# Influence of the surface states on the nonlinear Hall effect in Weyl
semimetals

Diego García Ovalle  email<EMAIL_ADDRESS>Aix-
Marseille Université, CNRS, CINaM, Marseille, France.  Armando Pezo  email:
<EMAIL_ADDRESS>Aix-Marseille Université, CNRS,
CINaM, Marseille, France.  Aurélien Manchon  email: aurelien.manchon@univ-
amu.fr Aix-Marseille Université, CNRS, CINaM, Marseille, France.

###### Abstract

We investigate the influence of surface states on the nonlinear Hall response
of non-centrosymmetric time-reversal invariant Weyl semimetals. To do so, we
perform a tomography of the Berry curvature dipole in a slab system using a
minimal two-band model. We find that in the type-I phase, the nonlinear Hall
response is not particularly sensitive to the presence of Fermi arcs or other
trivial surface states. However, in the type-II phase, we find that these
surface states contribute substantially to the Berry curvature dipole, leading
to a strong thickness dependence of the nonlinear Hall response. This feature
depends on the nature of the surface states and, henceforth, on the slab
geometry adopted. In order to assess the validity of this scenario for
realistic systems, we performed Berry curvature dipole calculations by first
principles on the WTe2, confirming the dramatic impact of surface states for
selected slab geometries. Our results suggest that surface states, being
topological or not, can contribute much more efficiently to the nonlinear Hall
response than bulk states. This prediction is not limited to topological
semimetals and should apply to topologically trivial non-centrosymmetric
materials and heterostructures, paving the way to interfacial engineering of
the nonlinear Hall effect.

††preprint: APS/123-QED

##  I Introduction

Conventional wisdom inherited from Hall’s foundational work [1, 2] states that
Hall currents flowing transverse to the injected charge direction are only
permitted as long as time-reversal symmetry is globally broken. In other
words, ordinary, anomalous or topological Hall effects only exist either in
the presence of an external magnetic field or in magnetic materials displaying
a net magnetization [3]. However, this long-lived statement has been recently
challenged by two important observations. First, it has been realized that
time-reversal does not need to be broken globally, but only locally. In fact,
as long as no crystal symmetry reinstalls time-reversal effectively, Hall
currents are allowed. This is particularly true for certain classes of
antiferromagnets with a non-collinear magnetic configuration [4, 5, 6, 7] (see
also Ref. [8]). Second, the time-reversal symmetry breaking necessary to
obtain Hall effect is not required anymore at the second order in the electric
field. Under certain conditions, anomalous Hall effect can appear in non-
magnetic materials to the second order of the electric field [9, 10, 11].
Whereas the anomalous Hall effect in ferromagnetic and antiferromagnetic
compounds is associated with the Berry curvature of the material’s ground
state, the nonlinear anomalous Hall effect (NLHE) is rather associated with
the Berry curvature dipole (BCD). From a symmetry standpoint, the minimal
requirement is inversion symmetry breaking which ensures that the Berry
curvature does not vanish, but this is not sufficient: mirror symmetry also
needs to be broken to obtain a finite BCD. Sodemann and Fu [9] identified the
crystallographic point groups that possess the minimal requirements for the
observation of NLHE, a study recently refined by Du et al. [12] to include
both intrinsic (BCD related) and extrinsic mechanisms allowed by symmetry.

Different material candidates have been explored experimentally and
theoretically as suitable options to obtain NLHE. From an experimental point
of view, quadratic responses in the electric field have been detected, among
others, in bilayers and few layers of WTe2 [10, 11] which have $C_{2v}$ point
group. From a theoretical perspective, other possibilities for large BCD
values have been proposed, including transition metal dichalcogenides [13, 14,
15, 16, 17, 18], graphene [19, 20, 21], and especially Dirac [22] and Weyl
semimetals (WSM) [23]. Ab initio simulations [24] have stimulated further
theoretical studies of BCD in 3D WSMs, because of their rich geometrical
features and their potential benefits to create highly efficient electronic
transport devices. NLHE in WSMs has also been verified analytically by
applying perturbation theory [25], suggesting that the transport is sensitive
to intraband processes, the chemical potential and the tilting of the Weyl
nodes. Besides, similar optical effects such as second harmonic generation can
be confirmed by applying Floquet [26] and many-body quantum formalisms [27].
Last but not least, a full Green function theory of the NLHE has been proposed
recently [12], pointing out the differences between extrinsic and intrinsic
contributions. In this article, we are interested on the nonlinear response
arising on the intrinsic mechanism driven by the BCD.

Nonmagnetic WSMs such as TaAs or WTe2 are particularly interesting platforms
for the realization of NLHE because inversion symmetry is necessarily broken
and Berry curvature diverges at the Weyl nodes. WSMs possess pairs of doubly
degenerate linearly dispersive states, forming Weyl cones at Fermi level [28].
According to the Nielsen-Ninomiya theorem [29], each pair of nodes carries
Berry curvature monopoles of opposite chirality which are connected via Fermi
arcs lying at opposite surfaces of the slab [30]. Type I WSMs, such as
elemental Tellurium [31], Janus superlattices [32] and Ta or As compounds [33,
34, 35], are characterized by point-like Fermi surface in the bulk and
vanishing density of states. Type II WSMs, such as MoTe2 [36, 37] and WTe2
[38, 39] but also the magnetic candidate Co3Sn2S2 [40, 41, 42, 43, 44, 45],
offer a slightly different paradigm as the Weyl cone spectrum is tilted in
momentum space, breaking Lorentz invariance. As a result, the Weyl points
arise at the boundary between electron and hole pockets. Notice that certain
compounds can support type I as well as type II Weyl nodes [46, 47].

A remarkable aspect of WSMs is the nature of their surface states. As
mentioned above, alike topological insulators WSMs possess topologically
protected surface states in the form of spin-momentum locked Fermi arcs that
connect bulk Weyl nodes of opposite chirality. In type I WSMs, the Fermi arcs
coexist with the projection of electron (or hole) pockets when the chemical
potential lies away from the neutrality point. In type II WSMs, the Fermi arcs
coexist with projected electron and hole pockets irrespective of the value of
the chemical potential, as well as with trivial closed loops called ”track
states” [48]. As a consequence, surface states of WSMs can be rich, resulting
in enhanced Edelstein effect [49], and unconventional patterns in quantum
oscillation experiments [50, 51, 52] (see also Ref. [53]). Previous works
pointed out that topological materials defined in slab geometries can exhibit
interesting transport properties due to finite size effects and the behavior
of surface states inside the samples. In this context, it has been shown that
the anomalous Hall conductivity is highly influenced by surface states such as
Fermi arcs in Weyl systems without time-reversal symmetry [54], even in
presence of disorder [55]. Additional studies have been performed in confined
geometries to clarify, among others, the behavior of chiral magnetic effects
[56], the magnetoresistance [57] and the quantum Hall effect in Dirac
semimetals [58].

In this work, we seek to understand how the surface states of nonmagnetic WSMs
influence the NHLE response driven by the BCD. To do so, we consider a minimal
2-band model of a time-reversal invariant WSM with inversion symmetry breaking
in a slab geometry, so that bulk and surface states are treated on equal
footing [48]. This model exhibits four Weyl points: the minimum number of
degeneracies due to time-reversal symmetry. These four points are associated
with local divergencies of the Berry curvature, as depicted on Fig. 1. In this
sense, after neglecting the vanishing components of the BCD tensor due to
mirror symmetries in the 3D lattice, Zeng et al. [23] recently reported that
NLHE requires Weyl cone tilting and an asymmetric Fermi surface when the nodes
lie at the same energy. The NLHE is also influenced by the distance between
nodes. Accordingly, our study gives further insight about the implications of
the Fermi arc configurations on the BCD. Remarkably, it also complements a
recent study that comprises a surface BCD due to the projection of Fermi arcs
in type I WSMs [59].

Our paper is organized as follows. In Section II, we recall the general
formalism for nonlinear Hall transport driven by BCD and basic physical
considerations for our model of interest. In Section III, we provide and
discuss our results regarding the connection between surface states and NLHE,
its dependency on the Weyl cones tilting and its layer decomposition. In
Section IV, we also collate these outcomes with realistic numerical
simulations on a WTe2 slab with different cuts. Finally, we summarize and
state our main conclusions in Section V.

Figure 1: (Color online) Berry curvature of the Hamiltonian (4) in the
($k_{x},\;k_{z}$) plane with $k_{y}=0$ and intrinsic parameters
$k_{0}=\frac{\pi}{2}$, $m=2$, $t_{x}=\frac{1}{2}$, $\gamma=1$ and $t=1$. Note
that the Berry curvature of the two-band model is not sensitive to the value
of $\gamma$ [48]. The solid lines represent the Berry curvature vector field
$(\Omega_{x},\;\Omega_{z})$, whereas the blue (red) dots represent the local
positive (negative) divergencies taking place at the Weyl nodes.

##  II General Theory and Model

Let us begin by recalling basic elements of NLHE theory [9]. We start with a
nonmagnetic crystal and its corresponding Bloch Hamiltonian
$\mathcal{H}_{\mathbf{k}}$, whose eigenstates $\ket{u_{n\mathbf{k}}}$ satisfy
Schrödinger equation
$\mathcal{H}_{\mathbf{k}}\ket{u_{n\mathbf{k}}}=\epsilon_{n\mathbf{k}}\ket{u_{n\mathbf{k}}}$
with eigenenergies $\epsilon_{n\mathbf{k}}$. If the crystal is subjected to a
sufficiently small electric field
$\mathbf{E}=$Re$\left[\mathbf{\mathcal{E}}\exp(i\omega t)\right]$,
$\mathbf{\mathcal{E}}\in\mathbb{C}^{3}$, such that the adiabatic limit is
still valid, and assuming weak disorder, the first order Hall current is
induced by the Berry curvature [3]

$\displaystyle\mathbf{\Omega}_{n\mathbf{k}}=i\sum_{m\not=n}\frac{\braket{u_{n\mathbf{k}}}{\mathbf{\mathbf{\hat{v}}_{\mathbf{k}}}}{u_{m\mathbf{k}}}\times\braket{u_{m\mathbf{k}}}{\mathbf{\mathbf{\hat{v}}_{\mathbf{k}}}}{u_{n\mathbf{k}}}}{(\epsilon_{n\mathbf{k}}-\epsilon_{m\mathbf{k}})^{2}},$
(1)

where
$\mathbf{\hat{v}}_{\mathbf{k}}=\partial_{\mathbf{k}}\mathcal{H}_{\mathbf{k}}$
is the velocity operator. Upon time reversal operation $\mathcal{T}$ we have
$\mathcal{T}\mathbf{\Omega}_{n\mathbf{k}}\mathcal{T}^{-1}=-\mathbf{\Omega}_{n-\mathbf{k}}$,
and thus the anomalous Hall effect vanishes. Nonetheless, when inversion
symmetry $\mathcal{P}$ is further broken, then
$\mathcal{P}\mathbf{\Omega}_{n\mathbf{k}}\mathcal{P}^{-1}\not=\mathbf{\Omega}_{n-\mathbf{k}}$,
and one can show that to the lowest order in scattering time $\tau$, the
(rectified) second order Hall current reads [9]

$\displaystyle
j_{a}^{0}=\left(\frac{e^{3}\tau}{2\hbar^{2}}\right)\epsilon_{adc}D_{bd}\mathcal{E}_{b}\mathcal{E}_{c}^{*}.$
(2)

Here, latin indices refer to the components of the usual cartesian basis and
$\epsilon_{adc}$ is the Levi-Civita tensor. $D_{bd}$ is the BCD defined as [9]

$\displaystyle D_{bd}=\int_{BZ}\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\left[\sum_{n}(\mathbf{v}_{n\mathbf{k}})_{b}(\mathbf{\Omega}_{n\mathbf{k}})_{d}\frac{\partial
f_{n\mathbf{k}}}{\partial\epsilon_{n\mathbf{k}}}\right],$ (3)

with $\mathbf{v}_{n\mathbf{k}}=\partial_{\mathbf{k}}\epsilon_{n\mathbf{k}}$
the eigenvalues of the velocity operator and $f_{n\mathbf{k}}$ the Fermi
distribution function. In the zero-temperature limit,
$\partial_{\epsilon_{n\mathbf{k}}}f_{n\mathbf{k}}\to-\delta(\epsilon_{n\mathbf{k}}-\mu)$,
setting $\mu$ as the chemical potential. Equation (2) neglects the intrinsic
contribution to NLHE that appears when time-reversal is broken as well as
higher order extrinsic contributions (i.e., side-jump and skew scattering)
[12].

The nonmagnetic WSM slab is built on the spinless 2-band model in a cubic
lattice introduced in Ref. [48]. The bulk Hamiltonian reads

$\displaystyle\mathcal{H}_{B}=\gamma(\cos 2k_{z}-\cos k_{0})(\cos k_{x}-\cos
k_{0})\hat{\sigma}_{0}$ $\displaystyle-\left[m(1-\cos^{2}k_{x}-\cos
k_{y})+2t_{x}(\cos k_{z}-\cos k_{0})\right]\hat{\sigma}_{1}$
$\displaystyle-2t\sin k_{y}\hat{\sigma}_{2}-2t\cos k_{x}\hat{\sigma}_{3},$ (4)

with $\hat{\sigma}_{i}$, $i=1...3$ being the $2\times 2$ Pauli matrices,
$\hat{\sigma}_{0}=\mathbbm{1}_{2\times 2}$ and the cubic Brillouin zone is
$\mathcal{C}=\left[-\pi,\pi\right]^{3}$. The parameters of Eq. (4) are set to
$k_{0}=\frac{\pi}{2}$, $m=2$, $t_{x}=\frac{1}{2}$ and $t=1$. This WSM
possesses four Weyl nodes of zero energy located at
$\mathbf{k}^{*}=\pm\frac{\pi}{2}(\hat{x}+\hat{z})$ (see Fig. 1). Importantly,
for the parameters adopted in this work, $\gamma=2$ sets the transition point
between type I ($\gamma<2$) and type II ($\gamma>2$) WSM phases. In the
remaining of this paper, the transport properties will be investigated as a
function of the chemical potential $\mu$ as well as of the tilting of the Weyl
cones controlled by $\gamma$. Interestingly, Eq. (4) is constrained by the
following mirror symmetries [23]

$\displaystyle\mathcal{M}_{x}^{{\dagger}}\mathcal{H}_{B}(k_{x},k_{y},k_{z})\mathcal{M}_{x}$
$\displaystyle=\mathcal{H}_{B}(-k_{x},k_{y},k_{z}),$ (5)
$\displaystyle\mathcal{M}_{z}^{{\dagger}}\mathcal{H}_{B}(k_{x},k_{y},k_{z})\mathcal{M}_{z}$
$\displaystyle=\mathcal{H}_{B}(k_{x},k_{y},-k_{z}),$ (6)

which impose that the Fermi arcs connecting the Weyl nodes lie on the
($z,\;x$) surfaces. In addition, these symmetries constrain the BCD tensor, so
that the only non-vanishing elements are $D_{zx}$ and $D_{xz}$. In other
words, a second order Hall current can only be obtained in the ($y,\;z$)
($\equiv D_{zx}$) and ($x,\;y$) ($\equiv D_{xz}$) planes. Consequently, the
Fermi arcs cannot contribute to the NLHE response. Nevertheless, as discussed
below, other trivial surface states can substantially impact the second order
response.

We now design the slabs by discretizing Eq. (4) along a given direction,
$\hat{x}$, $\hat{y}$ or $\hat{z}$, of the cubic Brillouin Zone (see, e.g.,
Ref. [60]). The Hamiltonian loses periodicity along the chosen axis. The new
Hamiltonian $\mathcal{H}_{S}^{\hat{n}}$ of size $2L\times 2L$, with
$\hat{n}=\hat{x},\;\hat{y},\;\hat{z}$ depending on the growth direction, and
$L$ the number of layers, is given by

$\displaystyle\mathcal{H}_{S}^{\hat{n}}=\left(\begin{array}[c]{cccccc}\mathcal{H}_{0}^{\hat{n}}&\mathcal{H}_{1}^{\hat{n}}&\mathcal{H}_{2}^{\hat{n}}&0&...&0\\\
\mathcal{H}_{1}^{\hat{n}{\dagger}}&\mathcal{H}_{0}^{\hat{n}}&\mathcal{H}_{1}^{\hat{n}}&\mathcal{H}_{2}^{\hat{n}}&\ddots&\vdots\\\
\mathcal{H}_{2}^{\hat{n}{\dagger}}&\mathcal{H}_{1}^{\hat{n}{\dagger}}&\ddots&\ddots&\ddots&0\\\
0&\mathcal{H}_{2}^{\hat{n}{\dagger}}&\ddots&\ddots&\ddots&\mathcal{H}_{2}^{\hat{n}}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\mathcal{H}_{1}^{\hat{n}}\\\
0&\ldots&0&\mathcal{H}_{2}^{\hat{n}{\dagger}}&\mathcal{H}_{1}^{\hat{n}{\dagger}}&\mathcal{H}_{0}^{\hat{n}}\end{array}\right).$
(13)

For the block matrices in Eq. (13), $\mathcal{H}_{0}^{\hat{n}}$ is the
intralayer Hamiltonian that retains in-plane periodicity after the cut, while
$\mathcal{H}_{1}^{\hat{n}}$ and $\mathcal{H}_{2}^{\hat{n}}$ are the nearest
neighbor and second nearest neighbor interlayer Hamiltonian, respectively. For
the cut along $\hat{x}$, the block matrices are given by

$\displaystyle\mathcal{H}_{0}^{\hat{x}}$ $\displaystyle=-\gamma\cos k_{0}(\cos
2k_{z}-\cos k_{0})\hat{\sigma}_{0}$
$\displaystyle-\left[m\left(\frac{1}{2}-\cos k_{y}\right)+2t_{x}\left(\cos
k_{z}-\cos k_{0}\right)\right]\hat{\sigma}_{1}$ $\displaystyle-2t\sin
k_{y}\hat{\sigma}_{2},$ (14) $\displaystyle\mathcal{H}_{1}^{\hat{x}}$
$\displaystyle=\frac{\gamma}{2}\left(\cos 2k_{z}-\cos
k_{0}\right)\hat{\sigma}_{0}-t\hat{\sigma}_{3},$ (15)
$\displaystyle\mathcal{H}_{2}^{\hat{x}}$
$\displaystyle=\frac{m}{4}\hat{\sigma}_{1}.$ (16)

A cut along $\hat{y}$ leads to

$\displaystyle\mathcal{H}_{0}^{\hat{y}}$ $\displaystyle=\gamma(\cos
2k_{z}-\cos k_{0})(\cos k_{x}-\cos k_{0})\hat{\sigma}_{0}$
$\displaystyle-\left[m\sin^{2}k_{x}+2t_{x}(\cos k_{z}-\cos
k_{0})\right]\hat{\sigma}_{1}$ $\displaystyle-2t\cos k_{x}\hat{\sigma}_{3},$
(17) $\displaystyle\mathcal{H}_{1}^{\hat{y}}$
$\displaystyle=\frac{m}{2}\hat{\sigma}_{1}+it\hat{\sigma}_{2}.$ (18)

Finally, a cut along $\hat{z}$ gives

$\displaystyle\mathcal{H}_{0}^{\hat{z}}$ $\displaystyle=-\gamma\cos k_{0}(\cos
k_{x}-\cos k_{0})\hat{\sigma}_{0}$ $\displaystyle-\left[m(1-\cos^{2}k_{x}-\cos
k_{y})-2t_{x}\cos k_{0}\right]\hat{\sigma}_{1}$ $\displaystyle-2t\sin
k_{y}\hat{\sigma}_{2}-2t\cos k_{x}\hat{\sigma}_{3},$ (19)
$\displaystyle\mathcal{H}_{1}^{\hat{z}}$
$\displaystyle=-t_{x}\hat{\sigma}_{1},$ (20)
$\displaystyle\mathcal{H}_{2}^{\hat{z}}$
$\displaystyle=\frac{\gamma}{2}\left(\cos k_{x}-\cos
k_{0}\right)\hat{\sigma}_{0}.$ (21)

Figure 2: (Color online) Band structures of the WSM slab of 25 layers, with
$\gamma=1$ (a-c) and $\gamma=3$ (d-f) along the high symmetry path of the
cubic lattice for different cuts: (left panels) $\hat{x}$ for $k_{y}=0$ and
$k_{z}\in[-\pi,\pi]$, (center) $\hat{y}$ along the path
$A_{1}=(-\pi,0,-\pi)\to\Gamma=(0,0,0)\to A_{2}=(\pi,0,\pi)$ and (right)
$\hat{z}$ for $k_{y}=0$ and $k_{x}\in[-\pi,\pi]$. The central panels clearly
displays type I (b) and type II (e) Weyl cones. The surface states at the top
and bottom surfaces are represented with a solid red line. The flat red bands
in (a), (c), (d) and (f) are surface states at zero energy that are highly
localized at the corresponding surfaces.

We illustrate the slab band structures for each cut in Fig. 2, for $\gamma=1$
(type I WSM) and $\gamma=3$ (type II WSM). Since the slab Hamiltonians retain
a Berry curvature perpendicular to the plane of the slab, we define
$D_{zx}^{\hat{x}}$ as the BCD associated with $\mathcal{H}_{S}^{\hat{x}}$ when
the electric field is applied along $\hat{z}$, and $D_{xz}^{\hat{z}}$ is the
BCD associated with $\mathcal{H}_{S}^{\hat{z}}$ when the electric field is
along $\hat{x}$. These are the only relevant BCD components for the slab
geometries presented above due to the symmetry restrictions. We now move
forward to the method for calculating the layer decomposition of transport
coefficients in the slabs. Since the periodic part of the Bloch function can
be written in terms of the complete layer basis as
$\ket{u_{n\mathbf{k}}}=\sum_{l=1}^{L}\ket{\omega_{l\mathbf{k}}}\braket{\omega_{l\mathbf{k}}}{u_{n\mathbf{k}}}$,
one can extract information about layer $l$ by applying the projection
operator
$\mathcal{S}_{l}=\ket{\omega_{l\mathbf{k}}}\bra{\omega_{l\mathbf{k}}}$,
$l\in[1,L]$ to an observable $\mathcal{O}$. In this way, it is easy to see
that $\mathcal{O}=\sum_{l=1}^{L}\mathcal{S}_{l}\mathcal{O}$ and
$\sum_{l=1}^{L}\mathcal{S}_{l}=\mathbbm{1}_{2L\times 2L}$. We use this
description to separate the contributions of the density of states for a slab
with a fixed number of layers,

$\displaystyle\mathcal{D}_{l\mathbf{k}}$
$\displaystyle=-\frac{1}{\pi}\operatorname{Im}\left[\operatorname{Tr}\left(\mathcal{S}_{l}G^{R}_{\mathbf{k}}\right)\right],$
(22)

with the Green’s function

$\displaystyle G^{R}_{\mathbf{k}}$ $\displaystyle=\lim_{\eta\to
0^{+}}\left[(\epsilon+i\eta)\mathbbm{1}-\mathcal{H}_{S}^{\hat{n}}\right]^{-1}.$
(23)

This decomposition provides an insight about the influence of the surface
states for each growth direction. Additionally, in order to obtain the
contribution of layer $l$ to the NLHE response along, say, $\hat{y}$, one
simply needs to perform the substitution
$\hat{v}_{y}\to\mathcal{S}_{l}\hat{v}_{y}$ into Eq. (1) and then into (3).

##  III Results and Discussion

###  III.1 Band structure and surface states

Let us first consider how the Weyl point tilting impacts the slab band
structure and its surface states. Since only $D_{zx}^{\hat{x}}$ and
$D_{xz}^{\hat{z}}$ are non-vanishing, and because the Weyl nodes are located
in the $(k_{x},\;k_{z})$ plane, we focus on slabs composed of $L=25$ layers
and normal to the $\hat{x}$ and $\hat{z}$ directions. Without loss of
generality, we fix $\epsilon=0.2$ and select the exemplary cases
$\gamma=1,\;3$ to illustrate the differences between a type I and a type II
WSM, respectively. To better understand how bulk and surface states evolve
upon tilting the Weyl cones, we represent the density of states in the 2D
Brillouin zone and projected on a set of representative layers, ranging from
the bottommost surface to the topmost surface. Our results are depicted in
Fig. 3 for the $\hat{x}$-cut and Fig. 4 for the $\hat{z}$-cut.

Figure 3: (Color online) Projected density of states (DOS) for
$\hat{n}=\hat{x}$ as a function of the momentum coordinates for selected
layers $l$ for the type I ($\gamma=1$) (a) and type II ($\gamma=3$) phase (b).
In the type I phase, the Weyl nodes dominate in the bulk and get connected by
degenerate Fermi arcs at the surface, whereas in the type II phase, large
trivial Fermi pockets appear across the slab.

Figures 3 and 4 show different behaviors of the bulk and surface states when
tuning the tilting parameter. When $\hat{n}=\hat{x}$ (Fig. 3), we notice two
distinct situations depending on the value of $\gamma$: if $\gamma=1$ [type I,
Fig. 3(a)], the density of states is dominated by the Weyl nodes across the
whole slab, with a surface state composed of the projected nodes connected by
degenerate Fermi arcs. When $\gamma=3$, [type II, Fig. 3(b)], the density of
states of the central layer is composed of electron and hole pockets touching
at the Weyl nodes, as expected. These pockets result in large trivial surface
Fermi pockets enclosing the nodes, whose connections change direction from
$\hat{z}$ (type I) to $\hat{x}$ (type II) WSMs [48].

Figure 4: (Color online) Projected density of states (DOS) for
$\hat{n}=\hat{z}$ as a function of the momentum coordinates for selected
layers $l$ for the type I ($\gamma=1$) (a) and type II ($\gamma=3$) phase (b).
In the type I phase, the Weyl nodes remain disconnected across the slab,
whereas in the type II phase, trivial Fermi pockets dominate in the bulk with
track states and projected Weyl nodes at the surface.

On the other hand, for $\hat{n}=\hat{z}$ (Fig. 4), we obtain a different
behavior. Whereas for $\gamma=1$ [type I, Fig. 4(a)], the Weyl nodes remain
disconnected throughout the slab, for $\gamma=3$ [type II, Fig. 4(b)], a
surface contribution appears due the emergence of track states [48].
Nonetheless, the surface Fermi pockets associated with the projection of the
bulk electron and hole pockets remain very small and the surface states are
dominated by the Weyl nodes, in sharp contrast with the $\hat{x}$-cut
discussed in Fig. 3. We therefore expect the surface states to have completely
different impact on the NLHE signal in these two different slab geometries.

###  III.2 Nonlinear Hall response

Let us now turn our attention towards the NLHE response for the two slabs. We
compute the BCD components given by Eq. (4) in both slab geometries, (a)
$D_{zx}^{\hat{x}}/L$ ($\hat{x}$-cut) and (b) $D_{xz}^{\hat{z}}/L$
($\hat{y}$-cut), as well as in the 3D bulk structure, $D_{zx(xz)}^{\rm 3D}$
(black symbols), in order to better identify the impact of the surface states.
Notice that the BCD calculated in the slab geometry is normalized by the
number of layers $L$ to allow for a quantitative comparison with the BCD
calculated in the bulk. By definition, the BCD is therefore unitless. In the
following, we set the chemical potential to $\mu=0.2$ and represent the BCDs
as a function of the tilting parameter $\gamma$, as displayed in Fig. 5. In
addition, the corresponding ratio between the BCD and the density of states is
reported on (c) and (d) and discussed further below.

Figure 5: (Color online) BCD components as a function of the tilting parameter
$\gamma$ for slab and bulk systems. (a) Slab ($D_{zx}^{\hat{x}}/L$) and bulk
($D_{zx}^{\rm 3D}$) components for the $\hat{x}$-cut, for different slab
thicknesses. (b) Slab ($D_{xz}^{\hat{z}}/L$) and bulk ($D_{xz}^{\rm 3D}$)
components for the $\hat{z}$-cut, for different slab thicknesses.

For the $\hat{x}$-cut [Fig. 5(a)], the bulk BCD ($D_{zx}^{\rm 3D}$, black
symbols) displays a nonlinear dependence as a function of the tilting
parameter, as observed by Zeng et al. [23]: it first increases smoothly with
$\gamma$, reaches a maximum and decreases for large $\gamma$. We attribute
this tendency to the contribution of the Fermi surface between the Weyl nodes,
as also mentioned by Zeng et al. [23], and the upper limit that should reach
the tilting parameter regarding the inclination of the Weyl nodes. In
contrast, the slab BCD ($D_{zx}^{\hat{x}}$, colored symbols) increases sharply
and reaches a plateau at large $\gamma$. Furthermore, the value of this
plateau is substantially larger than the maximum value obtained in the bulk
although it slightly decreases when increasing the number of layers,
suggesting that surface states play a major role in the NLHE response. Since
the transition between type I and type II regimes produces a change in the
configuration of the surface states, with the apparition of Fermi pockets Fig.
3(b)], it is clear that the mismatch between the slab and bulk responses arise
from the influence of the Fermi pockets’ projections on the BCD across the
slab.

Conversely, for the $\hat{z}$-cut [Fig. 5(b)], the bulk and slab BCDs show a
similar behavior, increasing continuously and displaying local plateaus at the
type I-type II transition, as well as for large tilting parameter $\gamma$.
Based on Fig. 4(b), we infer that the surface track states do not affect
qualitatively the behavior of BCD in finite slab samples. In fact, in the
$\hat{z}$-cut, the slab BCD converges faster towards the bulk BCD when
increasing the number of layers than in the $\hat{x}$-cut configuration.

Since the density of states of carriers also changes when tuning the band
structure, we report the corresponding ratio between the BCD and the density
of states in Figs. 5(c) and (d). In Fig. 5(c), for a slab normal to $\hat{x}$,
the renormalized BCD exhibits a qualitatively similar trend across the type
I-to-type II transition, although the slab BCD is markedly larger than the
bulk BCD when approaching the type II regime. Therefore, the Fermi pockets’
projections on this slab enhance the value of the BCD due to the
reconfiguration of the surface states and the presence of more local states
within the unit cell. In contrast, for a slab normal to $\hat{z}$ Fig. 5(d)
shows a smaller discrepancy between the slab and bulk calculations suggesting
that the BCD is weakly impacted by the track states.

Figure 6: (Color online) Layer dependent contribution for the BCD normalized
by the number of layers (in this case, $L=25$). We show the tomography for a
type I WSM with $\gamma=1$ in (a) and for a type II WSM with $\gamma=3$ in
(b).

To complement this analysis, we compute the layer decomposition of the NLHE in
slab geometries, for $\gamma=1,\;3$ and $L=25$. The result is reported in Fig.
6 for (a) $\gamma=1$ and (b) $\gamma=3$ for $\hat{x}$\- (black) and
$\hat{z}$-cut (red). As we can see from Fig. 6(a), in a type I WSM the NLHE
response strongly differs at the edges, i.e, at the top ($l$=1) and bottom
($l$=25) layers. From Figs. 3(a) and 4(a) we attribute this behavior the
surface states driven by the degenerate Fermi arcs that only appear in the
$\hat{x}$-cut. In the case of the type II WSM, Fig. 6(b), we note that the
magnitude of the central layer ($l=13$) in the $\hat{x}$-cut is larger than
that in the $\hat{z}$-cut. Actually, the main contributions stem from layers
located underneath the surfaces ($l=5$, $l=21$) rather than on the surface
layers. This fact corroborates our claim that in type II WSMs a decisive
factor to enhance NLHE is the presence of Fermi pockets’ projections at the
surface rather than track states.

##  IV Nonlinear Hall effect in WTe2 slabs

Let us now consider a realistic system, WTe2 in its orthorhombic phase, and
compute the NLHE from first principles. WTe2 is a well-known type II WSM [38,
39], in which NLHE has been originally reported [10, 11]. For the density
functional theory simulations [61, 62], we used the Perdew-Burke-Ernzerhof
[63, 64] exchange-correlation functional. The geometry optimizations were
performed using a plane-wave basis as implemented in the Vienna Ab-initio
Simulation Package (VASP) [65, 66]. We have employed 400 eV for the plane-wave
expansion cutoff with a force criterion of 5 $\mu$ev/Åand a reciprocal space
sampling containing $16\times 16\times 14$ $\mathbf{k}$-points within the
Brillouin zone. The ionic potentials were described using the projector
augmented-wave (PAW) method [67], post-processing calculations were performed
using WannierTools [68]. The band structure is displayed in Fig. 7 with the
inset showing the unit cell. The band crossings are located within the ${\rm
X}-\Gamma$ path in momentum space, such that the Fermi level was set to zero
near this region. With this consideration, we can project the Weyl points on
selected surfaces, i.e., $\hat{z}$ (corresponding to the (001) direction) and
$\hat{x}$ (corresponding to the (100) direction).

Figure 7: (Color online) Bulk band structure of WTe2 type II WSM obtained from
density functional theory simulations. The inset displays the unit cell.

Let us first look at the density of states in the bulk and at the surface.
Fig. 8 displays the projected density of states in the bulk (a,c) and at the
surface (b,d) for a slab cut along the $\hat{z}$ direction (a,b), and for a
slab cut along the $\hat{x}$ direction (c,d). As can be observed from Figs.
8(a,b), in the $\hat{z}$-cut the resulting density of states of the slab is
larger at the surface than in the bulk, especially near the origin
($\bar{\Gamma}$ point). On the other hand, the opposite situation happens when
the slab is normal to $\hat{x}$, as can be noticed from Figs. 8(c,d). From
these results and comparing to the lattice model in Figs. 3 and 4, we deduce
that the surface states highlighted in Fig. 8(b) should enhance the value of
the BCD for the slab normal to $\hat{z}$, being strongly dependent on the
thickness of the slab, whereas the geometry normal to $\hat{x}$ should be less
sensitive to the Fermi arc diversity on the sample.

Figure 8: (Color online) WTe2 density of states at Fermi level, projected on
the bulk (a,c) and top surfaces (b,d) for a slab geometry containing 25
layers. Panels (a,b) correspond to a cut along $\hat{z}$ whereas panels (c,d)
correspond to a cut along $\hat{x}$.

We now move on to the computation of the surface contribution of the NLHE for
the two cuts considered above. For the calculation of the BCD, we have used
Eq. (3) performing the sum over a sample Brillouin zone containing $500\times
500$ k-points. Figures 9(a) and (b) display the band structure and the BCD,
respectively, computed for a slab cut along the $\hat{z}$ direction. The color
bar in (a) represents the projections on the top and bottom surfaces of a slab
containing 25 layers. One can distinguish both surfaces by the dark and yellow
lines near the Fermi level. In panel (b), the BCD is computed as a function of
the energy for slabs containing an increasing number of layers, from 15 to 25.
It is clear that the energy profile of the BCD strikingly depends on the slab
thickness. This thickness dependence reflects the influence of the surface
states. For the sake of comparison, we also reported the value of the BCD
computed in the bulk (dashed line). For an infinitely thick slab, the peaks
present below the Fermi level and associated with the surface states
disappear. Similarly, Figs. 9(c,d) display the band structure and BCD,
respectively, calculated for a slab cut along the $\hat{x}$ direction. In
contrast to the $\hat{z}$-cut discussed above, the surface states cannot be
clearly identified in the band structure that is instead dominated by bulk
states. In panel (d), the BCD is computed for three different slab
thicknesses. Interestingly, the qualitative behavior of the BCD is similar,
displaying a peak whose position is weakly influenced by the slab thickness.
For an infinitely thick slab, the bulk BCD (dashed line) conserves the overall
structure, with a slight reduction in magnitude. These calculations show that
surface states do substantially impact the NLHE in realistic materials.

Figure 9: (Color online) Band structure of a WTe2 slab containing 25 layers
and cut along $\hat{z}$ (a) and $\hat{x}$ (c). For this case the color bar
represents the projections on the bottom (-1) and top (+1) layers. (b,d)
Corresponding BCD for bulk and slabs containing 15, 20 and 25 layers.

##  V Conclusions

We have investigated the influence of surface states on the NLHE response of
non-centrosymmetric time-reversal invariant WSMs. Using both a model
Hamiltonian and realistic first principles calculations, we have demonstrated
that depending on the direction of the cut, surface states emerge that can
substantially contribute to the NLHE of the slab. Notice that the topological
nature of the surface state (topologically protected arcs, or track states)
has no impact on the overall BCD, only the relative number of states occupying
the surface and the bulk matters. We emphasize that the relative contribution
of the surface states with respect to the bulk states is in fact very large,
leading to dramatic thickness-dependence of the NLHE response, in particular
in type-II WSMs. This observation, confirmed by first principles calculations
on WTe2 slab geometries, suggests that surface states can contribute much more
efficiently to NLHE than their bulk counterpart. These conclusions are not
limited to WSMs and should apply to other topological systems and
topologically trivial classes of non-centrosymmetric materials.

###### Acknowledgements.

D.G.O thanks to Andrés Saúl and Carlos Quezada for their help and fruitful
discussions related to the computational implementation of the work. Besides,
D.G.O and A.M acknowledge support from the Excellence Initiative of Aix-
Marseille Université - A*Midex, a French ”Investissements d’Avenir” program.

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