Abstract
Under broken time reversal symmetry such as in the presence of external magnetic field or internal magnetization, a transverse voltage can be established in materials perpendicular to both longitudinal current and applied magnetic field, known as classical Hall effect. However, this symmetry constraint can be relaxed in the nonlinear regime, thereby enabling nonlinear anomalous Hall current in timereversal invariant materials – an underexplored realm with exciting new opportunities beyond classical linear Hall effect. Here, using group theory and firstprinciples theory, we demonstrate a remarkable ferroelectric nonlinear anomalous Hall effect in timereversal invariant fewlayer WTe_{2} where nonlinear anomalous Hall current switches in oddlayer WTe_{2} except 1T′ monolayer while remaining invariant in evenlayer WTe_{2} upon ferroelectric transition. This evenodd oscillation of ferroelectric nonlinear anomalous Hall effect was found to originate from the absence and presence of Berry curvature dipole reversal and shift dipole reversal due to distinct ferroelectric transformation in even and oddlayer WTe_{2}. Our work not only treats Berry curvature dipole and shift dipole on an equal footing to account for intraband and interband contributions to nonlinear anomalous Hall effect, but also establishes Berry curvature dipole and shift dipole as new order parameters for noncentrosymmetric materials. The present findings suggest that ferroelectric metals and Weyl semimetals may offer unprecedented opportunities for the development of nonlinear quantum electronics.
Introduction
In classical linear Hall effect, a transverse voltage can be developed in materials with broken timereversal symmetry only (e.g. in the presence of external magnetic field or internal magnetization) due to Onsager’s relation. Second and higher order conductivity tensors, however, are not subject to this constraint, thereby enabling nonlinear anomalous Hall effect (NAHE) in timereversal invariant system.^{1,2,3,4} NAHE was observed very recently in fewlayer tungsten ditelluride (WTe_{2}),^{5,6,7,8,9,10,11} a layered material which also holds rich physics including hightemperature quantum spin Hall phase^{12,13,14,15} and electrostatic gating induced superconductivity^{16,17} in its 1T′ monolayer and typeII Weyl semimetallicity,^{18} large nonsaturating magnetoresistance^{19} and ultrafast symmetry switching^{20} in its bulk phase.
Monolayer 1T′ WTe_{2} is centrosymmetric with vanishing evenorder nonlinear current response, however vertical electric field can break its twofold screw rotation symmetry, generate Berry curvature dipole (BCD), and induce secondorder nonlinear anomalous Hall current.^{5,6,7,8} In contrast to monolayer WTe_{2}, bilayer WTe_{2} is naturally noncentrosymmetric due to the loss of twofold screw rotation symmetry, resulting in intrinsic nontrivial BCD in bilayer WTe_{2}.^{9,10,11} Surprisingly, ferroelectric switching was recently discovered in semimetallic bilayer and fewlayer WTe_{2},^{21} quite unusual as ferroelectricity and semimetallicity normally do not coexist in the same material.^{22} The subtlety lies in the reduced screening along the outofplane direction which gives rise to finite outofplane ferroelectric polarization while preserving inplane semimetallic nature. Conductance hysteresis persisting up to 300 K shows its great potential for room temperature device application. These recent studies combined reveal a striking feature of noncentrosymmetric fewlayer WTe_{2} – the coexistence of ferroelectricity and NAHE within a single material, enkindling a few fundamentally and technologically important questions: what’s the fundamental correspondence between NAHE and ferroelectricity in ferroelectric metals and Weyl semimetals? Compared to ferroelectric semiconductors,^{23} what are the unique advantages of ferroelectric metals^{21} and ferroelectric Weyl semimetals?^{24,25}
Here using firstprinciples approach and group theoretical analysis we show an intriguing ferroelectric nonlinear anomalous Hall effect (FNAHE) in timereversal invariant fewlayer WTe_{2}. In particular, while both bilayer and trilayer WTe_{2} possess switchable outofplane electric polarization, nonlinear transverse Hall current only switches in trilayer WTe_{2} upon ferroelectric switching. The microscopic origin of FNAHE in trilayer WTe_{2} is found to be rooted in the reversal of Berry curvature dipole and shift dipole upon ferroelectric transition, which reveals an exciting yet unexplored realm of ferroelectric metals and Weyl semimetals with potential applications in nonlinear electronics.
Results and discussion
Secondorder dc current
Consider an oscillating electric field \({\boldsymbol{E}}\left( {{\boldsymbol{r}},t} \right) = {\boldsymbol{E}}\left( \omega \right)e^{i\left( {{\boldsymbol{k}} \cdot {\boldsymbol{r}}  \omega t} \right)} + {\boldsymbol{E}}\left( {  \omega } \right)e^{  i\left( {{\boldsymbol{k}} \cdot {\boldsymbol{r}}  \omega t} \right)}\) with E(ω) = E^{*}(−ω) (e.g. under AC electric field or upon coherent light illumination), the secondorder nonlinear dc current under minimal coupling approximation was derived by Sipe et al.^{26}, i.e., \(j_a^0 = \chi _{abc}\left( {0;\omega ,  \omega } \right)E_b\left( \omega \right)E_c(  \omega )\), where χ_{abc} are the dc photocurrent susceptibility. In general \(j_a^0\) consists of two parts depending on the polarization of electric field/incident light, including linear photogalvanic effect (LPGE) and circular photogalvanic effect (CPGE),^{2,26,27} i.e., \(j_a^0 = j_a^L + j_a^C\). BCDinduced nonlinear photocurrent current was generalized to the multipleband case by Morimoto et al.^{28} using Floquet theory and Rostami et al.^{29} using density matrix beyond semiclassical Boltzmann theory. Nonlinear photocurrent originating from CPGE is also known as injection current.^{26}
Both LPGE and CPGE have intraband and interband contributions. For the sake of completeness we include all the terms as follows,
Here τ is relaxation time and \({\it{\epsilon }}_{adc}\) is the Levi–Civita symbol. \(D_{ab}^{{\mathrm{intra}}}\) is the wellknown BCD for intraband nonlinear process.^{2} \(D_{ab}^{{\mathrm{C}},{\mathrm{inter}}}\) is BCD for interband process associated with CPGE.^{9} \(D_{a,bc}^{{\mathrm{L}},{\mathrm{inter}}}\) is shift dipole (SD), originated from the simultaneous displacement of wavepacket upon excitation. More specifically, they are given by
Here, \(\hbar \omega _n({\boldsymbol{k}})\), \(v_n^b\left( {\boldsymbol{k}} \right)\), and f_{n}(μ) are band energy, group velocity, and chemicalpotential μ dependent FermiDirac distribution, respectively. f_{nm}(μ) ≡ f_{n}(μ) − f_{m}(μ), and [dk] ≡ d^{d}k/(2π)^{d} for ddimension integral. \({\mathrm{\Delta }}_{nm}^a \equiv v_n^a  v_m^a\) is the group velocity difference between two bands. \(r_{nm}^a\) is interband Berry connection or dipole matrix element. \({\mathrm{\Omega }}_{nm}^c\left( {\boldsymbol{k}} \right)\) is the interband Berry curvature between two bands, defined as \({\mathrm{\Omega }}_{nm}^c\left( {\boldsymbol{k}} \right) \equiv i{\it{\epsilon }}_{abc}r_{nm}^ar_{mn}^b.\) \({\mathrm{\Omega }}_n^c\left( {\boldsymbol{k}} \right)\) is the intraband Berry curvature for band n, given by \({\mathrm{\Omega }}_n^c\left( {\boldsymbol{k}} \right) = \mathop {\sum}\nolimits_{n \ne m} {{\mathrm{\Omega }}_{mn}^c} \left( {\boldsymbol{k}} \right)\). In addition, \(\left\{ {r_{nm}^b,r_{mn}^c} \right\} \equiv r_{nm}^br_{mn}^c + r_{mn}^cr_{nm}^b\). \(R_{mn}^a\) is shift vector, given by \(R_{mn}^a \equiv  \frac{{\partial \phi _{mn}\left( {\mathbf{k}} \right)}}{{\partial k^a}} + r_{mm}^a\left( {\boldsymbol{k}} \right)  r_{nn}^a\left( {\boldsymbol{k}} \right)\), where ϕ_{mn}(k) is the phase factor of the interband Berry connection and \(r_{nn}^a\) is intraband Berry connection. \({\mathrm{\Omega }}_{nm}^b\left( {\boldsymbol{k}} \right)\), \({\mathrm{\Omega }}_n^b\left( {\boldsymbol{k}} \right)\) and \(R_{mn}^a\) are all gauge invariant. For linearly polarized incident light/electric field, E_{b} = E_{c}, hence we denote \(D_{ab}^{{\mathrm{L}},{\mathrm{inter}}} \equiv D_{a,bc}^{{\mathrm{L}},{\mathrm{inter}}}\). The intraband and interband BCDs (\(D_{ab}^{{\mathrm{intra}}}\), \(D_{ab}^{{\mathrm{C}},{\mathrm{inter}}}\)) as well as SD \((D_{ab}^{{\mathrm{L}},{\mathrm{inter}}})\) have the same units of L^{3−d} for ddimensional system. Thus, BCD and SD have units of length in 2D, but become dimensionless in 3D.
The appearance of relaxation time τ in the dc current from interband LPGE \((j_{a,{\mathrm{inter}}}^L)\) seems different from the widelyused τindependent shift current formula by Sipe et al.^{26}, however the latter was derived for the clean limit when relaxation time τ approaches to infinite. In fact, as τ → ∞, \(\tau {\mathrm{Re}}\left( {\frac{1}{{1  i\left( {\omega  \omega _{mn}} \right)\tau }}} \right) \to \pi \delta \left( {\omega _{mn}  \omega } \right)\), and the original τtime independent shift current susceptibility can be exactly recovered from the above \(j_{a,{\mathrm{inter}}}^L\) formula. In reality, quasiparticles do have finite relaxation time, thus \(j_{a,{\mathrm{inter}}}^L\) shall depend on relaxation time. More detailed derivation about the SD and BCD can be found in the Supplementary Information.
Moreover, it is worth to classify the contributions of LPGE/CPGEinduced dc current at the low/high frequency region. At the low frequency limit, ωτ → 0, hence \(\frac{\tau }{{1  i\omega \tau }} \to \tau\). In this case, the photocurrent due to both intraband and interband CPGE as well as interband LPGE will vanish, however a dc current from intraband LPGE will remain finite which is perpendicular to the applied electric field, thereby inducing static NAHE. At high frequency, CPGE (i.e. injection current) and interband LPGE (i.e. shift current) will have nontrivial contribution to total nonlinear photocurrent, referred as to dynamic NAHE.
It is important to note that the direction of nonlinear photocurrent induced by CPGE and LPGE have very different symmetry properties. The LPGEinduced dc current cannot flow normal to a mirror plane, however it is allowed for the CPGEinduced dc current. This distinct symmetric property of CPGE and LPGEinduced dc current can be used to help distinguish different contributions. In fact, this is what we will see in bilayer and trilayer WTe_{2}.
More importantly, nonlinear dc current may switch their direction upon certain ferroelectric transition, giving rise to FNAHE which is the focus of this work. Below we will first reveal the fundamental difference between ferroelectric transitions in bilayer and trilayer WTe_{2}, then demonstrate a striking evenodd anomaly of NAHE, i.e. FNAHE, in bilayer and trilayer WTe_{2} and provide an explanation using group theoretical analysis as well as its implication for potential FNAHEbased quantum devices.
Ferroelectric transition in bilayer and trilayer WTe_{2}
Both bilayer and trilayer WTe_{2} were found to exhibit ferroelectric switching, however their transformation is fundamentally different, which plays a key role in their distinct NAHE. Crystal structures of monolayer, bilayer, and trilayer WTe_{2} are shown in Fig. 1. Monolayer 1T′ WTe_{2} has a C_{2h} point group with a mirror plane symmetry \({\cal{M}}_y\) perpendicular to yaxis and a twofold screw rotation symmetry C_{2y}. This leads to inversion symmetry \({\cal{I}} = {\cal{M}}_yC_{2y}\) or \(C_{2y}{\cal{M}}_y\). Upon van der Waals (vdW) T_{d} stacking, multilayer noncentrosymmetric T_{d} WTe_{2} possesses mirror plane symmetry \({\cal{M}}_y\) only, and no longer holds C_{2y} symmetry as the rotation axes of different layers are not related by any symmetry operation in the point group. Consequently, multilayer T_{d} WTe_{2} loses inversion center with C_{s} point group.
Ferroelectric transition pathways of bilayer and trilayer T_{d} WTe_{2} are shown in Fig. 2. In both cases, two opposite ferroelectric (FE) states (Fig. 2a, c for bilayer, Fig. 2d, f for trilayer) can switch to each other by a small inplane shift between adjacent layers along x by 2d_{x} (d_{x} ~ 20pm), passing through an intermediate paraelectric (PE) state. The intermediate PE state in bilayer WTe_{2} (Fig. 2b) has a C_{2v} point group with additional \(\left\{ {{\cal{M}}_{z}{{}}\frac{1}{2}a} \right\}\) symmetry, thus its outofplane electric polarization P_{z} vanishes. While the ferroelectric transition is achieved by inplane 2d_{x} shift, two FE states are related by a glide plane operation \(\left\{ {{\cal{M}}_z{{}}t_a} \right\}\) consisting of a mirror symmetry operation followed by a translation along x by a fractional translation t_{a} where \(t_{a} = \frac{1}{2}a\). For this reason, we denote the two FE states of bilayer WTe_{2} by −mFE and +mFE (Fig. 2a, c). In contrast, the two opposite FE states in trilayer WTe_{2} are related by an inversion operation \({\cal{I}}\), denoted by −iFE and +iFE (Fig. 2d, f). Furthermore, its intermediate PE state has a C_{2h} point group with inversion symmetry, hence the outofplane polarization P_{z} of the PE state in trilayer WTe_{2} vanishes as well.
Next, we calculate total electric polarization by summing the ionic and electronic contributions. Since we are interested in the polarization along the out of plane direction P_{z}, we can directly integrate the product between charge density/ionic charge and their corresponding position to obtain P_{z} without using Berry phase approach. More specifically, \(P_{z} = \frac{1}{S}( {{\sum\nolimits_I} Q_{I} \cdot \left( {z_{I}  {\boldsymbol{R}}_{0}^z} \right)  e{\int_V} \rho \left( {\boldsymbol{r}} \right)\left( {z  {\boldsymbol{R}}_0^z} \right)d^3r})\), where S is the inplane area of the unit cell, Q is ionic charge, ρ is electronic charge density, and R_{0} is a reference point which is set to the origin of the unit cell in the present case. The equilibrium electronic charge density ρ(r) was obtained from firstprinciples density functional theory (DFT)^{30,31} as implemented in the Vienna Ab initio Simulation Package (VASP).^{32} The calculated total electric polarization P_{z} is ±1.67 × 10^{−2} nm μC/cm^{2} for ±mFE in bilayer, and ±0.81 × 10^{−2} nm μC/cm^{2} for ±iFE in trilayer. This is in good agreement with experimentally measured vertical polarization in bilayer WTe_{2} of ~10^{4} e cm^{−1} (i.e. 1.60 × 10^{−2} nm μC/cm^{2}).^{21} Additionally, the intermediate PE state was recently observed in experiments.^{20} In brief, the results from the DFT calculations confirmed the ferroelectricity in both bilayer and trilayer WTe_{2}, however the symmetry relations between the two FE states are very different in the bilayer and trilayer cases, i.e. −mFE ↔ PE ↔ +mFE and −iFE ↔ PE ↔ +iFE, which is essential for understanding their distinct NAHE upon ferroelectric switching we will discuss shortly.
NAHE in bilayer and trilayer WTe_{2} upon ferroelectric switching
Now we proceed to discuss NAHE in fewlayer WTe_{2}, in particular ferroelectric switching of NAHE (i.e. FNAHE) in oddlayer WTe_{2}, and reveal the intriguing connection between BCD/SD and ferroelectric order. We compute their electronic structure by firstprinciples DFT using hybrid exchangecorrelation functional with spinorbit coupling taken into account. Quasiatomic spinor Wannier functions and tightbinding Hamiltonian were obtained by rotating and optimizing the Bloch functions with a maximal similarity measure with respect to pseudoatomic orbitals.^{33,34} Subsequently, firstprinciples tightbinding approach was applied to compute all the physical quantities such as band structure, BCD, SD, Berry curvature etc. More calculation details can also be found in Methods Section.
Electronic band structure of bilayer WTe_{2} is presented in Fig. 3a, colorcoded by total intraband Berry curvature of all occupied bands, that is, \({\mathrm{\Omega }}_{{\mathrm{occ}}}^{z,{\mathrm{intra}}}\left( {\boldsymbol{k}} \right) = \mathop {\sum }\limits_n f_n{\mathrm{\Omega }}_n^z\left( {\boldsymbol{k}} \right)\). It shows bilayer WTe_{2} is a small gap insulator, and the intraband Berry curvature is odd with respect to Γ due to the presence of timereversal symmetry. The kdependent intraband Berry curvature \({\mathrm{\Omega }}_{{\mathrm{occ}}}^{z,{\mathrm{intra}}}\left( {\boldsymbol{k}} \right)\) are shown in Fig. 3c, d at two different chemical potentials of μ = ±50 meV. Alternatively, one may use the Kubo formula with the sumoverstates approach for Berry curvature (see Supplementary Fig. 2). Similarly, interband Berry curvature \({\mathrm{\Omega }}_{nm}^z\left( {\boldsymbol{k}} \right)\) with at frequency ω = 120 meV is displayed in Fig. 3e, f for two sets of occupied and unoccupied bands around the Fermi energy, \({\mathrm{\Omega }}_{{\mathrm{VBM}}  1,{\mathrm{CBM}}}^{z,{\mathrm{inter}}}\left( {\boldsymbol{k}} \right)\) and \({\mathrm{\Omega }}_{{\mathrm{VBM}},{\mathrm{CBM}}  1}^{z,{\mathrm{inter}}}\left( {\boldsymbol{k}} \right)\), respectively. VBM refers to valence band maximum, and CBM refers to conduction band minimum. The Berry curvature distribution plots confirm the presence of mirror symmetry \({\cal{M}}_{y}\) and timereversal symmetry \({\cal{T}}\). Thus, the integral of the intraband Berry curvature over the full Brillouin zone vanishes, and linear anomalous Hall effect is absent. Furthermore, Fig. 3b shows the calculated BCD and SD tensor elements – \(D_{yz}^{{\mathrm{intra}}}\), \(D_{yz}^{{\mathrm{C}},{\mathrm{inter}}}\), and \(D_{xy}^{{\mathrm{L}},{\mathrm{inter}}}\) – the key physical quantities governing NAHE. It clearly demonstrates the presence of finite BCD and SD and thus NAHE in bilayer WTe_{2}. The calculated BCD varies between 0 and 0.4 Å depending on the chemical potential, which is in nice agreement with the experimental values of 0.1–0.7 Å by Kang et al.^{10}. Moreover, upon ferroelectric transition between −mFE and +mFE state, the Berry curvature, BCD and SD remain unchanged, thus nonlinear anomalous Hall current will not switch direction upon ferroelectric transition in bilayer WTe_{2}. Similarly, the outofplane spin polarization remains unflipped, while the inplane spin polarization is expected to reverse (see Supplementary Figs. 3 and 4). Furthermore, Du et al. recently studied NAHE in bilayer WTe_{2} using a model Hamiltonian and found that, as the SOC strength evolves, BCD becomes strong near tilted band anticrossings and band inversions.^{11} Our firstprinciples results also show large Berry curvature near band anticrossings which is consistent with the conclusion from Du et al.’s analysis. The magnitude of the calculated Berry curvature is similar to that in Ma et al.^{9} for bilayer WTe_{2} in the absence of electric field. The difference in the detailed band structure is mainly due to the electronic structure sensitive to DFT exchangecorrelation functional, vdW functional, and the Wannier function construction. Nevertheless, both our results and the work by Ma et al.^{9} show the nontrivial BCD contribution to NAHE.
Trilayer WTe_{2} is quite different from bilayer WTe_{2}. Figure 4a, b show its electronic band structure of −iFE and +iFE state, respectively. In contrast to the bilayer case, intraband Berry curvature changes sign upon ferroelectric transition. The similar sign change is also evidenced in the opposite kdependent intraband and interband Berry curvature \({\mathrm{\Omega }}_{{\mathrm{occ}}}^{z,{\mathrm{intra}}}\left( {\boldsymbol{k}} \right)\) and \({\mathrm{\Omega }}_{{\mathrm{VBM}}  1,{\mathrm{CBM}}}^{z,{\mathrm{inter}}}\left( {\boldsymbol{k}} \right)\) as displayed in Fig. 4e–h. Consequently, the sign of BCD and SD flips upon ferroelectric transition between −iFE and +iFE, demonstrated in Fig. 4c, d. Therefore, in direct contrast to bilayer WTe_{2}, the nonlinear dc current in trilayer WTe_{2} will switch its direction upon ferroelectric transition. The calculated BCD ranges from 0 to 0.7 Å depending on the chemical potential, also in good agreement with experiment.^{10} Moreover, there is a clear plateau in \(D_{yz}^{{\mathrm{C}},{\mathrm{inter}}}\) marked by purple arrow in Fig. 4c. It is originated from the large joint density of state around 120 meV indicated by purple arrow in Fig. 4a, which remains constant when the chemical potential is located between the energy window. It is also worth to note that, like the bilayer case, the integral of Berry curvature of trilayer WTe_{2} is also zero due to the presence of timereversal symmetry, hence the linear anomalous Hall effect is absent. Both inplane and outofplane spin polarizations are reversed (see Supplementary Figs. 5 and 6). Finally, the dc current susceptibility of bilayer and trilayer WTe_{2} will be reversed in the trilayer case only. Figure 5 shows the interband LPGE susceptibility σ_{abc} of bilayer and trilayer WTe_{2} at μ = 0, which is about 10 times higher than that in monolayer group IV monochalcogenides.^{23} It is clear that in bilayer WTe_{2} the two independent susceptibility tensor elements σ_{xxx} and σ_{xyy} of the ±mFE states remain invariant upon ferroelectric transition, while for trilayer WTe_{2} both σ_{xxx} and σ_{xyy} of the ±iFE states flip the sign.
The above electronic structure results demonstrate a striking difference between bilayer and trilayer WTe_{2}, that is, nonlinear anomalous Hall current flips its direction upon ferroelectric switching in trilayer WTe_{2}, but remains unchanged in bilayer WTe_{2}.
Group theoretical analysis of NAHE in bilayer and FNAHE in trilayer WTe_{2}
Here we provide a group theoretical analysis of NAHE in addition to the above firstprinciples calculations. Both bilayer and trilayer WTe_{2} have C_{s} point group with a mirror symmetry M_{y}. For circularly polarized incident light propagating along z, (E(ω) × E(−ω))_{z}, shares the same A″ representation as axial vector R_{z}. Therefore, \(\Gamma _{j_y} \otimes \Gamma _{{\boldsymbol{R}}_{{\boldsymbol{x}},{\boldsymbol{z}}}} = A^{\prime\prime} \otimes A^{\prime\prime} = A^{\prime}\), suggesting \(\Gamma _{j_y} \otimes \Gamma _{{\boldsymbol{R}}_{{\boldsymbol{x}},{\boldsymbol{z}}}}\) includes total symmetric irreducible representation, and hence nonlinear CPGE current can be induced along y, i.e., perpendicular to the xz mirror plane. Furthermore, \(\Gamma _{j_{x,z}} \otimes \Gamma _{{\boldsymbol{R}}_{{\boldsymbol{x}},{\boldsymbol{z}}}} = A^{\prime\prime}\), thus no CPGE current can be induced along x. In contrast, for linearly polarized incident light/electric field with inplane polarization, we have \({\mathrm{\Gamma }}_{j_x} \otimes {\mathrm{\Gamma }}_{E_x} \otimes {\mathrm{\Gamma }}_{E_x} = A^{\prime} \otimes A^{\prime} \otimes A^{\prime} = A^{\prime}\), and \({\mathrm{\Gamma }}_{j_x} \otimes {\mathrm{\Gamma }}_{E_y} \otimes {\mathrm{\Gamma }}_{E_y} = A^{\prime} \otimes A^{\prime\prime} \otimes A^{\prime\prime} = A^{\prime}\), indicating that the LPGE current can be induced along x. However, \({\mathrm{\Gamma }}_{j_y} \otimes {\mathrm{\Gamma }}_{E_x} \otimes {\mathrm{\Gamma }}_{E_x} = {\mathrm{\Gamma }}_{j_y} \otimes {\mathrm{\Gamma }}_{E_y} \otimes {\mathrm{\Gamma }}_{E_y} = A^{\prime\prime}\), thus no LPGE current can be induced along y. This leads to a contrasting CPGE and LPGEbased nonlinear anomalous Hall current in fewlayer WTe_{2} with C_{s} point group, that is, linearly polarized light/electric field with inplane polarization will generate nonlinear anomalous Hall current along x only \((j_x^L\, \ne\, 0,j_y^L = 0)\), while circularly polarized light propagating along z axis will generate nonlinear anomalous Hall current along y only \((j_x^C = 0,j_y^C \,\ne\, 0)\).
The correlation between the irreducible representations of parent group C_{2h} and its noncentrosymmetric subgroups C_{2}, C_{s}, and C_{1} is summarized in Supplementary Table 1. We start from monolayer 1T′ WTe_{2} which has point group of C_{2h}, whose second order nonlinear current response vanishes due to the presence of inversion symmetry. Upon vdW stacking (e.g. fewlayer and bulk T_{d} WTe_{2}), C_{2y} is broken with M_{y} left unchanged, which breaks the inversion symmetry and results in subgroup C_{s}. Consequently, as we analyzed above, \(j_x^C = 0\), but \(j_y^C \,\ne\, 0\) under circularly polarized light, while \(j_x^L \,\ne\, 0\) but \(j_y^L = 0\) under linearly polarized light/electric field with inplane polarization. However, if M_{y} is broken with C_{2y} being preserved, it will fall into subgroup C_{2}. In this case, \(j_x^C \,\ne\, 0\) and \(j_y^C = 0\) under circularly polarized light, while \(j_x^C = 0\) and \(j_y^C \,\ne\, 0\) under linearly x/ypolarized light/electric field. Furthermore, if both M_{y} and C_{2y} are broken, it will end up with subgroup C_{1}, and enable all possible LPGE and CPGE current responses along different directions.
We now discuss the fundamental difference of NAHE between bilayer and trilayer WTe_{2} upon ferroelectric switching. A general symmetry operator in Seitz notation is given by g = {Rt_{R}}, where R is point group symmetry operation and t_{R} is a translational vector. A timereversal antisymmetric pseudovector (e.g. Berry curvature and spin polarization) transforms under operator g as follows, m’(k) = gm(k) = P_{R}P_{T}Rm(k), where P_{R} and P_{T} are spatial and temporal parity associated with g, respectively. P_{T} = ±1 when Rk = ±k + K, where K is multiples of reciprocal lattice vector. For bilayer WTe_{2}, as aforementioned, two ferroelectric states can be related by a glide plane operation \(\left\{{\mathcal{M}}_{z}  t_{a} \right\}\), where t_{a} refers to a fractional translation along x. Thus, P_{R} = −1, P_{T} = 1, and (m_{x}, m_{y}, m_{z})^{+mFE} = (−m_{x}, −m_{y}, m_{z})^{−mFE}. For trilayer WTe_{2}, the two ferroelectric states are related by an inversion operation \(\left\{{\mathcal{I}}  0 \right\}\), thus P_{R} = P_{T} = −1, subsequently (m_{x}, m_{y}, m_{z})^{+iFE} = (−m_{x}, −m_{y}, −m_{z})^{−iFE}. The above two conclusions are applicable to any timereversal antisymmetric pseudovectors such as Berry curvature and spin polarization. For example, for intraband and interband Berry curvature, \({\mathcal{M}}_{z} {\mathrm{\Omega}}^{z}\left( {k_{x},k_{y}} \right) = {\mathrm{\Omega}}^{z}\left( {k_{x},k_{y}} \right)\) in bilayer WTe_{2}, and \({\mathcal{I}}{\mathrm{\Omega}}^{z}\left( {k_{x},k_{y}} \right) = {\mathrm{\Omega}}^{z}\left( { k_{x},  k_{y}} \right)\mathop { \to }\limits^{TRI}  {\Omega}^{z}\left( {k_{x},k_{y}} \right)\) in trilayer WTe_{2}, indicating that the sign of intraband and interband BCD (\(D_{ab}^{{\mathrm{intra}}}\), \(D_{ab}^{{\mathrm{C}},{\mathrm{inter}}}\)) flips only in trilayer WTe_{2} upon ferroelectric transition. This is in excellent agreement with the firstprinciples calculations shown in Fig. 3c–f and Fig. 4e–h. In addition, the inplane spin polarization switches in both cases, and the outofplane spin polarization becomes reversed in trilayer WTe_{2} while remaining unflipped for bilayer WTe_{2}, which also agrees with the calculations (Supplementary Figs. 3–6). Different from pseudovectors, polar vector p such as electric polarization and shift vector transforms as follows: p′ = Rp Therefore, both mirror \({\mathcal{M}}_{z}\) and inversion \(\cal{I}\) operation will lead to vertical polarization reversal, i.e. \(p_{z}^{\prime} = {\mathcal{M}}_{z}p_{z} =  p_{z}\) and \(p_{z}^{\prime} = {\cal{I}}p_{z} =  p_{z}\), i.e. the outofplane electric dipole flips sign in both bilayer and trilayer WTe_{2} upon ferroelectric transition. In addition, for inplane shift vector \(R_{mn}^{a}\) with a ∈ {x, y}, \(\left(R_{mn}^{a}\right)^{\prime} = {\mathcal{M}}_{z} R_{mn}^{a} = R_{mn}^{a}\), and \(\left(R_{mn}^{a}\right)^{\prime} = {\mathcal{I}}R_{mn}^{a} =  R_{mn}^{a}\), indicating that the inplane shift vector \(R_{mn}^{a}\) and thus SD \(D_{yz}^{{\mathrm{L}},{\mathrm{inter}}}\) will flip only in trilayer WTe_{2} upon ferroelectric transition. Consequently, the total \(j_{x}^{L}\) and \(j_{y}^{C}\) from CPGE and LPGE will switch direction upon ferroelectric transition, provoking FNAHE in timereversal invariant semimetals. Moreover, it suggests that the BCD and SD can serve as distinct order parameters for noncentrosymmetric semimetals. Figure 6a presents an illustrative summary of the transformation of Berry curvature, spin polarization, and electric polarization under different symmetry operation, while Fig. 6b, c show the ferroelectric switching of nonlinear current in the −iFE and +iFE state of trilayer WTe_{2}. Upon the outofplane polarization switching, nonlinear Hall current \(j_{x}^{L}\) generated via LPGE switches between −x and +x direction under the same external electric field with inplane linear polarization. Moreover, nonlinear Hall current \(j_{x}^{C}\) induced by CPGE switches between −y and +y direction under circularlypolarized light with normal incidence. It’s worth to emphasize that the intermediate PE state in bilayer and trilayer WTe_{2} has noncentrosymmetric C_{2v} and centrosymmetric C_{2h} point group, respectively. Thus, despite that the outofplane electric polarization vanishes in both cases, nonlinear anomalous Hall current of the PE state vanishes in trilayer, but remains finite in bilayer.
The present work considers intrinsic NAHE due to BCD. Disorder however can play an important role in NAHE as pointed out by Du et al.^{35} and Isobe et al.^{36} particularly in the dc limit due to side jump and skew scattering.^{37} As disorder scattering depends on scattering potential and defect density, further experimental studies are required to understand the nature of defects in bilayer and trilayer WTe_{2}. The transformation behavior of BCDinduced NAHE in multilayer WTe_{2} upon ferroelectric transition may be utilized to distinguish itself from the disorder scatteringinduced NAHE. For example, upon ferroelectric transition, the change in defect scattering potential may behave very differently from the change in crystal structure, thereby potentially helping differentiate the two NAHE contributions.
In conclusion, using firstprinciples calculations and group theoretical analyses we investigated the NAHE in bilayer and trilayer WTe_{2} and, more importantly, the underlying microscopic origin of FNAHE (i.e., ferroelectric switching of NAHE) in trilayer WTe_{2}. Although both bilayer and trilayer WTe_{2} exhibit ferroelectric transition with similar electric polarization, they behave very differently in NAHE. In the trilayer case, the nonlinear anomalous Hall current flips direction upon ferroelectric switching due to the reversal of BCD and SD under an effective inversion operation of the two ferroelectric states. In contrast, the two ferroelectric states in bilayer WTe_{2} are related effectively by a glide plane operation which does not flip the BCD/SD, thus its nonlinear anomalous Hall current will not flip upon ferroelectric switching. In addition, NAHE is expected to vanish in the PE state of trilayer WTe_{2}, but remains nontrivial for the PE state of the bilayer case. The above conclusions are applicable to any even and odd layer WTe_{2} (except monolayer 1T′ WTe_{2} as it is centrosymmetric with vanishing second order NAHE) as long as the two opposite ferroelectric states have the same relationship as the bilayer and trilayer case. The theoretical approaches presented here can also be applied to other materials such as Weyl semimetals.^{24,25}
More importantly, our results imply that BCD and SD can serve as new order parameters for noncentrosymmetric materials, which opens up the possibility to explore nonlinear multiferroicity based on the coupling of BCD/SD and ferroelectric order. Ferroelectric metals may be advantageous as their vanishing bandgap will not only bring intraband contributions to nonlinear anomalous Hall current that is absent in semiconductors/insulators, but also significantly enhance the interband contributions due to the reduced gap of nonlinear interband processes. For example, the calculated nonlinear anomalous Hall current from interband LPGE in bilayer and trilayer WTe_{2} is about one order of magnitude higher than that in ferroelectric GeS.^{23} Moreover, FNAHE provides a facile approach for direct readout of ferroelectric states, which, combined with vertical ferroelectric writing, may allow for realizing nonlinear multiferroic memory. In addition, the distinct ferroelectric transformation pathway may provide potential routes to realizing nonabelian reciprocal braiding of Weyl nodes.^{38} The present findings therefore reveal an underexplored realm beyond classical linear Hall effect and conventional ferroelectrics with exciting new opportunities for FNAHEbased nonlinear quantum electronics using ferroelectric metals and Weyl semimetals.
Methods
Firstprinciples calculations of atomistic and electronic structure
Firstprinciples calculations for structural relaxation and electric polarization were performed using densityfunctional theory^{30,31} as implemented in the Vienna Ab initio Simulation Package (VASP)^{32} with the projectoraugmented wave method.^{39} We employed the generalizedgradient approximation of exchangecorrelation functional in the Perdew–Burke–Ernzerhof form,^{40} a planewave basis with an energy cutoff of 300 eV, a Monkhorst–Pack kpoint sampling of 6 × 12 × 1 for the Brillouin zone integration, and optB88vdW functional^{41} to account for dispersion interactions. Ground state crystal structures were obtained by fully relaxing both atomic positions and inplane lattice parameters while keeping a large vacuum region of ~20 Å along the outofplane direction to reduce the periodic image interactions. The convergence criteria for maximal residual force was <0.005 eV/Å, and the convergence criteria for electronic relaxation is 10^{−6} eV. We have tested higher energy cutoff of 400 eV, and the difference in the lattice constants is <0.04%. Crystal structures of bilayer and trilayer WTe_{2} can be found in the Supplementary Information. In addition, total electric polarization was calculated by directly integrating the product of charge density/ionic charge and their corresponding position without using Berry phase approach.
Firstprinciples electronic structure calculations of NAHE
To compute the NHLErelated quantities, we first construct quasiatomic spinor Wannier functions and tightbinding Hamiltonian from Kohn–Sham wavefunctions and eigenvalues under the maximal similarity measure with respect to pseudoatomic orbitals.^{33,34} Spinorbit coupling is taken into account, and hybrid exchangecorrelation energy functional HSE06^{42} is employed with the rangeseparation parameter λ = 0.2 (see Supplementary Information for more details). Total 112 and 168 quasiatomic spinor Wannier functions were obtained for bilayer and trilayer WTe_{2}, respectively. Using the developed tightbinding Hamiltonian we then compute CPGE and LPGE susceptibility tensor with a modified WANNIER90 code^{43} using a dense kpoint sampling of 600 × 600 × 1 for both bilayer and trilayer WTe_{2}. A small imaginary smearing factor η of 0.05 eV is applied to fundamental frequency, and Sokhotski–Plemelj theorem is employed for the Dirac delta function integration. In addition, we tested the range separation parameter λ in the hybrid HSE functional. Although the values of Berry curvature etc. can change with respect to λ, the presence (absence) of Berry curvature switching in trilayer (bilayer) remains the same (see Supplementary Fig. 3 for HSE functional with λ = 0.4). We also checked the convergence of the kpoint sampling by increasing it to 1000 × 1000 × 1 (Supplementary Fig. 7). Finally, since fewlayer WTe_{2} is either semimetallic or having very small gap, the dielectric screening is large, thus the effect of the Coulombic interaction between electrons and holes is negligible.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.
References
 1.
Moore, J. E. & Orenstein, J. Confinementinduced berry phase and helicitydependent photocurrents. Phys. Rev. Lett. 105, 026805 (2010).
 2.
Sodemann, I. & Fu, L. Quantum Nonlinear Hall effect induced by berry curvature dipole in timereversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015).
 3.
Low, T., Jiang, Y. & Guinea, F. Topological currents in black phosphorus with broken inversion symmetry. Phys. Rev. B 92, 235447 (2015).
 4.
Deyo, E., Golub, L., Ivchenko, E. & Spivak, B. Semiclassical theory of the photogalvanic effect in noncentrosymmetric systems. Preprint at arXiv:0904.1917 (2009).
 5.
Xu, S.Y. et al. Electrically switchable Berry curvature dipole in the monolayer topological insulator WTe_{2}. Nat. Phys. 14, 900–906 (2018).
 6.
Zhang, Y., van den Brink, J., Felser, C. & Yan, B. Electrically tuneable nonlinear anomalous Hall effect in twodimensional transitionmetal dichalcogenides WTe_{2} and MoTe_{2}. 2D Mater. 5, 044001 (2018).
 7.
You, J.S., Fang, S., Xu, S.Y., Kaxiras, E. & Low, T. Berry curvature dipole current in the transition metal dichalcogenides family. Phys. Rev. B 98, 121109 (2018).
 8.
Shi, L.K & Song, J. C. W. Symmetry, spintexture, and tunable quantum geometry in a WTe_{2} monolayer. Phys. Rev. B 99, 035403 (2019).
 9.
Ma, Q. et al. Observation of the nonlinear Hall effect under timereversalsymmetric conditions. Nature 565, 337–342 (2019).
 10.
Kang, K., Li, T., Sohn, E., Shan, J. & Mak, K. F. Nonlinear anomalous Hall effect in fewlayer WTe_{2}. Nat. Mater. 18, 324–328 (2019).
 11.
Du, Z. Z., Wang, C. M., Lu, H.Z. & Xie, X. C. Band signatures for strong nonlinear Hall effect in Bilayer WTe_{2}. Phys. Rev. Lett. 121, 266601 (2018).
 12.
Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in twodimensional transition metal dichalcogenides. Science 346, 1344–1347 (2014).
 13.
Tang, S. et al. Quantum spin Hall state in monolayer 1T′WTe_{2}. Nat. Phys. 13, 683–687 (2017).
 14.
Fei, Z. et al. Edge conduction in monolayer WTe_{2}. Nat. Phys. 13, 677–682 (2017).
 15.
Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76 (2018).
 16.
Sajadi, E. et al. Gateinduced superconductivity in a monolayer topological insulator. Science 362, 922–925 (2018).
 17.
Fatemi, V. et al. Electrically tunable lowdensity superconductivity in a monolayer topological insulator. Science 362, 926–929 (2018).
 18.
Soluyanov, A. A. et al. TypeII Weyl semimetals. Nature 527, 495–498 (2015).
 19.
Ali, M. N. et al. Large, nonsaturating magnetoresistance in WTe_{2}. Nature 514, 205–208 (2014).
 20.
Sie, E. J. et al. An ultrafast symmetry switch in a Weyl semimetal. Nature 565, 61–66 (2019).
 21.
Fei, Z. et al. Ferroelectric switching of a twodimensional metal. Nature 560, 336 (2018).
 22.
Anderson, P. W. & Blount, E. I. Symmetry considerations on martensitic transformations: “Ferroelectric” metals? Phys. Rev. Lett. 14, 217–219 (1965).
 23.
Wang, H. & Qian, X. Ferroicitydriven nonlinear photocurrent switching in timereversal invariant ferroic materials. Sci. Adv. 5, eaav9743 (2019).
 24.
Li, R. et al. Weyl ferroelectric semimetal. Preprint at arXiv:1610.07142 (2016).
 25.
Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transitionmetal monophosphides. Phys. Rev. X 5, 011029 (2015).
 26.
Sipe, J. E. & Shkrebtii, A. I. Secondorder optical response in semiconductors. Phys. Rev. B 61, 5337–5352 (2000).
 27.
Tsirkin, S. S., Puente, P. A. & Souza, I. Gyrotropic effects in trigonal tellurium studied from first principles. Phys. Rev. B 97, 035158 (2018).
 28.
Morimoto, T., Zhong, S., Orenstein, J. & Moore, J. E. Semiclassical theory of nonlinear magnetooptical responses with applications to topological Dirac/Weyl semimetals. Phys. Rev. B 94, 245121 (2016).
 29.
Rostami, H. & Polini, M. Nonlinear anomalous photocurrents in Weyl semimetals. Phys. Rev. B 97, 195151 (2018).
 30.
Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. B 136, B864–B871 (1964).
 31.
Kohn, W. & Sham, L. J. Selfconsistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).
 32.
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
 33.
Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).
 34.
Qian, X. et al. Quasiatomic orbitals for ab initio tightbinding analysis. Phys. Rev. B 78, 245112 (2008).
 35.
Du, Z. Z., Wang, C. M., Li, S., Lu, H.Z. & Xie, X. C. Disorderinduced nonlinear Hall effect with timereversal symmetry. Nat. Commun. 10, 3047 (2019).
 36.
Isobe, H., Xu, S.Y. & Fu, L. Highfrequency rectification via chiral electrons in nonlinear crystals. Preprint at arXiv:1812.08162 (2018).
 37.
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).
 38.
Bouhon, A., Slager, R.J. & Bzdušek, T. Nonabelian reciprocal braiding of Weyl Nodes. Preprint at arXiv:1907.10611 (2019).
 39.
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953–17979 (1994).
 40.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
 41.
Klimeš, J., Bowler, D. R. & Michaelides, A. Chemical accuracy for the van der Waals density functional. J. Phys. Condens. Matter 22, 022201 (2010).
 42.
Krukau, A. V., Vydrov, O. A., Izmaylov, A. F. & Scuseria, G. E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys. 125, 224106 (2006).
 43.
Mostofi, A. A. et al. An updated version of wannier90: a tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 185, 2309–2310 (2014).
Acknowledgements
This work was supported by the National Science Foundation (NSF) under award number DMR1753054. Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing.
Author information
Affiliations
Contributions
X.Q. conceived the project. H.W. and X.Q. developed firstprinciples tightbinding approach for computing nonlinear susceptibility tensor. H.W. performed the calculations. Both H.W. and X.Q. analyzed the results and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wang, H., Qian, X. Ferroelectric nonlinear anomalous Hall effect in fewlayer WTe_{2}. npj Comput Mater 5, 119 (2019). https://doi.org/10.1038/s4152401902571
Received:
Accepted:
Published:
Further reading

Large family of twodimensional ferroelectric metals discovered via machine learning
Science Bulletin (2021)

Strain Tunable Berry Curvature Dipole, Orbital Magnetization and Nonlinear Hall Effect in WSe2 Monolayer*
Chinese Physics Letters (2021)

Electrically and magnetically switchable nonlinear photocurrent in РТsymmetric magnetic topological quantum materials
npj Computational Materials (2020)

Semiclassical theory of the circular photogalvanic effect in gyrotropic systems
Physical Review B (2020)

Electrical detection of ferroelectriclike metals through the nonlinear Hall effect
Physical Review B (2020)