# First Principles Study of the Electronic Structure of the Ni2MnIn/InAs and
Ti2MnIn/InSb interfaces

Brett Heischmidt  School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA  Maituo Yu  Department of Materials Science and
Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA  Derek
Dardzinski  Department of Materials Science and Engineering, Carnegie Mellon
University, Pittsburgh, PA 15213, USA  James Etheridge  School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA  Saeed
Moayedpour  Department of Chemistry, Carnegie Mellon University, Pittsburgh,
PA 15213, USA  Vlad S. Pribiag  School of Physics and Astronomy, University of
Minnesota, Minneapolis, MN 55455, USA  Paul A. Crowell  School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA  Noa Marom
Corresponding author<EMAIL_ADDRESS>Department of Materials Science
and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

###### Abstract

We present a first-principles study of the electronic and magnetic properties
of epitaxial interfaces between the Heusler compounds Ti2MnIn and Ni2MnIn and
the III-V semiconductors, InSb and InAs, respectively. We use density
functional theory (DFT) with a machine-learned Hubbard $U$ correction
determined by Bayesian optimization. We evaluate these interfaces for
prospective applications in Majorana-based quantum computing and spintronics.
In both interfaces, states from the Heusler penetrate into the gap of the
semiconductor, decaying within a few atomic layers. The magnetic interactions
at the interface are weak and local in space and energy. Magnetic moments of
less than 0.1 $\mu_{B}$ are induced in the two atomic layers closest to the
interface. The induced spin polarization around the Fermi level of the
semiconductor also decays within a few atomic layers. The decisive factor for
the induced spin polarization around the Fermi level of the semiconductor is
the spin polarization around the Fermi level in the Heusler, rather than the
overall magnetic moment. As a result, the ferrimagnetic narrow-gap
semiconductor Ti2MnIn induces a more significant spin polarization in the InSb
than the ferromagnetic metal Ni2MnIn induces in the InAs. This is explained by
the position of the transition metal $d$ states in the Heusler with respect to
the Fermi level. Based on our results, these interfaces are unlikely to be
useful for Majorana devices but could be of interest for spintronics.

††preprint: APS/123-QED

##  I Introduction

Majorana zero modes (MZMs) have been at the forefront of device physics for
over a decade due to the tantalizing possibility of using them to achieve
topological, fault tolerant quantum computing [1, 2, 3, 4, 5]. Thus far, the
most pursued scheme for MZM realization has been based on a semiconducting
nanowire with strong spin-orbit coupling (SOC) interfaced with a
superconductor [6, 7, 8, 9]. Obtaining Majoranas in semiconductor nanowires
requires a helical gap, which arises from the combined effect of the Rashba
SOC and a sufficient Zeeman interaction [10, 11]. In this system, the spin
channels of the quasi-1D channel’s first mode are shifted due to Rashba SOC.
Then, the degeneracy near $k=0$ is lifted by Zeeman splitting, induced by an
external magnetic field. This scheme further requires superconducting pairing
with a chemical potential inside this Zeeman gap, giving rise to a spinless
p-wave superconducting state [12, 13]. After careful fine tuning, the
(necessary but not sufficient) MZM signature of zero bias conductance peaks
has been observed in such superconductor-semiconductor systems [14, 15, 16].
Recently, these results have come under scrutiny with multiple alternative
explanations [17, 18, 19, 20, 21, 22], in particular, stemming from system
disorder. This has motivated several studies of these materials systems and
their interfaces.

Besides the challenge of mitigating defects and disorder, from a technological
standpoint, utilizing a global external magnetic field further constrains
device geometry due to its necessary orientation orthogonal with the Rashba
field. A recent avenue of research thus attempts to replace the external
magnetic field with local proximity-induced magnetic interactions [23]. If
successful, this could potentially remove restrictions on device structure and
scaling. The main criterion is that the induced Zeeman splitting in the
semiconductor should be on the order of 10 $\mu$eV or larger, in order to
generate a substantial helical gap. A second, but equally important criterion
is that the induced magnetic interactions should be present across the
diameter of the wire, not just at the interface with the ferromagnet because
an interfacial effect alone would not open a helical gap in the bulk of the
wire and would hence be inadequate for Majoranas. Third, the ferromagnet
should be insulating, in order to avoid shunting the current flowing through
the Majorana device and in order to avoid screening the electrostatic gate
acting on the nanowire.Work in this direction has focused primarily on the
ferromagnetic insulator EuS interfaced with InAs and Al [24, 23]. The
observation of zero bias peaks in this system has sparked an active debate
regarding their origin [25, 26, 27, 28, 29]. In particular, experimental [30]
and computational [31] studies of the EuS/InAs interface have suggested that
the proximity-induced magnetic moment in InAs may be too small and too
localized to produce sufficiently strong exchange coupling. Exploration of
alternatives to EuS may advance the understanding of these proposed mechanisms
as they relate to the atomistic structure and electronic properties of the
interface and ultimately lead to the discovery of new materials platforms for
the realization of MZMs.

We have chosen to investigate Heusler compounds as alternative magnetic
materials to EuS. The Heuslers boast a range of intriguing properties. For
example, the Heusler family includes topological superconductors [32], Weyl
materials [33], and skyrmionic materials [34, 35, 36]. Heuslers with Weyl
points have a large anomalous Hall angle [37, 33] and negative
magnetoresistance under alignment of electric and magnetic fields, known as
the chiral anomaly [38, 39, 40, 41, 42]. The sheer number of compounds within
the Heusler family [43] has prompted several computational high-throughput
screening efforts of bulk materials [44, 45, 46, 47, 48], based on density
functional theory (DFT) with computationally efficient semi-local exchange-
correlation functionals.

The Heusler family of materials has also long attracted interest in the
spintronics community, thanks to a high Fermi level spin polarization in
certain cases, including some half-metallic compounds [49, 50, 51, 52]. In the
context of spin injection, a main problem for epitaxially grown ferromagnetic
materials on substrates is misfit dislocations due to lattice mismatch, which
reduces the activation volume [53]. Mitigation of this effect is an ongoing
effort. Further, although there has been some exploration of using non-
ferromagnetic materials for spintronic applications [54], this research avenue
is still nascent, particularly in context of spin injection utilizing Heusler
compounds [55, 56, 57, 58, 59]. This motivates further study of interfaces
between semiconductors and lattice matched Heusler compounds, especially those
with high Fermi-level spin polarization. It has been demonstrated, for
example, that Heusler compounds such as Co2FeSi and Co2MnSi show higher spin-
injection efficiency than Fe [58]. Moreover, spin transport properties of
epitaxial interfaces show marked energy dependence near zero bias [60, 61, 62,
63], suggesting that interface states can have a significant impact on the
performance of spintronic devices. In extreme cases, the sign of the injected
polarization can reverse upon changes in interface conditions induced by
annealing [64]. These observations motivate a study of Heusler compounds
combined with an effort to model realistic interfaces with semiconductors.

In this work, we focus on Heuslers likely to form epitaxial interfaces with
the semiconductors most utilized in the Majorana community, i.e. InAs and InSb
with their relatively heavy elements and thus large SOC. Ni2MnIn and Ti2MnIn
are lattice matched to InAs and InSb to within 0.1% and 0.2%, respectively.
Ni2MnIn is a ferromagnetic metal with a moderate magnetic moment of 4.40
$\mu$B [65]. A DFT study using the generalized gradient approximation (GGA) of
Perdew, Burke, and Ernzerhof (PBE)[66, 67] has reported a Fermi level spin
polarization of 32% [68]). The interface of Ni2MnIn with InAs(001) has been
studied experimentally. The relatively disordered B2 state of Ni2MnIn forms at
low temperatures. L21 ordering (Cu2MnAl-type structure) and increased Curie
temperature can be achieved by increasing the growth temperature or a
subsequent anneal, but at the cost of interface mixing. Point contact Andreev
reflectometry (PCAR) further showed a spin polarization around 17% in a 155 nm
Ni2MnIn layer, although with a relatively rough surface and possibly B2
ordering [69, 70, 71, 72]. PCAR measurements on Ni2MnIn grown on InAs(110)
showed spin polarization of 28% [73, 74]. Ref. [75] used DFT with the local
density approximation (LDA) to the exchange-correlation functional to
calculate the band structure of bulk Ni2MnIn. Band matching considerations
were then used to predict the spin transport at the interface with InAs,
assuming Fermi level pinning slightly above the InAs conduction band minimum.
A free-electron model, described below, gives transmittance (T) along three
high symmetry directions as T[100]=0.75(0.19), T[110]=0.82(0.19), and
T[111]=0.99(0.39) for the minority (majority) spin bands. Subsequently, Ref.
[76] conducted a DFT LDA study of the (100) interface using a periodic
supercell consisting of eight atomic layers of Ni2MnIn and two layers of InAs.
They reported a magnetic moment of 0.01-0.03 $\mu_{B}$ induced in the InAs,
and a possible spin injection channel near the $\Gamma$ point.

To our knowledge, no experimental work has been done on Ti2MnIn to date. We
note that it is possible that Ti2MnIn has not been studied experimentally
because it is metastable. However, metastable phases can be grown by epitaxial
templating on lattice matched substrates (e.g., [77, 78, 79]). The properties
of Ti2MnIn have been studied computationally, using DFT with the PBE
functional [80, 81, 82]. Ti2MnIn has been predicted to be more stable in the
CuHg2Ti-type structure than in the Cu2MnAl-type structure of Ni2MnIn [80]. In
terms of its magnetic properties, Ti2MnIn has been predicted to be a half
metal with a gap of 0.4 eV in one spin channel and 0.04 eV in the other spin
channel, and ferrimagnetic with a small overall magnetic moment of 0.4
$\mu_{B}$ [81]. Ref. [81] and [82] reported similar magnetic moments, whereas
those from Ref. [80] are slightly larger. Notably, neither Ref. [80] nor Ref.
[81] considered SOC. Ref. [82] found that SOC breaks the degeneracy of the
Weyl points. Importantly, no work exists as of yet for the Ti2MnIn/InSb
interface.

In the following, we present a DFT analysis of the electronic structure and
magnetic properties of Ni2MnIn and Ti2MnIn, as well as their interfaces with
InAs and InSb, respectively. The local and semi-local functionals, which have
been used in several studies of Heuslers, cited above, have limited accuracy.
Due to the self-interaction error (SIE) [83, 84, 85], the spurious repulsion
of an electron from itself, (semi-)local functionals severely underestimate
the band gap, to the extent that some semiconductors are erroneously predicted
to be metallic [86, 87, 88]. In addition, the highly localized $d$-states are
destabilized by the SIE, pushing the $d$-bands to higher energies, which may
affect their hybridization with other bands in compounds [89]. As a result of
these limitations, (semi-)local functionals have failed to correctly predict
the half-metallic character and magnetic moments for some Heuslers [90, 91,
92]. Hybrid functionals contain a fraction of exact (Fock) exchange, which
mitigates the effect of the SIE. Hybrid functionals have been shown to provide
a better description of the electronic structure of Heuslers [93, 94].
However, the relatively high computational cost of hybrid functionals impedes
their application for simulations of large interface models with several
hundred atoms. The DFT+U approach provides a good compromise between accuracy
and efficiency by adding a Hubbard U correction to a (semi-)local functional
for the orbitals most affected by SIE [95]. Here, we use DFT+U with $U$ values
determined by Bayesian optimization (BO) with an objective function designed
to reproduce as closely as possible the band structures obtained from a more
accurate hybrid functional [96].

We find that adding the Hubbard $U$ correction improves upon the performance
of semi-local DFT and produces results closer to a hybrid functional.
Importantly, the computational cost of DFT+U(BO) enables simulating large
interface models. For both the Ni2MnIn/InAs and Ti2MnIn/InSb interfaces,
states from the Heusler penetrate into the gap of the semiconductor, decaying
within a few atomic layers. The magnetic interactions at the interface are
localized in space and energy. Weak magnetic moments of less than 0.1
$\mu_{B}$ are induced in the two atomic layers of the semiconductor closest to
the interface. Based on this, we consider these interfaces unlikely to be
useful for Majorana physics, at least via an exchange mechanism. The induced
spin polarization around the Fermi level of the semiconductor depends mainly
on the spin polarization around the Fermi level of the Heusler, rather than
the overall magnetic moment. The ferrimagnetic narrow-gap semiconductor
Ti2MnIn is more spin polarized around the Fermi level than the ferromagnetic
metal Ni2MnIn, owing to the position of the transition metal $d$ states.
Consequently, the Ti2MnIn induces stronger spin polarization around the Fermi
level of the InSb than the Ni2MnIn induces in the InAs. This has implications
for the potential for spin transport through the interface. In particular, in
the Ti2MnIn/InSb interface it may be possible to achieve spin switching by
applying a bias, which makes it of potential relevance for spintronics.

##  II Methods

DFT calculations were performed using the Vienna Ab initio Simulation Package
(VASP) [97] with the projector augmented wave method (PAW) [98, 99]. The
generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)
[66, 67] was employed to describe the exchange-correlation interactions among
electrons with a Hubbard $U$ correction [100] determined by Bayesian
optimization (BO).[96] We note that although the DFT+U method is commonly used
to mitigate the self-interaction error for localized $d$ and $f$ electrons, it
has been found that in some cases, it is necessary to apply a Hubbard $U$
correction to $s$ and $p$ states, in particular when there is a strong
hybridization [101, 95, 102, 103]. We have found that for InAs and InSb, which
the PBE functional erroneously predicts to be metallic, applying a negative
Hubbard $U$ correction to the $p$ orbitals opens a gap [96]. The $U$ values of
-0.5 eV for In-$p$ and -7.5 eV for As-$p$ for InAs and -0.2 eV for In-$p$ and
-6.1 eV for InSb-$p$ , obtained in Ref. [96] were used here (see also Table S1
in the SI).

For Ni2MnIn and Ti2MnIn the optimal $U$ values were found as detailed below.
PBE calculations with and without $U$ for both bulk materials used a kinetic
energy cutoff value of 350 eV and a k-point mesh of $20\times 20\times 20$.
The Heyd, Scuseria, and Enzerhof hybrid functional (HSE) [104, 105] was used
to calculate the bulk band structures of Ni2MnIn and Ti2MnIn to serve as a
reference for BO. For these calculations, the kinetic energy cutoff was set to
350 eV for Ni2MnIn and 400 eV for Ti2MnIn. The primitive cell Brillouin zone
was sampled using the Monkhorst-Pack scheme with a $9\times 9\times 9$ k-point
mesh for Ni2MnIn and an $8\times 8\times 8$ k-point mesh for Ti2MnIn.

For PBE+U(BO) calculations, the $U$ value applied to the $p$ orbitals of In
was kept the same as for InAs for Ni2MnIn and the same as for InSb for
Ti2MnIn. Two-dimensional BO was used to determine the $U$ values applied to
the $d$ orbitals of Ni, Ti, and Mn, as described in Ref. [96]. The
hyperparameters of our BO implementation are the coefficients $\alpha_{1}$ and
$\alpha_{2}$, which assign different weights to the band gap vs. the band
structure in the objective function, the number of valence and conduction
bands used for the calculation of the objective function, $N_{b}$, and the
parameter $\kappa$, which controls the balance between exploration and
exploitation in the upper confidence bound acquisition function. Because
Ni2MnIn is a metal, only the band shape was considered in the optimization,
i.e. $\alpha_{1}$ was set to zero. $\kappa$ was set to 10. A range of $N_{b}$
values was tested with the best results obtained for $N_{b}$=4 and $N_{b}$=8,
which gave similar $U$ values. $N_{b}$=8 was chosen because it provides
sampling both close to the Fermi level and into the $d$ manifold. This
produced values of $U_{eff}^{Ni,d}$ = 7.24 eV and $U_{eff}^{Mn,d}$ = 2.14 eV.
For Ti2MnIn $\alpha_{1}$ = 0.90 and $\alpha_{2}$ = 0.10 were used to balance
the contributions of the band gap and band shape to the objective function.
$N_{b}$ was set to 5 and $\kappa$ was set to 7.5. This produced values of
$U_{eff}^{Ti,d}$ = 1.04 eV and $U_{eff}^{Mn,d}$ = 1.19 eV. Notably, any $U$
values greater than 2 produced no band gap, thus constraining our parameter
space. We further note a strong dependence of the $U$ values on the BO
hyperparameters, as discussed in Ref. [106]. $U_{eff}$ values used and
Bayesian optimization convergence plots are shown in Table S1 and Figure S1,
respectively.

All interface calculations employed the PBE+U(BO) approach, utilizing k-point
meshes of $12\times 12\times 1$ for Ni2MnIn/InAs and $9\times 9\times 1$ for
Ti2MnIn/InSb. The kinetic energy cutoff was 450 eV for Ni2MnIn/InAs and 350 eV
for Ti2MnIn/InSb. For interface calculations, dipole corrections were employed
[107] and symmetry simplifications were turned off. A specific set of mixing
parameters were found to be necessary for the convergence of the Ni2MnIn/InAs
interface: IMIX=1, AMIX=0.02, BMIX=1.0, AMIX-MAG=0.2, and BMIX-MAG=2.0.
Structural relaxation was performed for three layers of the Heusler and six
layers of the semiconductor until the Hellman-Feynman forces acting on ions
were below 1e-03 eV/Å. To describe the van der Waals interactions at the
interface, the Tkatchenko-Scheffler (TS) pairwise dispersion method [108] was
used in the structural relaxation. SOC [109] was applied throughout with the
$z$ spin quantization axis, perpendicular to the interface. Spin-polarized
density of states and band structure plots were produced by projecting the
contributions of the majority and minority spin channels onto the SOC basis
vectors.

In the following, we provide a brief description of the free electron spin
injection analysis introduced in Ref. [75]. The model is based on a free
electron in one dimension encountering a finite step potential, and continuing
past the potential freely but with a decreased momentum. Reflection and
transmission coefficients are given by R=$(k_{1}-k_{2})^{2}/(k_{1}+k_{2})^{2}$
and T=1-R, where the subscripts 1 and 2 indicate the two sides of the step
potential. Ref. [75] applies this to the interface of Ni2MnIn/InAs, with the
step potential occurring as an itinerant electron passes from Ni2MnIn to InAs.
Because the bands in Ni2MnIn and InAs are approximately parabolic near the
Fermi level, the group velocity at the Fermi level goes like the crystal
momentum. The Fermi level in InAs is pinned slightly above the conduction band
minimum in agreement with the experimental observation when InAs is interfaced
with Au [110]. The momentum values when the bands cross the Fermi level are
used as the momenta on either side of the step potential. The corresponding
crystal momenta in Ni2MnIn and InAs give $k_{1}$ and $k_{2}$, respectively.
This model assumes an interface orientation commensurate with the transmission
direction, and a perfectly normal incidence. A benefit of this analysis is in
its simplicity. However, because this analysis is based entirely on the bulk
band structure of both materials, it does not consider the effect of the
atomistic arrangement of the interface on the electronic structure, including
band bending, metal-induced gap states, interface states, and proximity-
induced magnetism.

##  III Results and Discussion

###  III.1 Bulk Ni2MnIn and Ti2MnIn

Figure 1: Ti2MnIn band structures obtained with different DFT functionals: PBE
band structure projected on (a) the two spin channels and (b) atomic orbitals;
HSE band structure projected on (c) the two spin channels and (d) atomic
orbitals; PBE+U(BO) band structure projected on (e) the two spin channels and
(f) atomic orbitals. Red and blue represent spin majority and minority,
respectively. Navy, green, and yellow represent Ti(d), Mn(d), and In(p)
orbitals, respectively.  Figure 2: Ni2MnIn band structures obtained with
different DFT functionals: PBE band structure projected on (a) the two spin
channels and (b) atomic orbitals; HSE band structure projected on (c) the two
spin channels and (d) atomic orbitals; PBE+U(BO) band structure projected on
(e) the two spin channels and (f) atomic orbitals. Red and blue represent spin
majority and minority, respectively. Navy, green, and yellow represent Ni(d),
Mn(d), and In(p) orbitals, respectively.

Figures 1 and 2 show the bulk band structures of Ti2MnIn and Ni2MnIn, obtained
with different functionals. Density of states (DOS) plots are provided in
Figures S2 and S3. Figure 1a,b shows the band structure of Ti2MnIn obtained
with the PBE functional. In agreement with previous results [80, 81, 111], we
find a ferrimagnetic material with a small overall magnetic moment of 0.04
$\mu_{B}$, and asymmetric gaps in the two spin channels. We obtain a gap in
the spin majority band of 0.339 eV, whereas the spin minority band has a gap
of 0.128 eV (additional details are are reported in Table S2). These values
differ slightly from the majority and minority gaps of 0.394 and 0.042 eV,
respectively, reported in [81]. Ref [81] did not consider SOC, although we
find that adding SOC to the PBE calculation has a minor effect on the gaps
(see Table S2 in the SI). Our two calculations differed slightly in the plane-
wave energy cutoff, Brillouin zone sampling, and lattice constant, which
explains the small difference in the results.

Figure 1c,d shows the band structure of Ti2MnIn obtained with the HSE
functional. We find notable differences between PBE and HSE. The HSE
functional inverts the relative sizes of the gaps in the two spin channels,
with the majority gap of 0.237 eV now smaller than the minority gap of 0.414
eV. This result suggests that Ti2MnIn is not a nearly spin-gapless
semiconductor, but rather a conventional semiconductor with a small gap. This
decreases its relevance in the context of spin injection. Additional
differences between PBE and HSE are found in the features of the band
structure. The Mn and Ti $d$-bands in the HSE calculation are shifted farther
away from the Fermi level compared to PBE, with the Mn(d) states affected more
strongly by the addition of exact exchange. As a result of the shift in the
position of the $d$ bands, the local valence band maxima change positions
around the W and K points, and the conduction band minima change from near
degenerate minima near W and K to a clear minimum along $\Gamma$-X. In
addition, the character of the valence band local maximum at the $\Gamma$
point changes from predominantly Ti(d) with PBE to predominantly In(p) with
HSE and the character of the conduction band minimum changes from Mn(d) to
Ti(d). The magnetic moments on individual atoms are larger for HSE compared to
PBE (see Supplementary Table S3), although the total magnetic moment of 0.085
$\mu_{B}$, obtained with HSE, remains relatively close to zero.

Figure 1e,f shows the band structure of Ti2MnIn obtained with PBE+U(BO). The
valence band maximum and conduction band minimum have the same locations as in
the HSE calculation. The relative sizes of the gaps in the two spin channels
resemble the HSE result, although the magnitudes of 0.071 eV in the majority
channel and 0.192 eV in the minority channel are both smaller. The Mn(d) and
Ti(d) bands have an intermediary position between the PBE and HSE results
relative to the Fermi level. In the valence band, the Ti(d) states roughly
center around -0.8 eV, -1.5 eV, and -1.2 eV, respectively, with PBE, HSE, and
PBE+U(BO). The Mn(d) states are centered around -2.5 eV, -4.5 eV, and -3.5 eV,
respectively for PBE, HSE, and PBE+U(BO). The Ti(d) character of the
conduction band minimum reproduces that of HSE. In contrast, at the local
valence band maximum, the prominent Ti(d) character is similar to the PBE
result. The magnetic moments on the individual atoms also have intermediary
values between HSE and PBE (see additional details in Table S3), although the
overall moment of 0.015 $\mu_{B}$ is closer to zero than either.

Figure 2a,b shows the band structure of Ni2MnIn obtained with the PBE
functional. In agreement with previous results, Ni2MnIn is a metallic
ferromagnet [75, 68]. The overall moment of 4.08 $\mu_{B}$ compares favorably
with 3.91 $\mu_{B}$ reported in [112] using the von Barth-Hedin
parameterization of the LDA without SOC [113] and 4.16 $\mu_{B}$ reported in
[75] using the Perdew-Zunger parameterization of the LDA [114] with SOC. These
values agree reasonably well with the experimental value of 4.34 $\mu_{B}$
[115]. The percentage of spin polarization is extracted from the ratio between
the minority and majority DOS at the Fermi level (see panels (a) and (b) of
Figure S5 in the SI). We find that without SOC, the Fermi level spin
polarization, extracted from the DOS, is 32% in agreement with Ref. [68]. SOC
increases the Fermi level spin polarization to 71%.

Figure 2c,d shows the band structure of Ni2MnIn obtained with the HSE
functional. Similar to the case of Ti2MnIn, the most significant difference
between the PBE and HSE band structures is the shift of the Ni(d) and Mn(d)
bands away from the Fermi level in the HSE result. The band character in the
minority (down) channel is generally preserved around the $\Gamma$ point and
near the Fermi level, although the Ni(d) bands now cross the Fermi level in
contrast to the Mn(d) bands in the PBE band structure. The HSE magnetic moment
of 5.81 $\mu_{B}$ is substantially larger than both the PBE (4.08 $\mu_{B}$)
and experimental (4.4 $\mu_{B}$) values. This is consistent with the tendency
of hybrid functionals to favor high-spin states [116, 117]. The PBE band
structure has several minority spin bands around the Fermi level, driving the
polarization toward minority. In contrast, in the HSE band structure many of
the same Ni and Mn $d$ bands are pushed away from the Fermi level, inflecting
the 30% polarization at the Fermi level toward the majority spin channel.

Figure 2e,f shows the band structure of Ni2MnIn obtained with PBE+U(BO).
Similar to the case of Ti2MnIn, the PBE+U(BO) result is intermediate between
PBE and HSE. The Ni(d) bands are partially shifted toward the HSE result,
although the character around the $\Gamma$ point and Ef more closely resembles
that of PBE. The Fermi level position is more centred between the Mn(d) and
Ni(d) bands around $\Gamma$. The magnetic moment of 5.03 $\mu_{B}$ similarly
has an intermediate value between PBE and HSE. The Fermi level spin
polarization is 85% minority, higher than both the PBE and HSE results.

We next comment on the free electron spin injection analysis performed by
Killian and Victora in [75], and follow their model of momentum-accepting
states in InAs. A full analysis is reported in Table S5 and the main results
are summarized here. The transmission coefficient analysis is based on the
band structure, therefore changes to the band structures resulting from the
inclusion of SOC or from using different functionals indirectly affect the
resulting transmission coefficients. Our PBE results reproduce the
transmission coefficients obtained in by Killian and Victora, although we
observe an increase in T[110]min to 0.94, which we attribute to considering
SOC. In the HSE band structure the Ni(d) bands are at the Fermi level,
replacing the Mn(d) bands as those pertinent for this analysis. The crossings
are thus closer to the $\Gamma$ point, giving higher transmission coefficients
of T[100]min=0.96, T[110]min=0.90, and T[111]min=0.98, noting the final result
is from the second crossing in that direction. In the PBE+U(BO) band
structure, the Mn(d) bands cross the Fermi level, as in PBE, but the crossing
is farther away from $\Gamma$. As a result, the transmission coefficients are
lower: T[100]min=0.53, T[110]min=0.71, and T[111]min=0.95. HSE and PBE+U(BO)
generally drive the majority bands away from the Fermi level, stretching the
downward-dispersing bands and pushing them closer to $\Gamma$. As a
consequence, these functionals produce an increased transmission in the
majority spin channel. In particular, both the HSE and PBE+U(BO) yield
transmission coefficients of 0.75 in the [111] direction. These results
diverge significantly from those reported in Ref. [75], where the largest
majority transmission coefficient was 0.39 in the [111] direction. The
minority transmission remains robustly high in all channels even when a
different band is sampled. The majority transmission, however, changes
qualitatively, depending on the DFT functional and may exceed 50%. This has
implications for the extent to which a spin polarized current could be
injected from Ni2MnIn into InAs. This further justifies the need for more
detailed simulations of the interface.

###  III.2 Interface model construction

Figure 3:  Potential energy surface of (a) the Ni2MnIn/InAs, Ni-As interface
and (b) the Ti2MnIn/InSb, TiMn-Sb interface, obtained by shifting the Heusler
film on top of the III-V substrate in the $xy$ plane at the optimal
interfacial distance. The minimum, which corresponds to the most stable
configuration, is marked by a star and the maximum is marked by an X. The
corresponding structures are illustrated on the right.

Slab models of the Ni2MnIn/InAs and Ti2MnIn/InSb interfaces were constructed
with a (001) orientation. The interface models were constructed as periodic
heterostructures with no vacuum region. The bulk lattice constants of 6.058
$\AA$ (ICSD 24518) for InAs and 6.48 $\AA$ for InSb (ICSD 24519) were used
under the assumption that epitaxially grown Heusler films would conform to the
substrate. The InAs and InSb substrates were As and Sb terminated,
respectively. This is experimentally realizable with As(Sb)-rich growth
conditions. For each interface, two terminations are possible for the Heusler.
Ni2MnIn has alternating planes of Ni and MnIn along the [001] direction
whereas Ti2MnIn has alternating planes of TiMn and TiIn. For both terminations
of each material, the optimal interface distance was determined using
dispersion-inclusive DFT. Curves of the energy as a function of the
interfacial distance are shown in Figure S7. The minima for the Ni-As, MnIn-
As, TiMn-Sb, and TiIn-Sb interfaces are obtained at 1.0, 1.8, 1.8, and 1.9
$\AA$, respectively. The optimal interface registry was found by translation
of the Heusler slab with respect to the III-V semiconductor parallel to the
interface for a model comprising six layers of semiconductor and four layers
of Heusler.The results for the Ni2MnIn/InAs, Ni-As interface and the
Ti2MnIn/InSb, TiMn-Sb interface are shown in Figure 3. Results for the other
two interface terminations are provided in Figure S8. At the optimal
configuration the Heusler atoms nestle in the pockets created by the group V
atoms of the semiconductor. Finally, three layers of the Heusler and six
layers of the semiconductor were relaxed on either side of the interface.

Figure 4: Density of states of the middle two layers of a surface slab model
with $L$ atomic layers and a vacuum region of 60 $\AA$, compared to the bulk
density of states for (a) Ti2MnIn using a 11x11x1 k-point grid and (b) Ni2MnIn
using a 9x9x1 k-point grid.

To avoid size dependence due to the quantum size effect, once the interface
configuration was optimized, the thickness of each material was increased
until convergence of the electronic properties to their bulk values was
achieved. Once the minimal thickness required to achieve convergence is
reached, any further changes to the electronic structure with increasing
thickness are insignificant. 47 atomic layers of the semiconductor were used,
based on previous band gap convergence studies [118, 119, 31]. The Heusler
thickness was converged with respect to the density of states and magnetic
moment, which should approach the bulk limit at the center of the slab [106].
For Ti2MnIn, the convergence test was performed with the TiIn termination. We
note that the surface termination should not affect the convergence of the DOS
and magnetic moment at the center of the slab. To avoid spurious states in the
band gap of Ti2MnIn, dangling bonds on the surface were passivated by pseudo-
hydrogen atoms [120]. Charges of 1.0 $e$ and 1.25 $e$ were used to terminate
the surface Ti and In atoms, respectively. A k-point grid of 11x11x1 was used.
As shown in Figure 4a, the density of states in the middle of the Ti2MnIn slab
reaches the bulk limit by 23 layers. For the interface model reported below,
25 layers are used. The density of states for Ni2MnIn, terminated with Ni, as
a function of the number of layers is shown in Figure 4b for a 9x9x1 k-point
mesh. The DOS for 23 and 31 layers are closer to the bulk limit than those for
7 and 15 layers. The interface model reported below comprises 25 layers. The
DOS obtained from a similar calculation using a 21x21x1 k-point mesh, shown in
the SI, is closer to the bulk DOS. However, owing to the high computational
cost of the interface slab calculation with a total of 99 atoms, a 9x9x1
k-point mesh is used below. For both materials, the magnetic moments approach
the bulk values with increasing thickness, as shown in Figure S9. The number
of atomic layers, 47 for the semiconductor and 25 for the Heusler, was chosen
such that the construct was symmetric under a quarter rotation about the layer
in the middle of each material. This ensured that the two interfaces in each
periodic heterostructure were identical.

Figure 5:  Electronic structure of the Ti2MnIn/InSb TiIn-Sb interface. (a)
Illustration of the interface model. (b) Density of states in the InSb as a
function of atomic layer, where the layers are numbered based on distance from
the interface, which is located at zero. (c) Local density of states for
selected layers, indicated by boxes and lines in the same colors in panels (a)
and (b), respectively.  Figure 6:  Electronic structure of the Ti2MnIn/InSb
TiMn-Sb interface. (a) Illustration of the interface model. (b) Density of
states in the InSb as a function of atomic layer, where the layers are
numbered based on distance from the interface, which is located at zero. (c)
Local density of states for selected layers, indicated by boxes and lines in
the same colors in panels (a) and (b), respectively.

###  III.3 Ti2MnIn/InSb

Figures 5 and 6 show the density of states as a function of distance from the
interface for the InSb side of the Ti2MnIn/InSb TiIn-Sb interface and TiMn-Sb
interface, respectively. Similar plots for the Ti2MnIn side of the interface
are provided in Figures S10-S11. For both interface terminations, the InSb
does not exhibit significant band bending at the interface. For the TiIn-Sb
interface, the band gap of the InSb is above the Fermi level in the middle of
the construct, whereas for the TiMn-Sb interface the Fermi level is in the
middle of the InSb gap. Bader analysis [121] shows a charge of 0.028 $e$ per
supercell transferred from Ti2MnIn to InSb for the TiIn-Sb interface and 0.019
$e$ per supercell transferred from Ti2MnIn to InSb for TiMn-Sb interface.
Although this charge transfer is expected to raise the Fermi level in InSb,
comparison to the Ni2MnIn/InAs interface below, shows the charge transfer is
relatively small, thus not significantly affecting the Fermi level position.
Although the Ti2MnIn is not metallic, because its band gap is smaller than
that of InSb, states from the Ti2MnIn penetrate into the InSb, similar to
metal induced gap states (MIGS). These states decay with the distance from the
interface and completely vanish by layers 17-18.

Figure 7: The induced magnetic moments in the III-V semiconductor as function
of number of atomic layers from the interface: (a) InSb in Ti2MnIn/InSb and
(b) InAs in Ni2MnIn/InAs.

Figure 7a shows the magnetic moment in the InSb as a function of distance from
the interface. Weak spin polarization is induced in the first few layers of
InSb. The proximity induced magnetic moments for the TiIn interface of 0.000
$\mu_{B}$ on the first Sb layer and -0.098 $\mu_{B}$ on the second In layer
from the interface are smaller by orders of magnitude than the magnetic
moments of -1.693 $\mu_{B}$ on Ti in the first layer of Ti2MnIn (the In moment
in the first layer of Ti2MnIn is -0.116 $\mu_{B}$). Similarly, for the TiMn
interface, the induced moments in the semiconductor are -0.040 (-0.075)
$\mu_{B}$ for Sb (In), compared to the magnetic moments in the Heusler of
-1.632 (3.627) $\mu_{B}$ for Ti (Mn). Magnetic moment values across the
interface are tabulated in Tables S6-S9. The interface configuration does not
have a significant effect on the magnetic moment induced in the semiconductor,
as the moments in the TiIn (TiMn) layers of Ti2MnIn are ferromagnetically
(antiferromagnetically) coupled and thus have a similar net moment. The
induced magnetic moments found here are of the same order of magnitude
reported previously for the EuS/InAs interface [31] and the Fe/InSb interface
[122].

Figure 8:  Spin polarization, defined as the difference between the majority
DOS and minority DOS [n($\uparrow$) - n($\downarrow$)], as a function of the
distance from the interface for the InSb side of the Ti2MnIn/InSb interface
for (a) the TiIn termination and (b) the TiMn termination.

Spin polarized transport is important for spintronics. In Ref. [64] the
computed spin polarization at the Fermi level was correlated with the
experimentally observed spin polarization of electrons injected from Fe into
GaAs through an interface phase of Fe3Ga. Figure 8 shows the spin
polarization, i.e., the difference between the majority DOS and minority DOS,
as a function of the distance from the interface for the InSb side of the
Ti2MnIn/InSb interface. For this paper, ”majority” spin in the semiconductor
refers to spins with the same orientation as those in the majority channel in
the Heusler compound. For both the TiIn and TiMn terminations spin
polarization is induced in the InSb in the vicinity of the interface,
vanishing by around six layers into the semiconductor. The spin polarized
region spatially overlaps with the region in which states from the Ti2MnIn
penetrate into the InSb (see Figures 5 and 6). Owing to the localized nature
of the proximity induced magnetism, differences are observed between the two
interface terminations. The TiIn interface exhibits majority spin polarization
above the Fermi level and minority spin polarization below the Fermi level.
The TiMn interface exhibits minority spin polarization around the Fermi level.
The spin polarization around the Fermi level in Ti2MnIn is spin down, as shown
in both the primitive cell DOS and the layer-resolved spin polarization plots
in Figure S16. Based on this, we surmise that minority spin transport,
possibly with high transmission, may occur across the TiIn-Sb interface. For
the TiMn-Sb interface, it may be possible apply a bias to tune into a highly
transmissive minority transport regime or into a majority transport regime
with the interface acting as a spin filter, similar to the case of the Fe/GaAs
interface, as discussed in [64]. We note that both interface terminations may
be present in a real system, which could make it impossible to resolve the
differences in their transport behavior.

Figure 9:  Electronic structure of the Ni2MnIn/InAs Ni-As interface. (a)
Illustration of the interface model. (b) Density of states in the InAs as a
function of atomic layer, where the layers are numbered based on distance from
the interface, which is located at zero. (c) Local density of states for
selected layers, indicated by boxes and lines in the same colors in panels (a)
and (b), respectively. The lines indicating the band bending are a guide to
the eye.  Figure 10:  Electronic structure of the Ni2MnIn/InAs MnIn-As
interface. (a) Illustration of the interface model. (b) Density of states in
the InAs as a function of atomic layer, where the layers are numbered based on
distance from the interface, which is located at zero. (c) Local density of
states for selected layers, indicated by boxes and lines in the same colors in
panels (a) and (b), respectively. The lines indicating the band bending are a
guide to the eye.

###  III.4 Ni2MnIn/InAs

Figures 9 and 10 show the density of states as a function of distance from the
interface for the InAs side of the Ni2MnIn/InAs interface for the Ni-As
interface and the MnIn-As interface, respectively. As expected, both
interfaces exhibit band bending in the InAs [110] with the Fermi level
shifting toward the conduction band in the vicinity of the interface. Similar
to the Ti2MnIn/InSb interface, the Heusler termination affects the Fermi level
position and the charge transfer across the interface. The Fermi level lies in
the valence band deep in the InAs when interfaced with Ni, whereas it is in
the middle of the band gap when interfaced with MnIn. Bader analysis shows a
charge of 0.16 $e$ per supercell transferred from Ni2MnIn to InAs for the MnIn
interface and a charge of 0.243 $e$ per supercell transferred from InAs to
Ni2MnIn for the Ni-As interface. The latter indicates a depletion of charge in
the semiconductor and lowering of the Fermi level. MIGS penetrate from the
Ni2MnIn into the InAs. These decay with the distance from the interface and
completely vanish by layers 17-18.

Figure 11: Spin polarization, defined as the difference between the majority
DOS and minority DOS [n($\uparrow$) - n($\downarrow$)], as a function of the
distance from the interface for the InAs side of the Ni2MnIn/InAs interface
for (a) the NiAs termination and (b) the MnIn termination.

The induced magnetic moments in the InAs as a function of the distance from
the interface are shown in Figure 7b. Tabulated values are provided in Tables
S6-S9. More significant differences between the two terminations of the
Heusler are found for the Ni2MnIn/InAs interface than for the Ti2MnIn/InSb
interface. When adjacent to Ni, the first layer of As from the interface holds
a moment of -0.033 $\mu_{B}$. For the MnIn interface, the magnetic moment of
-0.002 $\mu_{B}$ on the first As layer is close to zero. The situation is
reversed for the second In layer from the interface. For the MnIn interface
the second In layer has a magnetic moment of 0.031 $\mu_{B}$, whereas the In
layer closest to the Ni interface has a nearly zero moment of 0.008 $\mu_{B}$.
The significant differences between the two terminations may be explained by
the relatively high magnetic moment of about 4 $\mu_{B}$ on the Mn atoms in
the Heusler compared to the much smaller magnetic moments of 0.024 $\mu_{B}$
and -0.114 $\mu_{B}$ on the first layer of Ni.

Figure 11 shows the spin polarization , i.e., the difference between the
majority DOS and minority DOS, as a function of the distance from the
interface for the InAs side of the Ni2MnIn/InAs interface. Both interface
terminations exhibit a slight spin polarization at the Fermi level next to the
interface, which decays within about five layers, or a couple of nanometers.
The spin polarized region overlaps with with the location of the MIGS. For
both Ni2MnIn terminations majority spin polarization is found around the Fermi
level in the InAs. The spin polarization for the Heusler side of this
interface, provided in Figure S16, shows minority spin polarization around the
Fermi level, which is opposite to that found in the InAs. In correspondence
with the discussion of the Fe/GaAs interface in [64], this interface may act
as a spin filter, rotating the spins as the interface is traversed.
Surprisingly, for both Ni2MnIn terminations, the magnetic behavior induced in
the first few layers of InAs is weaker than in the InSb, both in terms of the
magnetic moment and the spin polarization, despite the ferromagnetic nature of
Ni2MnIn compared to the ferrimagnetic Ti2MnIn. The induced magnetic moments on
atoms may be explained by the highly localized nature of the magnetic
interactions at the interface. The difference in the induced spin polarization
around the Fermi level may be explained by the fact that despite being overall
ferromagnetic, the Ni2MnIn is weakly spin polarized around the Fermi level
(see Figure S16) because the Mn $d$ states are relatively far from the Fermi
level (see Figure 2), whereas some of the Mn $d$ states in Ti2MnIn are in the
vicinity of the Fermi level. This demonstrates the importance of considering
the atomistic details of the interface, not just the bulk properties of the
isolated materials.

##  IV Conclusion

We have presented a first-principles study of epitaxial Heusler/III-V
interfaces of Ti2MnIn/InSb and Ni2MnIn/InAs using DFT with a machine-learned
Hubbard U correction determined by Bayesian optimization. For the bulk
materials, adding the Hubbard U correction improves upon the performance of
the semilocal PBE functional and produces band structures closer to the HSE
hybrid functional. The main effect of the addition of exact exchange in the
hybrid functional and of the Hubbard U correction is a shift of the transition
metal $d$ bands away from the Fermi level. This has qualitative implications
for the electronic structure of Ti2MnIn. With PBE it has a near zero gap in
the minority spin channel, however with HSE and PBE+U(BO) it is a conventional
narrow-gap semiconductor. Ni2MnIn remains a ferromagnetic metal but HSE and
PBE+U(BO) change the character and curvature of the states around the Fermi
level, which may have implications for transport through the interface.
Importantly, the computational cost of PBE+U(BO) enables us to study large
interface models, which would not be feasible with a hybrid functional.

Both the Ti2MnIn/InSb and Ni2MnIn/InAs interfaces exhibit states penetrating
from the Heusler into the gap of the semiconductor, which decay within 18
atomic layers. As expected from branching point theory [123, 124], the InAs
shows significant band bending whereas the InSb does not. Both interfaces
exhibit weak and localized proximity induced magnetism in the semiconductor.
The magnetic moments induced on the atoms in the second layer of the
semiconductor from the interface are less than 0.1 $\mu_{B}$. This is of the
same order of magnitude found previously for the for the EuS/InAs interface
[31] and the Fe/InSb interface [122]. For both interfaces, spin polarization,
i.e., a difference between the majority and minority DOS, is induced around
the Fermi level of the semiconductor. The spin polarized region spatially
overlaps with the MIGS and decays within a few atomic layers. Counter-
intuitively, although Ni2MnIn is ferromagnetic and Ti2MnIn is ferrimagnetic,
the magnetism induced in the first few layers of InAs is weaker than in the
InSb, both in terms of the magnetic moment and the spin polarization. This may
be explained by the fact that, despite being overall ferromagnetic, the
Ni2MnIn is weakly spin polarized around the Fermi level because the Mn $d$
states are relatively far from the Fermi level. In comparison, some of the Mn
$d$ states in Ti2MnIn are in the vicinity of the Fermi level, producing a more
significant spin polarization in that energy range.

For the Ni2MnIn/InAs interface, the InAs is slightly spin polarized at the
Fermi level with majority polarization, opposite to the minority polarization
around the Fermi level in the Ni2MnIn. This means that the interface may act
as a spin polarizer, similar to the Fe/GaAs interface [64]. For the
Ti2MnIn/InSb interface the sign of the induced spin polarization depends on
the termination of the Heusler. For the TiMn termination, minority spin
polarity is induced in the InSb, which is the same as the polarization of the
Ti2MnIn around the Fermi level. This means the interface may be highly
transmissive. For the TiIn termination the induced polarity is majority above
the Fermi level and minority below the Fermi level. As a result, it may be
possible to switch the spin polarization of a current going through the
interface by applying a voltage bias. We note that it is possible that both
terminations would be present simultaneously in a realistic system, depending
on whether the termination can be controlled by modifying the growth
conditions. Our results demonstrate that owing to the localized nature of the
magnetic interactions at the interface, it is important to consider the
atomistic details of the structure and the electronic structure around the
Fermi level, rather than the overall properties of the bulk materials.

In the context of Majorana devices, the criteria described in the introduction
are not met. The few nanometer range of the induced magnetism in the
semiconductor is smaller by two orders of magnitude than the typical InAs and
InSb nanowire diameter of around 100 nm. We interpret this as signifying that
the weak and local magnetic interactions at the interface are likely
insufficient to drive proximity-induced magnetism over the body of the wire.
These results are in agreement with previous work on EuS/InAs [31]. Moreover,
even if the relevant states to produce a Majorana zero mode were confined to
the region of the wire that is close to the interface, the presence of MIGS,
which render the semiconductor gapless, may preclude tuning the wire into a
single subband limit [10]. Therefore, we conclude that both the Ti2MnIn/InSb
and Ni2MnIn/InAs interfaces are of limited usefulness in the context of
Majorana-based quantum computing devices. Nevertheless, these interfaces and
other Heusler/III-V interfaces may be of interest for spintronics because they
offer the possibility of manipulating spin currents through ”band structure
engineering” by controlling the local structure and composition of the
interface. In particular, our findings demonstrate that hitherto overlooked
ferrimagnetic Heusler materials may still be capable of inducing spin
polarization in a semiconductor, depending on the extent of spin polarization
at the Fermi level. This calls for further computational and experimental
exploration of Heusler/semiconductor interfaces.

##  V  Acknowledgments

We thank Chris Palmstrøm from the University of California Santa Barbara for
helpful discussions. This research was funded by the Department of Energy
through grant DE-SC0019274. This research used resources of the National
Energy Research Scientific Computing Center (NERSC), a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under contract no. DE-AC02-05CH11231.

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