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 375-774 nm and 313-539 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 blue-to-violet sources with 2/3D confinement (NWs/QDs). MAPbBr$_{3}$ is also suitable for green-to-violet 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$_{\mathrm{x}}$Br3-x 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 (18-19 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$_{\mathrm{3-x}}$ 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$_{\mathrm{3-x}}$ 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$_{\mathrm{3-x}}$ 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 above-mentioned 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 non-negligible red-shift ^(9,10). The blue dotted curves in Fig. 4, compared to the red curves (i.e. results of uniform distributions), also support this red-shift 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 red-shift reduces as the fraction of I atoms (i.e., halides of lower electro-negativity) increases, regardless of confinement sizes. So, at a confinement size of 30 ucs, the red-shift 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 red-shift is observed with 20% of I, and this red-shift reduces to 0.04% with 80% of I. It should be noted that confinement sizes themselves also have non-negligible connections to the red-shift. When we focus on the cases where 20% of I is included, for example, the red-shift 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 well-known phenomenon caused by phase separation mainly stemming from lights (photo-irradiation). While there are studies that focus on physical origins of light-induced 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 red-shift 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$_{\mathrm{3-x}}$ 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$_{\mathrm{3-x}}$ 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 I-Br 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.