# Electrostatic effects of the MnBi2Te4-superconductor hetero-structures in
chiral Majorana search

Li Chen  State Key Laboratory of Low Dimensional Quantum Physics, Department
of Physics, Tsinghua University, Beijing, 100084, China  Frontier Science
Center for Quantum Information, Beijing 100184, China  Zhan Cao  Beijing
Academy of Quantum Information Sciences, Beijing 100193, China  Ke He  State
Key Laboratory of Low Dimensional Quantum Physics, Department of Physics,
Tsinghua University, Beijing, 100084, China  Frontier Science Center for
Quantum Information, Beijing 100184, China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China  Xin Liu<EMAIL_ADDRESS>School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei
430074, China  Dong E. Liu<EMAIL_ADDRESS>State Key Laboratory
of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing, 100084, China  Frontier Science Center for Quantum
Information, Beijing 100184, China  Beijing Academy of Quantum Information
Sciences, Beijing 100193, China

###### Abstract

The realization of chiral Majorana modes is a challenging task. We aim to
comprehend the phase diagrams and parameter control capabilities of the actual
devices used in the chiral Majorana search. Beyond the well-known minimal
models, we develop a numerical simulation scheme using a self-consistent
Schrödinger-Poisson approach to study, as an example, the MnBi2Te4 thin film
coupled to an s-wave superconductor. We show that both the superconducting
proximity effect and the tunability of the chemical potential for the
topological surface states are significantly influenced by the gate-induced
electrostatic potential. This complicates the implementation in experiments,
and the actual topological region will be narrowed in stark contrast to those
predicted in the previous minimal models. Nevertheless, we demonstrate that
the chiral Majorana mode still exists in a wide range of experimental
parameters with practical tunability.

##  I Introduction

The chiral Majorana modes (CMMs) Read and Green (2000); Schnyder  _et al._
(2008); Qi  _et al._ (2009, 2010) can be considered as the one-dimensional
homologous counterpart of Majorana zero modes Read and Green (2000); Kitaev
(2001), and are potentially useful for quantum information processing Nayak
_et al._ (2008); Lian  _et al._ (2018); Beenakker  _et al._ (2019). Pioneering
theoretical proposals Qi  _et al._ (2010); Wang  _et al._ (2015); Wang (2016)
predict that CMMs can be realized in hybrid systems that combine quantum
anomalous Hall insulators (QAHI) (please refer to the theories Haldane (1988);
Onoda and Nagaosa (2003); Qi  _et al._ (2006, 2008); Liu  _et al._ (2008); Yu
_et al._ (2010) and experiments Chang  _et al._ (2013)) with superconductors.
The half-quantized conductance plateau was proposed to be an evidence for CMMs
Chung  _et al._ (2011); Wang  _et al._ (2015); He  _et al._ (2017). However, a
controversy arises because certain non-Majorana trivial mechanisms can also
generate similar signatures, especially in disordered samplesHuang  _et al._
(2018); Ji and Wen (2018); Kayyalha  _et al._ (2020). In contrast to
magnetically doped topological insulators, the recent discovered MnBi2Te4
(MBT) family of materials promises a bigger magnetic exchange gap and less
disorders Li  _et al._ (2019a); Zhang  _et al._ (2019); Gong  _et al._ (2019);
Yan  _et al._ (2019a); Otrokov  _et al._ (2019); Chen  _et al._ (2019a, b); Li
_et al._ (2019b); Lee  _et al._ (2019); Vidal  _et al._ (2019); Deng  _et al._
(2020); Shikin  _et al._ (2020a); Nevola  _et al._ (2020a); Liu  _et al._
(2020); Nevola  _et al._ (2020b); Liu  _et al._ (2021); Ge  _et al._ (2022),
which is proposed as an potential platforms to realize CMMs Peng and Xu
(2019); Zhang and Liu (2021).

Another serious problem is that the proposed systems require the coexistence
of superconductivity and magnetism, and we may wonder if the CMMs phase can be
realized via a feasible parameter control of the device. Previous theoretical
works Wang  _et al._ (2015); Wang (2016); Peng and Xu (2019); Yan  _et al._
(2019b); He  _et al._ (2019); Yan (2019); Sun  _et al._ (2020); Zhang and Liu
(2021); Högl  _et al._ (2020); Luo  _et al._ (2021); Zhang and Das Sarma
(2021) only considered the oversimplified minimal models, which regard
phenomenological parameters, such as chemical potential and induced
superconducting (SC) gap, as independently adjustable parameters. Actually,
these crucial parameters are highly correlated and cannot be freely tuned in
real experiments by controlling the electrostatic environment Vuik  _et al._
(2016); Antipov  _et al._ (2018); Mikkelsen  _et al._ (2018). This could
greatly narrow the topological region and complicates the experimental
implementations. Thus to understand the device control capability, we need to
develop a more reliable numerical simulation scheme for realistic experimental
setups, especially for treating both SC proximity effect and the magnetism.

Figure 1:  A MnBi2Te4 thin film is coupled to a s-wave SC on the top surface.
The magnetic gap of top surface state is always below the Fermi level during
the gate tuning. CMMs will exist if the Fermi level is tuned in the magnetic
gap of bottom surface states.

Figure 2: (a) The error of Schrödinger-Poisson equations as a function of the
number of iterations $n$. The distribution of (b) electrostatic energy
$-e\phi(z)$ and (c) carrier density $\rho(z)$ as the the number of iterations
increase. The convergence occurs when the iterations number $n>50$ with the
error $\sigma<10^{-7}$ eV, see the black solid and dash lines in panels (b)
and (c).

In this work, we consider the debate about CMM realizations and the actual
device tunability. Beyond the well-known minimal models, we developed a self-
consistent Schrödinger-Poisson scheme Tan  _et al._ (1990); Luscombe  _et al._
(1992); Ambrosetti (2008) to include a key focus—the electrostatic effects
induced by the actual gate control. We apply this method to study an MBT thin
film coupled to an s-wave superconductor (SC) as an example. We find that the
band bending effect Bardeen (1947); Heine (1965); Gotoh  _et al._ (2003);
Akhgar  _et al._ (2022) at the MBT-SC interface severely restricts the
tunability of top surface states, and the corresponding magnetic Dirac gap is
always below the Fermi level during the gate tuning. Our results also show
that the induced superconductivity varies considerably as tuning the gate
voltage. These constraints, which are not properly considered in previous
works, are naturally thought to be detrimental to the realization of CMMs.
However, we show that the CMMs can be realized in a reasonable range of
experimental parameters. The key point is to tune the Fermi level of the
bottom surface state into the magnetic Dirac gap, which is ensured by the high
tunability. Remarkably, the required proximity superconductivity will not be
reduced by the presence of the magnetization in the topological regime, and
ensured a large topological gap. In addition, the previously predicted CMMs
phase with Chern number $\mathcal{C}=2$ Qi  _et al._ (2010); Wang  _et al._
(2015) can not be realized in a real MBT-SC device.

The rest of the paper is organized as follows. In Sec. II, we construct a
model Hamiltonian and calculate the electrostatic potential using Schrödinger-
Poisson method. In Sec. III, we investigate the proximity effect in MBT-SC
hybrid system. In Sec. IV, we discuss device control capabilities in the
chiral Majorana search, and demonstrate that the key point for achieving CMMs
is to tune the chemical potential of BSSs in their magnetic gap. Finally, we
conclude in Sec. V.

##  II Model Hamiltonian and electrostatic potential

We consider a two dimensional (2D) MBT thin film coupled to an s-wave SC, as
shown in Fig. 1. The antiferromagnetic ordering and the magnetization
direction are both assumed to be along the $z$ direction. A back-gate voltage
$V_{g}$ is applied at the bottom surface to control the Fermi level. The
Hamiltonian of 2D MBT thin films reads Sun  _et al._ (2020)

$\displaystyle H_{\textrm{TI}}(\bm{k})$ $\displaystyle=$
$\displaystyle\epsilon_{0}(\bm{k})+\begin{bmatrix}M(\mathbf{k})&-iA_{1}\partial_{z}&0&A_{2}k_{-}\\\
-iA_{1}\partial_{z}&-M(\mathbf{k})&A_{2}k_{-}&0\\\
0&A_{2}k_{+}&M(\mathbf{k})&iA_{1}\partial_{z}\\\
A_{2}k_{+}&0&iA_{1}\partial_{z}&-M(\mathbf{k})\end{bmatrix}$ (1)
$\displaystyle-e\phi(z)+H_{X}(z),$

The translational invariance in the $x-y$ plane allows us to consider a fixed
in-plane wave vector $\bm{k}=(k_{x},k_{y})$ of magnitude $k=|\bm{k}|$. And
$k_{\pm}=k_{x}\pm ik_{y}$,
$\epsilon_{0}(\bm{k})=C_{0}-D_{1}\partial_{z}^{2}+D_{2}(k_{x}^{2}+k_{y}^{2})$
and $M(\bm{k})=M_{0}-B_{1}\partial_{z}^{2}+B_{2}(k_{x}^{2}+k_{y}^{2})$. In our
calculations, $C_{0}$, $D_{i}$, $M_{0}$, $B_{i}$ and $A_{i}$ with $i=1,2$, are
model parameters adopted from _ab initio_ calculations, see Appendix A.
$H_{X}$ is the spatial profile of the exchange field in the antiferromagnetic
MBT. For simplicity, we consider $H_{X}$ in terms of the sinusoidal function,
which takes the form Sun  _et al._ (2020)

$H_{X}(z)=-m_{0}\sin{\left(\frac{\pi}{d}z\right)}s_{z}\sigma_{0},$ (2)

where $m_{0}$ is the amplitude of the intralayer ferromagnetic order, $d$ is
the thickness of a septuple layer (SL), $s_{j}$ and $\sigma_{j}$ ($j=0,x,y,z$)
are Pauli matrix acting in spin and orbital space, respectively. $\phi(z)$ is
electrostatic potential, which is obtained by Schrödinger-Poisson (SP) method
Vuik  _et al._ (2016); Antipov  _et al._ (2018); Mikkelsen  _et al._ (2018).

Figure 3: (a)-(c) Energy bands of MBT with different gate voltage $V_{g}$. The
blue, red, and black curves correspond to BSSs, TSSs, and bulk states,
respectively. (d) Gate voltage dependence of the eigen-energies of TSSs and
BSSs, see $E_{b}$ and $E_{t}$ in panels (a)-(c). Note the three different
corresponding markers. (e) The distribution of electrostatic potential energy
$-e\phi(z)$. The right side is fixed by the band bending strength between MBT
and SC. (f) The distribution of carrier charge density $\rho(z)$.

Firstly, we set a initial potential $\phi_{0}(z)$ and take it into the
Hamiltonian $H_{\textrm{TI}}$. In our calculations, we choose $\phi_{0}(z)$ to
be a constant function $\phi_{0}(z)=V_{g}$. Then we solve the Schrödinger-
Poisson,

$H_{\textrm{TI}}(\textbf{k},\phi_{0}(z))\Psi_{n,\textbf{k}}(z)=E_{n,\textbf{k}}\Psi_{n,\textbf{k}}(z),$
(3)

producing a set of eigenenergies $E_{n,\textbf{k}}$, and a corresponding set
of eigenstates $\Psi_{n,\textbf{k}}(z)$. $n$ is the index of the transverse
eigen-functions. Since the electrostatic screening of MBT is only weakly
modified by the SC Mikkelsen  _et al._ (2018), we solve the Schrödinger
equation only in the MBT region. It means that we treat the SC only as a
boundary condition for $\phi(z)$ with a band offset $W$ at the interface
between the MBT and SC. The charge density with the potential profile
$\phi_{0}(z)$ is obtained by integrating over the occupied eigenstates and
minus the density stems from the whole valence band $\rho_{\textrm{val}}(z)$

$\rho_{1}(z)=\frac{-e}{2\pi}\int_{0}^{\infty}\bigl{[}\sum_{n}|\Psi_{n,\mathbf{k}}(z)|^{2}f_{T}(E_{n,\mathbf{k}})-\rho_{\textrm{val}}(z)\bigr{]}kdk,$
(4)

where $f_{T}(E_{n,\mathbf{k}})=1/(e^{E_{n,\mathbf{k}}/T}+1)$ is Fermi
distribution. Because $H_{\textrm{TI}}$ is a four band $k\cdot p$ Hamiltonian,
we choose $\rho_{\textrm{val}}(z)=2$. A new potential $\phi_{1}(z)$ is
obtained by solving the Poisson equation

$\frac{d^{2}\phi(z)}{dz^{2}}=-\frac{\rho(z)}{\epsilon_{r}\epsilon_{0}},$ (5)

where $\epsilon_{r}$ denotes the dielectric constant of the MBT. As discussed
previously, the boundary conditions of Eq. (LABEL:eq-sp-p) are $\phi(0)=V_{g}$
and $\phi(L_{z})=W$. Usually, $\phi_{1}(z)$ is not consistent with the initial
potential $\phi_{0}(z)$. The error is defined as

$\sigma_{1}=\frac{\sum_{m}\left[\phi_{1}(z_{m})-\phi_{0}(z_{m})\right]^{2}}{N_{m}},$
(6)

where subscript of $\sigma_{1}$ represents the number of iterations. $m$ is
the site index and $N_{m}$ is the number of sites.

The SP problem requires a self-consistent solution of two iterative equations
Eq. (3) and Eq. (5) until the error of $i$-th iteration $\sigma_{i}$ is
smaller than the critical value $\sigma_{c}$. And the output $\phi_{i}(z)$ is
the final self-consistent potential. The most straightforward iterative method
is to replace the potential in Eq. (3) directly with the newly obtained
potential in Eq. (5). However, this usually leads to divergence of the
iterations, and requires the suitable choice of initial potential
$\phi_{0}(z)$. Thus, we employ a mixing scheme Mikkelsen  _et al._ (2018),
where the input potential used in each iteration is a mixing of the input and
output potential of the previous iteration:

$\phi_{i}^{\textrm{in}}(z)=\kappa\phi_{i-1}^{\textrm{out}}(z)+(1-\kappa)\phi_{i-1}^{\textrm{in}}(z).$
(7)

In our calculations we set $\kappa=0.1$ and $\sigma_{c}=10^{-8}$ eV. In Fig. 2
we show an iterative procedure when we calculate the potential with
$V_{g}=0.2$ eV. As shown in Fig 2(a), the iteration error decreases sharply as
the number of iterations increase. The convergence of the potential [Fig 2(b)]
and charge density [Fig 2(c)] occurs when the iterations number $n>50$ with
the error $\sigma<10^{-7}$ eV, see the black solid and dash lines. In Fig. 2
we show an iterative procedure when we calculate the potential with
$V_{g}=0.2$ eV. As shown in Fig 2(a), the iteration error decreases sharply as
the number of iterations increase. The convergence of the potential [Fig 2(b)]
and charge density [Fig 2(c)] occurs when the iterations number $n>50$ with
the error $\sigma<10^{-7}$ eV, see the black solid and dash lines.

Figure 4:  (a)-(d) The energy bands of MTB-SC system with four typical gate
voltages $V_{g}$ as marked in panel (e). (e) $\Delta_{\textrm{bss}}$,
$\Delta_{\textrm{tss}}$ and $\Delta_{\textrm{cond}}$ represents the induced
gap of BSSs, TSSs and conduction bands ($N=4$). $\Delta_{\textrm{ind}}$ is the
minimum of the gaps of all occupied states. The Chern number $\mathcal{C}=1$
in the gray region, otherwise $\mathcal{C}=0$. (f) The polar representation of
the SC gap $\Delta_{\textrm{ind}}$ along the whole Fermi surface of MBT. The
radial length represents the amplitude of $\Delta_{\textrm{ind}}$. (g)
$\Delta_{\textrm{ind}}$ as a function of the thickness of superconductor
$L_{s}$. The size of the induced gap oscillates with a period set by the Fermi
wavelength of the superconductor $\lambda_{s}$ (for our parameters
$\lambda_{s}=1.94$ nm), which is consistent with results in Ref. Legg  _et
al._ (2022a). (h) $\Delta_{\textrm{ind}}$ as a function of coupling strength
$t_{c}$. $t_{s}=\hbar^{2}/2m_{s}a_{s}^{2}$ is the hopping magnitude in the
superconductor. In all the panels, the blue, black and red curves correspond
to the cases with gate voltage $V_{g}=-0.1$ eV, $V_{g}=-0.028$ eV and
$V_{g}=0.1$ eV respectively.

The chemical potential of MBT can be obtained from $\phi(z)$ solution with
different gate voltage $V_{g}$, as shown in Figs. 3(a)-(c). Here we choose the
septuple layer number of MBT $N=4$ (with the full thickness $L_{0}=Nd$). The
inhomogeneous electrostatic potential breaks inversion symmetry, which lifts
the degeneracy of the surface states. Obviously, the bottom surface states
(BSSs) and the top surface states (TSSs) have totally different electrostatic
environments because they couple to back-gate and SC, respectively. Therefore,
the two surface states have different tunability with the change of gate
voltage $V_{g}$. The Fermi level of TSSs can be well controlled by the
different gate voltage, see Figs. 3(a)-(c). However, the magnetic Dirac gap of
TSSs is always below the Fermi level during the gate control. In Fig. 3(d), we
calculate the eigen-energies of TSSs and BSSs at $k_{x}=0$ (see $E_{t}$ and
$E_{b}$ labeled in Figs. 3 (a)-(c)) as a function of gate voltage $V_{g}$.
Note that the Fermi level of TSSs (red line) is nearly unaffected by the
change of $V_{g}$. The different tunability between BSSs and TSSs stems from
the non-uniform distribution of the electrostatic potential in MBT. As shown
in Fig. 3(e), the potential energy $-e\phi(z)$ at the SC-MBT boundary (right
side) is fixed at $W=-0.3$ eV, which is the band bending strength between MBT
and SC (see Appendix B for details). Nevertheless, the potential energy close
to the MBT-substrate boundary (left side) varies with the gate voltage.
Because TSSs distribute locally near the interface between MBT and SC, the
tunability is greatly limited by the band bending effect. This constraint
about the chemical potential tunability also manifest in the charge density
distribution [Fig. 3(f)]. The type of carrier near the left side is electron
(hole) when $V_{g}$ is positive (negative). While the carrier near the right
side is nearly unchanged with $V_{g}$. Actually, these results, which cannot
be obtained in previous minimal models Wang (2016); Peng and Xu (2019); Yan
_et al._ (2019b); He  _et al._ (2019); Yan (2019); Sun  _et al._ (2020); Zhang
and Liu (2021); Högl  _et al._ (2020); Luo  _et al._ (2021); Zhang and Das
Sarma (2021), could highly narrow the regions of parameter for achieving CMMs.

##  III Superconducting proximity effect

When the superconducting shell is considered, the Bogoliubov-de Gennes (BdG)
Hamiltonian of MBT-SC hybrid system takes the form

$\displaystyle H_{\textrm{BdG}}=\begin{pmatrix}H_{\rm
TI}+H_{s}+H_{t}&is_{y}\Delta(z)\\\ -is_{y}\Delta(z)&-(H_{\rm
TI}+H_{s}+H_{t})^{*}\end{pmatrix}$ (8)

We include an $s$-wave pairing potential only in the SC part, i.e.,
$\Delta(z)=\Delta_{0}$ for $z>L_{0}$, and $\Delta(z)=0$ for $z<L_{0}$. The
normal state of the SC has the form
$H_{s}=\frac{\hbar^{2}\bm{k}^{2}}{2m_{e}}-\mu_{s}$, where $\mu_{s}$ is the
chemical potential and the effective mass $m_{s}$ is taken to be infinite in
the direction parallel to the interface Vaitiekėnas  _et al._ (2020). It is
noted that the calculated self-consistent electrostatic potential $\phi(z)$ is
included in $H_{\rm TI}$. The coupling between the MBT and SC at the interface
takes the from Legg  _et al._ (2022b)

$H_{t}=\sum_{\langle
z,z^{{}^{\prime}}\rangle}\bigl{[}-t_{c}c_{z,k}^{\dagger}d_{z^{{}^{\prime}},k}+\textrm{H.c.}\bigr{]},$
(9)

where $\langle z,z^{{}^{\prime}}\rangle$ denotes the hopping between the
nearest sites. $t_{c}$ is the coupling strength. The operator
$c_{z,k}(d_{z,k})$ annihilates a state of momentum $k$ at site $z$ within the
MBT (SC).

Figure 5: (a) Topological phase diagram as a function of gate voltage $V_{g}$
and the layers of MBT $N$. Topological regions which stem from surface states
and bulk states are enclosed by the black and white dash lines, respectively.
(b) Thickness dependence of the induced gap. For each layer number $N$, we
calculate the largest SC gap in the topological region enclosed by black dash
lines in panel (a). When $N$ increases up to six SLs, the SC gap decrease
drastically because the Fermi level crosses the conduction bands. Inset: The
electron band of MBT when $N=4$ (left) and $N=8$ (right). (c) The Fermi level
of TSSs can not be tuned in the magnetic Dirac gap. This narrows the
topological regions in contrast to those predicted in previous works, as
indicated by the slash.

The induced SC gap of MBT is highly dependent on the types of the bands
crossing the Fermi level [Fig. 4(e)]. We use $\Delta_{\textrm{bss}}$,
$\Delta_{\textrm{tss}}$ and $\Delta_{\textrm{cond}}$ to represents the induced
gap of BSSs, TSSs and conduction bands. The induced gap of MBT is defined as
the minimum of all the gaps
$\Delta_{\textrm{ind}}=\textrm{min}\\{\Delta_{\textrm{bss}},\Delta_{\textrm{tss}},\Delta_{\textrm{cond}}\\}$.
In Figs. 4(a)-(d), we calculate the SC bands with four typical $V_{g}$, as
marked in Fig. 4(e). When MBT is bulk insulating and Fermi level only crosses
both BSSs and TSSs [Fig. 4(a)], TSSs opens a finite SC gap with
$\Delta_{\textrm{tss}}/\Delta_{0}\approx 0.462$. Arguably, superconductivity
at the BSSs may be strongly suppressed (see a recent experiments Hlevyack  _et
al._ (2020)) because of the very short penetration depth, about 1.62 nm
Hlevyack  _et al._ (2020); Sun  _et al._ (2020); Shikin  _et al._ (2020a). In
our calculations, the thickness of MBT is 5.48 nm (4 SLs). Thus, the
suppression of superconductivity on BSSs limits the SC proximity gap, except
for the region where the Fermi level is tuned in the magnetic gap of BSSs
[Fig. 4(b)]. In this case, MBT has largest induced gap because Fermi level
only cross TSSs [Fig. 3(b)], and $\Delta_{\textrm{ind}}$ is dominated by
$\Delta_{\textrm{tss}}$. When the Fermi level moves toward the bottom of
conduction bands, $\Delta_{\textrm{bss}}$ gradually increases because of the
increase of penetration depth [Fig. 4(c)]. When Fermi level crosses the
conduction bands [Fig. 4(d)], $\Delta_{\textrm{bss}}$ is still suppressed.
This is because the positive gate-induced electrostatic potential will change
the electrons’ confinement, and pull electron density away from the interface
between MBT and SC Antipov  _et al._ (2018). This in turn strongly suppresses
the SC proximity effect for states in MBT [Fig. 4(e)].

In our calculations, $\Delta_{\textrm{ind}}$ is obtained from the
superconducting band with $k_{y}=0$ because it is isotropic. In Fig. 4(f), we
plot the polar representation of $\Delta_{\textrm{ind}}$ along the whole Fermi
surface. The blue, black, and red curves correspond to the cases with
different gate voltage $V_{g}$. And the radial length represents the amplitude
of $\Delta_{\textrm{ind}}$ along the Fermi surface. Clearly,
$\Delta_{\textrm{ind}}$ is isotropic because of the circle shape. On the hand,
the SC gap also depends on the thickness of the superconductor $L_{s}$. In
Fig. 4(g), we calculate $\Delta_{\textrm{ind}}$ as a function of $L_{s}$. We
find that the size of the induced gap oscillates with a period set by the
Fermi wavelength of the superconductor $\lambda_{s}$ (for our parameters
$\lambda_{s}=1.94$ nm), which is consistent with results in Ref. Legg  _et
al._ (2022a). Another significant parameter affecting the proximity effect is
the coupling strength between the MBT and SC $t_{c}$ [Fig. 4(h)]. Certainly,
the SC gap gradually increase with the increases of $t_{c}$. Importantly, our
results change qualitatively when varying the amplitude of $t_{c}$ and
$L_{s}$. Thus, the induced SC gap of MBT is highly dependent on the gate
voltage. These results about the gate tunability cannot be captured by
previous minimal models.

##  IV chiral Majorana mode

As discussed above, we mainly have two constraints that limit the realization
of CMMs. Firstly, the tunability of TSSs is greatly limited by the band
bending effect. Secondly, BSSs exhibit a giant attenuation of surface
superconductivity. Nevertheless, we demonstrate that CMMs can still be
achieved in a reasonable range of experimental parameters. The key requirement
for realizing CMMs is to achieve superconductivity and magnetization on TSSs
and BSSs, respectively. As shown in Figs. 3(a)-(c), the magnetic Dirac gap of
TSSs is always much below the Fermi level during the gate tuning. This fact
protects the induced superconductivity of TSSs from the destruction of the
magnetization Chen  _et al._ (2021). Thus, the key point for achieving CMMs is
to tune the Fermi level into the magnetic gap of BSSs, which is enabled by the
high tunability [Figs. 3(d)]. We further calculate the Chern number
$\mathcal{C}$ (see Appendix C) as a function of gate voltage. We have
$\mathcal{C}=1$ in the gray region of Fig. 5(e) where the Fermi level locates
in the magnetic gap of BSSs, otherwise $\mathcal{C}=0$.

Fig. 5(a) shows the topological phase diagram as a function of the two
experimentally relevant and tunable quantities – gate voltage $V_{g}$ and the
layer number of MBT $N$, rather than more phenomenological parameters. Note
that most of the topological regions, which are enclosed by the black dash
lines, are concentrated in the range of $V_{g}\in(-0.06~{}0.03)$ eV. When the
thickness of MBT increases up to six SLs, the superconducting gap in the
topological regions decreases drastically because the Fermi level also crosses
the conduction bands [Fig. 5(b)]. Another remarkable result is that we also
have additional topological regions stemming from the bulk states of MBT
(Appendix D), which are enclosed by the white dash lines in Fig. 5(a). The
formation of these topological regions originates from two major effects on
bulk states: induced finite spin-orbital coupling due to the applied electric
field and the magnetization effects. Because of the antiferromagnetic
structure of MBT, these usually occur when the layer number is odd or the gate
voltage is negative. Nevertheless, the induced superconducting gaps in these
topological regions are very small, which is not favorable for achieving CMMs.
Notably, the obtained topological regions in Fig 5(a) are greatly narrowed
compared with those predicted in previous works (Appendix E). This is because
the Fermi level of TSSs can not be tuned into the magnetic Dirac gap, i.e.,
CMMs with Chern number $\mathcal{C}=2$ can not be realized, as illustrated in
Fig 5(c). To further confirm the system is exactly in the topological phase
under such conditions, we consider the MBT-SC system with open boundary
condition in the $y$ direction. In Fig. 6(a), we calculate the eigenenergy
with $k_{x}=0$ as a function of $V_{g}$. The gap closes when $V_{g}$
approaches -0.048 eV. Then a pair of zero modes emerges in the gap, which is
the crossing point of the two CMBSs at $k_{x}=0$ [Fig. 6(b)]. The distribution
of local density of CMMs at $k_{x}=0$ in the $y$-$z$ cross section (the top SC
part is not shown) is given in Fig. 6(c). As expected, CMMs mainly distribute
in the two edges of the MBT-SC slab and gradually decay into the bulk.

Figure 6:  (a) The eigen-energies at $k_{x}=0$ as a function of gate voltage
$V_{g}$. The red curves correspond to the CMMs. (b) The spectrum shows that
CMMs (red curves) appear in the SC gap. (c) The distribution of local density
of states of CMMs at $k_{x}=0$ in the $y-z$ cross section of MBT. The SC part
of MBT-SC is not shown. We choose $N=4$, $V_{g}=-0.028$ eV and
$L_{y}=1.5~{}\upmu$m.

##  V Conclusion

We consider a two dimensional MBT thin film in proximity to an s-wave SC.
Beyond the well-known minimal models, we includes a key focus—the
electrostatic effects induced by the actual gate control and solve the
problems self-consistently in a Schrödinger-Poisson scheme. We find that the
band bending effect at MBT-SC interface severely restricts the tunability of
top surface states, and the corresponding magnetic Dirac gap is always below
the Fermi level during the gate tuning. Moreover, we find that the induced SC
gap of MBT is highly dependent on the types of the bands crossing the Fermi
level. Arguably, superconductivity at the BSSs may be strongly suppressed
especially when bulk is insulating. These results, which cannot be obtained in
previous minimal models, could highly narrow the regions of parameter for
achieving CMMs. Nevertheless, we demonstrate that the CMMs can still be
realized via the control of the gate voltage. The key point is to tune the
Fermi level of the bottom surface state into the magnetic Dirac gap. Our
method provides a more accurate prediction about the topological phase and
device control capability. This is in stark contrast to those previous
oversimplified minimal models.

###### Acknowledgements.

Acknowledgments -  Authors thank Yayu Wang, Yang Feng, and Gu Zhang for
helpful discussions. This work was supported by the National Natural Science
Foundation of China (Grants No. 11974198, No. 12004040,No. 12074133), the
Innovation Program for Quantum Science and Technology (Grant No.
2021ZD0302400, No. 2021ZD0302700), and Tsinghua University Initiative
Scientific Research Program.

##  Appendix A Parameters used in this work

The parameters of the $k\cdot p$ Hamiltonian of MBT in Eq. (1) are adopted
from _ab initio_ calculations Zhang  _et al._ (2019):
$C_{0}=-0.05~{}\textrm{eV}$, $M_{0}=-0.117~{}\textrm{eV}$,
$D_{1}=2.72~{}\textrm{eV \AA${}^{2}$}$, $D_{2}=1.2~{}\textrm{eV \AA${}^{2}$}$,
$B_{1}=11.9~{}\textrm{eV \AA${}^{2}$}$, $B_{2}=9.40~{}\textrm{eV
\AA${}^{2}$}$, $A_{1}=2.7~{}\textrm{eV \AA}$, $A_{2}=3.2~{}\textrm{eV \AA}$.
The other parameters used in this work are given in Table 1. $a_{x,y,z}$ is
the lattice constant of MBT in the tight-binding calculations. In our
calculations, the choice of the superconducting (SC) materiel is NbSe2, which
has been widely used in experiments Xu  _et al._ (2014); Wang  _et al._
(2012); Xu  _et al._ (2015); Sun  _et al._ (2016). $a_{s}$ is the lattice
constant of SC. Because the dielectric constant of MBT has not been studied
experimentally, we set it equal to the value of Bi2Se3. The details about band
bending are in Appendix B.

Table 1: Parameters used for the calculations in this work.  $m_{0}$ |  $d$ |  $\Delta_{0}$ |  $\mu_{s}$ |  $L_{s}$
---|---|---|---|---
0.1 eV Shikin  _et al._ (2020b) |  1.37 nm Gong  _et al._ (2019) |  1.5 meV Clayman (1972) |  0.4 eV Yokoya  _et al._ (2001) |  10 nm
$\epsilon_{r}$ |  $a_{s}$ |  $a_{x(y)}$ |  $a_{z}$ |  $m_{s}$
25 Stordeur  _et al._ (1992) |  0.4 nm |  1 nm |  0.7 nm |  $m_{e}$
$W$ |  $t_{c}$ |  |  |
0.3 eV Xu  _et al._ (2014) |  0.05 eV |  |  |

##  Appendix B Effect of band bending strength

Figure 7: (a) The Chern number $\mathcal{C}$ (blue curves) and induced SC gap
$\Delta_{\textrm{ind}}$ (blue curves) as a function of gate voltage $V_{g}$
when the band bending strength $W=0.1$ eV. (b)-(d) The energy band structure
of MBT. The blue, red, and black curves correspond to BSSs, TSSs, and bulk
states, respectively. We choose the gate voltage of panels (b)-(d) as marked
in panel (a). In panels (b) and (d), the Fermi level is tuned in the magnetic
gap of the BSSs and TSSs, respectively. In panel (c), the Fermi level is tuned
in the trivial gap stemming from the coupling between BSSs and TSSs. (d)-(f)
The cases when $W=0.45$ eV. In panels (f) and (h), the Fermi level is tuned in
the magnetic gap of the BSSs and first lowest bulk band, respectively. Due to
the large band bending effect, the Fermi level can not be tuned in the
magnetic gap of TSSs. The layers number of MBT $N$ is fixed to 3 in all the
panels.

Figure 8: (a) The Chern number $\mathcal{C}$ as a function of gate voltage
$V_{g}$ when the layer number of MBT $N=3$. There exist two topological
regions with $\mathcal{C}=1$, which stem from surface states and bulk states
respectively. (b)-(c) The band structure of MBT. The blue, red, and black
curves correspond to BSSs, TSSs, and bulk states, respectively. We choose the
gate voltage of panels (b) and (c) as marked in panel (a). In panels (b) and
(c), the Fermi level is tuned in the magnetic gap of the BSSs and first lowest
bulk band, respectively. (d)-(f) The cases when $N=4$. Due to the
antiferromagnetic properties of MBT, the magnetization of bulk states is small
when $N$ is even. So the corresponding topological region is small. In
addition, the signs of the $C$ stem from surface states and bulk states are
different. This is because the gate-induced electrostatic potential will
confine the bulk states closer to the top surface of MBT. Thus, the BSSs and
bulk states have opposite magnetization.

The bend bending strength $W$ exists at the interface between MBT and SC
because of their work function imbalance Kiejna and Wojciechowski (1996). In
this section, we discuss the effect of band bending strength on our result. We
consider two limiting cases: a small band bending strength with $W=0.1$ eV and
a very large value, $W=0.45$ eV.

In Fig. 7(a), we calculate the Chern number $\mathcal{C}$ (blue curves) and
induced SC gap $\Delta_{\textrm{ind}}$ (blue curves) as a function of gate
voltage $V_{g}$ when the band bending strength $W=0.1$ eV. In Figs. 7(b)-(d),
we plot the energy bands of MBT with three typical different gate voltages as
marked in Fig. 7(a). Because of the small band bending strength, the Fermi
level can be tuned in the magnetic gap of TSSs [Fig. 7(b)]. Additional
topological regions arise in this case, but the corresponding SC gap is very
small because the superconductivity of BSSs is suppressed [Fig. 7(a)]. When
$W$ is very large [Figs. 7(e)-(h)], the Fermi level can not be tuned in the
magnetic gap of TSSs, which is consistent with the result in the main text.

The exact value of $W$ is unknown, depending on the choice of superconducting
materials in the experiment. In addition, $W$ also depends on the plane of SC
crystal Lang and Kohn (1971); Eastment and Mee (1973), as well as the
thickness of SC Qi  _et al._ (2007). The work functions of MBT, Bi2Te3 and
several SCs are given in Table 2. The band offset between Bi2Te3 and NbSe2 is
about 0.15-0.2 eV according to the experiments in Ref. Xu  _et al._ (2014).
Obviously, the band banding strength in the MBT-NbSe2 system is larger because
of the smaller work function of MBT. In the main text, we set $W=0.3$ eV.
Nevertheless, our results do not change qualitatively when varying the
amplitude of $W$, see Fig 7.

Table 2: The work functions of MBT, Bi2Te3 and several SCs.  MnBi2Te4 |  Bi2Te3 |  NbSe2 |  NbN |  Al |  Pb
---|---|---|---|---|---
4.0 eV Akhgar  _et al._ (2022) |  5.3 eV Shih  _et al._ (2010) |  5.9 eV Shimada  _et al._ (1994) |  4.7 eV Gotoh  _et al._ (2003) |  4.26 eV Eastment and Mee (1973) |  4.25 eV Lang and Kohn (1971)

##  Appendix C Numerical calculation of Chern number

For the calculation of Chern number $\mathcal{C}$, we don’t utilize the
Hamiltonian $H_{\textrm{BdG}}$ (Eq. (8) in the main text). This is because the
bands which stem from the superconductor are trivial. And the dimension of the
superconductor Hamiltonian is very large, which increased computational
effort. Thus, we treat the superconductor as the self-energy
$\Delta_{\textrm{ind}}$ Stanescu  _et al._ (2010) and consider the Hamiltonian

$\displaystyle H_{\textrm{BdG}}=\begin{pmatrix}H_{\rm
TI}&is_{y}\Delta_{\textrm{ind}}\\\ -is_{y}\Delta_{\textrm{ind}}&-H_{\rm
TI}^{*}\end{pmatrix}.$ (C1)

As discussed in the main text, $\Delta_{\textrm{ind}}$ is highly dependent on
the gate voltage. And top surface states (TSSs), bottom surface states (BSSs),
and bulk states have totally different induced SC gaps. In our calculations of
Chern number, we set $\Delta_{\textrm{ind}}=\Delta_{tss}$, where
$\Delta_{tss}$ is the SC gap of TSSs. This is because the topological phase
transition is mainly related to $\Delta_{tss}$ Qi  _et al._ (2010). The Chern
number $\mathcal{C}$ of Hamiltonian Eq. (C1) is Thouless  _et al._ (1982)

$\displaystyle\mathcal{C}=\frac{1}{2\pi}\int d^{2}kF_{12}(k),$ (C2)

where the Berry connection $A_{\mu}(k)~{}(\mu=1,2)$ and the associated field
strength $F_{12}(k)$ are given by

$\displaystyle A_{\mu}(k)$ $\displaystyle=$
$\displaystyle-i\sum_{E_{n}<0}\bra{\psi_{n}(k)}\partial_{\mu}\ket{\psi_{n}(k)},$
(C3) $\displaystyle F_{12}(k)$ $\displaystyle=$
$\displaystyle\partial_{1}A_{2}(k)-\partial_{2}A_{1}(k),$

where $\psi_{n}(k)$ and $E_{n}$ is the $n$th eigen-function and eigen-values
of Eq. (C1). We calculate the Chern number $\mathcal{C}$ numerically according
to the method proposed in Ref. Fukui  _et al._ (2005).

We consider lattice points $k_{l}~{}(l=1,...,N_{1}N_{2})$ on the two
dimensional discrete Brillouin zone as

$k_{l}=(k_{j_{1}},k_{j_{2}}),~{}~{}k_{j_{\mu}}=\frac{2\pi
j_{\mu}}{N_{\mu}},~{}~{}(j_{\mu}=0,...,N_{\mu}-1),$ (C4)

The occupied multiplet of Hamiltonian $H_{\textrm{BdG}}$ is
$\Psi=(\ket{\psi_{1}},...,\ket{\psi_{M}})$. The number of the occupied states
$M$ is half of the dimension of $H_{\textrm{BdG}}$ because of the particle-
hole symmetry. The $U(1)$ link variable is defined as

$U_{\mu}(k_{l})=\frac{1}{\mathcal{N}_{\mu}(k_{l})}\textrm{det}[\Psi^{\dagger}(k_{l})\Psi(k_{l}+\hat{\mu})],$
(C5)

where $\hat{\mu}$ is a vector in the direction $\mu$ with the magnitude
$\frac{2\pi}{N_{\mu}}$,
$\mathcal{N}_{\mu}(k_{l})=|\textrm{det}[\Psi^{\dagger}(k_{l})\Psi(k_{l}+\hat{\mu})]|$
is the normalization constant. Then the lattice field strength is

$F_{12}(k_{l})=\textrm{ln}\left[U_{1}(k_{l})U_{2}(k_{l}+\hat{1})U_{1}(k_{l}+\hat{2})^{-1}U_{2}(k_{l})^{-1}\right].$
(C6)

And the Chern number $\mathcal{C}$ is the summation of the lattice field
$F_{12}$

$\mathcal{C}=\frac{1}{2\pi i}\sum_{l}F_{12}(k_{l}).$ (C7)

##  Appendix D Topological regions stem from bulk states

The applied electric field will induce finite spin-orbital coupling on bulk
states. And topological regions will exist as long as the magnetization of
bulk states is enough large, i.e., satisfying the topological phase transition
condition
$M_{n,\textrm{bulk}}^{2}>\Delta_{n,\textrm{bulk}}^{2}+\mu_{n,\textrm{bulk}}^{2}$.
Here $M_{n,\textrm{bulk}}$ $\mu_{n,\textrm{bulk}}$, and
$\Delta_{n,\textrm{bulk}}$ is the magnetization, chemical potential and
induced SC gap of the $n$th bulk bands, respectively.

In Fig. 8(a), we calculate the Chern number $\mathcal{C}$ as a function of
gate voltage $V_{g}$ when the layer number of MBT $N=3$. Note that there exist
two topological regions with $\mathcal{C}=1$, which stem from surface states
and bulk states respectively. To see it more clearly, we choose the two gate
voltage as marked in Fig. 8(a), and calculate the corresponding band structure
of MBT [Fig. 8(b)(c)]. The blue, red, and black curves correspond to BSSs,
TSSs, and bulk states respectively. In Fig. 8(b) and (c), the Fermi level is
tuned in the magnetic gap of BSSs and the first lowest bulk band,
respectively. This indicates that the nonzero Chern number in these two cases
stem from surface states and bulk states, respectively. Due to the
antiferromagnetic properties of MBT, the magnitude of $M_{n,\textrm{bulk}}$
highly depends on the parity of the layer number of MBT $N$.
$M_{n,\textrm{bulk}}$ is usually very small when $N$ is even [Fig. 8(f)]. This
makes the corresponding topological region also very small, about 0.87 meV
[Fig. 8(d)]. We don’t plot this topological region in Fig. 3(e)(f) of the main
text. It is noted that the signs of the $C$ stem from surface states and bulk
states are different when $N$ is even. This is because the gate-induced
electrostatic potential will confine the bulk states closer to the top surface
of MBT. Thus, the BSSs and bulk states have opposite magnetization (The sign
of the Chern number is determined by the direction of magnetization). We also
find that the magnetization of bulk states changes with the gate voltage
because of the non-uniform distribution of the electrostatic potential in MBT.
When the gate voltage is very negative, bulk states can also have large
magnetization even for even $N$. As shown in the Fig. 5(a), the induced
superconducting gaps in topological regions stemming from bulk states are very
small, which is not favorable for achieving CMMs.

##  Appendix E The electrostatic potential narrows the topological regions

For simplicity, we consider the 2D effective Hamiltonian of MBT which consists
of the Dirac-type surface states only. It takes the form
$\mathcal{H}(\bm{k})=\sum_{\bm{k}}\psi_{\bm{k}}H_{sf}(\bm{k})\psi_{\bm{k}}^{\dagger}$
with

$\displaystyle H_{sf}(\bm{k})$ $\displaystyle=$ $\displaystyle
v_{F}k_{y}\sigma_{z}s_{x}-v_{F}k_{x}\sigma_{z}s_{y}+m_{k}\sigma_{0}s_{z}$
$\displaystyle+$ $\displaystyle M\sigma_{z}s_{z}+V\sigma_{z}s_{0},$

where the field operator
$\psi_{\bm{k}}=(c_{t\uparrow},c_{t\downarrow},c_{b\uparrow},c_{b\downarrow})^{T}$,
$t$ and $b$ denote the TSSs and BSSs. $\uparrow$ and $\downarrow$ represent
spin-up and spin-down, respectively. $\bm{k}=(k_{x},k_{y})$. $v_{F}$ is the
Fermi velocity of surface states. $\sigma_{i}$ and $s_{i}~{}(i=x,y,z)$ are the
Pauli matrix acting on layer and spin space, respectively.
$m_{k}=m_{0}+m_{1}(k_{x}^{2}+k_{y}^{2})$ describes the tunneling effect
between TSSs and BSSs and set $m_{1}>0$. $M$ is the magnetization of surface
states. Here layer number of MBT is even (For even layers, the exchange field
term changes as $M\sigma_{0}s_{z}$). $V$ is structure inversion asymmetry
imposed by the gated induced electrostatic states Wang (2016).

The Bogoliubov-de Gennes (BdG) Hamiltonian for the s-wave superconductor
proximity coupled MBT is
$\mathcal{H}_{\textrm{BdG}}(\bm{k})=\sum_{\bm{k}}\Psi_{\bm{k}}H_{\textrm{BdG}}(\bm{k})\Psi_{\bm{k}}^{\dagger}/2$,
with $\Psi_{\bm{k}}=(\psi_{\bm{k}},\psi_{\bm{-k}}^{\dagger})^{T}$ and

$H_{\textrm{BdG}}(\bm{k})=\begin{pmatrix}H_{sf}(\bm{k})-\mu&\Delta(\bm{k})\\\
\Delta^{\dagger}(\bm{k})&-H_{sf}(-\bm{k})^{*}+\mu\end{pmatrix},$ (E2)

where $\mu$ is chemical potential, $\Delta(\bm{k})$ is the pairing function
given by

$\Delta(\bm{k})=\begin{pmatrix}i\Delta_{t}s_{y}&0\\\
0&i\Delta_{b}s_{y}\end{pmatrix},$ (E3)

where $\Delta_{t(b)}$ is the SC gap of TSSs (BSSs).

We consider a limiting cases with $m_{k}=0$. The Hamiltonian
$H_{\textrm{BdG}}(\bm{k})$ is decoupled into two parts which contains BSSs and
TSSs, respectively. The Chern number of these two surface states
$\mathcal{C}_{t(b)}$ is determined by

$\mathcal{C}_{t(b)}=\begin{cases}\textrm{Sign}(M)&M^{2}>\Delta_{t(b)}^{2}+(\mu\mp
V)^{2}\\\ 0&\textrm{Otherwise}\end{cases},$ (E4)

The sign of $\mathcal{C}_{t(b)}$ is opposite (the same) in even (odd) layers
of MBT. In experiments, we usually have $V\gg M,~{}\Delta_{t(b)}$ Zhang  _et
al._ (2010). This makes the topological region stemming TSSs and BSSs well
separated. As discussed in the main text, the Fermi level of TSSs can not be
well tuned because of the band bending effect. This makes the magnetic gap of
TSSs always below the Fermi level during the gate tuning. Thus, we always have
$\mathcal{C}_{t}=0$, which highly narrows the topological regions, as show in
Fig. 5(c).

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