# Large Gap Quantum Anomalous Hall Effect in a Type-I Heterostructure Between
a Magnetically Doped Topological Insulator and Antiferromagnetic Insulator

Anh Pham  Center for Nanophase Materials Sciences, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Ling-Jie Zhou  Department of
Physics, The Pennsylvania State University, University Park, Pennsylvania
16802, USA  Yi-Fan Zhao  Department of Physics, The Pennsylvania State
University, University Park, Pennsylvania 16802, USA  Cui-Zu Chang  Department
of Physics, The Pennsylvania State University, University Park, Pennsylvania
16802, USA  Timothy Charlton  Spallation Neutron Source, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  P. Ganesh<EMAIL_ADDRESS>Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA

###### Abstract

Heterostructures between topological insulators (TI) and magnetic insulators
represent a pathway to realize the quantum anomalous Hall effect (QAHE). But
major challenges remain, typical heterostructures have weak magnetic exchange
interactions across the interface, resulting in small inverted band gaps.
Magnetic doping can further increase the band gap due to the additional Zeeman
effect, but gaps remain small at low concentrations of magnetic dopants, with
higher concentrations resulting in shifting of the Fermi-level, that can
render the heterostructure metallic. In addition, chemical bonding across
interfaces can lead to imperfect interfaces that might induce metallicity or
further lower the gap size. Using density functional theory based systematic
screening and investigation of thermodynamic, magnetic and topological
properties of heterostructures, we demonstrate that forming a type-I
heterostructure between a wide gap antiferromagnetic insulator Cr2O3 and a TI-
film, such as Sb2Te3, can lead to pinning of the Fermi-level at the center of
the gap, even when magnetically doped. Cr-doping in the heterostructure
increases the gap to $\sim$ 64.5 meV, with a large Zeeman energy from the
interfacial Cr dopants, thus overcoming potential metallicity due to band
bending effects. By fitting the band-structure around the Fermi-level to a
4-band k.p model Hamiltonian, we show that Cr doped Sb2Te3/Cr2O3 is a Chern
insulator with a Chern number C = -1. Transport calculations further show
chiral edge-modes localized at the top/bottom of the TI-film to be the
dominant current carriers in the material. Our predictions of a large
interfacial magnetism due to Cr-dopants, that coupled antiferromagnetically to
the AFM substrate is confirmed by our polarised neutron reflectometry
measurements on MBE grown Cr doped Sb2Te3/Cr2O3 heterostructures, and is
consistent with a positive exchange bias measured in such systems recently.
Consequently, Cr doped Sb2Te3/Cr2O3 heterostructure represents a promising
platform for the development of functional topological magnetic devices, with
high tunability.

††preprint: APS/123-QED

##  I  Introduction

The quantum anomalous Hall effect is characterized by a chiral edge state in
an insulating topological system with broken time-reversal symmetry. These
edge states carry a dissipationless current with zero longitudinal resistance
and quantized Hall resistance [1-3]. As a result, the QAHE has been proposed
to have a broad range of applications in electronic and spintronic devices as
well as topological quantum computing [4]. To realize the QAHE, it is expected
that a material system has strong spin-orbit coupling (SOC) and magnetic
exchange energy to induce nontrivial topology in the bulk band structure
[1-3]. Furthermore, due to the breaking of time reversal symmetry an exchange
gap can be opened on the surface of the topological insulator(TI).
Consequently, early theoretical studies suggested that the TI thin films doped
with transition metal elements in a diluted concentration can be used a
platform for the realization of the QAHE. In theory, the QAHE appears in TI
thin films with Cr or Fe doping, but it is absent in TI thin films with V or
Ti doping. The absence of the QAHE in the latter two systems is due to the
formation of the impurity states located at the magentic exchange gap, which
hinders the appearance of the insulating state [5,6]. This prediction was
subsequently confirmed in experimental studies of doped Bi(Sb)-based
chalcogenides thin films with V and Cr dopants [7-13]. However, the observing
temperature of the QAHE in magnetically doped topological insulator
films/heterostructures has been less than 1K thus limiting its functional
applications.
The low observing temperature of the QAHE in magnetically doped TI
films/heterostructures has been attributed to the inhomogeneity of magnetic
dopants, i.e. chemical potential fluctuation. This inhomogeneity can break the
inversion symmetry in the TI thin film, resulting in metallic bulk bands [14].
In addition, the inhomogeneity can also reduce the effective magnetic exchange
energy gap since the gap opening at the Dirac point due to time reversal
symmetry breaking and the magnetic interaction between TM dopants are strongly
dependent on the spatial distribution of magnetic defects [15]. Another draw
back of conventional doping technique to realize the QAHE is that only diluted
doping is allowed so that the Fermi level lies inside the bulk gap, while high
doping is not favourable because it can lead to the weakening of spin-orbit
coupling that will eventually destroy the nontrival surface states of the TI
thin films [16,17], even though the high doping level of transition metal
dopant can enhance the Curie temperature to achieve a larger magnetic exchange
gap. To overcome this fundamental material limitation, the modulation doping
technique [18] has been demonstrated to increase the observing temperature of
the QAHE by increasing the magnetic doping concentration on the two surface
layers of the magnetically doped sandwich heterostruture samples [19,20].
In addition to the magnetic doping, the QAHE has also been proposed to be
realized in different heterostructure configurations between a TI and a
ferromagnetic/antiferromagnetic insulator. Different ferromagnetic insulating
materials such as EuS [21,22], GdN [23], BaFe12O19 [24], Cr2Ge2Te6 [25],
ferrimagnet iron garnets (YIG/TIG) [26,27], and CrI3 [28,29] have all
demonstrated their ability to induce the QAHE on the surface of a TI. In these
heterostructures, the magnetic exchange is induced on the TI’s surface via
magnetic proximity effect. However, the induced moment tends to be small and
the effect can be short-range thus limiting the value of the gap opening due
to time-reversal symmetry breaking. Antiferromagnetic (AFM) insulators with
uncompensated intralayer interactions such as Cr2O3 [30] and LaCoO3 [31] have
also been reported to induce the proximity induced magnetic order when
interfacing with a TI thin film. AFM insulating substrates have several
notable advantages, they lack stray dipole fields and are robust against
external perturbing fields. Further, AFM insulators such as Cr2O3 also show
ultrafast dynamics [32]. In addition, since the Néel temperature in many AFM
insulators is much higher than the Curie temperature in FM insulators, this
can enhance the Curie temperature of the proximity induced FM order in the TI
[30].
Since Cr2O3 is a wide band gap AFM insulator with a high Néel temperature (TN
$\sim$307 K) [30], we systematically examine the effect of Cr doped TI in the
heterostructure configuration with TI using first principles calculation.
Experimentally many Cr2O3 thin films tend to be fabricated on substrate like
Al2O3 [33], we set our in-plane lattice constant to the value of Al2O3 (4.756
Å) [34] similar to our recent experimental fabrication of Cr doped
Sb2Te3/Cr2O3 on Al2O3 substrates [30]. We screened several TI materials to
assess lattice mismatch with the AFM insulator. As shown in Table 1, both
Sb2Te3 and Bi2Se3 have the lowest lattice mismatch with the Cr2O3 substrate.
But we find that Bi2Se3 and Cr2O3 interface to be highly reactive, with strong
Se-Cr bonding, resulting in a metallic ground state, making it unsuitable for
the realization of the QAHE. Screening over magnetic dopants in tellurides
suggests magnetic Cr-atoms to be ideal substitutional dopants at the metal
sites [35] similar to our earlier studies on SnTe [36]. We hence perform
computations for Cr doped Sb2Te3, due to low lattice mismatch, and favourable
defect properties. It forms an ideal interface with Cr2O3 with the Fermi level
pinned inside the bulk gap due to a type-I band alignment between Cr2O3 and Cr
doped Sb2Te3. We also explore the possibility and effect of modulation doping.
We find the gap opening in the bulk is strongly enhanced via modulation doping
of Cr dopants at high concentrations at the interface with Cr2O3 with band
gaps as large as 64.5 meV, exhibiting a nonzero Chern number of C = -1. As a
result, the modulation doped Sb2Te3:Cr/Cr2O3 heterostructure represents a
promising platform to realize a QAHE with potentially high temperature
functionality. Interfacing such a heterostructure with an s-wave
superconductor should lead to a large-gap topological superconductivity [37].

Table 1: Lattice constants and strain of different TI layer

TI | Lattice(Å) | d(Å)∗ | Mismatch($\%$)
---|---|---|---
Sb2Te3 | 4.25 | 2.603 | -1.308
Bi2Se3 | 4.138 | 2.404 | -0.463
Bi2Te3 | 4.383 | N/A | -2.19

*Distance between the 4 quintuple layer TI thin films and Cr2O3

Figure 1: Crystal structure of Sb2Te3/Cr2O3 heterostructure. a) Pristine
heterostructure. b) Structure with one Cr dopant substituting Sb at the
interface. c) Structure with two Cr dopants substituting Sb at the interface.

##  II Experimental and computational methods

Density functional calculation (DFT): The calculations were done using the
Perdew-Burke-Ernzerhof (PBE) [38] functional with projected augmented wave
(PAW) [39] method implemented in the VASP code. An effective Hubbard-U value
[40] of U = 4.0 eV and J = 0.58 eV [41] as tested in previous study of Cr2O3
substrate is added to the Cr-${\sl d}$ orbitals to account for the correlation
effects. To simulate the Cr2O3 substrate, a symmetric slab consisting of 24
layers of O and 16 layers of Cr was used with Cr-termination since it was
determined to be the most stable structure in a previous theoretical study
[41]. We consider 4 quintuple layers (QL) for Sb2Te3 and Bi2Se3 TI thin-films.
The in-plane lattice of the heterostructure is set to the experimental lattice
of Al2O3 (a = b = 4.756 Å) [34]. To minimize the strain between the TI films
and the Cr2O3 layers, different supercell configurations of AFM insulator and
TI films were constructed as shown in Table 1. Specifically, the smallest
minimally strained Sb2Te3/Cr2O3 heterostructure consists of a 3 $\times$ 3
$\times$ 1 supercell of Sb2Te3 and a $\sqrt{7}$ $\times$ $\sqrt{7}$ $\times$ 1
supercell of Cr2O3 resulting in a lattice mismatch of 1.3$\%$ for the TI
layer. The Bi2Se3/Cr2O3 heterostructure consists of a 2 $\times$ 2 $\times$ 1
supercell of Bi2Se3 and a $\sqrt{5}$ $\times$ $\sqrt{5}$ $\times$ 1 supercell
of Cr2O3 with a lattice mismatch of 0.46$\%$. While the Bi2Te3/Cr2O3
heterostructure consists of a 4 $\times$ 4 $\times$ 1 of Bi2Te3 and
$\sqrt{13}$ $\times$ $\sqrt{13}$ $\times$ 1 of Cr2O3 with a lattice mismatch
of 2.19$\%$. We focus our study primarily on the Sb2Te3/Cr2O3 and Bi2Se3/Cr2O3
since they contain the lowest lattice mismatch. The distance between the TI
thin film and the Cr2O3 layers was optimized as shown in Table 1 by
determining the minimal energy configuration as the distance between the TI
and the AFM layer was varied. An energy cutoff of 400 eV and 320 eV was used
for the Bi2Se3 and Sb2Te3 heterostructures respectively. A gamma-center k
point of 2 $\times$ 2 $\times$ 1 and 1 $\times$ 1 $\times$ 1 was used to
optimize the heterostructures of Bi2Se3 and Sb2Te3 respectively. To minimize
the interaction along the z-direction, a vacuum of 20 Å was used for all the
structures.
Synthesis of Cr doped Sb2Te3/Cr2O3 hetero-structures: The Cr2O3 films were
grown on heat-treated sapphire (0001) substrates in a pulsed laser deposition
(PLD) chamber and then transferred into a molecular beam epitaxy (MBE) chamber
(Omicron Lab 10) with a vacuum $\sim$ 2 $\times$ 10-10 mbar. High-purity Sb,
Cr and Te were evaporated from Knudsen effusion cells. During growth of the
Cr-doped Sb2Te3 film, the substrate was maintained at $\sim$ 240∘C. The flux
ratio of $\frac{Te}{(Sb+Cr)}$ was set to be greater than 10 to prevent Te
deficiency in the samples. Following the growth, the magnetic TI films were
annealed at $\sim$ 240∘C for 30 minutes to improve the crystal quality before
being cooled down to room temperature. Prior to the removal from the MBE
chamber, a 10 nm thick Te layer was deposited at room temperature on top of
the magnetic TI films for PNR measurements. See more details in our prior
study [30].
Polarized neutron reflectometry (PNR) measurement: The samples were measured
at 5 K in a saturating field to establish a base line. The magnetic field was
then reduced to near remanence for a second measurement. The sample was
subsequently warmed above the Néel temperature of Cr2O3 which showed no
interface exchange coupling to allow us to better constrain the model fitting
parameters. In all cases, the measurements were done from 0.01-0.15 1/Å to
distinguish the exchange at the interface and the bulk.

Figure 2: Thermodynamic properties of Cr dopants in the heterostructure. a)
Formation energy of a single Cr dopant substituting Sb as a function of
distance from the Cr2O3 substrate, with respect to the lowest formation
energy. The formation energy is defined as Ef = Edoped \- Epristine \-
n$\mu_{Cr}$ \+ m$\mu_{Sb}$, where Edoped is the total energy of the Cr doped
heterostructure, Epristine is the total energy of the pristine
heterostructure, $\mu_{Cr}$ is the chemical potential of bulk Cr and
$\mu_{Sb}$ is the chemical potential of Sb, and n and m are the numbers of Cr
added and Sb atoms removed from the heterostructure. b) Total energy
difference of 2 Cr dopants as a function of Cr-Cr separation distance.

##  III Results and Discussions

Before considering the effect of Cr dopants in the TI layer, we first consider
the interfacial effect between the TIs (Bi2Se3 and Sb2Te3) that showed small
lattice mismatch with Cr2O3. In the case of the Bi2Se3 interface, the TI thin
film exhibits strong chemical bonding with the antiferromagnetic substrate as
demonstrated by the small distance between the TI layer and Cr2O3 layers as
illustrated in Table 1. The relaxed crystal structure of Bi2Se3/Cr2O3 showing
the interfacial bonding is shown in Fig. S1. Due to the strong interaction
between Bi2Se3 and Cr2O3, the electronic ground state of this heterostructure
is half-metallic where the spin-up channel is metallic [Fig. S2b]. When the
SOC is turned on, the heterostructure becomes full metallic [Fig. S2c]. As
such this system cannot host a QAH phase. On the other hand, in the
Sb2Te3/Cr2O3 heterostructure the TI layer shows no such chemical bonding with
the Cr2O3 layer. This is also reflected in the larger interfacial distance
between Sb2Te3 and the AFM substrate, compared to Bi2Se3, as shown in Table 1.
Consequently, the Sb2Te3 TI potentially represents a higher quality interface
with magnetic materials such as Cr2O3. Notice, that due to an A-type AFM
ordering, the top-layer of the AFM insulator remains uncompensated [41], and
can effectively induce spin-polarization in the TI layer due to the proximity
coupling effect. However, since there is weak coupling between the TI and the
AFM layers, the induced moments on the interfacial Te/Sb atoms were negligible
($\textless$ 0.01 $\mu B$/atom). To further increase the magnetic exchange
interaction between the TI layer and the AFM substrate, we investigate doping
the TI-layer with magnetic dopants.
Among a wide range of possible magnetic dopants, we find Cr atoms that
substitutionally dope the Sb-site to be the most favourable and study the
electronic, magnetic and topological properties of the Cr-doped Sb2Te3/Cr2O3
heterostructure. To assess dopant segregation to surfaces/interfaces, we
compute the total energy of the substitutional Cr-dopant across the TI film.
As shown in Fig. 2a, Cr-dopant has lowest formation energy at the interface
with the AFM insulator. This suggests that Cr-dopants are likely to form at
the TI/AFM-substrate interface. To see if the dopants would cluster, we next
compare the total energy of a Cr-Cr dimer at the interface, as a function of
distance (Fig. 2b). We find that Cr-dopants have a tendency to cluster at the
interface. Further, this does not appear to introduce any unfavourable
structural distortions or chemical bonding. This opens the possibility of
increasing Cr-concentrations at the TI/AFM interface, achieving modulation
doping. As mentioned earlier, such modulation doping would result in a more
homogeneous exchange gap.
We next explore the nature of the magnetic exchange interaction of Cr-dopants
in a cluster, as well as with the AFM substrate. When one Cr dopant is
introduced in the TI layer at the interface, the magnetic interaction favors
an antiferromagnetic exchange with the uncompensated top magnetic layer in the
Cr2O3 substrate [Fig. 3a], with an exchange energy of $\sim$ 60 meV. This
magnetic exchange coupling with Cr2O3 decreases quickly as the dopant moves
farther away. To induce a QAHE, it is important that the magnetic coupling in
the TI layer prefers the out-of-plane direction, thus we also calculated the
magnetic anisotropy (MAE) by comparing the out-of-plane magnetic direction and
the in-plane direction of the Cr dopants to determine the most favourable
magnetic axis. According to Fig. 3a, at the interface the Cr-dopants strongly
prefer to orient along the out-of-plane direction, while the Cr-dopants
farther away from the interface, that are weakly coupled to the AFM substrate,
fluctuate between an in-plane and an out-of-plane direction. This further
suggests that Cr-dopants when at the interface will lead to a higher
temperature QAHE, if their clusters are ferromagnetically coupled.

Figure 3: Cr dopant-dopant and dopant-substrate magnetic exchange interaction
in Cr doped Sb2Te3 and Cr2O3 heterostructure. a) Total energy difference of a
single Cr-dopant in the TI-layer which is ferromagnetically or
antiferromagnetically coupled to the Cr2O3 substrate. A lower energy for the
AFM arrangement suggests that Cr dopants at the interface are
antiferromagnetically coupled to the top layer of Cr2O3 substrate. The
magnetic anisotropy ($E_{MAE}$) of the Cr dopant in Sb2Te3 is also plotted as
a function of distance from the Cr2O3 substrate. b) Different magnetic
configurations considered between the two Cr dopants in Sb2Te3 interfacial
layer and the Cr2O3 substrate. c) We plot $\Delta E_{Cr-
Cr}^{ex}=E_{AFM-2}-E_{AFM-1}$ and $\Delta E_{Cr-
substrate}^{ex}=E_{AFM-2}-E_{FM}$ as a function of Cr-Cr separation distance.
This suggests that Cr-dopants prefer to cluster at the interface TI-layer.
Figure 4: Polarized neutron measurement of four quintuple Sb1.8Cr0.2Te3/Cr2O3,
showing the spin asymmetry (SA=$(R^{+}-R^{-})/(R^{+}+R^{-})$) for the field
cooled (Bottom) and zero field cooled (Top) cases. Here the data have been
artificially shifted and individual “zero” lines drawn to allow easy
comparison.

In addition to the coupling between the magnetic dopant and the
antiferromagnetic substrate, to facilitate the QAH phase, the magnetic dopants
need to couple ferromagnetically in the TI layer to induce a bulk
magnetization in the TI-film, thereby breaking time-reversal symmetry. To
consider the Cr-Cr magnetic interaction in the TI layer, we consider three
different magnetic coupling between the two Cr dopants and the Cr2O3 layer as
shown in Fig. 3b at different distances between the Cr dopants: i) (AFM-1) AFM
coupling between the two Cr dopants and AFM coupling with the Cr2O3 layer, ii)
(AFM-2) FM coupling between two Cr dopants and AFM coupling with the Cr2O3
layer, and iii) (FM) FM coupling between the two Cr dopants and FM coupling
with the Cr2O3 layer. It should be noted that in our calculations we always
kept one substituting Cr at the interface while varying the position of the
other Cr dopants. According to Fig. 3c, the AFM-2 structure had the lowest
energy. When the Cr-dopants are clustered at the interface, they couple
antiferromagnetically to the substrate, while remaining ferromagnetically
coupled to each other. Previous theoretical study of Cr doped Sb2Te3 [35] has
suggested that the FM coupling can survive even under a large separation of Cr
dopants as d${}_{Cr_{Sb}-Cr_{Sb}}$ $\approx$ 12 Å. As a result, our results
suggest this long-range ferromagnetism is unaffected in heterostructuring
process with Cr2O3, which is also in good agreement with the recent
observation of an enhanced ferromagnetism in the Cr doped Sb2Te3 layer when
interfacing with Cr2O3 [26]. Furthermore, the preferred AFM coupling between
Cr dopants at the interface and the Cr2O3 layer also explains the observation
of the positive exchange bias between Cr doped Sb2Te3 and the Cr2O3 substrate
[30].
In addition to our DFT modeling of the magnetic exchange interaction at the
interface, the magnetism in our 4QL Sb1.8Cr0.2Te3/Cr2O3 heterostructure was
also investigated experimentally using polarized neutron reflectometry (PNR).
This technique is depth sensitive, thus allowing us to probe the magnetic
fluctuation at different heights inside the doped TI thin film. As shown in
Fig. 4, the spin asymmetry (SA) shows two different magnetic signatures for
the field-cooled and zero field-cooled measurements showing the different
magnetic layers spatially separated in the doped TI thin film along the
z-axis. Following a field cool measurement, an oscillation is observed in the
SA immediately after the total reflection region (Fig. 4 blue curve) that
indicates the presence of ferromagnetism throughout the bulk of the film. In
the zero-field cooled case (Fig. 4 red curve), clear oscillations appear with
a maximum near 0.05 1/Å and again near 0.15 1/Å indicating the presence of
magnetic layers near the interface. Consequently, these results confirm our
theoretical prediction of a stronger magnetic coupling at the interface
between the doped TI and the antiferromagnetic Cr2O3 substrate, possibly due
to dopant clustering.
Transition metal doping with Cr has been predicted to open a gap on the Dirac
surface of Sb2Te3 thin film up to a concentration of 10% while maintaining a
bulk insulating state [42-44]. To understand what other effects are introduced
by the Cr2O3 surface on the magnetically doped-TI, we calculated the
electronic structures for different doping concentrations of Cr in the
heterostructure with and without the Cr2O3 layer (Fig. 4-top-row(without) and
4-bottom-row(with)]. As shown in Fig. 4a, due to the small compressive strain
and the hybridization effect between the top and bottom surfaces a significant
inverted energy gap of 20.7 meV is observed in strained undoped free-standing
4 quintuple layer Sb2Te3 [Fig. 4a]. In the presence of Cr2O3, the energy gap
at the $\Gamma$ point in the Sb2Te3 layer significantly reduces due to band-
bending effects which also breaks the inversion symmetry of the free-standing
TI-film. With the introduction of Cr in free-standing Sb2Te3, a metallic
ground state is observed with increasing concentrations of Cr. In addition,
Cr-dopants also induce a Zeeman gap which further increases the size of the
inverted band gap at the $\Gamma$ point from 20.7 meV to 81.3 meV. The
introduction of Cr at the surface of the free-standing Sb2Te3 also breaks
inversion symmetry which induces pronounced asymetric Rashba splitting around
the $\Gamma$ point as shown in Figs. 4b and 4c. This asymmetric Rashba
splitting would also lead to inhomogeneous gaps. The metallic ground state can
be attributed to the compressive strain which has been known to narrow the
band bulk structure of Cr doped Sb2Te3 [45].
On the other hand, an insulating ground state is recovered in the
heterostructure configuration between Cr doped Sb2Te3 and Cr2O3 [Figs. 4e and
4f] with a maximum inverted band gap of 67.5 meV. Most importantly, the Fermi
level is inside the band gap. Furthermore, the direct gap at the $\Gamma$
point is shown to increase monotonically with increasing Cr-dopant
concentration without shifting the Fermi-level i.e. keeping the Fermi-level in
the middle of the band gap. Interestingly, as the Cr doping concentration
increases at the interface the band structure reveals an increased
contribution of a Rashba-like spin-orbit coupling to the band gap, in addition
to the Zeeman splitting. This is due to an increasing contribution of
inversion symmetry breaking with a higher concentration of defects. To
interrogate the importance of Cr-dopants at the interface, we also looked at
the nature of the gap as the Cr-dopant moves away from the interface. As shown
in Fig. S4 the band structure undergoes a gap closing and opening transition
with increasing Cr dopant distance from the interface. This is the signature
of a topological transition with the non-trivial phase only emerging when Cr-
dopants are at the interface, suggesting that likely Cr-dopants at
surfaces/interfaces of TI-films induce non-trivial topology. Recent
experimental studies have also suggested that the magnetic dopants at surfaces
can exhibit different exchange coupling to the bulk, with the possibility of
increasing the magnetization via modulation doping, thus potentially
increasing the band gap [46]. Our results strongly support the idea of
modulation doping in TI heterostructures with AFM-substrates to induce a large
inverted band gap.

Figure 5: Electronic band structures of Sb2Te3 and Sb2Te3/Cr2O3
heterostructure. a), b) and c) show the band-structure around the Fermi-level
without the Cr2O3 substrate for pristine, 1 Cr-dopant and 2 Cr-dopants at the
interface TI-layer, respectively. d), e) and f) show the band-structure around
the Fermi-level with the Cr2O3 substrate for the same three cases. The
difference between the top and bottom panels clearly suggests non-trivial role
of the Cr2O3 substrate on the electronic properties of the Cr-doped TI thin-
film.

Band offset has been known to play an important role in pinning the Fermi
level of Dirac points in TI/magnetic material heterostructure [47]. Based on
our results outlined in Fig. 4, it is suggested that the Fermi level is pinned
inside the band gap regardless of the doping concentration. Thus, to reveal
the role of band offset in the heterostructure configuration, we plot the
layer resolved density of states of Cr doped Sb2Te3/Cr2O3. As shown in Fig.
S3, regardless of the doping concentration, the contributions from the TI’s
orbitals are always located inside the band gap of Cr2O3, thus indicating a
signature of a type-I band offset. As a result, rather than inducing the
magnetization on the TI layer as previous magnetic material/TI heterostructure
Cr2O3 modifies the band structure of the doped Sb2Te3 via a type-I band offset
to pin the Fermi level inside the bulk gap.

Figure 6: Transport obtained by discretizing the k.p Hamiltonian on a square
lattice using parameters from table S1 for our heterostructure system
containing a low concentration (i.e. 1 Cr-dopant in the supercell) of dopant
at the interface. a) 2D square lattice with dimension 100L X 300W used for the
transport calculation. b) Conductance measurement for the system showing a
quantized value ($e^{2}/h$) indicating C = -1 expected for a QAH phase. c)
Current direction corresponding to the top edge. d) Current direction for the
bottom edge. Here the parameters were divided by an effective lattice of
12.584 Å so that all the fitting parameters in the kwant program have the same
unit of eV.

Since the Cr doped Sb2Te3 represents an ideal interface with Cr2O3, we
characterize the topological nature of the heterostructure at the two
interfacial doping concentrations, using a 4 band ${\sl k.p}$ model [14,28] as
described below by considering the states from both the top and bottom
surfaces with different basis such as $|t\uparrow\rangle$,
$|t\downarrow\rangle$, $|b\uparrow\rangle$, $|b\downarrow\rangle$.
$H_{total}$ = $H_{surface}$ \+ $H_{Zeeman}$ \+ $H_{interface}$
$H_{surface}$ =
$\begin{pmatrix}A(k_{x}^{2}+k_{y}^{2})&iv_{\alpha}k_{-}&M_{k}&0\\\
-iv_{\alpha}k_{+}&A(k_{x}^{2}+k_{y}^{2})&0&M_{k}\\\
M_{k}&0&A(k_{x}^{2}+k_{y}^{2})&-iv_{\alpha}k_{-}\\\
0&M_{k}&iv_{\alpha}k_{+}&A(k_{x}^{2}+k_{y}^{2})\\\ \end{pmatrix}$
where $M_{k}=M-B(k_{x}^{2}+k_{y}^{2})$ is the hybridization between the top
and bottom surfaces in the different spin direction,
$v_{\alpha}=v_{F}[1+\alpha(k_{x}^{2}+k_{y}^{2})]$ is the second order
correction to the Fermi velocity $v_{F}$ [48], and $k_{\pm}=k_{x}\pm ik_{y}$
is the Dirac dispersion.
$H_{Zeeman}$ = $\begin{pmatrix}\Delta&0&0&0\\\ 0&-\Delta&0&0\\\
0&0&\Delta&0\\\ 0&0&0&-\Delta\\\ \end{pmatrix}$
where $\Delta$ is the strength of Zeeman coupling originated from the magnetic
dopant.
$H_{interface}$ = $\begin{pmatrix}V&0&0&0\\\ 0&V&0&0\\\ 0&0&-V&0\\\
0&0&0&-V\\\ \end{pmatrix}$
where $V$ is potential due to the inversion symmetry breaking (i.e. band-
bending term).
We fit the band structures from the DFT data of the configurations consisting
one Cr-dopant and two-atom Cr-cluster at the interface [Fig S5] to the model
Hamiltonian $H_{total}$ around the $\Gamma$ point. The fitting parameters are
presented in table S1. Based on the fitting parameters, the Chern numbers from
the system were calculated to be -1 by using the Z2pack [49] which tracks the
evolution of the Wannier charge center (WCC) as shown in Fig. S5. Further
confirmation of the Chern insulating state was done by calculating the
transport properties using the ${\sl kwant}$ package [50] as shown in Fig. 6.
The conductance shows a quantized value of $e^{2}/h$ [Fig. 6b], and we also
observe chiral edge currents [Fig. 6c and 6d], with opposite chirality at the
top edge vs the edge at the interface with the AFM substrate. This is
consistent with what we would expect in a QAH phase, characterized by a Chern
number of C = -1. In addition, the results from ${\sl k.p}$ model suggest that
the Cr doped Sb2Te3/Cr2O3 always exhibits a large Zeeman splitting which
overcomes the effect of band bending effects that originates from inversion
asymmetry in the TI layer due to the presence of Cr-dopants and an interface.
Compared to previous studies of other interfaces such as CrI3/Bi2Se3 [28] the
Zeeman coupling is shown to be an order of magnitude larger in this sytem,
while the band gap of 64.5 meV is comparable to the recent experimental
discovery of a high temperature QAHE in MnBi2Te4/Bi2Te3 [51]. Consequently,
the Cr doped Sb2Te3/Cr2O3 represents a promising heterostructure systems which
can facilitate the realization of a high temperature QAHE.

##  IV Conclusions

In conclusion, we have demonstrated that Cr doped Sb2Te3/Cr2O3 is a promising
heterostructure to realize functional high temperature QAHE due to a
combination of i) a large Zeeman energy from Cr dopants, that prefer to
segregate to the interface and cluster, overcoming potential metallicity due
to band bending effect; ii) Fermi level pinning inside the bulk gap due to a
type-I band offset between the TI-films and the wide band gap AFM insulating
substrate and iii) a nontrivial topological ground state protected by a Chern
number of -1. Particularly, a low carrier concentration due to Fermi-level
pinning in the center of the band gap makes it an ideal system to realize QAHE
experimentally. Our study also opens up the possibility to utilize the AFM
Cr2O3 substrate as a heterostructure with the recently discovered magnetic TI
materials with ordered magnetic atoms, such as MnBi2Te4 [52,53] or to grow
multilayers achieving higher-order Chern numbers [20]. Also, growing
conventional s-wave superconductors on top of this heterostructure should also
induce topological superconductivity in these quantized edge-states at the
interface.

## Acknowledgement

A.D.P. was financially supported by the Oak Ridge National Laboratory’s
Laboratory Directed Research and Development project (Project ID 7448, PI:
P.G.). The DFT calculations used resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under Contract No. DE-AC02-05CH11231, via the
Center for Nanophase Materials Sciences, a US Department of Energy Office of
Science User Facility and the Oak Ridge Leadership Computing Facility at the
Oak Ridge National Laboratory, which is supported by the Office of Science of
the U.S. A portion of this research used resources at the Spallation Neutron
Source, a DOE Office of Science User Facility operated by the Oak Ridge
National Laboratory. These measurement are associated with proposal number
22026. C.-Z.C are supported by the Gordon and Betty Moore Foundation’s EPiQS
Initiative (Grant GBMF9063 to C.Z.C.) and the ARO Young Investigator Program
Award (W911NF1810198).

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