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 ½Šd n fnðÞvn ðÞ nðÞðh nð ÞÀ Þ <> BZ R P BZ  C;inter a b τ D ðÞ¼μ; ω ½dkŠ 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 ½dkŠ 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 . The switching, however their transformation is fundamentally differ- ; ; intraband and interband BCDs (Dintra, DC inter) as well as SD ðDL interÞ ent, which plays a key role in their distinct NAHE. Crystal structures − ab ab ab have the same units of L3 d for d-dimensional system. Thus, of monolayer, bilayer, and trilayer WTe2 are shown in Fig. 1. ′ BCD and SD have units of length in 2D, but become Monolayer 1T WTe2 has a C2h point group with a mirror plane dimensionless in 3D. symmetry My perpendicular to y-axis and a two-fold screw The appearance of relaxation time τ in the dc current from rotation symmetry C2y. This leads to inversion symmetry I¼ L τ MyC2y or C2yMy. Upon van der Waals (vdW) Td stacking, interband LPGE ðja;interÞ seems different from the widely-used - independent shift current formula by Sipe et al.26, however the latter multilayer noncentrosymmetric Td WTe2 possesses mirror plane symmetry M only, and no longer holds C symmetry as the was derived for the clean limit when relaxation time τ approaches to y 2y 1 rotation axes of different layers are not related by any symmetry infinite. In fact, as τ → ∞, τRe ω ω τ ! πδðÞ ωmn À ω ,andthe 1ÀiðÞÀ mn operation in the point group. Consequently, multilayer Td WTe2 original τ-time independent shift current susceptibility can be loses inversion center with Cs point group. L exactly recovered from the above ja;inter formula. In reality, Ferroelectric transition pathways of bilayer and trilayer Td WTe2 fi L are shown in Fig. 2. In both cases, two opposite ferroelectric (FE) quasiparticles do have nite relaxation time, thus ja;inter shall depend on relaxation time. More detailed derivation about the SD and BCD states (Fig. 2a, c for bilayer, Fig. 2d, f for trilayer) can switch to each can be found in the Supplementary Information. other by a small in-plane shift between adjacent layers along x by Moreover, it is worth to classify the contributions of LPGE/CPGE- 2dx (dx ~ 20pm), passing through an intermediate paraelectric (PE) induced dc current at the low/high frequency region. At the low state. The intermediate PE stateÈÉ in bilayer WTe2 (Fig. 2b) has a C2v τ 1 ωτ → τ point group with additional Mzj a symmetry, thus its out-of- frequency limit, 0, hence 1Àiωτ ! . In this case, the 2 photocurrent due to both intraband and interband CPGE as well plane electric polarization Pz vanishes. While the ferroelectric as interband LPGE will vanish, however a dc current from transition is achieved by in-plane 2dx shift, two FE states are intraband LPGE will remain finite which is perpendicular to the related by a glide plane operation fgMzjta consisting of a mirror applied electric field, thereby inducing static NAHE. At high symmetry operation followed by a translation along x by a 1 frequency, CPGE (i.e. injection current) and interband LPGE (i.e. fractional translation ta where ta ¼ 2 a. For this reason, we denote shift current) will have nontrivial contribution to total nonlinear the two FE states of bilayer WTe2 by −mFE and +mFE (Fig. 2a, c). photocurrent, referred as to dynamic NAHE. In contrast, the two opposite FE states in trilayer WTe2 are related It is important to note that the direction of nonlinear by an inversion operation I, denoted by −iFE and +iFE (Fig. 2d, f). photocurrent induced by CPGE and LPGE have very different Furthermore, its intermediate PE state has a C2h point group with

npj Computational Materials (2019) 119 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences H. Wang and X. Qian 3

Ferroelectric Transition in Bilayer WTe2 NAHE in bilayer and trilayer WTe2 upon ferroelectric switching Now we proceed to discuss NAHE in few-layer WTe2, in particular (a) -mFE (C s ) (b) PE (C 2v ) (c) +mFE (C s ) ferroelectric switching of NAHE (i.e. FNAHE) in odd-layer WTe2, and reveal the intriguing connection between BCD/SD and ferro- electric order. We compute their electronic structure by first- + 1 z 2 a principles DFT using hybrid exchange-correlation functional with spin-orbit coupling taken into account. Quasiatomic spinor Wannier functions and tight-binding Hamiltonian were obtained -dx -Pz d x =0 P z =0 +dx +Pz by rotating and optimizing the Bloch functions with a maximal similarity measure with respect to pseudoatomic orbitals.33,34 Subsequently, first-principles tight-binding approach was applied Ferroelectric Transition in Trilayer WTe 2 to compute all the physical quantities such as band structure, BCD, (d) -iFE (Cs ) (e) PE (C 2h ) (f) +iFE (C s ) SD, Berry curvature etc. More calculation details can also be found in Methods Section. Electronic band structure of bilayer WTe2 is presented in Fig. 3a, color-coded by total intrabandP Berry curvature of all occupied Ωz;intra k Ωz k bands, that is, occ ðÞ¼ fn nðÞ. It shows bilayer WTe2 is a small gap insulator, and then intraband Berry curvature is odd with respect to Γ due to the presence of time-reversal symmetry. The k- Ωz;intra k dependent intraband Berry curvature occ ðÞare shown in Fig. 3c, d at two different chemical potentials of μ = ±50 meV. Alternatively, one may use the Kubo formula with the sum-over- -dx -Pz d x =0 P z =0 +dx +Pz states approach for Berry curvature (see Supplementary Fig. 2). Similarly, interband Berry curvature Ωz ðÞk with at frequency ω = – nm Fig. 2 Ferroelectric transition in bilayer and trilayer WTe2.ac 120 meV is displayed in Fig. 3e, f for two sets of occupied and ; ferroelectric transition in bilayer WTe2. The two opposite ferro- Ωz inter k unoccupied bands around the Fermi energy, VBMÀ1;CBMðÞand electric states (−mFE and +mFE) in bilayer are transformed through Ωz;inter k VBM;CBM 1ðÞ, respectively. VBM refers to valence band max- a glide plane operation fgMzjta , that is, a mirror operation Mz À followed by an in-plane shift along x by t with t ¼ 1 a. The imum, and CBM refers to conduction band minimum. The Berry a a 2 curvature distribution plots confirm the presence of mirror intermediate PE state C2v point group, thus its out-of-planeÈÉ 1 symmetry My and time-reversal symmetry T . Thus, the integral polarization with vanishes due to the glide plane Mzj a 2 of the intraband Berry curvature over the full Brillouin zone symmetry. d–f ferroelectric transition in trilayer WTe2. The two opposite ferroelectric states (−iFE and +iFE) in trilayer are related to vanishes, and linear anomalous Hall effect is absent. Furthermore, – intra each other through an inversion operation fgIj0 . The intermediate Fig. 3b shows the calculated BCD and SD tensor elements Dyz , C;inter L;inter – PE state of trilayer WTe2 has C2h point group, thus its out-of-plane Dyz , and Dxy the key physical quantities governing NAHE. It polarization also vanishes due to inversion symmetry. The red and clearly demonstrates the presence of finite BCD and SD and thus green vertical dashed lines show the relative shift ±d between x NAHE in bilayer WTe2. The calculated BCD varies between 0 and adjacent WTe2 layers. The corresponding in-plane shift is very small ≈ 0.4 Å depending on the chemical potential, which is in nice (dx 20 pm), therefore it is exaggerated in the above plots for – illustrative purpose. 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 inversion symmetry, hence the out-of-plane polarization Pz of the PE state in trilayer WTe vanishes as well. direction upon ferroelectric transition in bilayer WTe2. Similarly, 2 fl Next, we calculate total electric polarization by summing the ionic the out-of-plane spin polarization remains un ipped, while the in- and electronic contributions. Since we are interested in the plane spin polarization is expected to reverse (see Supplementary Figs. 3 and 4). Furthermore, Du et al. recently studied NAHE in polarization along the out of plane direction Pz, we can directly integrate the product between charge density/ionic charge and bilayer WTe2 using a model Hamiltonian and found that, as the their corresponding position to obtain P without using Berry phase SOC strength evolves, BCD becomes strong near tilted band Pz ÀÁR anticrossings and band inversions.11 Our first-principles results approach. More specifically, P ¼ 1 ð Q Á z À Rz À e ρðÞr ÀÁ z S I I I 0 V also show large Berry curvature near band anticrossings which is z À Rz d3rÞ,whereS is the in-plane area of the unit cell, Q is 0 consistent with the conclusion from Du et al.’s analysis. The ionic charge, ρ is electronic charge density, and R is a reference 0 magnitude of the calculated Berry curvature is similar to that in Ma point which is set to the origin of the unit cell in the present case. 9 fi ρ r et al. for bilayer WTe2 in the absence of electric eld. The The equilibrium electronic charge density ( )wasobtainedfrom difference in the detailed band structure is mainly due to the first-principles density functional theory (DFT)30,31 as implemented 32 electronic structure sensitive to DFT exchange-correlation func- in the Vienna Ab initio Simulation Package (VASP). The calculated tional, vdW functional, and the Wannier function construction. −2 μ 2 total electric polarization Pz is ±1.67 × 10 nm C/cm for ±mFE in Nevertheless, both our results and the work by Ma et al.9 show the −2 μ 2 bilayer, and ±0.81 × 10 nm C/cm for ±iFE in trilayer. This is in nontrivial BCD contribution to NAHE. good agreement with experimentally measured vertical polarization Trilayer WTe is quite different from bilayer WTe . Figure 4a, b 4 −1 −2 μ 2 21 2 2 in bilayer WTe2 of ~10 ecm (i.e. 1.60 × 10 nm C/cm ). show its electronic band structure of −iFE and +iFE state, Additionally, the intermediate PE state was recently observed in respectively. In contrast to the bilayer case, intraband Berry 20 experiments. In brief, the results from the DFT calculations curvature changes sign upon ferroelectric transition. The similar fi con rmed the ferroelectricity in both bilayer and trilayer WTe2, sign change is also evidenced in the opposite k-dependent Ωz;intra k however the symmetry relations between the two FE states are intraband and interband Berry curvature occ ðÞ and very different in the bilayer and trilayer cases, i.e. −mFE ↔ PE ↔ Ωz;inter k – VBMÀ1;CBMðÞas displayed in Fig. 4e h. Consequently, the sign +mFE and −iFE ↔ PE ↔ +iFE, which is essential for understanding of BCD and SD flips upon ferroelectric transition between −iFE their distinct NAHE upon ferroelectric switching we will discuss and +iFE, demonstrated in Fig. 4c, d. Therefore, in direct contrast shortly. to bilayer WTe2, the nonlinear dc current in trilayer WTe2 will

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 119 H. Wang and X. Qian 4

2 (a)Å (b) 0.4 0.2 1000 intra D yz 0.3 C,inter D yz L,inter 0.1 500 0.2 D xy

Å) 0.1

0 0 0

-0.1 Energy (eV) BCD, SD ( -0.1 -500 -0.2

-0.3 -0.2 -1000 -Y Γ Y -0.4 k -0.2 -0.1 0 0.1 0.2 y (eV)

(c)z, intra 2 (d) z, intra Ω (-50 meV) 200 Å Ω (+50 meV) 2 k occ occ 200 Å x Γ Y

2 2 -200 Å -200 Å X k y

(e) z, inter 2 (f) z, inter 2 Ω 400 Å Ω 400 Å k VBM-1,CBM VBM,CBM+1 x

2 2 -400 Å -400 Å

Fig. 3 Electronic structure and NAHE of bilayer WTe2.aBand structure of bilayer WTe2 in ±mFE state color-coded by the z-component of Ωz k intraband Berry curvature nðÞ. Spin-orbit coupling is included and hybrid HSE06 functional is employed. b BCD and SD tensor elements intra μ C;inter μ; ω L;inter μ; ω μ ω k Dyz ðÞ, Dyz ðÞ, and Dxy ðÞas function of chemical potential . For interband BCD and SD, is set to 120 meV.c, d -dependent Ωz;intra k μ = k distribution of intraband Berry curvature occ ðÞat ±50 meV, respectively. e, f -dependent distribution interband Berry curvature Ωz;inter k = = + − nm ðÞfor (n, m) (VBM-1, CBM) and for (n, m) (VBM-1,CBM), respectively. The results are the same for both mFE and mFE state, suggesting that nonlinear anomalous Hall current can be induced in bilayer WTe2, but it will not switch sign upon ferroelectric transition.

switch its direction upon ferroelectric transition. The calculated (−ω))z, shares the same A″ representation as axial vector Rz. Γ Γ 00 00 0 Γ Γ BCD ranges from 0 to 0.7 Å depending on the chemical potential, Therefore, jy Rx;z ¼ A A ¼ A , suggesting jy Rx;z includes also in good agreement with experiment.10 Moreover, there is a total symmetric irreducible representation, and hence nonlinear C;inter clear plateau in Dyz marked by purple arrow in Fig. 4c. It is CPGE current can be induced along y, i.e., perpendicular to the xz Γ Γ 00 originated from the large joint density of state around 120 meV mirror plane. Furthermore, jx;z Rx;z ¼ A , thus no CPGE current indicated by purple arrow in Fig. 4a, which remains constant when can be induced along x. In contrast, for linearly polarized incident the chemical potential is located between the energy window. It is light/electric field with in-plane polarization, we have Γ Γ Γ 0 0 0 0 Γ Γ Γ also worth to note that, like the bilayer case, the integral of Berry jx Ex Ex ¼ A A A ¼ A , and jx Ey Ey ¼ 0 00 00 0 curvature of trilayer WTe2 is also zero due to the presence of time- A A A ¼ A , indicating that the LPGE current can be Γ Γ Γ Γ Γ Γ 00 reversal symmetry, hence the linear anomalous Hall effect is induced along x. However, jy Ex Ex ¼ jy Ey Ey ¼ A , absent. Both in-plane and out-of-plane spin polarizations are thus no LPGE current can be induced along y. This leads to a reversed (see Supplementary Figs. 5 and 6). Finally, the dc current contrasting CPGE- and LPGE-based nonlinear anomalous Hall susceptibility of bilayer and trilayer WTe will be reversed in the current in few-layer WTe2 with Cs point group, that is, linearly 2 fi trilayer case only. Figure 5 shows the interband LPGE susceptibility polarized light/electric eld with in-plane polarization will σ of bilayer and trilayer WTe at μ = 0, which is about 10 times generate nonlinear anomalous Hall current along x only abc 2 L ≠ ; L higher than that in monolayer group IV monochalcogenides.23 It is ðjx 0 jy ¼ 0Þ, while circularly polarized light propagating along clear that in bilayer WTe the two independent susceptibility z axis will generate nonlinear anomalous Hall current along y only 2 C ; C ≠ tensor elements σ and σ of the ±mFE states remain invariant ðjx ¼ 0 jy 0Þ. xxx xyy The correlation between the irreducible representations of upon ferroelectric transition, while for trilayer WTe2 both σxxx and parent group C2h and its noncentrosymmetric subgroups C2,Cs, σxyy of the ±iFE states flip the sign. The above electronic structure results demonstrate a striking and C1 is summarized in Supplementary Table 1. We start from monolayer 1T′ WTe which has point group of C , whose second difference between bilayer and trilayer WTe , that is, nonlinear 2 2h 2 order nonlinear current response vanishes due to the presence of anomalous Hall current flips its direction upon ferroelectric inversion symmetry. Upon vdW stacking (e.g. few-layer and bulk switching in trilayer WTe2, but remains unchanged in bilayer WTe2. Td WTe2), C2y is broken with My left unchanged, which breaks the inversion symmetry and results in subgroup Cs. Consequently, as Group theoretical analysis of NAHE in bilayer and FNAHE in trilayer C C ≠ we analyzed above, jx ¼ 0, but jy 0 under circularly polarized WTe2 L ≠ L light, while jx 0 but jy ¼ 0 under linearly polarized light/electric Here we provide a group theoretical analysis of NAHE in addition field with in-plane polarization. However, if My is broken with C2y fi C ≠ to the above rst-principles calculations. Both bilayer and trilayer being preserved, it will fall into subgroup C2. In this case, jx 0 and C C C ≠ WTe2 have Cs point group with a mirror symmetry My. For jy ¼ 0 under circularly polarized light, while jx ¼ 0 and jy 0 circularly polarized incident light propagating along z, (E(ω)×E under linearly x/y-polarized light/electric field. Furthermore, if

npj Computational Materials (2019) 119 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences H. Wang and X. Qian 5

-iFE (C s ) +iFE (C ) 2 s 2 Å Å (a)0.2 1000 (b) 0.2 1000

0.1 500 0.1 500

0 0 0 0 Energy (eV) Energy (eV) -0.1 -500 -0.1 -500

-0.2 -1000 -0.2 -1000 -YΓ Y -YΓ Y

0.8 0.8 (c)intra (d) D yz 0.6 0.6 C,inter D yz 0.4 0.4 L,inter Å) D xy Å) 0.2 0.2

0 0

-0.2 -0.2 BCD, SD ( -0.4 BCD, SD ( -0.4

-0.6 -0.6

-0.8 -0.8 -0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2 (eV) (eV) k k y y

(e) 2 (f) 2 z, intra 200 Å z, intra 200 Å Ω (+50 meV) Ω (+50 meV) k occ k occ x x Γ Y

2 2 -200 Å -200 Å X

k k y y

2 2 (g)Ω z, inter 400 Å (h) Ω z, inter 400 Å k VBM-1,CBM k VBM-1,CBM x x

2 2 -400 Å -400 Å

Fig. 4 Electronic structure and FNAHE of trilayer WTe2.a, b Band structure of trilayer WTe2 in −iFE and +iFE state, respectively. Both are Ωz k intra μ C;inter μ; ω color-coded by the z-component of intraband Berry curvature nðÞ. c, d BCD tensor elements Dyz ðÞand Dyz ðÞ, and SD tensor L;inter μ; ω μ − + ω element Dxy ðÞas function of chemical potential for iFE and iFE state, respectively. For interband BCD and SD, is set to 120 meV. k Ωz;intra k μ = − + k e, f -dependent distribution of intraband Berry curvature occ ðÞat ±50 meV for iFE and iFE state, respectively. g, h -dependent Ωz;inter k − + distribution interband Berry curvature nm ðÞbetween (VBM-1,CBM) around the Fermi surface for iFE and iFE state, respectively. The results clearly show that nonlinear anomalous Hall current in trilayer WTe2 will switch sign upon ferroelectric transition, in direct contrast to the bilayer case.

both My and C2y are broken, it will end up with subgroup C1, and related by a glide plane operation fgMzjta , where ta refers to a enable all possible LPGE and CPGE current responses along fractional translation along x. Thus, PR = −1, PT = 1, and (mx, my, +mFE −mFE different directions. mz) = (−mx, −my, mz) . For trilayer WTe2, the two ferro- We now discuss the fundamental difference of NAHE between electric states are related by an inversion operation fgIj0 , thus PR +iFE −iFE bilayer and trilayer WTe2 upon ferroelectric switching. A general = PT = −1, subsequently (mx, my, mz) = (−mx, −my, −mz) . symmetry operator in Seitz notation is given by g = {R|tR}, where R The above two conclusions are applicable to any time-reversal is point group symmetry operation and tR is a translational vector. antisymmetric pseudovectors such as Berry curvature and spin A time-reversal antisymmetric pseudovector (e.g. Berry curvature polarization. For example,ÀÁ forÀÁ intraband and interband Berry and spin polarization) transforms under operator g as follows, curvature, M Ωz k ; k ¼ Ωz k ; k in bilayer WTe , and m’ k = m k = m k ÀÁz ÀÁx y x ÀÁy 2 ( ) g ( ) PRPTR ( ), where PR and PT are spatial and z z TRI z IΩ kx; ky ¼ Ω Àkx; Àky ! ÀΩ kx; ky in trilayer WTe2, indi- temporal parity associated with g, respectively. PT = ±1 when k = k + K K intra C;inter R ± , where is multiples of reciprocal lattice vector. For cating that the sign of intraband and interband BCD (Dab , Dab ) bilayer WTe2, as aforementioned, two ferroelectric states can be flips only in trilayer WTe2 upon ferroelectric transition. This is in

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 119 H. Wang and X. Qian 6

250 150 (a)σ (b) σ xxx xxx ()+m FE () +i FE σ σ xyy 200 xyy ()+m FE 100 ()+i FE xxx ) σ xxx() -m FE ) σ () -i FE 2 150 σ 2 σ xyy xyy () -m FE 50 () -i FE

A/V 100 A/V  

. . 0 50 -50

(nm 0 (nm σ σ -50 -100

-100 -150 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ω (eV) ω (eV)

Fig. 5 Interband LPGE susceptibility σabc of bilayer and trilayer WTe2 at μ = 0. a σabc for the ±mFE states of bilayer WTe2. b σabc for the ±iFE state of trilayer WTe2. These two plots demonstrate that, under ferroelectric transition, the sign of LPGE-induced dc current susceptibility switches in the trilayer case, but remains invariant in the bilayer case.

excellent agreement with the first-principles calculations shown in (a) Fig. 3c–f and Fig. 4e–h. In addition, the in-plane spin polarization switches in both cases, and the out-of-plane spin polarization becomes reversed in trilayer WTe2 while remaining unflipped for bilayer WTe2, which also agrees with the calculations (Supple- mentary Figs. 3–6). Different from pseudovectors, polar vector p such as electric polarization and shift vector transforms as follows: p′ = Rp Therefore, both mirror Mz and inversion 0 I operation will lead to vertical polarization reversal, i.e. pz ¼ 0 Mzpz ¼Àpz and pz ¼Ipz ¼Àpz, i.e. the out-of-plane electric dipole flips sign in both bilayer and trilayer WTe2 upon a ferroelectric transition.ÀÁ In addition, for in-planeÀÁ shift vector Rmn ∈ a 0 a a a 0 a a (b) (c) with a {x, y}, Rmn ¼MzRmn ¼ Rmn, and Rmn ¼IRmn ¼ÀRmn, a L;inter indicating that the in-plane shift vector Rmn and thus SD Dyz will flip only in trilayer WTe2 upon ferroelectric transition. Conse- L C quently, the total jx and jy from CPGE and LPGE will switch direction upon ferroelectric transition, provoking FNAHE in time- reversal invariant semimetals. Moreover, it suggests that the BCD and SD can serve as distinct order parameters for noncentrosym- metric semimetals. Figure 6a presents an illustrative summary of Fig. 6 Transformation of pseudovectors and polar vectors under the transformation of Berry curvature, spin polarization, and different symmetry operation and ferroelectric switching of LPGE electric polarization under different symmetry operation, while and CPGE nonlinear current. a Transformation of Berry curvature, spin polarization, shift vector, BCD and SD as well as LPGE and CPGE Fig. 6b, c show the ferroelectric switching of nonlinear current in nonlinear current under different symmetry operation between two the −iFE and +iFE state of trilayer WTe . Upon the out-of-plane 2 ferroelectric states in time-reversal invariant few-layer WTe2. Berry L polarization switching, nonlinear Hall current jx generated via curvature and spin polarization transform as time-reversal antisym- LPGE switches between −x and +x direction under the same metric pseudovectors. Under a mirror symmetry operation Mz for external electric field with in-plane linear polarization. Moreover, the ferroelectric states in bilayer WTe2, most quantities remain C − invariant, except in-plane spin and Berry curvature component. nonlinear Hall current jx induced by CPGE switches between y Under inversion symmetry operation I for the −iFE and +iFE state and +y direction under circularly-polarized light with normal states in trilayer WTe2, all quantities, including Berry curvature, spin incidence. It’s worth to emphasize that the intermediate PE state polarization, shift vector, BCD and SD, flip the sign in the presence of in bilayer and trilayer WTe2 has noncentrosymmetric C2v and time-reversal symmetry, giving rise to FNAHE in trilayer WTe2. b, c − + centrosymmetric C2h point group, respectively. Thus, despite that Ferroelectric switching of nonlinear current in the iFE and iFE state of trilayer WTe2, respectively. Upon the polarization Pz the out-of-plane electric polarization vanishes in both cases, L nonlinear anomalous Hall current of the PE state vanishes in switching, nonlinear anomalous Hall current jx from LPGE switches between −x and +x direction under external field with in-plane trilayer, but remains finite in bilayer. linear polarization, while nonlinear anomalous Hall current jC from The present work considers intrinsic NAHE due to BCD. Disorder y CPGE switches between −y and +y direction under circularly- however can play an important role in NAHE as pointed out by Du polarized light with normal incidence. et al.35 and Isobe et al.36 particularly in the dc limit due to side 37 jump and skew scattering. As disorder scattering depends on In conclusion, using first-principles calculations and group scattering potential and defect density, further experimental theoretical analyses we investigated the NAHE in bilayer and studies are required to understand the nature of defects in bilayer trilayer WTe2 and, more importantly, the underlying microscopic and trilayer WTe2. The transformation behavior of BCD-induced origin of FNAHE (i.e., ferroelectric switching of NAHE) in trilayer NAHE in multilayer WTe2 upon ferroelectric transition may be WTe2. Although both bilayer and trilayer WTe2 exhibit ferroelectric utilized to distinguish itself from the disorder scattering-induced transition with similar electric polarization, they behave very NAHE. For example, upon ferroelectric transition, the change in differently in NAHE. In the trilayer case, the nonlinear anomalous defect scattering potential may behave very differently from the Hall current flips direction upon ferroelectric switching due to the change in crystal structure, thereby potentially helping differenti- reversal of BCD and SD under an effective inversion operation of ate the two NAHE contributions. the two ferroelectric states. In contrast, the two ferroelectric states

npj Computational Materials (2019) 119 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences H. Wang and X. Qian 7 in bilayer WTe2 are related effectively by a glide plane operation the Dirac delta function integration. In addition, we tested the range which does not flip the BCD/SD, thus its nonlinear anomalous Hall separation parameter λ in the hybrid HSE functional. Although the values current will not flip upon ferroelectric switching. In addition, NAHE of Berry curvature etc. can change with respect to λ, the presence (absence) of Berry curvature switching in trilayer (bilayer) remains the same is expected to vanish in the PE state of trilayer WTe2, but remains λ = nontrivial for the PE state of the bilayer case. The above (see Supplementary Fig. 3 for HSE functional with 0.4). We also checked the convergence of the k-point sampling by increasing it to 1000 × 1000 × conclusions are applicable to any even and odd layer WTe2 ′ 1 (Supplementary Fig. 7). Finally, since few-layer WTe2 is either (except monolayer 1T WTe2 as it is centrosymmetric with semimetallic or having very small gap, the dielectric screening is large, vanishing second order NAHE) as long as the two opposite thus the effect of the Coulombic interaction between electrons and holes ferroelectric states have the same relationship as the bilayer and is negligible. 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 DATA AVAILABILITY as new order parameters for noncentrosymmetric materials, which The datasets generated during and/or analyzed during the current study are available opens up the possibility to explore nonlinear multiferroicity based from the corresponding author upon reasonable request. on the coupling of BCD/SD and ferroelectric order. Ferroelectric metals may be advantageous as their vanishing bandgap will not Received: 22 August 2019; Accepted: 12 November 2019; 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 REFERENCES fi in bilayer and trilayer WTe2 is about one order of magnitude 1. Moore, J. E. & Orenstein, J. Con nement-induced berry phase and helicity- higher than that in ferroelectric GeS.23 Moreover, FNAHE provides dependent photocurrents. Phys. Rev. Lett. 105, 026805 (2010). a facile approach for direct readout of ferroelectric states, which, 2. Sodemann, I. & Fu, L. Quantum Nonlinear Hall effect induced by berry curvature combined with vertical ferroelectric writing, may allow for dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015). 3. Low, T., Jiang, Y. & Guinea, F. Topological currents in black phosphorus with realizing nonlinear multiferroic memory. In addition, the distinct broken inversion symmetry. Phys. Rev. B 92, 235447 (2015). ferroelectric transformation pathway may provide potential routes 4. Deyo, E., Golub, L., Ivchenko, E. & Spivak, B. Semiclassical theory of the photo- 38 to realizing non-abelian reciprocal braiding of Weyl nodes. The galvanic effect in non-centrosymmetric systems. Preprint at arXiv:0904.1917 present findings therefore reveal an underexplored realm beyond (2009). classical linear Hall effect and conventional ferroelectrics with 5. Xu, S.-Y. et al. Electrically switchable Berry curvature dipole in the monolayer – exciting new opportunities for FNAHE-based nonlinear quantum topological insulator WTe2. Nat. Phys. 14, 900 906 (2018). electronics using ferroelectric metals and Weyl semimetals. 6. Zhang, Y., van den Brink, J., Felser, C. & Yan, B. Electrically tuneable nonlinear anomalous Hall effect in two-dimensional transition-metal dichalcogenides WTe2 and MoTe2. 2D Mater. 5, 044001 (2018). METHODS 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). First-principles calculations of atomistic and electronic structure 8. Shi, L.-K & Song, J. C. W. Symmetry, spin-texture, and tunable quantum geometry

First-principles calculations for structural relaxation and electric polariza- in a WTe2 monolayer. Phys. Rev. B 99, 035403 (2019). tion were performed using density-functional theory30,31 as implemented 9. Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal- in the Vienna Ab initio Simulation Package (VASP)32 with the projector- symmetric conditions. Nature 565, 337–342 (2019). augmented wave method.39 We employed the generalized-gradient 10. Kang, K., Li, T., Sohn, E., Shan, J. & Mak, K. F. Nonlinear anomalous Hall effect in approximation of exchange-correlation functional in the few-layer WTe2. Nat. Mater. 18, 324–328 (2019). Perdew–Burke–Ernzerhof form,40 a plane-wave basis with an energy cutoff 11. Du, Z. Z., Wang, C. M., Lu, H.-Z. & Xie, X. C. Band signatures for strong nonlinear of 300 eV, a Monkhorst–Pack k-point sampling of 6 × 12 × 1 for the Brillouin Hall effect in Bilayer WTe2. Phys. Rev. Lett. 121, 266601 (2018). zone integration, and optB88-vdW functional41 to account for dispersion 12. Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in two-dimensional tran- interactions. Ground state crystal structures were obtained by fully relaxing sition metal dichalcogenides. Science 346, 1344–1347 (2014). ′ both atomic positions and in-plane lattice parameters while keeping a 13. Tang, S. et al. Quantum spin Hall state in monolayer 1T -WTe2. Nat. Phys. 13, large vacuum region of ~20 Å along the out-of-plane direction to reduce 683–687 (2017). – the periodic image interactions. The convergence criteria for maximal 14. Fei, Z. et al. Edge conduction in monolayer WTe2. Nat. Phys. 13, 677 682 (2017). residual force was <0.005 eV/Å, and the convergence criteria for electronic 15. Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a relaxation is 10−6 eV. We have tested higher energy cutoff of 400 eV, and monolayer crystal. Science 359, 76 (2018). the difference in the lattice constants is <0.04%. Crystal structures of 16. Sajadi, E. et al. Gate-induced superconductivity in a monolayer topological insulator. Science 362, 922–925 (2018). bilayer and trilayer WTe2 can be found in the Supplementary Information. In addition, total electric polarization was calculated by directly integrating 17. Fatemi, V. et al. Electrically tunable low-density superconductivity in a monolayer the product of charge density/ionic charge and their corresponding topological insulator. Science 362, 926–929 (2018). position without using Berry phase approach. 18. Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015). 19. Ali, M. N. et al. Large, non-saturating magnetoresistance in WTe2. Nature 514, 205–208 (2014). First-principles electronic structure calculations of NAHE 20. Sie, E. J. et al. An ultrafast symmetry switch in a Weyl semimetal. Nature 565, To compute the NHLE-related quantities, we first construct quasiatomic 61–66 (2019). spinor Wannier functions and tight-binding Hamiltonian from Kohn–Sham 21. Fei, Z. et al. Ferroelectric switching of a two-dimensional metal. Nature 560, 336 wavefunctions and eigenvalues under the maximal similarity measure with (2018). respect to pseudoatomic orbitals.33,34 Spin-orbit coupling is taken into 22. Anderson, P. W. & Blount, E. I. Symmetry considerations on martensitic trans- account, and hybrid exchange-correlation energy functional HSE0642 is formations: “Ferroelectric” metals? Phys. Rev. Lett. 14, 217–219 (1965). employed with the range-separation parameter λ = 0.2 (see Supplemen- 23. Wang, H. & Qian, X. Ferroicity-driven nonlinear photocurrent switching in time- tary Information for more details). Total 112 and 168 quasiatomic spinor reversal invariant ferroic materials. Sci. Adv. 5, eaav9743 (2019). Wannier functions were obtained for bilayer and trilayer WTe2, respec- 24. Li, R. et al. Weyl ferroelectric semimetal. Preprint at arXiv:1610.07142 (2016). tively. Using the developed tight-binding Hamiltonian we then compute 25. Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in CPGE and LPGE susceptibility tensor with a modified WANNIER90 code43 noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 using a dense k-point sampling of 600 × 600 × 1 for both bilayer and (2015). trilayer WTe2. A small imaginary smearing factor η of 0.05 eV is applied to 26. Sipe, J. E. & Shkrebtii, A. I. Second-order optical response in semiconductors. Phys. fundamental frequency, and Sokhotski–Plemelj theorem is employed for Rev. B 61, 5337–5352 (2000).

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 119 H. Wang and X. Qian 8 27. Tsirkin, S. S., Puente, P. A. & Souza, I. Gyrotropic effects in trigonal tellurium AUTHOR CONTRIBUTIONS studied from first principles. Phys. Rev. B 97, 035158 (2018). X.Q. conceived the project. H.W. and X.Q. developed first-principles tight-binding 28. Morimoto, T., Zhong, S., Orenstein, J. & Moore, J. E. Semiclassical theory of approach for computing nonlinear susceptibility tensor. H.W. performed the nonlinear magneto-optical responses with applications to topological Dirac/Weyl calculations. Both H.W. and X.Q. analyzed the results and wrote the manuscript. semimetals. Phys. Rev. B 94, 245121 (2016). 29. Rostami, H. & Polini, M. Nonlinear anomalous photocurrents in Weyl semimetals. Phys. Rev. B 97, 195151 (2018). COMPETING INTERESTS 30. Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. B 136, The authors declare no competing interests. B864–B871 (1964). 31. Kohn, W. & Sham, L. J. Self-consistent equations including exchange and corre- lation effects. Phys. Rev. 140, A1133–A1138 (1965). 32. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy ADDITIONAL INFORMATION calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996). Supplementary information is available for this paper at https://doi.org/10.1038/ 33. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized s41524-019-0257-1. Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012). 34. Qian, X. et al. Quasiatomic orbitals for ab initio tight-binding analysis. Phys. Rev. B Correspondence and requests for materials should be addressed to X.Q. 78, 245112 (2008). 35. Du, Z. Z., Wang, C. M., Li, S., Lu, H.-Z. & Xie, X. C. Disorder-induced nonlinear Hall Reprints and permission information is available at http://www.nature.com/ effect with time-reversal symmetry. Nat. Commun. 10, 3047 (2019). reprints 36. Isobe, H., Xu, S.-Y. & Fu, L. High-frequency rectification via chiral electrons in nonlinear crystals. Preprint at arXiv:1812.08162 (2018). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims 37. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall in published maps and institutional affiliations. effect. Rev. Mod. Phys. 82, 1539–1592 (2010). 38. Bouhon, A., Slager, R.-J. & Bzdušek, T. Non-abelian reciprocal braiding of Weyl Nodes. Preprint at arXiv:1907.10611 (2019). 39. Blöchl, P. E. Projector augmented-wave 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). Open Access This article is licensed under a Creative Commons š 41. Klime , J., Bowler, D. R. & Michaelides, A. Chemical accuracy for the van der Waals Attribution 4.0 International License, which permits use, sharing, density functional. J. Phys. Condens. Matter 22, 022201 (2010). adaptation, distribution and reproduction in any medium or format, as long as you give fl 42. Krukau, A. V., Vydrov, O. A., Izmaylov, A. F. & Scuseria, G. E. In uence of the appropriate credit to the original author(s) and the source, provide a link to the Creative exchange screening parameter on the performance of screened hybrid func- Commons license, and indicate if changes were made. The images or other third party tionals. J. Chem. Phys. 125, 224106 (2006). material in this article are included in the article’s Creative Commons license, unless fi 43. Mosto , A. A. et al. An updated version of wannier90: a tool for obtaining maximally- indicated otherwise in a credit line to the material. If material is not included in the – localised Wannier functions. Comput. Phys. Commun. 185,2309 2310 (2014). 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. ACKNOWLEDGEMENTS org/licenses/by/4.0/. This work was supported by the National Science Foundation (NSF) under award number DMR-1753054. Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing. © The Author(s) 2019

npj Computational Materials (2019) 119 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences