#  Quantum Anomalous Hall Effect in Three-dimensional Topological
Insulator/Thin-film Ferromagnetic Metal Bilayer Structure

Katsuhiro Arimoto<EMAIL_ADDRESS>Department of Physics,
Tohoku University, Sendai, Miyagi, 980-8578, Japan  Institute for Materials
Research, Tohoku University, Sendai, Miyagi, 980-8577, Japan  Takashi
Koretsune  Department of Physics, Tohoku University, Sendai, Miyagi, 980-8578,
Japan  Kentaro Nomura<EMAIL_ADDRESS>Institute for Materials
Research, Tohoku University, Sendai, Miyagi, 980-8577, Japan  Center for
Spintronics Research Network, Tohoku University, Sendai, Miyagi, 980-8577,
Japan

###### Abstract

We theoretically show that the three-dimensional (3D) topological insulator
(TI)/thin-film ferromagnetic metal (FMM) bilayer structure is possible to be a
quantum anomalous Hall (QAH) insulator with a wide global band gap. Studying
the band structure and the weight distributions of eigenstates, we demonstrate
that the attachment of a metallic thin-film on the 3DTI can shift the
topologically non-trivial state into the metal layers due to the hybridization
of bands around the original Dirac point. By introducing the magnetic exchange
interaction in the thin-film layers, we compute the anomalous Hall
conductivity and magnetic anisotropy of the bilayer structure to suggest the
appearance of wider gap realizing QAH effect than usual materials, such as
magnetically doped thin-films of 3DTI and 3DTI/ferromagnetic insulator
heterostructures. Our results indicate that the 3DTI/thin-film FMM bilayer
structure may implement the QAH effect even at room temperature, which will
pave a way to the experimental realization of other exotic topological quantum
phenomena.

##  I Introduction

Three-dimensional (3D) topological insulators (TIs) [1, 2] are materials which
exhibit novel conductive state only on their surfaces. The surface state
corresponds to linear-dispersing energy band with helical spin texture, called
as Dirac surface state (DSS). The peculiar energy band connects topologically
non-trivial bulk valence band and conduction band, which occurs due to a
strong spin-orbit interaction in the 3DTI. The gapless DSS are robust against
any perturbations unless time reversal symmetry in the 3DTI is broken or the
bulk band gap is closed [3, 4]. Several actual materials, such as
Bi_$1-x$Sb_$x$ alloy and Bi2Se3-family tetradymite chalcogenides, have been
experimentally proved to be 3DTI [5, 6, 7, 8, 9].

Quantum anomalous Hall (QAH) effect [10, 11] is a two-dimensional phenomenon
which gives discrete Hall conductivity in a unit of $e^{2}/h$ without external
magnetic field. The quantized Hall conductivity directly reflects the Chern
number of filled energy bands, which equals to the number of channel of chiral
edge conduction when the bulk is insulating. This phenomenon typically
requires strong spin-orbit interaction and ferromagnetic order to break time-
reversal symmetry, which indicates that the TIs with ferromagnetism are
candidate materials for QAH insulators [12]. When the chemical potential lies
in the gap of the Dirac dispersion induced by breaking of time-reversal
symmetry, called as surface gap, the QAH effect occurs. Emergence of the
energy gap indicates a possibility to realize other exotic topological
phenomena, such as topological electromagnetic effect [13, 14, 15, 16] and
image magnetic monopole effect [17], both of which were theoretically
predicted from the topological field theory [18]. The first observation of the
QAH effect is in Cr-doped thin-film (Bi,Sb)_2Te_3 [19, 20, 21, 22]. In this
case, the long range ferromagnetic order induced by the dopant opens the
surface gap. However, the doping engenders spatial inhomogeneity of electronic
potentials and exchange coupling by magnetic dopants. This inhomogeneity may
cause spatial variation and diminishment of the surface gap [23, 24] and
residual resistance by dissipative conductive states in bulk, both of which
can make observable temperature of the QAH effect lower than the Curie
temperature as much as two order of magnitude [25]. For experimental
confirmation of the exotic topological quantum phenomena, these disorder
effects originating from the magnetic impurity should be solved and a wider
surface gap in the whole Brillouin zone (BZ) needs to be induced.

Since the first observation of QAH effect in the thin-film 3DTI with magnetic
dopants, there have been many related experiments with various geometries
consisting of a 3DTI and ferromagnetic order. Mogi et al. has reported that
the surface gap can be enhanced by introducing Cr-doped thin layers at the
vicinity of both surfaces of a thin-film 3DTI due to suppression of the
disorder on the surfaces [26]. To realize the QAH insulator with the 3DTI
bearing no disorder, 3DTI/ferromagnetic insulator heterostructures have been
investigated [27, 28]. In this structure, it had been anticipated that the
strong exchange interaction acts on the DSS by penetration of the
magnetization into the 3DTI surfaces with no disorder, resulting in a wider
surface gap to the Dirac dispersion. However, the experimental results show
non-quantized anomalous Hall conductivity, which indicates that the exchange
interaction between the DSS and the penetrating magnetization is too small to
open a considerable surface gap. Recently, Deng et al. has observed the QAH
effect in intrinsic magnetic TI, an odd-layer flake of MnBi2Te4 [29]. Though
MnBi2Te4 shows antiferromagnetism in 3D bulk [30, 31, 32, 33, 34], its odd-
layer film cause the QAH effect. However, the observable temperature is one
order lower than the Néel temperature. It still remains challenging to manage
the wide surface gap for high-temperature QAH effect by operating significant
exchange interaction to the DSS.

In this paper, we propose the possibility of high-temperature QAH effect
arising in 3DTI/thin-film ferromagnetic metal (FMM) bilayer structure. This
study is motivated by the experimental discovery of topological proximity
effect; migration of the DSS into an attached thin-film metal [35]. From this
discovery, we anticipate in the 3DTI/FMM bilayer structure that the DSS
migrates into the FMM layer and is directly influenced by the strong exchange
interaction. By calculating the energy band structure, the anomalous Hall
conductivity, and the magnetic anisotropy, we will demonstrate this
anticipation by showing the emergence of a wide QAH gap with no magnetic
dopant in the 3DTI. The result suggests that the 3DTI/FMM bilayer structure
can realize higher-temperature QAH effect and other exotic effects, which is
anticipated to be applicable for low-power consumption devices and spintronics
[36].

This paper is organized as follows. In Sec. II, we introduce a model
Hamiltonian of a 3DTI/thin-film FMM bilayer structure shown in Fig. 1. In Sec.
III, we analyze how the attachment of thin-film FMM acts the exchange
interaction on the DSS in detail to clearly understand a mechanism of the
appearance of the QAH gap in the bilayer structure. At first, in Sec. III.1,
we discuss that the attachment of a thin-film normal metal (NM) on the 3DTI
can shift the DSS on the 3DTI surface into the metal layer by computing the
band structures and the layer-resolved weight distributions. Next, in Sec.
III.2, we demonstrate the QAH effect in the 3DTI/thin-film FMM bilayer
structure. Then in Sec. III.3, we calculate the magnetic-anisotropy energy to
discuss the uniform magnetic configuration in this system from the viewpoint
of band topology. We also compute variation of the QAH-gap size by changing
the position of energy band of thin-film FMM in Sec. III.4. We discuss that
the Dirac dispersion of the DSS and FMM band should initially possess
energetical degeneracies near to the Dirac point to arise the wide QAH gap.
Sec. IV is devoted to conclusion.

##  II Model Hamiltonian

Figure 1: Schematic picture of the 3DTI/thin-film FMM bilayer. The red arrows
represent magnetization vectors.

In this section, we construct a model Hamiltonian describing the structure
shown in Fig. 1. Our model Hamiltonian consists of three terms,

$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{\mathrm{TI}}+\hat{\mathcal{H}}_{\mathrm{FMM}}+\hat{\mathcal{H}}_{\mathrm{IH}},$
(1)

where $\hat{\mathcal{H}}_{\mathrm{TI}}$, $\hat{\mathcal{H}}_{\mathrm{FMM}}$
and $\hat{\mathcal{H}}_{\mathrm{IH}}$ are 3DTI term, thin-film FMM term, and
interface-hopping term between the 3DTI surface and thin-film FMM layer,
respectively. Let us give detailed expressions for them in the following.

We start with the 3DTI term. In this paper, we consider Bi2Se3-family as a
concrete material of the 3DTI. Electronic states around the Fermi level and
the point in these TIs are described by an effective tight-binding Hamiltonian
derived in Ref. [37]. The effective Hamiltonian forms the highest valence band
and the lowest conduction band, both of which have two-fold degeneracy due to
the presence of time-reversal symmetry and inversion symmetry. In bulk of the
3DTI, the effective Hamiltonian is explicitly written as

$\hat{\mathcal{H}}^{\mathrm{bulk}}_{\mathrm{TI}}=\sum_{\boldsymbol{k}\in\mathrm{BZ}}c^{{\dagger}}_{\boldsymbol{k}}H^{\mathrm{bulk}}_{\mathrm{TI}}(\boldsymbol{k})c_{\boldsymbol{k}},$
(2)

where
$c^{{\dagger}}_{\boldsymbol{k}}=\left[\begin{array}[]{cccc}\hat{c}^{{\dagger}}_{\boldsymbol{k}-\uparrow}&\hat{c}^{{\dagger}}_{\boldsymbol{k}-\downarrow}&\hat{c}^{{\dagger}}_{\boldsymbol{k}+\uparrow}&\hat{c}^{{\dagger}}_{\boldsymbol{k}+\downarrow}\end{array}\right]$
is a 4-component spinor of creation operators of electrons in the 3DTI, and
$H^{\mathrm{bulk}}_{\mathrm{TI}}(\boldsymbol{k})$ is a $4\times 4$ matrix
given as,

$H^{\mathrm{bulk}}_{\mathrm{TI}}(\boldsymbol{k})=\epsilon_{\mathrm{TI}}(\boldsymbol{k})1_{4}+\sum_{i=1}^{3}R_{i}(\boldsymbol{k})\alpha_{i}+m(\boldsymbol{k})\beta.$
(3)

Here, $\boldsymbol{k}=\left(k_{x},k_{y},k_{z}\right)$ is the Bloch wave
vector, $\uparrow$ ($\downarrow$) represents $z$ component of the total
angular momentum $j_{z}=+1/2$ ($-1/2$), and $\pm$ means sign of the parity
eigenvalue. In Eq. (3), $1_{4}$ is the $4\times 4$ identity matrix, and
$\alpha_{i}$ and $\beta$ are Dirac matrices which can be explicitly written as

$\alpha_{i}=\left[\begin{array}[]{cc}0&\sigma_{i}\\\
\sigma_{i}&0\end{array}\right],\quad\beta=\left[\begin{array}[]{cc}1_{2}&0\\\
0&-1_{2}\end{array}\right],$ (4)

where $\sigma_{i}$ are the Pauli matrices. We consider a 3DTI on a hexagonal
lattice with nearest neighbor hoppings and thus the coefficients of these
matrices in Eq. (3) can be written as

$\displaystyle\epsilon_{\mathrm{TI}}(\boldsymbol{k})$
$\displaystyle=\epsilon_{0,\mathrm{TI}}+\frac{2\epsilon_{2}}{c^{2}}\left\\{1-\cos\left[k_{z}c\right]\right\\}$
$\displaystyle\quad+\frac{4\epsilon_{1}}{3a^{2}}\left\\{3-2\cos\left[\frac{\sqrt{3}}{2}k_{x}a\right]\cos\left[\frac{1}{2}k_{y}a\right]-\cos\left[k_{y}a\right]\right\\},$
(5) $\displaystyle R_{1}(\boldsymbol{k})$
$\displaystyle=\frac{2A_{1}}{\sqrt{3}a}\sin\left[\frac{\sqrt{3}}{2}k_{x}a\right]\cos\left[\frac{1}{2}k_{y}a\right],$
(6) $\displaystyle R_{2}(\boldsymbol{k})$
$\displaystyle=\frac{2A_{1}}{3a}\left\\{\cos\left[\frac{\sqrt{3}}{2}k_{x}a\right]\sin\left[\frac{1}{2}k_{y}a\right]+\sin\left[k_{y}a\right]\right\\},$
(7) $\displaystyle R_{3}(\boldsymbol{k})$
$\displaystyle=A_{2}\sin\left[k_{z}c\right],$ (8) $\displaystyle
m(\boldsymbol{k})$
$\displaystyle=m_{0}+\frac{2m_{2}}{c^{2}}\left\\{1-\cos\left[k_{z}c\right]\right\\}$
$\displaystyle\quad+\frac{4m_{1}}{3a^{2}}\left\\{3-2\cos\left[\frac{\sqrt{3}}{2}k_{x}a\right]\cos\left[\frac{1}{2}k_{y}a\right]-\cos\left[k_{y}a\right]\right\\}$
(9)

Here, $a$ and $c$ are lattice constants in $xy$ plane and along $z$ axis,
respectively, and $m_{0}$ corresponds to the strength of the spin-orbit
interaction. $a$, $c$, $\epsilon_{0,\mathrm{TI}}$, $\epsilon_{1}$,
$\epsilon_{2}$, $A_{1}$, $A_{2}$, $m_{0}$, $m_{1}$, $m_{2}$ are material-
dependent parameters. We determine these values by fitting the energy spectrum
of this effective Hamiltonian to that of Bi2Se3 obtained by ab initio
calculation : $a=4.134~{}\left[$\mathrm{\SIUnitSymbolAngstrom}$\right]$,
$c=28.63~{}\left[$\mathrm{\SIUnitSymbolAngstrom}$\right]$,
$\epsilon_{1}=20.17~{}\left[$\mathrm{e}\mathrm{V}$$\mathrm{\SIUnitSymbolAngstrom}$^{2}\right]$,
$\epsilon_{2}=19.67~{}\left[$\mathrm{e}\mathrm{V}$$\mathrm{\SIUnitSymbolAngstrom}$^{2}\right]$,
$A_{1}=4.175~{}\left[$\mathrm{e}\mathrm{V}$$\mathrm{\SIUnitSymbolAngstrom}$\right]$,
$A_{2}=9.162~{}\left[$\mathrm{e}\mathrm{V}$$\mathrm{\SIUnitSymbolAngstrom}$\right]$,
$m_{0}=-0.28~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$m_{1}=58.28~{}\left[$\mathrm{e}\mathrm{V}$$\mathrm{\SIUnitSymbolAngstrom}$^{2}\right]$,
$m_{2}=117.1~{}\left[$\mathrm{e}\mathrm{V}$$\mathrm{\SIUnitSymbolAngstrom}$^{2}\right]$
[38, 39]. We took the value of $\epsilon_{0,\mathrm{TI}}$ as
$\epsilon_{0,\mathrm{TI}}=-0.3111~{}\left[$\mathrm{e}\mathrm{V}$\right]$ so
that the Dirac point on the 3DTI surface lies at the criterion of energy
$E=0$.

From Eq. (2), we implement a slab of the 3DTI with $N_{z}^{\mathrm{TI}}$
layers along the $z$ direction to construct the 3DTI/FMM bilayer structure. To
obtain this open boundary geometry, we rewrite the $k_{z}$ dependence of the
Eqs. (2) and (3) by truncating the hopping outside of
$n_{z}=1,2,\ldots,N^{\mathrm{TI}}_{z}$ layers in the tight-binding model. As a
result of this procedure, the explicit form of
$\hat{\mathcal{H}}_{\mathrm{TI}}$ in Eq. (1) is given as

$\hat{\mathcal{H}}_{\mathrm{TI}}=\sum_{k_{x},k_{y}\in\mathrm{SBZ}}c^{{\dagger}}_{k_{x}k_{y}}H_{\mathrm{TI}}\left(k_{x},k_{y}\right)c_{k_{x}k_{y}},$
(10)

where
$c^{{\dagger}}_{k_{x}k_{y}}=\left[\begin{array}[]{cccc}c^{{\dagger}}_{k_{x}k_{y}1}&c^{{\dagger}}_{k_{x}k_{y}2}&\ldots&c^{{\dagger}}_{k_{x}k_{y}N^{\mathrm{TI}}_{z}}\end{array}\right]$
and
$c^{{\dagger}}_{k_{x}k_{y}n_{z}}=\left[\begin{array}[]{cccc}\hat{c}^{{\dagger}}_{k_{x}k_{y}n_{z}-\uparrow}&\hat{c}^{{\dagger}}_{k_{x}k_{y}n_{z}-\downarrow}&\hat{c}^{{\dagger}}_{k_{x}k_{y}n_{z}+\uparrow}&\hat{c}^{{\dagger}}_{k_{x}k_{y}n_{z}+\downarrow}\end{array}\right]$
are matrices of the creation operators, SBZ means surface BZ, and

$\displaystyle H_{\mathrm{TI}}\left(k_{x},k_{y}\right)$
$\displaystyle=1_{N^{\mathrm{TI}}_{z}}\otimes\left\\{\tilde{\epsilon}_{\mathrm{TI}}\left(k_{x},k_{y}\right)1_{4}+\sum_{i=1}^{2}R_{i}(\boldsymbol{k})\alpha_{i}\right.$
$\displaystyle\left.\quad\quad\quad\quad+\tilde{m}\left(k_{x},k_{y}\right)\beta\right\\}+H_{z}.$
(11)

Here,
$\tilde{\epsilon}_{\mathrm{TI}}\left(k_{x},k_{y}\right)=\epsilon_{\mathrm{TI}}(\boldsymbol{k})+2B_{2}\cos\left[k_{z}c\right]/c^{2}$,
$\tilde{m}\left(k_{x},k_{y}\right)=m(\boldsymbol{k})+2m_{2}\cos\left[k_{z}c\right]/c^{2}$,
and

$\displaystyle H_{z}$ $\displaystyle=\left[\begin{array}[]{cccc}0&1&&\\\
1&\ddots&\ddots&\\\ &\ddots&\ddots&1\\\ &&1&0\\\
\end{array}\right]\otimes\left(-\frac{B_{2}}{c^{2}}1_{4}-\frac{m_{2}}{c^{2}}\beta\right)$
(16) $\displaystyle\quad+\left[\begin{array}[]{cccc}0&-\mathrm{i}&&\\\
\mathrm{i}&\ddots&\ddots&\\\ &\ddots&\ddots&-\mathrm{i}\\\ &&\mathrm{i}&0\\\
\end{array}\right]\otimes\left(\frac{A_{2}}{2c}\alpha_{3}\right).$ (21)

In Eqs. (11) and (21), the left-side matrices of the direct products have
$N^{\mathrm{TI}}_{z}\times N^{\mathrm{TI}}_{z}$ size representing the degrees
of freedom of finite layers in $z$ direction.

Next, we introduce the thin-film FMM term. As a simple model for this
material, we employ a two-dimensional single-orbital tight-binding Hamiltonian
[40, 41]. For simplicity, we implement the FMM in a two-dimensional hexagonal
lattice whose lattice constant in $xy$ plane is the same as that of the 3DTI.
The tight-binding Hamiltonian is explicitly written as

$\hat{\mathcal{H}}_{\mathrm{FMM}}=\sum_{k_{x},k_{y}\in\mathrm{SBZ}}d^{{\dagger}}_{k_{x}k_{y}n_{z}}H_{\mathrm{FMM}}\left(k_{x},k_{y}\right)d_{k_{x}k_{y}n_{z}},$
(22)

where
$d^{{\dagger}}_{k_{x}k_{y}n_{z}}=\left[\begin{array}[]{cc}\hat{d}^{{\dagger}}_{k_{x}k_{y}n_{z}\uparrow}&\hat{d}^{{\dagger}}_{k_{x}k_{y}n_{z}\downarrow}\end{array}\right]$
is a 2-component spinor of electron in thin-film FMM of $n_{z}$-th layer, and

$\displaystyle H_{\mathrm{FMM}}\left(k_{x},k_{y}\right)$
$\displaystyle=\epsilon_{\mathrm{FMM}}\left(k_{x},k_{y}\right)1_{2}-J\boldsymbol{M}\cdot\boldsymbol{\sigma}+\lambda_{\mathrm{R}}\left\\{\boldsymbol{r}(\boldsymbol{k})\times\boldsymbol{\sigma}\right\\}\cdot\boldsymbol{e}_{z},$
(23) $\displaystyle\epsilon_{\mathrm{FMM}}\left(k_{x},k_{y}\right)$
$\displaystyle=\epsilon_{0,\mathrm{FMM}}-t\left\\{\cos\left[\frac{\left(\sqrt{3}k_{x}-k_{y}\right)a}{2}\right]\right.$
$\displaystyle\quad\left.+\cos\left[\frac{\left(\sqrt{3}k_{x}+k_{y}\right)a}{2}\right]+\cos\left[k_{y}a\right]\right\\}.$
(24)

In Eq. (23), the first term is the hopping term, the second term is the
exchange interaction between magnetization $\boldsymbol{M}$ and electron’s
spin, and the last term is the Rashba spin-orbit interaction which reflects
two-dimensional character of FMM. Here we introduce vectors of
$\boldsymbol{r}(\boldsymbol{k})=\left(R_{1}(\boldsymbol{k})/A_{1},R_{2}(\boldsymbol{k})/A_{1},R_{3}(\boldsymbol{k})/A_{2}\right)$
and $\boldsymbol{\sigma}=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)$.
$\boldsymbol{e}_{z}$ is the unit vector in $z$ direction.
$\epsilon_{0,\mathrm{FMM}}$, $t$, $\lambda_{\mathrm{R}}$ and $J$ are the
onsite energy, the hopping integral, the Rashba constant and the exchange
interaction constant, respectively. By adjusting these parameters, we change
the position of FMM band to investigate the requirement for wide QAH gap in
Sec. III.4.

Then, we define the interface-hopping term. We locate the two-dimensional FMM
on both sides of the 3DTI as shown in Fig. 1, i.e., we set the thin-film FMM
layers at $n_{z}=0,N_{z}^{\mathrm{TI}}+1$ layers. When the FMM is attached to
the surface of 3DTI, interface coupling emerges between the $n_{z}=0$
($N_{z}^{\mathrm{TI}}+1$) layer and the $n_{z}=1$ ($N_{z}^{\mathrm{TI}}$)
layer. Thus we explicitly model the interface-hopping term as

$\displaystyle\hat{\mathcal{H}}_{\mathrm{IH}}$
$\displaystyle=t_{\mathrm{IH}}\sum_{k_{x},k_{y}\in\mathrm{SBZ}}\left(d^{{\dagger}}_{k_{x}k_{y}0}\left[\begin{array}[]{cc}1_{2}&1_{2}\end{array}\right]c_{k_{x}k_{y}1}\right.$
(26)
$\displaystyle\quad\left.+d^{{\dagger}}_{k_{x}k_{y}N_{z}^{\mathrm{TI}}+1}\left[\begin{array}[]{cc}1_{2}&1_{2}\end{array}\right]c_{k_{x}k_{y}N_{z}^{\mathrm{TI}}}\right)+\mathrm{H.~{}c.},$
(28)

where $t_{\mathrm{IH}}$ is the interface-hopping integral. Here, for
simplicity, we assume that the interface coupling is independent of the
orbital and spin, and non-zero only between the states with the same angular
momentum.

Finally, we can rewrite the Eq. (1) in matrix form as

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\sum_{k_{x},k_{y}\in\mathrm{SBZ}}\left[\begin{array}[]{ccc}d^{{\dagger}}_{k_{x}k_{y}0}&c^{{\dagger}}_{k_{x}k_{y}}&d^{{\dagger}}_{k_{x}k_{y}N_{z}^{\mathrm{TI}}+1}\end{array}\right]$
(30)
$\displaystyle\quad\times\left[\begin{array}[]{ccc}H_{\mathrm{FMM}}\left(k_{x},k_{y}\right)&H_{\mathrm{IH}}&\\\
{H_{\mathrm{IH}}}^{{\dagger}}&H_{\mathrm{TI}}\left(k_{x},k_{y}\right)&{H_{\mathrm{IH}}}^{{\dagger}}\\\
&H_{\mathrm{IH}}&H_{\mathrm{FMM}}\left(k_{x},k_{y}\right)\end{array}\right]\left[\begin{array}[]{c}d_{k_{x}k_{y}0}\\\
c_{k_{x}k_{y}}\\\ d_{k_{x}k_{y}N_{z}^{\mathrm{TI}}+1}\end{array}\right]$ (37)
$\displaystyle=\sum_{k_{x},k_{y}\in\mathrm{SBZ}}\left[\begin{array}[]{ccc}d^{{\dagger}}_{k_{x}k_{y}0}&c^{{\dagger}}_{k_{x}k_{y}}&d^{{\dagger}}_{k_{x}k_{y}N_{z}^{\mathrm{TI}}+1}\end{array}\right]H\left(k_{x},k_{y}\right)\left[\begin{array}[]{c}d_{k_{x}k_{y}0}\\\
c_{k_{x}k_{y}}\\\ d_{k_{x}k_{y}N_{z}^{\mathrm{TI}}+1}\end{array}\right],$ (42)

where
$H_{\mathrm{IH}}=t_{\mathrm{IH}}\left[\begin{array}[]{cc}1_{2}&1_{2}\end{array}\right]$.

##  III Results

###  III.1 Topological proximity effect in 3DTI/NM bilayer

(a) $t_{\mathrm{IH}}=0~{}\left[$\mathrm{e}\mathrm{V}$\right]$

(b) $t_{\mathrm{IH}}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$

(c) $t_{\mathrm{IH}}=0.3~{}\left[$\mathrm{e}\mathrm{V}$\right]$

(d)

(e) $t_{\mathrm{IH}}=0~{}\left[$\mathrm{e}\mathrm{V}$\right]$

(f) $t_{\mathrm{IH}}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$

(g) $t_{\mathrm{IH}}=0.3~{}\left[$\mathrm{e}\mathrm{V}$\right]$

(h)

Figure 2: 2(a)-2(c) Band structures of $H\left(k_{x},k_{y}\right)$ on the
$k_{y}=0$ line, for different interface-hopping values. Red (blue)-colored
plots on the energy band represents that the weight by Eq. (43) is large at
FMM (3DTI surface) layers. The vertical axes of these figures have their
scales in common. 2(d) Schematic picture describing the shift of gapless Dirac
point due to the hybridization breaking the band intersection. The color of
lines corresponds to the spin direction shown at the bottom of figure.
2(e)-2(g) Weight distributions of the topologically protected Dirac points,
calculated with Eq. (43) and for different interface-hopping values. The blue
(gray) area represents the layers of the 3DTI (NM). The vertical axes of these
figures have their scales in common. 2(h) Variation of the weight of the Dirac
point in NM layer as a function of interface-hopping strength within the range
of $0<\left|t_{\mathrm{IH}}/t\right|<1$. Band structures calculated at certain
values of $\left|t_{\mathrm{IH}}/t\right|$ are shown in the insets, whose axes
have the same quantities and scales as Figs. 2(a)-2(c). We took the adjustable
parameters in Eqs. (23) and (24) as
$\epsilon_{0}=-2.95~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$t=-1.0~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$J=0~{}\left[$\mathrm{e}\mathrm{V}$\right]$, and
$\lambda_{\mathrm{R}}=0.2~{}\left[$\mathrm{e}\mathrm{V}$\right]$.

Before discussing the 3DTI/thin-film FMM geometry, here we consider 3DTI/thin-
film NM bilayer structure, that is, the case of $J=0$ in Eq. (23). Under this
condition, we numerically diagonalize the Hamiltonian
$H\left(k_{x},k_{y}\right)$ of Eq. (42), and obtain eigenvalues
$E_{n}\left(k_{x},k_{y}\right)$ and $4N_{z}^{\mathrm{TI}}+4$-component
eigenvectors $\boldsymbol{u}_{n}\left(k_{x},k_{y}\right)$, where
$n=1,2,\ldots,4N_{z}^{\mathrm{TI}}+4$ is index of energy bands. From the
eigenvectors, we can calculate the layer-resolved weight distribution of
electron density corresponding to each eigenvalue as

$\left|\psi_{nk_{x}k_{y}}\left(n_{z}\right)\right|^{2}=\begin{cases}\displaystyle\sum_{i=1}^{2}\left|u_{n}^{(i)}\left(k_{x},k_{y}\right)\right|^{2}&\left(n_{z}=0\right)\\\
\displaystyle\sum_{i=4n_{z}-1}^{4n_{z}+2}\left|u_{n}^{(i)}\left(k_{x},k_{y}\right)\right|^{2}&\left(1\leq
n_{z}\leq N_{z}^{\mathrm{TI}}\right)\\\
\displaystyle\sum_{i=4N^{\mathrm{TI}}_{z}+3}^{4N^{\mathrm{TI}}_{z}+4}\left|u_{n}^{(i)}\left(k_{x},k_{y}\right)\right|^{2}&\left(n_{z}=N_{z}^{\mathrm{TI}}+1\right)\end{cases},$
(43)

where $u_{n}^{(i)}\left(k_{x},k_{y}\right)$ represents $i$-th component of the
eigenvector. Figs. 22(a)-22(g) show the band structures and the layer-resolved
weight distributions of topologically non-trivial gapless state at the
$\bar{\Gamma}$ point for different values of $t_{\mathrm{IH}}$. The gapless
state connects the valence band of bulk 3DTI or NM and the conduction band. In
Figs. 22(a)-22(c), the color gradient represents the projected weight on the
surface layers of 3DTI (blue) or the NM layers (red) for each eigenstate.
Here, we took the adjustable parameters in Eqs. (23) and (24) as
$\epsilon_{0}=-2.95~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$t=-1.0~{}\left[$\mathrm{e}\mathrm{V}$\right]$ and
$\lambda_{\mathrm{R}}=0.2~{}\left[$\mathrm{e}\mathrm{V}$\right]$ so that the
original NM band and the linear dispersion of 3DTI intersect.

At first, we study the case of
$t_{\mathrm{IH}}=0~{}\left[$\mathrm{e}\mathrm{V}$\right]$ where the 3DTI and
the thin-film NM are spatially separated. In this setting, the Hamiltonian of
Eq. (42) can be individually diagonalized at the 3DTI part and the NM part,
and we obtain independent topologically trivial NM band (red lines) and non-
trivial linear dispersion (blue lines) as shown in Fig. 22(a). Here, we
determine the band connecting the conduction band of 3DTI and the valence
bands of 3DTI or NM as topologically non-trivial. At the $\bar{\Gamma}$ point,
the Dirac dispersion on 3DTI surface forms topologically protected Dirac
point. The band structure of 3DTI around $k_{x}=0$ agrees well with the one
obtained by ab initio calculation in Ref. [42]. The weight distribution of the
eigenstate at the Dirac point has a sharp maximum at the top layer of 3DTI
surface and has no amplitude in NM layer as Fig. 22(e) shows.

Next, we introduce non-zero value for the interface-hopping strength of
$t_{\mathrm{IH}}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$. Resulting band
structure is shown in Fig. 22(b). In the figure, one can see that the
topologically non-trivial Dirac dispersion appears above the topologically
trivial band, which is in sharp contrast to the case of $t_{\mathrm{IH}}=0$. A
mechanism of the formation of these bands can be understood as follows. Let us
start with the situation of Fig. 22(a). Once the 3DTI and the thin-film NM are
attached together, non-zero $t_{\mathrm{IH}}$ appears in non-diagonal part of
Eq. (42) and hybridizes the Dirac dispersion on 3DTI surface and the NM band
which intersect each other at some points. As schematically shown in Fig.
22(d), such a hybridization between the intersecting bands breaks the
degenerated points. As a result, the original topologically non-trivial
(trivial) band is pushed up (down) to reconstruct a distorted Dirac
dispersion. Such deformation of energy bands has been experimentally
established in Ref. [35]. To observe the eigenstates of the reconstructed
topologically-non-trivial Dirac dispersion in detail, we show layer-resolved
weight distribution of the topologically non-trivial Dirac point in Fig.
22(f). The weight has a maximum at the thin-film NM layer, which indicates
that the Dirac electrons migrate from the 3DTI surface into the NM layer due
to the attachment of the thin-film NM.

Then, we discuss the case where larger interface coupling of
$t_{\mathrm{IH}}=0.3~{}\left[$\mathrm{e}\mathrm{V}$\right]$ exists, as shown
in Fig. 22(c). Comparing with Fig. 22(b), one can easily see that stronger
interface hopping leads to a larger split between the energy bands within the
bulk gap of 3DTI. Because of this large split, the Dirac point of
topologically non-trivial band shifts close to the conduction band of bulk
3DTI. As a result, the weight distribution at the point moves back to the 3DTI
as Fig. 22(g) shows. To see a continuous variation of the weight distribution
with respect to the interface-coupling strength, we plot the weight in NM
layer at the Dirac point as a function of $t_{\mathrm{IH}}$ as shown in Fig.
22(h). As this figure shows, the topologically stable Dirac states strongly
localize at NM layer in weak-attached limit, but it rapidly penetrates back to
the 3DTI as the value of $t_{\mathrm{IH}}$ becomes large. This is because
strong interface coupling widely breaks the intersect between the linear
dispersion of 3DTI and the NM band and energetically pushes up the
reconstructed Dirac cone. The pushed-up Dirac cone incorporates with the
valence band of bulk 3DTI and eigenstates of the Dirac cone possess wide
weight in the 3DTI.

At the end of this section, we discuss the ideal condition of the topological
proximity effect and decide the value of $t_{\mathrm{IH}}$ which we use in the
later sections. To realize a wide band gap in the reconstructed topologically-
non-trivial Dirac dispersion by breaking the time-reversal symmetry, the
topologically non-trivial Dirac dispersion should be separated from the bands
of bulk 3DTI and other surface dispersion. In this paper, we have set the NM
band right above the original Dirac point of the linear dispersion on 3DTI
surface as Fig. 22(a) so that both energy bands intersect close to the Dirac
point. In this situation, from Fig. 22(b) and 22(h), we can see that the
reconstructed Dirac dispersion is isolated from both the other reconstructed
topologically-trivial Dirac band and the conduction band of bulk 3DTI at
around $\left|t_{\mathrm{IH}}/t\right|=0.1$. The topologically non-trivial
Dirac cone is located close to the topologically trivial one (the conduction
band of bulk 3DTI) when $\left|t_{\mathrm{IH}}/t\right|<0.1$
($\left|t_{\mathrm{IH}}/t\right|>0.1$) as shown in the insets of Fig. 22(h).
Therefore, we fix the value of interface hopping strength as
$\left|t_{\mathrm{IH}}/t\right|=0.1$ in the later sections as the ideal
condition for the emergence of wide band gap. In this setting, the eigenstate
of the topologically non-trivial Dirac point migrates into the NM layer as we
have shown in Fig. 22(f).

###  III.2 Quantum anomalous Hall effect in 3DTI/FMM bilayer

Figure 3: Band structure of the 3DTI/thin-film FMM bilayer structure at left
side, and corresponding anomalous Hall conductivity as a function of the
chemical potential at right side. We took the adjustable parameters in Eqs.
(23), (24) and (28) as
$\epsilon_{0}=-2.95~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$t=-1.0~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$JM_{z}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$ with
$\boldsymbol{M}=M_{z}\boldsymbol{e}_{z}$,
$\lambda_{\mathrm{R}}=0.2~{}\left[$\mathrm{e}\mathrm{V}$\right]$ and
$t_{\mathrm{IH}}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$.

Here we turn on the exchange interaction between the uniform magnetization and
electron spin in thin-film metal layers to study a 3DTI/thin-film FMM bilayer.
In addition to the band structure, we compute the anomalous Hall conductivity
$\sigma_{xy}$ from intrinsic mechanism using the Kubo formula,

$\sigma_{xy}=\frac{e^{2}}{h}\sum_{n}\int_{\mathrm{SBZ}}\frac{\mathrm{d}k_{x}\mathrm{d}k_{y}}{2\uppi}b^{(z)}_{n}\left(k_{x},k_{y}\right)f(E_{n}-\mu),$
(44)

where $n$ is the band index, $\mu$ is the chemical potential of whole system
and $f(E_{n}-\mu)$ is the Fermi-Dirac distribution function.
$b^{(z)}_{n}\left(k_{x},k_{y}\right)=\partial_{k_{x}}a^{(y)}_{n}\left(k_{x},k_{y}\right)-\partial_{k_{y}}a^{(x)}_{n}\left(k_{x},k_{y}\right)$
($\partial_{k_{\eta}}\equiv\partial/\partial k_{\eta}$ for $\eta=x,y$) is
$z$-component of the Berry curvature, where
$a^{(\eta)}_{n}\left(k_{x},k_{y}\right)=-\mathrm{i}\Braket{u_{n}\left(k_{x},k_{y}\right)}{\partial_{k_{\eta}}u_{n}\left(k_{x},k_{y}\right)}$
is the Berry connection and $\Ket{u_{n}\left(k_{x},k_{y}\right)}$ is the Bloch
eigenstate obtained by the diagonalization.

The computed band structure is shown at the left side of Fig. 3. Here, we set
the magnetization vector in thin-film FMM to be perpendicular to the surface,
i.e., $\boldsymbol{M}=M_{z}\boldsymbol{e}_{z}$ (we discuss reasonability of
this setting in Sec. III.3), and took the value of exchange-interaction
strength in FMM as $JM_{z}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$.
Comparing the band structure of Fig. 3 with Fig. 22(b), one can see that non-
zero exchange interaction in FMM opens surface gap to the Dirac points at
$\bar{\Gamma}$ point. Remarkably, the width of the surface gap is as wide as
about $60~{}\left[$\mathrm{m}\mathrm{e}\mathrm{V}$\right]$, which is larger
than the magnetically-doped thin-film TIs, though the exchange-interaction
strength is much smaller than the typical values of that in actual transition
ferromagnetic metals. In Sec. III.4, we study a condition for the appearance
of this wide surface gap.

At the right side of Fig. 3, we show the corresponding anomalous Hall
conductivity at the temperature $T=0$ as a function of the chemical potential.
Since the global surface gap exists over the whole SBZ, the 3DTI/FMM bilayer
structure possesses a constant quantized anomalous Hall conductivity when the
chemical potential lies in the range of the surface gap. This quantization can
be qualitatively understood by considering how the exchange interaction works
to the Dirac electrons migrating into the FMM layers. Let us focus on one side
of the thin-film FMM. We adopt a simple $2\times 2$ Dirac Hamiltonian,

$H_{\mathrm{2D}}\left(k_{x},k_{y}\right)=\hbar
v\left(\boldsymbol{\sigma}\times\boldsymbol{k}\right)\cdot\boldsymbol{e}_{z},$
(45)

to describe the two-dimensional spin-momentum locked eigenstates around the
new Dirac point shown in Fig. 22(d). Here, $v$ is a constant which has
dimension of velocity. The exchange interaction for the Dirac electrons is
described as $H_{\mathrm{ex}}=-JM_{z}\sigma_{z}$, which works as a mass term
for the Dirac Hamiltonian. Thus the exchange interaction opens a mass gap to
the Dirac dispersion as shown at the left side of Fig. 3. Assuming that the
chemical potential is in the gap, one can easily calculate the Berry curvature
and the anomalous Hall conductivity of (44) from
$H_{\mathrm{2D}}\left(k_{x},k_{y}\right)+H_{\mathrm{ex}}$ as

$\sigma_{xy}=\mathrm{sgn}\left[JM_{z}\right]\frac{e^{2}}{2h},$ (46)

at $T=0$. This Hall conductivity is a contribution from one side of the FMM
layers, and the one from the other side is calculated similarly. Therefore, we
obtain the quantized Hall conductivity shown in Fig. 3 as contribution from
the whole of 3DTI/FMM bilayer structure.

###  III.3 Magnetic anisotropy of the 3DTI/FMM system

(a)

(b)

(c) $(\theta,\phi)=(0,0)$

(d) $(\theta,\phi)=(\uppi/2,0)$

Figure 4: 4(a) Definition of two angles representing the tilting of
magnetization, $\theta$ and $\phi$. 4(b) Relative variation $\Delta
u=u(\theta,\phi)-u(\theta=\phi=0)$ of Eq. (47) as a function of $\theta$ and
$\phi$. 4(c), 4(d) Band structure when the magnetization angle points 4(c)
out-of-plane direction and 4(d) in-plane direction against the surface. The
vertical axes of these figures have their scales in common.

Here we study magnetic anisotropy of the bilayer structure and discuss
possibility of the appearance of QAH effect. To study the anisotropy, we
compute the total energy of electrons in the system,

$u(T,\boldsymbol{M})=\frac{1}{N_{\mathrm{FMM}}}\sum_{n}\int_{\mathrm{SBZ}}\mathrm{d}k_{x}\mathrm{d}k_{y}E_{n}\left(k_{x},k_{y}\right)f(E_{n}-\mu),$
(47)

at $T=0$. Here, we divide the total energy by $N_{\mathrm{FMM}}$, the total
number of unit cells in two FMM layers, so that we compute the intensive
energy of magnetic anisotropy per one magnetized unit cell. The magnetization
vector is expressed with polar coordinates as
$\boldsymbol{M}(\theta,\phi)=|\boldsymbol{M}|\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)$
where $\theta$ and $\phi$ are defined as Fig. 44(a). We fix $|\boldsymbol{M}|$
with $J$ to be $J|\boldsymbol{M}|=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
the same value as Sec. III.2. The chemical potential $\mu$ in Eq. (47) is
determined so that the valence bands below the global surface gap in Fig. 3 is
completely filled at $T=0$. We derive relative variation from Eq. (47) as a
function of $\theta$ and $\phi$ independently, taking the value of
$u(\theta=\phi=0)$ as the criterion for the total energy. Fig. 44(b) shows the
calculated relative variation $\Delta u$. From the figure, one can easily see
that the total energy as a function of $\theta$, $\Delta u(\theta,0)$, has
minima at $\theta=0$ and $\uppi$, while a function of $\phi$, $\Delta
u(\uppi/2,\phi)$, is constant for all $\phi$ values in the range of
$0\leq\phi\leq 2\uppi$. These behaviors clearly indicate that the easy axis of
the magnetization in the thin-film FMM is perpendicular to the surface. In
other words, once macroscopic magnetization appears in the FMM layers at a
finite temperature, it points to the direction perpendicular to the surface,
and the system can become QAH insulator with wide surface gap.

This character of magnetic anisotropy can be understood by considering
topological properties of band structure as follows. As discussed in Sec.
III.2, the surface gap appearing as Fig. 44(c) corresponds to the
topologically-stable quantized anomalous Hall conductivity of
$\sigma_{xy}=e^{2}/2h$ for both the top and the bottom surfaces. Similarly,
for $\theta=\uppi$, the surface gap opens with $\sigma_{xy}=-e^{2}/2h$ for
each surface. The Hall conductivity does not depend on any parameters of the
system, and thus can be interpreted to be a topological number as a unit of a
physical constant $e^{2}/h$. These situations of $\theta=0$ and $\theta=\uppi$
accord each other by adiabatically rotate the magnetization vector. Therefore,
the topological number should jump from one to the other along the adiabatic
rotation, indicating that the global surface gap should close during the
adiabatic change. To close the surface gap, the filled surface valence band
adiabatically shifts upward to touch the empty surface conduction band as
shown in Fig. 44(d), and the energy levels of electrons occupying the surface
valence band increase. As a result, the total energy of electrons increases by
rotating the magnetization in FMM layers to the in-plane direction, and
therefore the magnetization vector prefers to point to the direction
perpendicular to the surface.

###  III.4 Condition to the wide quantum anomalous Hall gap

(a)

(b)

(c)

Figure 5: 5(a) Variation of the width of the global surface gap by shifting
the position of FMM band and changing the strength of exchange interaction in
thin-film FMM. We took the adjustable parameters in Eqs. (23), (24) and (28)
as $t=-1.0~{}\left[$\mathrm{e}\mathrm{V}$\right]$,
$\boldsymbol{M}=M_{z}\boldsymbol{e}_{z}$,
$\lambda_{\mathrm{R}}=0.2~{}\left[$\mathrm{e}\mathrm{V}$\right]$ and
$t_{\mathrm{IH}}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$. The label of
horizontal axis $\epsilon_{\mathrm{FMM},\downarrow}(\bar{\Gamma})$ represents
the value of upper FMM band at the $k_{x}=k_{y}=0$, i.e.,
$\epsilon_{\mathrm{FMM},\downarrow}(\bar{\Gamma})=\epsilon_{0,\mathrm{FM}}-3t+JM_{z}$
from Eqs. (23) and (24). 5(b), 5(c) Band structure in case of setting the FMM
band 5(b) far from or 5(c) near to the original Dirac point localized on
isolated 3DTI surface. We took the strength of exchange interaction as
$JM_{z}=0.1~{}\left[$\mathrm{e}\mathrm{V}$\right]$ and the onsite energy in
the thin-film FMM layers as 5(b)
$\epsilon_{0,\mathrm{FMM}}=-3.3~{}\left[$\mathrm{e}\mathrm{V}$\right]$ or 5(c)
$\epsilon_{0,\mathrm{FMM}}=-3~{}\left[$\mathrm{e}\mathrm{V}$\right]$. The
vertical axes of these figures have their scales in common.

Finally, we discuss the requirement to induce a wide QAH gap in the 3DTI/FMM
bilayer structure. For this purpose, we compare the width of the surface gap
by shifting the position of FMM band within the energy range of the bulk gap
of the 3DTI and changing the exchange-interaction strength in the FMM layer
independently. Fig. 55(a) shows the width of the surface gap by changing the
positions of the FMM band and different values of the exchange-interaction
strength. The label $\epsilon_{\mathrm{FMM},\downarrow}(\bar{\Gamma})$ of the
horizontal axis in the figure represents the energy value of upper FMM band at
the $k_{x}=k_{y}=0$ point in SBZ;
$\epsilon_{\mathrm{FMM},\downarrow}(\bar{\Gamma})=\epsilon_{0,\mathrm{FM}}-3t+JM_{z}$.
Here we fixed the magnetization vector in the thin-film FMM as
$\boldsymbol{M}=M_{z}\boldsymbol{e}_{z}$. We determined each width of the
surface gap as an energy range where the anomalous Hall conductivity is
constant and quantized as a unit of $e^{2}/h$ and the density of states
vanishes. From Fig. 55(a), we can anticipate that a sufficiently wide QAH gap
of $80$ \- $100~{}\left[$\mathrm{m}\mathrm{e}\mathrm{V}$\right]$ may appear in
the 3DTI/thin-film FMM bilayer structure because the strength of exchange
interaction can reach about $10^{0}~{}\left[$\mathrm{e}\mathrm{V}$\right]$ as
order. On the other hand, when the original FMM band and the conduction band
of bulk 3DTI intersect, which corresponds to the interval of
$\epsilon_{\mathrm{FMM},\downarrow}(\bar{\Gamma})\gtrsim
0.3~{}\left[$\mathrm{e}\mathrm{V}$\right]$ in Fig. 55(a), the surface gap
sharply vanishes because the FMM band completely covers the bulk gap of 3DTI
after the two materials are attached.

Fig. 55(a) also shows that the width of the surface gap becomes maximum when
the original FMM band lies on the original Dirac point which is initially
localized at the 3DTI surface. We qualitatively discuss the mechanism for such
variation of the surface gap in Fig. 55(a) by considering the band structure
and the weight distribution for two types of FMM band. One is the case where
the original FMM band lies far from the Dirac point on 3DTI surface at the
$\bar{\Gamma}$ point, and the other is the one where the FMM band and the
Dirac point intersect initially. Fig. 55(b) and 55(c) show the obtained band
structure with black lines and the weight distribution with color points on
the lines for the two setting types of the FMM band. Fig. 55(b) clearly shows
that the eigenstates around the gapped-out Dirac cone is still strongly
localized at the top surface of 3DTI when the FMM band is initially separated
from the original Dirac point. On the other hand, Fig. 55(c) shows that when
the FMM resides close to the Dirac point, the prominent migration of the DSS
occurs as we have shown in Sec. III.1, which leads to the wide surface gap.
This result indicates that intense exchange interaction on the DSS requires
the migration of DSS realized by the hybridization between the thin-film FMM
band and the DSS only around the original Dirac point. Once the DSS migrates
into the FMM layer by such hybridization through the the interface coupling, a
wide QAH gap opens because the exchange interaction can strongly acts on the
DSS in the thin-film FMM layer.

##  IV Conclusion

In this work, we have suggested possibility of the occurrence of the QAH
effect in the 3DTI/thin-film FMM bilayer structure and revealed its mechanism.
When the surface dispersion of 3DTI and the band of thin-film FMM initially
intersects near to the original Dirac point, the hybridization of the
degenerated states cause the shift of the DSS into the FMM layer. This
proximity effect allows the exchange interaction in the FMM to strongly act on
the migrated DSS, which brings the appearance of the wide gap at the migrated
Dirac dispersion. When the chemical potential lies in the surface gap, the
anomalous Hall conductivity can be quantized due to the topological character
of the non-trivial helical DSS. The realization of wide surface gap in the
3DTI/thin-film FMM bilayer structure leads to the easy observation of the QAH
effect, e.g., at the room temperature.

Our computed results demonstrate that the width of the surface gap reach about
$90~{}\left[$\mathrm{m}\mathrm{e}\mathrm{V}$\right]$ by setting practical
material-dependent parameters which are expected from the result of ab initio
calculation in the 3DTI and the FMM of our model. Because the 3DTI/thin-film
FMM bilayer structure can influence strong magnetic coupling in FMM to the DSS
of 3DTI without any impurities, it can experimentally procure the wide QAH gap
by solving not only the issue of disorder in the magnetically-doped TI but
also the torment of suppressed magnetic penetration in 3DTI/ferromagnetic
insulator heterostructures. Due to this advantage, we conclude that the
3DTI/thin-film FMM bilayer structure is one of the promising candidates to
experimentally realize the QAH insulator within a wide range of temperature,
which pave a way to further researches of other exotic topological phenomena
and application of the less-dissipative edge conduction.

###### Acknowledgements.

We are grateful to Koji Kobayashi for many helpful discussions. This work was
supported by JSPS KAKENHI Grants No. JP20H01830 and No. 19H00650, JST CREST
Grant No. JPMJCR18T2, and JST-Mirai Program Grant No. JPMJMI20A1.

## References

  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Shindou and Murakami [2009] R. Shindou and S. Murakami, Phys. Rev. B 79, 045321 (2009).
  * Kobayashi  _et al._ [2013] K. Kobayashi, T. Ohtsuki, and K. I. Imura, Phys. Rev. Lett. 110, 236803 (2013).
  * Hsieh  _et al._ [2008] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 452, 970 (2008).
  * Xia  _et al._ [2009] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 398 (2009).
  * Sato  _et al._ [2010] T. Sato, K. Segawa, H. Guo, K. Sugawara, S. Souma, T. Takahashi, and Y. Ando, Phys. Rev. Lett. 105, 136802 (2010).
  * Kuroda  _et al._ [2010] K. Kuroda, M. Ye, A. Kimura, S. V. Eremeev, E. E. Krasovskii, E. V. Chulkov, Y. Ueda, K. Miyamoto, T. Okuda, K. Shimada, H. Namatame, and M. Taniguchi, Phys. Rev. Lett. 105, 146801 (2010).
  * Chen  _et al._ [2010a] Y. L. Chen, Z. K. Liu, J. G. Analytis, J. H. Chu, H. J. Zhang, B. H. Yan, S. K. Mo, R. G. Moore, D. H. Lu, I. R. Fisher, S. C. Zhang, Z. Hussain, and Z. X. Shen, Phys. Rev. Lett. 105, 266401 (2010a).
  * Nagaosa  _et al._ [2010] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * Liu  _et al._ [2016] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. Rev. Condens. Matter Phys. 7, 301 (2016).
  * Yu  _et al._ [2010] R. Yu, W. Zhang, H. J. Zhang, S. C. Zhang, X. Dai, and Z. Fang, Science 329, 61 (2010).
  * Essin  _et al._ [2009] A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102, 146805 (2009).
  * Li  _et al._ [2010] R. Li, J. Wang, X. L. Qi, and S. C. Zhang, Nat. Phys. 6, 284 (2010).
  * Wang  _et al._ [2016] J. Wang, B. Lian, and S. C. Zhang, Phys. Rev. B 93, 045115 (2016).
  * Sekine and Nomura [2020] A. Sekine and K. Nomura, Axion electrodynamics in topological materials (2020), arXiv:2011.13601 [cond-mat.mes-hall] .
  * Qi  _et al._ [2009] X. L. Qi, R. Li, J. Zang, and S. C. Zhang, Science 323, 1184 (2009).
  * Qi  _et al._ [2008] X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B 78, 195424 (2008).
  * Chang  _et al._ [2013] C. Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L. L. Wang, Z. Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S. C. Zhang, K. He, Y. Wang, L. Lu, X. C. Ma, and Q. K. Xue, Science 340, 167 (2013).
  * Checkelsky  _et al._ [2014] J. G. Checkelsky, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, Y. Kozuka, J. Falson, M. Kawasaki, and Y. Tokura, Nat. Phys. 10, 731 (2014).
  * Kou  _et al._ [2014] X. Kou, S. T. Guo, Y. Fan, L. Pan, M. Lang, Y. Jiang, Q. Shao, T. Nie, K. Murata, J. Tang, Y. Wang, L. He, T. K. Lee, W. L. Lee, and K. L. Wang, Phys. Rev. Lett. 113, 137201 (2014).
  * Kou  _et al._ [2015] X. Kou, Y. Fan, M. Lang, P. Upadhyaya, and K. L. Wang, Solid State Commun. 215-216, 34 (2015).
  * Nomura and Nagaosa [2011] K. Nomura and N. Nagaosa, Phys. Rev. Lett. 106, 166802 (2011).
  * Lee  _et al._ [2015] I. Lee, C. K. Kim, J. Lee, S. J. Billinge, R. Zhong, J. A. Schneeloch, T. Liu, T. Valla, J. M. Tranquada, G. Gu, and J. C. Davis, Proc. Natl. Acad. Sci. U. S. A. 112, 1316 (2015).
  * Chen  _et al._ [2010b] Y. L. Chen, J. H. Chu, J. G. Analytis, Z. K. Liu, K. Igarashi, H. H. Kuo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu, M. Hashimoto, T. Sasagawa, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, Science 329, 659 (2010b).
  * Mogi  _et al._ [2015] M. Mogi, R. Yoshimi, A. Tsukazaki, K. Yasuda, Y. Kozuka, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Appl. Phys. Lett. 107, 182401 (2015).
  * Alegria  _et al._ [2014] L. D. Alegria, H. Ji, N. Yao, J. J. Clarke, R. J. Cava, and J. R. Petta, Appl. Phys. Lett. 105, 053512 (2014).
  * Mogi  _et al._ [2019] M. Mogi, T. Nakajima, V. Ukleev, A. Tsukazaki, R. Yoshimi, M. Kawamura, K. S. Takahashi, T. Hanashima, K. Kakurai, T. H. Arima, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 123, 016804 (2019).
  * Deng  _et al._ [2020] Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Science 367, 895 (2020).
  * Otrokov  _et al._ [2019] M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Estyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. Wolter, A. V. Koroleva, A. M. Shikin, M. Blanco-Rey, M. Hoffmann, I. P. Rusinov, A. Y. Vyazovskaya, S. V. Eremeev, Y. M. Koroteev, V. M. Kuznetsov, F. Freyse, J. Sánchez-Barriga, I. R. Amiraslanov, M. B. Babanly, N. T. Mamedov, N. A. Abdullayev, V. N. Zverev, A. Alfonsov, V. Kataev, B. Büchner, E. F. Schwier, S. Kumar, A. Kimura, L. Petaccia, G. Di Santo, R. C. Vidal, S. Schatz, K. Kißner, M. Ünzelmann, C. H. Min, S. Moser, T. R. Peixoto, F. Reinert, A. Ernst, P. M. Echenique, A. Isaeva, and E. V. Chulkov, Nature 576, 416 (2019).
  * Swatek  _et al._ [2020] P. Swatek, Y. Wu, L. L. Wang, K. Lee, B. Schrunk, J. Yan, and A. Kaminski, Phys. Rev. B 101, 161109(R) (2020).
  * Chen  _et al._ [2019] Y. J. Chen, L. X. Xu, J. H. Li, Y. W. Li, H. Y. Wang, C. F. Zhang, H. Li, Y. Wu, A. J. Liang, C. Chen, S. W. Jung, C. Cacho, Y. H. Mao, S. Liu, M. X. Wang, Y. F. Guo, Y. Xu, Z. K. Liu, L. X. Yang, and Y. L. Chen, Phys. Rev. X 9, 041040 (2019).
  * Hao  _et al._ [2019] Y. J. Hao, P. Liu, Y. Feng, X. M. Ma, E. F. Schwier, M. Arita, S. Kumar, C. Hu, R. Lu, M. Zeng, Y. Wang, Z. Hao, H. Y. Sun, K. Zhang, J. Mei, N. Ni, L. Wu, K. Shimada, C. Chen, Q. Liu, and C. Liu, Phys. Rev. X 9, 041038 (2019).
  * Mong  _et al._ [2010] R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev. B 81, 245209 (2010).
  * Shoman  _et al._ [2015] T. Shoman, A. Takayama, T. Sato, S. Souma, T. Takahashi, T. Oguchi, K. Segawa, and Y. Ando, Nat. Commun. 6, 6547 (2015).
  * Tokura  _et al._ [2019] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nat. Rev. Phys. 1, 126 (2019).
  * Liu  _et al._ [2010] C. X. Liu, X. L. Qi, H. J. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Phys. Rev. B 82, 045122 (2010).
  * Kurebayashi and Nomura [2014] D. Kurebayashi and K. Nomura, J. Phys. Soc. Jpn 83, 063709 (2014).
  * Sasaki  _et al._ [2011] S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, and Y. Ando, Phys. Rev. Lett. 107, 217001 (2011).
  * Zhang  _et al._ [2016] J. Zhang, J. P. Velev, X. Dang, and E. Y. Tsymbal, Phys. Rev. B 94, 014435 (2016).
  * Hsu  _et al._ [2017] Y. T. Hsu, K. Park, and E. A. Kim, Phys. Rev. B 96, 235433 (2017).
  * Zhang  _et al._ [2009] H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Nat. Phys. 5, 438 (2009).