Defect and carrier characteristics of chalcogenide perovskite BaZrS₃ under thermodynamic stability: a first-principles study for photovoltaic application

Abstract

BaZrS₃ is reported to be an exceptional chalcopyrite perovskite with remarkable stability, making it a key material for the next-generation perovskite-inspired technologies. In this study, the thermodynamic stability of BaZrS₃ was investigated while referencing all potential secondary phases in the Materials Project database. Furthermore, the influence of element doping on both the bands and thermodynamic stability of BaZrS₃ was also studied, the band adjustment efficiency was validated, and experimental synthesis challenges were elucidated. Based on a large supercell, Heyd–Scuseria–Ernzerhof hybrid functional and finite-size effect correction, the concentration properties of defects and charge carriers of BaZrS₃ were then determined in the thermodynamically stable region, from the characteristics of defect formation energy with varied Fermi levels. The results indicate that degenerated semiconductor characteristics exist in BaZrS₃ in the thermodynamically stable region, which should be avoided. The defect transition energy level characteristics of BaZrS₃ were also calculated and identified. Additionally, the properties of deep-level defects S_i with high concentration were analyzed. While S_i (+2/0) exhibits deep-level defect states, S_i (+1/0) does not. The non-radiative and radiative capture coefficients of deep-level defect S_i (+2/0) were calculated at 1.28682 × 10⁻²⁵ and 3.51 × 10⁻¹⁵ cm³ s⁻¹, respectively. The obtained defect and carrier qualities of BaZrS₃ were applied to evaluate its photoelectric conversion efficiency. The manuscript investigates, for the first time, how defects evolve with changes in chemical potential in stable BaZrS₃, providing valuable guidance for the experimentally controllable preparation of high-quality BaZrS₃ films or devices.

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1 Introduction

Perovskite materials have been the subject of a lot of research over the past decade and have been gaining prominence in the fields of photovoltaic,^1–5 photodetection^6–8 and light-emitting applications.^9–11 In recent years, halides have made up the majority of the semiconductor perovskite materials that have been thoroughly studied. Halide perovskites exhibit synergistically reduced band gaps suitable for photovoltaic applications, which can be attributed to the low conduction band resulting from the strong spin–orbit coupling in cations, such as lead, and the high valence band contributed by anions, such as the iodine 6p orbitals.^12,13 Another crucial physical property of halide perovskites is the defect tolerance, which allows these materials to be readily prepared for device fabrication with less need for high purity and high crystallinity, while keeping production costs low. Currently, halide perovskites face two main obstacles to larger-scale deployment: (i) long-term stability problems, partially due to the inherent instability of their organic components against moisture, heat, light and electric fields; and (ii) toxicity, primarily arising from the use of lead.^14–17 These drawbacks have severely impeded their commercial application.

Since their initial synthesis in 1957,¹⁸ chalcogenide perovskites (CPs with the formula ABX₃; A, B = metals with a combined valence of 6; X = S, Se or other chalcogen elements) have demonstrated the anticipated benefits of outstanding thermal and aqueous stability in addition to their nontoxic elemental composition. Although many CPs have been documented since then, research on their physical characteristics has remained intermittent.^19–21 Inspired by the success of halide perovskites, CPs have been proposed for photovoltaic applications in recent years. Their tunable band gaps (in the range 0.5 to 2.2 eV), strong light absorption coefficients (at 10⁻⁵ cm⁻¹), small effective masses of charge carriers (m_e, m_h < 1) and superior environmental stability in comparison to halide perovskites have made them promising candidates for next-generation optoelectronic devices.^22–27 Among these, barium zirconium sulfide (BaZrS₃) is notable for its ideal direct band gap (1.7 to 1.9 eV),excellent dielectric response and defect tolerance.^{23,24,28–34} Additionally, BaZrS₃ is found to exist preferentially in the distorted perovskite phase with 3D-connected corner-sharing BX₆ octahedra.³⁵ While other chalcogenide perovskites preferentially form phases with either edge-sharing or isolated BX₆ octahedra (so-called “needle-like”^36,37 and “hexagonal” phases^38,39). Because of the absence of connected octahedra along certain directions of the crystal structure, these structures are predicted to show more localized conduction and valence band edges, which will result in heavy electron and hole masses. Consequently, BaZrS₃ is anticipated to be more appropriate for optoelectronic devices from the perspective of carrier mobility. As a result, BaZrS₃ is a preferred material to evaluate whether CPs are viable options for applications to optoelectronic devices both theoretically and experimentally. For optoelectronic materials, one of the most important properties is their carrier concentration. A high carrier concentration is essential for achieving a high short-circuit current.^40–42 However, the carrier concentration of CPs, including BaZrS₃, has been scarcely reported in the literature, especially under thermodynamically stable conditions. Another crucial aspect for exceptional photoelectric materials is their defect characteristics, which affect the total performance of manufactured devices like solar cells. For example, the open-voltage, short-circuit current and fill factor of the solar cells are directly related to how many charge carriers can ultimately be transferred to the external circuit after carrier recombination by defects.⁴³ In optoelectronic materials, recombination centers (RCs), which typically manifest as deep-level defects, should be avoided as they compete with the desired carriers. It has been demonstrated that intrinsic defects in halide perovskites are mostly shallow defects, which do not serve as effective RCs.^42,44–48 This characteristic enables low-cost material synthesis and low-cost device fabrication without requiring complex defect management. Defect tolerance has been identified as one of the special qualities of halide perovskites in terms of their achievements,^{43,44,49–51} allowing for effective solar cell performance even in the presence of high defect concentration. Then, what are the characteristics of defects in BaZrS₃?

According to recent hybrid density functional theory (DFT) computational investigations conducted by Sunfs group, BaZrS₃ has a distinct self-passivation process in which the majority of vacancies and antisite defects generate shallow transition levels, as opposed to deep-level RCs. Although it has been claimed that several types of defects, such as S_i and S_Ba, are deep-level defects. It is supposed that, for deep-level defects including S_i and S_Ba, the formation of S clusters (dimer, trimer and tetramer) tends to remove mid-gap states in the neutral charge state and leaves behind a relatively clean band gap.²⁹ They also conducted calculations with the PBEsol + U functional, and the results show that the intrinsic defects of BaZrS₃ are all shallow-level defects, despite the underestimated band gap. In addition, Yan's group also studied the defect transition levels of BaZrS₃ with the GGA + U functional.²² Comparison of the findings from these investigations, however, shows they have very different defect transition characteristics, particularly for the locations of deep-level defects in the band gap.^22,29 Experimental validations also highlight discrepancies and synthesis-dependent defect kinetics. Compared to epitaxial thin films or single crystals, solution-synthesized BaZrS₃ nanoparticles exhibit larger defect concentrations, resulting in reduced carrier lifetime.^30,34,52 Therefore, such findings highlight the need to bridge theoretical defect characteristics with practical synthesis protocols, using a better understanding of the defect characteristics of BaZrS₃ under thermodynamic stability. However, the quantitative data that can adequately support the defect characteristics of BaZrS₃ is currently missing.

In this study, based on a large nearly cubic supercell of 240 atoms, the Materials Project database, the Vienna Ab initio Simulation Package (VASP) and the Defect and Dopant Ab initio Simulation Package (DASP),^53–56 the optical properties and thermodynamic stability of BaZrS₃ were studied first. The chemical potential of the stable BaZrS₃ compound was determined while fully considering the competitive secondary phases. Then, the defect formation energy, ionization energy, carrier concentration, Fermi energy level and deep-level defects, along with the non-radiation/radiation recombination characteristics of BaZrS₃, were systematically investigated in the thermodynamically stable region. Finally, the quantitative defect and carrier characteristics of BaZrS₃ were obtained, and their effects on the photoelectric properties of BaZrS₃ were examined.

2 Method of calculation

The electronic structure of the orthorhombic chalcogenide perovskite BaZrS₃ (pnma, NO. 62) was examined in the study by first-principles calculations, as mentioned above. The total energy calculations were performed in VASP using density functional theory with a plane wave energy cutoff of 336 eV (about 1.3 times the ENMAX in the pseudopotential files in the projector-augmented wave (PAW) method).^57,58 To determine the stable chemical potential range, the structures (along with the reciprocal space meshes) and PAW potentials (GGA-PBE) of the elemental components and secondary phases were all taken from the data provided in the Materials Project database. At first, the relaxation of the structure was done at the PBE level to obtain a converged WAVECAR file, and then the calculations were performed using the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06) with the converged WAVECAR. The defect characteristics of BaZrS₃ were studied using a large supercell of 240 atoms with a nearly cubic structure. The Γ point was used to represent the Brillouin zone. The PAW potentials for BaZrS₃ were kept the same. The forces were fully relaxed to below 0.01 eV Å⁻¹, and defect corrections (including potential alignment for charged defects) were performed.⁵⁹ HSE06 and finite-size electrostatic correction schemes were employed to determine the formation energy and transition energy levels of the defects.^60–62 The band gap of BaZrS₃ was fixed at a value of 1.732 eV, with a Hartree–Fock exchange value α of about 0.2163.

For BaZrS₃, the considered intrinsic defects in this study include the Ba, Zr and S vacancies (V_Ba, V_Zr and V_S), interstitials (Ba_i, Zr_i and S_i) and antisites (Ba_Zr, Ba_S, Zr_Ba, Zr_S, S_Ba and S_Zr). Taking S vacancies in the charge state q as an example, its defect formation energy ΔE_f(V^q_S) can be calculated from eqn (a):^63–65

ΔE_f(V^q_S) = E(V^q_S) − E(bulk) + E(S) + µ_S + q(E_F + E_VBM) + E_corr (a)

where the total energy of the supercell with and without an S vacancy is denoted by E(V^q_S) and E(bulk). µ_S is the elemental chemical potential of S referenced to its total energy E(S). E_F is the Fermi level referenced to the eigenvalue of the valence band maximum (VBM) level of the bulk defect-free supercell of BaZrS₃. E_corr is the correction for charged defects to account for the finite-size effect, with the FNV (Freysoldt–Neugebauer–Van de Walle) method.⁶² The formation energy of V_S in BaZrS₃ was calculated in the supercell structure, as mentioned above. The charge transition level is equal to the Fermi-level position, where the formation energies in two charge states are the same. For neutral and 1+ charge states, the α(0/1+) charge transition level with respect to the VBM (E_F = 0) can be calculated with eqn (b):

α(0/1+) = ΔE_f(V⁰_S; E_F = 0) − ΔE_f(V⁺_S; E_F = 0) (b)

For the calculation of the non-radiative recombination characteristics of deep-level defects with high concentration, the excited-state carrier dynamics properties based on the phonon spectrum and electron–phonon coupling were studied. As an example, the hole capture coefficient C_p by V⁰_S can be calculated using the one-dimensional static coupling approximation, as in eqn (c):⁶⁶

where the electron–phonon coupling matrix element W_if was determined using VASP via bulk/single-particle wave functions under related approximations, methods and theories in ref. 66 and 67. ω_m is the thermal occupation of the initial vibrational state m, χ_im and χ_fn are the vibrational wave functions and E_im and E_fn are the related eigenvalues. ΔQ is the difference in the equilibrium geometries of the initial and final state along the configuration coordinate. The carrier transition energy was denoted by ΔE; for the case of hole capture by V⁰_S, ΔE is exactly the same as the α(0/1+) transition level. More details of the element or parameter calculation in eqn (e) for carrier capture can be found in ref. 66 and 67, as indicated above.

3 Results and discussion

3.1 Optical characteristics of BaZrS₃

The optical characteristics of BaZrS₃ were investigated using standard DFT calculations with the HSE06 hybrid functional. First-principles calculations reveal the distinctive optical characteristics of BaZrS₃, as displayed in Fig. S1(a). The optical absorption spectrum exhibits three prominent peaks at 2.25, 2.57 and 3.87 eV, which could be assigned to direct allowed transitions between hybridized Zr-d and S-p orbitals.⁶⁸ The absorption coefficient exceeds 10⁵ cm⁻¹ near the band edge,⁶⁸ demonstrating light-harvesting capabilities on a par with or even better than those of CH₃NH₃PbI₃ and GaAs (at 10⁴ cm⁻¹),² and significantly higher than that of silicon.⁶⁹ The calculated characteristic curve is generally consistent with that from experimental-quality BaZrS₃ thin films.²³ The optical absorption edge of BaZrS₃ is at 1.96 eV from optical calculation, as indicated by the tangent in Fig. S1(a). Furthermore, the remarkable light-harvesting ability of BaZrS₃ single crystals at 10⁶ cm⁻¹ has also been asserted,³⁴ suggesting their potential as superior optoelectronic materials.

Fig. S1(b) illustrates the dielectric response properties of BaZrS₃. The real component exhibits a characteristic dispersion with a static dielectric constant of 10.2, in good agreement with earlier hybrid functional predictions.²⁹ The dielectric constant of BaZrS₃ is a little lower than that of silicon (11.7)⁷⁰ and GaAs (12.88) but higher than that of CH₃NH₃PbI₃ (6.5).⁷¹ The imaginary part onset at 1.91 eV, as displayed in the inset to Fig. S1(b), coincides with the absorption edge in Fig. S1(a).

3.2 Thermodynamic stability of BaZrS₃

The stability of BaZrS₃ is a prerequisite for studying its properties. During the crystal growth of BaZrS₃, although the abundance of a certain element can be controlled by changing the environment, the changes in the elemental chemical potentials are not actually unlimited, because they should satisfy a series of thermodynamic conditions to make the pure-phase crystalline semiconductor stable. Under equilibrium growth conditions, the weighted sum of the chemical potentials of its component elements should be equal to the formation energy of BaZrS₃, as illustrated in eqn (d). Here µ_i is the chemical potential of the element, where i refers to the most stable phase of the involved elements. The elemental chemical potentials are referenced to uncorrected DFT energies at 0 K. µ_BaZrS3 represents the formation energy of BaZrS₃ that can be calculated from the total energy of BaZrS₃ and the elemental standard states. Additionally, to ensure that the synthesized sample is pure-phase BaZrS₃, the formation or coexistence of the competing secondary phases, such as the binary and ternary compounds ZrS₂, ZrS₃, Zr₃S₄ and Ba₃Zr₂S₇, and the elemental phases Cu, Zr, S, should be avoided. Therefore, the weighted sum of the chemical potentials of their component elements should be lower than their corresponding formation energies (the formation energies of elemental phases are 0). As a result, the relationships in eqn (e) must also be satisfied, as detailed in the equations below. Here µ_ZrS2, µ_ZrS3, µ_Zr3S4 and µ_Ba3Zr2S7 represent the formation energies of ZrS₂, ZrS₃, Zr₃S₄ and Ba₃Zr₂S₇, respectively, which can also be calculated from the total energy of ZrS₂, ZrS₃, Zr₃S₄ and Ba₃Zr₂S₇, and the elemental standard states.

µ_Ba + µ_Zr + 3µ_S = µ_BaZrS3 = −9.9535 eV (d)

µ_Zr + 2µ_S < µ_ZrS2 = −5.1287 eV, µ_Zr + 3µ_S < µ_ZrS3 = −5.4155 eV, 3µ_Zr + 4µ_S < µ_Zr3S4 = −12.1999 eV, 3µ_Ba + 2µ_Zr + 7µ_S < µ_Ba3Zr2S7 = −24.4885 eV. (e)

As discussed above, the three constituent elements, Ba, Zr, S, and four additional compounds, ZrS₂, ZrS₃, Zr₃S₄ and Ba₃Zr₂S₇, are considered competitive secondary phases of BaZrS₃. (It should be noted that, when µ_Ba = 0, µ_Zr = 0 and µ_S = 0, it means that these elements are so rich that their pure solid phase can be formed). Under these constraints, according to the calculations, the chemical potential range of Ba, Zr and S that stabilizes the BaZrS₃ compound is bound to a polyhedron in the µ_Ba, µ_Zr and µ_S space, as depicted in Fig. 1. Table 1 lists the chemical potential boundary points p1–p4 that correspond to Fig. 1. As indicated in Table 1, we focus the discussion in the following sections on the stable region of BaZrS₃ with boundary points p1–p4.

Fig. 1 The calculated chemical potential region where BaZrS₃ is stable against the formation of three constituent elements and four competitive compounds (i.e., the allowed chemical potential region is bounded by Ba₃Zr₂S₇, Zr₃S₄, ZrS₃ and ZrS₂).

Table 1 The chemical potential boundary points p1–p4 that correspond to Fig. 1

Position	Ba/eV	Zr/eV	S eV^−1
p2	−2.6228	−1.4553	−1.9585
p3	−3.2318	−1.9425	−1.5931
p4	−4.5381	−4.5551	−0.2868
p1	−4.5381	−5.2859	−0.0432

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Researchers additionally investigated the features of BaZrS₃ doped by Ti or Se to lower its band gap to an ideal value of around 1.34 eV,⁷² which allows single-junction solar cells to maximize the utilization of incident sunlight. The superior optoelectronic characteristics of BaZrS₃ provide an alluring incentive for these investigations. Although there are many studies showing the characteristics of Ti or Se doping in regulating the band gap values of BaZrS₃, there has been little discussion of how this doping affects the stability of BaZrS₃. The stability of compounds is a prerequisite and foundation for their practical applications. Here, we carried out a supplementary study. In BaZrS₃, there are two different kinds of S that are not equal, and substituting Ti or Se for S begins with the lower-energy doped compound. As can be seen from Fig. S2(a), Se doping generally reduces the band gap of BaZrS₃ as the Se doping content increases. The band gap of BaZrS₃ can be adjusted starting from 1.732 to 1.623, 1.481, 1.623, 1.483, 1.481, 1.488, 1.558, 1.438, 1.425, 1.298, 1.334 and 1.2370 eV with Se doping contents of 0, 1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 8/12, 9/12, 10/12, 11/12 and 12/12. It should be noted, nonetheless, that Se doping has a very low modulating effect on the band gap value of BaZrS₃. Fig. S2(b) shows the band gap and energy above hull (e_above_hull, eV/per atom) of BaZrS₃ with varying Ti doping. It has been indicated that the modulation efficiency of Ti doping on the band gap value of BaZrS₃ is much higher that of Se,³¹ which may be due to the larger differences in atomic radius between S and Ti than between S and Se. The band gap of BaZrS₃ can be adjusted starting from 1.732 to 1.580, 1.505, 1.316, 1.025 and 0.171 eV with Ti doping contents of 0, 1/16, 2/16, 4/16, 8/16 and 16/16. If the Ti doping content is 4/16, or 25%, the band gap of BaZrS₃ can be tuned to 1.316 eV. Additionally, the band gap of BaZrS₃ can decrease from 0.152 eV to 1.580 eV with only 6.25% Ti doping content, in line with the previously published calculation results.⁷³ Therefore, for the band gap modulation of BaZrS₃, Ti doping can be an effective method, as confirmed by published experimental data.⁷⁴ Next, using the e_above_hull criterion for the doped compounds, the impact of Ti doping on the stability of BaZrS₃ is investigated. The e_above_hull of the compounds changes from −0.0162 to 0.0053, −0.0015, 0.0283, 0.0655 and 0.0534 eV/per atom with Ti doping contents of 0, 1/16, 2/16, 4/16, 8/16 and 16/16. The stability of Ti-doped BaZrS₃ is not as good, because the majority of the e_above_hulls of the doped compounds are positive, even if their e_above_hulls are not very high and all of them are below 7 meV/per atom.^74,75 Therefore, experimental synthesis of these doped compounds with high stability could be challenging, as has been mentioned in published work.^22,74 Nevertheless, the prototypical BaZrS₃ with e_above_hull of −1.62 meV and good characteristics is worth further comprehensive investigation.

3.3 Intrinsic defects in BaZrS₃

In a ternary BaZrS₃ compound, the sulfur chemical potential has a significant impact on the stability and properties of the compound.^34,52,76 Furthermore, the chemical potential of sulfur (or experimental conditions) has a direct correlation with the defect features of BaZrS₃. This study conducts calculations for the stable chemical potential region of BaZrS₃ within the boundary points p1–p4 (calculation in the sequence p2–p3–p4–p1, 300 data calculation points between each boundary position). In order to provide more effective references for the experiment, the calculation results are analyzed in terms of the chemical potentials in the sequence p2–p3–p4–p1 for the chemical potential transition of S from poor to rich, as illustrated in Fig. 2. It is observed that, during this transition, the chemical potential of Ba and Zr generally decreases inside the compound stable region of BaZrS₃. The differential change trend between the chemical potential of the constituent elements S, Ba and Zr could have a significant impact on the defect characteristics of BaZrS₃, which should be noted in the study.

Fig. 2 The calculation results are analyzed in terms of the chemical potentials in the sequence of p2–p3–p4–p1, for the transition of S from poor to rich.

Fig. S3 shows the formation energy of the intrinsic defects as a function of the Fermi energy (E_F, also Fermi level) in the sequence p2–p3–p4–p1 (where defects related to nonequivalent S are denoted by subscripts 1 and 2, respectively). At the starting point of p2, S is poor, but Zr is rich. Zr_i is the most likely defect to form in this situation if it is evaluated solely in terms of chemical trends. The second potential defect is V_S when E_F is on the conduction band minimum (CBM) side. When switching to p3, with S becoming somewhat richer, the defect characteristics of the BaZrS₃ compound did not show any significant alteration. Arriving at p4, S is rich at a chemical potential of −0.2868 eV, but at the same time, the chemical potential of Zr and Ba becomes low. In this case, Zr_i is still the most likely defect when E_F is on the VBM side. However, V_S is non-ignorable, which may be attributed to the volatile nature of S (no matter whether it is rich or not, as shown over the whole stable chemical potential region in Fig. S3). It is worth noting that V_Zr could become dominant when E_F is on the CBM side. Finally, at p1, S becomes richest at a chemical potential of −0.0432 eV with Zr becoming poorer (Ba is kept at the same chemical potential as that of p4). The defect characteristics of the BaZrS₃ compound do not change significantly, except that V_Zr becomes the most popular defect when E_F is at the CBM side. It should be noted that Zr_i and V_S would contribute electron carriers, but V_Zr might serve as a carrier RC in the n-type BaZrS₃ compound, depending on where it is in the band.^31,77 These findings show that the defect features of BaZrS₃ are strongly influenced by the chemical potentials of the composition elements.

3.4 The photoelectric properties of BaZrS₃

The concentration of a defect α with charge state q, under thermodynamic equilibrium, is determined by its formation energy,^63,78 as indicated in eqn (f):

n(α, q) = N_sitesg_qe^{(−ΔE_f/k_BT)} (f)

where N_sites is the concentration of possible defect sites, g_q is the charge-dependent degeneracy factor, and ΔE_f is the defect formation energy, which can be determined from the given Fermi energy and elemental chemical potentials, as can be seen in Fig. S3.

In non-ionic materials, carriers are produced by all the ionized defects with charge state q ≠ 0. The electron carriers are produced by positively charged donor defects with q > 0, and their summed charge is . Conversely, the hole carriers are produced by negatively charged acceptor defects with q < 0, and their summed charge is . Both the thermal excitation and the ionization of all these defects (dopants) contribute to the final concentrations of electron and hole carriers. The equilibrium-state Fermi energy can be calculated by solving for the charge neutrality using eqn (g):

Here, n₀ and p₀ are free carrier concentrations, which can be defined using eqn (h) and (i):where g_C(E) and g_V(E) are the density of states for the conduction and valence bands, respectively, and f(e) is the Fermi–Dirac occupation function. Both g_C(E) and g_V(E) can be calculated from the parabolic band approximation, as mentioned in ref. 78. Semiconductors are usually grown at high temperature and then put through a rapid annealing process to a lower working (measuring) temperature. Therefore, defects are usually formed at high temperature and then the densities of different charge states will be redistributed during the rapid annealing. Our calculation method is in accordance with such a fabrication process.^78–80 First, eqn (g) is solved at a high growth temperature (it is set at 1000 K in this study), and then the Fermi level and the densities of each defect in different charge states q can be obtained at this high temperature. Afterwards, eqn (7) is solved again at the lower measuring temperature (it is set at 300 K in this study), but now the densities of defects in different charge states do not follow eqn (f). In the second step, the density summation for each defect over all charge states is fixed at the value calculated in the first step, and the density of each charge state will undergo a redistribution according to the Fermi–Dirac occupation. Then, a new Fermi level can be obtained at the working (measuring) temperature, and the redistributed defect densities and carrier densities can be calculated.

After the defect formation energy ΔE_f has been obtained, the carrier concentration can be computed using the above-described theory, and the result is displayed in Fig. 3(a). As S gradually changes from poor to rich, for the chemical potential transformation sequence p2–p3–p4–p1, the carrier characteristics also change. Yet BaZrS₃ always remains n-type over the whole stable chemical potential region, which could be due to the donor defects of V_S and Zr_i, as shown in Fig. S3. At p2, the n₀ and p₀ of BaZrS₃ are at 5.422919 × 10¹⁹ and 7.453176 × 10⁻¹⁴ cm⁻³, respectively. At p1, the n₀ of BaZrS₃ decreases to 6.966194 × 10¹⁶ cm⁻³, while p₀ rises comparatively to 3.279799 × 10⁻⁹ cm⁻³, even though it is still much lower than n₀. The increased p₀ may come from the acceptor V_Zr. The properties of the carrier concentration align with the experimental findings.²³ The E_F of BaZrS₃ was also calculated and is displayed in Fig. 3(b), which demonstrates the changes in the semiconductor type more clearly. It is evident that BaZrS₃ is an n-type semiconductor over the whole stable chemical potential region. At p2 and p3, E_F is located above the CBM, indicating degenerated semiconductor formation, which might not be suitable for use in solar cells. At p4 and p1, E_F is below the CBM but close to it, which is beneficial for the generation of electron carriers. The n-type chalcogenide perovskite BaZrS₃ with a high intrinsic carrier concentration could have great advantages for application in semiconductors and electronics.

Fig. 3 The calculated carrier concentrations (a) and Fermi energy level (b) of BaZrS₃ in the sequence p2–p3–p4–p1.

The transition levels of the intrinsic defects in BaZrS₃ were also extracted from Fig. 3S (where the defect charge state determines the slope of the line, and the turning points represent the transition energy levels between different charge states for a given defect), as shown in Fig. 4. Compared to earlier references, the results based on a large supercell of 240 atoms with an HSE06 hybrid functional and finite-size effect correction directly display a considerably clearer band gap.^22,29 According to the published work, even when deep-level defects like V_Zr and S_Zr are present, their formation of S clusters (dimer, trimer and tetramer) tends to remove mid-gap states in the neutral charge state and leaves behind a relatively clean band gap.²⁹ Therefore, some deep-level defects mentioned in the published work might not actually exist in BaZrS₃ in its thermodynamically stable region. As displayed in Fig. 4, from the standpoint of carrier features and the degree of impact of the defects, the decisive defects are expected to be V_S, Zr_i, V_Zr, S_i and S_Ba in BaZrS₃. As can be seen in the figure, V_S and Zr_i are donor defects, and all of the defects are located on the CBM side. Defect S_i (+1/0) does not exist. Meanwhile, S_i (2+/0) is a donor defect and is located around the middle of the band at 0.77016 eV above the VBM, which is considered a deep-level defect. S_Ba (1+/0), located at 0.23845 eV above the VBM, is also a donor defect and might behave as a deep-level defect in BaZrS₃. From Fig. 3(a) and 4, the donor defects V_S and Zr_i could be the main contributors to n-type BaZrS₃, and the acceptor defect V_Zr could be the main contributor to the p₀ of BaZrS₃, even if it is quite low. Deep-level defects may act as carrier RCs, and they may have a significant impact on the performance of the fabricated device. The concentrations of deep-level defects in BaZrS₃ are illustrated in Fig. 5. The concentrations of defects S_i⁰ and S_Ba⁰ are high, whereas those of S_i²⁺ and S_Ba¹⁺ are low, which is consistent with the results shown in Fig. S3 when E_F is at the CBM side. The concentrations of defects S_i⁰ and S_Ba⁰ at p1 are 2.09804 × 10¹⁷ and 6.16878 × 10¹¹ cm⁻³, respectively. Given the extremely high concentration of deep-level defect S_i⁰, knowledge of the carrier recombination properties of the defect is essential.

Fig. 4 The transition levels of intrinsic defects (deep-level defects are also indicated by the red arrows) in BaZrS₃.

Fig. 5 The concentrations of deep-level defects inside the band gap of BaZrS₃ in the sequence p2–p3–p4–p1.

The non-radiative recombination characteristic of the deep-level defect S_i with high concentration was then investigated. The carrier capture coefficients for each process were calculated in order to accurately evaluate the effect of carrier recombination characteristics of n-type BaZrS₃. However, it is vital to understand the recombination cycle in this instance prior to the quantitative evaluation. In n-type BaZrS₃, the Fermi level approaches the CBM, causing a high concentration of S_i⁰ but a relatively low concentration of S_i²⁺, which can be analyzed from the formation energy diagram in Fig. S3 and the concentrations of deep-level defects inside the band gap of BaZrS₃ in Fig. 5. As a result, the recombination process normally starts from S_i⁰, first capturing two holes (it should be an excess carrier under steady-state illumination) from the VBM, becoming 2+ charged. S_i²⁺ would then absorb two electrons from the CBM to return to neutral. One rate-limiting step is the minority (hole) carrier capture by S_i⁰. Thus, the overall recombination rate will be determined by hole capture between the neutral (q0) and 2+ charge (q2) states. Therefore, for the BaZrS₃ system, the donor defect S_i (2+/0) with capture of minority carrier holes from the neutral (q0) to the 2+ charge (q2) state was studied extensively. The results are shown in Fig. 6.⁶⁶ We first calculated the configuration coordinate (CC) diagram for the 0/2+ transition of S_i⁰, as shown in Fig. 6(a), where the red curve is the ground-state potential energy surface (PES) of S_i⁰, and the blue curve is its excited-state PES. The CC diagram in Fig. 6(a) demonstrates the configuration-dependent features of the non-radiative recombination dynamics of defect S_i. According to the calculation, the ΔQ between two structures is 10.3942 amu^1/2 Å. The barrier for the non-radiative process shifted upward according to the neutral (q0) and 2+ charge (q2) state transition level is 1.04037 eV, which is reasonable for the transition level value of 0.77 eV (as mentioned above for Fig. 4) of S_i (2+/0). Using a configuration coordinate diagram, according to the theory and method described in ref. 66 and 67, the W_if and other parameters in eqn (c) can be calculated and the temperature-dependent capture coefficient for minority (hole) carrier could be obtained, as shown in Fig. 6(b). It can be observed from the results that the carrier capture coefficient for minority (hole) carrier at 300.0 K is 1.28682 × 10⁻²⁵ cm³ s⁻¹, which is extremely small. It can be predicted that the electron capture coefficient should likewise be minimal. The calculated result shows that the carrier capture coefficient for the majority (electron) carrier at 300.0 K is 4.37809 × 10⁻²³ cm³ s⁻¹, as the concentration of electron carriers at p1 is very high at 6.966194 × 10¹⁶ cm⁻³, much higher than that of holes with a concentration of 3.279799 × 10⁻⁹ cm⁻³. Therefore, for the deep-level defect of S_i, the carrier recombination characteristics are still decided by the minority carrier hole, which shows an extremely low non-radiative carrier capture coefficient. The carrier capture coefficient of another deep-level defect, S_Ba, at 300.0 K was also calculated, and it is also very low at 1.52933 × 10⁻²⁰ cm³ s⁻¹. Therefore, the chalcogenide perovskite BaZrS₃ could in fact be defect tolerant,^{43,48,49,81–83} which is of crucial importance for its further development and application in photoelectric devices.

Fig. 6 (a) The configuration coordinate (CC) diagram for non-radiative calculation (b) and corresponding temperature-dependent hole capture coefficient of S_i from q0 to q2 in BaZrS₃.

Fig. 7 displays the results of an investigation into the radiative recombination characteristic of the deep-level donor defect of S_i (2+/0). The radiative recombination process shows clear configuration–lattice coupling features, as can be seen from the CC diagram in Fig. 7(a). The ΔQ between two structures is also the same at 10.3942 amu^1/2 Å. According to the calculations, the lattice relaxation energy is 1.5691 eV. The emission energy is 0.7991 eV. Therefore, the result is consistent with the transition level value 0.77 eV of S_i (2+/0). The PL spectrum in Fig. 7(b) indicates that the position of the peak in the lineshape is 0.98 eV. Compared to the transition level value 0.77 eV of S_i (2+/0), it is slightly larger but reasonable for radiative recombination of this defect. The full width at half maxima of the lineshape is 0.16 eV. The volume of this supercell was assessed at around 6.08 × 10⁻²¹ cm⁻³. Therefore, the radiative carrier capture coefficient was calculated to be 3.51 × 10⁻¹⁵ cm³ s⁻¹, which is also quite good.⁸²

Fig. 7 (a) The configuration coordinate (CC) diagram for radiative calculation (b) and corresponding PL lineshape spectrum of S_i from q0 to q2 in BaZrS₃.

Using a solar cell capacitor simulator from the University of Gent in Belgium, the photoelectric conversion properties of BaZrS₃ were computed by employing Poisson's equation and continuity equations for electron and hole carriers.⁸⁴. [C₆H₄N(C₆H₂(CH₃)₃)C₆H₄]_n (PTAA) shows good transparency for visible light, mechanical flexibility, conductivity and stability. Additionally, it has received certification for highly effective perovskite solar cells. In this study, the effects of the carrier and defect characteristics on the performance of the strong n-type BaZrS₃ were simulated, based on a BaZrS₃-PTAA single-junction solar cell without an electron transport layer (ETL), which is of great significance to the simple and controllable preparation of BaZrS₃ solar cells. First, the p1 point with rich S was studied. The thin-film thickness of PTAA was set at 10 nm, and other essential input parameters were obtained from the above calculations and published ref. 85 and 86. To examine the effect of the thickness of BaZrS₃, the thin-film thickness of BaZrS₃ was varied from 100 to 1000 nm in steps of 100 nm, while the other parameters were fixed. It was found that BaZrS₃, with its previously mentioned superior light absorption, might absorb most of the incident sunlight (standard AM1.5) with a thickness of less than 100 nm. Thus, in order to find the optimal value, the BaZrS₃ thin-film thickness was further varied from 10 to 100 nm in steps of 10 nm. This revealed that, BaZrS₃ with a thickness of only 80 nm, can absorb the majority of incident sunlight with a conversion efficiency of 22.04%, as shown in Fig. S4(a).

The photoelectric performances of BaZrS₃ at stable chemical potential points p2, p3 and p4 were also studied (solar cell parameters were all set the same, but without those related to chemical potential), and the results are displayed in Fig. S4(b) and Table S1, together with that of p1. As can be seen from the results, p2 and p3 with very high n₀ (see Fig. 3), which lead E_F into the CBM, turn out to show low conversion efficiency, which could be due to the degenerated semiconductor characteristics. Arriving at p4 with n₀ falling and E_F sliding to below the CBM, the conversion efficiency of BaZrS₃ quickly rises to 21.96%. At p1, the conversion efficiency of BaZrS₃ becomes 22.04%. The high conversion efficiency of BaZrS₃ could be attributed to the high n₀ and excellent defect tolerance, which indicates that BaZrS₃ is an excellent photovoltaic material. Moreover, the conversion efficiency of BaZrS₃ could readily approach 25.46%, as shown by p1⁺ in Fig. S4(b) and Table S1, when the ΔVBM between BaZrS₃ and PTAA was decreased to 0.1 eV by modifying the VBM value of PTAA, as this can further improve the V_oc of the BaZrS₃ solar cell from 1.23 to 1.45 eV. All these results demonstrate the superior photovoltaic capabilities of BaZrS₃.

4 Conclusion

The defect and carrier characteristics of BaZrS₃ were studied by first-principles calculations using a 240-atom supercell with HSE06 hybrid functional and finite-size effect correction. The stable chemical potential range was determined, where BaZrS₃ is thermodynamically stable against the formation of its three constituent elements and four competitive secondary phase compounds (i.e., Ba₃Zr₂S₇, Zr₃S₄, ZrS₃ and ZrS₂). This indicated that BaZrS₃ with e_above_hull of −1.62 meV has good thermodynamic stability. The chemical potential boundary points were indicated as p1 (µ_Ba −4.5381, µ_Zr −5.2859 and µ_S −0.0432 eV), p2 (µ_Ba −2.6228, µ_Zr −1.4553 and µ_S −1.9585 eV), p3 (µ_Ba −3.2318, µ_Zr −1.9425 and µ_S −1.5931 eV) and p4 (µ_Ba −4.5381, µ_Zr −4.5551 and µ_S 0.2868 eV). The formation energy of intrinsic defects in BaZrS₃ was studied according to the chemical potential transition of S from poor to rich in the sequence p2–p3–p4–p1. As the chemical potential of S increases in the sequence p2–p3–p4–p1, the carrier characteristics in BaZrS₃ also change. Yet BaZrS₃ always remains n-type over the whole stable chemical potential region, which could be due to the donor defects of V_S and Zr_i. At p2, the n₀ and p₀ of BaZrS₃ are 5.422919 × 10¹⁹ and 7.453176 × 10⁻¹⁴ cm⁻³, respectively. At p1, the n₀ of BaZrS₃ decreases to 6.966194 × 10¹⁶ cm⁻³, while p₀ rises comparatively to 3.279799 × 10⁹ cm⁻³, even though it is still much lower than n₀. The increased p0 may come from the acceptor V_Zr. The properties of the carrier concentration align with the experimental findings. The transition levels of the intrinsic defects in BaZrS₃ show a much cleaner band gap than that in previously reported references. S_i (2+/0) and S_Ba (1+/0) are shown to be deep-level donors, located around the middle of the band at 0.77016 eV and 0.23845 eV above the VBM, respectively. The donor defects V_S and Zr_i could be the main contributors to n-type BaZrS₃, while the deep-level defects may behave as carrier recombination centers in BaZrS₃. The concentrations of defects S_i⁰ and S_Ba⁰ at p1 are 2.09804 × 10¹⁷ and 6.16878 × 10¹¹ cm⁻³, respectively. The concentration of deep-level defect S_i⁰ is very high, and the carrier recombination characteristics of the defect were thus further studied. However, the results show that its non-radiative capture coefficient for the minority (hole) carrier at 300.0 K is extremely small, at 1.28682 × 10⁻²⁵ cm³ s⁻¹. Its radiative capture coefficient is also likewise quite good at 3.51 × 10⁻¹⁵ cm³ s⁻¹, indicating its defect tolerance characteristics.

Based on the carrier and defect characteristics, the photoelectric properties of BaZrS₃ were further examined with PTAA as its partner hole transport layer, excluding the electron transport layer. This indicated that p2 and p3, with very high n₀ that leads E_F into the CBM, turn out to show low conversion efficiency, which could be due to the degenerate semiconductor characteristics. At p1, the results show that BaZrS₃ with just 80 nm can absorb most of the incident sunlight and shows a conversion efficiency of 22.04%, which is sufficient for commercial use. The high conversion efficiency of BaZrS₃ could be attributed to the high n₀ and excellent defect tolerance. Moreover, the conversion efficiency of BaZrS₃ could approach 25.46% when the ΔVBM between BaZrS₃ and PTAA is lowered to 0.1 eV, demonstrating the exceptional photovoltaic capabilities of BaZrS₃.

This work has elucidated the influence of defects and carriers on the photoelectric properties of BaZrS₃, which can help clarify the true upper limit of conversion efficiency and aid in the experimental, controllable preparation of high-quality BaZrS₃ films and devices. This work might also open a path to a direct connection between practical synthesis protocols and crystal defect features, which is of crucial importance for the further development and application of relevant photoelectric devices.

Author contributions

Qinmiao Chen: funding acquisition, conceptualization, investigation, writing – original draft. Yi Ni: investigation, formal analysis. Yufei Wang: methodology, investigation, writing – review & editing.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The data supporting this article have been included as part of the manuscript and supplementary information (SI). Other datasets that may support the current study are available from the corresponding author on reasonable request. Supplementary information: optical characteristics of BaZrS₃; influence of element doping on the band and thermodynamically stability of BaZrS₃; formation energy of intrinsic defects in BaZrS₃ as a function of the Fermi energy; photoelectric conversion characteristics of the modeled BaZrS₃ solar cell. See DOI: https://doi.org/10.1039/d5ra07387a.

Acknowledgements

The project is supported by the National Natural Science Foundation of China (No. 11604097).

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