Spin-orbit-splitting-driven nonlinear Hall effect in NbIrTe4

The Berry curvature dipole (BCD) serves as a one of the fundamental contributors to emergence of the nonlinear Hall effect (NLHE). Despite intense interest due to its potential for new technologies reaching beyond the quantum efficiency limit, the interplay between BCD and NLHE has been barely understood yet in the absence of a systematic study on the electronic band structure. Here, we report NLHE realized in NbIrTe4 that persists above room temperature coupled with a sign change in the Hall conductivity at 150 K. First-principles calculations combined with angle-resolved photoemission spectroscopy (ARPES) measurements show that BCD tuned by the partial occupancy of spin-orbit split bands via temperature is responsible for the temperature-dependent NLHE. Our findings highlight the correlation between BCD and the electronic band structure, providing a viable route to create and engineer the non-trivial Hall effect by tuning the geometric properties of quasiparticles in transition-metal chalcogen compounds.

simulations, the direct experimental confirmation and comprehensive understanding of the BCD based on electronic structures have not yet been fully attained.This achievement would provide crucial insights into the underlying mechanism and controllability of the BCD.
In this paper, we report the room-temperature NLHE in NbIrTe4 thin flakes which exhibit a frequency-doubled Hall conductivity proportional to the square of the driving current.We also demonstrate that the sign change in the NLHE at 150 K is induced by the sign change of the BCD because of the chemical potential shift at high temperatures.It is unambiguously evidenced by direct observation of a rigid shift in the temperature-dependent band dispersion using ARPES and calculated BCDs.Investigation on the electronic structures using ARPES and density functional theory (DFT) also indicates that the main contributor of BCDs is a partial occupation of spin-orbit split bands.Our findings provide important insights into the momentum texture of the Berry curvature and into controlling the Berry curvature dipole hosting the NLHE which can be utilized for NLHE-based devices.

Prediction of the nonlinear Hall effect in NbIrTe4
The crystal structure of bulk NbIrTe4 (Fig. 1a) has an orthorhombic unit cell with space group Pmn21 23 .The experimental lattice constants are a = 3.77 Å, b = 12.51 Å, and c = 13.12Å, where the two-dimensional planes are stacked along the c-axis.The material has a nonmagnetic metallic phase, and no anomalous Hall effect is expected.Moreover, because of the combination of mirror ({Ma|(0,0,0)}) and glide mirror ({Mb|(1/2,0,1/2)}) operations, no NLHE is expected within the ab plane [23][24][25] .However, in the slab geometry, the symmetry is lowered to Pm space group due to the breaking of the translation along the c-axis, leaving only identity and mirror ({Ma|(0,0,0)}) operations (Fig. 1b).The resultant symmetry allows nonlinear Hall current along the b-axis when there is a driving current along the a-axis, parallel to the direction of the BCD (Fig. 1b) 24,26 .The devices were fabricated into a Hall bars pattern (Fig. 1c), which is aligned in the crystallographic direction by angle-resolved polarized Raman spectroscopy (Fig. 1d).Crystallographic directions were confirmed via high intensity along the a-axis and low intensity along the b-axis, which is evidence of broken inversion symmetry in NbIrTe4, consistent with the previous report 27 .In this system, the interplay between spin-orbit coupling and the presence of broken inversion symmetry leads to the emergence of a prominent BCD hot spot within spin-orbit-split bands (Fig. 1e), as elaborated upon in the subsequent theoretical section.

Observation of nonlinear Hall effect in NbIrTe4
To investigate the NLHE in NbIrTe4, nonlinear transport measurements on a 15 nm-thick Hall bar device fabricated from a NbIrTe4 flake were performed under zero magnetic field (Fig. 2).The second-harmonic transverse voltage (# !"# ) under zero magnetic field in a 15 nm-thick NbIrTe4 flake at 2 K responds quadratically to the current $ # along the a-axis, indicating the presence of the NLHE on NbIrTe4 flakes (Fig. 2a).Additionally, we have verified both the direction and frequency of driving AC current-dependent nonlinear Hall responses (see Fig. S2 and S3 of SI), thereby confirming the absence of experimental measurement artifacts.The second-harmonic transverse voltage (# !"# ) of a NbIrTe4 flake gradually decreases as the temperature increases from 2 to 150 K (nearly decay), followed by a sign change with an increasing magnitude as the temperature further increases to 300 K (Fig. 2b), consistent with the NLHE observed in TaIrTe4 6 .However, the slight difference in the sign-changing temperature and the magnitude of NLHE between the two materials are due to the band structure changes associated with increased ionic radius and the spin-orbit coupling strength from Nb to Ta 12,15 .

Temperature-dependent chemical potential shift nature of NbIrTe4
Since the NLHE is derived from BCD, understanding the nature of BCD from the perspective of electronic band structures enables both understanding and effective control of the NLHE.To investigate the electronic bands that dominantly contribute to NLH behavior, we explored the electronic band structure using ARPES experiments and DFT calculations.The Fermi surface of a NbIrTe4 single crystal consists of elliptical-shaped electron pockets around the Γ point (A in Fig. 3b) and semi-elliptical-shaped broad bulk hole pockets along the Γ-X directions (B in Fig. 3b).Both Fermi surface features (Fig. 3b) and the band dispersions along the X-Γ-X and S-Y-S directions (Fig. 3c) are in good agreement with the DFT calculation results shown by red dotted lines in Fig. 3b and c and those reported previously 8,25,[29][30][31] .
To further elucidate the origin of the temperature dependence of the NLHE behavior (Fig. 2b), ARPES spectra along the X-Γ-X direction were acquired at various temperatures (Fig. S6 in SI).The energy distribution curves (EDCs) at the Γ point as a function of temperature (Fig. density, increasing as a function of temperature obtained from the reported transport results 28,30 .
The ARPES and transport results suggest that there is substantial change in the band structures with increasing temperature.This could be related to the sign change in BCD which suggests the further investigation through DFT analysis.

Origin of the BCD of NbIrTe4
We investigated the mechanism that induces the nonlinear Hall conductivity and the origin of the sign change by first-principles calculations.The current induced by the NLHE in twodimensional systems is expressed as 4 where ℏ is Planck's constant, 1̂ is the direction normal to the plane parallel to the c-axis, τ is We find that the BCD is dominantly contributed from the partially occupied bands with spin-orbit induced splitting by analyzing the momentum texture of BC, integrated around the energy at which BCD changes abruptly.Fig. 4a shows the energy dependence of 7 -(/), where a large increase in Da is observed at around −10 meV.The momentum-resolved BC integrated in the energy range from −12 to −7 meV (red shaded area in Fig. 4a) shows that the BC is concentrated in small areas around the Γ point, with a clear sign change in Ω .$,0across the Ma mirror axis (Fig. 4b), which dominantly contributes to a sharp decrease in Da.The bands contributing to the BC around −10 meV are presented in Fig. 4c, which corresponds to the region indicated by the black-dotted square in Fig. 4b.The one-dimensional cut along the X-Γ-X line reveals the characteristics of these bands (Fig. 4d), where large Rashba spin-orbit splitting is observed, consistent with the absence of inversion symmetry.The BC calculated along the X-Γ-X line (bottom panel of Fig. 4d) shows that partially occupied spin-split bands from spin-orbit coupling contribute largely to the BC (Ωc) (blue shaded area in Fig. 4d).Thus, we attribute the large change in the BCD to a shift of the EF as the occupation of the spin-orbit split bands changes accordingly.We note that the BCD and momentum-resolved BC of the bilayer slab are insensitive to the choice of the local density approximation (LDA) or generalized gradient approximation (GGA) for the exchange-correlation potential (see Fig. S12 in SI).
The sign change around the Fermi energy of the calculated BCD in a thicker slab (Fig.  The positive BCD at −20 meV can be explained on the basis of an overall negative-to-positive sign change in the integrated BC across the mirror axis, giving a net positive value.The sign of BCD changes around EF as a result of the addition of the opposite component of the BCD shown in the middle panel with the opposite sign change in the BC across the mirror axis.Thus, the BCD at EF has a net negative sign with the BC texture, which is the sum of those corresponding to Da at −20 meV and ∆Da.From the analysis of the bilayer case, we expect that BCD of 12-layer slab can be also mainly contributed from the partially occupied spin-split bands.Our analysis reveals the mechanism of the temperature-dependent sign change in the NLHE observed in the transport measurements.The shift in EF, as observed in the temperaturedependent ARPES measurements, changes the distribution of the BC sensitive to band filling, resulting in the opposite sign of the BCD as the temperature increases.

Conclusions
We report the NLHE in non-magnetic NbIrTe4 flakes.Our transport measurements show that the magnitude of the Hall voltage at a doubled frequency increases proportional to the square of the driving current, indicating the occurrence of the NLHE.Moreover, the measured Hall voltage shows a sign change with increasing temperature that persists up to room temperature.ARPES measurements and first-principles calculations reveal that the BCD in NbIrTe4 originates mainly from partially filled spin-orbit split bands.Furthermore, we identified the mechanism of the sign changes in the NLHE on the basis of the chemical potential shift and associated change in the distribution of the BC in momentum space, which is supported by temperature-dependent energy shift in the ARPES results and sign-changing BC texture in the slab calculation.The identified mechanism suggests that the NLHE can be electronically controlled by changing the momentum-dependent texture of the BC, which can be used to enhance the efficiency of BCD-based devices such as memories and rectifiers.

Methods
Device fabrication and electrical measurements.NbIrTe4 samples were mechanically exfoliated from a single crystal NbIrTe4 grown as described elsewhere 32 and transferred onto SiO2/Si substrates.Hall bar device geometries were patterned with Ti(10 nm)/Au(100 nm) electrodes on NbIrTe4 crystals.To remove the oxidation layer on the surface of the NbIrTe4 flakes, Ar + -ion sputtering at 200 V was employed for 300 s.To perform nonlinear transport measurements, the device was biased with a harmonic current along the a-axis at a fixed frequency (/ = 117.77Hz) and measured both the first-harmonic frequency (/) and secondharmonic frequency (2/) using a lock-in amplifier.
Raman spectroscopy.A linearly polarized 514 nm laser was focused to a spot of approximately 1-2 µm onto the nanoflakes at room temperature.The laser power was limited to less than 200 µW.A polar plot of angle-resolved measurements was obtained by fixing the Raman modes at 151.3 cm −1 and comparing the intensity as a function of angle 33 .ARPES measurements.ARPES measurements were performed at Beamline 10.0.1, Advanced Light Source, Lawrence Berkeley National Laboratory.The ARPES system was equipped with a Scienta R4000 electron analyzer.The photon energy was set to be 60 eV, with an energy resolution of 20 meV and an angular resolution of 0.1 degrees.
First-principles calculations.We used the first-principles DFT to calculate the electronic structures and nonlinear Hall conductivity.The calculations were performed using the Vienna Ab-initio Simulation Package (VASP).The Ceperley-Alder (CA) 34,35 and Perdew-Burke-Ernzerhof (PBE) 36,37 parameterizations were used for the local density approximation (LDA) and the generalized gradient approximation (GGA), respectively.The projector augmentedwave method 38 was used with an energy cut-off of 500 eV.Spin-orbit coupling was included.
For bulk and slab NbIrTe4, k-point sampling grids of 3×12×3 and 3×12×1 were used, respectively.The experimental atomic structures were used to construct the bulk and slab geometry of 2-and 12-layer NbIrTe4 with a vacuum layer of 16 Å.For the calculations of the BC and the BCD, the Wannier90 code 39 was used to construct the tight-binding Hamiltonian using Nb-d, Ir-d, and Te-p derived bands.For the bilayer, the tight-binding Hamiltonian was constructed from the bilayer-vacuum configuration.For the thicker slab, the tight-binding Hamiltonian was constructed by making a supercell of the bulk tight-binding Hamiltonian using the Python Tight Binding (PythTB) code 40 .The slab band structures of the supercell tight-binding Hamiltonian and those of DFT using vacuum-slab geometry were found to be in good agreement (Fig. S11 in SI).The calculations with a small hole doping of 0.025 h/f.u. were done by reducing the number of electrons with a compensating uniform background charge.The BC and BCD were calculated using the Wannier-Berri code 41,42 and the surface bands were calculated using WannierTools code 43 .indicating the disappearance of the linear Hall response (V ⊥ ω ) due to preserved time-reversal symmetry.The green crossbars illustrate the experimental geometries.Vǁ ω and V ⊥ ω represent the first-harmonic voltage parallel and perpendicular to the applied current direction (four probe IV measurement and Hall measurement), respectively.The applied current is injected from the source (S) electrode to the drain (D) and the voltage is measured between A and B electrodes.V ω (mV) We performed NLHE measurements on NbIrTe4 flakes with two different thicknesses, 15 nm and 48 nm, to compare the NLHE in different thickness.The strength of the nonlinear Hall conductivity, $ $" , indicates a quadratic coefficient of the parabolic fitting function of !# $" vs. # " .The sign of the NLH signal changes at ~150 K irrespective of the sample thickness, whereas the NLH signal $ $" for 15 nm-thick flakes is substantially higher than that for 48 nm-thick flakes, supporting the previous argument that the NLH signal can only originate from the surface because of the absence of the glide mirror plane on the surface 2 .

First harmonic Hall voltage of NbIrTe4
We show representative temperature-dependent ARPES E-k cuts along Γ-X direction at temperature of 20 K, 70 K, 180 K, and 240 K, respectively.In Fig. 3 As the Berry curvature dipole (BCD) is calculated in a slab geometry with a thickness smaller than the thickness of an experimental sample, we checked the dependence of the calculated BCD as a function of slab thickness.Fig. S8 shows the BCD calculated for slabs thicknesses of 8, 10, and 12 layers; further increasing the number of slabs is challenging because of the large numbers of Wannier functions required for calculating the BCD ( 1056Wannier functions for 12 layers).We find that the key feature in the BCD (i.e., a positive-tonegative sign change as the chemical potential is increased) is maintained with increasing thickness.Thus, we expect that the same feature will be maintained for the thicker slab.

Obtaining Berry curvature dipole from the transport data
We estimate the BCD using the following formula from Eq (1): where ℏ is Planck's constant, τ is the scattering time, Ea(ω) is an external electric field along the a-axis, Da(ω) is the BCD along the a-axis, and jb(2ω) is the second-harmonic current density along the b-axis.We assume the static limit (ω τ << 1) 3 and express Da(ω) as where n is the charge density, t and W are the thickness and width of the sample, respectively, Vb(2ω) is the second-harmonic Hall voltage along the b-axis, 6 * is the effective mass that we approximate as the electron mass (me), Ia(ω) is the current along the a-axis, and ρ0 is the resistivity in the static limit, given as In addition to identifying the surface termination, we check whether there are surfacedependent changes in the band structures induced by the chemical potential shift using theoretical calculations.The panels (d-i) in Fig. S11 present the bulk and surface band structures for undoped and hole-doped NbIrTe4, in which the hole-doping effect is included by reducing the number of electrons (0.025 h/f.u.) with a compensating uniform background charge.The amount of doping is chosen such that the chemical potential shift of the bulk band structures is about 20 meV, close to the estimated value from ARPES (panels d and e).
We find that the surface band structures along the Γ-X high-symmetry line for both S1 and S2 terminations exhibit uniform upward shifts with amounts almost identical to the bulk bands.Thus, the temperature-dependent shift around -0.3 eV that we identify with ARPES spectra can be interpreted as the approximately same chemical potential shift, regardless of the surface termination.
projected bands among the three regions, consistent with the relatively weak inter-layer van der Waals coupling.More importantly, we find no noticeable changes in the projected bands with respect to doping, suggesting that the main effect of doping is a rigid shift in the band structures.

Figure 1 .
Figure 1.Crystal structure symmetry and band model of NbIrTe4 a. Crystal structure of NbIrTe4 exhibiting broken inversion symmetry with a mirror plane (Ma) illustrated as a yellow plane.Top view (b) of the crystal structure of NbIrTe4 with mirror axis Ma (yellow dashed line).JNLHE is the generated nonlinear Hall current parallel to Ma, and Eext is the applied electric field perpendicular to Ma. c. Illustrations of Hall devices with optical image (top left inset).Iω, V2ω, Eω, and J2ω represent the current, voltage, electric field, and generated nonlinear Hall current, respectively, at frequencies ω and 2ω.The scale bar in the optical image (top left inset) is 1 μm.d.Polar plots of Raman modes at 151.33 cm −1 .The results are shown as intensity versus angle configuration.e.A schematic model picture of band structures with Berry curvature (BC) Ω(k) evolution including spin-orbit coupling (SOC).The lower panels display momentumresolved BC, highlighting the emergence of a large dipole hotspot of BC in presence of SOC.

Figure 2 .
Figure 2. Nonlinear Hall response in NbIrTe4.a. Second-harmonic # !"# as a function of the AC current amplitude $ ω scaling quadratically in a 15 nm-thick flake of NbIrTe4 at 2 K.The black solid line is a quadratic fit to the data.The current is along the a-axis.Green cross bars in the inset represent the geometry of the measurements.The applied current is injected from the source (S) electrode to the drain (D) electrode and the voltage is measured between A and B electrodes.A sign change occurs upon simultaneous reversal of both the applied current

Figure 3 .
Figure 3. Electronic structures and evidence for the temperature-dependent chemical potential shift of NbIrTe4.a.The surface Brillouin zone of NbIrTe4 with high-symmetry points marked as red points.The blue arrow represents the BCD (D).b.ARPES and calculated (red dotted lines) Fermi surface (FS) of NbIrTe4 through two Brillouin zones (green solid lines) at 20 K. Ma is a mirror plane.c.ARPES intensity plots with the calculated band structure (red dotted lines) and corresponding second-derivatives ARPES spectra for the enhanced visibility along the X-Γ-X and S-Y-S directions at 20 K. d.The energy distribution curves (EDCs) at the Γ point as a function of temperature along with black fitted curves.The multi-peak fits are 3d) demonstrate that, as the temperature increases, the two peaks at E − EF = −0.3,−1.0 eV of the EDCs shifts systematically to a lower binding energy.In Fig.3e, we show the temperaturedependent energy shift (∆E) for the peaks of EDCs at the Γ point (blue, ∆E = E − ET=20 K), obtained from multi-peak fits with multiplication of Fermi-Dirac distribution (FD) function and convolution of instrumental resolution as detailed in Fig.S7.It is clearly observed that the ∆E shifts about 15~20 meV as the temperature increases from 10 K to 280 K which implies the chemical potential shifts down to higher binding energy which is consistent with our simulated ARPES spectra (See Fig.S11in SI).The increase of the hole pockets in the Fermi surface associated with the chemical potential shift is consistent with the behavior of the hole carrier

Figure 4 .
Figure 4. Calculated Berry curvature dipole and momentum-dependent Berry curvature.a.The BCD (Da) contributing to the in-plane NLHE for bilayer NbIrTe4, plotted as a function of the chemical potential.b.Momentum-resolved BC (Ωc(k) in Å 2 ) of bilayer NbIrTe4 integrated from −12 to −7 meV corresponding to the red shaded area in panel a. c.Band structures in the area denoted as the black dotted square in panel b.d.Band dispersion around the k-points contributing large Berry curvatures of bilayer NbIrTe4.The top panel shows bands near the Fermi energy expressed as solid magenta lines along the X-Γ-X cut (dotted magenta box in panel c.The black dashed lines are the bands calculated without spin-orbit coupling.The bottom panel is Ωc(k) integrated up to −10 meV relative to EF (red dashed line).e.Chemical potential versus the component of the Da contributing to the in-plane NLHE for 12layer NbIrTe4.f.Integrated momentum-resolved BC (Ξ $ (+) for 12-layer NbIrTe4 (arb.units) integrated from −40 meV to EF, with the left, middle, and right panels corresponding to the energy range marked with red, magenta, and green arrows in panel e, respectively.The black lines denote iso-energy surfaces at EF − 20 meV (left), EF − 10 meV (middle), and EF (right).

,
the scattering time, E(ω) is the external electric field, and D(ω) is the BCD vector.Given the symmetry of the NbIrTe4 slab (space group Pm), D(ω) is written as 7 -(/)8 9, where 8 9 is the unit vector along the a-axis, which gives the nonlinear current in the b-axis under electric fields applied along the a-axis.7 -(/) is obtained by the derivative of the out-of-plane BC along the a-axis, expressed as7 -(/) = : ; d " = 4? " @(ℏ/ − C .$ ) DΩ .wheren is the band index, C .$ denotes the single-particle energy, @(ℏ/ − C .$ ) is the Fermi-Dirac distribution function, and Ω .$,0 is the Berry curvature along the c-axis.From the expression of BCD, the important parts of the Brillouin zone are regions with a large change in the BC along the a-axis.To identify the characteristics of the band structures that are the main contributors to the BCD, we first calculate 7 -(/) for a bilayer NbIrTe4 slab.Compared with a thick slab geometry, which is difficult to analyze because of dense sub-bands, the bilayer system having a similar energy dependence of BCD with 12-layer NbIrTe4 is suitable to pinpoint the hotspots of BC that mainly contribute to the BCD.
4e) provides clear evidence that the sign inversion of the Hall conductivity can be induced by the Fermi energy shift.The calculated BCD for a 12-layer slab, which is sufficiently thick to represent the experimental geometry (see Fig.S8in SI) shows a nonzero value of about Da = −60 Å at the Fermi energy.More importantly, the BCD changes sign as E − EF decreases, showing a positive peak about Da = 40 Å around −20 meV, which is comparable to the peak shift observed in the ARPES results (Fig.3d).Thus, we propose that the observed sign change in the NLHE with increasing temperature is induced by a negative chemical potential shift, as evidenced by a sign change in the calculated BCD.Reducing the number of electrons corresponding to a 0.025 h/f.u.doping induces the chemical potential shifts about -20 meV both for bulk and slab geometry (see Fig.S12in SI).We note that the Da from the experimental data is estimated to be −348 Å at low temperatures (see a section of 'Obtaining Berry curvature dipole from the transport data' in SI) which is similar to the theoretical value of −55 Å, as explained in the following discussion.

Figure 4f shows the
Figure4fshows the change in the momentum texture of the BC of the 12-layer slab

First 1 .Figure S2 .
Figure S2.The direction of driving AC current direction-dependent nonlinear Hall voltage (V 2ω ) in NbIrTe4.(a) The in-plane crystal structure with unit cell of square box.The mirror axis is parallel to b-axis and perpendicular to a-axis.The AC current is applied along a-and b-axis, labeled as Va-bb 2ω and Vb-aa 2ω .(b) The second-harmonic transverse voltage depending on driving

Figure S8 .
Figure S7.Multi-peak fits of energy distribution curves (EDCs) at the Γ, which is fitted with Lorentzian peaks, incorporating the multiplication of the Fermi-Dirac distribution (FD) function and convolution with instrumental resolution.The energy range of fitting is from -1.77 eV to Fermi energy (EF) with temperatures ranging from 20 K to 240 K.

4 * 5 '
Figure S9.Comparison of slab band structures of supercell tight-binding (TB) and DFT calculations.The panel (a), (b), and (c) are TB band structures of two, four, and six-layer slabs calculated from supercell TB Hamiltonian, respectively, constructed from the bulk TB parameters.The panel (d), (e), and (f) are DFT band structures of two, four, and six-layer slabs, respectively, calculated with vacuum-slab geometry.