Band signatures for strong nonlinear Hall effect in bilayer WTe$_2$

TL;DR

This paper investigates the origins of the nonlinear Hall effect in bilayer WTe$_2$, linking strong signals to specific band-structure features like tilted anticrossings and inversions, advancing understanding of emergent material responses.

Contribution

It reveals the band-structure signatures associated with strong nonlinear Hall signals in 2D WTe$_2$, providing a theoretical framework for analyzing nonlinear Hall effects in low-symmetry materials.

Findings

Nonlinear Hall signals are prominent near tilted band anticrossings.

Band inversions are linked to strong nonlinear Hall responses.

The study explains experimental observations in 2D WTe$_2$.

Abstract

Unconventional responses upon breaking discrete or crystal symmetries open avenues for exploring emergent physical systems and materials. By breaking inversion symmetry, a nonlinear Hall signal can be observed, even in the presence of time-reversal symmetry, quite different from the conventional Hall effects. Low-symmetry two-dimensional materials are promising candidates for the nonlinear Hall effect, but it is less known when a strong nonlinear Hall signal can be measured, in particular, its connections with the band-structure properties. By using model analysis, we find prominent nonlinear Hall signals near tilted band anticrossings and band inversions. These band signatures can be used to explain the strong nonlinear Hall effect in the recent experiments on two-dimensional WTe$_{2}$. This Letter will be instructive not only for analyzing the transport signatures of the nonlinear Hall effect but also for exploring unconventional responses in emergent materials.

Tables

Table 1. Table 1: Comparison of the linear and nonlinear Hall effects. The Berry dipole in Eq. ( 2 ) can be written as a Fermi-sea integral by using the integration by parts.

	Linear	Nonlinear
Broken symmetry	Time reversal	Inversion
I-V relation	$\sim V_{\omega }^y/I_{\omega }^x$	$\sim V_{2\omega }^y/I_{\omega }^x{I}_{\omega }^x$
Fermi-sea integral of	Berry curvature ${\Omega }_{𝐤}$	$\partial {\Omega }_{𝐤}/\partial 𝐤$
In-gap signal	Possible (e.g., QHE)	No
Angular dependence	No	Eq. (Band signatures for strong nonlinear Hall effect in bilayer WTe2)

Equations

$${\chi }_{abc}^{(0)}={\chi }_{abc}^{(2)}={\epsilon }^{acd}{D}_{bd}{e}^3\tau /[2{\hslash }^2(1+i\omega \tau )],$$

$$D_{bd}$$

$${\Omega }_{i\text{k}}^a$$

$${\hat{\mathcal{H}}}_d=t{k}^x+v(k^y{\sigma }_x+\eta k^x{\sigma }_y)+(m/2-\alpha k^2){\sigma }_z,$$

$$\displaystyle\hat{\mathcal{H}}_{\mathrm{B}}=\left(\begin{array}[]{cccc}\hat{\mathcal{H}}_{d_{1}}&\hat{\mathcal{P}}&0&\gamma\\ \hat{\mathcal{P}}&\hat{\mathcal{H}}_{d_{1}}&\gamma&0\\ 0&\gamma&\hat{\mathcal{H}}_{d_{2}}&\hat{\mathcal{P}}\\ \gamma&0&\hat{\mathcal{P}}&\hat{\mathcal{H}}_{d_{2}}\\ \end{array}\right),$$

$$\displaystyle\hat{\mathcal{P}}=\left(\begin{array}[]{cc}\nu_{x}k_{x}-i\nu_{y}k_{y}&0\\ 0&-\nu_{x}k_{x}-i\nu_{y}k_{y}\\ \end{array}\right),$$

$${\chi }_{xxy}$$

$${\chi }_{xyy}$$

$$\frac{V_x^{2\omega }}{(V_y^{\omega }{)}^2}$$

Full text

Introduction.– The Hall effects are among the most paradigmatic phenomena in condensed matter physics because of their deep connections with the geometry and topology Klitzing et al. (1980); Cage et al. (2012); Xiao et al. (2010). All known measurable Hall effects need magnetic fields or magnetic dopants to break time-reversal symmetry Klitzing et al. (1980); Cage et al. (2012); Nagaosa et al. (2010); Yasuda et al. (2016). Recently, a new Hall effect, i.e., the nonlinear Hall effect, was proposed Sodemann and Fu (2015), which does not need time-reversal symmetry breaking, but inversion symmetry breaking. To measure it, the zero- or double-frequency component of the Hall conductance is rectified in response to a driving electric field oscillating at a low frequency (Fig. 1). This measurement is highly accessible to known experimental conditions, and thus opens new avenues of probing the spectral, symmetry, and topological properties of a number of emergent materials. More importantly, the idea can be generalized to other unconventional responses upon breaking discrete and crystal symmetries, leading to a promising and unknown territory. Despite the symmetry argument, how a considerably strong nonlinear Hall signal can be measured remains less known and is the focus of recent explorations.

In this Letter, we investigate the relation between the nonlinear Hall effect and signatures of energy bands in 2D systems. The nonlinear Hall response is proportional to the so-called Berry dipole Sodemann and Fu (2015). By analyzing a generic model of tilted massive Dirac fermions, we find that the Berry dipole is strong near tilted band anticrossings and band inversions (Fig. 2). To give an example of broad interest, we calculate the nonlinear Hall response in the bilayer WTe2 (Fig. 3), in which spin-orbit coupling modulates band anticrossings and band inversions to give divergences of the Berry dipole and strong nonlinear Hall responses. Our results are consistent with the strong nonlinear Hall response and its angular dependence (Fig. 4) in recent experiments Ma et al. (2018); Kang et al. and will inspire more experiments on the nonlinear Hall effects and related novel phenomena.

Review of the nonlinear Hall effect. – The nonlinear Hall effect originates from the dipole moment of the Berry curvature in momentum space, or, in short, the Berry dipole. When applying an oscillating electric field $\mathbf{E}(t)=\mathrm{Re}\{\mathcal{E}e^{i\omega t}\}$ with the amplitude vector $\mathcal{E}$ and frequency $\omega$, the nonlinear current in response to the electric field can be formally decomposed into the dc and double-frequency components J_{a}=\mathrm{Re}\big{\{}J^{(0)}_{a}+J^{(2)}_{a}e^{i2\omega t}\big{\}}, with $J^{(0)}_{a}=\chi^{(0)}_{abc}\mathcal{E}_{b}\mathcal{E}^{*}_{c}$ and $J^{(2)}_{a}=\chi^{(2)}_{abc}\mathcal{E}_{b}\mathcal{E}_{c}$, respectively. For time-reversal symmetric systems, their nonlinear Hall coefficients have the form Sodemann and Fu (2015)

[TABLE]

where $-e$ is the electron charge and $\tau$ is the momentum relaxation time. According to the experiments Ma et al. (2018); Kang et al. , the typical frequency of the ac electric field is about 10-1000 Hz and $\tau$ is approximately picoseconds; thus the frequency dependence in the denominator can be neglected due to $\omega\tau\ll 1$, which is the key difference from nonlinear optics. The Berry dipole can be found as

[TABLE]

where $\varepsilon^{abc}$ is the Levi-Civita symbol, $a,b,c,d\in\{x,y,z\}$, $f_{\textbf{k}}$ is the Fermi distribution, and the Berry curvature can be found as Xiao et al. (2010)

[TABLE]

where $|i\rangle$ refers to the eigenstate in band $i$ with energy $\epsilon^{i}_{\mathbf{k}}$ for a given wave vector $\mathbf{k}$. The derivative of the Fermi distribution in this definition of the Berry dipole infers that the states close to the Fermi surface mainly contribute to the nonlinear Hall response. Note that, under time reversal, $\Omega^{a}_{i}(-\textbf{k})=-\Omega^{a}_{i}(\textbf{k})$ and $\partial\epsilon^{i}_{\mathbf{k}}/\partial(-k_{b})=-\partial\epsilon^{i}_{\mathbf{k}}/\partial k_{b}$; thus the integral in Eq. (2) can survive in time-reversal symmetric systems. Meanwhile, inversion symmetry must be broken to support finite Berry curvature so that the integral in Eq. (2) does not vanish. Table 1 compares the differences between the linear and nonlinear Hall effects.

Tilted 2D massive Dirac model.– According to Eq. (2), the nonlinear Hall effect requires anisotropic bands with finite Berry curvature, so that $\Omega(\mathbf{k})\neq\Omega(-\mathbf{k})$. The tilted 2D massive Dirac model can give the ingredients

[TABLE]

where $(k_{x},k_{y})$ are the wave vectors, $(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices, $\eta=\pm 1$, $m$ is the gap, and $t$ tilts the Dirac cone along the $x$ direction. Compared to Ref. Sodemann and Fu (2015), $\alpha$ is introduced to regulate topological properties as $k\rightarrow\infty$ Shen (2017). The time reversal of the model contributes equally to the Berry dipole, so it is enough to study this model only. The model describes two energy bands (denoted as $\pm$) with the band dispersions $\varepsilon^{\pm}_{\mathbf{k}}=tk_{x}\pm[v^{2}k^{2}+(m/2-\alpha k^{2})^{2}]^{1/2}$, where $k^{2}\equiv k^{2}_{x}+k^{2}_{y}$. In the $x-y$ plane, the Berry curvature behaves like a pseudoscalar, with only the $z$ component $\Omega^{z}_{\pm\mathbf{k}}=\pm\eta v^{2}(m/2-\alpha k^{2})/\{2[v^{2}k^{2}+(m/2-\alpha k^{2})^{2}]^{3/2}\}$. Correspondingly, the Berry dipole behaves as a pseudovector constrained in the $x-y$ plane, e.g., $D_{xz}\rightarrow D_{x}$ and $D_{yz}\rightarrow D_{y}$.

Berry dipole near tilted band anticrossing. – The band dispersion, Berry curvature, and Berry dipole of the Dirac cone in Eq. (4) is shown in Fig. 2. If there is no tilt ($t=0$), Fig. 2(a) shows that the Berry curvature is symmetrically concentrated around the band edges so there is no Berry dipole. As the Dirac cone is tilted, the Berry curvature no longer symmetrically distributes, so the peaks of the Berry dipole appear near the band edges, as shown in Fig. 2(b)-(d). Here the tilt along the $x$ direction does not break the $y$-direction mirror reflection symmetry ($k_{y}\rightarrow-k_{y}$), so only the $x$ component of the Berry dipole is nonzero. Figure 2(e) shows that the Berry dipole vanishes at zero tilt ($t=0$), grows with increasing tilt, and reaches a maximum near the critical point ($t=v$) beyond which the Dirac cone is overtilted ($t>v$).

The Berry dipole vanishes in the gap due to the absence of carriers. Thus, unlike the Berry curvature in the intrinsic anomalous Hall effect, the Berry dipole does not build up across the band gap in the nonlinear Hall effect. The Berry dipole of the conduction and valence bands have opposite signs, because the bands of the model are particle-hole symmetric and the Berry curvature changes sign across the gap. The Berry curvature usually reaches maximum at the band edges, but the Berry dipole peaks do not lie exactly at the band edges because the group velocity vanishes at the band edges (see details in Sec. SII of Sup ).

Divergence of Berry dipole near band inversions.– By varying $m$ in model (4) from the same sign as $\alpha$ to the opposite sign, the system undergoes a band inversion, accompanied by a topological phase transition from the quantum anomalous Hall insulator to a trivial 2D insulator Bernevig et al. (2006); Lu et al. (2010). During the transition, the Berry curvature of the conduction and valence bands exchanges sign Lu et al. (2013) and the linear Hall conductance changes by $e^{2}/h$. Consequently, one can expect that the Berry dipole also changes sign during this topological phase transition. Figure 2(f) shows that divergences appear at the critical point where $m$ changes sign. The divergences near the gap closing indicate that the nonlinear Hall response can be significantly enhanced near the critical point of the topological phase transition. The divergences of the Berry dipole can serve as a probe for the topological phase transition of 2D tilted Dirac fermions.

Bilayer WTe2.– Now we show that the above-mentioned band signatures, i.e., the tilted band anticrossing and band inversion, can give strong nonlinear Hall signals in bilayers of WTe2. According to the above analysis, a time-reversal symmetric system needs inversion symmetry breaking to have large Berry curvature and large band tilt near the Fermi surface, in order to show a strong nonlinear Hall response. A number of materials meet the requirements Wang et al. (2016a); Autes et al. (2016); Ma et al. (2017); Zhang et al. (2018); Facio et al. (2018). Among them, the most promising candidate is WTe2, which has attracted tremendous attention due to its properties associated with a significant distribution of the Berry curvature Ali et al. (2014); Soluyanov et al. (2015); Wang et al. (2016b); Qian et al. (2014); Zheng et al. (2016); Fei et al. (2017); Tang et al. (2017); Wu et al. (2018); Xu et al. (2018); MacNeill et al. (2017a, b); Fei et al. (2018), e.g., the bulk WTe2 is a type-II Weyl semimetal Soluyanov et al. (2015); Wang et al. (2016b), its monolayer can host the high-temperature quantum spin Hall effect Qian et al. (2014); Zheng et al. (2016); Fei et al. (2017); Tang et al. (2017); Wu et al. (2018) and electrically switchable circular photogalvanic effect Xu et al. (2018). For the nonlinear Hall effect, the WTe2 bilayer has several advantages. It can be naturally exfoliated from bulk crystals. It has no inversion symmetry MacNeill et al. (2017a, b), while monolayers require gating to break inversion symmetry. It exhibits a spontaneous out-of-plane electric polarization Fei et al. (2018), which can help tuning spin-orbit coupling through gating.

In the absence of spin-orbit coupling, the bilayer WTe2 can be described by four tilted Dirac fermions Muechler et al. (2016). In the presence of spin-orbit coupling, we propose an effective model of the bilayer WTe2, which couples four pairs of tilted Dirac cones and reads

[TABLE]

where $\hat{\mathcal{H}}_{d_{i}}$ describes the tilted Dirac cone located at $\mathbf{K}_{i}$ (see Sec. SIII of Sup ),

[TABLE]

and $\nu_{x}$ and $\nu_{y}$ measure the spin-orbit coupling strength along the $x$ and $y$ directions, respectively, and can be tuned by a gate voltage applied along the $z$ direction. This model describes only the negative half of the Brillouin zone, which contributes equally to the Berry dipole as the positive half does. This model can qualitatively capture many key features of the bilayer WTe2 including the band structure evolution with spin-orbit coupling and the nonlinear Hall effect.

The evolution of the band structure with spin-orbit coupling is shown in the left panels of Fig. 3. According to the first-principles calculations Muechler et al. (2016), in the absence of spin-orbit coupling the bilayer WTe2 is a semimetal with a gap opened by the interlayer coupling, and each band is twofold degenerate. We can identify two tilted Dirac cones (marked by the ovals) in Fig. 3. The Berry curvature of the two Dirac cones are opposite because the two layers of the bilayer WTe2 are related through a mirror reflection. The experiments have observed that the bilayer WTe2 becomes insulating at low temperatures, implying a possible gap formation due to spin-orbit coupling Fei et al. (2017). As spin-orbit coupling is turned on, the degeneracy of the bands is lifted. The band structure shown in Fig. 3(f) is well gapped due to spin-orbit coupling and is likely the case in the experiments. As spin-orbit coupling decreases, Fig. 3 shows that the band gaps shrink and the system undergos two band inversions [indicated by the arrows in Fig. 3(b) $\leftrightarrow$ (d) and (c) $\leftrightarrow$ (e)], at which the conduction band exchanges the sign of the Berry curvature with the valence band. This band structure evolution is consistent with the first-principles calculations and experiments Muechler et al. (2016); Fan et al. (2015).

The corresponding Berry dipole results of these band structures are shown in the right panels of Fig. 3. As spin-orbit coupling decreases, the magnitude of the nonlinear Hall response increases with the shrinking gaps. In particular, the divergences appear at two band inversions. These divergences at the band inversions follow the mechanism shown in Fig. 2(f). The Berry dipole behavior is qualitatively consistent with the giant nonlinear Hall effect observed in the experiments, in which the nonlinear Hall response shows many sign changes and sharp resonancelike peaks as the gate voltage aligns the Fermi energy with the band signatures, such as the tilted band anticrossings and band inversions Ma et al. (2018).

Angular dependence of the nonlinear Hall voltage.– For systems without highfold rotation axis perpendicular to the 2D plane, it is known that the linear longitudinal and transverse resistances show twofold angular dependence as $\rho_{xx}=\rho_{\parallel}\cos^{2}\theta+\rho_{\perp}\sin^{2}\theta$ and $\rho_{xy}=(\rho_{\parallel}-\rho_{\perp})\sin\theta\cos\theta$. Here $\rho_{\parallel}$, $\rho_{\perp}$ refer to the resistivity along the principal axes and $\theta$ is the angle between the driving current and the $||$ principal axis. The nonlinear Hall coefficients also show angular dependence, but different from the linear case, the angular dependence is onefold

[TABLE]

where $\chi_{\parallel}$ and $\chi_{\perp}$ represent the nonlinear Hall coefficients along the principal axes (see details in Sec. SIV of Sup ). All these coefficients change sign when the current direction is reversed. The experiments measure only the Hall voltages, so we transform the angular dependence from the nonlinear Hall coefficients to that of the Hall voltages, and for 2D WTe2 it reads (see Sec. SIV of Sup )

[TABLE]

where we have assumed that the driving current is along the $y$ direction and the nonlinear Hall voltage is measured along the $x$ direction.

This means that the line shape of the angular dependence of $V^{2\omega}_{x}/(V^{\omega}_{y})^{2}$ depends on the resistance anisotropy ratio $\rho_{\parallel}/\rho_{\perp}$, as shown in Fig. 4. For $\rho_{\parallel}/\rho_{\perp}=0.268$, we can quantitatively reproduce the case measured in the experiment Kang et al. . In the isotropic case ($\rho_{\parallel}/\rho_{\perp}=1$), the angular dependence is simply a sine function. For $\rho_{\parallel}/\rho_{\perp}>1$, the peaks of the sine function are flattened, which is not the case in the experiment Kang et al. . If the resistance anisotropy ratio is large enough, say $\rho_{\parallel}/\rho_{\perp}=3$, the maxima at $\theta=\pi/2,~{}3\pi/2$ turn into minima, giving a double-peak line shape. This double-peak angular dependence can be observed by modifying the resistance anisotropy through tuning the gate voltages Ma et al. (2018).

We thank insightful discussions with Huimei Liu. This work was supported by the Guangdong Innovative and Entrepreneurial Research Team Program (2016ZT06D348), the National Basic Research Program of China (2015CB921102), the National Key R & D Program (2016YFA0301700), the National Natural Science Foundation of China (11534001, 11474005, 11574127), and the Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217, JCYJ20170412152620376).