Ryu Hoon^{1}

(Division of National Supercomputing R&D, Korea Institute of Science
and Technology Information, Daejeon 34141, Korea)
Copyright © The Institute of Electronics and Information Engineers(IEIE)
Index Terms
Metal halide perovskites, realistically
sized nanostructures, band gap engineering, halide
phase separation, tightbinding simulations
I. INTRODUCTION
Implementation of optoelectronic devices using metal halide perovskite (MHP) materials
has been keen interest to researchers ever since the successful demonstration of efficient
light absorbers for photovoltaic cells ^{(1,}^{2)}. Lightemitting diode (LED) has been particularly a focal target application of MHPs
^{(3}^{5)} due to promising properties such as narrow emission bands and a flexible tunability
of band gap energies ^{(6}^{8)}. Upon the first demonstration of roomtemperature working MHP LEDs ^{(3)}, their quantum efficiencies have increased exceeding 20% these days ^{(5)}.
Precise control of band gap energies is critical to design qualified LEDs. As the
emission spectra of MHP samples heavily depend on their confinement structures and
halide types ^{(6,}^{7)}, modelingbased prediction including realistic conditions of MHP samples, e.g. lithographically
feasible sizes and uncertainty of atomic positions in alloys, etc., are thus important
for procurement of critical design factors, which is in principle hard to be covered
by experimental efforts due to the wide range of control variables. Band gap engineering
with halide mixtures, particularly, is the focal research topic these days as it has
opened the feasibility of fine control of emission wavelengths in a visible spectrum
^{(9,}^{10)}. While MHP band structures have been theoretically studied with the stateoftheart
Density Function Theory (DFT) simulations ^{(11}^{13)}, focuses are limited to bulks or ultrathin layers that are simulated using extremely
tiny supercells (including < 100 atoms) with periodic boundary conditions.
In this work, we expand the simulation scopes of MHP samples to confined structures
in over 10nm scales. With an atomistic tightbinding (TB) model that uses parameter
sets fitted to match bulk band structures known by either experiments or DFT studies
^{(14,}^{15)}, the effects of size and composition engineering of band gaps are understood for
MHP nanostructures including up to ~100K atoms. Using methylammonium lead iodide and
bromide (MAPbI$_{3}$/Br$_{3}$) as target materials, we examine the tunability of band
gaps with size/composition engineering, presenting a complete map of band gap energies
that are theoretically achievable. Also, the wellknown redshift issue stemming from
phase separation in halide mixtures ^{(7,}^{8,}^{16,}^{17)}, is investigated and strategies to reduce the redshift are proposed. Sound connections
to recent experimental studies on bulk MHPs are also presented to validate our approach,
establishing a modeling framework for realistically sized, confined MHP structures.
II. METHODOLOGY
Four types of supercells are considered to understand confinement effects on band
gap energies, e.g. cubic (box) and sphere quantum dots (QDs), square nanowires (NWs),
and quantum wells (QWs), as described in Fig. 1. All the supercells are assumed to be in a cubic phase according to the crystalline
characteristics known at T=300K ^{(18,}^{19)}. Confinement sizes are varied within 330 unitcells (ucs), so the largest supercell
of modeling targets has 1.08x10$^{5}$ atoms. An 8band $sp^3$ TB model is employed
to represent an atom in electronic structures with spinorbit coupling. The TB parameter
sets reported by BoyerRichard et al. and Ashhab et al. ^{(20,}^{21)} are utilized to model of MAPbI$_{3}$ and MAPbBr$_{3}$, respectively, where the two
previous works focused on bulks ^{(20)} or small supercells consisting up to 8$^{3}$ cubic unitcells ^{(21)}. MAPbI$_{3}$ and MAPbBr$_{3}$ bulk band gap energy and band offset computed with
our parameters are presented in Table 1.
Table 1. Bulk MAPbI$_{3}$/Br$_{3}$ band gap energies and offsets that are obtained
with simulations using an 8band $sp^3$ TB model
MHP
type

Band gap
energy (eV)

Offset w.r.t. MAPbI$_3$ (eV)

Conduction band

Valence band

MAPbI$_3$

1.60





MAPbBr$_3$

2.30

+0.01

0.69

Fig. 1. Simulated supercell structures. Confinement size (L) of 330 unitcells (ucs)
are taken into consideration in this work.
TB simulations of electronic structures involve a largescale diagonalization problem,
as the size of Hamiltonian matrices are linearly proportional to the number of atoms
residing in supercells to be modelled. Workloads of these diagonalizations are processed
with our inhouse solvers, which has demonstrated the capability to solve electronic
structures for supercells consisting of up to 4 billion atoms with a parallel computing
^{(22}^{24)}. The NURION high performance computing cluster ^{(25)} is extensively utilized for all the simulations.
III. RESULTS AND DISCUSSION
1. Band Gap Energies with Homogeneous Halides
The feasibility of size and shape engineering as control factors for band gap energies
are examined for supercells shown in Fig. 1. Simulation results are summarized in Fig. 2, where band gap energies are converted to wavelengths (in a nm unit) for convenience
of discussion with respect to the color spectrum that is included as an inset of Fig. 2. At a room temperature, band gap energy of bulk MAPbI$_{3}$ and MAPbBr$_{3}$ are
about 774 nm and 540~nm, respectively. As confinement effects prevail with reduction
of supercell sizes, however, energies start to decrease. In overall, our results observe
that band gap energies of 375774 nm and 313539 nm are theoretically achievable with
MAPbI$_{3}$ and MAPbBr$_{3}$ supercells simulated in this work. Confinement effects
on band gap energies are much larger in MAPbI$_{3}$ than MAPbPb$_{3}$, so MAPbI$_{3}$
can be considered for designs of green light sources with 1D confinement (QWs), and
bluetoviolet sources with 2/3D confinement (NWs/QDs). MAPbBr$_{3}$ is also suitable
for greentoviolet light sources, but is not feasible to cover higher wavelengths
since the maximally achievable value is the bulk value (540 nm).
Fig. 2. Band gap energies of various nanostructures calculated for MAPbI3 (top) and
MAPbBr$_3$ (bottom). Energies are shown as wavelength (nm) for easier connection to
the color spectrum.
Another interesting point that is worthy of discussion is when confined supercells
would lose bulk characteristic, since the answer can be used as a practical initial
guideline to those who want to reduce the size of absorption layers with no changes
in band gap energies known for bulks. To understand this, deviation of band gap energies
against the bulk value is calculated for all the supercells. Results are plotted in
Fig. 3 as a function of confinement sizes, where 20 nm (a ballpark value of emission band
demonstrated for MAPbI$_{\mathrm{x}}$Br3x bulk samples by Yoon et al. ^{(8)}) is taken as a criterion to determine whether the supercells have bulk characteristic.
Here, results reveal that bulk characteristic is lost even in supercells of confinement
sizes larger than 10nm. Sphere QDs, which would be most severely affected by quantum
confinement than other types of supercells at the same confinement size, already experience
a deviation of 18.6~nm (MAPbBr$_{3}$) and 47.6 nm (MAPbI$_{3}$) even when the confinement
size of 30 ucs (1819 nm). Confinement effects of cube QDs and NWs are weaker than
sphere QDs, but in the case of MAPbBr$_{3}$ that generally exhibits smaller shift
of band gap energies compared to MAPbI$_{3}$, NWs and cube QDs lose their bulk characteristics
at a confinement size of 15 ucs (9 nm) and 19 ucs (11 nm), respectively.
Fig. 3. Deviation of band gap energies calculated for confined supercells with respect
to the bulk value. 20 nm, a ballpark value of emission band known for MAPbI$_{\mathrm{x}}$Br$_{\mathrm{3x}}$
bulk samples, is used as a criterion to determine whether a supercell is a bulk.
2. Band Gap Energies with Mixed Halides
As known by previous studies for bulk MHP samples ^{(9,}^{10)}, composition ratio in mixed halides can be utilized as a control factor of band gap
energies. The random nature of halide distributions, however, would drive unexpectedness
of band gap energies and is not easy to be avoided during lithographical processes.
To explore the effects of random distribution on band gap energies in confined structures
of halide mixtures, electronic structures of MAPbI$_{\mathrm{x}}$Br$_{\mathrm{3x}}$
cube QDs are simulated. For the four composition ratios (20, 40, 60, 80% of Iodide),
100 cases of uniform distributions are considered per each composition ratio. In Fig. 4, results of selective confinement sizes (05, 10, 15, 30 ucs) are shown with red curves,
where the error bar plotted on top of each red curve indicates energy fluctuation
stemming from 100 random distributions of I and Br atoms. Here, our results reveal
that band gap fluctuations due to the randomness in halide distributions increases
as confinement sizes reduce, so, when the confinement size is 5 ucs, band gap energies
fluctuates up to on the order of the latest reported emission band (~20nm). In supercells
of larger confinement sizes, however, effects of random halide placement on band gap
energies are not remarkable, as long as we do not consider the extreme cases where
halide atoms are placed such that they cannot be considered to be uniformly distributed.
Fig. 4. Band gap energy simulated for MAPbI$_{\mathrm{x}}$Br$_{\mathrm{3x}}$ cube
QDs. From the top left, results of confinement size of 5, 10, 15, and 30 ucs are shown
in a clockwise manner. Conceptual Illustration of uniform and segregated distribution
of halide atoms are also presented in bottom.
Because simulations of 100 random cases would not be sufficient enough to capture
the abovementioned extreme cases of halide distributions, we simulated one more cases
by artificially arranging halide positions such that I and Br are completely segregated
as illustrated in the right bottom in Fig. 4, and results are presented with blue dotted curves in Fig. 4. In reality, when the sample are exposed to light, I and Br atoms in halide mixtures
are strongly segregated, causing a nonnegligible redshift ^{(9,}^{10)}. The blue dotted curves in Fig. 4, compared to the red curves (i.e. results of uniform distributions), also support
this redshift against the mean value of uniformly distributed cases at the same confinement
size and composition ratio. One interesting feature that can be found here, is that
the redshift reduces as the fraction of I atoms (i.e., halides of lower electronegativity)
increases, regardless of confinement sizes. So, at a confinement size of 30 ucs, the
redshift with 20% of I is about 20% of the average of uniformly distributed cases,
while it can be reduced to 6% by including 80% of I. The situation is still same in
QDs of a confinement size of 5 ucs, so 4% of the redshift is observed with 20% of
I, and this redshift reduces to 0.04% with 80% of I. It should be noted that confinement
sizes themselves also have nonnegligible connections to the redshift. When we focus
on the cases where 20% of I is included, for example, the redshift is about 20% at
a confinement size of 30 ucs, but it reduces to 11%, 7%, and 4% at a confinement size
of 15, 10, and 5 ucs, respectively.
Halide segregation is a wellknown phenomenon caused by phase separation mainly stemming
from lights (photoirradiation). While there are studies that focus on physical origins
of lightinduced phase separation ^{(7,}^{8)}, remarkable solutions to get rid of this issue is not available yet. In this situation,
we note the messages in the above paragraph are critical, since they reveal the potential
for huge reduction of the redshift through size and composition engineering, which
eventually accelerate the feasibility of stable light source designs with halide mixtures.
3. Validation: Connections to Experiments
As a final point of discussion, we validate our modeling approach, particularly by
analyzing simulation results with respect to the three experimental works ^{(6}^{8)}. In the top subfigure of Fig. 5, we compared the dependency of band gap energies of MAPbBr$_{3}$ cube QDs on confinement
sizes, to the results measured for colloid QDs by Protesescu et al. ^{(6)}, where the size of a single cubic unitcell is assumed to be 5.9Å ^{(26)} to convert QD sizes to a nm unit. Here, it is quite obvious that calculated band
gap energies precisely match the measured data such that the maximum deviation between
modeling and experiment data is just about 4 nm, which is just about 0.7% of the measured
band gap energy. It should be noted that the experimental work used Cesium (Cs) instead
of MA (CH$_{3}$NH$_{3}$) as a cation. The comparison is however still fair enough
since it is well known that the electronic states near band edges of XPbBr$_{3}$ do
not have remarkable correlations to the cation material X ^{(20)}.
Fig. 5. Connection of modeling to experiment. MAPbBr$_{3}$ cube QDs (Top) and MAPbI$_{\mathrm{x}}$Br$_{\mathrm{3x}}$
QDs (Bottom) are simulated and band gap energies are compared to experimental results.
The next connection between theory and experiment is established for bulk MAPbI$_{\mathrm{x}}$Br$_{\mathrm{3x}}$
samples, where we used modeling results of cube QDs of the largest confinement size
(30 ucs). The comparison is shown in the bottom sub figure of Fig. 5, where the point of analysis is the position of photoluminescence (PL) peaks. From
the experimental data, it is clear that photoinduced phase separation moves the position
of PL peaks to a longer wavelength in both samples that include 10% (MAPbI$_{\mathrm{0.3}}$Br$_{\mathrm{2.7}}$)
^{(6)} and 43% of I (MAPbI$_{\mathrm{1.2}}$Br$_{\mathrm{1.8}}$) in IBr mixtures ^{(7)}. The two positions of PL peaks measured for each sample are 535/683 nm (I= 10) and
656/757 nm (I=43%), and are marked with block and green boxes in the bottom subfigure
of Fig. 5, where error bars indicate full widths at half maximum (FWHMs) of PL peaks. Modeling
results here, shown with red and blue curves, also have nice accuracy so that the
deviation calculated with respect to the median of experimental data is smaller than
50 nm (< 10% of experimental data) for the two composition ratios of interest. Note
that this deviation becomes even smaller if FWHMs are considered.
IV. CONCLUSIONS
Composition and size engineering of band gap energies are theoretically studied with
tightbinding simulations for MAPbI$_{\mathrm{x}}$Br$_{\mathrm{3x}}$ nanostructures
of physically realizable sizes. With a map of theoretically achievable band gap energies
that can be used an initial guideline for those who want to design materials aiming
for a specific range of emission wavelengths, this work present strategies to overcome
the redshift problem induced by phase separation in halide mixtures, which serves
as a critical bottleneck for stable light source designs.
ACKNOWLEDGMENTS
This work has been supported by the Korea Institute of
Science and Technology Information (KISTI)
institutional R&D program (K19L02C07), and the
NURION high performance computing cluster has been
extensively used for all the simulations.
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Author
Hoon Ryu received B.S. degree from
Seoul National University, Korea,
M.S. degree from Stanford University,
and Ph.D. degree from Purdue
University, USA.
Dr. Ryu was with
System LSI division in Samsung
Electronics, and, is now a principal
researcher at Korea Institute of Science and Technology
Information, where he works nanoelectronics modeling
for advanced materials and devices design with aids of
numerical analysis.
His research interests include
modeling studies on properties of semiconductor
materials and devices, and performance optimization of
largescale numerical problems in high performance
computers.