, it was pointed out that flat bands with nontrivial Chern numbers can be generated in twisted bilayer WSe_{2} (tWSe_{2}) and a couple of topological insulating phases were predicted at a wide range of twist angles^{13}. As in the case of TBG, the topology of the bands would affect the nature of the insulating phase when the bands are half filled and the interaction effects are strong. In this work, we propose that the nonlinear response in electric field can be used to unveil the topological properties of the flat bands and identify the topological phase transition point through the measurements of the nonlinear Hall (NLH) effect.
The NLH effect is a fascinating phenomenon recently proposed by Sodemann and Fu^{19}, and experimentally observed in bilayer and multilayer WTe_{2}^{20,21}. It is the generation of a transverse DC current and a transverse voltage with frequency 2ω when an AC current of frequency ω is applied, which has potential applications in rectifications and second harmonic generations. The effect originates from the nonvanishing dipole moment of the Berry curvature of the bands which characterizes the secondorder nonlinear hall susceptibility. In pristine tWSe_{2}, regardless of how large the Berry curvatures of the bands are, the three fold rotational symmetry forces the Berry curvature dipole to be zero and the NLH effect vanishes. However, we demonstrate here that a small strain breaks the threefold rotational symmetry and generates a large Berry curvature dipole. Importantly, these symmetry breaking strain effects have been observed recently in experiments^{22}. Moreover, the Berry curvature dipole is strongly enhanced and has opposite signs across the topological phase transition point when the top two valence bands touch each other and exchange Berry curvatures^{23}. As the topological phase transition can be tuned by a vertical displacement field, the measurement of the NLH effect can serve as a probe of the topological phase transition in tWSe_{2}. Moreover, the strong enhancement and the highly tunability of the NLH effect by a displacement field renders tWSe_{2} and related moire materials available platforms for rectification and second harmonic generations applications.
The rest of our paper is organized as follows. First, we present the continuum model of tWSe_{2} which takes into account a uniaxial strain in the bottom layer induced by substrate or external modulation, which breaks the threefold rotational symmetry such that nonzero Berry curvature dipole can be created. Second, we calculate the Berry curvature dipole in the presence of strain. We show that the strained tWSe_{2} exhibits large nonlinear Hall response. Third, we explore the behavior of the NLH response near the topological phase transition induced by a displacement field. We find that the Berry curvature dipole is strongly enhanced near the topological phase transition point and it changes sign across the transition point.
Results
Continuum model of strained tWSe_{2}
We consider a AA stacking bilayer WSe_{2} with lattice constant a_{0} in a single layer and at a twist angle θ. A schematic figure is shown in Fig. 1a. The moiré superlattice, which has a moiré lattice constant \({L}_{M}={a}_{0}/2\sin \frac{\theta }{2}\), folds the energy bands and gives rise to the socalled moiré Brillouin zone (see Fig. 1b). These moiré energy bands originating from the K_{+} or K_{−} valleys can be described by the continuum Hamiltonian^{13}\(H={\sum }_{\xi }\int d{{{{{{{\boldsymbol{r}}}}}}}}{\psi }_{\xi }^{{{{\dagger}}} }({{{{{{{\boldsymbol{r}}}}}}}}){\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{\xi }({{{{{{{\boldsymbol{r}}}}}}}}){\psi }_{\xi }({{{{{{{\boldsymbol{r}}}}}}}})\) with
Here, ξ = ± is the valley index denoting whether the bands originate from the K_{+} or K_{−} valleys of the monolayer Brillouin zone. The electron creation operators are denoted as \({\psi }_{\xi }^{{{{\dagger}}} }=({\psi }_{b,\xi }^{{{{\dagger}}} },{\psi }_{t,\xi }^{{{{\dagger}}} })\), where t and b label the top and bottom layers respectively. It is important to note that due to the large Ising spinorbit coupling in 2Hstructure WSe_{2}, the top valence band of monolayer WSe_{2} K_{+} and K_{−} valleys are fully spin polarized and have opposite spin. Therefore, the spin and valley indices are locked together and the spinindex is dropped. As a result, the Hamiltonian of a single layer at valley ξ can be written as
where l = + 1(−1) labels the bottom(top) layer, m* is the effective mass of valence band, and V_{z} is the staggered layer potential generated by the vertical displacement field. The intralayer moiré potential and the coupling between the two layers are denoted as Δ_{l}(r) and T_{ξ}(r) respectively which can be written as:
Here, V and ψ, characterize the amplitude and phase of the moire potentials respectively, w characterizes the tunneling strength between the top and bottom layers, and the moiré reciprocal lattice vectors are \({{{{{{{\boldsymbol{{g}}}}}}}_{i}}}=\frac{4\pi }{\sqrt{3}{L}_{M}}\left(\cos \frac{2(i1)\pi }{3},\sin \frac{2(i1)\pi }{3}\right)\). We adopt the model parameters from ref. ^{13} (a_{0}, m*, w, V, ψ) = (3.32Å, 0.44m_{e}, 9.7meV, 8.9 meV, 91^{∘}), which are estimated from the first principle calculations.
It is important to note that the Berry curvature dipole transforms as a pseudovector which vanishes in the presence of a threefold rotational symmetry as discussed in detail in the next section. In realistic systems, the threefold symmetry in tWSe_{2} can be broken by the substrate induced or externally applied strain. The strain induced symmetry breaking has been observed experimentally^{22}. In general, the physical properties of tWSe_{2} only depend on the relative deformation between the two layers. Therefore for all the calculations in this work, the strain is only applied to the bottom layer of WSe_{2}, while the strain in the top layer is set to zero. Specifically, the strain tensor \({{{{{{{\mathcal{E}}}}}}}}\) is a twodimensional matrix, which can be written as^{24}:
The angle φ denotes the inplane direction of the uniaxial strain direction with respect to the zigzag edge of the sample and ϵ characterizing the strength of strain, and ν = 0.19 is the poisson ratio for WSe_{2}^{25}.
In general, the strength ϵ can be estimated to 0.5%~1%, but the direction φ can be random and difficult to determine. We highlight the breaking of the C_{3} rotational symmetry due to strain is the key of the nonlinear Hall response, while the details of the strain are not essential for our discussion. For simplicity, we use the model that there is no strain in the top layer and the strain is applied to the bottom layer.The strain has two important effects shifting the Dirac point for bottom layer WSe2 to \({{{{{{{{\boldsymbol{D}}}}}}}}}_{b,\xi }=(I{{{{{{{{\mathcal{E}}}}}}}}}^{T}){{{{{{{{\boldsymbol{K}}}}}}}}}_{b,\xi }\xi {{{{{{{\boldsymbol{A}}}}}}}}\) and generating an effective gauge field \({{{{{{{\boldsymbol{A}}}}}}}}=\frac{\sqrt{3}}{2{a}_{0}}\beta ({\epsilon }_{xx}{\epsilon }_{yy},2{\epsilon }_{xy})\). Here, β is taken as 2.30 in our calculations according to previous first principle calculations for strained monolayer WSe_{2}^{24,26}. As a result, the continuum Hamiltonian of this strained tWSe_{2} is obtained by replacing \({\hat{H}}_{b}\) in Eq. (1) as
To estimate the range of strain ϵ within which our model is valid, we expect that the strain induced shifting of the Dirac point would be smaller than the separation of the Dirac points due to twisting. With a twist angle θ, the separation of the Dirac point K in momentum space is ΔK = ∣K∣θ, and the shift of the K point by uniaxial strain is ΔK_{s} ≈ ϵ∣K∣. Therefore, the strain effect can be treated as a perturbation to the moiré superlattice when ϵ ≪ θ. In our calculations below, we assume that the strain induces a 0.5% change in the lattice constant of the bottom layer WSe2 along the direction of the strain such that ϵ ≈ 0.2θ.
In Fig. 2a, we show the moiré energy bands of tWSe_{2} with twist angle θ = 1. 4^{∘} with and without strain. The top two valence bands originating from the K_{+} valley of the monolayer Brillouin zones carry finite Chern numbers C = 1 and C = − 1 respectively when V_{z} = 0. Due to timereversal symmetry, the top two valence bands originating from the K_{−} valleys carry Chern numbers C = − 1 and C = 1 respectively. As the two valleys do not couple in momentum space, one can define a Z_{2} topological invariant as \({Z}_{2}= ({C}_{{K}_{+}}{C}_{{K}_{}})/2\) to describe the topological properties of the bulk bands, where C_{K} is the Chern number of the top valence band originating from the K valleys of the monolayer Brillouin zone. In Fig. 2b, a staggered potential of V_{z} = 5 meV induces a topological phase transition and the system becomes topologically trivial. The band structure with and without strain are depicted. It is important to note that the strain with ϵ = 0.5% is not sufficient to close the band gap and the topological properties of the flat bands are not changed by strain. Moreover, the energy dependence of the density of states for V_{z} = 0, 5 meV are also shown on the right side of Fig. 2, which is important to determine the occupation number of holes n_{h} as discussed below.
Nonlinear Hall response
In this section, we consider the NLH response for tWSe_{2}. The NLH effect is characterized by the generation of a transverse voltage using a charge current in timereversal invariant systems without external magnetic fields or magnetic orders. Moreover, this effect is nonlinear in nature and exhibits a quadratic currentvoltage relation. More specifically, when an electric field \({{{{{{{\boldsymbol{E}}}}}}}}(t)=\frac{1}{2}(\varepsilon {e}^{i\omega t}+{\varepsilon }^{* }{e}^{i\omega t})\) with the amplitude vector ε and frequency ω is applied, the Hall current has both rectified and secondharmonic components \({J}_{y(x)}^{(0)}={\chi }_{yxx(xyy)}{\varepsilon }_{x(y)}{\varepsilon }_{x(y)}^{* }\) and \({J}_{y(x)}^{(2)}={\chi }_{yxx(xyy)}{\varepsilon }_{x(y)}{\varepsilon }_{x(y)}\), where χ is the nonlinear Hall susceptibility. As shown in ref. ^{19,27,28}, the nonlinear Hall susceptibility can be written as:
where τ is the relaxation time. The Berry curvature dipole D_{i} are elements of the Berry curvature dipole pseudovector. It is important to note that in a twodimensional system, the Berry curvature dipole transforms as a pseudovector as described by the vector D_{i} (i = x, y). In other words, D_{i} must be invariant under crystal point group operations. For twisted tWSe_{2} with or without the displacement field, the system respects a threefold rotational symmetry C_{3z} inherited from monolayer 2Hstructure WSe2. Due to the timereversal symmetry, Ω_{n}(k + K) = − Ω_{n}( − k − K), v_{n}(k + K) = − v_{n}( − k − K), each valley has an equal contribution, allowing us to consider the K_{+} valley for the sake of simplicity. For K_{+} valley, C_{3z} symmetry ensures \({\Omega }_{n}({{{{{{{\boldsymbol{k}}}}}}}})={\Omega }_{n}({C}_{3}{{{{{{{\boldsymbol{k}}}}}}}})={\Omega }_{n}({C}_{3}^{2}{{{{{{{\boldsymbol{k}}}}}}}})\) and \(\mathop{\sum }\nolimits_{i = 0}^{2}{{{{{{{{\boldsymbol{v}}}}}}}}}_{n}({C}_{3}^{i}{{{{{{{\boldsymbol{k}}}}}}}})=0\). Therefore, D_{i} vanishes if C_{3z} symmetry is preserved. One way to obtain finite D_{i} is to take into account the strain effects which break C_{3z}. The strain can be induced by the substrate which couples to tWSe_{2} or it can be induced externally as shown in the above sections. As we will see in the next section, only a very small strain is needed to induce a strong NLH response in tWSe_{2}.
Topological phase transition and NLH effect
With the formalism discussed above, we can now explore the NLH effects in tWSe_{2}. By applying a unaxial strain with ϵ = 0.5% along the direction of the zigzag edge and at twist angle 1. 4^{∘}, D_{x} as a function of the vertical displacement field V_{z} and the occupation number of holes per moire unit cell n_{h} are depicted in Fig. 3a. The occupation number n_{h} is obtained by integrating the density of states from the top of the valence to the Fermi energy including two valleys. It is clear from Fig. 3a that the Berry curvature dipole is generally large, in the order of 1Å.
Importantly, the Berry curvature dipole is strongly enhanced near the bottom of the top valence band when n_{h} is close to 2 (this happens when the Fermi energy is near the bottom of the top valence band) and V_{z} close to 4.2 meV. Specifically, the bands originating from the K_{+} (or K_{−}) valley have nontrivial topological invariant Z_{2} = 1. By applying a displacement field which introduces a potential difference between the top and the bottom layers, the top valence band and the nearby valence band can touch near the K points and exchange Berry curvatures, resulting in a change of Z_{2} invariant from 1 to 0. The V_{z} dependence of the Berry curvature dipole across the phase transition with fixed n_{h} is shown in Fig. 3b. It is clear that Berry curvature dipole is strongly enhanced near the critical V_{z} ≈ 4.2 meV when n_{h} is close to 2. The enhancement of Berry curvature dipole comes from two contributions, one is the enhancement of density of states due to the band flatness, and the other is the large Berry curvature near band edge. This value is several times larger than the optimal Berry curvature dipole measured experimentally in other systems which is about 1Å^{20,21,29}. Moreover, the Berry curvature dipole changes sign across the phase transition point. This is caused by the exchange of Berry curvatures near the K points before and after the topological phase transition as shown in Fig. 3d, e. To illustrate the sign change of Berry dipole more clearly, we show the Berry dipole flow v_{i}Ω in Supplementary note 1.
Furthermore, the V_{z} dependence of the Berry curvature dipole with fixed n_{h} is shown in Fig. 3c when strain is applied along the armchair edge direction. Similar enhancement and sign change in the Berry curvature dipole near the topological phases transition can be observed. In the supplementary note 2, the V_{z} and n_{h} dependence of the Berry curvature dipole with different twist angles and strain are presented and behaviors similar to Fig. 3a are observed. Therefore, the NLH measurements can be used to indicate possible topological phase transitions in tWSe_{2}. Note that in practice to confirm the sign switching of NLH signal is induced by the topological phase transition rather than other mechanisms, it must be accompanied by other measurements that can probe the bulk topology or edge excitations.
Discussion
It is important to note that the appearance of NLH effect studied in this work is very general. We expect a finite NLH response as long as the threefold rotational symmetry is broken intrinsically by the substrate^{22} or by an externally applied strain. The calculated Berry dipole in tWSe_{2} is about 1 ~ 3Å, which is relatively large in usual materials. In other strained systems, such as strained MoS2 and strained bilayer graphene^{30,31}, the straininduced Berry dipole is only 10^{−2}Å. For bilayer and fewlayer WTe_{2}^{28,32}, the Berry dipole is about 0.2Å, 1Å respectively.
Furthermore, we point out that strained tWSe_{2} and other moiré materials are excellent candidate materials for studying NLH effects due to their low symmetry, nontrivial Berry curvature and high tunability of experimental parameters^{33,34}. Especially, the C_{3z} symmetry breaking strain have been widely observed in moiré materials. In our other recent work collaborating with experimentalists^{35}, it was found that the twisted monolayermonolayer, bilayerbilayer and trilayertrilayer WSe_{2} samples, all of them exhibit strong nonlinear hall response, roughly 1000 times larger than those of bilayer or fewlayer WTe_{2}^{20,21}. This clearly shows that the moiré superlattices behave as an ideal setup in realizing the nonlinear Hall effects, although the understanding of such a giant nonlinear hall response in the moiré superlattices is still open and worth further works in the future.
Our calculations are based on the Boltzmann approach in ref. ^{19}, which requires that the inelastic scattering time τ_{in} and the band width W should satisfy W ≫ ℏ/τ_{in}. Although the bandwidth is reduced at orders of meV by the moiré potential in tWSe_{2}, we assume the Boltzmann approach is still valid in this case. This assumption is based on the observation that τ_{in} is usually around 10 ps for 2D van der Waals materials^{36}, corresponding to an energy scale ℏ/τ_{in} ~0.07 meV that is still much smaller than the moiré band width.
The electronelectron interaction in general is expected to play an important role in moiré superlattices. Recently, correlated insulating phase is observed when the filling factor n_{h} is close to 1^{14}. In principle, the interaction can modify the singleparticle band structures and further change the Berry curvature dipole. In the experiment^{35}, the nonlinear Hall response is strongly enhanced at half filling of the first moiré band, which cannot be explained in singleparticle level and correlation effects must have contributions to the NLH response. Near halffilling of the first moiré band, the nonlinear Hall signal shows a sharp peak which can originate from a massdiverging type continuous Mott transition. Also the symmetrybreaking ground states as discussed in^{6} may also get involved and interplay with NLH. For example, the C_{3z} rotational symmetry can be broken in the presence of the nematic phase, which is essential in generating a finite NLH in tWSe_{2}. The interplay between the interaction effects and nonlinear Hall effect is of great interest and is worth a more detail study in the future.
In addition, the topological phase transition happens near n_{h} ≈ 2. For a fixed displacement field V_{z}, the Berry curvatures near the band edge of top two valence bands are opposite and as a result, the Berry dipoles change sign when filling factor n_{h} across 2. For a fixed n_{h}, by applying V_{z} which introduces a potential difference between the top and the bottom layers, the top valence band and the nearby valence band can touch near the K points and exchange Berry curvatures, resulting in the sign change of D_{i}. This is the origin of four wings of the "butterfly" in Fig. 3a. Near the phase transition regime, we expect the system to be metallic and can be described by a Fermi liquid even in the presence of electronelectron interaction. Nevertheless, the NLH measurements can reveal the topological properties of the bands away from the Fermi energy that shed light on the nature of correlated phases.
Methods
Berry curvature and dipole
In strictly two dimensions, the Berry curvature dipole is given by
where n as the band index and v^{i} as the Fermi velocity of the Bloch state, and the Berry curvature can be obtained from Bloch wavefunctions as:
The Berry curvature dipoles should be integrated over whole moiré Brillouin zone and both valleys are involved in the result. Also the Chern number for an isolated band can be evaluated from Berry curvature:
Data availability
All essential data are available in the paper. Additional data are given in the supplementary file. Further supporting data can be provided from the corresponding author upon request.
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Acknowledgements
We are grateful to the illuminating discussions with Meizhen Huang, Zeifei Wu and Ning Wang. We thank the support of the Croucher Foundation and HKRGC through 16324216, 16307117 and 16309718.
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K.T.L. conceived the project. J.X.H. performed the major part of the calculations and analysis. J.X.H., Y.M.X. and K.T.L. wrote the manuscript with contributions from all authors. C.P.Z. contributed to part of the calculations. All authors are involved in the discussions.
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Hu, JX., Zhang, CP., Xie, YM. et al. Nonlinear Hall effects in strained twisted bilayer WSe_{2}. Commun Phys 5, 255 (2022). https://doi.org/10.1038/s42005022010347
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DOI: https://doi.org/10.1038/s42005022010347
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