††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work††thanks<EMAIL_ADDRESS>
# Topological Superconductivity in Monolayer T${}_{\textrm{d}}$-MoTe2

Xin-Zhi Li  School of Physical Science and Technology, ShanghaiTech
University, Shanghai 201210, China  Zhen-Bo Qi  School of Physical Science and
Technology, ShanghaiTech University, Shanghai 201210, China  Quansheng Wu
Beijing National Laboratory for Condensed Matter Physics and Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  University of
Chinese Academic of Sciences, Beijing 100049, China  Wen-Yu He  School of
Physical Science and Technology, ShanghaiTech University, Shanghai 201210,
China

###### Abstract

Topological superconductivity has attracted significant attention due to its
potential applications in quantum computation, but its experimental
realization remains challenging. Recently, monolayer T${}_{\textrm{d}}$-MoTe2
was observed to exhibit gate tunable superconductivity, and its in-plane upper
critical field exceeds the Pauli limit. Here, we show that an in-plane
magnetic field beyond the Pauli limit can drive the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 into a topological superconductor. The topological
superconductivity arises from the interplay between the in-plane Zeeman
coupling and the unique _Ising plus in-plane SOC_ in the monolayer
T${}_{\textrm{d}}$-MoTe2. The _Ising plus in-plane SOC_ plays the essential
role to enable the effective $p_{x}+ip_{y}$ pairing. Importantly, as the
essential _Ising plus in-plane SOC_ in the monolayer T${}_{\textrm{d}}$-MoTe2
is generated by an in-plane polar field, our proposal demonstrates that
applying an in-plane magnetic field to a gate tunable 2D superconductor with
an in-plane polar axis is a feasible way to realize topological
superconductivity.

##  I Introduction

Topological superconductors hosting unpaired Majorana zero modes are known as
a potential platform for topological quantum computation Kitaev1 ; Kitaev2 ;
ReadGreen ; Ivanov ; Sarma1 ; Alicea1 ; Alicea3 ; Kane1 ; Shoucheng2 ; Franz ;
Sato2 , which have stimulated the intense pursuit of topological
superconductivity (TSC) in real materials LiangFu1 ; Xiaoliang ; Sato ; Sau ;
Sarma2 ; Oppen ; Alicea2 ; Patrick1 ; Shoucheng ; EuaKim ; Jinfeng ; DingHong
; Kezilebieke ; Yulin ; Sau2 ; Stern ; FengLiu . The key element to realize
TSC is to have the $p$-wave pairing in the Fermi surfaces Kitaev1 ; ReadGreen
; LiangFu1 , but the intrinsic $p$-wave superconductors are extremely rare and
yet to confirm. In the past decades, intensive efforts have been made
attempting to induce the effective $p$-wave pairing via the proximity effect
with an $s$-wave superconductor LiangFu1 ; Xiaoliang ; Sato ; Sau ; Sarma2 ;
Oppen ; Alicea2 ; Patrick1 ; Jinfeng ; Sau2 ; Stern . Among a wide variety of
proposals, one ingenious setup is the hybrid superconductor-semiconductor
system with strong spin-orbit coupling (SOC) subjected to a Zeeman field Sato
; Sau ; Sarma2 ; Oppen ; Alicea2 ; Patrick1 , where the Zeeman coupling
results in odd number of Fermi surfaces and makes the proximity pairing in one
single Fermi surface become the effective $p$-wave pairing. The complexity of
the hybrid superconductor-semiconductor structure and its reliance on the
pairing proximity effect, however, largely limits the experimental realization
of TSC. Recently, several two dimensional (2D) materials are found to exhibit
the gate tunable superconductivity Justin1 ; Justin2 ; Fai1 ; Iwasa ; Barrera
; Fatemi ; Sajadi ; Caoyuan ; Park ; Andrea0 ; Andrea1 ; Andrea ; Park2 ;
ZhangYiran ; Holleis ; Pantaleon ; Faxian ; Guangtong ; Rhodes ; Jindal ; Smet
; Wakamura ; YuanGan . Therefore, seeking an intrinsic 2D superconductor that
is highly tunable to support the TSC phase becomes an alternative approach
that bypasses the unfavorable conditions of the hybrid superconductor-
semiconductor device.

MoTe2 is a layered transition metal dichaocogenide that exhibits rich
topological and superconductivity properties Guangtong ; Rhodes ; Jindal ;
Smet ; Wakamura ; YuanGan ; Zhijun ; Yanpeng ; Wudi . The MoTe2 of
T${}_{\textrm{d}}$ structure is an intrinsic superconductor in the bulk
Yanpeng ; Wudi , and the superconductivity gets greatly enhanced as the
T${}_{\textrm{d}}$-MoTe2 is exfoliated down to the monolayer Rhodes ; Jindal ;
Smet ; Wakamura ; YuanGan . In the dual-gate device of monolayer
T${}_{\textrm{d}}$-MoTe2, superconductivity has been achieved in both the
electron and hole dominated region Rhodes ; Smet . Owing to the gate
tunability, the electron and hole superconducting states can be converted to
each other Rhodes ; Smet . According to the first principle band structure
calculations, the Fermi surfaces of monolayer T${}_{\textrm{d}}$-MoTe2 are
composed of one hole pocket centred at $\Gamma$ along with two flanking
electron pockets Guangtong ; Rhodes ; Junwei ; Mengli . Given the condition
that the chemical potential is gate tuned to be around the energy of $\Gamma$,
an in-plane magnetic field induced Zeeman gap at $\Gamma$ would leave one
single hole Fermi surface, which creates the prerequisite to realize the
effective $p$-wave pairing in the monolayer T${}_{\textrm{d}}$-MoTe2.

The monolayer T${}_{\textrm{d}}$-MoTe2 involves a special type of SOC that is
a mixture of the Ising SOC and in-plane SOC Guangtong ; Rhodes . The SOC
arises from the gate induced out of plane electric field JustinSong  and the
inversion symmetry breaking of the crystal structure Suyang . Importantly, for
the hole superconducting states in the monolayer T${}_{\textrm{d}}$-MoTe2, the
_Ising plus in-plane SOC_ resembles the out of plane Dresselhaus SOC and in-
plane Rashba SOC in the hybrid superconductor-$\left(110\right)$-grown-
semiconductor system Alicea2 , which can be driven into a topological
superconducting state by applying an in-plane magnetic field. In the
superconducting T${}_{\textrm{d}}$-MoTe2 of atomic thickness, it has been
demonstrated that the _Ising plus in-plane SOC_ gives rise to an enhanced in-
plane upper critical field with an emergent two fold symmetry Guangtong .
Since the Ising SOC component protects the pairing from the in-plane magnetic
field Justin2 ; Fai1 ; Iwasa ; Barrera , it becomes promising that the gate
tunable superconducting monolayer T${}_{\textrm{d}}$-MoTe2 in an in-plane
magnetic field can enter the TSC phase.

Here, we propose that the gate tunable superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 can be driven into a topological superconductor by
applying an in-plane magnetic field. We show that the unique _Ising plus in-
plane SOC_ in the monolayer T${}_{\textrm{d}}$-MoTe2 is essential to the in-
plane magnetic field induced TSC phase. For the monolayer
T${}_{\textrm{d}}$-MoTe2 with a polar axis tilted to its atomic layer
Guangtong ; Jindal ; Rhodes ; JustinSong , we point out that for the TSC phase
transition, the preferred direction of the in-plane magnetic field is along
the in-plane projection of the polar axis. By performing the mean field
calculation, we demonstrate that the in-plane magnetic field induced TSC phase
occupies a considerable region in the phase diagram of the superconducting
monolayer T${}_{\textrm{d}}$-MoTe2. As entering the TSC phase is always
accompanied by the in-plane magnetic field driven gap closing, the tunneling
spectroscopy to measure the density of states (DOS) is suggested to
experimentally detect the phase transition to the TSC. It is worth emphasizing
the key advantages held in our scheme of TSC realization in the monolayer
T${}_{\textrm{d}}$-MoTe2: i) the TSC phase is derived from an intrinsic 2D
superconductor; ii) the monolayer nature circumvents the undesired orbital
depairing effect of the in-plane magnetic field; iii) the monolayer
T${}_{\textrm{d}}$-MoTe2 is highly tunable through the dual-gate and in-plane
magnetic field.

Figure 1: The schematic diagram of realizing topological superconductivity in
systems with SOC. (a) The schematic band dispersions with Rashba SOC (upper
panel) and the Rashba SOC induced spin texture in the Fermi surface (lower
panel). The system with Rashba SOC has a $z$-directional polar axis. Applying
a magnetic field parallel to the $z$-directional polar axis introduces a
Zeeman gap at $\bm{k}=\bm{0}$. (b) The schematic band dispersions and the
corresponding spin textures derived from a 2D system with an in-plane mirror
symmetry (the C1v symmetry). The mirror plane contains an in-plane polar axis
that lies in the atomic plane. The resulting in-plane polar field generates
the unique _Ising plus in-plane_ SOC, where its $z$-directional and in-plane
components of the spin texture are plotted in the upper and lower panel
respectively. Similar to the case of Rashba SOC, an in-plane magnetic field
parallel to the in-plane polar axis leads to a Zeeman gap at $\bm{k}=\bm{0}$.
(c) In the superconducting state, the effective $p_{x}+ip_{y}$ pairing is
induced by the magnetic field applied parallel to the polar axis. The upper
and lower panel corresponds to the effective $p_{x}+ip_{y}$ pairing in the 2D
system with the Rashba SOC and the _Ising plus in-plane SOC_ respectively.

##  II Results

###  II.1 Realization of TSC in systems with SOC

To illustrate the essential role of the _Ising plus in-plane SOC_ in our TSC
realization in the monolayer T${}_{\textrm{d}}$-MoTe2, we first start with
reviewing the seminal scheme of effective $p_{x}+ip_{y}$ pairing in the Rashba
superconductor Sato ; Sau ; Sarma2 ; Oppen . For a crystalline system with
SOC, its general Hamiltonian in the Bloch basis
$\left[c_{\bm{k},\uparrow},c_{\bm{k},\downarrow}\right]$ can be written as

$\displaystyle
H\left(\bm{k}\right)=\epsilon_{\bm{k}}-\mu+\bm{g}_{\bm{k}}\cdot\bm{\sigma}+\frac{1}{2}g\mu_{\textrm{B}}\bm{B}\cdot\bm{\sigma},$
(1)

with $\epsilon_{\bm{k}}$ being the band energy in terms of the Bloch wave
vector $\bm{k}$, $\mu$ being the chemical potential, $\bm{g}_{\bm{k}}$
denoting the SOC, and
$\bm{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ being the Pauli
matrices in the spin space. The last term in Eq. 1 represents the magnetic
field induced Zeeman coupling, where $g=2$ is the Landé $g$ factor and
$\mu_{\textrm{B}}$ is the Bohr magneton. For a 2D electron gas with a
$z$-directional polar axis, an electric field along the polar axis generates
the Rashba SOC $\bm{g}_{\bm{k}}=\alpha\left(k_{y},-k_{x},0\right)$. The Rashba
SOC pins the electrons’ spins to the in-plane direction and gives a helical
spin texture as shown in Fig. 1 (a). Given an isotropic band dispersion
$\epsilon_{\bm{k}}=\frac{\hbar^{2}\bm{k}^{2}}{2m}$, the Zeeman effect of a
magnetic field parallel to the $z$-directional polar axis lifts the Kramers
degeneracy at $\bm{k}=\bm{0}$ and opens the Zeeman gap. When the chemical
potential is tuned into the Zeeman gap, there is only one single Fermi surface
present in the system. Crucially, due to the Rashba SOC, the $z$-directional
magnetic field cannot fully polarize the electrons’ spins to the
$z$-direction, so the electrons with canted spins in the single Fermi surface
are allowed to have finite singlet pairing Sato ; Sau ; Sarma2 ; Oppen . Under
the condition of weak pairing
$\sqrt{\Delta^{2}+\mu^{2}}<\frac{1}{2}g\mu_{\textrm{B}}|\bm{B}|$, the pairing
in the single Fermi surface becomes the effective $p_{x}+ip_{y}$ pairing,
making the Rashba superconductor become a topological superconductor Sato ;
Sau ; Sarma2 ; Oppen .

Table 1: The explicit forms of the SOC $\bm{g}_{\bm{k}}\cdot\bm{\sigma}$ in a quasi-2D system. The quasi-2D system respects the C1, $C_{1v}$, C2 or C2v point group symmetry so that the SOC pseudo-vector $\bm{g}_{\bm{k}}$ has a nonzero $z$ component. The terms $\bm{g}_{\bm{k}}\cdot\bm{\sigma}$ are expanded relative to $\Gamma$ at $\bm{k}=\bm{0}$. The coordinate system used in the expansion is fixed by the in-plane polar axis direction. Point group | $\bm{g}_{\bm{k}}\cdot\bm{\sigma}$ | In-plane polar axis
---|---|---
C1 | $\sum_{i=x,y,z;j=x,y}\lambda_{i,j}\sigma_{i}k_{j}$ | arbitrary
C1v | $\lambda_{x}\sigma_{x}k_{y}+\lambda_{y}\sigma_{y}k_{x}+\lambda_{z}\sigma_{z}k_{y}$ | $x$
C2 | $\lambda_{x}\sigma_{x}k_{x}+\lambda_{y}\sigma_{y}k_{y}+\lambda_{z}\sigma_{z}k_{y}$ | $x$
C2v | $\lambda_{z}\sigma_{z}k_{y}$ | $x$

One long standing challenge for the TSC realization in the Rashba
superconductor is known to be the orbital depairing effects introduced by the
$z-$directional magnetic field Alicea2 . A modification to the Rashba
superconductor scheme is to introduce an extra out of plane Dresselhaus SOC so
that the transition to the TSC phase can be induced by an in-plane magnetic
field Alicea2 . However, an intrinsic 2D superconductor with such in-plane and
out of plane SOC that enables the in-plane magnetic field driven topological
phase transition remains yet to know. Microscopically, the SOC arises from the
relativistic interaction between the electron’s spin and the crystal electric
field $\bm{E}\left(\bm{r}\right)$. The approximate form of the SOC field takes
$\bm{g}_{\bm{k}}\propto\bm{E}\left(\bm{r}\right)\times\left(\hbar\bm{k}+\hat{\bm{p}}\right)$
with $\hat{\bm{p}}$ being the momentum operator Winkler . For the 2D Rashba
superconductor, it is clear that its Rashba SOC is produced by the
$z$-directional polar field. In a 2D system, the $z$-directional SOC that
resembles the out of plane Dresselhaus SOC can only be generated by an in-
plane polar field. As a result, in a 2D system, a polar axis with in-plane
projection is required for the $z$-directional SOC. Such in-plane projection
of the polar axis is referred as the in-plane polar axis.

In the crystalline system, the SOC field $\bm{g}_{\bm{k}}$ is known to respect
the symmetry constraint Samokhin ; Agterberg

$\displaystyle\bm{g}_{\bm{k}}=\det\left(\hat{R}\right)\hat{R}\bm{g}_{\hat{R}^{-1}\bm{k}},$
(2)

with $\hat{R}$ being the orthogonal matrix that represents the point group
transformation. In the quasi-2D case of atomic thickness, it has been pointed
out that only C1, C1v, C2 and C2v symmetries can have the in-plane polar axis
and allow the nonzero $z$-component of $\bm{g}_{\bm{k}}$ up to the linear
order of $\bm{k}$ wenyu1 . The explicit forms of $\bm{g}_{\bm{k}}$ expanded
from $\Gamma$ are listed in Table 1. It is clear from Table 1 that
$\bm{g}_{\bm{k}}$ derived from the C1, C1v or C2 symmetry is the _Ising plus
in-plane SOC_ that involves both the $z$-directional and in-plane components,
while that from the C2v symmetry is the pure Ising SOC Tongzhou  with only the
$z$ component of $\bm{g}_{\bm{k}}$ nonzero. Superconductors with the pure
Ising SOC can have the nodal TSC phase driven by an in-plane magnetic field
wenyu2 . Since the mechanism of the effective $p_{x}+ip_{y}$ pairing in the
Rashba superconductor tells that the realization of the fully gapped TSC phase
requires $\bm{g}_{\bm{k}}$ to have at least two nonzero components, the
$\bm{g}_{\bm{k}}$ in the quasi-2D system with the C1, C1v or C2 symmetry meets
the criteria. As a result, the seek for the in-plane magnetic field driven TSC
phase focuses on the intrinsic 2D superconductivity with the C1, C1v or C2
symmetry.

Importantly, the _Ising plus in-plane SOC_ in the quasi-2D system with the C1v
symmetry, as can be seen from the spin texture of
$\bm{g}_{\bm{k}}=\left(\lambda_{x}k_{y},\lambda_{y}k_{x},\lambda_{z}k_{y}\right)$
in Fig. 1 (b), is very similar to the combination of the out of plane
Dresselhaus SOC and in-plane Rashba SOC in the hybrid
superconductor-(110)-grown-semiconductor system Alicea2 . The C1v group only
has one in-plane mirror symmetry, so the in-plane polar axis is defined along
the intersection of the mirror plane and the layer plane. For an intrinsic 2D
superconductor with C1v symmetry that can have its chemical potential gate
tuned to be around the energy of $\Gamma$, it is expected that applying an in-
plane magnetic field parallel to the in-plane polar axis can lead to the TSC
phase in the similar mechanism as the hybrid superconductor-(110)-grown-
semiconductor system Alicea2 . As is schematically shown in Fig. 1 (c), the
in-plane magnetic field parallel to the in-plane polar axis induces the
effective $p_{x}+ip_{y}$ pairing, similar to the effect of the $z$-directional
magnetic field parallel to the $z$-directional polar axis in the Rashba
superconductor. Since it is promising to engineer a 2D superconductor with the
C1v symmetry into a topological superconductor, the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 that belongs to the C1v point group turns out to be a
suitable candidate for TSC.

Figure 2: The crystal structure and electronic band structure of monolayer Td-
MoTe2. (a) The top view of the crystal structure for monolayer Td-MoTe2. The
red dashed line labeled by $M_{y}$ denotes the in-plane mirror symmetry. (b)
Due to the nonalignment of the Te atoms in the vertical direction, an out-of-
plane electric field $E_{z}$ induces a net in-plane polar field. (c) The
electronic band dispersions along Y-$\Gamma$-X-S for the monolayer Td-MoTe2. A
$z$-directional electric field $E_{z}=0.1~{}\textrm{V/\AA}$ is considered in
the DFT calculations. (d) The spin textures of the $\Gamma$ centered hole
pocket and the two flanking electronic pockets.

###  II.2 Electronic structure of the monolayer T${}_{\textrm{d}}$-MoTe2

The crystal structure of the monolayer T${}_{\textrm{d}}$-MoTe2 hosts only one
in-plane mirror symmetry along $x$ as is shown in Fig. 2 (a). Isolated from
the inversion symmetry breaking bulk T${}_{\textrm{d}}$-MoTe2, the inversion
symmetry of the monolayer T${}_{\textrm{d}}$-MoTe2 gets slightly broken Rhodes
; Jindal ; Smet ; Suyang . In the dual-gate device, an out of plane electric
field further enhances the inversion symmetry breaking JustinSong : the
applied $E_{z}$ field induces a tilted dipole moment as illustrated in Fig. 2
(b). The in-plane component of the $E_{z}$ field induced dipole moment gives
rise to the $z$-directional Ising SOC, while its out of plane component
generates the in-plane SOC. Under the assumption of the applied
$E_{z}=0.1~{}\textrm{V/\AA}$, we perform the first principle density
functional theory (DFT) calculation to obtain the band structure of the
monolayer T${}_{\textrm{d}}$-MoTe2 (see the methods section). The results are
depicted in Fig. 2 (c). Consistent with the previous calculations Guangtong ;
Rhodes ; Junwei ; Mengli , the Fermi surfaces are composed of one hole pocket
at $\Gamma$ and two electron pockets at either side of $\Gamma$. Such metallic
band structure of monolayer T${}_{\textrm{d}}$-MoTe2 is supported by the
recent transport measurements Rhodes ; Pawlik . Deviating from $\Gamma$, spin
splittings due to the SOC are observed to appear. On the Fermi surfaces, the
_Ising plus in-plane SOC_ is clearly demonstrated by the spin textures plotted
in Fig. 2 (d).

Since the TSC phase arises from the effective $p_{x}+ip_{y}$ pairing in the
Fermi surface that encloses the time reversal invariant momenta Sato ; Sau ;
Sarma2 ; Oppen ; Alicea2 , our primary focus is on the hole pocket centred at
$\Gamma$. In the Bloch basis of
$\left[c_{\bm{k},\uparrow},c_{\bm{k},\downarrow}\right]$ near $\Gamma$, an
effective Hamiltonian respecting the C1v symmetry can be constructed to
describe the hole bands as

$\displaystyle H_{0}\left(\bm{k}\right)=$
$\displaystyle-\frac{\hbar^{2}k_{x}^{2}}{2m_{x}}-\frac{\hbar^{2}k_{y}^{2}}{2m_{y}}-\mu$
$\displaystyle+\lambda_{x}k_{y}\sigma_{x}+\lambda_{y}k_{x}\sigma_{y}+\lambda_{z}k_{y}\sigma_{z},$
(3)

with $m_{x}=0.75m_{e}$, $m_{y}=1.07m_{e}$ being the effective mass (here
$m_{e}$ is the electron mass), and $\lambda_{x}=2.94\times 10^{-2}$eV$\cdot$Å,
$\lambda_{y}=4.13\times 10^{-2}$eV$\cdot$Å, $\lambda_{z}=7.50\times
10^{-2}$eV$\cdot$Å denoting the strength of SOC. In experiment, the SOC is
tunable by the gate induced $E_{z}$ field. In the experimental accessible
range of $E_{z}\in\left[-0.1,0.1\right]\textrm{V/\AA}$ Jindal ; ZhangYiran ;
Holleis , our first principle DFT calculations show that the _Ising plus in-
plane SOC_ in the effective Hamiltonian in Eq. II.2 is sufficient to capture
the spin texture in the hole pocket (see the Supplementary Note 1 and
Supplementary Note 2).

###  II.3 TSC realization in the monolayer T${}_{\textrm{d}}$-MoTe2

The monolayer T${}_{\textrm{d}}$-MoTe2 is found to enter the superconductivity
phase with the transition temperature being 5K$\sim$8K Rhodes ; Jindal ; Smet
. Distinctly different from its isostructural monolayer
T${}_{\textrm{d}}$-WTe2 where the superconductivity occurs only in the
electron type carriers Fatemi ; Sajadi , the monolayer
T${}_{\textrm{d}}$-MoTe2 hosts intrinsic superconductivity in both the
electron and hole dominated region Rhodes ; Jindal ; Smet , indicating that
there exists finite pairing in the hole pocket at $\Gamma$. Importantly, due
to the protection of the Ising SOC component, the in-plane upper critical
field of the superconducting monolayer T${}_{\textrm{d}}$-MoTe2 gets strongly
enhanced: for the hole superconducting states with $T_{\textrm{c}}\approx 7$K,
the measured in-plane upper critical field $B_{\textrm{c}2}$ is extrapolated
to reach $B_{\textrm{c}2}/B_{\textrm{P}}\approx 2\sim 3$ Rhodes ; Smet , where
$B_{\textrm{P}}\approx 1.84T_{\textrm{c}}$ is the Pauli paramagnetic limit
Clogston ; Chandrasekhar . Since the monolayer T${}_{\textrm{d}}$-MoTe2 has an
atomic thickness of 0.35 nm that is far smaller than the measured coherence
length of $8\sim 16$ nm Rhodes ; Smet , the orbital magnetic field effect gets
suppressed and the Zeeman coupling with the superconducting states plays the
dominate role. In the Nambu basis of
$\left[c_{\bm{k},\uparrow},c_{\bm{k},\downarrow},c^{\dagger}_{-\bm{k},\uparrow},c^{\dagger}_{-\bm{k},\downarrow}\right]$,
the Bogliubov-de Gennes (BdG) Hamiltonian for the hole pocket pairing in the
presence of an $x$-directional Zeeman coupling takes the form:

$\displaystyle H_{\textrm{BdG}}\left(\bm{k}\right)=$
$\displaystyle\left(-\frac{\hbar^{2}k_{x}^{2}}{2m_{x}}-\frac{\hbar^{2}k_{y}^{2}}{2m_{y}}-\mu\right)\tau_{z}+\frac{1}{2}g\mu_{\textrm{B}}B\tau_{z}\sigma_{x}$
$\displaystyle+\lambda_{x}k_{y}\sigma_{x}+\lambda_{y}k_{x}\tau_{z}\sigma_{y}+\lambda_{z}k_{y}\sigma_{z}+\Delta\tau_{y}\sigma_{y},$
(4)

where $\tau_{z}$ is the Pauli matrix in the particle-hole space and $\Delta$
denotes the pairing order parameter. Similar to the Rashba superconductor, the
superconducting state in Eq. II.3 undergoes the topological phase transition
and enters the TSC phase for
$\frac{1}{2}g\mu_{\textrm{B}}|B|>\sqrt{\Delta^{2}+\mu^{2}}$. Specifically, at
$\mu=0$, the topological phase transition is estimated to occur around
$B=\sqrt{2}B_{\textrm{P}}$ wenyu2 , which is smaller than the in-plane
$B_{\textrm{c}2}$ measured in the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 Rhodes ; Smet . It indicates that the hole pocket
pairing in the superconducting monolayer T${}_{\textrm{d}}$-MoTe2 can survive
to become the effective $p_{x}+ip_{y}$ pairing after the in-plane magnetic
field driven topological phase transition.

The above analysis of the in-plane magnetic field driven TSC phase based on
the effective Hamiltonian of the $\Gamma$ pocket gets further supported by the
mean field phase diagram calculation. At a given temperature $T$, the
evolution of the superconducting pairing order parameter $\Delta$ with an
applied in-plane magnetic field can be determined by minimizing the free
energy density (see the methods section)

$\displaystyle F=$
$\displaystyle\frac{|\Delta|^{2}}{U}-\frac{1}{2\beta\Omega}\sum_{\bm{k},\nu}\log\left(1+e^{-\beta\xi_{\nu,\bm{k}}}\right),$
(5)

with $U$, $\beta=1/\left(k_{\textrm{B}}T\right)$ and $\Omega$ denoting the
intra-band attractive interaction, the thermodynamic beta and the volume of
the system respectively. Here $\xi_{\nu,\bm{k}}$ are the eigenvalues of the
BdG Hamiltonian that is derived from a twelve-band tight binding model of the
monolayer T${}_{\textrm{d}}$-MoTe2 (see the methods section). In the
calculation, the intra-band attractive interaction $U$ is determined by
fitting the superconducting critical temperature at zero magnetic field to be
7.6K. The chemical potential $\mu$ is fixed to be at the energy of $\Gamma$
that locates at the top of the hole bands. The resulting $B$-$T$ phase diagram
of the superconducting state is shown in Fig. 3 (a). It clearly shows that the
in-plane $B_{\textrm{c}2}$ is enhanced to reach 3$B_{\textrm{P}}$. By solving
$|\Delta\left(T,B\right)|=\frac{1}{2}g\mu_{\textrm{B}}|B|$, we obtain the
topological phase transition line. Between the in-plane $B_{\textrm{c}2}$ and
the topological phase transition line, it can be seen that there exists a
considerable superconducting region supporting the TSC phase. At $T=0$K, given
an in-plane magnetic field in the TSC region, the BdG quasiparticle energy
spectrum is plotted in Fig. 3 (b), where a chiral Majorana edge state appears
as expected for the TSC phase.

Figure 3: The phase diagram of monolayer Td-MoTe2 in an $x$-directional
magnetic field. In the region between the in-plane upper critical field
$B_{\textrm{c}2}\left(T\right)$ (red dashed line) and the topological phase
transition line (black solid line) in (a), the superconducting monolayer Td-
MoTe2 becomes a topological superconductor. (b) The BdG quasiparticle energy
spectrum of a monolayer Td-MoTe2 ribbon along $x$. The applied $x$-directional
magnetic field is $B=2.1\Delta/\left(g\mu_{\textrm{B}}\right)$.

In reality, the gate tunability in the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 facilitates our proposal of the in-plane magnetic
field driven TSC. First, the gate tunability enables the conversion of the
superconducting state between the electron type and the hole type Rhodes ;
Smet , indicating that the gate can tune the chemical potential close to the
energy of $\Gamma$ at the top of the hole bands. In the presence of an in-
plane magnetic field, opening a Zeeman gap at $\Gamma$ reduces the DOS, so the
magneto-conductance exhibits a dip once the chemical potential is tuned to lie
within the Zeeman gap (see the Supplementary Note 2 and Supplementary Fig. 2).
Second, the gate induced $E_{z}$ field can tune the strength of the SOC. For
the _Ising plus in-plane SOC_
$\bm{g}_{\bm{k}}=\left(\lambda_{x}k_{y},\lambda_{y}k_{x},\lambda_{z}k_{y}\right)$,
its $y$ and $z$ components that are perpendicular to the $x$-directional
magnetic field play the major role to enhance the in-plane $B_{\textrm{c}2}$
and promote the effective $p_{x}+ip_{y}$ pairing, while its $x$ component
suppresses the pairing in the presence of an $x$-directional magnetic field.
By varying $E_{z}$ in our first principle DFT calculation, we show that
$E_{z}\in\pm[0.05,0.07]\textrm{V/\AA}$ can reduce the $x$ component of
$\bm{g}_{\bm{k}}$ so that the in-plane $B_{\textrm{c}2}$ can be further
enhanced and the TSC region gets enlarged (see the Supplementary Note 2 and
Supplementary Fig. 3). For the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 with dual-gate, the chemical potential, the _Ising
plus in-plane SOC_ , and the in-plane magnetic field are all individually
tunable. Such high tunability makes the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 a promising platform to realize the long sought
topological superconductivity.

Given a 2D TSC phase in the monolayer T${}_{\textrm{d}}$-MoTe2, one can
further cut it into a quasi-1D wire with its width smaller than the decay
length of the Majorana edge state. The decay length of the Majorana edge state
is estimated to be about 300 nm for the pristine topological superconducting
monolayer T${}_{\textrm{d}}$-MoTe2 at
$B=2.1\Delta/\left(g\mu_{\textrm{B}}\right)$. In such a quasi-1D
superconducting wire, a pair of robust Majorana zero modes live in the ends of
the wire Potter01 , indicating that the quasi-1D wire of topological
superconducting monolayer T${}_{\textrm{d}}$-MoTe2 can serve as the building
block for the Majorana based topological quantum computation Sarma1 ; Alicea1
; Alicea3 .

###  II.4 Detection

For our proposed in-plane magnetic field driven TSC phase in the
superconducting monolayer T${}_{\textrm{d}}$-MoTe2, we show that one can
measure the evolution of DOS with $\bm{B}$ to identify the topological phase
transition. It is known that the topological phase transition must undergo the
gap close and reopen in the BdG quasiparticle energy spectrum. Given a small
$\bm{B}$, the gap in the BdG energy spectrum is not closed yet, so the
electronic DOS exhibits a U-shape with zero DOS in the gap. At the critical
value of $\bm{B}$ that enables the topological phase transition, the BdG
quasiparticle has the Dirac cone like dispersion with the gap closing at
$\bm{k}=\bm{0}$, so the resulting DOS becomes the V-shape. After the
topological phase transition, the gap reopens so the electronic DOS turns back
to the U-shape. In Fig. 4 (a), we present the local electronic DOS in the bulk
sample of superconducting monolayer T${}_{\textrm{d}}$-MoTe2. Here the
electronic DOS is calculated by
$N\left(E\right)=-\frac{1}{\pi}\textrm{Im}G^{\textrm{R}}\left(E\right)$ with
$G^{\textrm{R}}\left(E\right)$ being the retarded Green’s function (see the
Supplementary Note 3). Clearly, the electronic DOS evolution in Fig. 4 (a)
shows the gap close and reopen, consistent with our expected signature of the
in-plane magnetic field induced topological phase transition.

After the topological phase transition, the key feature of the TSC phase is
known to be the chiral Majorana edge state that appears within the gap. At the
edges of the sample, the local electronic DOS in the gap is expected to get
strongly enhanced by the chiral Majorana edge states. By calculating the local
electronic DOS at the edge of the supeconducting monolayer
T${}_{\textrm{d}}$-MoTe2 (see the Supplementary Note 3), we find that after
the topological phase transition, a plateau in the DOS emerges within the gap
of the BdG quasiparticle energy spectrum as shown in Fig. 4 (b). The plateau
appearing within the gap signifies the chiral Majorana edge states with the
linear energy dispersion in the TSC phase. Since both the local electronic DOS
in the bulk and at the edge can be detected by the tunneling spectroscopy
Bruus ; wenyu3 ; Costanzo , we suggest to carry out tunneling experiments to
probe the in-plane magnetic field driven TSC phase.

Figure 4: The signatures of the in-plane magnetic field driven superconducting
topological phase transition. (a) The electronic DOS in the bulk at different
in-plane magnetic fields. In the magnetic field driven topological phase
transition, the electronic DOS shows the gap closing and reopening. At the
topological phase transition with the gap closing at $\bm{k}=\bm{0}$, the DOS
exhibits the V-shape (red colored) as shown in the inset. During the
topological phase transition, the electronic DOS change from the U-shape to
the V-shape and then back to the U-shape. (b) The surface electronic DOS
calculated at the edge along $x$ direction. After the topological phase
transition, a plateau centered at the zero energy emerges, which corresponds
to the chiral Majorana edge states with the linear energy dispersion in the
topological superconductivity. The surface electronic DOS after the
topological phase transition are colored in red.

##  III Discussion

In the above sections, we have proposed that applying an in-plane magnetic
field to the superconducting monolayer T${}_{\textrm{d}}$-MoTe2 is promising
to achieve the TSC phase. In the proposal, the _Ising plus in-plane SOC_ plays
the crucial role in the TSC realization. The Ising SOC component is
responsible for the enhancement of the in-plane $B_{\textrm{c}2}$ to guarantee
the finite superconducting pairing after the magnetic field driven topological
phase transition. It is evident that a larger Ising SOC component is more
favorable for the TSC realization. For the moderate Ising SOC component in the
monolayer T${}_{\textrm{d}}$-MoTe2, one can consider to put it in contact with
another 2D material with strong Ising SOC so that the Ising SOC component in
the monolayer T${}_{\textrm{d}}$-MoTe2 can get proximity-enhanced ZhangYiran ;
Holleis ; Wakatsuki . Since the Ising SOC arises from the in-plane polar
field, it is also possible to add uniaxial strain to increase the Ising SOC
strength Manzeli ; Martin . With the Ising SOC component enhanced, the TSC
region is expected to get further enlarged in the phase diagram of the
monolayer T${}_{\textrm{d}}$-MoTe2.

In our proposal, for the _Ising plus in-plane SOC_
$\bm{g}_{\bm{k}}=\left(\lambda_{x}k_{y},\lambda_{y}k_{x},\lambda_{z}k_{y}\right)$
in the monolayer T${}_{\textrm{d}}$-MoTe2, we suggest to apply the in-plane
magnetic field along the in-plane mirror axis, namely the $x$ direction, to
realize the TSC phase. In the case of applying an in-plane magnetic field
along the $y$ direction, the whole system respects the in-plane mirror
symmetry, so the $y$-directional magnetic field can only lead to the mirror
symmetry protected nodal point degeneracies at $k_{y}=0$ plane. It indicates
that the fully gapped TSC phase requires the applied in-plane magnetic field
to have a nonzero $x$-directional component. For the applied in-plane magnetic
field slightly deviating from the $x$ direction, the magnetic field can still
induce the topological phase transition, but the resulting topological BdG
energy spectrum is a bit tilted in the $k_{x}$ direction (see the
Supplementary Note 3 and Supplementary Fig. 5). The tilted BdG energy spectrum
indicates that the magnetic field makes the centre of the Fermi surface get
shifted away from $\bm{k}=\bm{0}$, so the pairing in the Fermi surface is
possible to gain finite momentum under appropriate conditions WeiYi ; Chuanwei
; Yuanfanqi .

The crucial element in our proposal is the _Ising plus in-plane SOC_ in a gate
tunable 2D superconductor. For the monolayer T${}_{\textrm{d}}$-MoTe2, its
_Ising plus in-plane SOC_ arises from the C1v symmetry. Apart from the
monolayer T${}_{\textrm{d}}$-MoTe2, the bilayer T${}_{\textrm{d}}$-MoTe2 also
belongs to the C1v point group and exhibits gate tunable superconductivity
Guangtong . The Fermi surfaces of the bilayer T${}_{\textrm{d}}$-MoTe2 include
two hole pockets at $\Gamma$ Guangtong ; Jindal , so it is possible that
applying an in-plane magnetic field to the superconducting bilayer
T${}_{\textrm{d}}$-MoTe2 gives rise to a topological superconductor with the
Chern number $C=2$. The seek for the intrinsic 2D superconductors with _Ising
plus in-plane SOC_ can extend beyond the 4 point groups listed in Table 1. For
example, the monolayer 2M-WS2 is a superconductor whose crystal structure
respects the C2h symmetry Yulin . In the presence of gate induced out of plane
electric field, the C2h symmetry is reduced to the C1v symmetry so that a gate
tunable _Ising plus in-plane SOC_ can be generated in monolayer 2M-WS2.
Similar to our proposal of TSC realization in monolayer
T${}_{\textrm{d}}$-MoTe2, we expect that an in-plane magnetic field can induce
the TSC in monolayer 2M-WS2 as well.

In a brief summary, we propose that for the gate tunable superconducting
monolayer T${}_{\textrm{d}}$-MoTe2 with the C1v symmetry, the TSC phase can be
realized by applying an in-plane magnetic field. Based on the symmetry
analysis, we point out that 2D superconductors with an in-plane polar axis are
good platforms to host the TSC driven by an in-plane magnetic field.

##  IV Methods

###  IV.1 The First principle calculation and effective tight binding model

The band structure of monolayer T${}_{\textrm{d}}$-MoTe2 is calculated by
first-principle DFT as implemented in the Vienna Ab initio Simulation Package
(VASP) using the projected-augmented wave method (PAW) VASP_PAW . We adopt the
Perdew-Burke-Ernzerhof’s (PBE) form of exchange-correlation functional within
the generalized-gradient approximation (GGA). The Brillouin zone is sampled in
a Monkhorst-Pack k-point of $9\times 15\times 1$ and a large vacuum slab of 30
Å along the $z$ direction is used to reduce image interactions under periodic
boundary condition in the calculation. The monolayer crystal structure is
fully relaxed with a maximum residual force of less than 0.01 eV/Å. The energy
cutoff of the plane wave basis is set to be 520 eV. To take into account the
dual-gate induced $z$-directional electric field, we have considered an out of
plane electric field in the range of [-0.1, 0.1]V/Å in the calculation.

After the first principle DFT calculation is done, a twelve-band tight binding
model of monolayer T${}_{\textrm{d}}$-MoTe2 is built through Wannier function
construction by using the Wannier 90 package wannier90 . At the Fermi level
around the $\Gamma$ pocket, the dominant orbitals are the $d_{x^{2}-y^{2}}$
type from the Mo atom and $p_{y}$ type from the Te atom. We project twelve
bands near the Fermi level into these atomic orbitals. By a Wannier
interpolation wannier90  of these twelve DFT bands, we construct a basis of
localized Wannier functions, comprising four $d_{x^{2}-y^{2}}$-type orbitals
that are derived from the Mo atoms and eight $p_{y}$-type orbitals derived
from Te atoms. The spin degrees are already involved in the twelve orbitals.
The hopping matrix of the twelve-band tight binding Hamiltonian is obtained in
this way. After the Fourier transformation, we get the twelve-band tight
binding Hamiltonian of the monolayer T${}_{\textrm{d}}$-MoTe2 in the $\bm{k}$
space.

###  IV.2 The mean field free energy calculation for the superconducting
pairing state

Given the twelve-band tight binding model of monolayer
T${}_{\textrm{d}}$-MoTe2 in the Wannier basis, the Hamiltonian in the presence
of an effective pairing attractive interaction can be written as

$\displaystyle\mathcal{H}=$
$\displaystyle\sum_{\bm{k},i,o,o^{\prime},s,s^{\prime}}c^{\dagger}_{\bm{k},o,s}[h_{os,o^{\prime}s^{\prime}}(\bm{k})-\mu+\frac{1}{2}g\mu_{\textrm{B}}B_{i}\sigma_{i,ss^{\prime}}]c_{\bm{k},o^{\prime},s^{\prime}}$
$\displaystyle-\dfrac{U}{\Omega}\sum_{\bm{k},\bm{k}^{\prime}}\sum_{o,o^{\prime}}c^{\dagger}_{\bm{k},o,\uparrow}c^{\dagger}_{-\bm{k},o,\downarrow}c_{-\bm{k}^{\prime},o^{\prime},\downarrow}c_{\bm{k}^{\prime},o^{\prime},\uparrow},$
(6)

where $c^{\dagger}_{\bm{k},o,\sigma}$ is the creation operator and
$h_{os,o^{\prime}s^{\prime}}\left(\bm{k}\right)$ denotes the hopping matrix.
Here $o$ is the orbital index, $s=\uparrow/\downarrow$ represents the spin
index, and $i=x,y,z$ denotes the spatial components of magnetic field. In the
mean field approximation, the partition function for the superconducting
pairing state can be obtained through the path integral formalism Altmanbook

$\displaystyle\mathcal{Z}\approx$
$\displaystyle\int\mathcal{D}\left[\psi^{\dagger}_{\bm{k}},\psi_{\bm{k}}\right]e^{-\frac{\beta\Omega|\Delta|^{2}}{U}+\frac{1}{2}\sum_{\bm{k},n}\psi^{\dagger}_{\bm{k},n}\left[i\omega_{n}-\mathcal{H}_{\operatorname*{BdG}}(\bm{k})\right]\psi_{\bm{k},n}}$
$\displaystyle=$ $\displaystyle
e^{-\frac{\beta|\Omega|^{2}}{U}+\frac{1}{2}\sum_{\bm{k},n}\operatorname*{Tr}\log\left[-\beta
G^{-1}(\bm{k},i\omega_{n})\right]},$ (7)

where
$\psi^{\dagger}_{\bm{k}}=\left[c^{\dagger}_{\bm{k},o,\uparrow},c^{\dagger}_{\bm{k},o,\downarrow},c_{-\bm{k},o,\uparrow},c_{-\bm{k},o,\downarrow}\right]^{\textrm{T}}$
is the Nambu spinor,
$G^{-1}\left(i\omega_{n},\bm{k}\right)=i\omega_{n}-h_{\textrm{BdG}}\left(\bm{k}\right)$
is the Matsubara Green’s function with the BdG Hamitonian taking the form

$\displaystyle
h_{\textrm{BdG}}\left(\bm{k}\right)=\begin{pmatrix}\tilde{h}\left(\bm{k}\right)-\mu&-i\sigma_{y}\Delta\\\
i\sigma_{y}\Delta&-\tilde{h}^{\ast}\left(-\bm{k}\right)+\mu\end{pmatrix}.$ (8)

Here
$\tilde{h}\left(\bm{k}\right)=h\left(\bm{k}\right)+\frac{1}{2}g\mu_{\textrm{B}}\bm{B}\cdot\bm{\sigma}$
is the twelve-band tight binding Hamiltonian matrix that takes into account
the Zeeman coupling. The free energy density
$F=-\frac{1}{\beta\Omega}\log\mathcal{Z}$ gives Eq. 5 with $\xi_{\nu,\bm{k}}$
being the eigenvalues of $h_{\textrm{BdG}}\left(\bm{k}\right)$ in Eq. 8. The
self-consistent mean field equation of the pairing order parameter is given by
$\frac{\partial F}{\partial\Delta}=0$. In the calculation, at $B=0\textrm{T}$,
we set $T_{\textrm{c}}=7.6\textrm{K}$ and then the effective attraction $U$
can be determined by solving $\frac{\partial F}{\partial\Delta}=0$. By
minimizing the free energy density at given $T$ and in-plane magnetic field,
the phase diagram of the superconducting pairing state can be obtained.

## Acknowledgment

The authors are grateful for the helpful discussions with Patrick A. Lee.
W.-Y. H. acknowledges the support from the National Natural Science Foundation
of China (No. 12304200), the BHYJRC Program from the Ministry of Education of
China (No. SPST-RC-10), and the start-up funding from ShanghaiTech University.
Q. W. acknowledges the support from the National Key R&D Program of China
(Grant No. 2023YFA1607400).

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Supplementary Material: Topological Superconductivity in Monolayer Td-MoTe2

## Supplementary Note 1: The Band Structures and Spin Textures under Different
Electric Fields

In the presence of an out of plane electric field at $E_{z}$=0.05, 0.1, and
0.15 V/Å , we perform the first principle DFT calculations and use the Wannier
90 package to obtain the band structures as depicted in Supplementary Fig. 1.
The out of plane electric field breaks the inversion symmetry and induces the
spin splittings in the bands. Different values of $E_{z}$ slightly changes the
spin splittings in the bands, but the spin textures of the Fermi pockets can
all be described by the effective Hamiltonian in Eq. 3 in the main text. The
monolayer T${}_{\textrm{d}}$-MoTe2 with a gate induced $z$-directional
electric field belongs to the C1v group, so all the spin textures manifest the
unique _Ising plus in-plane SOC_ in the monolayer T${}_{\textrm{d}}$-MoTe2.
Importantly, comparing the spin textures in Fig. 1 (e) with those in Fig. 1
(d) and (f), one can identify a sign change in the $x$ component of
$\bm{g}_{\bm{k}}=\left(\lambda_{x}k_{y},\lambda_{y}k_{x},\lambda_{z}k_{y}\right)$.
The sign change of $g_{x,\bm{k}}$ induced by the $E_{z}$ field indicates that
the $x$ component of
$\bm{g}_{\bm{k}}=\left(\lambda_{x}k_{y},\lambda_{y}k_{x},\lambda_{z}k_{y}\right)$
can be reduced to zero by changing the value of $E_{z}$.

Supplementary Fig. 1: The band structure of monolayer T${}_{\textrm{d}}$-MoTe2
calculated at $E_{z}=0.05\textrm{ V/\AA}$ for (a), $E_{z}=0.1\textrm{ V/\AA}$
for (b) and $E_{z}=0.15\textrm{ V/\AA}$ for (c). The corresponding spin
textures in the Fermi pockets are calculated through the twelve-band effective
tight binding model. The results are present in (d), (e) and (f) respectively.

## Supplementary Note 2: The High Tunability of Dual-gate Device

The dual-gate setup of monolayer T${}_{\textrm{d}}$-MoTe2 is schematically
plotted in Supplementary Fig. 2 (a). One can regard the dual-gate setup as two
quantum capacitors in series. Through tuning the top and bottom gate, one can
tune the charge density $n_{\textrm{2D}}$ in the monolayer
T${}_{\textrm{d}}$-MoTe2 and the effective $z$-directional displacement field
individually as gatezheng ; gatecao :

$\displaystyle
n_{\rm{2D}}=\frac{\epsilon_{0}\epsilon_{b}V_{bg}}{ed_{b}}+\frac{\epsilon_{0}\epsilon_{t}V_{tg}}{ed_{t}},\quad\quad
D_{z}=\frac{1}{2}\left(\frac{\epsilon_{0}\epsilon_{b}V_{bg}}{d_{b}}-\frac{\epsilon_{0}\epsilon_{t}V_{tg}}{d_{t}}\right),$
(1)

where $\epsilon_{b/t}$ denotes the relative perimitivity in the bottom/top
region, and the distances $d_{b/t}$ are denoted in Fig. 2 (a). We know that
the chemical potential of a two-dimensional electronic system is determined by
the electron density. We also know that a $D_{z}$ field creates an electric
field $E_{z}=D_{z}/\epsilon_{0}$. As a result, the dual-gate setup can tune
the chemical potential and $E_{z}$ field individually by tuning the top and
bottom gate voltage. Given $E_{z}=0.1\textrm{ V/\AA}$, the in-plane magnetic
field induced Zeeman coupling opens a Zeeman gap at $\Gamma$ as shown in
Supplementary Fig. 2 (b) and (c). If the chemical potential is tuned into the
Zeeman gap at $\Gamma$, the electronic DOS gets reduced and manifests as a dip
as shown in Fig. 2 (c). Such in-plane Zeeman coupling induced dip in the
electronic DOS is expected to get identified through a magneto-conductance
measurement.

In the experimental accessible range of $E_{z}\in\left[-0.1,0.1\right]$ V/Å,
we calculate the band structures of the monolayer T${}_{\textrm{d}}$-MoTe2 and
fit them by the effective hole pocket Hamiltonian around the $\Gamma$ in Eq.
3. The strength of the _Ising plus in-plane SOC_ as a function of the applied
$E_{z}$ field is plotted in Supplementary Fig. 3. It shows that the SOC field
$\bm{g}_{\bm{k}}=(\lambda_{x}k_{y},\lambda_{y}k_{x},\lambda_{z}k_{y})$ can be
modulated by varying the $E_{z}$ field. Importantly, due to
$\bm{g}_{\bm{k}}\propto\bm{E}\left(\bm{r}\right)\times\left(\hbar\bm{k}+\hat{\bm{p}}\right)$,
the $x$ component of the SOC $g_{x,\bm{k}}=\lambda_{x}k_{y}$ is greatly
affected by the applied $E_{z}$ field. Within the range of
$E_{z}\in\pm[0.05,0.07]$ V/Å, $\lambda_{x}$ is found to reverse its sign,
while $\lambda_{y}$ and $\lambda_{z}$ keep increasing with $E_{z}$. Our
calculation results indicate that in the dual-gate device, tuning the $E_{z}$
field can reduce the $x$ component of $\bm{g}_{\bm{k}}$ and make
$\bm{g}_{\bm{k}}$ more favorable to the in-plane magnetic field driven TSC.

Supplementary Fig. 2:  (a) The schematic showing of a dual-gate device with a
monolayer T${}_{\textrm{d}}$-MoTe2 crystal involved. (b) and (c) demonstrate
the band structure around the $\Gamma$ without/with an in-plane magnetic
field. Here we consider the applied electric field in the calculation to be
$E_{z}=0.1$ V/Å. (d) The electronic DOS for the band structures in (b) and
(c). An in-plane magnetic field introduces a Zeeman gap at $\Gamma$ and
reduces the DOS. Supplementary Fig. 3: The strength of the _Ising plus in-
plane SOC_ as a function of the applied $E_{z}$ field. Within the range of
$E_{z}\in\pm\left[0.05,0.07\right]$ V/Å, the $x$ component of
$\bm{g}_{\bm{k}}$ is shown to get reduced.

## Supplementary Note 3: The Calculations for the Density of States and the
Energy Spectrum at the Edge

The electronic DOS in the bulk of the superconducting monolayer
T${}_{\textrm{d}}$-MoTe2 is given by

$\displaystyle
N\left(E\right)=-\frac{1}{\pi}\textrm{Im}G^{\textrm{R}}\left(E\right).$ (2)

Here the regarded Green’s function $G^{\textrm{R}}\left(E\right)$ can be
derived as:

$\displaystyle
G^{\textrm{R}}\left(E\right)=\frac{1}{\Omega}\sum_{\bm{k}}\frac{1}{E+i0^{+}-h_{\textrm{BdG}}\left(\bm{k}\right)},$
(3)

which is obtained through the analytic continuation of
$G\left(i\omega_{n},\bm{k}\right)$. In the basis of
$\left[c^{\dagger}_{\bm{k},o,\uparrow},c^{\dagger}_{\bm{k},o\downarrow},c_{-\bm{k},o,\uparrow},c_{-\bm{k},o,\downarrow}\right]^{\textrm{T}}$,
the specific form of the BdG Hamiltonian for the monolayer superconducting
T${}_{\textrm{d}}$-MoTe2 with an in-plane magnetic field reads

$\displaystyle h_{\textrm{BdG}}\left(\bm{k}\right)=$
$\displaystyle\begin{pmatrix}h\left(\bm{k}\right)-\mu
I_{12}+\frac{1}{2}g\mu_{\textrm{B}}\bm{B}\cdot\bm{\sigma}\otimes
I_{6}&-i\sigma_{y}\Delta\otimes I_{6}\\\ i\sigma_{y}\Delta\otimes
I_{6}&-h^{\ast}\left(-\bm{k}\right)+\mu
I_{12}-\frac{1}{2}g\mu_{\textrm{B}}\bm{B}\cdot\bm{\sigma}^{\ast}\otimes
I_{6}\end{pmatrix},$ (4)

with $I_{n}$ denoting the $n\times n$ identity matrix. Here the in-plane
magnetic field takes the form $\bm{B}=B\left(\cos\theta,\sin\theta,0\right)$.

By applying the periodic boundary condition solely in the $x$ direction, one
can derive the BdG Hamiltonian
$h^{\textrm{ribbon}}_{\textrm{BdG}}\left(k_{x}\right)$ of a monolayer
T${}_{\textrm{d}}$-MoTe2 ribbon along $x$ through the Fourier transformation.
By applying the surface Green’s function method surfacegf , one can obtain the
retarded Green’s function $G^{\textrm{R},\textrm{ribbon}}\left(E,k_{x}\right)$
at the edge of a monolayer T${}_{\textrm{d}}$-MoTe2 ribbon that is semi-
infinite in the $y$ direction. The electronic DOS at the edge of the
topological superconducting monolayer T${}_{\textrm{d}}$-MoTe2 ribbon can then
be calculated as

$\displaystyle
A\left(E,k_{x}\right)=-\frac{1}{\pi}\textrm{Im}G^{\textrm{R},\textrm{ribbon}}\left(E,k_{x}\right).$
(5)

Performing a summation over $k_{x}$ gives the electronic DOS at the edge:

$\displaystyle
N_{\textrm{edge}}\left(E\right)=\frac{1}{L}\sum_{k_{x}}A\left(E,k_{x}\right),$
(6)

with $L$ denoting the length of the ribbon. The result of
$A\left(E,k_{x}\right)$ and $N_{\textrm{edge}}\left(E\right)$ at
$B=1.1\Delta/\mu_{\textrm{B}}$ is plotted in Supplementary Fig. 4. It is clear
that the chiral Majorana edge state contributes to a plateau in the electronic
DOS at the energy within gap of the BdG quasiparticle spectrum.

Supplementary Fig. 4: The electronic DOS at the edge of the topological
superconducting monolayer T${}_{\textrm{d}}$-MoTe2 ribbon. The ribbon respects
the periodic boundary condition along $x$ and is semi-infinite along $y$. The
left panel is $A\left(E,k_{x}\right)$ and the right panel corresponds to
$N_{\textrm{edge}}\left(E\right)=\frac{1}{L}\sum_{k_{x}}A\left(E,k_{x}\right)$.
The in-plane magnetic field applied is set to be
$B=1.1\Delta/\mu_{\textrm{B}}$.

In the main text, we mainly focus on the in-plane magnetic field applied along
the $x$ direction. In Supplementary Fig. 5, we show how the BdG quasiparticle
energy spectrum evolve as the direction of the applied in-plane magnetic field
deviates from the $x$ direction. It is clear that the whole spectrum get
tilted in the $k_{x}$ direction. As long as the bulk gap does not close, the
topology of the superconducting state does not change. It indicates that our
proposal of the in-plane magnetic field driven topological superconducting
state in the monolayer T${}_{\textrm{d}}$-MoTe2 is not sensitive to the
direction of the applied in-plane magnetic field.

Supplementary Fig. 5: The BdG quasiparticle energy spectrum with a magnetic
field $B=1.1\Delta/\mu_{\textrm{B}}$ along different in-plane directions: (a)
$\theta=0^{\circ}$, (b) $\theta=20^{\circ}$, (c) $\theta=40^{\circ}$ and (d)
$\theta=60^{\circ}$. Here $\theta$ is the angle between the direction of the
in-plane magnetic field and the positive direction of the $x$-axis.

## References

  * (1) Zheng, Z. et al. Unconventional ferroelectricity in moiré heterostructures. _Nature_ 588, 71-76 (2020).
  * (2) Cao, Y. et al. Tunable correlated states and spin-polarized phases in twisted bilayer-bilayer graphene. _Nature_ 583, 215-220 (2020).
  * (3) Datta, S. _Quantum Transport_ (Cambridge Unniversity Press, Cambridge, New York, 2005).