# Accelerated Band Offset Prediction in Semiconductor Interfaces with DFT and
Deep Learning

Kamal Choudhary  Material Measurement Laboratory, National Institute of
Standards and Technology, Maryland, 20899, USA.  Electronic mail:
<EMAIL_ADDRESS>0000-0001-9737-8074 Kevin F. Garrity
0000-0003-0263-4157

###### Abstract

We introduce a computational framework to predict band offsets of
semiconductor interfaces using density functional theory (DFT) and graph
neural networks (GNN). As a first step, we benchmark DFT based work function
and electron affinity values for surfaces against experimental data with
accuracies of 0.29 eV and 0.39 eV. Similarly, we evaluate band offset values
using independent unit (IU) and alternate slab junction (ASJ) models leading
to accuracies of 0.45 eV and 0.22 eV respectively. During ASJ structure
generation, we use Zur’s algorithm along with a unified GNN force-field to
tackle the conformation challenges of interface design. At present, we have
300 surface work functions calculated with DFT, from which we can compute
44850 IU band offsets as well as 250 directly calculated ASJ band offsets.
Finally, as the space of all possible heterojunctions is too large to simulate
with DFT, we develop generalized GNN models to quickly predict band edges with
an accuracy of 0.26 eV. We use such models to predict relevant quantities
including ionization potentials, electron affinities, and IU-based band
offsets. We establish simple rules using the above models to pre-screen
potential semiconductor devices from a vast pool of nearly 1.4 trillion
candidate interfaces.

## Introduction

Interfaces are critical for a variety of technological applications including
semiconductor transistors and diodes, solid-state lighting devices, solar-
cells, data-storage and battery applications [1, 2, 3, 4, 5, 6]. While
interfaces are ubiquitous, predicting even basic interface properties from
bulk data or chemical models remains challenging. Furthermore, the continued
scaling of devices towards the atomic limit [7] makes interface properties
even more important. There have been numerous scientific efforts to model
interfaces with a variety of techniques including density functional theory
(DFT)[8, 9, 10, 11, 12, 13, 14, 15, 16, 17], force-field[18, 19, 20, 21],
tight-binding [22, 23, 24, 25] and machine learning techniques [16, 26, 27,
28]. However, to the best of our knowledge, there is no systematic
investigation of interfaces for a large class of structural variety and
chemical compositions. Most of the previous efforts focus on a limited number
of interfaces, and hence there is a need for a dedicated infrastructure for
data-driven interface materials design.

Some of the key quantities for determining interface properties are:
equilibrium geometries, energetics, work functions, ionization potentials,
electron affinities, band offsets, carrier effective masses, mobilities, and
thermal conductivities. Calculations of band offsets and band-alignment at
semiconductor heterojunctions are of special interest for device design.
Semiconductor device transport and performance depend critically on valence
band offsets ($\Delta E_{v}$) and conduction band offsets ($\Delta E_{c}$), as
well as interfacial roughness and defects [29, 30, 31]. Based on the band-
alignment, heterostructures can be categorized into three classes: (i) type-I
(straddling gap), (ii) type-II (staggered gap), and (iii) type-III (broken
gap). The type-I heterostructures are used for transistors, lasers and light-
emitting diode (LED) applications, type-II are used for photoabsorbers and
photocatalysts, and type-III are used for tunneling field effect transistors.

Experimentally, band offsets can be measured using optical spectroscopy, X-ray
photoelectron spectroscopy (XPS), ultraviolet photoelectron spectroscopy
(UPS), and electrical measurements [3]. However, these experiments can be
quite time and resource consuming. Additionally, the variability across
multiple reported measurements can be reasonably high. For example, the
reported AlN/GaN interface $\Delta E_{v}$ varies from 0.57 eV to 1.36 eV with
reported uncertainties of up to 0.24 eV[31]. In this respect, the computation
of band offsets can serve as a complementary tool to experimental analysis.
Nevertheless, the calculation of band offsets is rather challenging [32] and
has been an area of research for about a century [33, 34, 35]. Density
functional theory calculations are one of the most widely used techniques for
predicting band offsets, as they can describe the electronic and atomic
structure at the interface in a self-consistent manner. There are two main
approaches to predicting band offsets using DFT. The first is to directly
simulate the interface using either an alternating slab-junction
(ASJ)/superlattice or surface terminated junction (STJ)/slab vacuum geometry,
either of which requires a computationally expensive calculation for each pair
of materials. Alternatively, the independent unit (IU)/electron
affinity/Anderson’s model [13, 36] requires only independent surface
calculations of each material, greatly reducing computational cost but
ignoring specific interface effects. ASJ models were shown to be most accurate
in Ref. [13], but IU models are surprisingly competitive.

Importantly, the generation of an atomistic interface geometry is a
challenging task due to the high number of possible conformations and
configurations. There are several important factors determining an interface
such as: the selection of the lattice alignment, the relative
orientation/displacement between surfaces, the separation distance, point/line
defects, and the presence of interfacial charge transfer. Several previous
tools have attempted to address this challenge, including MPInterfaces [15],
TribChem [37] and QuantumATK [38] packages.

Moreover, DFT calculations of interfaces require initial pre-relaxed bulk
structures which in this work are obtained from the Joint Automated Repository
for Various Integrated Simulations (JARVIS)-DFT [39, 40] database containing
nearly 80000 bulk 3D and 1100 2D materials. The JARVIS-DFT originated about 5
years ago and contains millions of properties materials and has carefully
converged atomic structures with tight convergence parameters, various
exchange-correlation functionals such as OptB88vdW[41], TBmBJ[42], R2SCAN [43]
and HSE06 [44]. JARVIS-DFT contains metallic, semiconducting, insulator,
superconductor , high-strength, topological, solar , thermoelectric,
piezoelectric , dielectric, two-dimensional, magnetic, porous, defect and
various other classes of bulk materials [45, 46]. We have also previously
looked at the band alignment of layered two dimensional materials using
JARVIS-DFT [16]. However, three dimensional systems with chemical bonding
between the materials require much greater effort, as the interfacial bonding
has a much greater effect of the interface properties, and the determination
of even a single interface structure is a challenging task. Out of above
material class combinations, semiconductor-semiconductor are of special
interest for this work. As DFT calculations can be time-consuming for surfaces
and interfaces machine-learning (ML)/deep learning (DL) techniques based on
DFT data can be used to accelerate atomistic predictions [47, 48]. Such models
have often been applied for bulk property predictions and their applicability
for defects and interfaces remains an open question. Several machine learning
tools available in JARVIS such as classical force-field inspired descriptors
(CFID) [40], atomistic line graph neural network (ALIGNN) [49], computer
vision for atomistic images (AtomVision) [50] and natural language processing
for chemistry (ChemNLP) [51] can be used in this regards to accelerate the
interface design tasks. In particular, ALIGNN has been used to develop several
fast surrogate models for property predictions as well as a unified force-
field for fast structure optimizations.

Most importantly, for all the above predictions, it is important to benchmark
and quantify error with respect to experimental data to gain confidence in
predictions. This work addresses the above challenges and provides a
streamlined framework for semiconductor interface design (InterMat). Although
focusing on semiconductors, this work has relevance to other applications such
as battery, data-storage, and solar-cell devices. We believe that this work
will be a precursor to more thorough theoretical and experimental
investigations of semiconductor interfaces.

## Results and discussion

Figure 1: Schematic overview of the workflow. The number of possible
semiconductor-semiconductor interfaces is exceedingly large. The workflow aims
to provide a toolkit to generate interface structures and use multi-fidelity
methods to accelerate interface/heterostructure design. Figure 2: ALIGNN based
regression models for a) VBM for JARVIS-DFT 3D/bulk materials test set data,
b) CBM for bulk materials test set data, c) slab surface VBMs not part of the
training set to evaluate extrapolation strength, d) slab surface CBMs not part
of the training set to evaluate extrapolation strength. Figure 3: A few
examples of band alignment based on the ionization potential (IP) and electron
affinity (EA) for Anderson’s/independent unit (IU) model. a) Density
functional theory based alignments can be used to obtain band offsets. b) Fast
ALIGNN based band alignment predictions. The numbers in the green bars
represent electron affinity while that in the blue bars represent ionization
potentials. The trained models based band alignment will be available at
JARVIS-Heterostructure website soon.

A schematic overview of the combinatorial problem of interfaces and our
approach is shown in Fig. 1. Initial atomic structures are obtained from the
JARVIS-DFT repository. Out of 80000 materials available in JARVIS-DFT, we
select 20901 semiconductors with OptB88vdW bandgap thresholds from 0.1 eV and
6 eV. Using a maximum Miller index (M) of 1, 2, and 3, the number of
symmetrically distinct surface slabs are 186847, 591639 and 1642584,
respectively. Using these surfaces, the number of binary interface systems
that can be generated are 17.5 billion, 175 billion and 1.4 trillion.
Including different surface terminations and defects at the interfaces can
lead to further possible combinations. It is unrealistic to analyze such a
large search space to find combinations for devices by using conventional
experimental or computational techniques. We will instead use the ALIGNN
models to guide and prioritize DFT calculations among the vast pool of
candidates. We develop machine learning models for fast predictions of valence
band maxima (VBM) and conduction band minima (CBM) using ALIGNN. These
predictions can be further used for fast IU based band alignment using
electron affinity/Anderson’s rule [36]. To assess the strengths and
limitations of such models, we develop a surface dataset for IU models and an
interface datasets for ASJ models using DFT. We particularly focus on
industrially relevant semiconductors including group IV (C, Si, Ge etc.), III-
IV (AlN, GaN, GaAs, GaP, InSb etc.), II-VI (CdS, CdSe, ZnO, ZnS etc.). We also
assess the strengths and limitations of IU and ASJ models against experimental
measurements. The DFT dataset hence generated for surfaces and interfaces can
then be fed back to fine tune the ALIGNN models to further improve accuracy.

### DFT dataset of work function, electron affinity, ionization potential and
surface energy

Table 1: Work function ($\phi$, eV), electron affinity ($\chi$, eV) and surface energy ($\gamma$, $Jm^{-2}$) of a few unreconstructed non-polar surface slabs from OptB88vdW (OPT) against experimental data. The IDs represnt JARVIS-DFT identifiers. System | IDs | Miller |  $\phi$ (OPT) |  $\phi$ (Exp) | $\chi(OPT)$ |  $\chi$(Exp) |  $\gamma$ (OPT) |  $\gamma$ (Exp)
---|---|---|---|---|---|---|---|---
Si | 1002 | 111 | 5.00 | 4.77[52] | 4.10 | 4.05[53] | 1.60 | 1.14[54]
Si | 1002 | 110 | 5.30 | 4.89[52] | 4.10 | - | 1.66 | 1.9[54]
Si | 1002 | 001 | 5.64 | 4.92[52] | 3.60 | - | 2.22 | 2.13[55]
C | 91 | 111 | 4.67 | 5.0[56] | -2.9 | - | 5.27 | 5.50[57]
Ge | 890 | 111 | 4.87 | 4.80[58] | 5.2 | 4.13[29] | 0.99 | 1.30[55]
SiGe | 105410 | 111 | 4.93 | 4.08[59] | 4.5 | - | 1.36 | -
SiC | 8118 | 001 | 5.26 | 4.85[60] | 1.3 | - | 3.51 | -
GaAs | 1174 | 110 | 4.89 | 4.71[61] | 4.40 | 4.07[53] | 0.67 | 0.86[54]
InAs | 1186 | 110 | 4.85 | 4.90[61] | 4.9 | 4.9[29] | 0.57 | -
AlSb | 1408 | 110 | 5.11 | 4.86[61] | 3.70 | 3.65[29] | 0.77 | -
GaSb | 1177 | 110 | 4.48 | 4.76[61] | 3.70 | 4.06[29] | 0.71 | -
AlN | 39 | 100 | 5.56 | 5.35[60] | 1.3 | 2.1 [62] | 2.27 | -
GaN | 30 | 100 | 5.74 | 5.90[63] | 2.8 | 3.3[64] | 1.67 | -
BN | 79204 | 110 | 6.84 | 7.0[65] | 1.4 | - | 2.41 | -
GaP | 1393 | 110 | 5.31 | 6.0[60] | 4.0 | 4.3[53] | 0.88 | 1.9[54]
BP | 1312 | 110 | 5.61 | 5.05[66] | 2.8 | - | 2.08 | -
InP | 1183 | 110 | 5.17 | 4.65[61] | 4.10 | 4.35[53] | 0.73 | -
CdSe | 1192 | 110 | 5.70 | 5.35[67] | 6.4 | - | 0.38 | -
ZnSe | 96 | 110 | 5.67 | 6.00[68] | 5.4 | - | 0.44 | -
ZnTe | 1198 | 110 | 5.17 | 5.30[69] | 4.10 | 3.5[29] | 0.36 | -
Al | 816 | 111 | 4.36 | 4.26[56] | - | - | 0.82 | -
Au | 825 | 111 | 5.5 | 5.31[56] | - | - | 0.90 | -
Ni | 943 | 111 | 5.35 | 5.34[56] | - | - | 2.02 | 2.34[70]
Ag | 813 | 001 | 4.5 | 4.2[56] | - | - | 0.99 | -
Cu | 867 | 001 | 4.7 | 5.1[56] | - | - | 1.47 | -
Pd | 963 | 111 | 5.54 | 5.6[56] | - | - | 1.57 | -
Pt | 972 | 001 | 5.97 | 5.93[56] | - | - | 1.94 | -
Ti | 1029 | 100 | 3.84 | 4.33[56] | - | - | 2.27 | -
Mg | 919 | 100 | 3.76 | 3.66[56] | - | - | 0.35 | -
Na | 931 | 001 | 2.97 | 2.36[56] | - | - | 0.10 | -
Hf | 802 | 111 | 3.7 | 3.9[56] | - | - | 2.02 | -
Co | 858 | 001 | 5.22 | 5.0[56] | - | - | 3.49 | -
Rh | 984 | 001 | 5.4 | 4.98[56] | - | - | 2.46 | -
Ir | 901 | 100 | 5.85 | 5.67[56] | - | - | 2.77 | -
Nb | 934 | 100 | 3.87 | 4.02[56] | - | - | 2.41 | -
Re | 981 | 100 | 4.96 | 4.72[56] | - | - | 2.87 | -
Mo | 21195 | 100 | 4.17 | 4.53[56] | - | - | 3.30 |
Zn | 1056 | 001 | 4.27 | 4.24[71] | - | - | 0.36 | -
Bi | 837 | 001 | 4.31 | 4.34[56] | - | - | 0.65 | 0.43[72]
Cr | 861 | 110 | 5.04 | 4.5[56] | - | - | 3.31 | -
Sb | 993 | 001 | 4.64 | 4.7[56] | - | - | 0.67 | -
Sn | 1008 | 110 | 4.82 | 4.42[56] | - | - | 0.91 | -
MAE | - | - | 0.29 | - | 0.39 | - | 0.34 | -

We develop a dataset of non-polar unreconstructed slab surfaces using the
JARVIS-DFT workflow. We carefully benchmark work function ($\phi$), electron
affinity ($\chi$) and surface energy ($\gamma$) values against experimental
measurements from the literature. Recently, it was shown that vacuum and slab
thicknesses of at least 10 Å are sufficient for surface models [72]. Usually,
both of these parameters should be converged for each surface, but here for
computational cost reasons, we use a fixed value for the thickness and vacuum
padding. We use a 1.6 nm thickness and 1.2 nm vacuum in this work. During the
DFT calculations, the converged K-point [73] values from the relevant bulk
calculations are used for surfaces also. We optimize the internal coordinates
of these surfaces keeping the cell volume constant. We obtain the OptB88vdW
VBM and vacuum level (from the maximum value of average electrostatic
potential) of surface slabs using DFT. Subtracting the vacuum level from the
VBM provides ionization potential (IP) information. Then, we add the bandgap
($E_{g}$) of the material to the ionization potential to get the electron
affinity (EA, $\chi$). In order to benchmark our surface dataset, we compare
work functions of these surfaces in Table. 1 which shows excellent agreement
with experiments with a mean absolute error value of 0.29 eV, consistent with
previous benchmarking effortsLABEL:PhysRevB.94.235418. Similarly, we obtain a
mean absolute error of 0.39 eV and 0.34 $Jm^{-2}$ for electron affinity and
surface energy data respectively. Currently, there are 300 surface data
calculated using the workflow and the dataset is still growing. Using, these
300 surfaces, 44850 IU-band offsets can be predicted. Also, we plan to include
reconstructured and polar surfaces in the future.

### ALIGNN models for bulk and surface band edges

We seek to accelerate the prediction of band edges using ML models, but the
absolute prediction of band edges relative to vacuum requires DFT calculations
with surfaces, which significantly limits our potential dataset. The JARVIS-
DFT dataset contains a much larger dataset of three dimensional materials with
CBM and VBM values using the VASP convention that the average electrostatic
potential of a unit cell is set to zero. We first train an ML model using
ALIGNN based on these bulk quantities, and then below we will show that the
model is useful for predictions of relative work function and band edges.

In ALIGNN, a crystal structure is represented as a graph using atomic elements
as nodes and atomic bonds as edges. Each node in the atomistic graph is
assigned 9 input node features based on its atomic species: electronegativity,
group number, covalent radius, valence electrons, first ionization energy,
electron affinity, block and atomic volume. The inter-atomic bond distances
are used as edge features with radial basis function up to 8 Å cut-off and a
12-nearest-neighbor ($N$). This atomistic graph is then used for constructing
the corresponding line graph using interatomic bond-distances as nodes and
bond-angles as edge features. ALIGNN uses edge-gated graph convolution for
updating nodes as well as edge features using a propagation function ($f$) for
layer ($l$), atom features ($h$), and node ($i$), details of which can be
found in Ref. [49, 74]:

$h_{i}^{(l+1)}=f(h_{i}^{l}{\\{h_{j}^{l}}\\}_{{}_{i}})$ (1)

We split each dataset into 90:5:5 train:validation:test parts. We train on 90
% train data and evaluate the validation and test data using ALIGNN. We find a
mean absolute error (MAE) of 0.28 ev for CBM and VBM while the data can vary
from -10 to 10 eV suggesting that the model should be reasonable for
predictions. The mean absolute deviation (MAD) for the CBM and VBM are 2.08 eV
and 2.67 eV, respectively, so the MAD:MAE is nearly 10 relative to a trivial
baseline model. Out of several other material properties trained using ALIGNN
[49], CBM/VBM models has one of the higher MAD:MAE ratios, especially compared
to other electronic properties like the band gap. We show the CBM and VBM
prediction models in Fig.2a and Fig.2b respectively. We believe with more data
using the active learning loop we can further increase the MAD:MAE in future
[47].

Next, we evaluate the bulk-trained ALIGNN models on the DFT surface dataset
and show results in Fig.2c and Fig.2c for CBM and VBM respectively.
Interestingly, most of the data points are on x = y line suggesting excellent
agreement. We find MAE values of 0.55 eV and 0.96 eV for VBM and CBM
respectively. It is interesting to note that the error in CBM is higher than
that of VBM, perhaps an explanation why predicting bandgaps of materials using
machine learning is ever-standing difficult problem. We only use the VBM
predicted from the ALIGNN model for the slab and add the bandgap value of the
bulk material from JARVIS-DFT to get the CBM. Note the AI model was not
trained on any surface structures suggesting that our models have high
extrapolation power. Similar successful extrapolations for bulk-trained models
were observed in Ref. [75] for vacancies.

In order to translate the results from the ALIGNN model to absolute band
energies relative to vacuum, we need access to the vacuum levels.
Interestingly, we find that for the fixed slab and vacuum thicknesses used in
this work, the vacuum levels are clustered around 5 eV even across different
material chemistries. Because the vacuum values are reasonably constant, it is
possible to predict relative band alignments using the ALIGNN model. Relative
band alignments are more important than absolute values for classification
into the three heterostructure types.

### DFT and ALIGNN based IU band alignment

IU band alignment, also known as Anderson’s rule [36], predicts semicondcutor
band offsets at interfaces using only the IP and EA data from independent
surface calculations. For a semiconductor heterojunction between A and B, the
conduction band offset can be given by:

$\Delta E_{c}=\chi_{B}-\chi_{A}$ (2)

Similarly, the valence band offset is given by:

$\Delta E_{v}=(\chi_{A}+E_{gA})-(\chi_{B}+E_{gB})$ (3)

In Fig.3a, we show the DFT-based IU band aligments for a set of well-known
semiconductor surfaces. We also include dotted lines for the energy levels of
H2 and H2O, which are relevant for photo-catalyst applications. We compare the
DFT based IU band offsets for 8 interfaces in Table 2 against experiments. We
find a mean absolute error of 0.45 eV which is similar to a value of 0.32 eV
as found in ref. [13] for different systems.

We visualize IU based band alignment using ALIGNN model in Fig. 3b. We observe
that the overall trends of DFT and ALIGNN closely resemble each other. For
these surfaces, we calculate the classification accuracy of the
heterostructures in type-I, type-II and type-III heterostructures. We find
precision scores of 66.7 %, 66.1 % and 58.2 % respectively. Precision is
defined as the fraction of relevant instances among all of the retrieved
instances. The classification precision scores are based on DFT optimized
surfaces, which are not available for all the materials in the database. So,
we generate structures directly from the bulk counterparts, relax them using
ALIGNN-FF and then predict the electron affinity and ionization potentials
using the procedure mentioned above. In this way, we find precision scores of
55.0 %, 63.4 % and 60.0 % respectively suggesting that structure optimization
of surfaces has an impact on the ALIGNN predictions. Still, sufficiently high
precision scores suggest that such models can be used for pre-screening
applications followed by density functional theory calculations and
experiments. Also, as the DFT bulk, surface and interface dataset is growing
continuously, there is plenty of scope to improve the model performance in the
future. For the 1.4 trillion semiconductor interfaces, we find 294 billion as
type-I, 322 billion as type-II and rest as type-III heterostructures using the
ALIGNN+IU model. The results suggest that finding type-I interfaces for
transistor applications is more challenging than other heterostructure
applications. Having such a large number of options and further screened for
desirable properties such as effective masses, dielectric, piezoelectric,
thermoelectric properties etc. can be helpful for technological applications.
We emphasize the point that AI models should be considered as a pre-screening
step only and would require thorough DFT and/or experimental validation.

### Interface geometry generation and alternate slab junction (ASJ) model

Figure 4: Structure generation criteria selection and initial xy-plane scan
with ALIGNN-FF for Si(110)/GaAs(110) interface. a) All the parameters such as
mismatch in x-direction (u), mismatch in y-direction (v) and maximum allowed
are optimized to generate suitable structures. b) To save computational cost,
a pre-screening of the optimum xy-plane scan is carried out using ALIGNN-FF
and then subjected to computational method such as DFT. c) electrostatic
potential of the system. d) atom projected density of states for the system,
separated along the z direction. Red, green and blue colors represent gallium,
arsenic and silicon atom contributions respectively.

To go beyond the limits of the IU model, we next consider explicit DFT
calculations of interfaces, which first require generating candidate interface
structures. This can be done in either of the following two ways: 1) by
attaching the two surface slabs together without vacuum padding, which leads
to superlattice or alternating slab junction (ASJ) models, or 2) by attaching
the two surface slabs with vacuum padding, surface terminated junctions (SJT).
Currently, we have interface properties calculated with the ASJ approach. We
first obtain the bulk atomic structures as well as convergence data from
JARVIS-DFT. We generate the interfaces following the Zur et. al. algorithm
[76]. The Zur algorithm generates a number of superlattice transformations
within a specified maximum surface area and also evaluates the length and
angle between film and substrate superlattice vectors to determine if they can
match within a tolerance. This algorithm is applicable to different crystal
structures and their surface orientations. We use a maximum lattice mismatch
of 8 %, maximum area of 300 ${\textup{\AA}}^{2}$, and maximum angle tolerance
of 1 degree. Note that in previous studies, lattice mismatch of 20 % has been
reported[11, 77]. An example of the application of the algorithm to the
Si(110)/GaAs(110) interface is shown in Fig. 4a. After eliminating structures
with area higher than max-area tolerance and structures with mismatch angle
more than the specified angle threshold, we then choose the remaining
structure (if any) with the minimum mismatch lattice vector lengths.

Thus far, we have determined a candidate unit cell, but the relative alignment
of the structures in the in-plane, as well as the terminations still need to
be decided. We perform a grid search of possible in-plane alignments with a
space of 0.05 fractional coordinates to determine the initial structure for
further relaxation. Doing such a large number of calculation with DFT would be
prohibitive, so we use ALIGNN-FF [74] to identify the starting in-plane
alignment. Additionally, ALIGNN-FF can be used to pre-relax the structure as
well. ALIGNN-FF is a universal force field ML model developed using JARVIS-DFT
data with 307113 structures and can be used to model combinations of 89
elements from the periodic table. An example of ALIGNN-FF predictions of an
in-plane grid serach is shown in Fig. 4b. Here high-peaks usually represent
too close atoms during the translation operations, which should be avoided.
After the ALIGNN-FF calculations, a full DFT relaxation is performed.

For computational purposes, it is important to have a unique identifier for an
interface. While generating the interfaces, we use a naming convention to
include a) material IDs (such as JVASP-1002 for Si and JVASP-1174 for GaAs),
b) film and substrate Miller indices (such as 110 for each), c) film and
substrate thickness values (such as 16 Å each), d) separation between these
two surface slabs (such as 2.5 Å for an ASJ model, 18 Å for STJ interface
models), e) relative displacement in xy-plane (such as a displacement vector
of [0.5, 0.2]), f) calculator method (such as DFT (VASP), ALIGNN-FF etc.)
giving rise to an interface with a name such as Interface-
JVASP-1002_JVASP-1174_film_miller_1_1_0_sub_miller_1_1_0_film_thickness_16_subs_thickness_16_separation_2.5_disp
_0.5_0.2_vasp. Such a scheme helps to reproduce the unique interfaces. Of
course, realistically, more complex parameters for an interface can be
important such as terminations, reconstructions, misfit-dislocation, vacancies
on the interface etc., but they can be easily included in the naming scheme as
well later.

After selecting a good guess of the interface using the above approach, DFT
calculations are performed to calculate the band offset value. During the DFT
calculations, maximum value of the two converged K-point and energy cutoff for
constituent bulk materials[73] are used. An example of the Si(110) and
GaAs(110) interface along with its DFT electrostatic potential is shown in
Fig. 4c. Furthermore, we can project the electron density of states across the
cell dimension in Fig.4d to show how electronic states are distributed along
the interface region. Clearly, at the interface region, the GaAs band gap
decreases due to the Si part. Such analysis can also help understand the band
alignment but from the atomic contribution of electronic states perspective
and over the length-scale of the device.

After determining the optimized geometric structure for the interface using
DFT, we obtain band offset data using formalism detailed in Ref. [8]. As an
example, we show a detailed analysis of Si(110)/GaAs(110) and
AlN(001)/GaN(001) in Fig. 5. In Fig. 5a, we show the atomic structure of the
ASJ based heterostructure of Si(110)/GaAs(110). The left side (with blue
atoms) represents the Si and the right side is the GaAs region. In Fig. 5c, we
show the electrostatic potential profile, averaged in-plane, of the interface.
The approximately sinusoidal profile on both regions represents the presence
of atomic layers. The cyan lines show the region used to define the repeat
distance, $L$, used for averaging in each material (see below). The red and
green lines show the average potential profiles for the left and right parts
using the repeat distance. The valence band offset ($\Delta E_{v}$) of an
interface between semiconductor A and B, $\Delta E_{v}$ is obtained using eq.
4. The difference in the averages for the left and right parts gives the
$\Delta V$ term. Now the bulk VBMs of the left and right parts are also
calculated to determine the $\Delta E$. The sum of these two quantities gives
the band offset that can be compared to experiments.

$\Delta E_{v}(A/B)=(E_{v}^{B}-E_{v}^{A})+\Delta V$ (4)

$\Delta V=\bar{\bar{V}}_{A}-\bar{\bar{V}}_{B}$ (5)

Figure 5: Atomic structures and band-alignment using the average electrostatic potential of metal-semiconductor and semiconductor-semiconductor interfaces. a) atomic structure view of Si/GaAs(110), b) atomic structure view of polar interface AlN/GaN(001), c) electrostatic potential profile for Si/GaAs(110), d) average electrostatic potential profile for polar surfaces of AlN/GaN (001). The cyan lines are used to find the repeat unit layers for the left and right parts. The linear parts of the red and green lines show the average potential profiles for the left and right parts using the repeat distance mentioned above. The dotted vertical blue line denotes the interface. Table 2: Band offsets (in eV) of a few independent unit (IU)/Anderson’s model and alternating slab-junction (ASJ) based semiconductor/semiconductor interfaces with OptB88vdW (OPT) and R2SCAN functionals in comparison to previously reported experiments. Here ID, Miller and P represent a JARVIS-DFT identifier, Miller index and polar surface interfaces respectively. System | ID | Miller | IU (OPT) | ASJ (OPT) | ASJ (R2SCAN) | Exp
---|---|---|---|---|---|---
AlP/Si | 1327/1002 | 110/110 | 1.24 | 0.88 | 1.04 | 1.35 [13]
GaAs/Si | 1174/1002 | 110/110 | 0.30 | 0.31 | 0.39 | 0.23 [78]
CdS/Si | 8003/1002 | 110/110 | 3.22 | 1.48 | 1.70 | 1.6 [79]
AlAs/GaAs | 1372/1174 | 110/110 | 0.60 | 0.48 | 0.50 | 0.55 [80]
CdS/CdSe | 8003/1192 | 110/110 | 0.35 | 0.10 | 0.11 | 0.55 [81]
InP/GaAs | 1183/1174 | 110/110 | 0.25 | 0.72 | 0.75 | 0.19[82]
ZnTe/AlSb | 1198/1408 | 110/110 | 0.8 | 0.25 | 0.33 | 0.35 [83]
CdSe/ZnTe | 1192/1198 | 110/110 | 1.8 | 0.58 | 0.67 | 0.64[84]
InAs/AlAs | 1186/1372 | 110/110 | - | 0.46 | 0.39 | 0.5 [85]
InAs/AlSb | 1186/1408 | 110/110 | - | 0.05 | 0.16 | 0.09 [86]
ZnSe/InP | 96/1183 | 110/110 | - | 0.13 | 0.18 | 0.41 [87]
InAs/InP | 1186/1183 | 110/110 | - | 0.11 | 0.09 | 0.31[82]
ZnSe/AlAs | 96/1372 | 110/110 | - | 0.38 | 0.45 | 0.4 [88]
GaAs/ZnSe | 1174/96 | 110/110 | - | 0.72 | 0.80 | 0.98 [89]
ZnS/Si | 10591/1002 | 001/001 | - | 0.92 | 1.16 | 1.52 [90]
Si/SiC | 1002/8118 | 001/001 | - | 0.51 | 0.47 | 0.5 [91]
GaN/SiC (P) | 30/8118 | 001/001 | - | 1.12 | 1.37 | 0.70 [92]
Si/AlN (P) | 1002/30 | 001/001 | - | 3.51 | 3.60 | 3.5 [93]
GaN/AlN (P) | 30/39 | 001/001 | - | 0.80 | 0.86 | 0.73 [94]
AlN/InN (P) | 39/1180 | 001/001 | - | 1.24 | 1.07 | 1.81 [95]
GaN/ZnO (P) | 30/1195 | 001/001 | - | 0.51 | 0.46 | 0.7 [96]
MAE | - | - | 0.45 | 0.22 | 0.23 | -

Here, ${E_{v}^{A}}$ (${E_{v}^{B}}$) represents the position of the VBM with
respect to the average electrostatic potential in the bulk material A (B), and
$\Delta V$ represents dipole potential or the difference between the
macroscopic-averaged electrostatic potential between A and B. Moreover,
$\bar{\bar{V}}$ is the average along the repeat unit $L$ of $\bar{V}$, which
is the planar averaged electrostatic potential:

$\bar{\bar{V}}(z)=\frac{1}{L}\int_{-L/2}^{L/2}\bar{V}(z+z^{{}^{\prime}})dz^{{}^{\prime}}$
(6)

$\bar{V}$ is given by:

$\bar{V}(z)=\frac{1}{S}\int_{S}V(x,y,z)dxdy$ (7)

where, $L$ is the distance between repeat units and $S$ is the area which is
parallel to the interface. The corresponding conduction-band offset can be
determined by using band-gap values from the respective bulk calculations, or
experimental data. We will use the convention that a positive value of the
valence-band offset at an interface A/B indicates that the VBM is higher in
material B. For the GaAs/Si interface we obtain $\Delta E_{v}$ of 0.31 eV and
0.39 eV using OptB88vdW and R2SCAN functionals, respectively, which is in
close agreement with the experimental value of 0.23 eV.

Next, we show a polar semiconductor heterojunction example for AlN
(001)/GAN(001) interface in Fig. 5b. The electrostatic potential profile is
shown in Fig. 5d. In contrast to flat average potential values in Fig. 5c, we
observe inclined profiles for this system representing indicating the presence
of a constant electric field. We fit lines for both sides and extrapolate to
the interface. The difference of the lines at the interface gives $\Delta V$.
The calculation of $\Delta E$ remains the same as the non-polar case. These
calculations are automated in the workflow, however, it is important to check
that the slab is thick enough to define a bulk-like region where
$\bar{\bar{V}}(z)$ is linear. Now, in the Table. 2 we compare some of the ASJ
based valence band offsets ($\Delta E_{v}$) with experimental measurements. We
find a mean absolute error of 0.22 eV and 0.23 eV for OptB88vdW and R2SCAN
respectively, which is comparable a value of 0.16 eV from to Liberto et. al.
for a smaller number of systems using the HSE06 functional. In future, we plan
to carry out HSE06 calculations for surfaces as well as interfaces to further
improve the quality of predictions. These benchmarks will also be available in
the JARVIS-Leaderboard platform [97] as well. Out of numerous possible
combinations, only 250 DFT calculations of ASJ-based interfaces are available
right now and the database is still growing.

In summary, we have provided a computational framework and dataset for
investigating interface systems using multi-fidelity computational approaches.
We have developed one of the largest datasets, containing 300 surfaces, 44850
IU-band offsets, and 250 ASJ interface band offset using DFT. Using universal
graph neural network models, we have quickly screened potential semiconductor
device candidates as transistors from a pool of 1.4 trillion possible
interfaces, which would not have been possible using conventional
computational or experimental techniques. Although we have applied this
framework for semiconductors, it can be useful for other technological
applications as well. After pre-screening, we have shown and benchmarked this
streamlined workflow for band offset predictions using independent unit and
alternate slab junction models. This work paves the way for the application of
materials design approach to interface systems. All of the tools and datasets
developed in this work will be distributed publicly in the spirit of open-
science.

## Methods

Graph neural networks are trained using Atomistic Line Graph Neural Network
(ALIGNN) framework [49] which uses PyTorch and deep graph library (DGL).
ALIGNN is trained for 500 epochs and with default parameters in the package.
We use a 90:5:5 training:validation:testing data split for the CBM and VBM for
the bulk materials dataset. DFT calculations were carried out using the Vienna
Ab-initio simulation package (VASP) software [98, 99] with OptB88vdW [41],
TBmBJ[42] and R2SCAN [43] functionals using the workflow given on our ‘jarvis-
tools’ GitHub page (https://github.com/usnistgov/jarvis ). Please note
commercial software is identified to specify procedures. Such identification
does not imply recommendation by National Institute of Standards and
Technology (NIST). We use the OptB88vdW functional, which gives accurate
lattice parameters for both vdW and non-vdW (3D-bulk) solids. The crystal
structure was optimized until the forces on the ions were less than 0.01 eV/Å
and energy less than $10^{-6}$ eV. Also, we calculate the local potential
containing ionic plus Hartree contributions to determine the vacuum potential
(VAC) of surface slabs. The VAC is subtracted from the valence band maxima
(VBM) and conduction band minima (CBM) to enable the comparison of band-
diagrams of individual slabs in band-alignment diagrams. The converged
k-points and cut-off for the bulk materials were also used for the
corresponding surface slab models. The ASJ based interface structures were
generated using Zur algorithm [76] available in the JARVIS-Tools. For a quick
scan of xy-displacements for surfaces in the interfaces, ALIGNN-FF was used.

## Acknowledgements

We thank computational resource from National Institute of Standards and
Technology (NIST).

## Author contributions statement

K.C. conceived the high-throughput workflow and conducted all calculations,
K.F.G helped in setting up calculations and analysing the results. All authors
reviewed the manuscript.

## Competing interests

The authors declare no competing interests.

## Data availability

The data generated by this work will be made publicly available at Figshare
and JAVRIS website (https://jarvis.nist.gov/jarvish/).

## Code availability

The code used in this work, InterMat will be made publicly available at :
https://github.com/usnistgov/intermat. It depends on closely related codes
available at https://github.com/usnistgov/jarvis and
https://github.com/usnistgov/alignn.

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