Introduction

Chalcogenide perovskites have gained increased attention from the scientific community due to their promising photovoltaic and thermoelectric properties. These perovskites form in the ABX 3 configuration with A and B being elements with combined valence of 6 (generally containing Group-IIA elements such as Ba, Ca and Sr and Group-IVB elements such as Hf, Zr, and Ti) and X typically being S or Se. As these constituent elements are considered less toxic and gentle on the environment relative to, for instance, Pb in organic-inorganic halide perovskites, some of these compounds have been synthesized experimentally, [1][2][3][4][5][6] mostly in the needle-like phase (NH 4 CdCl 3 ), 1,7 the hexagonal phase (BaNiO 3 ), 2,8 or as the orthorhombic distorted perovskite (GdFeO 3 ). 4 Among the recently examined chalcogenide perovskites, BaZrS 3 has demonstrated strong potential for photovoltaics since the calculated band gap of 1.76-1.82 eV 9,10 is in the neighborhood of B1.35 eV that is also considered optimum for energy-conversion applications. 11 While reports exist on BaZrS 3 for alloying and defect control, 12 stability and band gap tuning, 13 the effect of substituting S for O in BaZrO 3 and the subsequent band gap 14 and phonon transport properties, 15 efforts towards understanding the fundamental electron mediated transport mechanisms, and quantifying the electronic properties, are absent. Such knowledge is key to evaluate the applicability of BaZrS 3 for thermoelectric and optoelectronic devices. Here, we employ density functional theory (DFT) calculations together with Boltzmann transport equation to examine the electrical and electron thermal conductivities, in addition to the optical absorption spectrum of BaZrS 3 in its orthorhombic distorted perovskite structure. Our predictions, as elaborated below, reveal that strong anisotropic effects impact the conductivities along specific lattice directions under doping, and the material possesses high absorption coefficients (410 5 cm À1 ) with an early onset of photon absorption significantly prior to the fundamental band gap.

Results and discussion

The lattice constants of the BaZrS 3 structure are well reproduced from our first principles calculations, and in excellent agreement with experimental measurements. 4 The calculated lattice parameters of the distorted perovskite structure are listed in Table 1. Note that, in addition, a precise representation of the electronic band structure is required from the computations to accurately estimate the electronic properties. Our calculations predict a direct band gap of 1.79 eV at the G point (0.0, 0.0, 0.0). This result is in excellent agreement with experimentally measured values of 1.73-1.85 eV. 2,12,13,16 Additionally, our calculated band gap using the hybrid HSE06 functional concurs with other theoretical calculations that employ the same functional with estimates of 1.75, 9 1.76, 12 1.7, 16 1.7, 14 and 1.82 eV. 10 Calculations using other functionals tend to underestimate the band gap of BaZrS 3 with the PBE functional rendering a value of 1.02 eV, while the GGA+U functional with a U parameter of 4.5 eV yields a band gap of 1.72 eV. 12 These observations attest to the validity of the hybrid HSE06 functional used to predict the electronic band gap necessary for an accurate description of the charge transport mechanisms.

The electronic band structure and the density of states (DOS) are illustrated in Fig. 1. The band structure shows that the valence band maximum (VBM) and the conduction band minimum (CBM) are composed of S-3p states and Zr-4d states, respectively, at the G point.

A deeper scrutiny of the VBM and CBM suggests the bands as dispersive, as also noted from the DOS. This behavior is attributed to the more disperse and less localized Zr-4d states and the S-3p states. The conduction band reveals antibonding energy states between Zr-4d and S-3p orbitals, while the valence band suggests bonding states between Zr-4d and S-3p states. The dispersive nature of the band edges implies low carrier effective masses that contribute to efficient carrier transport. We calculate the electron (hole) effective masses along the G-X, G-Y and G-Z directions as 0.347 (À0.556), 0.218 (À0.253) and 0.422 (À0.747) fractions of m o (electron rest mass), respectively. Irrespective of whether electrons or holes act as the predominant carriers, enhanced transport properties are expected along the a and b lattice directions. We also expect notable carrier transport properties for BaZrS 3 given the calculated carrier effective mass is less than 1.0 m o . The optical absorption spectra presented in Fig. 2 provides insights into the ultraviolet-visible (UV-Vis) optical absorption of BaZrS 3 as well as the imaginary part of the dielectric constant, e 2 , that determines the absorption properties of the chalcogenide perovskite.

The optical absorption of BaZrS 3 initiates around photon energies of B0.5 eV, indicating that the perovskite has an early onset for optical absorption even before the fundamental band gap is attained. This early onset of optical absorption is dependent on the material, temperature and the range of wavelengths considered. This phenomenon is typically due to free-carrier absorption by either free electrons or free holes, inter-valence-band absorption by holes or inter-conductionband absorption by electrons. One or more of these mechanisms dominate depending on the relative concentration of electrons or holes in the material. For intrinsic materials, both the electron and hole contributions play an important role. 17 The distorted BaZrS 3 structure achieves its highest UV-Vis absorption at B4.8 eV of photon energy. As photons with energy less than 3.4 eV contribute to the majority of solar energy reaching the earth's surface, 9 BaZrS 3 holds promise as a photovoltaic material, and attains a highest optical absorption of B3 Â 10 5 cm À1 at B3 eV. The optical absorption properties are typically attributed to the d orbitals located at the band edges. 9 This large absorption coefficient suggests the photovoltaic efficiency of BaZrS 3 is not solely influenced by carrier mobility. Also, for the maximum solar spectral irradiance that occurs at an incident photon energy of B2.5 eV, 9 BaZrS 3 is estimated to possess optical absorption of 2 Â 10 5 cm À1 , which makes it a potential material for solar thin film applications. BaZrS 3 obtains its highest e 2 = 7.9 at photon energy of B4.3 eV, although e 2 = 4.5 acquired at 2.7 eV is higher than that estimated for advanced solar materials, such as CdTe and GaAs, at the same photon energy. 9 Solution to the Boltzmann transport equations through the BOLTZTRAP code 18 is used to predict the Seebeck coefficient (S), the electrical conductivity (s) and the electron thermal conductivity (k e ) along the a, b, and c lattice directions of the distorted perovskite for different chemical potentials (m) and temperatures (T). We employ the constant relaxation-time (t) approximation since it is difficult to estimate t given the occurrences of different scattering mechanisms (at the boundaries, between carriers, etc.) in the material. We calculate t = 10 À14 s, similar to previous reports. 19,20 Moreover, t = 10 À14 s was recently calculated for the chalcogenide perovskite CaZrSe 3 . 21 The upper limit of the thermoelectric figure of merit (ZT e ), however, does not depend on t.

The variation of the power factor (PF = S 2 s) and ZT e over a range of carrier concentrations from 10 15 -10 22 cm À3 across different temperatures are presented in Fig. S1 and S2 (ESI †), respectively. At lower temperatures (o600 K), higher PF values are noted for p-type doping, while n-type doping yields higher values at temperatures 4600 K. The highest PF values are achieved at 10 21 cm À3 for both p-and n-type doping at all , where e is the carrier charge, DOS(E) is the energy-dependent electronic density of states, n(E) is the energy dependent number of states, k B is the Boltzmann constant, and j is the energy dependent carrier mobility. S is inversely proportional to j and hence a high effective mass (i.e., low carrier mobility) yields higher PFs. 22 The average effective mass for holes is higher than that for electrons, resulting in an enhanced PF when placing higher emphasis on the contribution of S to PF. As the doping concentration increases, s increases and S decreases; thus, there exists an optimum PF at relatively high carrier concentrations in the proximity of the conduction band (here, that carrier concentration being 10 21 cm À3 ). 23 From Fig. S2 (ESI †), we find that higher ZT e is achieved for lower (10 15 -10 18 cm À3 ) rather than higher carrier concentrations. ZT e is rooted to the entropy production in a thermoelectric energy conversion process, and higher carrier concentration contributes to an increased entropy production that drastically reduces the ZT e . On the other hand, at higher temperatures, k e increases due to the relatively higher number of excited electrons, thereby decreasing ZT e . The calculated ZT e values are similar for p-and n-type doping (B1.0) between 100-700 K. The highest ZT e E 1.0 occurs at 300-700 K at carrier concentrations of 10 15 -10 18 cm À3 . ZT e decreases with increasing carrier concentration across all temperatures. From Fig. S3 (ESI †), the variation of carrier concentration from 1 Â 10 17 to 9 Â 10 17 cm À3 suggests that the highest ZT e for 100-700 K is achieved at 10 17 cm À3 . Combining the distributions for PF and ZT e against chemical potential and temperature, we hence predict the electronic transport properties at a carrier concentration of 10 17 cm À3 , as also suggested in the literature for optimal photovoltaic performance of BaZrS 3 . 12 The electron-phonon scattering rate as a function of energy across a temperature range from 300-800 K is obtained as the reciprocal of the relaxation time (1/t nk ), and presented in Fig. 3(a). The illustration depicts the shape of the DOS that demonstrates the available phase space for carrier scattering. This phase space provides the available volume and area within which three-phonon scattering processes such as absorption and annihilation occur as well as the electron-phonon coupling which results in scattering of the charge carriers. The density of states at the different energy levels shown in Fig. 3(a) provides an estimate of the available phase space at such energies where the electron-phonon coupling processes occurs. For instance, at 800 K, within an energy window of À1 to À5 eV, the available phase space is larger than the energy window of À2 to À3 eV. The resulting lower carrier relaxation times at higher temperatures is evidenced by the increase in scattering rate with temperature. In Fig. 3(b), the relaxation time for holes (electrons) is extracted from the valence band maximum (conduction band minimum) which suggests that electrons assume higher relaxation times relative to holes. Increased scattering at higher temperatures leads to decrease in the relaxation times. Fig. 3(b) yields a relaxation time in the order of 10 À14 s at different temperatures for BaZrS 3 .

The temperature and directional dependence for S, s, k e and PF in BaZrS 3 for p-type doping are presented in Fig. 4 (the corresponding variations for n-type doping are in Fig. S4 as ESI †). S increases steeply from 50 to 300 K, gradually from 300 to 850 K and decreases at T 4 850 K. This phenomenon is explained by the fact that S is a metric for the average entropy transported by a charge carrier per unit charge in the limit of thermal equilibrium; hence S decreases (or increases) with reducing (or rising) temperatures. 24 At 300 K, S E 820 mV K À1 for p-type doping, while S E 620 mV K À1 for n-type doping. The calculated S is higher for BaZrS 3 (p-type) relative to CaZrS 3 , but comparably lower for n-type doping, 21 because S is inversely proportional to the carrier mobility and higher carrier mobility are calculated for p-type doping in CaZrSe 3 compared to BaZrS 3 while the opposite is observed for n-type doping. Minor anisotropic effects are observed in S between 200-850 K, while no such effects are noted in case of n-type doping. This directional dependence can be utilized when designing a BaZrS 3 based thermoelectric device to yield optimum performance along specific crystallographic directions.

s decreases steeply at T o 300 K while a more gradual reduction is noted thereafter (Fig. 4(b)), in contrast to the trend observed for S. The decrease in s with increasing temperature occurs due to presence of more excited electrons and enhanced electron scattering. At 300 K, the average s E 0.20 O À1 cm À1 is lower than the mean value (0.68 O À1 cm À1 ) predicted for CaZrSe 3 , 21 and significantly lower than that of competing electronic materials, such as SnSe, GeS, GeSe and SnS. 20 However, s assumes a higher magnitude of 0.62 O À1 cm À1 under n-type doping because of the low carrier effective mass for electrons relative to that for the holes. s also depends on the choice of the carrier relaxation time t since a lower t will result in a higher s. The highest s is recorded along the a and b lattice directions for both p-type and n-type doping and for T 4 150 K. This finding is attributed to the lower carrier effective mass (high carrier mobility) along the a and b lattice directions (À0.556 m o and À0.253 m o , respectively) for p-type doping, and likewise (0.347 m o and 0.218 m o , respectively) for n-type doping. However, we note that there is a slight deviation in s at T o 150 K for p-type doping, possibly due to computational uncertainties. A similar observation is made for the phonon transport properties of BaZrS 3 along the different lattice directions. Though the magnitude of the anisotropy in the lattice thermal conductivity is minute, enhanced phonon transport properties are observed along the b-and a-lattice directions while lower lattice thermal conductivity is predicted along the c-direction. 15 Additionally, the BaZrS 3 lattice parameters along the a-and b-directions are higher than the c-direction, which leads to longer phonon mean free path along these directions before annihilation. As similar observations are made for the electron (hole) and phonon transport properties along the different lattice directions, we expect strong electron-phonon coupling along the a-and b-lattice directions and subsequently higher carrier scattering along these directions. When designing efficient photovoltaic and thermoelectric BaZrS 3 based devices, the different transport properties along the various lattice directions could be modified to achieve the desired efficiencies. For instance, BaZrS 3 samples prepared with the a-facets could be utilized in improving the optical absorption properties and the thermoelectric efficiencies as this lattice direction exhibits marginally lower lattice thermal conductivity, relatively lower electronic thermal conductivity, higher Seebeck coefficient but only slightly lower in electrical conductivity as compared to the b-facets of the crystals.

k e reveals directional dependence at T 4 150 K and exhibits low magnitudes relative to CaZrSe 3 and other similar compounds. 20 The directional dependence is akin to that observed for s. An ultra-low mean k e E 0.004 W m À1 K À1 is predicted at 300 K and is attributed to the low carrier mobility compared to those through SnSe, GeS, GeSe and SnS lattices. From Fig. 4(c) we note that k e increases rather linearly with temperature as more carriers are excited at higher T. A similar trend is observed under n-type doping (Fig. S4(c), ESI †), while the highest k e are obtained along the a and b lattice directions for both p-type and n-type doping, analogous to s.

The variation of PF with T presented in Fig. 4(d), shows that PF increases from 50 to 350 K and decreases at T 4 350 K. Nevertheless, PF increases from 50 to 1000 K for n-type doped BaZrS 3 , while the highest PF = 0.135 mW cm À1 K À2 is achieved at 350 K for p-type doping with room temperature PF E 0.133 mW cm À1 K À2 . These predictions are lower than that of other chalcogenide perovskites 21 due to the relatively low s of BaZrS 3 .

The variations of S, s, k e and PF with chemical potential m at different temperatures are reproduced in Fig. 5. m essentially mimics the intensity with which BaZrS 3 is doped, and also determines the center of the Fermi-Dirac distribution function that governs the probability of occupation of a quantum state with energy E. We predict a high S E 3000 mV K À1 at 300 K for m E 0.1 eV, which is higher than that for CaZrSe 3 (2300 mV K À1 ) 21 and significantly greater than that of other competing electronic materials 20 (highest S E 2000 mV K À1 for GeS).

As T increases from 300 to 600 K, S reduces to B1500 mV K À1 due to the increased entropy. Also, S being inversely proportional to T, as noted in the correlation between S and Peltier The evolution of the thermoelectricity relevant electronic transport properties, viz., S, s, and k e as a function of m at 300 K is illustrated in Fig. 6 to understand the underlying anisotropy in the different lattice directions for both p-and n-type doping. S exhibits no anisotropic effects, while s and k e reveal strong directional dependence for À1 o m 4 1. The highest s and k e are achieved along the a and b lattice directions for both p-and n-type doping. We corroborate that these variations are due to the low carrier effective mass along the corresponding lattice directions under doping.

The distribution of ZT e with m across different temperatures for both p-and n-type doping is presented in Fig. 7. While no significant differences in ZT e are noted, with respect to T at m E AE1 eV, we observe a high ZT e E 1 at m E AE1 eV. The highest ZT e = 1.08 is achieved at m E À4.0 eV as well as at higher m values for p-type doping. The high ZT e arises primarily due to a combination of high S and low k e of BaZrS 3 . This result identifies a potential range for m at which BaZrS 3 can be designed to achieve an optimum thermoelectric efficiency.

Conclusion

We examine the electronic transport and optical properties of the distorted orthorhombic perovskite BaZrS 3 . The material has an optical band gap of 1.79 eV that is in the neighborhood of the optimum band gap of 1.35 eV for single junction photovoltaic applications. At photon energies of 3 eV, we calculate a high optical absorption coefficient of 3 Â 10 5 cm À1 . Also, there is an early onset to optical absorption at 0.5 eV that is significantly below the optical band gap. The power factor PF and upper limit of the thermoelectric figure of merit ZT e are predicted at several carrier concentrations, while a carrier concentration of 10 17 cm À3 is identified to yield optimum properties, where we calculate the Seebeck coefficient S, electrical conductivity s, and electron thermal conductivity k e .

The highest S is observed for p-type doping relative to n-type doping while s produces high values for n-type doping compared to p-type doping. The electronic transport properties display strong directional dependence across all temperatures for both p-and n-type doping. The variation of the electronic properties with chemical potential m for both p-and n-type doping reveals that BaZrS 3 exhibits high S E 3000 mV K À1 when doped within a narrow chemical potential range of 0 AE 0.3 eV. The highest s are observed when BaZrS 3 is doped at m E À2.8 eV and 1.8 eV while the highest PF values are achieved at chemical potentials m E À1 and À3.5 eV for p-type doping and m E 1.2 eV for n-type doping. Thus, the highest PF is observed for n-type doping due to the corresponding higher electrical conductivity (B2Â that of p-type doping). Strong anisotropic dependencies are observed along the a and b lattice directions for s and k e for p-type and n-type doping. The highest ZT e E 1.0 and 1.08 achieved at m E AE1 and À4 eV, respectively, together with lattice conductivities, 15 suggests that BaZrS 3 holds potential for thermoelectric applications, as has been also suggested previously, 42 in addition to its capability for high photovoltaic performance.

Methods

Mermin's finite temperature DFT 26 implemented in the Vienna ab initio Simulation Package (VASP) 27 is used to perform all the DFT calculations. Ba, Zr and S atoms electronic configurations are represented by [Xe]6s 2 , [Kr]4d 2 5s 2 and [Ne]3s 2 3p 4 respectively. The core part of the valence electron wavefunctions, which are constrained, as well as the valence electron wavefunctions are represented by the Projector Augmented Wavefunction (PAW) pseudopotentials. 28 The number of planewaves that are needed to effectively describe the electrons close to the nuclei are significantly reduced with the use of the PAW pseudopotentials. The unit cell and atomic positions are relaxed using the PBEsol 29 functional for solids while the Methfessel-Paxton 30 smearing scheme is utilized with the gamma parameter set to 0.1 eV. An energy cut-off of 600 eV is used for the planewave expansion. A Monkhorst-Pack 31 special grid sampling of the k-points for integration of the Brillouin zone yields 7 Â 5 Â 7 k-points representing 123 irreducible number of sampling points for all bulk calculations. Resolution of the Kohn-Sham equations is achieved by using the self-consistent field procedure and setting energy changes for each cycle at 10 À4 eV as the convergence criterion between two successive iterations. The distorted orthorhombic structure of BaZrS 3 is illustrated in Fig. 8. For the band gap and the optical absorption spectrum calculations, the hybrid HSE06 functional 32 is used. The mixing parameter a that controls the amount of the Hartree-Fock exchange energy is set to 0.25, while the long-range part of the exchange (o) is set to 0.2 Å À1 .

The transport properties are determined using the linearized Boltzmann transport equation in the relaxation time approximation as implemented in the BoltzTraP 18 code. The code uses a Fourier expansion of the electronic energies calculated from VASP for the optimized structures. A denser k-mesh of 1 20 000 points is used to ensure accurate transport properties are predicted. The thermoelectric figure of merit

T, where S is the Seebeck coefficient, s is the electrical conductivity, T is temperature, k e and k L are the electronic thermal and lattice thermal conductivities, respectively. S 2 s k e T ¼ ZT e denotes the upper limit of ZT. Although S is independent of the relaxation time t, s and k e are dependent on the choice of t. Here, we use t = 10 À14 s, which yields predictions that concur with experimental measurements. 19,20 The Sumo Python toolkit is used to characterize the electronic band structure and the UV-Vis absorption spectrum. 33 The carrier relaxation time is calculated using the electron-phonon (optical and acoustic) interaction as implemented in the Electron-Phonon Wannier (EPW) code. 34,35 All self-consistent (SCF), non self-consistent (non-SCF) and phonon calculations needed for the electron-phonon coupling matrix are performed with the Quantum-Espresso (QE) [36][37][38] package. Initially, we calculate the Kohn-Sham wavefunctions and Eigenvalues through the Brillouin zone of the crystal using an SCF, followed by a non-SCF which provides the DFT band structure with an under-estimated band gap as compared to the HSE06 band gap. This process was followed by a full phonon spectrum calculation using a 3 Â 3 Â 3 coarse q-grid, yielding 8 irreducible q-points. We scrutinize the phonon spectrum and find no imaginary frequency pockets as we have reported previously. 15 This data is used in constructing the dynamical matrix required for computing the electron-phonon coupling matrix. In the first part of the EPW calculation, the coarse 3 Â 3 Â 3 phonon q-grid is interpolated onto a fine 120 Â 120 Â 120 grid in combination with the k-point path employed to determine the electronic band structure. For obtaining the electron-phonon coupling constant, we use the Maximally Localized Wannier Functions (MLWF) [39][40][41] to interpolate the electronic band structure. The interpolation is performed by projecting the Wannier functions on the Zr-d and S-p orbitals since these contribute the most to the conduction and valence bands around the Fermi level as can be noted from Fig. 1. Additionally, in the Wannier interpolation scheme, we select a disentanglement window between 5.5 to À2.5 eV while the frozen window is selected to be between 5.0 to À1.5 eV. The electron self-energy S nk for band n and state k is derived from the interpolated dense electron-phonon coupling matrix elements, while the relaxation time t is calculated from the scattering rate using t nk ð Þ À1 ¼ 2 Im S nk ð Þ ½ = h, where h is the reduced Planck constant. The imaginary part of the electron self-energy is used in estimating the scattering rate at different temperatures. Thus, the relaxation time was calculated using the self-energy relaxation time approximation as implemented in the EPW code as scattering_serta.