# High-mobility two-dimensional electron gas in $\gamma$-Al2O3/SrTiO3
heterostructures

Xiang-Hong Chen  Tianjin Key Laboratory of Low Dimensional Materials Physics
and Preparing Technology, Department of Physics, Tianjin University, Tianjin
300354, China  Zhi-Xin Hu  Center for Joint Quantum Studies and Department of
Physics, Tianjin University, Tianjin 300354, China  Kuang-Hong Gao
<EMAIL_ADDRESS>Tianjin Key Laboratory of Low Dimensional Materials Physics
and Preparing Technology, Department of Physics, Tianjin University, Tianjin
300354, China  Zhi-Qing Li<EMAIL_ADDRESS>Tianjin Key Laboratory of Low
Dimensional Materials Physics and Preparing Technology, Department of Physics,
Tianjin University, Tianjin 300354, China

###### Abstract

The origin of the two-dimensional electron gas (2DEG) in the interface between
$\gamma$-Al2O3 (GAO) and SrTiO3 (STO) (GAO/STO) as well as the reason for the
high mobility of the 2DEG is still in debate. In this paper, the electronic
structures of [001]-oriented GAO/STO heterostructures with and without oxygen
vacancies are investigated by first-principle calculations based on the
density functional theory. The calculation results show that the necessary
condition for the formation of 2DEG is that the GAO/STO heterostructure has
the interface composed of Al and TiO2 layers. For the heterostructure without
oxygen vacancy on the GAO side, the 2DEG originates from the polar
discontinuity near the interface, and there is a critical thickness for the
GAO film, below which the 2DEG would not present and the heterostructure
exhibits insulator characteristics. For the case that only the GAO film
contains oxygen vacancies, the polar discontinuity near the interface
disappears, but the 2DEG still exists. In this situation, the critical
thickness of the GAO film for 2DEG formation does not exist either. When the
GAO film and STO substrate both contain oxygen vacancies, it is found that the
2DEG retains as long as the oxygen vacancies on the STO side are not very
close to the interface. The low-temperature mobilities of the 2DEGs in these
GAO/STO heterostructures are considered to be governed by the ionized impurity
scattering, and $\sim$3 to $\sim$11 times as large as that in LaAlO3/SrTiO3
heterojunction. The high mobility of the 2DEG is mainly due to the small
electron effective mass in GAO/STO heterostructure.

##  I INTRODUCTION

Since the report of existence of two-dimensional electron gas (2DEG) at the
LaAlO3/SrTiO3 (LAO/STO) interfaces [1], great attention has been paid on the
interfaces of STO-based heterostructures [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16]. The 2DEGs confined in the STO-based heterostructures could
reveal a range of unique phenomena including the existence of
superconductivity at low temperatures [17, 18], electric-field-tuned
metal–insulator and superconductor–insulator phase transitions [19, 20, 21,
22], and coexistence of ferromagnetism and superconductivity at the interfaces
[23, 24]. These phenomena make STO-based heterostructures not only have
potential applications in novel electronic devices but also be good model
systems for fundamental research. Recently, a 2DEG with a low-temperature
mobility of $\sim$$1.4\times 10^{5}$ cm2/Vs, being much higher than that in
LAO/STO interfaces, has been realized at the interface between STO and a
heteroepitaxial spinel $\gamma$-Al2O3 (GAO) thin film [11]. This finding shows
promise for the applications of the STO-based heterostructures in future
electronic devices. Thus it is crucial to understand the formation mechanisms
and the origin of the high mobility of the 2DEG at GAO/STO interfaces.
However, there has been no consensus for the formation mechanisms of 2DEG in
this heterostructure up to now. Some groups believe that the GAO in the
heterostructures has polarity, and the charge accumulation is caused by the
polar discontinuity at the interface [15, 14, 16]; while some groups argue
that the formation of 2DEG is caused by oxygen vacancies at the interface [25,
26, 27, 28]. Besides, the origin of the high mobility remains poorly
understood as well [26, 29]. To explore why high-mobility 2DEG can be formed
at the GAO/STO interfaces, we systematically investigated the electronic
structure of GAO/STO heterostructures via first-principles calculations. It is
found that the polar discontinuity is the main mechanism for the formation of
2DEG in GAO/STO heterojunction when there is no oxygen vacancy on the GAO
side; while the oxygen vacancies are the main source of the 2DEG for the
heterojunction that contains oxygen vacancies on the GAO side. The reasons for
the high mobility of the 2DEG in the GAO/STO heterojunctions are also
disclosed.

##  II CALCULATION METHODS AND HETEROJUNCTION CONSTRUCTION

All calculations are carried out in the frame work of density-functional
theory using a first-principle calculation software Vienna _ab initio_
simulation package. The generalized gradient approximation (GGA) parametrized
by Perdew-Burke-Ernzerhof (PBE) plus the on-site coulomb interaction approach
(GGA+$U$) was used for the exchange-correlation functional. For all
heterostructures, a Hubbard interaction energy $U=8.5$ eV for Ti 3$d$ orbitals
was found to be appropriate [30]. A cutoff energy of 500 eV for plane wave
basis set and a Monkhorst-Pack k-point mesh of $4\times 4\times 1$ were
employed. The electronic energy convergence was set to be 10-6 eV, and a
Gaussian smearing of 0.05 eV was employed for density of states (DOS)
calculations.

Figure 1: Schematic structures of (GAO)2.25/(STO)5 heterostructures with Al
and TiO2 layers terminations (a) before, and (b) after atomic optimization.
The atomic layers on the GAO side are labelled as L1, L2, $\cdots$, L9 (the
adjacent Al and AlO atomic layers share the same number), and the atomic
layers on the STO side are numbered as L$\bar{1}$, L$\bar{2}$, $\cdots$,
L$\bar{10}$. For clarity, only L1, L9, L$\bar{1}$, and L$\bar{10}$ are shown
in the diagram.

For GAO/STO heterostructures, considering the experimental results that the
2DEG can only be present in the interface of GAO and the TiO2 terminated (001)
STO substrate [16, 31], we construct a $2\times 2$ in-plane STO supercell with
TiO2 layer terminated on the GAO side and SrO layer terminated on the vacuum
side to simulate the [001]-oriented STO substrate. Along the [001] direction,
the supercell contains 5 TiO2 and 5 SrO layers, respectively (see Fig. 1).
Usually, GAO belongs to the defective spinel structure with the space group
$Fd\bar{3}m$. In an ideal spinel structure, a unit cell (uc) contains 24 Al
atoms and 32 oxygen atoms (8 Al atoms in the tetrahedral sites, 16 Al atoms in
the octahedral sites, and 32 O atoms forming a fcc array), and the ratio of
cation to anion is $3:4$. In order to obtain stoichiometric Al2O3, 8/3 Al
vacancies need to be created per spinel unit cell in average. Therefore, at
least 0.75 spinel unit cells are required for GAO (001) surface to match the
Al2O3 stoichiometric and vacancy distribution requirements (two Al vacancies).
In our calculations, the thickness of GAO slabs varies from 0.75 to 3 uc. For
the distribution of Al vacancies in GAO slab, three aspects are considered
[32, 33, 34, 35, 36]: (1) Al vacancies preferentially occupy the octahedral
positions; (2) the distance between Al vacancies is as far as possible to
ensure that the vacancies are distributed as homogeneously as possible in the
lattice; and (3) Al vacancies are distributed as deep as possible inside GAO
rather than on its surface.

Since the structure of GAO can be viewed as an alternating stacking sequence
of Al atomic layer and AlO atomic layer along the [001] direction, both
Al/TiO2 and AlO/TiO2 interfaces were considered, respectively. Finally, a
series of configurations of (GAO)n/(STO)5 ($n$=0.75, 1.5, 2.25, and 3) with
Al/TiO2 or AlO/TiO2 interfaces were constructed (i.e., there are eight
configurations). A vacuum layer with thickness of 15 ${\rm{\AA}}$ is inserted
between neighbor slabs. Structural optimization was done for all the
configurations. All atoms, including the surface layer, are optimized until
the residual stress is less than 0.03 eV/${\rm{\AA}}$. It is found that the
configurations with AlO/TiO2 interface were not significantly changed before
and after optimization. However, the case of the configurations with Al/TiO2
interface is different. Figure 1(a) and 1(b) show the interface geometry of
(GAO)2.25/(STO)5 heterostructure with Al/TiO2 interface before and after
optimization. For simplicity, we define (GAO)2.25/(STO)5 heterostructure with
Al/TiO2 interface as Configuration 1 (Conf. 1). We find that the structure of
GAO near the interface in the optimized Conf. 1 is distorted compared with the
structure before optimization. As a consequence, the positions of atoms in the
three Al layers and AlO layers near the interface have been changed. For
example, the Al atom (pointed out by arrow) in the L2 AlO layer should be at
the center of the octahedron, while it deviates from that position after
optimization. Although the structure distortion occurs in the GAO slab near
the interface, the alternating structure of octahedral cation and tetrahedral
cation along the [001] direction is still retained, which is consistent with
the experimental results [25, 37].

Figure 2: The DOS of (GAO)n/(STO)5 heterostructures with Al/TiO2 interface for
(a) $n=1.5$, and (b) $n=2.25$.  Figure 3: (a) The partial density of states
projected onto atomic planes in the STO and GAO of Conf. 1. (b) The plane
average electrostatic potential (solid line) and macroscopic average
electrostatic potential (dashed line) of Conf. 1. Inset: a sketch of the
GAO/STO heterostructure. (c) The orbital resolved partial DOS for L$\bar{3}$
TiO2 layer in STO.

##  III RESULTS AND DISCUSSIONS

###  III.1 GAO/STO heterostructures without oxygen vacancies

Firstly, we calculated the electron DOS of the GAO/STO heterostructures with
the eight configurations mentioned above. It is found that the Fermi levels of
the heterostructures with AlO/TiO2 interface lie in the bandgap no matter how
thick the GAO layer is. Thus the AlO-layer-terminated heterostructures are
always in the insulating state. The situation on GAO/STO heterostructures with
Al/TiO2 interface is different. Figure 2(a) and 2(b) show the total DOS of the
(GAO)n/(STO)5 heterostructures with Al/TiO2 interface for $n=1.5$ and 2.25,
respectively. The Fermi level for the $n=1.5$ heterostructure is in the
bandgap, indicating that the $n=1.5$ heterostructure is an insulator in the
band structure. The band structure of the $n=0.75$ heterostructure is similar
to that of the $n=1.5$. In contrast, the gap vanishes entirely and apparent
nonzero DOS can be observed near the Fermi level for the $n=2.25$
heterostructure [Fig. 2(b)]. That is, the $n=2.25$ heterostructure possesses
band structure of metals and similar phenomenon is also observed in the $n=3$
heterostructure. Therefore, the 2DEG will be formed in GAO/STO heterostructure
with Al/TiO2 interface when the thickness of GAO is greater than 1.5 uc
($n>1.5$), and the critical thickness of GAO for the presence of 2DEG in
GAO/STO heterostructure is $\sim$1.5 uc. Since the results in the $n=3$
heterostructure are similar to those in the $n=2.25$, we will systematically
present the results and discussions concerning the GAO/STO heterostructures
with Al/TiO2 interface and $n=2.25$ below.

Figure 3(a) shows the partial density of states projected onto atomic planes
in the STO and GAO of the $n=2.25$ heterostructure (Conf. 1). From this
figure, one can see that the states at the top of the valence band are mainly
contributed by the AlO layers, and the related valence-band edge gradually
shifts upward with increasing layer number from L1 to L5 layers (i.e., the
farther the AlO layer is from the interface, the greater the related valence-
band edge moves upward). On the other hand, the conduction-band edge of
L$\bar{1}$ to L$\bar{3}$ TiO2 layers is lower than the Fermi level, thus the
valence band of L5 to L9 AlO layers has an overlap with the conduction band of
L$\bar{1}$ to L$\bar{3}$ TiO2 layers. These band overlaps cause the electrons
to be transferred from the former to the latter, resulting in incompletely
filled bands, and hence the 2DEG near the interface.

As for the origin of the band overlap, the polar discontinuity at GAO/STO
interfaces is the most likely candidate. We have known that the GAO slab in
optimized Conf. 1 maintains the alternating structure of octahedral cation and
tetrahedral cation along the [001] direction [Fig. 1(b)]. The homogeneous
distribution of aluminum vacancies on the octahedral site leads to each AlO
layer having a $-1.5e$ charge, and then each tetrahedral layer containing only
aluminum cations has a charge of $+1.5e$. Therefore, the GAO can be
interpreted as stacking of atomic layers with an alternating charge of
$\sigma=\pm 1.5e$ [38] [see inset of Fig. 3(b)], and the GAO is a polar
crystal with nonvanishing electric dipole moment in the [001] direction [38,
39]. The main panel of Fig. 3(b) shows the plane average (solid curve) and
macroscopic average (dashed curve) electrostatic potential of Conf. 1 as a
function of the position in the direction normal to the interface. The
potential in GAO increases gradually along $z$ direction due to the
alternating polarity of the atomic layers, indicating the presence of a polar
electric field in GAO. Consequently, the polar electric field causes an upward
shift of the valence-band edge. By a linear fitting the macroscopic average
electrostatic potential [40], a polar electric field of 0.28 V/${\rm{\AA}}$ is
obtained. Using the electric field of 0.28 V/${\rm{\AA}}$ and the GGA+U band
gap of 3.0 eV for STO, one can obtain the critical thickness of GAO that is
$\sim$1.4 uc. This value is close to our result mentioned above. If the
electric field of 0.28 V/${\rm{\AA}}$ was completely compensated, 0.92
electrons per unit-cell area (e/A) (equivalently, carrier concentration value
is $1.47\times 10^{14}\,{\rm{cm}}^{-2}$) should be transferred from GAO to
STO. Integrating the DOS spectra from the bottom of the conduction band to the
Fermi level, one can obtain the carrier concentration $n$ of the
heterojunction. The carrier concentration in the slab with Conf. 1 is
$1.14\times 10^{14}\,{\rm{cm}}^{-2}$, which is close to the density of the
compensating electrons mentioned above. This result confirms that the polar
discontinuity induces the formation of 2DEG in Conf. 1, which is similar to
the case in LAO/STO heterostructures [41, 42]. The slight deviation between
the two values could be caused by the ionic polarization that partially
screens the polar electric field in GAO [43]. And we find that 66.4% of the
electrons occupy the Ti-3$d$ orbitals of the L$\bar{3}$ TiO2 layer. Therefore,
the metallic state of Conf. 1 is mainly contributed by the L$\bar{3}$ TiO2
layer.

Figure 4: Formation energies of oxygen vacancies at different atomic layers in
Conf. 1. The legend of 1VO@Conf. 1 represents one oxygen vacancy in the Conf.
1, the other follows the same convention. For the Conf. 1 containing two
oxygen vacancies, one is fixed in the L6 AlO layer and the other varies at
different atomic layers on the STO side. For the Conf. 1 containing one oxygen
vacancy, the circles mark the positions with the lower formation energy of the
oxygen vacancy on the STO side and the GAO side.

Figure 3(c) shows the orbital resolved partial DOS for the L$\bar{3}$ TiO2
layer. The electrons transferred from the AlO layer mainly occupy the
Ti-3$d_{xz}$ orbitals (about 77.1% of the electrons occupy these orbitals) and
partially occupy Ti-3$d_{xy}$ and Ti-3$d_{yz}$ orbitals. This is different
from LAO/STO heterostructures, where the preferential occupation of 2DEG is
the in-plane Ti-3$d_{xy}$ orbitals [44]. This anomalous behavior could be
attributed to the contact of the interfacial Ti cations with the tetrahedral
Al cations. The Al/TiO2 interface breaks the perovskite lattice symmetry and
changes the crystal field around the Ti ions near the interface. As a result,
the energy of $d_{xz}$/$d_{yz}$ orbitals is reduced and these orbitals become
the preferable states for the electrons at the spinel-perovskite interface.
The experimental results of Chikina  _et al._ [45] and Cao  _et al._ [46] also
indicate that the energy of the $d_{xz}/d_{yz}$ orbitals is less than that of
the $d_{xy}$ orbitals in GAO/STO heterostructures.

Figure 5: The partial density of states projected onto atomic planes in the
STO and GAO for (a) the ($\gamma$-Al2O3)2.25/(SrTiO2.95)5 heterostructures
with one oxygen vacancy on L$\bar{3}$ TiO2 layer (Conf. 2), and (b) the
($\gamma$-Al2O2.96)2.25/(SrTiO3)5 heterostructures with one oxygen vacancy on
L6 AlO layer (Conf. 3). The hollow arrow in (a) marks the electron
accumulation layer and the solid arrow points out the energy states of oxygen
vacancies.

###  III.2 Influence of oxygen vacancies on the electronic structure of
GAO/STO heterostructures

As mentioned in the Sec. I, some researchers have suggested that the formation
of 2DEG in GAO/STO heterostructures is caused by oxygen vacancies. Therefore,
the effect of oxygen vacancies on the formation of 2DEG in GAO/STO
heterostructures is further investigated. Considering that oxygen vacancies
could be located in AlO, SrO, and TiO2 layers, we construct
($\gamma$-Al2O2.96)2.25/(SrTiO3)5 heterostructures and
($\gamma$-Al2O3)2.25/(SrTiO2.95)5 heterostructures (each configuration
contains only one oxygen vacancy in the supercell). Firstly, we obtained the
optimized structures of these configurations. All crystal structures
experience similar structural distortions as that in Conf. 1. Considering the
GAO with Al layer termination at the interface, we calculate the formation
energy ($E_{f}$) of oxygen vacancies using the relation,

$E_{f}=E_{\rm{V}}-(E_{0}-n_{V}\mu_{\rm{O}}),$ (1)

where $E_{\rm{V}}$ and $E_{0}$ are the total energies of the interface
structure with and without oxygen vacancies, $n_{V}$ is the numbers of oxygen
vacancies, and $\mu_{\rm{O}}$ is the chemical potential of oxygen atom. Figure
4 shows the formation energy as a function of the position of the oxygen
vacancy. The square symbol represents the formation energy of the oxygen
vacancy for the supercell containing only one oxygen vacancy. Inspection of
Fig. 4 indicates that there are two energy minima, one lies in the L$\bar{3}$
TiO2 layer and the other in the L6 AlO layer. The partial density of states
projected onto atomic planes in the STO and GAO for oxygen vacancy located in
the L$\bar{3}$ TiO2 layer (designated as Conf. 2) and the L6 AlO layer
(designated as Conf. 3) are plotted in Fig. 5 (a) and 5 (b), respectively. For
Conf. 2, the top of the valence band also overlaps with the bottom of the
conduction band, which is similar to that in Conf. 1. Thus the 2DEG would also
appear in the heterostructure with structure of Conf. 2. However, the 2DEG in
Conf. 2 is mainly located in the L$\bar{7}$ TiO2 layer, which is different
from that in Conf. 1. In addition, the electrons transferred from the AlO
layer mainly occupy the Ti-3$d_{xy}$ orbitals. This is expected because the
L$\bar{7}$ TiO2 layer is far from the Al/TiO2 interface and located deeply in
the STO substrate. The carrier concentration of Conf. 2 is $5.18\times
10^{13}\,{\rm{cm}}^{-2}$, which is less than the concentration to fully
compensate the polar electric field of GAO ($1.47\times
10^{14}\,{\rm{cm}}^{-2}$). Two factors could be responsible for this
phenomenon: (1) the oxygen vacancies in the L$\bar{3}$ layer could affect the
lattice distortion near the interface and thus reduce the polar electric
field; (The calculated polar field of Conf. 2 is $\sim$0.24 V/${\rm{\AA}}$,
which is less than that of Conf. 1. Correspondingly the critical thickness of
GAO film for the existence of 2DES should be $\sim$1.6 uc.) (2) the electrons
released by the oxygen vacancies are found to form band gap states [marked by
the solid arrow in Fig. 5(a)] and consequently have no contribution to the
carrier concentration.

Table 1: Calculated carrier concentration $n$, average electron effective mass $\overline{m^{*}}/m_{e}$, $k\alpha$, and electron mobility $\mu$ of Conf. 1, 2, 3, 5, and (LAO)5/(STO)5 heterostructures. Configuration |  $n$ (cm-2) | $\overline{m^{*}}/m_{\rm{e}}$ | $k\alpha$ |  $\mu$ (cm2/Vs)
---|---|---|---|---
Conf. 1 | $1.14\times 10^{14}$ | 1.20 | 12.27 | $5.30\times 10^{4}$
Conf. 2 | $5.18\times 10^{13}$ | 0.69 | 14.22 | $1.52\times 10^{5}$
Conf. 3 | $2.05\times 10^{14}$ | 0.62 | 18.87 | $1.71\times 10^{5}$
Conf. 5 | $2.60\times 10^{14}$ | 0.60 | 19.94 | $1.80\times 10^{5}$
(LAO)5/(STO)5 | $1.32\times 10^{12}$ | 2.81 | 3.630 | $1.74\times 10^{4}$
Figure 6: (a) The band structure of ($\gamma$-Al2O2.96)2.25/ (SrTiO2.95)5
heterostructure with one oxygen vacancy in the L6 AlO layer and the other in
the L$\bar{3}$ TiO2 layer. (b) and (c) partial charge density isosurfaces for
the band 1 and band 2. (d) The band structure of
($\gamma$-Al2O2.96)2.25/(SrTiO2.95)5 heterostructure with one oxygen vacancy
in the L6 AlO layer and the other in the L$\bar{7}$ TiO2 layer. (e) and (f)
partial charge density isosurfaces for the band 1 and band 2. The isosurface
value is 0.045 e/${\rm{\AA}^{3}}$.

For Conf. 3, there is no any overlap between the top of the valence band and
the bottom of the conduction band [see Fig. 5(b)]. However, the Fermi level
shifts upward into the conduction band, and the slab still reveals metallic
state in band structure. Therefore, the 2DEG in Conf. 3 does not originate
from polar discontinuity at the GAO/STO interface but from the oxygen
vacancies in the GAO (similar to modulation doping). An oxygen vacancy in the
GAO/STO supercell could release one or two electrons to the STO [47, 48]. This
concentration of the transferred electron should vary from $1.57\times
10^{14}\,{\rm{cm}}^{-2}$ to $3.14\times 10^{14}\,{\rm{cm}}^{-2}$. The carrier
concentration of Conf. 3 is $2.05\times 10^{14}\,{\rm{cm}}^{-2}$ (Table 1),
indicating a partial ionization of the oxygen vacancies. The electrons
transferred from the oxygen vacancy in the GAO are mainly accumulated in the
L$\bar{3}$ TiO2 layer. The value of the carrier concentration in Conf. 3 is
greater than the concentration to fully compensate the polar electric field of
GAO ($1.47\times 10^{4}\,{\rm{cm}}^{-2}$, as mentioned above). Thus, the polar
electric field of GAO could be completely compensated. This suggests that the
formation of 2DEG in GAO/STO heterostructures would be irrelevant to the
thickness of the GAO slab. Then, the electronic structures of
($\gamma$-Al2O2.87)0.75/(SrTiO3)5 and ($\gamma$-Al2O2.94)1.5/(SrTiO3)5
(containing only one oxygen vacancy in the supercell on GAO side) are also
calculated, and both of them exhibit metallic characteristics. Thus, there
will be no critical thickness of GAO for the presence of 2DEG in the GAO/STO
heterostructures when the oxygen vacancies are located on the GAO side.
Experimentally, Chen  _et al._ performed during-growth transport measurements
for LAO/STO and related heterostructures and found that an amorphous film of
less than 0.04 nm monolayer coverage on STO can already generate a highly
conducting 2DEG [49]. The results demonstrate that the oxygen vacancies in GAO
film grown on STO play an important role in generating 2DEG. We also calculate
the electronic structures of the GAO/STO heterostructures with AlO and TiO2
terminations and one oxygen vacancy in the supercell, and find that all
configurations exhibit insulator characteristics in band structure.

Now, we consider the situation that in the supercell both the GAO and the STO
sides contain one oxygen vacancy, respectively. As the formation energy of
oxygen vacancy in the L6 AlO layer is the lowest (the filled square symbols in
Fig. 4), we fix one oxygen vacancy in the L6 AlO layer and change the position
of the other in the STO side for the ($\gamma$-Al2O2.96)2.25/(SrTiO2.95)5
heterostructures. The formation energy variation with the position of oxygen
vacancy is also shown in Fig. 4 (the filled circle symbols), from which one
can see that the formation energy does not change greatly with the change of
oxygen vacancy position. This means that oxygen vacancies can be formed in any
atomic layer on the STO side under low oxygen pressure.

Next, we focus on the electronic structures of
($\gamma$-Al2O2.96)2.25/(SrTiO2.95)5 heterostructures. It is found that the
($\gamma$-Al2O2.96)2.25/(SrTiO2.95)5 heterostructures show insulator
characteristics in band structure when the oxygen vacancy is located in
L$\bar{1}$, L$\bar{2}$, and L$\bar{3}$ layers, while for other cases, the
heterostructures exhibit metallic characteristics. We take two configurations,
in which the oxygen vacancy in the L$\bar{3}$ TiO2 layer (designated as Conf.
4) and L$\bar{7}$ TiO2 layer (designated as Conf. 5), respectively, as
examples to present the detailed results. The band structure of Conf. 4 along
high symmetry lines in the Brillouin zone is plotted in Fig. 6(a). The Fermi
level lies in the bandgap, indicating that the 2DEG cannot be present in this
configuration. Below the Fermi level, there are two discrete energy bands,
which are designated as band 1 and band 2 [as indicated in Fig. 6(a)] for
convenience. Figure 6(b) shows the k-resolved charge density isosurfaces for
band 1 in the entire two-dimensional Brillouin zone. The charge density of
band 1 exhibits a typical Ti-3$d$ $e_{g}$-based bonding orbital
characteristic, suggesting that the electrons are strongly localized at the
oxygen vacancy and its adjacent Ti atoms [see L$\bar{3}$ layer in Fig. 6(b)].
It is indicated that band 1 corresponds to the impurity band introduced by
oxygen vacancies, while band 2 is introduced by electrons transferred from GAO
to STO, and the flat band shows that the electron effective mass is large.
Figure 6(c) shows the k-resolved charge density isosurfaces for band 2. The
charge density is located on one of the Ti atoms in the L$\bar{1}$ TiO2 layer,
showing a rather complex $d$-orbital composition. It could be composed of
$d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{3z^{2}-r^{2}}$, and $d_{x^{2}-y^{2}}$
orbitals. Thus, the electrons transferred from GAO to STO form localized
states. In contrast to Conf. 2 with only one oxygen vacancy in L$\bar{3}$
layer, no 2DEG is formed in Conf. 4 with an additional oxygen vacancy in the
L6 layer. In fact, the additional oxygen vacancy could affect the crystal
field near the Al/TiO2 interface, which in turn causes the electrons
transferred from GAO to STO to be localized.

Figure 6(d) illustrates the band structure of Conf. 5. Unlike the electronic
structure of Conf. 4, the Fermi level for Conf. 5 shifts upward into the
conduction band, indicating that the 2DEG forms in heterostructures with this
configuration. The bands introduced by the oxygen vacancies in STO and the
electrons transferred from GAO to STO are also labeled as band 1 and band 2,
respectively. Figure 6(e) and 6(f) show the k-resolved charge density
isosurfaces for the two bands. From Fig. 6(e) one can see that the electrons
in L$\bar{7}$ are localized at the oxygen vacancy and its adjacent Ti atoms,
which is similar to that in L$\bar{3}$ layer of Conf. 4. Thus the oxygen
vacancies in L$\bar{7}$ layer of STO are deep donors. Inspection of Fig. 6(f)
indicates that the electrons transferred from GAO to STO are mainly located in
the L$\bar{3}$ TiO2 layer, occupy the Ti-3$d_{xy}$ orbitals, and are uniformly
distributed on each Ti atom. These electrons can only move freely in the
L$\bar{3}$ TiO2 layer, suggesting that the electrons transferred from GAO to
STO are confined in the L$\bar{3}$ TiO2 layer and form a 2DEG. It should be
noted that a few electrons are localized on a Ti atom in the L$\bar{1}$ layer.
The corresponding charge density isosurface region is far less than that of
Conf. 4. Compared with the oxygen vacancies in the L$\bar{3}$ layer, the
vacancies in the L$\bar{7}$ layer are farther away from the interface and have
a weak effect on the crystal field near the Al/TiO2 interface. Therefore, the
electronic structure of Conf. 5 is similar to that of Conf. 3. The above
results reveal that the formation of 2DEG in the GAO/STO heterostructures is
closely related to the position of oxygen vacancies.

###  III.3 The electron mobilities in GAO/STO heterostructures

In the framework of free-electron-like model, the mobility can be expressed as
$\mu=e\left\langle\tau\right\rangle/m^{*}$, where $e$, $\tau$, and $m^{*}$ are
the elementary charge, relaxation time, and electron effective mass,
respectively. Besides the relaxation time, the effective mass plays an
important role in determining the electron mobility. From the energy versus
wave vector dispersion near the bottom of the conduction band, we obtain the
electron effective masses along the $\Gamma$-M direction and the $\Gamma$-X
direction. Then the average effective masses can be obtained using
$\overline{m^{*}}=\sqrt{m_{\Gamma-{\rm{M}}}^{*}m_{\Gamma-{\rm{X}}}^{*}}$. The
average effective masses for Confs. 1, 2, 3, and 5 are listed in Table 1. For
comparison, we also calculated the band structure of n-type LAO/STO
heterostructure with configuration (LAO)5/(STO)5 (i.e., the five LAO layers
are stacked on the five STO layers). Our results concerning the energy band of
(LAO)5/(STO)5 are almost identical to those in Ref. [42]. The lowest band near
the $\Gamma$ point is composed of $d_{xy}$ orbitals, and $d_{yz}$ subbands
show small dispersions from $\Gamma$ to X points. The average effective mass
of (LAO)5/(STO)5 is also given in Table 1. For the GAO/STO heterostructures,
the effective mass of Conf. 1 is the largest, which is about 3/7 of that in
(LAO)5/(STO)5. While there is little difference in effective masses of Confs.
2, 3, and 5, which is about one-forth of that in (LAO)5/(STO)5. Hence, the
smaller effective masses must be one of the important reasons for the high
mobility of 2DEG in GAO/STO heterostructures.

According to Christensen  _et al._ [50], the electron mobility in GAO/STO
heterostructures is governed by ionized impurity scattering at low temperature
regime ($T<5$ K). The ionized impurity scattering has been treated
theoretically by Mansfield [51, 52, 53, 54]. When the screen radius ($\alpha$)
is far greater than the wavelength of electrons (i.e., $k\alpha\gg 1$ with $k$
being the wave vector), the ionized-impurities-scattering limited electron
mobility can be expressed as [51, 52, 53, 54]

$\mu=\frac{3\epsilon^{2}h^{3}n}{16\pi^{2}e^{3}Z^{2}N_{i}m^{*2}g(b)}$ (2)

with

$g(b)=\ln(1+b)-\frac{b}{1-b}$ (3)

and

$b=\frac{h^{2}\epsilon}{2e^{2}m^{*}}\left(\frac{3n}{8\pi}\right)^{1/3},$ (4)

where $\epsilon$ is the dielectric constant, $h$ is Planck’s constant, $n$ is
the carrier concentration, $Z$ is the charge of donors and $N_{i}$ is the
concentration of charged donor centers. The screening radius $\alpha$ can be
calculated by the Thomas-Fermi expression [54, 53, 55],

$\alpha=\left[\frac{\hbar^{2}\epsilon}{4m^{*}e^{2}}\left(\frac{\pi}{3n}\right)^{1/3}\right]^{1/2},$
(5)

where $\hbar$ is the Planck’s constant divided by 2$\pi$. The values of
$\alpha$ were obtained by substituting the $n$, $\overline{m^{*}}$, and
$\epsilon=18000$ (STO) into Eq. (5). Taking $k=k_{\rm{F}}$ with $k_{\rm{F}}$
being the Fermi wave vector, we obtain the values of $k\alpha$ and list them
in Table 1. Checking Table 1 indicates that the condition $k\alpha\gg 1$ is
satisfied for each listed configuration. Since the charge donors are partially
ionized, the values of $Z$ and ${N_{i}}$ are taken as 1 and $n$, respectively.
Thus the electron mobility for each configuration is obtained and listed in
Table 1. The electron mobility in Conf. 1 is about 3 times as large as that in
(LAO)5/(STO)5, while the electron mobilities in heterostructures with Confs.2,
3, and 5 are $\sim$9, 10, and 11 times as large as that in (LAO)5/(STO)5. In
addition, for the GAO/STO heterostructures, it is worth noting that there is a
certain distance between the electrons (forming 2DEG) and their donors in all
configurations with 2DEGs. For example, in Conf. 3, the 2DEG lies in the
L$\bar{3}$ TiO2 layer, which is 1.6 nm away from the oxygen vacancy lying in
the L6 AlO layer. This is equivalent to realizing the modulation doping [26],
which is a general way to suppress the donor scattering and improves the
mobility in semiconductor technology. Therefore, the electron mobility in each
GAO/STO heterostructure should be greater than the calculated value in Table
1. The modulation doping, together with the smaller effective masses, explains
why the higher mobility 2DEGs exist in the GAO/STO heterostructures (compared
to LAO/STO heterostructures).

##  IV CONCLUSIONS

In summary, the electronic structures of the [0 0 1]-oriented GAO/STO
heterostructures with TiO2 layer terminated on the interfaces are
systematically investigated by the first-principle calculations. It is found
that the necessary condition for the existence of 2DEG in the heterostructure
is the device possessing interface with TiO2 and Al layers terminations. For
GAO/STO heterojunction without oxygen vacancy on the GAO side, the polar
discontinuity near the interface is the main mechanism for the occurrence of
2DEG. When the oxygen vacancies are introduced into the GAO films, polar
electric field near the interface disappears and the oxygen vacancies become
the main source of the 2DEG. In GAO/STO heterostructure the ionized impurity
scattering is considered to govern the low-temperature mobility of the 2DEG,
and it is found that the mobility in GAO/STO heterostructure with proper
configurations can be one order of magnitude higher than that in LAO/STO
heterostructure. The high mobility of the 2DEG is mainly due to the small
electron effective mass in GAO/STO heterostructure.

###### Acknowledgements.

The calculation was conducted on the CJQS-HPC platform at Tianjin University.
This work is supported by the National Natural Science Foundation of China
through Grant Nos. 12174282 and 11774253.

## References

  * Ohtomo and Hwang [2004] A. Ohtomo and H. Y. Hwang, A high-mobility electron gas at the LaAlO3/SrTiO3 heterointerface, Nature 427, 423 (2004).
  * Nakagawa  _et al._ [2006] N. Nakagawa, H. Y. Hwang, and D. A. Muller, Why some interfaces cannot be sharp, Nat. Mater. 5, 204 (2006).
  * Herranz  _et al._ [2007] G. Herranz, M. Basletić, M. Bibes, C. Carrétéro, E. Tafra, E. Jacquet, K. Bouzehouane, C. Deranlot, A. Hamzić, J.-M. Broto, A. Barthélémy, and A. Fert, High mobility in LaAlO3/SrTiO3 heterostructures: origin, dimensionality, and perspectives, Phys. Rev. Lett. 98, 216803 (2007).
  * Kalabukhov  _et al._ [2007] A. Kalabukhov, R. Gunnarsson, J. Börjesson, E. Olsson, T. Claeson, and D. Winkler, Effect of oxygen vacancies in the SrTiO3 substrate on the electrical properties of the LaAlO3/SrTiO3 interface, Phys. Rev. B 75, 121404 (2007).
  * Bell  _et al._ [2009a] C. Bell, S. Harashima, Y. Kozuka, M. Kim, B. G. Kim, Y. Hikita, and H. Y. Hwang, Dominant mobility modulation by the electric field effect at the LaAlO3/SrTiO3 interface, Phys. Rev. Lett. 103, 226802 (2009a).
  * Caviglia  _et al._ [2010a] A. D. Caviglia, S. Gariglio, C. Cancellieri, B. Sacépé, A. Fête, N. Reyren, M. Gabay, A. F. Morpurgo, and J.-M. Triscone, Two-dimensional quantum oscillations of the conductance at LaAlO3/SrTiO3 interfaces, Phys. Rev. Lett. 105, 236802 (2010a).
  * Caviglia  _et al._ [2010b] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri, and J.-M. Triscone, Tunable rashba spin-orbit interaction at oxide interfaces, Phys. Rev. Lett. 104, 126803 (2010b).
  * Gunkel  _et al._ [2013] F. Gunkel, K. Skaja, A. Shkabko, R. Dittmann, S. Hoffmann-Eifert, and R. Waser, Stoichiometry dependence and thermal stability of conducting NdGaO3/SrTiO3 heterointerfaces, Appl. Phys. Lett. 102, 071601 (2013).
  * Perna  _et al._ [2010] P. Perna, D. Maccariello, M. Radovic, U. Scotti di Uccio, I. Pallecchi, M. Codda, D. Marré, C. Cantoni, J. Gazquez, M. Varela, S. J. Pennycook, and F. M. Granozio, Conducting interfaces between band insulating oxides: The LaGaO3/SrTiO3 heterostructure, Appl. Phys. Lett. 97, 152111 (2010).
  * Annadi  _et al._ [2012] A. Annadi, A. Putra, Z. Q. Liu, X. Wang, K. Gopinadhan, Z. Huang, S. Dhar, T. Venkatesan, and Ariando, Electronic correlation and strain effects at the interfaces between polar and nonpolar complex oxides, Phys. Rev. B 86, 085450 (2012).
  * Chen  _et al._ [2013] Y. Z. Chen, N. Bovet, F. Trier, D. V. Christensen, F. M. Qu, N. H. Andersen, T. Kasama, W. Zhang, R. Giraud, J. Dufouleur, T. S. Jespersen, J. R. Sun, A. Smith, J. Nygård, L. Lu, B. Büchner, B. G. Shen, S. Linderoth, and N. Pryds, A high-mobility two-dimensional electron gas at the spinel/perovskite interface of $\gamma$-Al2O3/SrTiO3, Nat. Commun. 4, 1371 (2013).
  * Christensen  _et al._ [2016] D. V. Christensen, F. Trier, M. von Soosten, G. E. D. K. Prawiroatmodjo, T. S. Jespersen, Y. Z. Chen, and N. Pryds, Electric field control of the $\gamma$-Al2O3/SrTiO3 interface conductivity at room temperature, Appl. Phys. Lett. 109, 021602 (2016).
  * Lee  _et al._ [2013] S. W. Lee, J. Heo, and R. G. Gordon, Origin of the self-limited electron densities at Al2O3/SrTiO3 heterostructures grown by atomic layer deposition – oxygen diffusion model, Nanoscale 5, 8940 (2013).
  * Schütz  _et al._ [2015] P. Schütz, F. Pfaff, P. Scheiderer, Y. Z. Chen, N. Pryds, M. Gorgoi, M. Sing, and R. Claessen, Band bending and alignment at the spinel/perovskite $\gamma$-Al2O3/SrTiO3 heterointerface, Phys. Rev. B 91, 165118 (2015).
  * Yang  _et al._ [2020] X. Yang, H. Su, and G. Wu, Orbital-adapted electronic structure and anisotropic transport in $\gamma$-Al2O3/SrTiO3 heterostructure, Phys. Rev. Mater. 4, 016001 (2020).
  * Wolff  _et al._ [2017] K. Wolff, R. Schäfer, M. Meffert, D. Gerthsen, R. Schneider, and D. Fuchs, Anisotropic electronic transport of the two-dimensional electron system in Al2O3/SrTiO3 heterostructures, Phys. Rev. B 95, 245132 (2017).
  * Reyren  _et al._ [2009] N. Reyren, S. Gariglio, A. D. Caviglia, D. Jaccard, T. Schneider, and J.-M. Triscone, Anisotropy of the superconducting transport properties of the LaAlO3/SrTiO3 interface, Appl. Phys. Lett. 94, 112506 (2009).
  * Reyren  _et al._ [2007] N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Ruetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J. Mannhart, Superconducting interfaces between insulating oxides, Science 317, 1196 (2007).
  * Caviglia  _et al._ [2008] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, Electric field control of the LaAlO3/SrTiO3 interface ground state, Nature 456, 624 (2008).
  * Cen  _et al._ [2008] C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C. S. Hellberg, J. Mannhart, and J. Levy, Nanoscale control of an interfacial metal–insulator transition at room temperature, Nat. Mater. 7, 298 (2008).
  * Bell  _et al._ [2009b] C. Bell, S. Harashima, Y. Kozuka, M. Kim, B. G. Kim, Y. Hikita, and H. Y. Hwang, Dominant mobility modulation by the electric field effect at the LaAlO3/SrTiO3 interface, Phys. Rev. Lett. 103, 226802 (2009b).
  * Thiel [2006] S. Thiel, Tunable quasi-two-dimensional electron gases in oxide heterostructures, Science 313, 1942 (2006).
  * Bert  _et al._ [2011] J. A. Bert, B. Kalisky, C. Bell, M. Kim, Y. Hikita, H. Y. Hwang, and K. A. Moler, Direct imaging of the coexistence of ferromagnetism and superconductivity at the LaAlO3/SrTiO3 interface, Nat. Phys. 7, 767 (2011).
  * Li  _et al._ [2011a] L. Li, C. Richter, J. Mannhart, and R. C. Ashoori, Coexistence of magnetic order and two-dimensional superconductivity at LaAlO3/SrTiO3 interfaces, Nat. Phys. 7, 762 (2011a).
  * Kormondy  _et al._ [2015] K. J. Kormondy, A. B. Posadas, T. Q. Ngo, S. Lu, N. Goble, J. Jordan-Sweet, X. P. A. Gao, D. J. Smith, M. R. McCartney, J. G. Ekerdt, and A. A. Demkov, Quasi-two-dimensional electron gas at the epitaxial alumina/SrTiO3 interface: Control of oxygen vacancies, J. Appl. Phys. 117, 095303 (2015).
  * Schütz  _et al._ [2017] P. Schütz, D. V. Christensen, V. Borisov, F. Pfaff, P. Scheiderer, L. Dudy, M. Zapf, J. Gabel, Y. Z. Chen, N. Pryds, V. A. Rogalev, V. N. Strocov, C. Schlueter, T.-L. Lee, H. O. Jeschke, R. Valentí, M. Sing, and R. Claessen, Microscopic origin of the mobility enhancement at a spinel/perovskite oxide heterointerface revealed by photoemission spectroscopy, Phys. Rev. B 96, 161409 (2017).
  * Lee  _et al._ [2012] S. W. Lee, Y. Liu, J. Heo, and R. G. Gordon, Creation and control of two-dimensional electron gas using Al-based amorphous oxides/SrTiO3 heterostructures grown by atomic layer deposition, Nano Lett. 12, 4775 (2012).
  * Ngo  _et al._ [2015] T. Q. Ngo, N. J. Goble, A. Posadas, K. J. Kormondy, S. Lu, M. D. McDaniel, J. Jordan-Sweet, D. J. Smith, X. P. A. Gao, A. A. Demkov, and J. G. Ekerdt, Quasi-two-dimensional electron gas at the interface of $\gamma$-Al2O3/SrTiO3 heterostructures grown by atomic layer deposition, J. Appl. Phys. 118, 115303 (2015).
  * Chen  _et al._ [2014] Y. Z. Chen, N. Bovet, T. Kasama, W. W. Gao, S. Yazdi, C. Ma, N. Pryds, and S. Linderoth, Room temperature formation of high-mobility two-dimensional electron gases at crystalline complex oxide interfaces, Adv. Mater. 26, 1462 (2014).
  * Lee and Demkov [2008] J. Lee and A. A. Demkov, Charge origin and localization at the n-type SrTiO3/LaAlO3 interface, Phys. Rev. B 78, 193104 (2008).
  * Mardegan  _et al._ [2019] J. R. L. Mardegan, D. V. Christensen, Y. Z. Chen, S. Parchenko, S. R. V. Avula, N. Ortiz-Hernandez, M. Decker, C. Piamonteze, N. Pryds, and U. Staub, Magnetic and electronic properties at the $\gamma$-Al2O3/SrTiO3 interface, Phys. Rev. B 99, 134423 (2019).
  * Gutiérrez  _et al._ [2001] G. Gutiérrez, A. Taga, and B. Johansson, Theoretical structure determination of $\gamma$-Al2O3, Phys. Rev. B 65, 012101 (2001).
  * Vijay  _et al._ [2002] A. Vijay, G. Mills, and H. Metiu, Structure of the (001) surface of $\gamma$ alumina, J. Chem. Phys. 117, 4509 (2002).
  * Pinto  _et al._ [2004] H. P. Pinto, R. M. Nieminen, and S. D. Elliott, Ab initio study of $\gamma$-Al2O3 surfaces, Phys. Rev. B 70, 125402 (2004).
  * Ouyang  _et al._ [2009] C. Y. Ouyang, Ž. Šljivančanin, and A. Baldereschi, First-principles study of $\gamma$-Al2O3 (100) surface, Phys. Rev. B 79, 235410 (2009).
  * Menéndez-Proupin and Gutiérrez [2005] E. Menéndez-Proupin and G. Gutiérrez, Electronic properties of bulk $\gamma$-Al2O3, Phys. Rev. B 72, 035116 (2005).
  * Lu  _et al._ [2016] S. Lu, K. J. Kormondy, T. Q. Ngo, T. Aoki, A. Posadas, J. G. Ekerdt, A. A. Demkov, M. R. McCartney, and D. J. Smith, Spectrum and phase mapping across the epitaxial $\gamma$-Al2O3/SrTiO3 interface, Appl. Phys. Lett. 108, 051606 (2016).
  * Christensen and Smith [2017] D. Christensen and A. Smith, Is $\gamma$-Al2O3 polar?, Appl. Surf. Sci. 423, 887 (2017).
  * Tasker [1979] P. W. Tasker, The stability of ionic crystal surfaces, J. Phys. C 12, 4977 (1979).
  * Harrison  _et al._ [1978] W. A. Harrison, E. A. Kraut, J. R. Waldrop, and R. W. Grant, Polar heterojunction interfaces, Phys. Rev. B 18, 4402 (1978).
  * Behtash  _et al._ [2016] M. Behtash, S. Nazir, Y. Wang, and K. Yang, Polarization effects on the interfacial conductivity in LaAlO3/SrTiO3 heterostructures: a first-principles study, Phys. Chem. Chem. Phys. 18, 6831 (2016).
  * Son  _et al._ [2009] W.-J. Son, E. Cho, B. Lee, J. Lee, and S. Han, Density and spatial distribution of charge carriers in the intrinsic n-type LaAlO3-SrTiO3 interface, Phys. Rev. B 79, 245411 (2009).
  * Li  _et al._ [2011b] Y. Li, S. N. Phattalung, S. Limpijumnong, J. Kim, and J. Yu, Formation of oxygen vacancies and charge carriers induced in the n-type interface of a LaAlO3 overlayer on SrTiO3 (001), Phys. Rev. B 84, 245307 (2011b).
  * Chikina  _et al._ [2018] A. Chikina, F. Lechermann, M.-A. Husanu, M. Caputo, C. Cancellieri, X. Wang, T. Schmitt, M. Radovic, and V. N. Strocov, Orbital ordering of the mobile and localized electrons at oxygen-deficient LaAlO3/SrTiO3 interfaces, ACS Nano 12, 7927 (2018).
  * Chikina  _et al._ [2021] A. Chikina, D. V. Christensen, V. Borisov, M.-A. Husanu, Y. Chen, X. Wang, T. Schmitt, M. Radovic, N. Nagaosa, A. S. Mishchenko, R. Valentí, N. Pryds, and V. N. Strocov, Band-order anomaly at the $gamma$-Al2O3/SrTiO3 interface drives the electron-mobility boost, ACS Nano 15, 4347 (2021).
  * Cao  _et al._ [2016] Y. Cao, X. Liu, P. Shafer, S. Middey, D. Meyers, M. Kareev, Z. Zhong, J.-W. Kim, P. J. Ryan, E. Arenholz, and J. Chakhalian, Anomalous orbital structure in a spinel–perovskite interface, npj Quantum Mater. 1, 16009 (2016).
  * Lin and Demkov [2013] C. Lin and A. A. Demkov, Electron correlation in oxygen vacancy in SrTiO3, Phys. Rev. Lett. 111, 217601 (2013).
  * Lopez-Bezanilla  _et al._ [2015] A. Lopez-Bezanilla, P. Ganesh, and P. B. Littlewood, Magnetism and metal-insulator transition in oxygen-deficient SrTiO3, Phys. Rev. B 92, 115112 (2015).
  * Chen  _et al._ [2019] Z. Chen, M. Zhang, T. Ren, and Y. Xie, Unravelling oxygen-vacancy-induced electron transfer at SrTiO3-based heterointerfaces by transport measurement during growth, J. Phys.: Condens. Matter 31, 505002 (2019).
  * Christensen  _et al._ [2018] D. Christensen, Y. Frenkel, P. Schütz, F. Trier, S. Wissberg, R. Claessen, B. Kalisky, A. Smith, Y. Chen, and N. Pryds, Electron mobility in $\gamma$-Al2O3/SrTiO3, Phys. Rev. Appl. 9, 054004 (2018).
  * Tufte and Chapman [1967] O. N. Tufte and P. W. Chapman, Electron mobility in semiconducting strontium titanate, Phys. Rev. 155, 796 (1967).
  * Chattopadhyay and Queisser [1981] D. Chattopadhyay and H. J. Queisser, Electron scattering by ionized impurities in semiconductors, Rev. Mod. Phys. 53, 745 (1981).
  * Frederikse and Hosler [1967] H. P. R. Frederikse and W. R. Hosler, Hall mobility in SrTiO3, Phys. Rev. 161, 822 (1967).
  * Verma  _et al._ [2014] A. Verma, A. P. Kajdos, T. A. Cain, S. Stemmer, and D. Jena, Intrinsic mobility limiting mechanisms in lanthanum-doped strontium titanate, Phys. Rev. Lett. 112, 216601 (2014).
  * March [1957] N. March, The Thomas-Fermi approximation in quantum mechanics, Adv. Phys. 6, 1 (1957).