# Tunable Topology and Berry Curvature Dipole in Transition Metal
Dichalcogenide Janus Monolayers

Nesta Benno Joseph  Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India.  Saswata Roy  Undergraduate
Programme, Indian Institute of Science, Bangalore 560012, India.  Awadhesh
Narayan<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India.

###### Abstract

Janus transition metal dichalcogenides, with intrinsic mirror asymmetry,
exhibit a wide array of interesting properties. In this work, we study Janus
monolayers derived from WTe2 using first-principles and tight-binding
calculations. We discover that WSeTe and WSTe are topologically trivial, in
contrast to the parent quantum spin Hall insulator WTe2. Motivated by the
growing interest in non-linear Hall effect, which also requires asymmetric
structures, we investigate the Berry curvature and its dipole in these Janus
systems and find that they exhibit strikingly large values of Berry curvature
dipole, despite being in the topologically trivial phase. We track down the
origin of this behaviour and put forth a low-energy massive Dirac model to
understand the central features of our ab inito computations. Our predictions
introduce Janus monolayers as promising new platforms for exploring as well as
engineering non-linear Hall effect.

##  I Introduction

Two dimensional materials have been a prime focus of research ever since the
discovery of graphene Novoselov  _et al._ (2004) and identification of its
novel properties such as high carrier mobility Bolotin  _et al._ (2008),
optical transparency Nair  _et al._ (2008) and high electrical and thermal
conductivity Balandin  _et al._ (2008). Transition metal dichalcogenides
(TMDCs), with an X-M-X sandwich structure, M being a transition metal within
group III - VI and X being either S, Se or Te, are one of the most intensively
studied two-dimensional materials Manzeli  _et al._ (2017). They exhibit a
staggeringly wide array of electronic and structural properties, and are
potential candidates for diverse applications Chhowalla  _et al._ (2013);
Bhimanapati  _et al._ (2015); Manzeli  _et al._ (2017). Depending on the M-X
coordination, TMDCs exist in three different phases - trigonal prismatic
coordinated 2H, octahedral coordinated 1T and the distorted octahedral
coordinated 1T’. Among the three phases, the ground state configuration of
each TMDC depends on its composition, with the stable 2H MoS2 most studied as
the representative member of the group Ganatra and Zhang (2014). This is
because the semiconducting material was found to display an indirect to a
direct bandgap transition when thinned down from bulk to monolayer, as a
result of quantum confinement Splendiani  _et al._ (2010). Most of the members
display ground state stability in either 2H or 1T coordination, the other
phases being a higher energy or a meta-stable state, with the exception of
WTe2. WTe2 is the only known TMDC to display ground state stability in its 1T’
phase and exists as a Weyl semimetal Soluyanov  _et al._ (2015) and quantum
spin Hall insulator Qian  _et al._ (2014) in its bulk and monolayer forms,
respectively.

The exploitation of electron spin for the development of spintronic devices
requires materials having an enhanced spin-orbit coupling interaction along
with broken inversion symmetry, leading to a large Rashba effect Manchon  _et
al._ (2015). Monolayer 2H phase of TMDCs became a viable candidate, but the
lack of Rashba splitting due to the out-of-plane mirror symmetry was a hurdle
to overcome. Janus monolayers of TMDCs, first proposed by Cheng et al. Cheng
_et al._ (2013), became a possible solution. In Janus TMDCs, one layer of
chalcogen in the MX2 sandwich monolayer is replaced by another, leading to an
asymmetric MXY structure. It was predicted that the broken out-of-plane
symmetry in these systems could lead to distinct and novel properties in the
Janus monolayers, compared to their parent materials Shi and Wang (2018); Wang
_et al._ (2018). Different experimental strategies have been devised to
synthesise such asymmetric structures, with the first Janus monolayer TMDC
MoSSe synthesised by Zhang et al. Zhang  _et al._ (2017) and Lu et al. Lu  _et
al._ (2017). After the synthesis of MoSSe, other two-dimensional Janus
structures were also predicted to exist. The half-metallic, ferromagnetic FeXY
Li  _et al._ (2021), TMDC oxides WXO Varjovi  _et al._ (2021), In2SSe, the
Janus analogue of binary compounds InSe and InS Kandemir and Sahin (2018a) and
Janus structures derived from metal monochalcogenides Liu  _et al._ (2021a)
are a few examples. The different Janus materials show several intriguing
properties Li  _et al._ (2018); Yagmurcukardes  _et al._ (2020); Ju  _et al._
(2021), including the aforementioned large Rashba effect Hu  _et al._ (2018),
tunable Dzyaloshinskii-Moriya interactions Liang  _et al._ (2020),
piezoelectricity and novel valleytronic effects Zhang  _et al._ (2019); Idrees
_et al._ (2019); Nandi  _et al._ (2021); Ahammed  _et al._ (2020). Another
major application of these materials is in water splitting and hydrogen
evolution reactions, as these are predicted to exhibit enhanced photocatalytic
activity Er  _et al._ (2018); Ju  _et al._ (2020); Peng  _et al._ (2019).

In a parallel, yet distinct, thread of developments, non-linear
generalizations of the Hall effect have been proposed. Following the
prediction of Sodemann and Fu, the first moment of the Berry curvature, termed
Berry curvature dipole (BCD), has been identified as an important quantity
responsible for non-linear Hall effects Sodemann and Fu (2015). Such a non-
linear Hall effect can also be generated in systems preserving time reversal
(TR) symmetry, in contrast to the linear Hall effect Sodemann and Fu (2015);
Du  _et al._ (2021); Ortix (2021). Diverse set of materials have been
predicted to show finite BCD and the resulting second order Hall effect,
including two-dimensional materials Low  _et al._ (2015); You  _et al._
(2018); Zhang  _et al._ (2018a); Joseph and Narayan (2021); Samal  _et al._
(2021), and Dirac and Weyl semimetals Du  _et al._ (2018); Xiao  _et al._
(2020); Singh  _et al._ (2020); Zhang  _et al._ (2018b); Roy and Narayan
(2021), strained bilayer and monolayer graphene Battilomo  _et al._ (2019) and
Rashba systems Facio  _et al._ (2018). Significant advances have also been
made in the experimental detection of such Hall signals in several material
platforms, starting with few-layer WTe2 Ma  _et al._ (2019); Shvetsov  _et
al._ (2019); Dzsaber  _et al._ (2021); Huang  _et al._ (2020); Kiswandhi and
Osada (2021); He  _et al._ (2021); Kumar  _et al._ (2021). Building on these
rapid developments, higher order Hall responses have also been predicted Liu
_et al._ (2021b); Zhang  _et al._ (2020), and very recently realized
experimentally Lai  _et al._ (2021).

Motivated by these advances, in this work, we investigate Janus monolayers
derived from 1T’ WTe2 using first-principles density functional calculations.
As mentioned above, unlike other TMDCs, the stable phase of WTe2 is 1T’.
Previous studies on Janus WXY concentrated on the 2H phase of the material
Kandemir and Sahin (2018b); Ju  _et al._ (2020), for example, with application
as photo catalysts in water splitting reactions Ju  _et al._ (2020). The
topologically non-trivial ground state of WTe2 make this material an
intriguing platform to look for tunable topology, in context of this study, as
a result of replacing one layer of Te with its lighter congeners. We show that
WSeTe and WSTe are topologically trivial, in contrast to WTe2, which is a
quantum spin Hall insulator. Using Wannier function based tight binding
models, we find that although the edge states in the three systems appear very
similar, a closer examination reveals their topologically distinct nature. We
investigate the BCD in these systems and find that all of them exhibit large
values of BCD, which are tunable in the Janus structures. Remarkably, the BCD
shows a pronounced peak near the Fermi level in the WSTe system, owing to a
reduced band gap which results in a larger Berry curvature. We supplement our
findings with an effective low-energy massive tilted Dirac model, which
captures the essential features of our first-principles results. Our studies
place Janus monolayers as promising platforms to engineer non-linear Hall
effects. We hope that our work motivates further investigations of topology
and Berry curvature dipole in transition metal dichalcogenide Janus
structures.

##  II  Computational Details

Our first-principles calculations were carried out based on the density
functional theory (DFT) framework as implemented in the quantum espresso code
Giannozzi  _et al._ (2017, 2009). A kinetic energy cut-off of $60$, $80$ and
$55$ Ry were considered for WTe2, WSeTe and WSTe respectively, using the
ultrasoft pseudopotentials Vanderbilt (1990) to describe the core electrons,
including spin-orbit coupling interactions. We used the Perdew-Burke-Ernzerhof
(PBE) form for the exchange correlation functional Perdew  _et al._ (1996).
The Janus structures were derived starting from WTe2, by replacing the top
layer Te with Se/S atoms and fully relaxing the resulting structures until
forces on each atom were less than at least $10^{-4}$ eV/Å. The Brillouin zone
was sampled over a uniform $\Gamma$-centered $k$-mesh of $12\times 8\times 1$.
The monolayers were modelled with at least 15 Å vacuum to avoid any spurious
interaction between the periodic images.

To study the topological properties, maximally localized Wannier functions
(MLWFs) were computed to derive a tight-binding model from the ab initio
calculations, with W $d$ orbitals, Te/Se $p$ orbitals, and S $s$ and $p$
orbitals as basis, using the wannier90 code Mostofi  _et al._ (2014). Two
methods were employed to characterise the topological nature of the monolayers
– direct calculation of the Z2 topological invariant and a computation of edge
states. The calculation and further analysis of edge spectrum was performed
using the WannierTools code Wu  _et al._ (2018), employing the method of
calculating surface Green’s function using the iterative Green’s function
approach Guinea  _et al._ (1983); Sancho  _et al._ (1984, 1985). A
computationally feasible method to calculate the $Z_{2}$ topological invariant
was proposed by Rui et al. Yu  _et al._ (2011) and Soluyanov et al. Soluyanov
and Vanderbilt (2011), using the notion of Wannier charge centers (WCCs). The
Wannier functions (WFs) associated with any unit cell are given by

$\big{|}R_{n}\big{\rangle}=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk~{}e^{-ik(R-x)}~{}\big{|}u_{nk}\big{\rangle},$
(1)

where $\big{|}u_{nk}\big{\rangle}$ is the periodic part of the Bloch state of
band $n$ at momentum $k$. The expectation value of the position operator in
the state $\big{|}0_{n}\big{\rangle}$ corresponding to WFs in the unit cell
$R=0$ gives the WCCs. The evolution of these WCCs of the MLWFs, along the TR
invariant plane $k_{z}=0$ is used to characterise the material as
topologically trivial or non-trivial.

For non-centrosymmetric systems, Hall-like currents have been predicted to be
present, even when the system preserves TR symmetry. This Hall effect, albeit
in the non-linear regime, arise due to the presence of BCD. For two-
dimensional systems, the Berry curvature, given by

$\Omega_{z}^{n}(\textbf{k})=2i\hbar^{2}\sum_{m\neq n}\frac{\langle
n|\hat{v}_{x}|m\rangle\langle
n|\hat{v}_{y}|m\rangle}{(\epsilon_{n}-\epsilon_{m})^{2}},$ (2)

has only an out-of-plane ($z$) component present. Here, $\epsilon_{m}$ and
$|m\rangle$ are the $m$-th energy eigenvalue and eigenvector of the
Hamiltonian, $\hat{v}_{x}$ and $\hat{v}_{y}$ are the velocity operators along
$x$ and $y$ respectively, and $\hbar$ is the reduced Planck’s constant. BCD is
a measure of the first order moment of the Berry curvature. For two-
dimensional materials, BCD behaves as a pseudo-vector and is given by Sodemann
and Fu (2015)

$D_{az}=\int_{k}~{}f_{n}^{0}(\textbf{k})~{}\frac{\partial\Omega^{n}_{z}}{\partial
k_{a}}d\textbf{k},$ (3)

with $f_{n}^{0}$(k) being the equilibrium Fermi-Dirac distribution and
$a\in\\{x,y\\}$ denotes the direction. The non-zero components of BCD are
further determined by the symmetries of the system. We used the Wannier-based
tight-binding Hamiltonian to compute the Berry curvature and BCD associated
with it using the wannier-berri code Tsirkin (2021).

##  III  Results and Discussion

Figure 1: Janus monolayers and their band structures. Structure of (a) monolayer 1T’ WTe2 and its associated Janus structures, (b) WSeTe and (c) WSTe, along with the respective electronic bands structures. The different types of W-X bonds present are marked $l_{1}$ – $l_{8}$. Here, the blue spheres depict the W atoms, while the orange, green and yellow spheres represent the Te, Se and S atoms respectively. The bands split due to the strongly broken out-of-plane symmetry and the bandgap along X-$\Gamma$ reduces as one layer of Te is replaced with its lighter congeners. |  $a$ (Å) |  $b$ (Å) | l1 | l2 | l3 | l4 | l5 | l6 | l7 | l8
---|---|---|---|---|---|---|---|---|---|---
WTe2 | 3.498 | 6.338 | 2.85 | 2.72 | 2.74 | 2.84 | 2.84 | 2.74 | 2.73 | 2.85
WSeTe | 3.406 | 6.154 | 2.80 | 2.72 | 2.73 | 2.83 | 2.64 | 2.58 | 2.54 | 2.72
WSTe | 3.344 | 6.064 | 2.79 | 2.72 | 2.73 | 2.83 | 2.50 | 2.46 | 2.41 | 2.63
Table 1: Structural parameters of WTe2 and associated Janus structures. The
lattice parameters $a$ and $b$ reduce on substitution of Te with Se/S. The
different W-X bond lengths are labelled l1–l8 [see Figure 1(b)]. All bond
lengths are given in Å.

###  III.1  Electronic and topological characterisation

In this section, we begin by describing the electronic properties of the Janus
monolayers and the associated topological characterisation. Figure 1 shows the
structure of pristine 1T’ WTe2, the Janus monolayers and their respective
electronic band structures. The top layer Te atoms of monolayer WTe2 were
replaced with its lighter congeners – Se and S – thereby constructing 1T’
Janus monolayers WSeTe and WSTe. We also carried out calculations of the
formation energy to compare the stability of the two Janus structures. The
formation energy, $E_{form}$, is given by $E_{form}=(E_{total}-E_{W}\times
N_{W}-E_{X}\times N_{X}-E_{Y}\times N_{Y})/N$. Here $E_{total}$ is the total
energy of the Janus WXY monolayer. $E_{W}$, $E_{X}$, $E_{Y}$ are the energies
of W, X and Y atoms in their respective bulk phases, $N_{W}$, $N_{X}$, $N_{Y}$
are the number of each of the atoms present in the unit cell and $N$ is the
number of formula units present. We find that the formation energy of WSeTe is
$\sim-0.52$ eV/f.u., while that of WSTe is $\sim-0.60$ eV/f.u. Therefore, the
formation of WSTe is energetically more favourable than WSeTe. The structural
parameters associated with the pristine WTe2 and the derived Janus structures
are given in Table 1. The band structure of both WSeTe and WSTe reveal
extensive splitting of band degeneracy due to the strongly broken out-of-plane
symmetry, when compared to WTe2. We find that all three materials have non-
zero direct bandgap at every point in the Brillouin zone. The electron and the
hole pockets present in WTe2 along the high symmetry direction X-$\Gamma$
vanish for both the Janus monolayers leaving the systems gapped, but the band
splitting leads to a reduced band gap. Along the same direction, the band gap
decreases from 143 meV in monolayer WTe2 to 32 meV in WSeTe and 6 meV in WSTe.
As we shall see, this reduction in gap has important consequences for Berry
curvature in these systems.

Figure 2: Topological characterisation and edge states. The Wannier Charge
Centres (WCCs) tracked along the $k_{z}=0$ plane for (a) WTe2, (b) WSeTe and
(c) WSTe. The line (red dashed) drawn parallel to the momentum axis intersects
the WCCs (black dotted) at exactly 1, 0 and 0 times for WTe2, WSeTe and WSTe,
respectively, indicating that the pristine monolayer is topologically non-
trivial while both the Janus structures are topologically trivial materials.
(d-f) The edge density of states (DOS) associated with the three structures,
along the two different types of edges present on construction of a one-
dimensional ribbon. Brighter bands indicate higher DOS along the edge. The
nature of the edge states is different in the Janus monolayers compared to
pristine WTe2, a manifestation of the different underlying topological
properties.

Employing the methods described in Section II, we calculated the Z2
topological invariant for the three monolayers. Figure 2 (a)-(c) present the
WCCs along the TR invariant plane $k_{z}=0$. The topological nature of the
material can be deduced by drawing a line parallel to the $k$ axis. For a two-
dimensional material, if any such line intersects the WCCs odd number of
times, it can be classified as topologically non-trivial. If the intersection
is such that the line crosses the WCCs even number of times, including zero
crossings, the material is trivial. In Figure 2 (a), the line (red dashed)
drawn parallel to $k$ axis intersects the WCCs (black dotted) an odd number of
times indicating that the pristine monolayer WTe2 is topologically non-
trivial, as expected and consistent with previous reports Qian  _et al._
(2014). However, as shown in Figure 2 (b) and (c), we notably find that there
exist zero or even number of such intersections, meaning both the Janus
monolayers WSeTe and WSTe are topologically trivial materials.

Computation of the respective edge spectrum further gives insight to this
difference in properties. Figure 2 (d)-(f) shows the density of states
projected onto the edges of the monolayers. The figure details two sets of
edge states [(d)-(f) top and bottom panels] for each monolayer, corresponding
to the two different (W terminated and chalcogen terminated) edges present on
the one dimensional ribbon. An initial look suggests the edge states on all
the three monolayers to be similar in nature, with a pair of bands on one edge
moving up, towards the conduction band manifold and another pair on the
opposite edge moving down, towards the valence band manifold. On closer
examination, we find that the origin of the bands in the Janus monolayers is
different from the pristine WTe2. In WTe2, for each pair of edge state, one of
them originates from the valence band, while the other from the conduction
band, leaving the one-dimensional ribbon gapless. In case of the Janus
monolayers, WSeTe and WSTe, both the pairs on either of the edges arise from
either the valence band or the conduction band manifolds. In both these cases,
any type of perturbation can easily gap the system, and the edge states will
not be robust as in the case of WTe2. This difference in the nature of origin
of the surface states between the Janus structures and WTe2 lead to their
respective trivial and non-trivial topological character, as obtained from our
topological invariant calculations too.

###  III.2  Berry curvature dipole

Figure 3: Berry curvature and Berry curvature dipole from first-principles.
(a) Berry curvature along the high symmetry $\Gamma$-X direction, where the
bandgap reduces as one layer of Te is replaced by its lighter congeners. The
largest values of the Berry curvature occur near band anti-crossings. Here the
red and blue colours indicate positive and negative values of the Berry
curvature, respectively. (b) BCD associated with the monolayers as a function
of the Fermi energy. Note the large peak in the vicinity of $E-E_{F}=0$ for
WSTe.

As we discussed, recently, it has been found that systems with lowered
symmetry and broken inversion display Hall-like currents in the non-linear
regime, even when the TR symmetry is preserved Sodemann and Fu (2015).
Breaking inversion leads to a separation of positive and negative values of
Berry curvature in the momentum space, resulting in a dipole. The 1T’
structure of TMDC monolayers already breaks inversion, with its distorted
octahedral metal-chalcogen coordination. The only other symmetry present in
these systems is a mirror plane along the $x$ axis ($M_{x}$), with the out-of-
plane symmetry further strongly broken by the construction of Janus
monolayers. Such a system – with broken inversion and lowered symmetry with a
single mirror plane – is ideal for the formation of BCD. Since the Berry
curvature only has an out-of-plane component and the BCD acts as a pseudo-
vector in two-dimensions, as discussed in Section II, due to the presence of
$M_{x}$ mirror plane, the only non-zero BCD component present is $D_{xz}$.

We next study the Berry curvature and BCD in our Janus structures. The
distribution of Berry curvature along the high symmetry $\Gamma$-X direction,
where the band gap closes due to substitution of the top layer chalcogen, is
shown in Figure 3 (a). The red and blue colors indicate the positive and
negative values respectively. Berry curvature increases in intensity as the
band gap is reduced, moving from Te to Se to S, taking up large values around
points of anti-crossing. Also notably, the Berry curvature becomes
significantly localized in momentum, specially near the Fermi level upon this
elemental substitution. We also highlight the strong splitting of the band
degeneracy in the Janus structures and find that the resulting split-off bands
carry opposite Berry curvatures.

Figure 3 (b) shows the non-zero $D_{xz}$ component of BCD as a function of the
Fermi energy, in pristine WTe2 and its associated Janus monolayers. The BCD is
a sensitive function of the Fermi level with peaks of varying intensities. We
note that the BCD value at the Fermi level increases as one goes from Te to Se
to S. This is a direct consequence of the increase in Berry curvature and its
momentum space localization in the Janus structures. The most noticeable
feature is the sharp peak in BCD at the Fermi level for WSTe. The value of BCD
at the Fermi level is nearly ten times larger in WSTe compared to WTe2.
Another intriguing feature occurs around $E-E_{F}=0.3$ eV, which is distinct
to the Janus monolayers. This peak-dip structure – with the BCD sharply
changing sign – is due to the splitting of the band degeneracy and creation of
nearly touching bands with opposite signs of Berry curvature away from the
Fermi level. Our obtained values of BCD and enhanced BCD around regions of
band anti-crossings are consistent with previous results on TMDCs, with the
dipole tunability realizable as a result of strain Son  _et al._ (2019) and
application of an electric field Zhang  _et al._ (2018a). In comparison,
twisted bilayer graphene displays a BCD two orders of magnitude higher than
TMDCs. This higher value, when compared to TMDCs, has also been attributed to
narrower bands and concentration of Berry curvature around the gapped Dirac
cone in this system Pantaleón  _et al._ (2021). It would further be
interesting to explore the strain-tuning of BCD in our Janus structures, since
it has been shown that Berry curvature in TMDCs can be efficiently controlled
using strain Wang  _et al._ (2019); Jena  _et al._ (2019). Our calculations,
therefore, demonstrate that Janus monolayers can be promising platforms to
engineer large values of BCD and resulting non-linear Hall effects.

###  III.3  Low-energy tilted massive Dirac model

Figure 4: Band Structure, Berry curvature and Berry curvature dipole in the
tilted massive Dirac model. Band structures for the tilted two-dimensional
massive Dirac Hamiltonian along $k_{x}$ (for $k_{y}=0$) for (a) $m=0.286$, (b)
$m=0.05$, (c) $m=0.012$. Value of the Berry curvature is superimposed on band
structure and corresponding color scales are shown. The corresponding BCD
component $D_{xz}$ is plotted for (d) $m=0.286$, (e) $m=0.05$, (f) $m=0.012$.
With decrease in $m$ the band gap decreases. The Berry curvature, which is
concentrated at the band edges, increases and gets concentrated in smaller
region with decrease in band gap. The BCD also increases with decreasing band
gap due to higher value of the underlying Berry curvature. The BCD peaks get
sharper and move closer to the Fermi energy with decreasing $m$. Other
parameters are chosen to be $\eta=-1$, $v=1$, $\alpha=1$, and $t=1$.

To gain further insights into our ab initio results, we consider a simple low
energy model for tilted two-dimensional massive Dirac cone. The Hamiltonian
reads Du  _et al._ (2018)

$H=tk_{x}+v(k_{y}\sigma_{x}+\eta k_{x}\sigma_{y})+(m/2-\alpha
k^{2})\sigma_{z},$ (4)

where ($\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$) are the Pauli matrices, $v$
is the effective velocity, $t$ is the tilt of the Dirac cones along $k_{x}$
direction, ($k_{x}$, $k_{y}$) are the wave vectors,
$k^{2}=k_{x}^{2}+k_{y}^{2}$, and $\eta=\pm 1$. Here $m$ is the mass parameter
which controls the gap of the system. For regulating the topological
properties as $k\rightarrow\infty$, the parameter $\alpha$ is introduced Shen
(2012). This Hamiltonian preserves TR symmetry, but inversion symmetry is
broken by the addition of the tilt term with strength $t$. The dispersion
relation is found to be

$E_{\pm}=tk_{x}\pm\sqrt{v^{2}k^{2}+(m/2-\alpha k^{2})^{2}}.$ (5)

Note that the mass parameter, $m$, determines the gap and can be used to tune
between our pristine and Janus monolayers. We can compute the Berry curvature,
which has only a non-zero $z$ component, as

$\Omega^{z}_{\pm}=\pm\eta v^{2}(m/2-\alpha k^{2})/\\{2[v^{2}k^{2}+(m/2-\alpha
k^{2})^{2}]^{3/2}\\}.$ (6)

As we discussed before, for our system the BCD tensor is a pseudo-vector in
the $xy$ plane. In Figure 4 (a)-(c), we show how the mass parameter, $m$,
controls the band dispersion, at a fixed tilt. Here we choose three different
values of $m$, which are in the ratio of the band gaps obtained from our
first-principles calculations for WTe2, WSeTe and WSTe. We find that with
decreasing $m$, the band gap decreases as expected. As a consequence, the
Berry curvature increases to very large values and gets concentrated in a
smaller and smaller momentum region near the band edges. Note the different
scales for the Berry curvature values in Figure 4(a) to (c).

We next calculated the BCD using the above expression for the Berry curvature.
The variation of $D_{xz}$ with the Fermi level is shown in Figure 4 (d)-(f)
for the above mentioned $m$ values. We find that as $m$ is reduced, the
maximum value of the BCD component increases sharply. The BCD peaks near the
band edges where the Berry curvature is concentrated. However it is not
exactly at the band edge since the group velocity
($v_{i}=\frac{\partial{E}}{\partial{k_{i}}}$) vanishes. Therefore, the peak
occurs at a point that is close to the band edges where the product of Berry
curvature and the group velocity is maximum. Furthermore, the peaks in the BCD
become narrower and move closer to the $E-E_{F}=0$ with decreasing $m$, due to
the Berry curvature becoming more localized in the momentum space. For the
smallest values of the mass parameter ($m=0.012$), we find a large peak near
$E-E_{F}=0$, closely resembling our DFT result for WSTe. It is remarkable that
such a simple two-band effective model captures the essential features of our
first-principles results.

##  IV Summary

We investigated 1T’ Janus monolayers derived from WTe2 parent structure using
a combination of density functional and tight-binding model computations. We
discovered that the Janus monolayers of WSeTe and WSTe are in the
topologically trivial phase, in contrast to the quantum spin Hall insulator
WTe2. We characterized their topology using a direct calculation of the
Wannier charge centers as well as an examination of the edge states. Motivated
by growing interest in non-linear Hall effects, we studied the Berry curvature
dipole in these systems and found that all of them exhibit large values. A
notable finding is a pronounced peak in the non-zero dipole component near the
Fermi level for WSTe monolayer, resulting from a reduced band gap which in
turn gives rise to a large Berry curvature. We further proposed a low-energy
massive Dirac model, which captures the main features of our first-principles
results. Our findings put forward Janus monolayers as promising platforms to
engineer tunable non-linear Hall effects, and we hope that this motivates
future studies of topology and Berry curvature dipole in Janus structures.

## Acknowldegments

We acknowledge discussions with D. Varghese, A. Bandyopadhyay, H. Liu, S. A.
Yang, S. Bhowal, N. A. Spaldin, N. P. Aetukuri and M. Jain. N.B.J.
acknowledges support from the Prime Minister’s Research Fellowship (PMRF). S.
R. is supported by the Kishore Vaigyanik Protsahan Yojana (KVPY) fellowship.
A.N. acknowledges support from the start-up grant (SG/MHRD-19-0001) of the
Indian Institute of Science and DST-SERB (project number SRG/2020/000153).

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