# Self-consistent quasi-particle $GW$ and hybrid functional calculations for
Al/InAs/Al heterojunctions: band offset and spin-orbit coupling effects

H. Ness  Department of Physics, Faculty of Natural and Mathematical Sciences,
King’s College London, Strand, London WC2R 2LS, UK  F. Corsetti  Microsoft
Azure Quantum, 2800 Lyngby, Denmark  D. Pashov  Department of Physics, Faculty
of Natural and Mathematical Sciences, King’s College London, Strand, London
WC2R 2LS, UK  G. W. Winkler  Microsoft Azure Quantum, Goleta, California
93111, USA  M. van Schilfgaarde  Department of Physics, Faculty of Natural and
Mathematical Sciences, King’s College London, Strand, London WC2R 2LS, UK
National Renewable Energy Laboratory, Golden, Colorado 80401, USA  R. M.
Lutchyn  Microsoft Azure Quantum, Goleta, California 93111, USA

###### Abstract

The electronic structure of surfaces and interfaces plays a key role in the
properties of quantum devices. Here, we study the electronic structure of
realistic Al/InAs/Al heterojunctions using a combination of density functional
theory (DFT) with hybrid functionals and state-of-the-art quasi-particle $GW$
(QS$GW$) calculations. We find a good agreement between QS$GW$ calculations
and hybrid functional calculations which themselves compare favourably well
with ARPES experiments. Our study confirm the need of well controlled quality
of the interfaces to obtain the needed properties of InAs/Al heterojunctions.
A detailed analysis of the effects of spin-orbit coupling on the spin-
splitting of the electronic states show a linear scaling in $k$-space, related
to the two-dimensional nature of some interface states. The good agreement by
QS$GW$ and hybrid functional calculations open the door towards trust-able use
of an effective approximation to QS$GW$ for studying very large
heterojunctions.

electronic structure of surfaces and interfaces, density functional theory,
hybrid functionals and many-body perturbation theory, quantum materials, III–V
semiconductors

###### pacs:

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##  I Introduction

Because of their unique combination of material parameters (i.e. large spin-
orbit coupling, small effective mass, large Lande g-factor), narrow-gap III-V
semiconductors (such as InAs or InSb) have generated considerable interest in
many technological applications.

Recently, these materials have been central to the experimental realisation of
the so-called Majorana zero modes Sau  _et al._ (2010); Alicea (2010); Lutchyn
_et al._ (2010); Oreg  _et al._ (2010); Alicea (2012); Leijnse and Flensberg
(2012); Beenakker (2013); S. Das Sarma and Nayak (2015); Aguado (2017);
Lutchyn  _et al._ (2018). In this devices, the main goal is to develop
topological $p$-wave superconductivity at the interface of a conventional
semiconductor and an $s$-wave superconductor. An exceptionally good control of
the interface properties is needed to realise topological superconducting
phases and to manipulate Majorana zero modes which are the key ingredient in
topological quantum computation proposals Nayak  _et al._ (2008); S. Das Sarma
and Nayak (2015); Lutchyn  _et al._ (2018); Aghaee and et. al. (2024). The
hybrid semiconductor-superconductor Majorana devices are required to have a
large g-factor, strong Rashba spin-orbit coupling and significant proximity-
induced superconducting gap. Recently, the proximity-induced superconductivity
has been studied in devices made of a semiconductor nanowire in contact with a
superconductor, including Al/InAs Krogstrup  _et al._ (2015); Suominen  _et
al._ (2017); Nichele  _et al._ (2017); Vaitiekénas  _et al._ (2018); Matsuo
_et al._ (2020); Ménard  _et al._ (2020); Vaitiekenas  _et al._ (2020),
Al/InSb de Moor  _et al._ (2018); Shen  _et al._ (2018); Anselmetti  _et al._
(2019); Op het Veld and et al. (2020) Pb/InAs Kanne  _et al._ (2021), and
Sn/InSb Pendharkar  _et al._ (2021). High-quality superconductor/semiconductor
interfaces (i.e. uniform and transparent) are required to optimise the
topological gaps in these heterostructures.

The geometry of the interface may give rise to (desirable or undesirable)
interface states, it may alter the band bending and band alignment, or affect
the magnitude of the proximity-induced gap and of the spin-orbit coupling.
Understanding the resulting surface/interface states and Fermi-level pinning
is important for engineering appropriate interface Hamiltonian and realising
topological superconductivity hosting Majoranas.

Band bending and surface states have been observed by angle-resolved
photoemission spectroscopy (ARPES)N. Tomaszewska and Kolodziej (2015);
Schuwalow  _et al._ (2021); Yang  _et al._ (2022) and scanning tunneling
microscopy and spectroscopy (STM/STS)Batabyal  _et al._ (2023). First
principles simulations based on density functional theory (DFT) can help
interpret experiments and resolve the effects of the interfaces. DFT studies
of InAs and InSb surfaces and interfaces have been limited because local
(local density approximation LDA) and semi-local exchange-correlation
functionals severely underestimate the band gap to the limiting point where it
reduces to zero Krukau  _et al._ (2006a). More accurate methods involving
quasi-particle self-consistent $GW$ (QS _GW_) approaches or hybrid functionals
provide results much closer to the experimental (bulk) gap (0.42 eV for bulk
InAs).

In this paper, we present calculations of realistic Al(111)/InAs(001)
heterojunctions using a QSGW method implemented in the Questaal package. The
QS _GW_ results are also compared with hybrid functionals DFT calculations. We
focus our attention on the effects of “disorder” (using numerical
“experiments”) on the electronic structure of realistic InAs/Al interfaces
described at the atomic scale. The disorder we consider is coming from: (i)
atomic relaxations (i.e. the atoms at the InAs/Al interfaces do not rest at
their correspond bulk atomic positions), (ii) substitution disorder which
mimics in a simple way potential atomic diffusion at the interface, and (iii)
rescaling the spin-orbit coupling strength on some atoms which mimics the
presence of some external electric fields at the interfaces.

The paper is organised as follows: In Section II, we present the InAs/Al
system we considered and the two software packages used for the electronic
structure calculations, namely the Questaal and QuantumATK packages. The
results of our calculations are shown and analysed in Section III, where we
extract the profiles of the valence band maxima (VBM) and conduction band
minim (CBm) along the InAs/Al heterojunction and study in details the effects
of spin-orbit coupling (SOC) on some specific bands. Conclusions are presented
in Section IV. Additional information are provided in the appendices: about
the implementation of the Questaal code on GPUs in Appendix A, about local
density of states, and bulk versus heterojunctions bands in subsequent
appendices.

##  II Calculations

First principles electronic structure calculations have been performed using
two different packages: the Questaal package que  and the pseudopotential
QuantumATK package Smidstrup  _et al._ (2019).

Questaal is an all-electron method, with an augmented wave basis consisting of
partial waves inside augmentation spheres based on the linear muffin-tin
orbital (LMTO) technique Pashov  _et al._ (2020). It includes conventional
DFT-based calculations, as well as many-body perturbation theory, especially
with its implementation of a quasi-particle self-consistent _GW_ (QS _GW_)
approach Faleev  _et al._ (2004); Kotani  _et al._ (2007).

We have considered InAs/Al heterojunctions for which the interface between the
two materials is built from the (001) surface for InAs and from the (111)
surface for Al, with As-terminated InAs surfaces in direct contact with the Al
surface (as suggested by the experiments in Schuwalow  _et al._ (2021)). We
took a low temperature lattice parameter of $a_{0}=6.06$ Å for InAl, and a
(001) surface supercell based on the two following (001) surface vectors
$u_{1}=[2,0]a_{0}$ and $u_{2}=[-1,3]a_{0}$. For this (001) supercell, there
are 6 atoms in each In (As) atomic planes perpendicular to the $z$-direction,
see Fig. 1. For this supercell, one can match the Al(111) surface rather well,
with a slight stretch (of 3%) in the $u_{2}$ direction, using a bulk lattice
parameter of $a_{\text{Al}}=4.05$ Å. Then, each Al atomic plane parallel to
the InAs/Al interface contains 15 atoms.

In order to minimise the computational cost, more specifically for the QS _GW_
calculations, we have considered the minimal possible size for the junctions.
We have found that to be able to keep the bulk-like character for the
electronic structure in the middle of the InAs slab, one needs to go beyond a
few layers of InAl: typically for 6 (7) atomic planes of In (As) (and beyond)
we recover the bulk-like density of states for the In (As) atoms in the center
of the InAs slab. As Al is a metal with a shorter screening length, fewer
atomic layers are needed (typically 4 atomic (111) planes are enough) to
obtain a bulk-like density of states in the central atomic layers.

Relaxation of the atomic positions have been performed within DFT-LDA. We have
allowed the atoms in the Al atomic layers next to the interfaces, and the
atoms of As and In in the two outmost atomic layers close to the InAs/Al
interfaces to relax until the force components are below 10 mRy/bohr (257
meV/Å).

We did not impose any symmetry during the atomic relaxation. Therefore the two
InAs/Al interfaces of the supercell are not equivalent. This allows us to
minimise the possible existence of unwanted electronic states that may have
arisen due to size and coupling effects between two perfectly symmetric
interfaces. To some extend, this can be also seen as a simplified case of
“geometric” disorder.

Figure 1:  (Top) Ball-and-stick representation of the relaxed
InAs(001)/Al(111) supercell, for different number of InAs layers. In atoms are
shown in grey, As atoms in yellow and Al atoms in purple. (Bottom) Side view
of the InAs(001)/Al(111) atomic planes. The largest supercell contains 138
atoms with 4, 6, 7 atomic planes of In, As, Al respectively. Each plane
contains either 6 atoms of In (As) or 15 atoms of Al.

In DFT-LDA and even in $GW$ calculations, band gaps are often underestimated
(LDA) or overestimated ($GW$). Indeed they should be because the RPA screened
coulomb interaction $W$ is not sufficiently screened. Improvement of $W$ by
the addition of ladder diagrams indeed does improve the gaps Cunningham  _et
al._ (2023); Kutepov (2017). In most of the cases, the dielectric constant of
semiconductors and insulators is about 80% of the experimental value. This is
because the ladder diagrams are missing in the RPA. However, these higher
order diagrams are computationally costly, and here we adopt a simpler
approach here. We have found that scaling the dielectric constant by 0.8, or
alternatively using a hybrid of 80% QS$GW$ and 20% LDA, we can mimic the
effect of the ladders. This eliminates most of the errors.

Hence, we have use hybrid of LDA and QS$GW$ functionals, $\Sigma^{\rm
scaled}=\Sigma_{\rm QSGW}\times 0.8+V_{\rm xc}^{\rm LDA}\times 0.2$, in the
calculations of our band structures.

The system we considered contains 138 atoms ($4\times 15$ atoms of Al,
$6\times 6$ of In and $7\times 6$ of As). To our knowledge, this is one of the
first time that self-consistent QS _GW_ calculations have been performed for
such a “large” system. The Questaal package was re-developed and optimised to
take advantage of GPU-based computing on a multi-petaflop modular
supercomputer, see Appendix A.

We are interested in determining the profile of the valence band maxima (VBM)
and of the conduction band minima (CBm) along the InAs/Al heterojunction.
There are different ways to find such a profile, for example by considering
the electrostatic potential of the heterojunction, or by considering the
change in energy position of deep electronic levels of the junction in
comparison to their bulk equivalent. Obviously, the profile of the VBM or CBm
obtained from such atomic scale systems (for example, the thickness of the
InAs slab in the 138 atom supercell is $\sim 26.5$ Å) will not reflect the
band bending of Schottky barriers expected from a continuum model of the
semiconductor/metal contact described on the micron scale. However our atomic
scale calculations incorporate the more local effects of the InAs/Al
interfaces against the bulk property of the materials.

To evaluate the profile of the valence (conduction) band maxima (minima), we
extract the energy position of the deep electronic levels on each In atom
(deep $d$-orbital) and each As atom (deep $s$-orbital). Assuming a rigid
energy shift of these deep electronic levels relative to the Fermi level
$E_{\text{F}}$ in both the bulk and the heterojunction systems, we can
determine the profile (averaged over the number of atoms in each atomic layer)
of the VBM (along the main direction of the junction) relative to the exact
QS$GW$ Fermi level of the junction. The profile of the CBm is obtained from a
rigid shift of the VBM by the bulk QS$GW$ band gap (0.47 eV in the present
case).

It is important to note that all the QS$GW$ calculations were performed in the
presence of spin-orbit coupling (SOC). Orbitals with $s$-like character are
not affected by the presence of SOC. However the $d$-orbitals are split by the
SOC. For bulk-like environment, the $d$-orbitals are split into two subsets
according to the crystal symmetry. For In atoms close the Al/InAs interfaces
the symmetry is reduced (further reduced by the atomic relaxations), and
different energy shifts occur for the different $d$-orbitals on these atoms.
There is more “fluctuation” of the energy shift for these atoms in comparison
to bulk-like In atoms in the center of the InAs slab.

As a final comparison, we also perform calculations on the same system using a
different methodology for correcting the band gap problem: hybrid functionals.
Traditional hybrid functionals which use a fixed global mixing fraction
$\alpha$ of Fock exchange can show limitations in the case of inhomogeneous
interface systems. More specifically if the different materials require
different values of $\alpha$ to recover the correct bulk electronic structure
and band gap. The situation is even more severe for the present case of
semiconductor/metal interfaces.

To overcome this problem, we build on a recently developed scheme for a local
(i.e., spatially-varying) mixing fraction, based on an estimator of the local
dielectric constant defined as a functional of the electronic density Marques
_et al._ (2011); Borlido  _et al._ (2018). In order to deal with the metallic
region in our system, we introduce a second, metallic estimator, which
determines locally if the material is a metal, and, if so, one sets $\alpha$
to zero. We apply this scheme to the HSE06 functional Krukau  _et al._
(2006b). We shall refer to these calculations as HSE06+DDH (dielectric-
dependent hybrid).

The HSE06+DDH calculations are performed using the QuantumATK package
Smidstrup  _et al._ (2019) (version T-2022.03) within a pseudopotential and
linear combination of atomic orbitals (LCAO) formalism. The calculations are
carried out with a spin-polarized non-collinear Hamiltonian. We use norm-
conserving pseudopotentials from the PseudoDojo van Setten  _et al._ (2018)
fully relativistic set and the medium basis set (LCAO-MSmidstrup  _et al._
(2019)) from QuantumATK. The auxiliary density matrix method (ADMM) Guidon
_et al._ (2010) is used to speed up the calculation of the exchange matrix.

The determination of the VBM/CBm is performed in the same way as for the
QS$GW$ calculations, i.e. by extracting the energy position of the core levels
relative to $E_{\text{F}}$. However, due to the configuration of the
pseudopotentials, we use the semi-core $d$-orbitals for both In and As. The
bulk InAs band gap obtained with HSE06+DDH is 0.47 eV, in agreement with
QS$GW$.

##  III Results

###  III.1 Band alignment

Figure 2 shows the profile of the VBM in the InAl/Al heterojunctions. It
corresponds to a shallow parabolic-like curve where the VBM is higher close to
the InAs/Al interfaces than in the bulk-like part of InAs. Such a profile does
not directly compare with conventional band-bending in Schottky barriers, the
latter occurs on much larger length scales ($\sim\mu\text{m}$) than the scale
corresponding to our atomic-scale calculations. However, the inflections of
the VBM reflects the effect of the interfaces against the bulk, most certainly
due to the presence of interface electric dipoles.

The dispersion in the energy shifts displayed by the symbols (red circles for
the In deep $d$-orbital, green up-triangles for the As deep $s$-orbital)
reflects that each atom in a given atomic layer are different. This is mostly
true for the atoms near the InAs/Al interfaces which have been allowed to
relax. Such a dispersion is minimal for the atoms in the center of the InAs
slab where their position correspond to bulk unrelaxed atomic positions.

Figure 2:  Profile of the local valence band maximum (VBM) for the InAs/Al
junction made of 138 atoms. Symbols represent the different shifted energy
levels (deep In-$d$ and As-$s$ orbitals) of the In and As atoms located in
each atomic plane along the main axis of the junction. Lines correspond to the
averaged value of the energy shift in each atomic plane. The black dashed and
dashed-dotted lines correspond to the QS$GW$ and LDA VBM respectively. The red
dashed line is the QS$GW$ conduction band minima (CBm), i.e. the VBM shifted
by the QS$GW$ bulk band gap. Blue line presents the Fermi level $E_{\text{F}}$
position. Both QS$GW$ and LDA have been shifted to correspond to this energy
reference $E_{\text{F}}=0$. The QS$GW$ CBm lies $\sim$80 meV below
$E_{\text{F}}$ in the center of the InAs slab.

The QS$GW$ calculations for bulk InAs provides a band gap 0.47 eV, close to
the experimental band gap of 0.42 eV at low temperature. From that value, we
can see that the CBm (obtained from rigid shift of the VBM) crosses the Fermi
level of the junction. The CBm lies below $E_{\text{F}}$, by an amount of
$\sim$80 meV, in the “bulk” part of the InAs slab.

This prediction of an accumulation layer in InAs is in agreement with ARPES
measurements Schuwalow  _et al._ (2021), although the band offset extracted
from experiment is larger than what is seen in our simulation cell. For
quantitative agreement, convergence to larger cells is probably needed.

For this, we turn to our hybrid functional calculations using the HSE06+DDH
method. This method represents a more empirical and less accurate approach
than QS$GW$, but also significantly less demanding of computational resources,
and therefore potentially able to scale to larger systems. Figure 3 shows the
comparison of the local band edges calculated with QS$GW$ and HSE06+DDH. The
overall shape of the band edges is well reproduced by HSE06+DDH, and there is
an excellent quantitative agreement in the bulk of the semiconductor (within
two layers of the interface). As an additional check, we have also compared
the LDA band edges calculated with the package Questaal (shown in Figure 2)
with QuantumATK (not shown) and recovered a similar good agreement. Therefore,
the HSE06+DDH method provides an effective approximation to the full QS$GW$
for this system, and might be used to explore much larger interface cells or
multiple different interface configurations at a lower computational cost.

Figure 3:  Profile of the local VBM and CBm for the InAs/Al junction obtained
from QS$GW$ (same values as in Figure 2) and HSE06+DDH. The agreement between
the two methods is particularly good.

In order to get a better understanding of the effects of disorder at the
InAs/Al interfaces on the energy level alignment, we have considered the
following “numerical experiment”.

We simulate the possible atomic diffusion at the interfaces by substituting
some atoms by others. From the valence properties of In, As and Al, it appears
reasonable to envisage substitutional disorder between In and Al atoms. We
have therefore considered InAs/Al interfaces for which some In atoms (in the
atomic layers closest to the interfaces) are replaced by Al atoms.

There are many possible combinations to realise such substitutions, and we
have considered only a few of them. We started by replacing only one In atom
by one Al atom in the In atomic layer located the closest to the the right
InAs/Al interface, see labels for the atomic planes in Figure 4. We have
performed calculations for only two cases over the six possible cases of one
atom substitution. The results for the VBM profile for one of this case is
shown in Figure 4. We have also considered one case in which two In atoms are
swapped by two Al atoms and we have found similar trends for the profile of
the VBM. Our calculations indicate that the VBM is pushed down, by a further
$\sim$100 meV, to lower energy in the case of the “dirty” interfaces compared
to the case of “perfect” interfaces.

Figure 4:  Profile of the local valence band maximum (VBM) for the InAs/Al
junction with similar notations as in Fig 2. Comparison between perfect and
“disordered” interfaces. One atom of In is replaced by one Al atom in the In
atomic plane closest to the right InAs/Al interface (see label InAl on the
horizontal axis). All calculations are performed with QS$GW$. The VBM/CBm are
pushed down to lower energy in the case of the “dirty” interface, i.e. compare
dashed-lines with dotted-lines (black for VBM, red for CBm). “Interface
disorder” seems to push the CBm down by a further $\sim$ 100 meV. A similar
behaviour has also been obtained in two different cases of one In $\leftarrow$
Al atom substitution (the Al atom is located at a different site in the
corresponding In atomic plane), as well as for a case of 2 In $\leftarrow$ Al
atoms substitution in the corresponding In atomic plane.

###  III.2 Spin-orbit coupling effects

The presence of Rashba-like spin-orbit coupling (SOC) in narrow-gap InAs
semiconductors is one of the central ingredients for inducing superconducting
property by proximity of an $s$-wave superconductor like Al. Once
superconducting, a InAs nanowire can eventually hosts a pair of Majorana state
at each of its ends where the superconducting order parameter vanishes.

In the previous section, we have studied how band alignment in InAs deviates
from pure bulk to InAs/Al interfaces, including some form of disorder of the
interfaces.

We now consider the possibility of another type of disorder and its effects on
the band structure of the InAs/Al junction. For that we now consider the
following numerical “experiment”: the strength of the SOC, on some atoms in
the system, is rescaled to larger values. The SOC rescaling is applied on
either all the In and As atoms or only on the In and As atoms close to the
InAs/Al interfaces (i.e. In atoms labelled $z=12,z=27$ and As atoms labelled
$z=0.3,z=10$ in panel (d) for Fig. 7). Note that a light element like Al does
not have strong SOC and rescaling the SOC on Al is not relevant. The increase
of the SOC can be seen as an indirect effect of the presence of an extra
external electric field (perpendicular to the InAs/Al interface in the case of
Rashba-like SOC) due to gating or other effects not taken into account in our
model of the InAs/Al heterojunction.

In Appendix B.3, we show how the band structure of the InAs/Al junction
differs strongly from the bulk InAs bands due to the coupling to the metallic
As states. Most of the bands in the junction come from a mixture of all In, As
and Al orbitals. However, we have identified an equivalent to the bottom of
the bulk InAs conduction band for the case of the heterojunction. And we
calculate how the energy difference $\Delta E(k)$ between the SOC spin-
splitted bands (around the $\Gamma$ point) varies with the rescaling to the
SOC.

(a) (b) (c) (d)

Figure 5:  QS$GW$ band structure of the InAs/Al heterojunction made of 138
atoms, for different rescaling of the SOC. Panel (a), (b), (c) and (d)
corresponds to rescaling the original SOC by $\times 2,\times 3,\times 4$ and
$\times 5$ respectively. Focusing on the bands around the $\Gamma$ point and
at $E\sim E_{\text{F}}-0.05$, one can see the spin-splitted bands (due to SOC)
along the $\Gamma-Qy$ direction. The splitting between the two bands increases
with increasing rescaling of the SOC. The band encircled in red corresponds to
the bottom of the bulk InAs conduction band in the case of the junction (see
Appendix B.3).

Figure 6 shows the energy difference $\Delta E(k)$ of the spin-splitted bands
around the $\Gamma$ point, bands encircled in red in Fig. 5, versus the
$k$-vector along the $\Gamma-Qy$ direction. The spin splitting is more
important (for small $k$ values away from $\Gamma$) when the SOC rescaling in
applied to all In and As atoms, instead of only on the interface In, As atoms.
However, it is clear that, in both cases, the spin splitting $\Delta E(k)$ is
linear in $k$ (for $k/Qy<0.1$), which is most probably the signature of the 2D
like character of the corresponding states parallel to the InAs/Al interface
Luo  _et al._ (2010).

(a) (b) (c)

Figure 6:  Energy difference $\Delta E(k)$ of the spin-splitted bands around
the $\Gamma$ point in the $\Gamma-Qy$ direction. The rescaling (soscl) of the
spin-orbit coupling is applied to all In and As atoms in panel (a), to only
the In and As interface atoms in panel (b). In both cases, the spin splitting
is linear in $k$ for $k/Qy<0.1$ around each band crossing, i.e. $\Delta
E(k)\equiv\alpha k$. Panel (c): dependence of the slope of $\Delta
E(k)\equiv\alpha k$ upon the rescaling soscl of the spin-orbit coupling.

It is also interesting to check how the linear dependence of the spin
splitting varies with the rescaling of the SOC. We determine the slope of the
linear relation $\Delta E(k)\equiv\alpha k$ for small $k$ values (for
$k/Qy<0.05$). The dependence of $\alpha(\text{soscl})$ on the SOC rescaling is
shown in panel (c) of Figure 6. We obtain a linear dependence on SOC rescaling
when the rescaling is applied to all In and As atoms, and a sublinear
dependence when the rescaling is only applied to the interface In,As atoms,
indicating different screening effects on the local SOC rescaling.

##  IV Conclusion

We have studied the electronic structure of realistic Al/InAs/Al
heterojunctions using a combination DFT with hybrid functionals and state-of-
the-art QS$GW$ calculations. The InAs/Al heterojunctions we considered are
central to superconducting induced properties in InAs and to the design of
topological quantum computation platforms. The InAs/Al heterojunctions are
described at the atomic level and include atomic relaxations at the InAs/Al
interfaces. Our study confirm the need of well controlled quality of the
interfaces to obtain the needed properties of InAs/Al heterojunctions. The
local band alignment (i.e. top of VB, bottom of CB) obtained from QS$GW$ for
semiconductor/metal interfaces can be well reproduced using dielectric-
dependent hybrid functional DFT with the novel metallic estimator which
automatically switches off the Fock exchange within the bulk of the metal. The
prediction of an accumulation layer for InAs/Al is in agreement with
experimental evidence. The HSE06+DDH method appears to provide an effective
approximation to the full QS$GW$ for this system, and open new paths for
exploring larger interface cells or multiple different interface
configurations in relation with experimental devices. Furthermore, a detailed
analysis of the effects of spin-orbit coupling on the spin-splitting of some
electronic states show a linear scaling in $k$-space. A behaviour most
probably related to the two-dimensional nature of the interface states. Our
work indicates the possibility of tailoring the properties of the electronic
states central to the realisation of topological computers from the quality of
the semiconductor/metal interface.

###### Acknowledgements.

This work was authored by the National Renewable Energy Laboratory, operated
by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy
(DOE) under Contract No. DE-AC36-08GO28308, funding from Office of Science,
Basic Energy Sciences, Division of Materials. We acknowledge the use of the
National Energy Research Scientific Computing Center, under Contract No. DE-
AC02-05CH11231 using NERSC award BES-ERCAP0021783 and we also acknowledge that
a portion of the research was performed using computational resources
sponsored by the Department of Energy’s Office of Energy Efficiency and
Renewable Energy and located at the National Renewable Energy Laboratory. For
early statges of this work, HN and MvS acknowledge financial support from
Microsoft Station Q via a sponsor agreement between KCL and Microsoft
Research. HN, DP and MvS acknowledge the Partnership for Advanced Computing in
Europe (PRACE) for awarding us access to Juwels Booster and Cluster (Jülich,
Germany).

##  Appendix A Questaal on GPU

Due to the relatively large simulation cell for QS$GW$ standards, together
with (i) relatively dense Brillouin zone sampling and (ii) high angular
momentum cutoffs required, the all-electron, the full frequency QS$GW$
calculations are rather difficult to achieve. They were only made practical
with the efficient use of new clusters with high-density, high-memory GPU
nodes and good interconnect.

Algorithmic improvements avoided most of the filesystem IO. Together with a
more flexible memory management, they allowed efficient parallelisation across
multiple levels of processes and threads enabling various launch
configurations. Nearly all of the remaining IO was moved to parallel HDF5
maintaining the same file layout independent of the parallelism.

The screened coulomb potential and the mixed product basis projectors occupy
the bulk of the memory available and in the present case could not fit
together entirely; fortunately the projectors can be generated and used in
piecewise fashion with little overhead.

In the GPU context, each device is handled by a thread allowing simple use of
multiple devices per process. The threads distribute batches of matrix
operations across dynamically estimated number of streams depending on
dimensions and the available memory. In this way host-device transfers and
kernel launches are overlapped through asynchronous executions, hiding latency
and maximising occupancy and efficiency.

Most of the compute routines make heavy use of the performance libraries
cuBLAS, cuSOLVER, cuFFT and cuSPARSE in this mode. Certain larger matrix
operations were done collectively with cuSOLVERMg (cu*Mp were not available at
the time).

The heaviest step in the computations is the calculation of the full off-
diagonal self-energy, it sustained close to 20 PFLOPS on the Juwels-Booster
cluster using 288 nodes.

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##  Appendix B Supplementary Material

###  B.1 Local density of states

(a) (b) (c) (d)

Figure 7:  Local density of states LDOS per atomic layers in the InAs/Al
heterojunction, panels (a) and (b), obtained from QS$GW$ (solid lines) and LDA
(dotted lines) calculations. Panel (c) shows the corresponding bulk LDOS,
panel (d) shows the labelling of the atomic layers in the $z$-direction. The
LDOS of the central In and As atomic layers, labelled $z=18,20,21$, are
similar to the bulk LDOS. The LDOS of the In and As atomic layers acquires a
stronger admixture with the Al states, the closer the layers are to the
InAs/Al interfaces. Note that in the bulk LDOS, the Fermi energy
$E_{\text{F}}$ is located at the top of the valence band as a convention for
any semiconductors at zero temperature. For the heterojunction, the position
of $E_{\text{F}}$ is governed by the metallic states of the Al slab.

###  B.2 Transition bulk to heterojunction

Figure 8:  Schematic representation of the modifications of the In and As deep
electronic levels upon changing atoms from a bulk configuration to a InAs/Al
heterojunction configuration.

###  B.3 Band structure

Figure 9:  QS$GW$ Bulk band structure of InAs. Energies are given in eV. For
allowing direct comparisons between bulk and heterojunction systems, the bulk
bands have been calculated for a cell having the same lattice vectors
$u_{1,2}$ in the (001) plane as for the InAs/Al heterojunction. $Qx,y,z$
represents the (x,y,z)-direction in the reciprocal space. The color scheme
represent the weight of the As $s$-orbital in the bands: red: large weight of
As $s$-orbital, grey/black: no As $s$-orbital weight. One can see that the
bottom of the conduction band is mostly consisting of As $s$-states. This
color scheme will be useful to identify the “bulk InAs conduction band” in the
band structure of the InAs/Al heterojunction.

(a) (b) (c)

Figure 10:  QS$GW$ band structure of the InAs/Al heterojunction made of 138
atoms. Energies are given in eV. The band structure look much more complex
than for bulk InAs due to the large size of the system, the presence of the Al
metallic states and the relaxation of the InAs/Al interfaces. There is clearly
no band gap anymore in the heterojunction. The bands in panels (a), (b), (c)
are colored in red, blue, green respectively according to the projection
weight of the states onto the bulk-like atomic layers of As $z=20$, In $z=18$,
In $z=21$. See panel (d) in Fig. 7 for the $z$-direction labelling. There is a
strong mixing of all As and In orbitals in the bands, as well as mixing with
Al orbitals. Note the two flat bands around $E_{\text{F}}$ along the $\Gamma-
Qz$ direction implying the existence of localized “interface” states. This
states are however delocalised in the ($xy$) planes of the InAs/Al interfaces.
A careful analysis of the composition of these states (not shown here) reveal
that they consist mostly of orbitals of the As $z=10$ and Al $z=9$ (and of the
As $z=0.3$ and Al $z=2$) atomic layers.

Figure 11:  QS$GW$ band structure of the InAs/Al heterojunction made of 138
atoms. Energies are given in eV. Zoom in energy of the bands shown in Fig.
(10). The bands are colored in red according to the projection weight of the
states onto the $s$-orbitals of the bulk-like atomic layers of As $z=20$. In
Fig. (9), we have seen that the bottom of the bulk InAs conduction band,
around the $\Gamma$ point, is mostly of As $s$-character. For the InAs/Al
heterojunction, we can see that some bands around the $\Gamma$ point, still
keep a non-negligeable weight of the bulk-like As $s$-orbitals; more specially
the bands $\sim$ 50 meV below $E_{\text{F}}$. This energy position below
$E_{\text{F}}$ corresponds well with our estimate of the energy shift of the
bulk-like conduction band minimum shown in Fig. (2). We therefore consider
that these bands are the equivalent of the bottom of the bulk-like conduction
band of InAs which strongly couples to the Al states in the InAs/Al
heterojunction.

(a) (b) (c) (d)

Figure 12:  QS$GW$ band structure of the InAs/Al heterojunction made of 138
atoms, for different rescaling of the SOC. Panel (a), (b), (c) and (d)
corresponds to rescaling the original SOC by $\times 2,\times 3,\times 4$ and
$\times 5$ respectively. Focussing on the bands around the $\Gamma$ point and
at $E\sim E_{\text{F}}-0.05$, one can see the spin-splitted bands (due to SOC)
along the $\Gamma-Qy$ direction. The splitting between the two bands increases
with increasing rescaling of the SOC, as expected.