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. 2023 Jul 26;62(31):12480–12492. doi: 10.1021/acs.inorgchem.3c01696

How the Temperature and Composition Govern the Structure and Band Gap of Zr-Based Chalcogenide Perovskites: Insights from ML Accelerated AIMD

Namrata Jaykhedkar †,*, Roman Bystrický †,‡, Milan Sýkora †, Tomáš Bučko §,‡,*
PMCID: PMC10410608  PMID: 37495216

Abstract

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The effects of temperature and composition on the structural and electronic properties of chalcogenide perovskite (CP) materials AZrX3 (A = Ba, Sr, Ca; X = S, Se) in the distorted perovskite (DP) phase are investigated using ab initio molecular dynamics (AIMD) accelerated by machine-learned force fields. Long-range van der Waals (vdW) interactions, incorporated into the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional using the DFT-D3 scheme, are found to be crucial for achieving correct predictions of structural parameters. Our calculations show that the distortion of the DP structure with respect to the parent cubic (C) phase, realized in the form of interoctahedral tilting, decreases with the increasing size of the A cations. The tendency for a gradual transformation of the DP-to-C phase with increasing temperature is shown to be strongly composition-dependent. The transformation temperature decreases with the size of cation A and increases with the size of anion X. Thus, within the range of the temperatures considered here (300–1200 K), a complete transformation is observed only for BaZrS3 (∼600 K) and BaZrSe3 (∼900 K). The computed band gap of CPs is shown to monotonically decrease with increasing temperature, and the magnitude of this decrease is found to be proportional to the extent of the thermally induced changes in the internal structure. Diverse factors affecting the magnitude of band gaps of CP materials are analyzed.

Short abstract

Ab initio molecular dynamics in the NPT ensemble is employed to investigate the structure and electronic properties of chalcogenide perovskites AZrX3 (A = Ca, Sr, Ba; X = S, Se) over the temperature range of 300−1200 K. The band gap of these materials is found to monotonously decrease but the magnitude of this change is shown to be strongly composition-dependent. Various factors influencing this trend, including charge transfer, volume expansion, and distortion of the ideal cubic framework, are analyzed.

1. Introduction

Mixed halide perovskites received enormous attention in the scientific community,1−6 thanks to the record power conversion efficiencies achieved by solar cells prepared from these materials using low-cost fabrication methods. However, the toxicity of one of its key components, Pb, and the limited thermal and chemical stability of halide perovskites represent major challenges to their effective application.7,8 Chalcogenide perovskites (CPs) have recently been recognized as a possible alternative with a similar electronic structure, better thermal and chemical stability compared to their halide counterparts, and generally lower toxicity.9−11 Although chemical methods for the preparation of CPs are known for more than 60 years,12−14 the development of practical applications of CPs has been slow because of the high temperatures and long reaction times required. Recently, several new methods for the preparation of CPs under more moderate conditions have been reported.15,16

Computational predictions also played a substantial role in the increased interest in developing CP materials. Recently, the density functional theory (DFT)-based high-throughput computational screening of varied compositions was performed to identify compounds that are thermodynamically stable and have electronic properties suitable for exploitation in photovoltaics and optoelectronics.17−19 Experimental and computational studies suggest that ABX3 chalcogenides can have distinct crystal structures with a different connectivity of the BX6 octahedra. Thermodynamically, the most stable CP phases are the GdFeO3-type structure, also known as the distorted perovskite (DP) phase, and the nonperovskite NH4CdCl3-type structure, also called the needle-like (NL) phase. Sun et al.19 proposed that CaTiS3, BaZrS3, CaZrSe3, and CaHfS3 in the DP phase are the most suitable CPs for exploitation in single-junction solar cells. Meng et al.20 suggested that compositions such as BaZrSe3 and BaZr1–xTixS3 have an optimal band gap for use in single-junction solar cell, with a theoretical maximum efficiency of up to 30%. Liu et al.21 reported theoretical investigations on AZrX3 (A = Ca, Sr, Ba; X = S, Se) at 0 K, suggesting selenide compositions as promising candidates for photovoltaic applications based on their stability, mechanical, electronic, and optical properties. However, such compositions have not yet been prepared experimentally.

Many of the reports on high-throughput material screening are of limited accuracy due to the large number of compositions taken into consideration. One of the typical simplifications is the neglect of the finite-temperature effects. In real situations, even moderate changes in temperature can cause phase transitions and significant structural distortions, especially in DP structures.22−25 Structural changes caused by thermal vibrations can significantly affect the physical and electronic properties of a material, such as thermodynamic stability, band gap, conductivity, etc. Predicting and understanding these changes are important prerequisites for the effective use of materials in practical applications. Accurate modeling of the dependence of material properties on temperature is also very valuable in identifying appropriate experimental conditions for synthesizing specific crystal structures with desired electronic properties. Such effects have been observed and systematically studied for halide perovskites.22,26−31

Current understanding of the influence of temperature on various CP compositions is, however, very limited.32 In our previous report,33 we began to address this deficiency by performing a theoretical investigation of the finite-temperature properties of a representative CP material, SrZrS3. To this end, a series of ab initio molecular dynamics simulations (AIMD) in the NPT ensemble, accelerated by machine learning, were performed. It was found that upon heating in the temperature range of 300–1500 K, the DP phase undergoes a gradual conversion into the undistorted parent cubic (C) phase. As a consequence, the electronic band gap (Eg) decreases by ∼0.8 eV with increasing temperature, shifting closer to the optimal photovoltaic range. Based on this analysis, we expected to observe similar effects for other Zr-based CPs in their DP phase, particularly within the series AZrX3 (A = Ca, Sr, Ba; X = S, Se). Our preliminary investigations based on static calculations33 suggested that different CP compositions will undergo the DP-to-C transformation at distinctly different temperatures and therefore also their Eg dependence on T is likely to be quite different. In this work, we use our established simulation protocol to conduct a comprehensive investigation and screening of the temperature-dependent properties of a series of AZrX3 (A = Ca, Sr, Ba; X = S, Se) compositions. Furthermore, several computational studies have shown that dispersive interactions are crucial in determining the lattice constants and mechanical properties of halide perovskites as they alter the energetics and geometry of the crystal.34−36 For instance, as reported by Li et al.,35 the stabilization of the inorganic framework and organic components in hybrid organic–inorganic halide perovskite is affected by van der Waals (vdW) interactions, resulting in smaller unit cell volumes and lattice constants than predicted by the semilocal functional PBE. Hence, in this study, we employ vdW-corrected AIMD simulations to examine the importance of long-range dispersion interactions on the structural parameters and consequently their effect on the electronic band gaps. Investigation of various factors that affect the properties of the selected series of CPs is reported. The role of the relative sizes of constituent cations and anions on the structure and band gap of the series of compositions is discussed. In-house experimental investigations, mainly the high-temperature X-ray diffraction (HTXRD) characterizations, are also performed to corroborate our theoretical predictions on structural phase transformation at high temperatures.

This paper is organized as follows: the outline of the model systems and the computational details of the methodology used in this work are described in Section 2, the role of vdW corrections on structural predictions of CPs is discussed in Section 3.1, followed by Section 3.2, wherein the factors influencing the band gap of CPs are analyzed. In Sections 3.3 and 3.4, we explore the thermal effects on the structure and the electronic band gap, respectively. A summary of the most important findings is provided in Section 4.

2. Materials and Methods

2.1. Computational Methods

The results presented in this study were obtained using the periodic DFT in the projector-augmented wave (PAW)37 method of Blöchl,38 as implemented in the Vienna Ab initio Simulation Package (Vasp).39,40 Generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) parameterization41 was used for the exchange–correlation functional. A kinetic energy cutoff of 400 eV was used to define the plane-wave basis set, and the electronic energies were converged to self-consistency with an accuracy of 10–6 eV/cell. A positive uniform background charge was considered in calculations of hypothetical negatively charged systems discussed in Section 3.2. To account for vdW interactions, dispersion corrections were employed using the zero damping DFT-D3 method proposed by Grimme et al.42,43

Building on the preliminary results from our previous report,33 we focus here on the most relevant phase of each of the ABX3 (A = Ca, Sr, Ba; X = S, Se) compositions, which is the DP phase that can gradually transform into the C phase upon heating (see Figure 1). The relation between the lattice geometries of the DP and C phases is explained in Section SII of the Supporting Information (SI). In structural optimizations, carried out using the external optimizer Gadget,44 conventional unit cells for DP and C structures, containing 20 atoms each, have been used. Brillouin zones of these unit cells were sampled by a Γ-centered mesh of 4 × 4 × 4 k-points. All structures were relaxed until the maximum components of forces acting on atoms were smaller than 5 × 10–3 eV/Å.

Figure 1.

Figure 1

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Conventional unit cell of the distorted perovskite phase (a, c) and cubic phase (b, d) of chalcogenide materials ABX3. Perspective (above) and top (below) views are shown. The navy blue, gray, and saffron spheres represent the A cation, the B cation, and the X anion, respectively.

The AIMD in the NPT ensemble at zero pressure and temperatures of 300, 600, 900, and 1200 K was performed with the supercells constructed as 2 × 2 × 2 multiples of a conventional cell of the DP structure. The atomic positions and lattice parameters were treated as dynamic variables. The simulation temperature was maintained using a Langevin thermostat45,46 with friction coefficients of 2 and 10 ps–1 for atomic and lattice degrees of freedom, respectively. The pressure was controlled by a Parrinello–Rahman barostat47,48 whereby the mass associated with the lattice degrees of freedom was set to 2 amu. The velocity Verlet algorithm with a time step of 3 fs was used to integrate the equations of motion of the ions. Consistent with the setting used in relaxations, the k-point grid was set to 2 × 2 × 2 in all MD simulations.

In order to accelerate the MD simulations, the machine-learned force field (MLFF) was employed, as recently implemented in Vasp.49,50 For the description of the machine-learned potential energy surface, descriptors based on the Gaussian representation of atomic distribution50,51 were used. The predicted target properties, i.e., energies, forces, and stress tensor components, were obtained via the Bayesian linear regression, allowing for reliable estimation of the quality of predictions.49,52 A separate training procedure was executed for each composition and temperature considered. Post training, a production run for ∼3 ns was performed at the MLFF level.

The calculation of temperature-dependent band gaps is detailed in Section SVI of the SI. The calculation of densities of states (DOS) and crystal orbital Hamilton population (COHP) was performed using the program LOBSTER.53−55 All of the figures of structures presented in this work were created using the VESTA program.56

2.2. Experimental Methods

2.2.1. Materials

BaZrS3 and SrZrS3 were prepared as described previously.15 Briefly, the corresponding ternary oxide (BaZrO3 or SrZrO3) was thoroughly mixed with solid sulfur and a small excess of boron in an agate mortar and pestle. The mixed powder was loaded into a quartz tube with an inner diameter of 15 mm. The tube was evacuated to 2 Pa and flame-sealed. The sample was placed in a furnace and heated at 1270 K for 36 h. The typical heating ramp rate was 5 K/min with a holding time of 5 h at 573 and 873 K, and the cooling rate was 5 K/min. After the cooling, the ampules were opened at ambient conditions. The product removed from the reaction ampule was transferred into a beaker and sonicated for 10 min with a small amount (∼20 mL) of HPLC-grade methanol to remove any soluble byproducts (e.g., B2O3).

2.2.2. HTXRD Measurements

High-temperature XRD measurements were performed on a Panalytical Empyrean diffractometer coupled with an Anton Paar high-temperature chamber HTK-16 N using a Pt heating strip in air and in the N2 atmosphere. The heating rate during nonisothermal segments was 8 K min–1. The duration of the isothermal segments in which the patterns were collected was 10 min. The temperature step for collecting the individual patterns was 20 K. Rietveld analysis for lattice parameter determination was done using HighScore plus on scans with 2θ ranging from 15 to 55° with a step of 0.026°.

3. Results and Discussion

3.1. Effect of Long-Range Dispersion Interactions on the CP Structure

It follows from the results presented in our previous work33 and other reports57 that the PBE functional tends to slightly overestimate the lattice parameters of SrZrS3 and other CPs.20,57 The likely reason for this problem is the incorrect treatment of long-range vdW interactions by semilocal functionals.58,59 Indeed, several computational studies demonstrated the crucial role of dispersive interactions for correct predictions of the structure, mechanical properties, and energetics of halide perovskites.34−36 In this section, we examine the performance of Grimme’s D342,43 correction to the PBE functional. Albeit several more sophisticated vdW correction schemes are available in the literature,60−64 the D3 method has a comparative advantage of being one of the most CPU effective schemes (and hence suitable for the use in MD simulations) while yielding a good accuracy for a wide range of materials.65

In order to compare the lattice geometries computed in relaxations (effectively corresponding to T = 0 K simulations) with known experimental results available for CaZrS3, SrZrS3, and BaZrS3, which were measured at room temperature, we extrapolated the latter to 0 K. To this end, the thermal expansion coefficients determined in our finite-T calculations presented in Section 3.3 were used. Note that it follows from the comparison of our results obtained for SrZrS3 with and without correction33 that the vdW interactions have a negligible effect on the result of extrapolation. On average, the uncorrected PBE overestimates the lattice parameters and volume of the sulfide series of CPs with the DP structure by ∼1.0 and ∼3.4%, respectively (see Table 1). The inclusion of the D3 correction leads to a systematic reduction in lattice parameters and volume, yielding predictions with significantly lower mean absolute errors (∼0.4 and ∼1.4% for the cell parameters and the cell volume, respectively). We notice that the quality of the PBE+D3 predictions worsens slightly with the increasing size of the cation A: while the error in V for CaZrS3 is only 0.8%, the value computed for BaZrS3 is 2.0% higher compared to the experiment.

Table 1. Lattice Constants (a, b, c), Cell Volume per Formula Unit (V1), and Transversal Displacements of the X Atoms (uT) Determined for the DP Phase of AZrX3 (A = Ba, Sr, Ca; X = S, Se) Compositions Relaxed with PBE and PBE+D3 Methodsa.

compound method a (Å) b (Å) c (Å) V1 (Å3/f.u.) uT (Å)
CaZrS3 exp.13 (298 K) 7.030 9.590 6.587 110.2 0.858
  exp.13 (0 K) 7.015 9.552 6.551 108.9 0.870
  PBE 7.071 (0.78) 9.644 (1.17) 6.576 (0.38) 112.1 (3.13) 0.857
  PBE+D3 7.023 (0.11) 9.574 (0.22) 6.531 (−0.31) 109.8 (0.83) 0.870
SrZrS3 exp.13 (298 K) 7.108 9.766 6.735 117.0 0.706
  exp.13 (0 K) 7.098 9.725 6.698 115.7 0.721
  PBE 7.167 (1.00) 9.827 (1.07) 6.784 (1.33) 119.4 (3.37) 0.709
  PBE+D3 7.121 (0.32) 9.762 (0.37) 6.746 (0.72) 117.2 (1.30) 0.725
BaZrS3 exp.13 (298 K) 7.060 9.981 7.025 123.8 0.451
  exp.13 (0 K) 7.075 9.941 6.962 122.5 0.490
  PBE 7.165 (1.26) 10.051 (1.11) 7.050 (1.28) 126.9 (3.76) 0.484
  PBE+D3 7.149 (1.06) 10.000 (0.60) 6.987 (0.33) 124.9 (1.96) 0.514
CaZrSe3 PBE 7.402 10.086 6.847 127.8 0.935
  PBE+D3 7.351 10.004 6.783 124.7 0.954
SrZrSe3 PBE 7.519 10.269 7.041 135.9 0.797
  PBE+D3 7.456 10.188 7.001 132.9 0.815
BaZrSe3 PBE 7.553 10.498 7.289 144.5 0.596
  PBE+D3 7.523 10.431 7.226 141.7 0.628
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a

The relative errors (in %) in computed lattice parameters and V1 with respect to the experimental reference are given in parentheses.

The dispersion interactions also tend to increase the distortion of the DP structures with respect to the parent cubic structures, again making the predictions closer to the experiment, which is obvious, e.g., from a slight increase of the transversal displacement (uT; see Figure 2) of X atoms from the ideal position in the middle of the Zr–Zr link (see Table 1). As is clear from the data presented in Table 2, the degree of distortion in relaxed distorted CP structures, as measured by uT, correlates with the stabilization of the parent C phase. The effect of including D3 correction on the structure of the selenide CP series is similar in magnitude as for their sulfide counterparts. However, since the experimental data are presently not available for these compounds, the accuracy of our theoretical predictions cannot be confirmed.

Figure 2.

Figure 2

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Illustration demonstrating the transversal displacement (uT) of X (saffron) atoms in the middle of the Zr–Zr link from their ideal positions in the undistorted phase.

Table 2. Lattice Constants (a, b, c), Cell Volume per Formula Unit (V1), and Relative Energies of the C Phase with Respect to the DP Phase (ΔEDP → C) of AZrX3 (A = Ba, Sr, Ca; X = S, Se) Compositions Relaxed with PBE and PBE+D3 Methods. The estimated temperatures at which transitions from the DP to C phase are completed (TDP → C) are also shown.

compound method a (Å) b (Å) c (Å) V1 (Å3/f.u.) ΔEDP → C (eV/f.u.) TDP → C (K)
CaZrS3 PBE 4.978 (7.040) (9.956) (7.040) 123.4 1.420 2197
  PBE+D3 4.958 (7.012) (9.916) (7.012) 121.9 1.547 2394
SrZrS3 PBE 5.006 (7.079) (10.011) (7.079) 125.4 0.704 1089
  PBE+D3 4.987 (7.052) (9.973) (7.052) 124.0 0.792 1226
BaZrS3 PBE 5.046 (7.136) (10.091) (7.136) 128.4 0.162 250
  PBE+D3 5.023 (7.103) (10.045) (7.103) 126.7 0.208 322
CaZrSe3 PBE 5.219 (7.381) (10.491) (7.381) 142.2 1.582 2447
  PBE+D3 5.195 (7.346) (10.389) (7.346) 140.2 1.745 2700
SrZrSe3 PBE 5.246 (7.418) (10.446) (7.418) 144.3 0.880 1361
  PBE+D3 5.223 (7.386) (10.446) (7.386) 142.5 0.990 1531
BaZrSe3 PBE 5.281 (7.468) (10.561) (7.468) 147.3 0.295 456
  PBE+D3 5.258 (7.436) (10.516) (7.436) 145.4 0.374 579
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Because the inclusion of D3 correction increases the relative stability of the DP phase with respect to the C phase by 46–163 meV/f.u. (see Table 2), which is related to the increase in distortion of the framework discussed above, one can expect that it will also affect the temperature (TDP→C) at which the thermally induced DP-to-C transition is completed. Due to the quasi-continuous nature of this transformation,33 a rough estimate of TDP→C can be made by equating the potential energy difference between the relaxed C and DP (ΔEDP→C = E(C) – E(DP)) phases and the kinetic energy of an NVT ensemble expressed via equipartition principle Inline graphic. The dispersion interactions are predicted to increase the transition temperature by ∼70 to 250 K. Due to the relation between the magnitude of distortion and relative stabilization of DP structures with respect to their cubic counterparts, the predicted transition temperatures also correlate with the uT values calculated for the relaxed CPs. The results summarized in Table 2 indicate that the phase transition temperature decreases with the size of the cation A and that the transition temperatures for the selenide series are ∼200 to 300 K higher compared to the corresponding sulfides.

Next, we analyzed the effect of the inclusion of the long-range interactions in the finite-temperature simulations of SrZrS3 for which the PBE results have been reported in our previous work33 (a detailed PBE+D3 level analysis of thermal effects on the sulfides and selenides series is presented in Section 3.3). Figure 3 compares scaled lattice parameters obtained in NPT MD simulations performed at PBE and PBE+D3 levels for the T range between 0 and 1200 K (the 0 K result corresponds to the result of static relaxation). As expected from the relaxations, the PBE+D3 values are systematically shifted toward lower values. Nevertheless, the two sets of curves are nearly parallel, with deviation in thermal expansion coefficients being within ∼10% (see Table S2 in the SI). At 1200 K, which is the highest temperature considered in this work, a phase transition takes place, which is evident from clear discontinuous changes (see Figure S3 in the SI) in lattice parameters. The fact that the average lattice parameters scaled as a′ = a/√2, b′ = b/2, c′ = c/√2 are identical after this transformation suggests that the newly formed structure corresponds to the cubic phase for which the identity a′ = b′ = c′ holds true (see Figures S1–S3 in the SI). Naturally, the structural changes due to the phase transition also affect the internal geometry of SrZrS3. This can be seen from the radial distribution function computed for the Zr–S (RDFZr-S ) pairs, where the initially well-resolved next-nearest neighbor peaks centered at r ≈ 5.1 and ≈ 6.0 Å tend to merge into a single broad band at ∼5.6 Å (corresponding to the next-nearest Zr–S in a perfect cubic structure) as T is increased (see Figure 4). A similar behavior is observed also for the Sr–S pairs (Figures S10 and S11 in the SI). Although the scaled lattice parameters determined for 1200 K can be considered identical within the uncertainty of our simulations (a′ = 4.799 Å, b′ = 4.796 Å, c′ = 4.799 Å), it is clear from the shape of RDFZr-S and a nonvanishing value of the transversal displacement of S atoms determined for averaged Cartesian coordinates (uT = 0.41 Å) that the transition to C is not complete at 1200 K. A visual inspection of the structure averaged over the whole MD run reveals that the internal structure of this pseudo-C phase resembles that of the tetragonal phase (see Figure S2 in the SI), albeit the cell geometry and volume are distinctly different. As shown in our previous work,33 the undistorted cubic lattice geometry, as well as the internal structure, is achieved at 1500 K. Importantly, the similarity of the RDFZr-S obtained at both levels of theory for the extreme temperatures considered in this work (i.e., 300 and 1200 K), evident from Figure 4, indicates that the D3 dispersion correction has only a small effect on the internal structure of SrZrS3. This small effect is due to the slight shortening of the beyond-nearest-neighbor interatomic distances resulting from an overall reduction in lattice parameters discussed above.

Figure 3.

Figure 3

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Comparison of temperature-dependent scaled lattice parameters obtained from PBE (indicated by a dashed line) and PBE+D3 (indicated by solid lines) calculations for SrZrS3. Note that the transformation used to obtain the scaled parameters are a′ = a/√2, b′ = b/2, c′ = c/√2, whereby identity a′ = b′ = c′ holds in the case of the undistorted cubic phase. This is indeed the case at 1200 K, where a transition to the pseudo-C phase occurs.

Figure 4.

Figure 4

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Comparison of radial distribution functions obtained for the Zr–S pairs (RDFZr-S) in SrZrS3 at 300 and 1200 K. The results from the simulations performed at the PBE (indicated by a dashed line) and PBE+D3 (indicated by solid lines) levels are compared. RDFZr-S obtained for T = 1500 K at the PBE level is also shown as an example of the distribution after a complete transformation to the cubic phase.

3.2. Factors Influencing the Band Gap of DPs

For the purposes of further discussion, it is useful to analyze the main factors influencing the electronic structure of CPs. These include (i) charge transfer from the A to ZrX3 framework, (ii) volume expansion, and (iii) distortion of the ideal cubic framework. To disentangle the contributions of these strongly intertwined factors, we analyzed the DOS and COHP for a series of hypothetical structures for which the individual effects are well defined. Since, at this stage, our goal is only a qualitative analysis, the electronic structure calculations presented here are restricted to the computationally efficient PBE+D3 method, which captures all of the essential trends discussed with sufficient accuracy. For brevity, only the AZrS3 systems are discussed; nevertheless, most of our conclusions also apply to the AZrSe3 CPs, as detailed in Section SIV of the SI.

We start our analysis by considering the neutral undistorted cubic structure ZrS3 without any cation A (system i). The cell vectors compatible with the DP setting were used, whereby the cell volume was set to that of the relaxed DP phase of SrZrS3 (117.2 Å3/f.u.). As evident from DOS, shown in Figure 5a, system i is metallic. It follows from the partial DOS computed for the S atoms and the COHP projected to the nearest neighbors Zr–S and S–S distances that the states near the Fermi level (ϵF) are dominated by the S(3p) orbitals, partly involved in the antibonding S–S interactions (Figures S5 and S6 in the SI). The states with energy below ϵF are hybrids resulting from mixing the S(3p) with Zr(4d) atomic orbitals, whereby the contribution of the latter increases with decreasing energy. According to the COHP analysis, these hybrid states originate from the bonding Zr–S interactions. The regions of the dominant 4dxy and 4dz2 (the lower part of the valence band) and 4dyz, 4dxz, and 4dx2–y2 (the central part of the band) states of Zr can be clearly distinguished (Figure S7 in the SI). The broad low-intensity band of unoccupied states starting at ∼1.5 eV above ϵF is almost exclusively due to the 4dyz, 4dxz, and 4dx2–y2 states of Zr. The same Zr(4d) orbitals (with only a modest participation of 4dxy and 4dz2 orbitals) hybridize with the S(p) states, giving rise to a peak centered at ∼4.0 eV. The COHP analysis (Figure S6 in the SI) shows that these hybrid states are involved in the antibonding Zr–S interactions.

Figure 5.

Figure 5

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Total and partial density of states projected over S and Sr atoms in (a) system i—neutral cubic structure [ZrS3]0 with no A cation, (b) system ii—a hypothetical system [ZrS3]−2 with undistorted cubic structure, (c) system iii—[ZrS3]−2 with the cell geometry and atomic positions fixed at those from the relaxed DP phase of SrZrS3, (d) system iv—cubic [ZrS3]−2 structure with a cell volume of 692.8 Å3 obtained from the unconstrained relaxation of the distorted structure, (e) system v—cubic SrZrS3 with a cell volume of 117.2 Å3, and (f) system vi—relaxed distorted structure of SrZrS3.

The formal charge of cation A in the AZrX3 structures is +2, i.e., formally each cation provides two electrons to the system. The effect of this charge transfer can be identified by considering a hypothetical system ZrS32– (system ii) that is isostructural with the system i discussed above. Supplementing the extra charge into the system causes only modest changes in the overall shape of the DOS and the nature of the individual features. Importantly, however, the Fermi level is shifted such that the valence band is fully occupied (see Figure 5b), which is the reason why system ii exhibits a small but nonzero band gap (Eg = 0.16 eV).

The fact that the antibonding states due to the S–S interactions contribute to the highest occupied states of the ZrS32– structure suggests that the system could be stabilized by increasing the S–S distances, e.g., via distortions occurring in the DP phase of the AZrX3 compounds. In the next step, we therefore considered the system ZrS3 with the cell geometry and atomic positions fixed at those from the relaxed DP phase of SrZrS3 (system iii). Compared to the parent cubic structure, the nearest neighbor S–S separations increased from 3.46 to at least 3.59 Å. As evident from COHP (see Figure S5 in the SI), the valence band contribution of the antibonding S–S interactions was significantly reduced. Compared to system ii, all five Zr(4d) orbitals now contribute more evenly to both the valence and the conduction bands (see Figure S7 in the SI); both bands are contracted, and the computed band gap increased significantly (from 0.16 eV for system ii to 1.07 eV for system iii). The overall effect of these changes is a relative stabilization of the structure by 0.46 eV/f.u.

In the absence of the A cation, however, the distorted ZrS32– structure is unstable. Indeed, the unconstrained relaxation leads to a cubic structure (system iv) with a very large cell volume (173.2 Å3/f.u.), whereby the total energy is lowered by as much as 3.15 eV/f.u. compared to system ii. This massive increase in volume, leading to elongation of the Zr–S and S–S distances (from 2.44 to 2.78 Å and from 3.46 to 3.94 Å, respectively), can be viewed as another way of eliminating the antibonding S–S interactions. As in the DOS computed for system iv (see Figure 5d), an overall contraction of the valence and conduction bands leads to a notable increase in the band gap (0.49 eV) relative to system ii (0.16 eV). The nature of the valence and conduction states, however, remains the same as in system ii.

Next, we compare the cubic SrZrS3 (system v) with a cell volume of 117.2 Å3/f.u. with its counterpart without cation A (i.e., system ii). We emphasize that the only difference between these two systems is the presence of cation A instead of the uniform background charge. The iterative Hirshfeld charge66,67 of Sr in SrZrS3 is 2.02 |e|, which is close to the formal charge of cation A in CPs. As shown in Figures 5e,b, S5, and S6 in the SI, the shape and nature of the valence and conduction features of the DOS closest to ϵF computed for both systems exhibit a striking similarity. This is not only because the amount of charge transferred to ZrS3 is similar in both cases but also because the contribution of atomic orbitals of Sr to these states is negligible. The computed Eg of SrZrS3 in this setting is 0.35 eV, which is not too different from the value of 0.16 eV obtained for ZrS32–. Replacing Sr by Ca or Ba in the same structure leads to a very modest change in Eg (the computed values are 0.33 (CaZrS3) and 0.34 eV (BaZrS3)), despite somewhat different amounts of charge transferred to ZrS3 (the iterative Hirshfeld charges of Ca and Ba in these structures are 1.73 and 2.11 |e|, respectively). The increase in volume leads to a modest monotonic increase in the band gap, as shown in Figure 6a. For SrZrS3, Eg increases from 0.27 to 0.57 eV over the interval ±9% around the ground-state value. A nearly identical dependence is found for CaZrS3 and BaZrS3. This is a remarkable result showing that the cation A affects the band gap of cubic AZrS3 rather indirectly via stabilization of low cell volumes through the Coulomb interactions with the ZrS3 framework. Indeed, the volumes of the relaxed cubic CaZrS3, SrZrS3, and BaZrS3 (121.9, 124.0, and 126.7 Å3/f.u., respectively) are all significantly lower compared to that of relaxed ZrS32– (173.2 Å3/f.u.), while their increase with the size of cation A is rather modest.

Figure 6.

Figure 6

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Variation in the electronic band gap as a function of volume for different cations A (A = Ca, Sr, Ba) in AZrS3 in C and DP phases with fixed fractional coordinates of atoms. (a) In the case of the C phase, the volume was varied in the interval ± 9% around the ground-state volume determined for each composition. In the case of the DP phase, the volume was varied in the interval ± 9% around the ground-state volume determined for SrZrS3, whereby the lattice geometry was fixed. (b) The volume for the DP phase was varied in the interval ± 9% around the ground-state volume determined for each composition, whereby the atomic positions and lattice geometries were relaxed.

Relatively small changes in DOS are observed also when replacing the uniform background charge in system iii with the cation Sr2+ (system vi). According to the iterative Hirshfeld analysis, the charge transferred from Sr to ZrS3 is 2.02 |e|, which is, coincidentally, identical to the value determined for the cubic structure discussed above. As in the case of the cubic structure, the band gap computed for the SrZrS3 (1.22 eV) is larger than that for the analogous system without A cation (1.07 eV). Replacing Sr by Ca or Ba while fixing the cell and atomic geometry leads, however, to a more pronounced variation in the band gap (Eg = 1.27 eV and 1.11 eV for CaZrS3 and BaZrS3, respectively) than observed for the cubic systems (Figure 6a). Still, this result shows that the direct effect of cation A on the band gap, i.e., the decrease in Eg with increasing size of A at a fixed structure, is rather modest. As in the cubic structures, the increase in volume leads to a monotonic increase in the band gap. If only volume is varied (i.e., fractional coordinates of atoms are fixed) over the interval ± 9% around the ground-state value, the band gap of SrZrS3 increases from 1.10 to 1.27 eV. The curves obtained for the same structures of CaZrS3 and BaZrS3 compositions are similarly shaped but, unlike in the case of cubic structures, they are shifted by up to ±0.1 eV relative to the SrZrS3. As expected, this similarity in the shape and position of the Egversus V dependencies is significantly reduced upon relaxation of the DP structures (see Figure 6b). This result shows, once again, that the structure of the ZrS3 framework is the key factor determining the band gap of AZrS3 compounds, while the role of cation A is mainly indirect, via its effect on the ZrS3 structure and volume.

3.3. Thermal Effects on the Crystal Structure

In this section, we investigate the thermal effects on the AZrX3 structures using ab initio NPT MD simulations accelerated by MLFF50 (see Section 2). The finite-temperature values of structural parameters for the DP and C phases are listed in Table 3, and the corresponding plots are shown in Figure 7. In addition, the thermal dependence of the linear and volume thermal expansion coefficients evaluated using eq S1 is presented in Figure S9 in the SI.

Table 3. Ensemble Averages of Lattice Constants (a, b, c), Cell Volume per Formula Unit (V1), and Transversal Displacement of X in Averaged Structures (uT) Calculated for AZrX3 (A = Ca, Sr, Ba; X=S, Se) in the Temperature Range between 0 and 1200 K Using the PBE+D3 Functionala.

compound T (K) a (Å) b (Å) c (Å) V1 (Å3/f.u.) uT (Å)
CaZrS3 0 7.023 9.573 6.531 109.8 0.870
  300 7.038 9.611 6.567 111.0 0.858
  600 7.053 9.651 6.605 112.4 0.846
  900 7.069 9.694 6.648 113.8 0.831
  1200 7.083 9.742 6.692 115.4 0.813
SrZrS3 0 7.121 9.761 6.746 117.2 0.725
  300 7.131 9.802 6.783 118.5 0.709
  600 7.138 9.846 6.827 119.9 0.692
  900 7.139 9.899 6.879 121.5 0.665
  1200 (c) 7.041 (4.979) 9.952 (4.976) 7.041 (4.979) 123.3 0.408
BaZrS3 0 7.150 10.001 6.985 124.9 0.514
  300 7.135 10.041 7.048 126.2 0.475
  600 (c) 7.126 (5.039) 10.069 (5.034) 7.126 (5.039) 127.8 0.087
  900 (c) 7.154 (5.059) 10.125 (5.062) 7.154 (5.059) 129.5 0.028
  1200 (c) 7.184 (5.080) 10.173 (5.086) 7.184 (5.080) 131.2 0.004
CaZrSe3 0 7.350 10.004 6.783 124.7 0.954
  300 7.372 10.044 6.821 126.2 0.942
  600 7.393 10.087 6.862 127.9 0.928
  900 7.414 10.136 6.905 129.7 0.913
  1200 7.434 10.186 6.956 131.6 0.910
SrZrSe3 0 7.456 10.187 7.003 133.0 0.815
  300 7.473 10.229 7.041 134.5 0.800
  600 7.486 10.277 7.086 136.2 0.782
  900 7.495 10.332 7.137 138.1 0.760
  1200 7.495 10.396 7.205 140.3 0.726
BaZrSe3 0 7.523 10.431 7.225 141.8 0.628
  300 7.516 10.481 7.286 143.5 0.600
  600 7.503 10.538 7.355 145.4 0.554
  900 (c) 7.480 (5.289) 10.566 (5.283) 7.477 (5.287) 147.7 0.108
  1200 (c) 7.513 (5.312) 10.637 (5.318) 7.513 (5.312) 150.0 0.041
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a

The symbol (c) indicates the cubic (or pseudo-cubic) phase for which the values in parentheses represent the scaled parameters a′ = a/√2, b′ = b/2, c′ = c/√2.

Figure 7.

Figure 7

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Scaled lattice parameters a′ = a/√2, b′ = b/2, c′ = c/√2 of the DP/C form of AZrX3 (A = Ca, Sr, Ba; X = S, Se) as functions of temperature. Note that the identity a′ = b′ = c′ holds in the case of the undistorted cubic phase. Triangle and star symbols represent the theoretical and experimental results, respectively. To facilitate a clearer comparison with theory, which tends to slightly overestimate the lattice parameters (see Table 3), all experimental data points were shifted by 0.02 Å.

All of the compositions in this study undergo a monotonic and positive volume thermal expansion upon heating. Close to 0 K, the magnitude of the volume thermal expansion coefficient is similar for all compositions examined, with a typical value of 33–40 × 10–6 K–1 (see Figure S9 in the SI). The temperature-induced changes in the lattice parameters are anisotropic and, as discussed in our previous work,33 they are related to the fact that increased temperature tends to reduce the distortion with respect to the cubic phase, which is eventually restored at a sufficiently high T. Hence, the lattice parameter c, with the value deviating more from that in the C phase than a and b (cf. Tables 1 and 2), changes most significantly with T. Interestingly, although the lattice parameters generally elongate upon heating, the overall tendency of distortion reduction with increasing T may also lead to a negative expansion in some cases. This is observed for the parameter a in SrZrS3 and SrZrSe3 at temperatures over 900 K.

At the internal structural level, the tendency to reduce the deformation with respect to the C structure upon heating is obvious from the radial distribution function for the Zr–X or A–X pairs, where the initially well-resolved next-nearest neighbor peaks tend to merge into a single band (Figures S10 and S11 in the SI). As a simple measure of the progress of the DP → C transformation, one can also use uT computed for the averaged Cartesian coordinates (see Table 3), which clearly decreases with T. Although this trend is general, its magnitude depends strongly on the composition of CP. As is clear from the data presented in Table 3, the magnitude of the T-induced reduction of uT in AZrX3 increases with the size of A cation and decreases with the size of X anion. Thus, BaZrS3 represents one extreme case, with uT dropping from 0.51 Å at 0 K to 0.00 Å at 1200 K. The other extreme case is CaZrSe3 with the change in uT being as small as 0.04 Å (from 0.95 Å to 0.91 Å) within the same temperature interval.

The lower the values of uT at 0 K (see Table 3), the lower the temperature at which the transformation to C is completed. In particular, the compound BaZrS3, with a low-temperature uT value of 0.51 Å, exhibits phase transition between 300 and 600 K, while the transition T for BaZrSe3 with nominal uT = 0.63 Å falls into the interval between 600 and 900 K. Furthermore, as discussed in Section 3.1, the lattice geometry of SrZrS3, which is the compound with the next lowest uT value at 0 K (0.73 Å), undergoes a change corresponding to a pseudo-cubic phase at 1200 K, but the RDF evaluated for the Zr–S and Sr–S pairs, as well as the relatively large uT value (0.41 Å), indicates that phase transition is not yet fully completed at this point (as shown in our previous work, a full conversion is achieved when T is increased to 1500 K). We note that these results are in qualitative agreement with the transition temperatures estimated from the potential energy differences between the relaxed DP and C structures (see Table 2). Thus, although the latter seem to significantly underestimate the actual TDP → C, the static approach proves to be useful for computationally inexpensive and fast transition temperature estimates.

Our theoretical predictions can be confronted with our in-house experimental measurements performed for SrZrS3 and BaZrS3. As shown in Figure 7, the qualitative agreement is reasonable. Indeed, the predicted reduction in separation between scaled lattice parameters of both compounds is confirmed by the experiment. Also, this separation determined for BaZrS3 is significantly smaller compared to that for SrZrS3 and, indeed, it tends to vanish at T ≈ 600 K in the former case. Finally, we remark that we did not consider in our simulations any processes requiring reconnection of the Zr–S bonds, such as, e.g., the transformation to the NL phase or structural disintegration due to thermal instability. Such investigations would require extremely time-consuming free-energy calculations, which are well beyond the scope of the present study. Indeed, a trace of structural transitions that cannot be explained by the quasi-continuous DP-to-C transition can be observed in our experimental data measured for BaZrS3 at 800 K, where discontinuous changes in lattice parameters take place.

3.4. Effect of Temperature on the Electronic Band Gap

Using the structural data generated in our NPT MD runs described in Section 3.3, the temperature-dependent band gaps were determined as ensemble averages (⟨Eg⟩). A detailed description of the procedure to obtain the electronic band gaps has been provided in Section SVI of the SI.

The T-dependence of band gaps for all compositions is shown in Figure 8. The corresponding numerical values for both the PBE+D3 and the HSE06 levels of theory are compiled in Table S3 of the SI. The general trend observed is a monotonic decrease of the ⟨Eg,HSE06⟩ with T. The slope of this dependence increases in absolute value with T in most cases. The exceptions are the compounds BaZrS3 and BaZrSe3 undergoing the DP-to-C transition below 600 and 900 K, respectively (see Section 3.3), where also the largest changes in the ⟨Eg,HSE06⟩ occur. At first sight, the Egversus T trend seems to disagree with the analysis provided in Section 3.2, where a small monotonous increase of Eg with volume (itself monotonously increasing with T; see Section 3.3) was demonstrated. However, based on our analysis from Section 3.2 and in agreement with our previous report,33 we show that the observed decrease in the Eg with T is due to a gradual DP → C transition causing changes in Zr(4d) orbital contributions dominating the conduction band. Indeed, the magnitude of the total band-gap decrease over the temperature range considered is the largest (0.65 eV) for BaZrS3 and the smallest (0.24 eV) for CaZrSe3, the systems with the largest and the smallest temperature-induced reductions in uT, respectively. From the data presented in Table S3 of the SI and Figure 8, one can deduce that |ΔEg| increases with the size of cation A and decreases with the size of anion X. All of the trends predicted by HSE06 are virtually identical to those obtained from the PBE+D3 calculations, with the ⟨Eg,PBE+D3⟩ values being upshifted by 0.5–0.8 eV.

Figure 8.

Figure 8

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Temperature dependence of the HSE06 electronic band gaps for DP/C phases of AZrX3 (A = Ba, Sr, Ca; X = S,Se) computed as ensemble averages from NPT MD performed at the PBE+D3 level. The dotted lines define the optimal range of the desired band gap for photovoltaics.68

The experimental information on the band gap is presently available only for the X = S series at room temperature.16,32 Our results are in very good agreement with these reports, with the error within 2–8%. For a fixed finite T, the general tendency is a decrease in the band gap with an increase in the size of A and X, with the results for CaZrSe3 and SrZrSe3 at 300 K being the only exceptions from this trend. These results are, again, in agreement with the conclusions made in Section 3.2. From the data presented in Figure 8, we conclude that the calculated band-gap values fall within the optimal range68 of single-junction photovoltaic devices for all compositions of selenides across all temperatures. While the band gaps of CaZrS3 and SrZrS3 are slightly above the desired range, the Eg value for BaZrS3 moves into the range above 400 K. Since BaZrX3(X = S, Se) is expected to show early phase transitions to the ideal cubic phase (600–900 K), we anticipate that the thermal band gaps could be eventually tuned to the optimal photovoltaic range by exploring the mixed BaZrS3–xSex compositions, as also proposed by Liu et al.21

4. Conclusions

A comprehensive DFT investigation of the effect of composition and temperature on the structure and electronic properties of a class of chalcogenide perovskites AZrX3 (A = Ca, Sr, Ba; X = S, Se), with a distorted perovskite structure, was performed. The long-range dispersion interactions, introduced by means of the D3 method of Grimme et al.,42,43 were shown to improve the quality of zero and finite-temperature structural predictions, reducing the average error in lattice parameters and volume, for compounds with experimentally known structures (CaZrS3, SrZrS3, and BaZrS3) from 1.0% and 3.4% to 0.4% and 1.4%, respectively.

Various factors affecting the electronic band gap of CPs were analyzed in detail in a series of zero-T calculations. It was shown that the cation A affects the Eg mainly indirectly by influencing the magnitude of structural distortion in CPs, while its direct effect, identified through the change of Eg for a fixed geometry upon changing A, is only modest (0.01–0.08 eV). For all compositions, a monotonous decrease of the band gap with temperature was observed. In contrast, the change of structure from a distorted perovskite to the cubic lattice leads to a large reduction of the Eg (by 0.5–0.7 eV), making the underlying structural transformation the key factor dictating trends in the band-gap changes due to various external stimuli. The importance of the symmetry-breaking distortions on the electronic structure was also demonstrated in our previous work33 for the specific case of SrZrS3 and in other reports for related materials.22,28,29

The finite-temperature effects on the structure of CPs were investigated using ab initio-quality MLFF-accelerated molecular dynamics in the NPT ensemble. Over the temperature range of 0–1200 K considered in this work, a monotonous expansion of volume and most of the lattice vectors was observed. For all compounds, the magnitude of distortions with respect to the parent cubic structure gradually decreased with increasing temperature. The magnitude of the decrease, measured by the parameter uT, however, was found to be strongly composition-dependent, whereby it increases with the size of cation A and decreases with the size of anion X. We found that BaZrS3 represents an extreme case for which uT was reduced from 0.51 Å at 0 K to 0.00 Å at 1200 K, indicating that the DP → C phase transition was complete. Another extreme case is CaZrSe3, with a very rigid structure for which uT barely changed (from 0.95 to 0.91 Å) over the same T interval.

The T-dependences of the electronic band gaps were computed at the HSE06 level as ensemble averages over the data generated in NPT MD. For all compositions, a monotonic decrease in the Eg with T was observed, with the magnitude of the decrease dictated by the underlying internal structure. Hence, ΔEg of −0.65 eV and −0.24 eV obtained for BaZrS3 and CaZrSe3, respectively, represent two extreme cases. The Eg values computed for sulfides series at T = 300 K are in reasonable agreement (relative error within 2–8%) with the available experimental data.16,32 To the best of our knowledge, our predictions for the selenide series represent the only finite-T report to date. In contrast to sulfides, the band gaps of selenides computed for a wide range of temperatures (including the room temperature) fall within the optimal range for photovoltaic applications (i.e., within 0.9–1.6 eV).68

Overall, we demonstrated that the primary effect of temperature on the structure of CPs is the change in the extent of distortion relative to the parent cubic phase. This is also the most important factor affecting the magnitude of their band gap. Of course, the structural distortions can also be achieved by other means, e.g., via partial substitutions of cations and/or anions by atoms of different sizes,21 which we plan to investigate in our future work. Also, the CP materials are known to exist in several competing phases, such as the orthorhombic NH4CdCl3-type and hexagonal structure.9,18 A correct ranking of the relative stability of the DP phase with respect to these alternative structures is a pertinent and challenging problem, requiring accurate free-energy calculations at a suitable level of theory to accurately predict energy differences that are sometimes extremely small. We shall address this problem in our future work.

Acknowledgments

This work was supported by the European Union’s Horizon 2020 research and innovation program under Grant Agreement 810701 and by the Slovak Research and Development Agency under Grant Agreement APVV-19-0410. N.J. acknowledges partial support from Project USCCCORD (ZoNFP: NFP313020BUZ3), cofinanced by the European Regional Development Fund within the Operational Programme Integrated Infrastructure. Part of the research was obtained using the computational resources procured in the national project National competence centre for high performance computing within the Operational programme Integrated infrastructure (project code: 311070AKF2).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.inorgchem.3c01696.

  • See the Supporting Information for additional information on the structure and electronic structure of materials discussed in the main text and a description of the procedure for obtaining temperature-dependent band gaps at the HSE06 level (PDF)

The authors declare no competing financial interest.

Supplementary Material

ic3c01696_si_001.pdf (1.5MB, pdf)

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