# Favorable band alignment for photocatalysis at the strontium germanate
interface with silicon

Tomáš Rauch  Institut für Festkörpertheorie und -optik, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany  European Theoretical
Spectroscopy Facility  Pavel Marton  Institute of Physics of the Czech Academy
of Sciences, Praha, Czech Republic  Institute of Mechatronics and Computer
Engineering, Technical University of Liberec, Studentská 2, 461 17 Liberec,
Czech Republic  Silvana Botti  Institut für Festkörpertheorie und -optik,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
European Theoretical Spectroscopy Facility  Jiří Hlinka  Institute of Physics
of the Czech Academy of Sciences, Praha, Czech Republic

###### Abstract

Photocatalytic water splitting is a promising strategy for large-scale clean
energy production. However, efficient and low-cost solid-state photocatalysts
are still lacking. We present here first-principles calculations to
investigate the suitability as photocathode of an epitaxial layer of strontium
germanate on a Si(100) single crystal. Conduction and valence bands offsets at
the interface between these two semiconductors were determined using state-of-
the-art approximations of density functional theory for the accurate
prediction of band alignments. The resulting type-III band line-up is also
confirmed by inspection of the spatially resolved density of states. It is
concluded that the electronic structure of the investigated heterostructure is
favorable for photocathodic functionality.

##  I Introduction

Solar water splitting in the presence of semiconductor photocatalysts is a
clean energy source that enables renewable energy production and storage
without dependence on fossil fuels and carbon dioxide emission. Owing to its
outstanding semiconductor properties and its wide use in the electronic
industry, crystalline silicon remains one of the most convenient bulk
materials for any optoelectronic devices. In many applications, the native
SiO2 oxide layer which is purposely grown or naturally formed on the silicon
surface acts as a useful mechanical protection or functional dielectric layer.

At the SiO2/Si interface, the band edges of Si electronic states are contained
inside the band gap of SiO2. This type of heterojunction is known as type-I
Franciosi and Van de Walle (1996) or straddling band lineupKroemer (2001).
However, this type of interface is unsuitable for highly demanded photo-
catalytic cells for hydrogen evolution reaction, as the desired efficient and
fast drift of photoexcited electrons towards the surface of the device in
contact with water is hindered by the energy barrier in the conduction band
and therefore another protective oxide is neededKumah  _et al._ (2020);
Spreitzer  _et al._ (2021); Stoerzinger  _et al._ (2018); Luo  _et al._
(2019). For example, it has been found that the interface between epitaxially-
grown SrTiO3 (STO) and a Si(100) crystalline substrate can significantly
reduce the conduction-band offsetKolpak and Ismail-Beigi (2012); Luo  _et al._
(2019). In particular, high-quality epitaxial films of SrTiO3 grown on a Sr-
passivated Si(100) surface were identified as a direct contact of the non-
dimerized (1x1) Si surface provided that no oxygen ions are contained in the
interface Sr monolayer Tong  _et al._ (2022). Their conduction band alignment
is indeed favorable for the drift of the photoexcited electrons from Si to STO
Tong  _et al._ (2022) and the catalytic performance of STO/Si based
photocathodes was already demonstrated in experimentsJi  _et al._ (2015).

We explore in this Article the opportunity that similar or improved
performance can be achieved with other perovskite protective oxides Abbasi
_et al._ (2022); Reiner  _et al._ (2010); Walter  _et al._ (2010). In this
respect, we consider the cubic perovskite form of SrGeO3 (SGO). Recent studies
showed that cubic SGO is a semiconductor with an indirect bandgap of about 2.7
eV and a direct gap of about 3.5 eVMizoguchi  _et al._ (2011), a remarkably
high electron mobility of the order of 400-500 cm2/VsRowberg  _et al._ (2020)
and a small effective mass at the lowest conduction band (about 0.2
$m_{e}$Niedermeier  _et al._ (2020)). Most interestingly, SGO belongs to the
very few discovered transparent conductive oxides Mizoguchi  _et al._ (2011);
Rowberg  _et al._ (2020). The underlying salient feature of the SGO electronic
structure is that its conduction band is formed by $s$ and $p$ orbitalsWang
_et al._ (2019); Niedermeier  _et al._ (2020), which are much less localized
than empty $d$ orbitals of STO-like perovksites.

Throughout this paper we use the conventions of Ref. Franciosi and Van de
Walle, 1996 to indicate band offsets at interfaces: A/B or A-B heterojunctions
assume an overlayer A grown on a substrate B, and the valence-band offset
(VBO) $\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}$ is positive if the valence-band
maximum in the overlayer (A) is lower in energy than in the substrate (B)
while the conduction-band offset (CBO) $\Delta^{\mathrm{A-B}}_{\mathrm{CBO}}$
is taken positive if the conduction-band minimum in the overlayer A is higher
in energy then the substrate, so that the algebraic sum of the offsets equals
the bandgap difference
$\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}+\Delta^{\mathrm{A-B}}_{\mathrm{CBO}}=E_{\rm
g}(A)-E_{\rm g}(B)$. With such a convention, all these three energies are
positive for the SiO2/Si case. For applications as photocathode the CBO at the
A/B interface should be zero or negative.

The central motivation for the present study is to check whether the
conduction band alignment between Si and SrGeO3 in SGO/Si may be favorable for
releasing to water electrons that are photoexcited in the bulk silicon
substrate through the protective oxide layer. Having in mind that the
character of the bonding and the electronic band structure of the two
materials at the heterojunction is quite different, we tested different
flavors of density functional theory. Since the SGO/Si system has not been
realized experimentally yet, and considering the similarity of SGO/Si with
STO/Si we assume the STO/Si interface structure proposed in Ref. Tong  _et
al._ (2022), with a passivating Sr mono-layer that separates the Si(100)
unreconstructed surface from epitaxially-grown SGO. Technical details and
information on the adopted methods are given in Section II. The optimized
geometry and calculated electronic properties of the interface are described
in Section III, along with the electronic structure of bulk Si and bulk SGO in
an adequately strained and polarized state. In Section IV the values of the
VBO and CBO are then derived independently using two different alignment
procedures: the average electrostatic potential alignmentVan de Walle and
Martin (1987); Hinuma  _et al._ (2013) (EPA) and the core level alignment Wei
and Zunger (1998) (CLA). These band offset calculations agree on a type-III
band line-up. We complement them with a direct inspection of the layer-
resolved projected density of states (PDOS) and local density of states
(LDOS), the evaluation of charge transfer at the interface, as well as by
considerations about the short-range band-bending effects. This analysis
allows us to conclude at the end the suitability of the proposed epitaxial
layer for application as a photocathode, as discussed in the concluding
Section V.

##  II Computational details

###  II.1 Ab initio calculations

All DFT calculations were performed with the Vienna Ab initio Simulation
Package (VASP) Kresse and Furthmüller (1996, 1996); Kresse and Hafner (1993)
(version 6.2.1) that implements the projector augmented-wave method Kresse and
Joubert (1999). For structural optimization we utilized the PBEsol Perdew  _et
al._ (2008) exchange-correlation (XC) functional. The electronic bandstructure
was then recalculated with the more accurate HSE06 hybrid functional Heyd  _et
al._ (2003, 2006), for which we used the mixing value $\alpha=0.33$ to better
reproduce the bulk band gap of SGO. Further explanation is provided in Sec.
III.2.1. The kinetic-energy cutoff was set to 500 eV (306.7 eV for bulk Si),
corresponding to the `PREC=High` setting of VASP. Spin-orbit coupling was
neglected in all calculations. We checked that its effect on the electronic
structure of both Si and SGO is negligible for our purposes.

For supercell calculations, we used a $\Gamma$-centered grid of
12$\times$6$\times$1 k points for ionic relaxation (using a 2$\times$1 in-
plane supercell) and 16$\times$16$\times$1 k points for electronic structure
calculations (using a 1$\times$1 in-plane supercell). For bulk electronic
structure calculations we used 12$\times$12$\times$12 and 8$\times$8$\times$8
k points for Si and SGO, respectively. The geometry optimization was stopped
when the forces on each atom were smaller than 15 meV/Å.

In the post-processing step, we used the codes pyPROCAR Herath  _et al._
(2020) to visualize the atom-projected band structure and density of states
(PDOS) and DensityTool Lodeiro and Rauch (2022) to obtain the local density of
states (LDOS).

The figures were created with Matplotlib v2.2.2, VESTA v3.4.6 Momma and Izumi
(2011), and Inkscape v0.92.

###  II.2 Interface model

To study the atomic and electronic structure of epitxially-grown SGO on a
(001) oriented Si substrate, we modeled the system as a finite slab surrounded
by vacuum in a supercell. We followed the modeling procedure proposed in
previous works on STO/Si heterostructures Kolpak and Ismail-Beigi (2012); Tong
_et al._ (2022), taking advantage of the similarly to SGO/Si. For the
substrate we used 9 atomic layers of Si terminated by H atoms at the interface
with vacuum to simulate a semi-infinite bulk. We observed that this thickness
is necessary to ensure a bulk-like behavior inside of the Si layer. The
Si(100) surface was passivated by a monolayer of Sr, bonded to the SGO layer,
that consisted of three unit cells terminated by an SrO layer. Based on
previous results for the similar STO/Si interface, we expect the atomic and
electronic structure to depend only weakly on the termination of the
perovskite layer Kolpak and Ismail-Beigi (2012). The in-plane lattice constant
was set to the equilibrium bulk value of Si (3.843 Å) Tong  _et al._ (2022),
to which the out-of-plane lattice constant of SGO has to accommodate for
epitaxial growth, resulting in a substrate-induced in-plane strain of 1.4 %.
We remark that the presence of a Sr monolayer at the STO/Si interface was
proved to be the most probable reconstruction Kolpak and Ismail-Beigi (2012);
Tong  _et al._ (2022). Finally, the slab was embedded in more than 15 Å of
vacuum.

To allow possible relaxations at the interface, we adopted a 2$\times$1 in-
plane supercell during the structural relaxation of the slab (with in-plane
lattice parameters $a=$ 7.686 Å and $b=$ 3.843 Å). As the final atomic
structure was still symmetric, we performed the successive electronic-
structure calculations with a 1$\times$1 slab model ($a=b=$ 3.843 Å).

###  II.3 Band alignment calculations

Various methods have been proposed to calculate the band alignment at the
interface of two materials, ranging from pure bulk-based methods Tersoff
(1984); Schleife  _et al._ (2009), or estimate from individual ionization
potentials and electron affinities that neglect the details of the common
interface Oba and Kumagai (2018), to supercell calculations Van de Walle and
Martin (1987). In this work, we employ the latter approach, since we want to
include structural details of the interface that can have important effects on
the final electronic structure.

Since the precise numerical values of band edges at different positions in the
supercell cannot be easily extracted from calculations of the electronic
structure of the supercell, we perform as well two auxiliary bulk calculations
(one for bulk Si and one for strained bulk SGO) and align them to chosen
reference levels according to their difference across the interface in the
supercell calculation.

For a sufficiently large supercell, the ionic positions are expected to
reproduce the bulk crystal structure in the central part of a material layer.
We assume the bond lengths in these region to build the unit cells for the
auxiliary bulk calculations. Note that these auxiliary structures can be
strained, and thus different from the equilibrium bulk structures due to the
lattice mismatch of the two interfaced materials. As it will be further
discussed in Sec. III.1, the Si substrate in our simulation adopts the
unstrained cubic structure of Si. On the other hand, the SGO layer is very
thin and it does not exhibit any periodic behavior in the direction
perpendicular to the interface. We will choose an auxiliary bulk structure
obtained by averaging the interlayer distances and oxygen-cation displacements
in the central region of the SGO part of the supercell.

The valence band offset $\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}$ at the
interface of a semiconductor A grown on a semiconducting substrate B is
calculated as Hinuma  _et al._ (2013)

$\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}=\Delta\epsilon^{\mathrm{B}}_{\mathrm{VBM-
Ref}}-\Delta\epsilon^{\mathrm{A}}_{\mathrm{VBM-
Ref}}-\Delta\epsilon^{\mathrm{A-B}}_{\mathrm{Ref}},$ (1)

where $\Delta\epsilon_{\mathrm{VBM-Ref}}$ is the difference between the
valence band maximum (VBM) and the reference level of one of the materials,
obtained from a calculation of the auxiliary periodic bulk system, and
$\Delta\epsilon^{\mathrm{A-B}}_{\mathrm{Ref}}$ is the difference of the
reference levels in the central regions of the two semiconductors, obtained
from a supercell calculation. We selected and compared two different reference
levels for the band alignment at the interface: the macroscopic average of the
electrostatic potential Van de Walle and Martin (1987); Hinuma  _et al._
(2013) is the chosen reference for the EPA and the average of the core level
positions is the reference used for the CLA Wei and Zunger (1998). This choice
is what defines the EPA and CLA approaches mentioned in the introduction. Both
methods were previously tested in Ref. Liberto and Pacchioni, 2021 and they
were found in that work to give comparable results.

In the EPA, $\Delta\epsilon^{\mathrm{A-B}}_{\mathrm{Ref}}=\Delta
V^{\mathrm{A-B}}$ where $V^{\mathrm{A}}$ is the macroscopic average
$\overline{\overline{V}}(z)$ of the planar average $\overline{V}(z)$ of the
electrostatic potential in the middle of the layer A of the supercell (denoted
by the coordinate $z^{\mathrm{A}}$). The macroscopic average is calculated as

$\overline{\overline{V}}(z)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\int\overline{V}(z^{\prime})e^{-\frac{(z-z^{\prime})^{2}}{2\sigma^{2}}}dz^{\prime}.$
(2)

The Gaussian average is necessary since the interlayer distance varies for
each pair of adjacent layers, in particular in the SGO part of the supercell.
$V^{\mathrm{A}}$ is calculated for a range of reasonable values
$\sigma\in(1.0,4.0)$Å. We set as final value for further calculations the
$\sigma$ for which the maximal deviation from nearby
$\overline{\overline{V}}(z)$ values (for
$z\in(z^{\mathrm{A}}-\sigma,z^{\mathrm{A}}+\sigma)$) is minimal. The same
procedure is followed for the layer B of the supercell.

To better understand the CLA, we rewrite Eq. (1) as

$\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}=\epsilon^{\mathrm{B}}_{\mathrm{VBM}}-\epsilon^{\mathrm{A}}_{\mathrm{VBM}}+\Delta\epsilon^{\mathrm{B}}_{\mathrm{Ref}}-\Delta\epsilon^{\mathrm{A}}_{\mathrm{Ref}},$
(3)

where $\epsilon^{\mathrm{A}}_{\mathrm{VBM}}$ is the VBM of material A in bulk
calculations and $\Delta\epsilon^{\mathrm{A}}_{\mathrm{Ref}}$ is the
difference of the reference levels (here the core levels) of the material A in
the supercell calculation and in the auxiliary bulk calculation. We calculated
$\Delta\epsilon_{\mathrm{Ref}}$ for single core levels (Si:$1s$,$2s$,$2p$,
Sr:$1s$,$2s$,$2p$,$3s$,$3p$,$3d$, Ge:$1s$,$2s$,$2p$,$3s$,$3p$, O:$1s$) of
atoms in the central parts of the Si and SGO layers composing the supercell
and we observed huge differences (up to 1 eV) in the calculated VBOs for
different choices of the reference core level. Therefore, the individual
results obtained for different core levels have to be averaged appropriately.
First, the results obtained for core levels of individual atoms are averaged.
For Si the central atom in the middle of the Si layer is considered. In the
case of SGO, we consider Ge and O atoms in the middle of the layer and the Sr
atoms in the atomic layer next to the central one. Finally, the results for
the individual atoms of SGO are averaged according to weights proportional to
their number in the bulk unit cell.

Within the EPA and CLA, we can obtain the CBO from the knowledge of the VBO
($\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}$) and of the band gaps of the two
constituents ($E^{\mathrm{A}}_{g}$ and $E^{\mathrm{B}}_{g}$) calculated in the
auxiliary bulk system:

$\Delta^{\mathrm{A-B}}_{\mathrm{CBO}}=E^{\mathrm{A}}_{g}-E^{\mathrm{B}}_{g}-\Delta^{\mathrm{A-B}}_{\mathrm{VBO}}.$
(4)

##  III Results

###  III.1 Atomic structure

We display the relaxed atomic structure for the chosen SGO/Si interface in
Fig. 1.

Figure 1: Relaxed atomic structure of the studied SGO/Si interface. Top: Ball-
and-stick model of the structure (Si: blue, Sr: green, Ge: grey, O: red, H:
beige.). Bottom: Calculated average interlayer distances (black) and the
oxygen-cation distances in SGO (red dots).

We can clearly observe that the average interlayer distances in the middle of
the Si part of the slab are close to the Si bulk value (1.359 Å). In the SGO
part on the other hand, no bulk-like behavior can be observed. While the
(GeO)-(SrO) distances are strongly influenced by the interface and surface,
for the central (SrO)-(SrO) distance we obtained 3.784 Å, which indicates a
strain in the SGO layer leading to a tetragonal $c/a$ ratio of 0.985. This is
in agreement with the expectation based on the lattice mismatch between bulk
cubic Si and cubic SGO. The average distance between the top Si layer and the
first GeO layer was 3.703 Å, which is close to the result obtained previously
for the same type of interface of STO/Si (3.56 Å) Kolpak and Ismail-Beigi
(2012).

Further, indicated by red dots in the bottom panel of Fig. 1 we show the
calculated oxygen-cation displacement $\delta z$ in each atomic layer. We
found $\delta z=$ 0.25 Å in the GeO-layer closest to the interface, which is
again close to 0.35 Å found in the first TiO layer in STO/Si for the same type
of interface Kolpak and Ismail-Beigi (2012). The value of $\delta z$ decreases
for deeper SGO layers and it changes sign in the surface SrO layer. Overall,
we can state that the oxygen-cation displacement in a thin layer of SGO on Si
substrate behaves qualitatively as in STO, but the out-of-plane strain of SGO
is opposite from the one in STO/Si, since the equilibrium lattice constant of
SGO (STO) is smaller (larger) than the lattice constant of Si.

Finally, to study the band alignment with the CLA or EPA, an auxiliary bulk
calculation for Si and SGO is necessary. In the middle of the Si region the
atomic arrangement is like the one in bulk Si and therefore the cubic
equilibrium structure of Si is assumed for the auxiliary bulk calculation.
Since bulk-like lattice parameters are not recovered in the thin SGO part of
the supercell, we assume a strained unit cell with $c/a=0.985$ as in the
center of the SGO slab. For the oxygen-cation displacement, we take the
supercell value in the central GeO layer $\delta z_{\mathrm{GeO}}$ = 0.095 Å
and the average of the two central SrO layers $\delta z_{\mathrm{SrO}}$ =
0.056 Å.

###  III.2 Electronic structure

####  III.2.1 Bulk Si and SGO

The electronic properties and particularly the width of the band gap are known
to depend strongly on the choice of the XC functional. In the past, the
hybrids PBE0 Perdew  _et al._ (1996) and HSE06 with the mixing parameter
$\alpha=0.33$ (called here HSE033) were employed to reproduce the experimental
band gap of bulk SGO Mizoguchi  _et al._ (2011); Wang  _et al._ (2016);
Rowberg  _et al._ (2020). Therefore, in Tab. 1 we show the band gaps
calculated with PBEsol, PBE0, HSE06, and HSE033, compared with experimental
values.

| PBEsol | PBE0 | HSE06 | HSE033 | exp
---|---|---|---|---|---
Si | 0.47 | 1.78 | 1.16 | 1.36 | 1.17
SGO | 0.34 | 2.85 | 2.16 | 2.77 | 2.70
Table 1: Band gaps in eV calculated with different XC functionals for bulk
cubic Si ($a=$ 3.843 Å) and SGO ($a=$ 3.79 Å). The experimental values are
from Ref. Sze and Ng, 2006 (Si) and Mizoguchi  _et al._ , 2011 (SGO).

We observe that PBEsol severely underestimates both band gaps, as expected
from a GGA XC functional. PBE0 reproduces well the experimental band gap of
SGO but overestimates the Si band gap by 0.61 eV, while HSE06 yields a very
good result for Si but underestimates the SGO band gap by 0.54 eV. Therefore,
HSE033 provides the best compromise for both Si and SGO and we used it for all
our electronic structure calculations in this work unless otherwise stated.

For the evaluation of the band alignment in the SGO/Si heterostructure with
EPA and CLA we used the strained and polarized crystalline auxiliary SGO unit
cell, as described in Sec. III.1. For this structure, we obtained for the band
gap 0.06 eV with PBEsol and 2.4 eV with HSE033. The band gap thus becomes
smaller with strain. We show the electronic band structure of the strained and
polarized bulk SGO calculated with HSE033 in Fig. 2.

Figure 2: Electronic structure of strained SGO ($c/a=0.985$) with oxygen-
cation displacements $\delta z_{\mathrm{GeO}}$ = 0.095 Å and $\delta
z_{\mathrm{SrO}}$ = 0.056 Å calculated with HSE033.

The band gap is indirect and the lowest conduction band at $\Gamma$ has a
small effective mass and DOS, as was found earlier in Ref. Rowberg  _et al._ ,
2020. The highest valence bands, on the other hand have a much larger DOS.

####  III.2.2 SGO/Si interface

In this section we finally present the main result of this work - the
calculated electronic structure of a thin SGO layer on a Si substrate with the
chosen model interface including a full Sr monolayer. We recall that in our
convention the VBO is positive when the VBM of SGO lies below the one of Si
and the CBO is negative when the conduction band minimum (CBM) of SGO lies
below the one of Si.

The calculations of the EPA and CLA were obtained by means of Eqs. (1), (2),
(3) and (4), using the auxiliary bulk band edge positions. We summarize our
final VBOs and CBOs, calculated with the PBEsol and HSE033 XC functionals, in
the first two columns of Tab. 2.

Valence band offsets
---
| EPA | CLA | PDOS
PBEsol | 2.09 | 2.18 | 2.12
HSE033 | 3.31 | 3.44 | 3.76
Conduction band offsets
| EPA | CLA | PDOS
PBEsol | -2.51 | -2.60 | -1.32
HSE033 | -2.26 | -2.39 | -1.89
$\epsilon^{\mathrm{SGO}}_{\mathrm{CBM}}-\epsilon^{\mathrm{Si}}_{\mathrm{VBM}}$
| EPA | CLA | PDOS
PBEsol | -2.03 | -2.13 | -0.63
HSE033 | -0.91 | -1.04 | -0.43
Table 2: Valence band offset ( $\Delta_{\mathrm{VBO}}$), conduction band
offset ($\Delta_{\mathrm{CBO}}$), and the distance between the CBM of SGO and
VBM of Si obtained with the EPA, CLA, and estimated from the PDOS (Fig. 3) in
eV.

In addition to the VBO and CBO, we also provide in Tab. 2 the calculated
difference of the CBM of SGO and VBM of Si.

##  IV Discussion

###  IV.1 Band alignment

We begin the discussion of our results with the calculated VBOs. The results
obtained with EPA and CLA using the same XC functional are in excellent
agreement. The average $\Delta_{\mathrm{VBO}}=$ 3.38 eV from HSE033 is
significantly larger than the average $\Delta_{\mathrm{VBO}}=$ 2.14 eV from
PBEsol, showing that the more precise hybrid XC functional is necessary for
this calculation.

Also in the case of CBO, the EPA and CLA results are very similar, with an
average value of $\Delta_{\mathrm{CBO}}=$ -2.33 eV (-2.56 eV) with HSE033
(PBEsol). Interestingly, we note that the PBEsol and HSE033 functionals yield
similar results for the conduction band offsets, even though the calculated
band gaps obtained with the different XC functionals differ considerably. We
will discuss in Sec. IV.2 that the CBO is almost independent of the XC
functional because the CBM of SGO is pinned to the Fermi level of the
supercell. A similar effect was found previously also for interface models of
STO/Si Kolpak and Ismail-Beigi (2012).

Comparing the calculated CBOs with the Si band gap, we observe that the CBM of
SGO ($\epsilon^{\mathrm{SGO}}_{\mathrm{CBM}}$) lies below the VBM of Si
($\epsilon^{\mathrm{Si}}_{\mathrm{VBM}}$), indicating an overall metallic
system. To obtain a measure for how far the system is from being an insulator,
we also present the values of
$\epsilon^{\mathrm{SGO}}_{\mathrm{CBM}}-\epsilon^{\mathrm{Si}}_{\mathrm{VBM}}$
in Tab. 2. Also here, EPA and CLA yield very close results, but the choice of
the XC functional changes the result by $\sim$1 eV. On average, the CBM of SGO
is calculated to be located 0.98 eV (2.08 eV) below the VBM of Si with HSE033
(PBSEsol).

###  IV.2 Local density of states

In order to visualize and interpret the local band structure of the
heterostructure near the interface, we calculated the spatially resolved
density of states in the form of PDOS for atoms belonging to chosen layers of
the supercell (shown in Fig. 3) and the in-plane integrated LDOS Lodeiro and
Rauch (2022) (shown in Fig. 4):

$\bar{L}(E,z)=\frac{\Omega_{\mathrm{cell}}}{(2\pi)^{3}}\sum_{n}\int_{\mathrm{BZ}}\delta(E-\epsilon_{n,\mathbf{k}})\bar{P}_{n,\mathbf{k}}(z)\
d^{3}k$ (5)

where $\Omega_{\mathrm{cell}}$ is the unit cell volume, $E$ the energy and
$\bar{P}_{n,\mathbf{k}}(z)$ the in-plane integrated partial charge density
$P_{n,\mathbf{k}}(\mathrm{r})=\left|\varphi_{n,\mathbf{k}}(\mathrm{r})\right|^{2}$
with the Kohn-Sham wave functions $\varphi_{n,\mathbf{k}}(\mathrm{r})$ and
eigenvalues $\epsilon_{n,\mathbf{k}}$.

Figure 3: PDOS of chosen atomic layers of the SGO/Si supercell calculated with
PBEsol (top) and HSE033 (bottom). Figure 4: Atomic structure and LDOS of the
SGO/Si supercell calculated with PBEsol (top) and HSE033 (bottom).

PDOS and LDOS are usually not suitable for a precise extraction of the band
offsets, since (i) states originating from one constituent, the interface, or
the surface often reach into the band gap of the second constituent, making
the identification of band edges complicated, (ii) the size of the local band
gaps in the slab is larger than in the auxiliary bulk calculation due to the
finite size of the supercell, and (iii) the DOS is broadened due to the finite
$\mathbf{k}$-point sampling for sums in the reciprocal space and due to the
practical substitution of the delta-function in Eq. (5) by a Gaussian.

Therefore, it is better to resort to EPA or CLA using the electronic structure
from auxiliary bulk calculations to obtain accurate band alignments for large
heterostructures. On the other hand, if one is interested in the local
electronic states close to the interface or within thin layers, a direct
inspection of PDOS or LDOS can be more instructive. We estimated the band-edge
positions from the PDOS of the supercell, knowing that the finite smearing can
introduce an error of $\sim$0.1-0.2 eV due to the choice of the smearing
parameter $\sigma=$ 0.05 eV both in the self-consistent calculation as well as
in the LDOS calculation with Eq. (5). We used the PDOS of the three central Si
layers (“bulk Si” in Fig. 3) and the central SGO layer (“SGO2” in Fig. 3) as
representative of the two materials forming the interface. The extracted local
band gaps of Si (PBEsol: $E_{g}^{\mathrm{Si,slab}}=$ 0.69 eV, HSE033:
$E_{g}^{\mathrm{Si,slab}}=$ 1.46 eV) are fairly close to the bulk value. On
the other hand, for SGO we obtained $E_{g}^{\mathrm{SGO,slab}}=$ 1.49 eV with
PBEsol and $E_{g}^{\mathrm{SGO,slab}}=$ 3.33 eV with HSE033. These values are
$\sim$1.0-1.5 eV larger than the band gap of the auxiliary bulk SGO structure
used in the EPA and CLA. A band gap broadening at the order of 1 eV is indeed
expected already from the simple model of a particle with an effective mass in
an infinite box potential with a width corresponding to the width of the thin
SGO layer Fox and Ispasoiu (2007). This shows that the effect of quantum
confinement is very strong for the thin SGO layer in our supercell and the
auxiliary bulk structure is not suitable to calculate the positions of the SGO
band edges in the thin SGO layer. The VBO, CBO, and
$\epsilon^{\mathrm{SGO}}_{\mathrm{CBM}}-\epsilon^{\mathrm{Si}}_{\mathrm{VBM}}$
obtained from the PDOS are listed in the last column of Tab. 2. We observe a
large discrepancy between the PDOS-derived values and those obtained with EPA
and CLA in case of CBO and
$\epsilon^{\mathrm{SGO}}_{\mathrm{CBM}}-\epsilon^{\mathrm{Si}}_{\mathrm{VBM}}$,
whereas all three methods agree for the VBO.

To gain more insight into the electronic structure of the supercell, we
present in Fig. 4 the calculated LDOS, that we plot together with the
structural model. While the general information is similar to the one offered
by the PDOS, the LDOS provides a much better spatial resolution. Most
importantly, we can see the metallic character of the interface, which is
caused by the fact that the lowest conduction band of SGO is located below
$E_{F}$ and thus partially occupied. This has to be counterbalanced by
unoccupied valence bands of Si, which is also visible in the LDOS in the Si
side of the interface. This band bending is particularly interesting in case
of SGO. In Fig. 5 we show the band structure of the whole supercell with a
color code indicating the projection of the Kohn-Sham wave functions onto
different SGO layers in the supercell.

Figure 5: Band structure of the SGO/Si supercell calculated with PBEsol (top)
and HSE033 (bottom). The color code shows the projection of the calculated
wave functions for bands close to the gap on atoms of chosen SGO layers in the
supercell.

The conduction band of SGO with the large band velocity (visible along
$\Gamma$-X also in the bulk band structure in Fig. 2) is located fairly low in
the first SGO layer, with its minimum in the vicinity of the local VBM with
PBEsol and $\sim$1.5 eV above it with HSE033. In the second and third SGO
layer, this SGO conduction band is located much higher in energy (close to
$E_{F}$) with both XC functionals. On the contrary, the VBM of SGO remains
almost at the same energy in all three SGO layers. It is thus only the CBM of
SGO which exhibits band bending.

###  IV.3 Interface charge transfer

As described above, the formation of the studied interface between the Si
substrate and the thin layer of SGO leads to the depletion of the Si valence
bands and occupation of SGO conduction bands. We calculated the magnitude of
the transferred charge and the generated interface dipole from the ab-initio
induced charge density

$\delta\rho(x,y,z)=\rho_{\mathrm{tot}}(x,y,z)-\rho_{\mathrm{Si/Sr}}(x,y,z)-\rho_{\mathrm{SGO}}(x,y,z)\,,$
(6)

where $\rho_{\mathrm{tot}}$ is the charge density of the full supercell model,
$\rho_{\mathrm{Si/Sr}}$ and $\rho_{\mathrm{SGO}}$ are the charge densities of
the separated Si substrate (including the interface Sr monolayer) and the thin
SGO layer, respectively. We used the same (arbitrary) choice for the plane
dividing the two constituents as in Ref. Kolpak and Ismail-Beigi, 2012. We
further integrate the induced charge density in the plane of the interface and
obtain $\Delta\rho(z)=\int\delta\rho(x,y,z)d\\!xd\\!y$. Finally, we calculate
the induced charge $\Delta
Q=\int^{\mathrm{vac.SGO}}_{\mathrm{int}}\Delta\rho(z)dz$ and interface dipole
$p_{\mathrm{int}}=\int^{\mathrm{vac.SGO}}_{\mathrm{vac.Si}}\Delta\rho(z)zdz$
where “int” indicates the origin ($z=0$) which is set to be the plane at the
interface for which $\Delta\rho$ crosses zero Wang  _et al._ (2019); Rauch
_et al._ (2021) and “vac. Si” and “vac. SGO” are positions in the vacuum
region on the Si and SGO side of the slab, respectively, where $\Delta\rho=0$.
The calculation of the transferred charge and interface dipole was done with
PBEsol. Finally, for the studied interface of SGO/Si, we obtained $\Delta Q=$
0.14 e and $p_{\mathrm{int}}=$ 0.19 eÅ per $1\times 1$ interface unit cell
(3.843 Å $\times$ 3.843 Å). The calculated interface dipole of SGO/Si is
considerably smaller than the calculated interface dipole of the same
interface model of STO/Si Kolpak and Ismail-Beigi (2012) (0.69 eÅ per $1\times
1$ interface unit cell), but it is in the general range of values common in
STO/Si interfaces.

###  IV.4 Photocatalytic properties

After the successful demonstration of STO/Si as a photocathode for water
reduction in Ref. Ji  _et al._ , 2015, we want to consider the SGO/Si platform
for the same application. We aligned therefore the band structure of the
supercell and the redox potentials of H+/H2 and O2/H2O against the vacuum
level. The PDOS result from the HSE033 calculation is schematically
represented in Fig. 6 as we believe it is the most accurate for the thin SGO
layer on Si substrate.

Figure 6: SGO/Si band edges aligned against vacuum potential together with the
redox potentials of water splitting reactions. The band edges from the PDOS
HSE033 result are shown.

We observe that both redox potentials lie inside the local band gap of SGO and
below the Fermi energy of the supercell. Therefore, photogenerated electrons
from the bulk of Si coming to the SGO-capped surface are expected to pass the
SGO/Si interface and participate in water reduction. We conclude that the thin
SGO layer with partially occupied lowest conduction band would play the role
of a metallic electrode in the system.

##  V Conclusions

We studied the structural and electronic properties of a thin SGO layer on a
(100)-terminated Si substrate. We found the oxygen-cation displacement in the
atomic layers of SGO qualitatively similar to the same interface structure of
STO/Si Kolpak and Ismail-Beigi (2012). We calculated the band alignment using
either a combination of a supercell calculation with auxiliary bulk
calculations (EPA, CLA) or directly from the electronic structure of the
supercell (PDOS, LDOS). All three methods agree in the identification of a
type III Franciosi and Van de Walle (1996) band alignment and the whole system
is thus metallic. A similar electronic structure with type III band alignment
has been recently measured experimentally for a STO/Si interface and reported
in Ref. Chrysler  _et al._ , 2021.

Due to the strong broadening effect of the quantum confinement on the local
band gap of the thin SGO layer, the PDOS is more appropriate to study the
quantitative band alignment of the studied system. With the custom HSE033
hybrid functional, we found a large VBO of 3.76 eV and CBM of SGO being
located 0.43 eV below the VBM of Si. We further found a charger transfer from
Si to SGO. This causes the VBM of Si to be bent up above $E_{F}$ and
depopulated at the interface, whereas the CBM of SGO is bent down and
populated at the interface, as can be seen in PDOS (Fig. 3), LDOS (Fig. 4),
and the projected band structure (Fig. 5).

Finally, the local band gap of the thin SGO layer contains both redox
potentials of water splitting, and thus photogenerated electrons from Si bulk
could pass the interface and participate in hydrogen production at the SGO
surface.

Our results demonstrate that SGO might be an alternative to STO as a capping
layer on Si with a CBO favorable for the transition of excited electrons from
Si to SGO. In order to further understand the structural and electronic
properties of real SGO/Si interfaces, experimental evidence for the interface
atomic structure, as well as further theoretical calculations are necessary.
Nevertheless, our work provides first hints that the properties of the SGO/Si
are similar to STO/Si and therefore this system has a great potential for
applications as photocathode.

###### Acknowledgements.

This work was supported by the Czech Science Foundation (project no.
21-20110K). T.R. and S.B. acknowledge funding from the Volkswagen Stiftung
(Momentum) through the project “dandelion”. J.H., T.R. and P.M. thank Dr. Wen-
Yi Tong and prof. Philippe Ghosez for their kind advice and sharing POSCAR
files from their recent work Tong  _et al._ (2022) prior its publication.
Computational resources were supplied by the project “e-Infrastruktura CZ”
(e-INFRA CZ LM2018140 ) supported by the Ministry of Education, Youth and
Sports of the Czech Republic.

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