www.nature.com/npjcompumats ARTICLE OPEN Ferroelectric nonlinear anomalous Hall effect in few-layer WTe2 Hua Wang1 and Xiaofeng Qian1* 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 time-reversal invariant materials – an underexplored realm with exciting new opportunities beyond classical linear Hall effect. Here, using group theory and first-principles theory, we demonstrate a remarkable ferroelectric nonlinear anomalous Hall effect in time-reversal invariant few-layer WTe2 where nonlinear anomalous Hall current switches in odd-layer WTe2 except 1T′ monolayer while remaining invariant in even-layer WTe2 upon ferroelectric transition. This even-odd 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 odd-layer WTe2. 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. 1234567890():,; npj Computational Materials (2019) 5:119 ; https://doi.org/10.1038/s41524-019-0257-1 INTRODUCTION what’s the fundamental correspondence between NAHE and In classical linear Hall effect, a transverse voltage can be ferroelectricity in ferroelectric metals and Weyl semimetals? 23 developed in materials with broken time-reversal symmetry only Compared to ferroelectric semiconductors, what are the unique 21 (e.g. in the presence of external magnetic field or internal advantages of ferroelectric metals and ferroelectric Weyl magnetization) due to Onsager’s relation. Second and higher semimetals?24,25 order conductivity tensors, however, are not subject to this Here using first-principles approach and group theoretical constraint, thereby enabling nonlinear anomalous Hall effect analysis we show an intriguing ferroelectric nonlinear anomalous 1–4 (NAHE) in time-reversal invariant system. NAHE was observed Hall effect (FNAHE) in time-reversal invariant few-layer WTe2.In 5–11 very recently in few-layer tungsten ditelluride (WTe2), a layered particular, while both bilayer and trilayer WTe2 possess switchable material which also holds rich physics including high-temperature out-of-plane electric polarization, nonlinear transverse Hall current 12–15 quantum spin Hall phase and electrostatic gating induced only switches in trilayer WTe2 upon ferroelectric switching. The 16,17 superconductivity in its 1T′ monolayer and type-II Weyl microscopic origin of FNAHE in trilayer WTe2 is found to be rooted semimetallicity,18 large non-saturating magnetoresistance19 and in the reversal of Berry curvature dipole and shift dipole upon ultrafast symmetry switching20 in its bulk phase. ferroelectric transition, which reveals an exciting yet unexplored Monolayer 1T′ WTe2 is centrosymmetric with vanishing even- realm of ferroelectric metals and Weyl semimetals with potential order nonlinear current response, however vertical electric field applications in nonlinear electronics. can break its two-fold screw rotation symmetry, generate Berry curvature dipole (BCD), and induce second-order nonlinear 5–8 anomalous Hall current. In contrast to monolayer WTe2, bilayer RESULTS AND DISCUSSION WTe2 is naturally noncentrosymmetric due to the loss of two-fold screw rotation symmetry, resulting in intrinsic nontrivial BCD in Second-order dc current 9–11 fi Er; E ω iðÞkÁrÀωt bilayer WTe2. Surprisingly, ferroelectric switching was recently Consider an oscillating electric eld ðÞ¼t ðÞe þ 21 E ω ÀiðÞkÁrÀωt E ω = E* −ω fi discovered in semimetallic bilayer and few-layer WTe2, quite ðÞÀ e with ( ) ( ) (e.g. under AC electric eld unusual as ferroelectricity and semimetallicity normally do not co- or upon coherent light illumination), the second-order nonlinear exist in the same material.22 The subtlety lies in the reduced dc current under minimal coupling approximation was derived by 26 0 χ ; ω; ω ω ω χ screening along the out-of-plane direction which gives rise to Sipe et al. , i.e., ja ¼ abcðÞ0 À EbðÞEcðÀ Þ, where abc are 0 finite out-of-plane ferroelectric polarization while preserving in- the dc photocurrent susceptibility. In general ja consists of two plane semimetallic nature. Conductance hysteresis persisting up parts depending on the polarization of electric field/incident light, to 300 K shows its great potential for room temperature device including linear photogalvanic effect (LPGE) and circular photo- 2,26,27 0 L C application. These recent studies combined reveal a striking galvanic effect (CPGE), i.e., ja ¼ ja þ ja . BCD-induced non- feature of noncentrosymmetric few-layer WTe2 – the coexistence linear photocurrent current was generalized to the multiple-band of ferroelectricity and NAHE within a single material, enkindling a case by Morimoto et al.28 using Floquet theory and Rostami et al.29 few fundamentally and technologically important questions: using density matrix beyond semiclassical Boltzmann theory. 1Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA. *email: [email protected] Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences H. Wang and X. Qian 2 Nonlinear photocurrent originating from CPGE is also known as (a)monolayer WTe 2 (b)bilayer WTe2 (c) trilayer WTe 2 injection current.26 Both LPGE and CPGE have intraband and interband contribu- tions. For the sake of completeness we include all the terms as y ( ) y follows, ( ÀÁ L e3 τ ϵ intra ω ω x ( ) j ; ¼À2 2 Re D ReðÞE ð ÞE ðÀ Þ L L L a intra h 1Àiωτ adc bd b c ja ¼ ja;intra þ ja;inter 3 ; jL 2 e τDL interRe E ω E ω a;inter ¼À h2 a;bc ðÞbð Þ cðÀ Þ (1) 2y ( ÀÁ z ( ) C e3 τ intra E ω ´ E ω j ; ¼À 2 Im D ImðÞðÞ ðÞÀ C C C a intra h 1Àiωτ ab b ja ¼ ja;intra þ ja;inter 3 ; (2) jC e τDC interIm E ω ´ E ω x ( ) a;inter ¼À2h2 ab ðÞðÞ ðÞÀ b 1T’ (C 2h ) T d (Cs ) T d (Cs ) Here τ is relaxation time and ϵ is the Levi–Civita symbol. Dintra adc ; ab is the well-known BCD for intraband nonlinear process.2 DC inter is Fig. 1 Crystal structure of monolayer, bilayer, and trilayer WTe2. ; ab ′ BCD for interband process associated with CPGE.9 DL inter is shift a Monolayer 1T WTe2 with centrosymmetric C2h point group. It has a;bc a mirror plane M and a screw rotation symmetry C , which leads dipole (SD), originated from the simultaneous displacement of y 2y to the inversion symmetry I¼MC . b, c Bilayer and trilayer T wavepacket upon excitation. More specifically, they are given by y 2y d 8 R R P WTe2 with Cs point group. C2y symmetry is broken, hence the intra μ μ ∂ Ωb k μ a k Ωb k δ ω k μ inversion symmetry I is also broken in bilayer and trilayer WTe2. > Dab ðÞ¼ f0ðÞa ¼ ½d n fnðÞvn ðÞ nðÞðh nð ÞÀ Þ <> BZ R P BZ C;inter a b τ D ðÞ¼μ; ω ½dk fnmðÞμ Δ ðÞk Ω ðÞk Re ω ω τ > ab BZ mn mn nm 1ÀiðÞÀ mn > R P ÈÉsymmetry properties. The LPGE-induced dc current cannot flow : L;inter a b c 1 D ðÞ¼μ; ω ½dk fnmðÞμ R ðÞk r ; r Re a;bc BZ mn mn nm mn 1ÀiðÞωÀωmn τ normal to a mirror plane, however it is allowed for the CPGE- (3) induced dc current. This distinct symmetric property of CPGE- and LPGE-induced dc current can be used to help distinguish different ω k b k μ Here, h nð Þ, vnðÞ, and fn( ) are band energy, group velocity, contributions. In fact, this is what we will see in bilayer and trilayer and chemical-potential μ dependent Fermi-Dirac distribution, d d WTe2. 1234567890():,; μ ≡ μ − μ k ≡ π respectively. fnm( ) fn( ) fm( ), and [d ] d k/(2 ) for d- More importantly, nonlinear dc current may switch their Δa a a dimension integral. nm vn À vm is the group velocity difference a direction upon certain ferroelectric transition, giving rise to FNAHE between two bands. rnm is interband Berry connection or dipole fi Ωc k which is the focus of this work. Below we will rst reveal the matrix element. nmðÞis the interband Berry curvature between fundamental difference between ferroelectric transitions in bilayer c a b c two bands, defined as Ω ðÞk iϵabcr r : Ω ðÞk is the nm nm mn n and trilayer WTe2, then demonstrate a striking even-odd anomaly intrabandP Berry curvature forÈÉ band n, given by of NAHE, i.e. FNAHE, in bilayer and trilayer WTe and provide an Ωc k Ωc k b ; c b c c b 2 nðÞ¼ n≠m mnðÞ. In addition, rnm rmn rnmrmn þ rmnrnm. explanation using group theoretical analysis as well as its ∂ϕ a a mnðÞk a k a k Rmn is shift vector, given by Rmn À ∂ka þ rmmðÞÀrnnðÞ, implication for potential FNAHE-based quantum devices. where ϕmn(k) is the phase factor of the interband Berry a Ωb k Ωb k connection and rnn is intraband Berry connection. nmðÞ, nðÞ Ferroelectric transition in bilayer and trilayer WTe2 a and Rmn are all gauge invariant. For linearly polarized incident Both bilayer and trilayer WTe2 were found to exhibit ferroelectric fi = L;inter L;inter light/electric eld, Eb Ec, hence we denote Dab Da;bc .