Abstract
Semiconductor device technology has greatly developed in complexity since discovering the bipolar transistor. In this work, we developed a computational pipeline to discover stable semiconductors by combining generative adversarial networks (GAN), classifiers, and highthroughput firstprinciples calculations. We used CubicGAN, a GANbased algorithm for generating cubic materials and developed a classifier to screen the semiconductors and studied their stability using first principles. We found 12 stable AA\({}^{\prime}\)MH_{6} semiconductors in the F43m space group including BaNaRhH_{6}, BaSrZnH_{6}, BaCsAlH_{6}, SrTlIrH_{6}, KNaNiH_{6}, NaYRuH_{6}, CsKSiH_{6}, CaScMnH_{6}, YZnMnH_{6}, NaZrMnH_{6}, AgZrMnH_{6}, and ScZnMnH_{6}. Previous research reported that five AA\({}^{\prime}\)IrH6 semiconductors with the same space group were synthesized. Our research shows that AA\({}^{\prime}\)MnH_{6} and NaYRuH_{6} semiconductors have considerably different properties compared to the rest of the AA\({}^{\prime}\)MH_{6} semiconductors. Based on the accurate hybrid functional calculations, AA\({}^{\prime}\)MH_{6} semiconductors are found to be widebandgap semiconductors. Moreover, BaSrZnH_{6} and KNaNiH_{6} are directbandgap semiconductors, whereas others exhibit indirect bandgaps.
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Introduction
Semiconductors are essential components of modern devices that use transistors, lightemitting diodes^{1}, integrated circuits^{2}, photovoltaic^{3}, solar cells^{4}, and so on^{5,6,7}. Semiconductors exhibit variable resistance since electron flow can be controlled by light and heat. Therefore, these materials can be used for energy conversion, and digital switching^{8}. The elemental semiconductors found from Group XIV in the periodic table, like Si and Ge, and the compounds of Ge are widely used in electronics, photovoltaic and optoelectronic devices. However, semiconductors with various properties are required for industrial applications^{8,9}. For instance, good thermal conductivity and electric field breakdown strength, and also wide bandgap of SiC semiconductor make it a suitable material for hightemperature, highpower, highfrequency, and highradiation conditions^{10}. Thus, computational approaches for exploring semiconductors are essential to enhance future technologies. Highthroughput screening with the aid of firstprinciples calculations was performed by several groups to discover optoelectronic semiconductors. Setyawan et al. and Ortiz et al. reported the highthroughput screening and datamining frameworks to investigate bandgap materials for radiation detection^{11,12,13}. High throughput material screening by Zhao et al. found that CuInbased Halide Perovskite as potential photovoltaic solar absorbers^{13,14}. Based on 4507 hypothetical materials, Li et al. suggest 23 candidates for lightemitting applications, and 13 potential compounds for solar cell technologies^{13,15}. Such examples indicate that highthroughput screening can now be used to explore promising semiconductor materials.
Generative adversarial networks (GANs) are a kind of generative models that learn patterns/distribution from input data^{16}. GANs use two submodels to train a generative model. The generator model generates fake data, and the discriminator model learns to tell fake data from real data. The two submodels are trained simultaneously to achieve a Nash Equilibrium: the generator can generate data that the discriminator can recognize half the chance. Wasserstein distance^{17} and gradient penalty^{18} are introduced during training in order to overcome mode collapse and improve the training stability in original GANs^{16}. There are a limited number of works that leverage GANs to generate crystal structures in material science. The reasons behind that are: 1) Crystal structures have so many formations, such as a different number of elements and number of atoms in a unit cell. It is hard to come up with a unified representation to make GANs learn from them like images or text; 2) GANs used in computer vision cannot generate crystal structures that satisfy physics or symmetric constraints. For instance, GANs easily generate materials that are not recognizable or that have crowd atoms in a unit cell. CrystalGAN^{19} is believed to be the first work that uses GANs to generate materials. It applies CyClyGAN^{20} to simple systems mapping ternary a hydride into another. In^{21}, Kim et al. use WGANGP^{18} to train a generative model to generate MgMnO systems with atom coordinates as the input. All the works above only consider a simple or specific family of materials at a limited scale. CubicGAN proposed by Zhao et al.^{22}, however, is the first work that generates materials at a large scale.
In this research, we developed a binary classifier to filter the semiconductors/Insulators (nonmetals) from the dynamically stable quaternary Cubic materials discovered using the CubicGAN model, where highthroughput calculations were done with the assistance of a GAN model and density functional theory (DFT). We studied the most important elemental and electronic properties, which are helpful to distinguish the nonmetals and metals using the machine learning models. In addition, we carried out DFT calculations for those semiconductors to corroborate the thermodynamic stability and semiconductor properties. As a result, we find that 12 cubic semiconductors of a particular class of materials, which we label as AA\({}^{\prime}\)MH_{6}, are thermodynamically stable against their competing phases. We further performed the DFT calculations to study their structural, mechanical, thermodynamic, and electronic properties. Our results show that AA\({}^{\prime}\)MnH_{6} and NaYRuH_{6} have higher C_{ii} (i = 1, 2, 3) elastic constants, bulk modulus, shear modulus, and Young’s modulus compared to the respective mechanical properties of the rest of the AA\({}^{\prime}\)MH_{6} materials. At temperatures less than 200 K, AA\({}^{\prime}\)MnH_{6} and NaYRuH_{6} have lower specific thermal capacity (C_{v}) relative to other AA\({}^{\prime}\)MH_{6} materials. The highest C_{v} at 300 K found in this work is from BaSrZnH_{6} (127.96 JK^{−1}mol^{−1}). Moreover, hybrid functional calculations show that all AA\({}^{\prime}\)MH_{6} materials are widebandgap semiconductors, which will be useful to develop optical and hightemperature power devices^{23,24}.
Results and discussion
Dataset of nonmetals and metals
As the CubicGAN model generates only ternary and quaternary materials, we first analyzed the number of nonmetals (semiconductors and insulators), and metals in the material project (MP) database^{25}, as shown in Table 1. We collected all the ternary and quaternary materials, where the bandgap details are available, using the Pymatgen code^{26}. It could be found that ≈44 % of the ternary materials are nonzero bandgap materials while ≈ 56 % are metals. However, ≈73 % of the quaternary materials are semiconductors or insulators, whereas only ≈27 % of them are metals. This indicates that the probability of finding a stable quaternary material with a nonzero bandgap is higher compared to finding that in a ternary material set. We also compared the same details of the cubic materials. Interestingly, ≈80 % of the cubic ternary materials are metals, and only ≈20 % of them are nonmetals. On the contrary, the quaternary cubic materials have 30 % more nonzero bandgap materials than the number of metals. It shows that there is a low probability of discovering a nonzero bandgap cubic ternary compound. Instead, in this project, we mainly focused on the quaternary cubic materials for finding stable semiconductors. In this way, by reducing the search space of the materials, we can shorten the computational time taken by the DFT calculations.
Feature importance
Understanding which features are significant during the classification will be vital for discovering semiconductors. In Section 2.1, we could show that quaternary materials have a higher percentage of semiconductors compared to the ternary materials. Next, we analyzed which features have higher importance than others for classifying a quaternary material as metal or nonmetal. Feature importance (FI) of random forest algorithm is defined as the mean of the impurity decrease within each tree. This builtin feature of the random forest makes it convenient and a widely used method to calculate FI. Here, we trained our RFC model for the whole quaternary materials data set. The classification report of this model is in Supplementary Information. Even though both Avg. and the maximum difference of each atomic/electronic property were considered for the RFC model, only three features related to maximum difference have FI greater than 1 %. This indicates that Avg. value of the properties plays a significant role when classifying a material as metal or nonmetal. The top features of FI ⪆ 2.0% are mentioned in Fig. 1. Avg. Availability of metallic elements has the highest FI, while Avg. availability of nonmetal also has a FI of around 2 %. This indicates that having a metallic or nonmetallic element is important for the material to be a metal or a semiconductor/insulator. It is generally accepted that metallic elements have a higher boiling point and higher density compared to that of nonmetals. It should be noted that the elemental properties like metallicity, being semiconductor/insulator, density, and boiling point are properties of the bulk material formed with a given element. Since the availability of metallic and nonmetallic elements plays a significant role, the boiling point and density of those elements also can become important features when classifying metals and nonmetals. It is also clear that electronic properties like Avg. number of unfilled orbitals, Avg. number of pvalence electrons, and Avg. availability of +2 and +3 oxidation states have high FI.
We also studied the descriptors to understand how the number of metals and nonmetals depends on the percentage availability of the metal (P_{M}), nonmetal (P_{NM}) and transitionmetal (P_{TM}) elements in the chemical formulas. We use M, NM, and TM to indicate the type of elements to avoid confusion between material class (metal or nonmetal) and element type (metal, nonmetal, transition metal). Figure 2 shows the violin plots with all the 39024 quaternary materials against those three atomic properties. Here, P_{M} = 100%, P_{NM} = 100%, and P_{TM} = 100% for a given chemical formula when all the elements are M, NM, and TM, respectively. Figure 2(a) clearly evidences that nonmetals dominate until P_{M} ≈ 60 %. The ratio between amounts of metals and nonmetals (metals : nonmetals) is around 1: 3 at P_{M} < 60 %. This becomes approximately 5: 1 after 60 %, showing the probability of finding a semiconductor/insulator decreases. On the contrary, Fig. 2(b) shows the opposite behavior of metals and nonmetals, while P_{NM} alters. Moreover, it is clear that semiconductors and insulators prefer a lower number of TM elements relative to the other element types. At P_{TM} > 30 %, number of metals become significant compared to that of nonmetals. When P_{TM} ≤ 5 %, metals : nonmetals ratio is 1: 6.
Predicting Semiconductors
We further analyzed the error of the DNN and RFC models trained with quaternary cubic materials data. The 10fold crossvalidation accuracy results for each training step of the DNN model are 0.86, 0.92, 0.91, 0.97, 0.94, 0.94, 0.88, 0.94, 0.94, 0.86. Those of the RFC model are 0.86, 0.89, 0.88, 0.9, 0.87, 0.88, 0.90, 0.90, and 0.88. Thus, the mean accuracy was obtained for the DNN (RFC) model as 0.92 ± 0.034 (0.88 ± 0.013). Figure 3 shows the normalized confusion matrices for the classifiers. It is apparent that 33 (32) % of the instances were classified as true metals while 65 (60) % of the materials were listed as true nonmetals by the DNN (RFC) classifier. The percentages of false metals and false nonmetals from the DNN (RFC) model were 9.8 (4.9) % and 1.2 (2.5) %, respectively. The classification report for the model is shown in Table 2. It is clear that the DNN (RFC) classifier predicts whether a quaternary material is a metal or nonmetal with 0.88 (0.91) accuracy. Precision is the matrix that compares the number of true positive instances with the number of predicted positive instances. In our work, the DNN (RFC) model classifies a material as a nonmetal with 0.76 (0.96) and metal with 0.76 (0.84) precision. The recall is a measure of the number of correctly predicted positive cases compared to the total number of positive cases in the dataset. Table 2 shows that there is 0.85 (0.91) recall for nonmetal, while there is 0.93 (0.93) recall for metals from the DNN (RFC) model. By combining precision and recall, F1score can be calculated as 0.90 (0.93) for nonmetal and 0.84 (0.88) for metal classes. Therefore, the predictions of semiconductors/insulators from our DNN and RFC models can be expected to be highly accurate.
As seen in Table 2, the RFC model exhibits a slight improvement over the DNN model. To show the methodology of finding stable semiconductors based on generative adversarial networks, we applied our RFC classifier on CubicGAN predicted mechanically and dynamically stable quaternary materials. Out of 323 quaternary materials predicted by the CubicGAN model, 137 compounds were classified as nonmetals.
Structure and thermodynamic stability
We carried out our DFT calculations on those nonmetals to find thermodynamically stable semiconductors. We discovered that 12 semiconductors, which have chemical formulas in the form of AA\({}^{\prime}\)MH_{6}, exhibit zero energyabovehull against the respective competing phases. Those are BaNaRhH_{6}, BaSrZnH_{6}, BaCsAlH_{6}, SrTlIrH_{6}, KNaNiH_{6}, NaYRuH_{6}, CsKSiH_{6}, CaScMnH_{6}, YZnMnH_{6}, NaZrMnH_{6}, AgZrMnH_{6}, and ScZnMnH_{6}. We also find that Kadir et al. reported 5 different AA\({}^{\prime}\)MH_{6} type semiconductors, where M = Ir^{27}. They were able to synthesize NaCaIrH_{6}, NaSrIrH_{6}, NaBaIrH_{6}, KSrIrH_{6}, and KBaIrH_{6} by direct combination of the alkali (Na and K), alkaline earth (Ca, Ba, and Sr) binary hydrides/deuterides with Ir powder. Their Xray and neutron powder diffraction studies confirm that those semiconductors have the space group symmetry F43m. Furthermore, the open quantum materials database (OQMD)^{28,29} contains the structural properties and band gaps of NaCaIrH_{6}, NaSrIrH_{6}, NaBaIrH_{6} semiconductors and the MP database has those information on NaCaIrH_{6}, and NaBaIrH_{6} semiconductors^{25} (See Supplementary Information).
CubicGAN generates conventional structures with cubic Bravais lattice with F43m (216) space group for AA\({}^{\prime}\)MH_{6} materials, which have 36 atoms. On the contrary, the primitive unit cell with hexagonal Bravais lattice has only 9 atoms. Therefore, we considered the hexagonal unit cell to lower the computational time of the DFT calculations. In the primitive unit cells (see Fig. 4), green and red sites are symmetrically equivalent, while grey sites are located in the right middle of the hexagonal unit cell. Thus, we label the green and red sites as A and A\({}^{\prime}\), while the middle site is M. Rest of the 6 sites are occupied by H atoms. In the research work of Kadir et al., they considered alkali atoms as A atoms, alkaline earth atoms as A\({}^{\prime}\) atoms, and M atoms as Ir. In this research, our findings show that both A and A\({}^{\prime}\) atoms can be alkali atoms (E.g., CsKSiH_{6}) or alkaline earth atoms (E.g., BaSrZnH_{6}). Moreover, the M atom can be a transition metal atom or even Al or Si. Therefore, our experiments show that those materials can have high chemical diversity.
The lattice parameters, AH, MH, AM, and AA\({}^{\prime}\) bond lengths, are mentioned in Table 3. The primitive hexagonal unit cells have a/c = 1 lattice parameter ratio making a = b = c. As shown in Table 3, Mnrelated AA\({}^{\prime}\)MH_{6} and NaYRuH_{6} structures have the shortest lattice parameters compared to the rest of the materials. They have lattice parameters less than 5.0 Å, while other materials have greater than 5.4 Å. All A, A\({}^{\prime}\) and M elements make bonds with H atoms. A and A\({}^{\prime}\) elements are bonded to twelve equivalent H atoms to form AH_{12} and A\({}^{\prime}\)H_{12} cuboctahedra. And also, M atoms make MH_{6} octahedra by making bonds with 6 H atoms. An AH_{12} (A\({}^{\prime}\)H_{12}) cuboctahedra shares corners with twelve equivalent AH_{12} (A\({}^{\prime}\)H_{12}) cuboctahedra. Moreover, they share faces with four MH_{6} octahedra^{30}. Due to symmetry, AH and A\({}^{\prime}\)H bond lengths are equal. MH bond lengths are the shortest compared to other bonds for a given compound. AA\({}^{\prime}\) of Mnrelated AA\({}^{\prime}\)MH_{6} and NaYRuH_{6} structures are less than 3.4 Å, and AM and A\({}^{\prime}\)M distances are less than 3.1 Å. It can cause strong interactions between those atoms. AA\({}^{\prime}\) distance for the rest of the materials is greater than 3.8 Å, and AM and A\({}^{\prime}\)M distances are greater than 3.3 Å, indicating relatively weaker interactions.
The thermodynamic stability of the AA\({}^{\prime}\)MH_{6} materials against their elements was studied using the formation energies, which were based on the following equation.
Here, E_{tot} is the total energy per unit formula of the material. x_{i} is the number of atoms of each element in the unit formula; i.e., 1 for A, A\({}^{\prime}\), M atoms and 6 for H. N = ∑x_{i}; i.e., 9 for AA\({}^{\prime}\)MH_{6}. To find the atomic energies (E_{i}), we collected the most stable structures of each element using the Pymatgen code^{26}. Same DFT settings were used to calculate the energy of each element. It is clear that all the six materials have negative formation energies, which confirms their stability. We also carried out spinpolarized calculations for the AA\({}^{\prime}\)MH_{6} semiconductors with transition metal atoms to reveal whether they form magnetism. We observed that those materials do not have magnetic groundstates. Thus, all the AA\({}^{\prime}\)MH_{6} semiconductors are nonmagnetic materials.
Mechanical properties and stability
Next, we studied the mechanical properties and stability of the AA\({}^{\prime}\)MH_{6} materials by calculating the elastic constants using the DFPT method. To analyse the mechanical properties, we used the Vaspkit code^{31}, which computes the elastic constants by considering the AA\({}^{\prime}\)MH_{6} cubic system. Since cubic unitcells has a = b = c lattice lengths and α = β = γ = 90^{0} lattice angles, C_{11} = C_{22} = C_{33}, C_{44} = C_{55} = C_{66}, and C_{12} = C_{13} = C_{23}^{32}. Therefore, we mention only the three independent elastic constants (C_{11}, C_{12} and C_{44}) in Table 4. It is clear that AA\({}^{\prime}\)MH_{6} materials have relatively higher C_{11} for AA\({}^{\prime}\)MnH_{6} and NaYRuH_{6}, compared to the other four materials in Table 4. As discussed before, the lattice constants and AA\({}^{\prime}\) bond lengths of AA\({}^{\prime}\)MnH_{6} and NaYRuH_{6} structures are considerably lower than that of the rest of the materials. As illustrated by Fig. 4, AA\({}^{\prime}\) bonds are aligned in a, b and c directions. C_{11}, C_{22}, and C_{33} are parallel to the a, b and c directions, respectively. Therefore, higher C_{ii} (i = 1, 2 and 3) can be mainly due to the strong interactions between the A and A\({}^{\prime}\) atoms. Born stability criteria for the cubic systems are C_{11} − C_{12} > 0, C_{11} + 2C_{12} > 0 and C_{44} > 0^{32}. It is clear from Table 4 that all the eight materials comply with the above requirements.
We also calculated the Bulk modulus (K), Young’s modulus (Y), and isotropic Poisson’s ratio (μ) based on the Hill approximation^{33} as mentioned in Table 4. The smallest K values were found from CsKSiH_{6} (16.615 GPa), while the largest value was calculated from AgZrMnH_{6} (120.755 GPa). SrTlIrH_{6} (21.915 GPa) provides the lowest Y, while NaZrMnH_{6} (156.876 GPa) exhibits the maximum Y. It is clear that NaYRuH_{6} and all the Mnbased materials have significantly larger K and Y values than that of the other six materials. This can be mainly because of high C_{ii} (i = 1, 2, and 3) formed due to strong AA\({}^{\prime}\) bonds. Because of low Y, NaYRuH_{6} and Mnbased AA\({}^{\prime}\)MH_{6} materials can be considered stiffer materials relative to the other six semiconductors. And also, they exhibit more resistance to compression due to high K. All the μ values of the AA\({}^{\prime}\)MH_{6} materials are between 0.2 and 0.4. maximum μ was found from SrTlIrH_{6}. Thus, SrTlIrH_{6} has considerably low Y and high μ. This indicates that SrTlIrH_{6} semiconductor is less stiff due to small Y and more deformable elastically at small strains due to large μ.
Thermodynamic properties and dynamical stability
The temperature of the highest normal mode of a crystal is known as the Debye temperature θ_{D}. This can be obtained by employing Debye sound velocity (ν_{D}) as explained by Eq. (2). Debye sound velocity can be calculated using the longitudinal and transverse sound velocities, which can be determined based on K and G as shown in Eq. (4)^{34}. Here, N, V_{0}, and ρ are the number of atoms, volume, and density of the unicell, respectively. And also, h is Plank’s constant, and k_{B} is Boltzmann’s constant.
Table 5 shows the respective ρ, ν_{l}, ν_{t}, ν_{D} and θ_{D} values for AA\({}^{\prime}\)MH_{6} crystals. Debye temperature of NaYRuH_{6} and Mnbased AA\({}^{\prime}\)MH_{6} materials are significantly higher than that of other AA\({}^{\prime}\)MH_{6} materials. As θ_{D} depends on K and G (see Eq. (4) and (2)), enhanced θ_{D} is due to the high K and G of those semiconductors.
We also plotted C_{v} as a function of temperature T using the Phonopy code^{35}. C_{v} can be determined based on the following expression,
where ω_{qj} is the phonon frequency for q wave vector at jth phonon band index and ℏ is the reduced Plank’s constant^{35}. The phonon frequency for each Kpoint is plotted in Fig. 5. As can be seen in Fig. 6, the C_{v} of NaYRuH_{6} and Mnbased AA\({}^{\prime}\)MH_{6} materials are plotted with broken lines, and that of the rest of the materials is indicated by solid lines. It is clear that the C_{v} of NaYRuH_{6} and Mnbased AA\({}^{\prime}\)MH_{6} materials are smaller than that of the other materials at the low temperatures (0 to 150 K). At the lowtemperature limit (T ≥ θ_{D}, θ_{D}/T < < 1), C_{v} is proportional to (T/θ_{D})^{3}. Since θ_{D} is higher compared to that of other materials, C_{v} is smaller at low temperatures for NaYRuH_{6} and Mnbased AA\({}^{\prime}\)MH_{6}.
Electronic Properties
As can be seen in Table 6, A, A\({}^{\prime}\) and M elements lose electrons (except in Ru, where it has small negative value), while H atoms gain electrons. Thus, we can expect an ionic character in AH, A\({}^{\prime}\)H, and MH bonds. Even though A and A\({}^{\prime}\) sites are symmetrically equivalent, the atoms at those sites can lose a different amount of electrons. This is mainly because atoms at those sites have different oxidation states. Based on Table 6, Na, K, and Cs alkali atoms have their usual oxidation state (+1), while alkaline earth atoms such as Ca, Sr, and Ba lose more than 1 electron as they can donate up to 2 electrons. Al, Si, and Tl exhibit their most common oxidation states, which are +3, +4, and +1, respectively. It is reported that firstprinciples computations provide only negligible changes in the local transitionmetal charge for semiconducting crystals^{36}. Therefore, we propose that we can consider MH\({}_{6}^{n}\) complex as a single unit since the MH bond lengths are very short compared to other Hrelated bonds. n can be found by computing Δq_{M} + 6 × Δq_{H}, which is greater than 2 for all the M atoms except for Ni and Si. For those two atoms, n ≈ 1.6. Therefore, we can expect MH\({}_{6}^{2}\) for Si and Ni complexes, while MH\({}_{6}^{3}\) for the rest of the complexes. Kadir et al. suggest that IrH\({}_{6}^{3}\) complexes exist in AA\({}^{\prime}\)IrH_{6} semiconductors^{27}. Therefore, MH\({}_{6}^{3}\) can be the common complex that exists in AA\({}^{\prime}\)MH_{6} materials.
Figures 7 and 8 show the band structures and partial density of states (PDOS) of the AA\({}^{\prime}\)MH_{6} materials. It is clear that all six AA\({}^{\prime}\)MH_{6} materials are semiconductors. The bandgap for each material is mentioned in Table 7. The DFT calculations with PBE exchangecorrelation functional underestimate the band gaps due to selfinteraction error. It has been shown that the HeydScuseriaErnzerhof (HSE) screened Coulomb hybrid functional calculations provide reasonable estimation for the band gaps of semiconductors^{37,38}. HSE06 uses \(\frac{1}{4}\) of exact exchange and \(\frac{3}{4}\) of PBE exchange. Based on our HSE06 computations, all the AA\({}^{\prime}\)MH_{6} semiconductors can have bandgaps greater than 2.00 eV (see Supplementary Information). The bandgap range of widebandgap semiconductors is considered as the range above 2 eV^{23}. Thus, we can identify that those materials are widebandgap semiconductors. As reported by Kadir et al., NaCaIrH_{6}, NaSrIrH_{6}, NaBaIrH_{6}, KSrIrH_{6} and KBaIrH_{6} have bandgaps between 2.91 and 3.33 eV^{27} (see Supplementary Information). Widebandgap semiconductors are vital for manufacturing optical devices emitting green, red, and UV frequencies and also power devices functioning at higher temperatures^{23,24}.
Other than in BaCsAlH_{6} and CsKSiH_{6}, all the AA\({}^{\prime}\)MH_{6} materials have their conduction band minimum (CBM) at X highsymmetric Kpoint. The CBM of BaCsAlH_{6} and CsKSiH_{6} are at Γ points. The valence band maximum (VBM) of BaNaRhH_{6}, SrTlIrH_{6}, YMnZnH_{6}, NaYRuH_{6}, and AgZrMnH_{6} exist at W Kpoint. BaSrZnH_{6}, KNaNiH_{6} and BaSrZnH_{6} have VBM at X, while that of CaScMnH_{6} and AgZrMnH6 is at K highsymmetric point in the reciprocal space. Thus, both CBM and VBM of BaSrZnH_{6} and KNaNiH_{6} reside at X Kpoint, indicating those materials are direct bandgap semiconductors. Direct bandgap semiconductors are preferred for LED and laser devices over their indirect counterparts. Widebandgap semiconductors with direct bandgap are widely investigated for solar cells due to optical transparency^{39}. BaNaRhH_{6}, KNaNiH_{6}, CaCsMnH_{6}, and NaYRuH_{6} materials have very flat bands near the Fermi level, which is indicated by zero energy. Relative to other materials, BaSrZnH_{6} contains narrow (less flat) bands near the Fermi level. As a result, this can lower the effective mass of the carriers. Some research has shown that low effective mass will help developing efficient thermoelectric devices^{40,41,42}. As shown by electronic band theory, the electron effective mass can be very high in the flat bands^{43}. It is also shown that flat bands at the bottom of the conduction bands can provide high thermoelectric power^{44}. YMnZnH_{6} and ScZnMnH_{6} materials also exhibit that the CBM are relatively flat. Moreover, as shown in Fig. 7, we can modulate the shape of the bands near the Fermi level using the chemical formula. As a result, the thermoelectric properties can be tuned. Therefore, we propose that AA\({}^{\prime}\)MH_{6} semiconductors should be investigated for thermoelectric applications. Our partial density of states (PDOS) studies reveal that dorbitals of transition metal atoms reside at the M site dominate in the valence region near the Fermi level. Even though the transition metal atoms can be found at A and A\({}^{\prime}\) sites, their pdos of dorbitals are not significant near the Fermi level.
Method
Generative adversarial network
The hypothetical materials used in our research are generated by our CubicGAN^{22}, a generative adversarial network (GAN) based model for generating cubic crystal structures in a highthroughput manner. Our GAN model consists of a generator network and a discriminator/critic network. The discriminator learns to tell real materials from fake materials generated by the generator. The generator learns how to generate samples with similar distribution as the training samples. After trained, we can sample from the generator to generate nonexisting materials. In CubicGAN, we focused on generating ternary and quaternary materials with the space groups 221, 225, and 216. Moreover, to simplify the problem, CubicGAN uses special fractional coordinates, all in the set of {0.0, 0.25, 0.5, 0.75}. The CubicGAN is trained using material data from OQMD^{45,46} and is evaluated on material data from Materials Project^{47} and ICSD^{48}. The main framework of CubicGAN and the postprocessing for the generated materials are shown in Fig. 9. It is notoriously hard to train the original GAN model because the adversarial loss is not continuous in the generator, which causes vanishing gradients and saturation in the discriminator. We take advantage of the Wasserstein GAN with gradient penalty by penalizing the norm of gradients of the critic with respect to the inputs^{18}. The critic takes real materials and fake materials generated by the generator and then outputs a score which can be interpreted as how real the input materials are. The score is used to update the parameters of the models of the generator and the critic. The adversarial loss is defined as:
where D means the score function from the critic. \(\hat{{{{\bf{x}}}}}\) is the linear interpolation between a real material x and the generated one \(\hat{{{{\bf{x}}}}}\) and \(\mathop{E}\limits_{\hat{{{{\bf{x}}}}}\sim {{\mathbb{P}}}_{\hat{{{{\bf{x}}}}}}}[{({\parallel {\nabla }_{\hat{{{{\bf{x}}}}}}D(\hat{{{{\bf{x}}}}})\parallel }_{2}1)}^{2}]\) is the gradient penalty which enforces gradients with the norm at most 1 everywhere. λ is set 10 by default in this work.
Conditioning on random noise, three or fourelement combinations, and space group, the generator not only generates materials with existing prototypes but also generates stable ones with nonexisting prototypes. When the CubicGAN generates 10 million materials, it can rediscover most of the cubic materials in Materials Project and ICSD. In CubicGAN, we only focus on the generated materials with prototypes, which are defined by the anonymous formula and the space group ID. In total, 24 and 1 nonexisting prototypes are found in 10 million generated ternary and quaternary materials, respectively. Subfigure (a) of Fig. 9 shows how to filter out the materials. On average, 90% of generated materials have readable CIFs, and we only select materials with neutral charge and negative formation energy predicted by CGCNN^{49}. After filtering down materials with nonexisting prototypes, we performed DFT calculations, and 36847 candidate materials have been relaxed successfully. Further, 506 stable materials are verified by phonon dispersion.
Nonmetal  metal classifier
To develop a nonmetal  metal classifier, we first collected the pretty formulas, Bravais lattice type, and bandgap details of all the cubic quaternary materials from the MP database. There were 2578 nonzero bandgap materials (semiconductors and insulators) and 1,438 metals in the collected dataset. We considered 55 elemental and electronic structure attributes, such as the first ionization energy, atomic volume, electronegativity, total number of valence electrons, and number of valence electrons in s, p, d, and f orbitals, to develop the feature set (see Supplementary Information). The weighted average (Avg.) and a maximum difference of those properties for a given chemical formula were added to the feature set. The Avg. of a property S of a quaternary compound A_{α}B_{β}C_{γ}D_{δ} was calculated based on the following expression,
where S_{A}, S_{B}, S_{C} and S_{D} are the property S of A, B, C, and D elements, respectively. Altogether, 119 features were considered for training the models.
We created the DNN classifier with two hidden layers using Keras^{50} on top of TensorFlow^{51}. The first and second hidden layers of DNN include 200, and 100 neurons, respectively. To include the nonlinearity in the system, we shifted the summed weighted inputs of each layer through the rectified linear unit (ReLu) activation function. We randomly dropped out 5% of the units of the hidden layers while training the models. This process is very important for limiting the overfitting of training data. Another useful approach to diminishing overfitting is weight regularization. We employed Ridge (L2) regularization method for adding penalties during updating weights. The adaptive moment estimation (Adam) optimizer with a 0.001 learning rate was considered with binary crossentropy as the loss function and the metric during the calculations. The optimized number of epochs and batch size are 500 and 1500, respectively.
We developed a random forest classifier (RFC) as the second model, which uses an ensemble technique. Here, data is divided randomly, which is known as bagging and carries out training with multiple decision trees. The final prediction is given by averaging the output of all the decision trees. The hyperparameter optimization was performed using GridSearchCV algorithm as implemented in the Scikitlearn code^{52}. The optimized number of decision trees, minimum samples split, minimum samples leaf, and maximum depth are 500, 10, 3, and 90, respectively. Furthermore, we used the RFC model to study the feature importance for whole quaternary materials data set. It will help discovering semiconductors in the future.
For both DNN and RFC models, the cubic quaternary materials dataset with 4016 materials was split randomly into 98 % and 2 % as the training and testing sets, respectively. The 10fold crossvalidation with accuracy as the scoring method was performed on the training set. Here, the training set was partitioned into 10 subsets, where 9 subsets were for training the model and the remaining subset was for validating.
Density functional theory (DFT) calculations
Density functional theory calculations were performed as implemented in the Vienna ab simulation package (VASP) code^{53,54,55,56}. The electron wave functions were described using the PAW pseudopotentials^{57,58}. The exchangecorrelation interactions were treated based on the generalized gradient approximation (GGA) within the PerdewBurkeErnzerhof (PBE) formulation^{59,60}. The energy threshold value of the planewave basis was set as 500 eV. In addition, the energy convergence criteria were set to 10^{−8} eV, and the force convergence criterion for the ionic steps is set to 10^{−2} eV/Å. The Brillouin zone integrations were performed using a dense kpoint mesh within the MonkhorstPack scheme for the structure optimizations, band structure, density of states, mechanical properties, and phonon calculations. For instance, a 14 × 14 × 14 Kmesh was used for BaNaRhH_{6} with 5.5105 Å lattice constant. The 2 × 2 × 2 supercells were employed for obtaining Phonon dispersions using the Phonopy code^{35}. The elastic constants were calculated by employing density functional perturbation theory (DFPT) as implemented in VASP^{61}. VASPKIT code^{31} was used to obtain the bulk modulus (K), Shear modulus (G), Young’s modulus (Y), and Poisson’s ratio (μ) of the materials based on the Hill method^{62}.
Data availability
The quaternary materials’ data used in this project is available at https://github.com/dilangaem/SemiconAI. The structures of the materials generated from CubicGAN model can be downloaded from Carolina Materials Database at http://www.carolinamatdb.org/.
Code availability
The classifier developed in this research work can be downloaded from https://github.com/dilangaem/SemiconAI. The CubicGAN model is available at https://github.com/MilesZhao/CubicGAN.
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Acknowledgements
The research reported in this work was supported in part by National Science Foundation under the grant and 1940099, 1905775, and 2110033. The views, perspectives, and content do not necessarily represent the official views of the NSF. We also would like to thank the support received from the department of computer science and engineering of the University of Moratuwa, Sri Lanka.
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Conceptualization, J.H. and E.S.; methodology, E.S., Y.Z.; software, J.H., Y.S.; resources, J.H., I.P.; writing–original draft preparation, E.S., Y.Z.; writing–review and editing, J.H., I.P., and E.S.; visualization, E.S. and Y.Z.; supervision, J.H.; funding acquisition, J.H.
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Siriwardane, E.M.D., Zhao, Y., Perera, I. et al. Generative design of stable semiconductor materials using deep learning and density functional theory. npj Comput Mater 8, 164 (2022). https://doi.org/10.1038/s41524022008503
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DOI: https://doi.org/10.1038/s41524022008503
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