A2AgCrBr6 (A = K, Rb, Cs) and Cs2AgCrX6(X = Cl, I) Double Perovskites: A Transition-Metal-Based Semiconducting Material Series with Remarkable Optics

With an interest to quest for transition metal-based halogenated double perovskites AB′B″X6 as high performance semiconducting materials for optoelectronics, this study theoretically examined the electronic structures, stability, electronic (density of states and band structures), transport (effective masses of charge carriers), and optical properties (dielectric function and absorption coefficients, etc.) of the series A2AgCrBr6 (A = K, Rb, Cs) using SCAN + rVV10. Our results showed that A2AgCrBr6 (A = Rb, Cs), but not K2AgCrBr6, has a stable perovskite structure, which was revealed using various traditionally recommended geometry-based indices. Despite this reservation, all the three systems were shown to have similar band structures, density of states, and carrier effective masses of conducting holes and electrons, as well as the nature of the real and imaginary parts of their dielectric function, absorption coefficient, refractive index, and photoconductivity spectra. The small changes observed in any specific property of the series A2AgCrBr6 were due to the changes in the lattice properties driven by alkali substitution at the A site. A comparison with the corresponding properties of Cs2AgCrX6 (X = Cl, I) suggested that halogen substitution at the X-site can not only significantly shift the position of the onset of optical absorption found of the dielectric function, absorption coefficient and refractive spectra of Cs2AgCrCl6 and Cs2AgCrI6 toward the high- and low-energy infrared regions, respectively; but that it is also responsible in modifying their stability, electronic, transport, and optical absorption preferences. The large value of the high frequency dielectric constants—together with the appreciable magnitude of absorption coefficients and refractive indices, small values of effective masses of conducting electrons and holes, and the indirect nature of the bandgap transitions, among others—suggested that cubic A2AgCrBr6 (A = Rb, Cs) and Cs2AgCrCl6 may likely be a set of optoelectronic materials for subsequent experimental characterizations.


Introduction
Innovative materials that display impressive geometrical, stability, electronic, transport, and optical characteristics in the infrared through visible to ultraviolet region of the electromagnetic spectrum are a special class of semiconductors for application in optoelectronic devices [1][2][3][4][5][6][7][8][9]. Halide based single [10][11][12][13][14][15][16][17] and double perovskites [1][2][3][4][5][6][7][8][9][18][19][20][21][22] known by the chemical formulae ABX 3 and AB B"X 6 , respectively, are examples of such compounds. Not only do they absorb light, but many of them also emit light in specific regions of the electromagnetic spectrum, depending on the nature of ingredients at the atomic scale with which they are built. In single halide perovskites with ABX 3 composition [10,11], A is the organic or inorganic species in the +1 oxidation state (viz. CH 3 NH 3 + , the traits of the first two heaviest members of the series, as commonly reported for similar systems with different B"-site species [1][2][3][4][5][6][7][8][9][18][19][20][21][22][23][24]. However, we extend our calculations for another lighter member of the series to deduce the extent to which the monovalent cation K at the A site can modify the geometric, electronic, and optical properties. The stability of the perovskite nature of A 2 AgCrBr 6 is investigated using a set of geometry-based indices such as the octahedral factor [48], Goldschmidt's tolerance factor [49,50], global instability index [51,52], and the newly proposed tolerance factor [39]. We investigate the nature of orbital contribution of each atom type leading to the formation of the valence band maximum (VBM) and conduction band minimum (CBM). In addition, we investigate the optical properties [53][54][55][56][57]-e.g., the spectra of dielectric function, absorption coefficient, electrical conductivity, complex refractive index, and the Tauc plot-and compare them with those reported for similar perovskite systems. Finally, we perform similar calculations for Cs 2 AgCrX 6 (X = Cl, I) to elucidate the character and magnitude of bandgap transitions and the extent to which halogen substitution at the X-site modifies the geometrical, stability, transport, and optoelectronic properties of Cs 2 AgCrBr 6 .

Computational Details
For reasons discussed in various previous studies [58][59][60][61][62][63], the geometries of A 2 AgCrBr 6 (A = Cs, Rb, K) and Cs 2 AgCrX 6 (X = Cl, I) were relaxed with SCAN + rVV10 [58]. The SCAN functional is "strongly constrained and appropriately normed", which is paired with the non-local correlation part from the rVV10 vdW density functional. It is one of the most suitable functionals recommended for the study the atomic structure of hybrid perovskite materials [63]. The k-point mesh 8 × 8 × 8 centered at Γ was used for sampling the first Brillouin zone. The projector augmented-wave (PAW) method [64] was used. The energy cut-off used for geometry relaxation was 520 eV. The cut-off in total energy for the relaxation of the electronic degrees of freedom was set to 10 −8 eV per cell, instead of the default value of 10 −4 . A negative value of −0.01 for EDIFFG was set that defines the break condition for the ionic relaxation loop. The average forces on ions were minimized to 0.005 eV/Å. Dynamical and mechanical stabilities of the studied systems were not examined, as this may be the subject of another extensive study. All spin-polarization calculations, without considering the effect of spin-orbit coupling, were performed using VASP 5.4 [65,66].
αβ (ω) was calculated using the Kubo-Greenwood formula given by Equation (3) [72], where subscripts in italics c and υ refer to conduction and valence band states, respectively, and u ck is the cell periodic part of the orbitals at the k-point k.
The calculation of dielectric function has often been performed with dense k-point meshes, which is arguably because of the convergence of dielectric constant with respect the number of k-points leading to greater chemical accuracy [73]. Because of limited computational resources, we carried out such calculations using a 4 × 4 × 4 k-point mesh, in which, the number of empty conduction band states used was doubled and the number of grid points on which to compute DOS was set to a recommended value of 2000 [74]. Since the calculation of the linear response properties cannot be performed with the meta-GGA functional (SCAN + rVV10) using VASP 5.4 [65,66], we used the PBESol functional [75], in combination with the density functional perturbation theory (DFPT) method [76][77][78], to calculate such properties. The SCAN + rVV10 relaxed geometries of A 2 AgCrBr 6 (A = Cs, Rb, K) and Cs 2 AgCrX 6 (X = Cl, I) were used. Optical spectral properties, viz. photoabsorption coefficient α(ω) [55], photoconductivity σ(ω) [57], refractive index n(ω) [54][55][56], and extinction coefficient κ(ω) [54][55][56] spectra were calculated using Equations (4)-(7), respectively.
In Equation (4), c and ω are the speed of light in vacuum and the frequency of light wave, respectively. In Equation (5), E is the energy of light in eV, ε 0 is the permittivity of vacuum in C 2 /Nm 2 , and e is the charge of electron in Coulomb. The calculated values of real and imaginary parts of ε(ω) were used for the calculation of optical constants given by Equations (4)-(7) [65,66]. Table 1 summarizes the calculated lattice constants (a, b, c, α, β, and γ), cell volume (V) and density (ρ) of A 2 AgCrBr 6 (A = Cs, Rb, K). Figure 1a shows the relaxed polyhedral model of Cs 2 AgCrBr 6 , which is analogous with that found for A 2 AgCrBr 6 (A = Rb, K). Because a = b = c and α = β = γ for each of them, each adopts a 3D elpasolite structure with space group Fm-3m. Table 1. Selected lattice properties (lattice constants, volume, and density) and bandgaps of A 2 AgCrBr 6 (A = K, Rb, Cs).   The lattice constant a is found to be the largest of 10.68 Å for Cs2AgCrBr6, and the smallest of 10.52 Å for K2AgCrBr6. The trend in the decrease of a in the series is consistent with the corresponding decrease in the cell volume V, with the V values ranging from 1217.10 Å 3 (Cs2AgCrBr6) to 1165.60 Å 3 (K2AgCrBr6). These changes are in agreement with the corresponding decrease in ρ, as well as that in the Ag-Cl, Cr-Cl, and A-Cl bond distances (Table 2), across the series passing from Cs2AgCrBr6 through Rb2AgCrBr6 to K2AgCrBr6. For comparison, our calculation on Cs2AgBiCl6 and Cs2AgBiBr6 yielded in lattice constants, density, and volumes that were not only close to experiment (within 2%), but also led to a suggestion that the Cr-substituted compounds are relatively lightweight, thus advocating the reliability of the accuracy of the DFT functional chosen.  The lattice constant a is found to be the largest of 10.68 Å for Cs 2 AgCrBr 6 , and the smallest of 10.52 Å for K 2 AgCrBr 6 . The trend in the decrease of a in the series is consistent with the corresponding decrease in the cell volume V, with the V values ranging from 1217.10 Å 3 (Cs 2 AgCrBr 6 ) to 1165.60 Å 3 (K 2 AgCrBr 6 ). These changes are in agreement with the corresponding decrease in ρ, as well as that in the Ag-Cl, Cr-Cl, and A-Cl bond distances (Table 2), across the series passing from Cs 2 AgCrBr 6 through Rb 2 AgCrBr 6 to K 2 AgCrBr 6 . For comparison, our calculation on Cs 2 AgBiCl 6 and Cs 2 AgBiBr 6 yielded in lattice constants, density, and volumes that were not only close to experiment (within 2%), but also led to a suggestion that the Cr-substituted compounds are relatively lightweight, thus advocating the reliability of the accuracy of the DFT functional chosen.

Geometrical Properties and Stability of Perovskite Structures
The geometrical stability of A 2 AgCrBr 6 was examined using the global instability index, GII, given where n is the number of ions and d is the bond discrepancy factor defined as the deviation of bond valence sum (BVS) from formal valence [51,52]. BVS was calculated using the sum of bond valences (s ij ) around any specific ion given by , l ij is a bond length, l 0 is the bond valence parameter empirically determined using experimental room-temperature structure data, and b is the bond softness parameter. For geometrically stable perovskite structures without steric distortions, GII equals 0.0 valence unit (v.u.); and for empirically unstable structures, GII > 0.2 v.u. [51,52]. Table 2. Comparison of selected bond distances r of A 2 AgCrBr 6 (A = K, Rb, Cs) with Cs 2 AgCrX 6 (X = Cl, I).

System
Bond Distances/Å The results of our calculation listed in Table 1 show that GII for Cs 2 AgCrBr 6 , Rb 2 AgCrBr 6 and K 2 AgCrBr 6 are 0.02, 0.06, and 0.08 v.u., respectively. This means that all the studied systems are associated with very marginal geometrical distortion compared to what might be expected of ideal perovskite structures. The largest instability is observed for K 2 AgCrBr 6, which is believed to be due to the small size of the K + cation that causes the lattice K 2 AgCrBr 6 to contract. Table 3 lists Shannon's ionic radii of atoms that were used to understand the geometrical stability of A 2 AgCrBr 6 . Specifically, we used them for the calculation of the octahedral factor µ (µ = r B /r X ), and Goldschmidt tolerance factor (t = r A +r X √ 2(r B +r X ) ). According to numerous previous demonstrations which have appeared in the literature [48][49][50], the combination (µ and t) should be suitable to assess the formability of perovskite structures provided µ and t values are in the ranges 0.414 < µ < 0.732 and 0.825 < t < 1.059, respectively. Our calculation gave a value of 0.45 for µ for all A 2 AgCrBr 6 . Similarly, the t values for these systems were in the range 0.90 < t < 0.96 (Table 3). Clearly, the combination (µ and t) recognizes the formability of A 2 AgCrBr 6 (A = Cs, Rb, K) as stable perovskites. For comparison, the most studied CH 3 NH 3 PbI 3 was reported to have a t of 0.91 (unstable with respect to tilting), whereas NH 4 PbI 3 has a t of 0.76 [79]. For the latter case, an alternative non-perovskite structure was suggested [79]. Table 3. Shannon's radii (r) of ions, octahedral factor (µ), Goldschmidt tolerance factor (t), and new tolerance factor (τ) for A 2 AgCrBr 6 (A = Cs, Rb, K). The corresponding properties for Cs 2 AgCrX 6 (X = Cl, I) are also included.  Table 3 also includes the τ (a newly proposed tolerance factor [39]) values for A 2 AgCrBr 6 . These were calculated using the relationship given by ), where n A is the oxidation state of A, r i is the ionic radius of ion i, and r A > r B by definition. The proposal to evaluate the geometrical stability of single and double perovskites (ABX 3 and A 2 B B"X 6 , respectively) using τ (τ < 4.18) was emerged after it was being realized that the recommended ranges for µ and t do not guarantee the formation of the perovskite structure since they give a high false-positive rate (51%) in the regions of t (0.825 < t < 1.059) and µ (0.414 < µ < 0.732). In fact, τ was used to generalize outside of a 1034 training set of experimentally realized single and double perovskites (91% accuracy), and that its application was also useful to identify 23,314 new double perovskites ranked by their probability of being stable as perovskite [39]. Based on the criterion that τ < 4.18 for geometrically stable perovskites, we found that A 2 AgCrBr 6 (A = Cs, Rb) are a set of such geometrically stable perovskites (4.04 < τ < 4.14), and K 2 AgCrBr 6 (τ = 4.22) is a partially unstable structure (a non-perovskite!), where the ionic radius of B was calculated as the arithmetic mean of the ionic radii of B and B". These results are not in exact agreement with that inferred from GII or the combination of µ and t values (see Tables 1 and 3). This may either mean that the value of τ (τ < 4.18 [39]) recommended for identifying unknown perovskite structures may be stringent, causing the rejection of K 2 AgCrBr 6 as a stable perovskite, or both the GII and the combination of µ and t mislead the stability features.
To understand the effect of halogen substitution at the X site of A 2 AgCrX 6 , we have carried out similar calculations only for Cs 2 AgCrI 6 and Cs 2 AgCrCl 6 . The calculated lattice constants, cell volume, and density were 11.48 Å, 1511.9 Å 3 , and 5.2 gcm −3 for Cs 2 AgCrI 6 , respectively. These were 10.13 Å, 1040.2 Å 3 , and 4.08 gcm −3 for Cs 2 AgCrCl 6 , respectively. In other words, the halogen substitution does not cause significant strain in the lattice that can lead to the change in the symmetry of the system from Fm-3m. However, the replacement of the Br atoms in Cs 2 AgCrBr 6 by the Cl and I anions has indeed resulted in lattice contraction and expansion, respectively, which are expected to affect the electronic, transport, and optical properties of the resulting systems (vide infra). The presence of lattice contraction and expansion is also evident of metal-halide and alkali-halide bond lengths that are decreasing across the series in this order: Cs 2 AgCrCl 6 < Cs 2 AgCrBr 6 > Cs 2 AgCrI 6 . (A comparison detail of the Ag-X, Cr-X, and Cs-X bond distances for Cs 2 AgCrX 6 (X = Cl, Br, I) is given in Table 2).
Moreover, the calculated µ and t values were 0.40 and 0.94 for Cs 2 AgCrI 6 ( Table 3), respectively. Based on the traditional arguments [48,50], one may not disapprove to call Cs 2 AgCrI 6 a stable perovskite. The same argument applies to Cs 2 AgCrCl 6 , as the combination of the µ and t values (Table 3) favors a perovskite structure. This result is consistent with that inferred from the GII values (0.10 v.u. for Cs 2 AgCrCl 6 and 0.00 v.u. for Cs 2 AgCrI 6 ). However, our calculated τ value of 4.31, far exceeding its recommended value of 4.18, rejects Cs 2 AgCrI 6 as a stable perovskite. This is analogously as τ rejected the K 2 AgCrBr 6 system to be called a perovskite. However, this is not the case with Cs 2 AgCrCl 6 (τ = 3.87), in agreement with that reported recently [39]. Clearly, the aforesaid mismatch between the results associated with different indices suggests that the geometrical properties of a large body of AB B"X 6 systems of different B and B"compositions, together with X = I and Br and A = K, need to be analyzed to reach any definitive conclusion on the predictability of GIIand τ-based stability features.

Density of States and Band Structures
Figure 2a-g shows the orbital-projected partial density of states of each atom type, and Figure 2h shows the atom projected partial density of states for Cs 2 AgCrBr 6 . From these, it may be said that the dispersion of the HOMO band (VBM) results collectively from contributions of e g and t 2g states of both Ag (4d) and Cr(3d), and the 4p states of Br. This is also reminiscent of the data provided in Table 4, in which, the VBM is mainly due to the 4p states of Br (88%), and that the contribution from the 3d (Cr) and 4d (Ag) states is very small.
On the other hand, the LUMO band (CBM) of Cs 2 AgCrBr 6 is predominantly of Cr (3d) character. It is substantially composed of the e g orbital energy states (~72.5%). This, together with the small contributions from the Ag (5s) and Br (4p) states, causes the dispersion of the band. Note that alkali substitution at the A site has very little effect on the orbital characters of VBM and CBM. However, when the Br atom at the X-site was replaced by the I and Cl atoms, this had a very marginal effect on the orbital character of the CBM, and significant effect on the VBM. In particular, the 3d (Cr) energy states dominate below the Fermi level for Cs 2 AgCrCl 6 (46%), which was comparatively larger than those arose from the 3p (35.2%) and 4d (17.0%) states of Cl and Ag, respectively. For Cs 2 AgCrI 6 , the I(5p) states dominate below the Fermi level, and the 3d (Cr) and 4d (Ag) states contribute marginally to the VBM. Figure 3 compares the nature of band dispersion of Cs 2 AgCrBr 6 with Cs 2 AgCrI 6 , and Figure 4 illustrates the orbital-and atom-projected partial density of states for Cs 2 AgCrI 6 . On the other hand, the LUMO band (CBM) of Cs2AgCrBr6 is predominantly of Cr (3d) character. It is substantially composed of the eg orbital energy states (~72.5%). This, together with the small contributions from the Ag (5s) and Br (4p) states, causes the dispersion of the band. Note that alkali substitution at the A site has very little effect on the orbital characters of VBM and CBM. However, when the Br atom at the X-site was replaced by the I and Cl atoms, this had a very marginal effect on the orbital character of the CBM, and significant effect on the VBM. In particular, the 3d (Cr) energy states dominate below the Fermi level for Cs2AgCrCl6 (46%), which was comparatively larger than those arose from the 3p (35.2%) and 4d (17.0%) states of Cl and Ag, respectively. For Cs2AgCrI6, the I(5p) states dominate below the Fermi level, and the 3d (Cr) and 4d (Ag) states contribute marginally to the VBM. Figure 3 compares the nature of band dispersion of Cs2AgCrBr6 with Cs2AgCrI6, and Figure 4 illustrates the orbital-and atom-projected partial density of states for Cs2AgCrI6.   From the bandgap (E g ) data listed in Table 1, it is apparent that the A 2 AgCrBr 6 systems are an indirect bandgap material. This is arguably because the VBM is located at the high symmetry L-valley (VBM) and the CBM is located at the Γ-valley (CBM); see Figure 3a for Cs 2 AgCrBr 6 . The calculated bandgaps (E g ) of A 2 AgCrBr 6 were found between 1.27 and 1.29 eV (Table 1), thereby suggesting the semiconducting nature of these materials. Small variations in E g across the alkali series indicate that alkali substitution at the A-site of A 2 AgCrCl 6 does not significantly affect the gap between CBM and Nanomaterials 2020, 10, 973 9 of 21 VBM, and is consistent with the nature and extent of orbital characters that were involved in the formation of these bands (Table 4).  Table 1 lists the Eg values for Cs2AgCrBr6. The conventional unit-cell geometries were used to calculate the band dispersion.  Table 1 lists the E g values for Cs 2 AgCrBr 6 . The conventional unit-cell geometries were used to calculate the band dispersion.
By contrast, Cs 2 AgCrI 6 has a bandgap of 0.43 eV, and is direct at the Γ-valley (Figure 3b). Obviously, the replacement of the Br atoms of Cs 2 AgCrBr 6 with iodine atoms not only causes the shift of the VBM from the L-valley to appear at the Γ-valley, but also significantly narrows the gap between the CBM and VBM and alters the character of the electronic transition between them. This is not the case with Cs 2 AgCrCl 6 since the bandgap for this system is increased giving rise to an E g value of 1.84 eV (SCAN + rVV10) and was indirect between the L-valley (VBM) and Γ-valley (CBM). These results lead to a conclusion that halogen substitution from Cl through Br to I at the X-site of Cs 2 AgCrX 6 not only triggers the nature of the bandgap from indirect to direct, but also adjusts its magnitude from 1.84 eV to 0.43 eV.
In any case, the charge transport properties of any semiconducting material are strongly dependent on the effective masses of the charge carriers [80][81][82]. Table 5 summarizes the effective masses of conduction electrons and holes for A 2 AgCrBr 6 , which were calculated by parabolic fitting of the lower conduction band and the upper valence band centered at the Γand L-valleys, respectively. Since the HOMO band is formed by several spin-up and spin-down channels (total eight) caused by the Cr 3+ ions, there are spin-up and spin-down holes along the crystallographic directions. The most important of these is the L→Γ direction, which is linked directly with the electronic transition between the VBM and CBM. As can be seen from Table 5, the effective masses of the spin-up holes (m h *(up)) are always lighter than that of the spin-down holes (m e *(down)), regardless of the nature of the A-site in A 2 AgCrBr 6 . This means that (m h *(down)) would play an insignificant role in determining the transport phenomena that are usually governed by the band structures at and around the close vicinity of the Γ-point. In contrary, there are only spin-up electrons that are associated with the bottom of the conduction band. They are indeed small (m e *(up) (values lie between 0.50 and 0.62 m 0 for all directions), but are not always heavier than those of m h *(up). These results suggest that the studied systems might be suitable as hole (and electron) transporting materials due to their high mobility [81][82][83]. A similar result was found for Cs 2 AgCrCl 6 (values not given). From the bandgap (Eg) data listed in Table 1, it is apparent that the A2AgCrBr6 systems are an indirect bandgap material. This is arguably because the VBM is located at the high symmetry L-valley (VBM) and the CBM is located at the Γ-valley (CBM); see Figure 3a for Cs2AgCrBr6. The calculated bandgaps (Eg) of A2AgCrBr6 were found between 1.27 and 1.29 eV (Table 1), thereby suggesting the semiconducting nature of these materials. Small variations in Eg across the alkali series indicate that alkali substitution at the A-site of A2AgCrCl6 does not significantly affect the gap between CBM and VBM, and is consistent with the nature and extent of orbital characters that were involved in the formation of these bands (Table 4).
By contrast, Cs2AgCrI6 has a bandgap of 0.43 eV, and is direct at the Γ-valley (Figure 3b). Obviously, the replacement of the Br atoms of Cs2AgCrBr6 with iodine atoms not only causes the shift of the VBM from the L-valley to appear at the Γ-valley, but also significantly narrows the gap between the CBM and VBM and alters the character of the electronic transition between them. This is not the case with Cs2AgCrCl6 since the bandgap for this system is increased giving rise to an Eg value of 1.84 eV (SCAN + rVV10) and was indirect between the L-valley (VBM) and Γ-valley (CBM). These results lead to a conclusion that halogen substitution from Cl through Br to I at the X-site of Cs2AgCrX6 not , and high dielectric constants, has led to a suggestion that this system is suitable for application as a top absorber of the tandem solar cell.

Optical Properties
The excitonic effect is generally approximated by solving the Bethe-Salpeter equation for the two-body Green's function, without or with considering local field effect [85]. Due to the high computational cost and limited computed resources, the frequency-dependent complex dielectric function was calculated within the framework of DFPT without taking into account the electron-hole coupling effect. Studies have shown that neglecting electron-hole coupling can yield reasonable results for semiconductors with small band gaps [86,87]. Nevertheless, Figure 5 shows the plot of the real and imaginary parts of the dielectric function as a function of photon energy for A 2 AgCrBr 6 . The first peak of the ε 2 curve is located at an energy higher than that of the ε 1 curve, and alkali substitution at the A-site causes a slight fluctuation in the high frequency dielectric behavior. For instance, these peaks on the ε 1 and ε 2 curves are positioned at energies of 1.45 and 1.63 eV for Cs 2 AgCrBr 6 , respectively. These are 1.53 and 1.72 eV for Rb 2 AgCrBr 6 , respectively, and are 1.55 and 1.74 eV for K 2 AgCrBr 6 , respectively. Similarly, the low frequency limit of the isotropically averaged value of ε 1 (ω = 0) = ε ∞ (called optical dielectric constant or high-frequency dielectric constant) is found to be 6.1 for Cs 2 AgCrBr 6 , 5.4 for K 2 AgCrBr 6 , and 5.8 for K 2 AgCrBr 6 . This result indicates that the contribution of electrons to the static dielectric constant is appreciable and that the studied systems contain ionic bonds. Zakutayev et al. [88], as well as others [84], have previously demonstrated that perovskites with appreciable dielectric constants are defect tolerant. For comparison, MAPbI 3 (X = Cl, Br, I), and other perovskite systems [73] were reported to have virtually similar optical dielectric constants (viz. ε ∞ = 6.5 for MAPbI 3 , 5.

Optical Properties
The excitonic effect is generally approximated by solving the Bethe-Salpeter equation for the two-body Green's function, without or with considering local field effect [85]. Due to the high computational cost and limited computed resources, the frequency-dependent complex dielectric function was calculated within the framework of DFPT without taking into account the electron-hole coupling effect. Studies have shown that neglecting electron-hole coupling can yield reasonable results for semiconductors with small band gaps [86,87]. Nevertheless, Figure 5 shows the plot of the real and imaginary parts of the dielectric function as a function of photon energy for A2AgCrBr6. The first peak of the ε2 curve is located at an energy higher than that of the ε1 curve, and alkali substitution at the A-site causes a slight fluctuation in the high frequency dielectric behavior. For instance, these peaks on the ε1 and ε2 curves are positioned at energies of 1.45 and 1.63 eV for Cs2AgCrBr6, respectively. These are 1.53 and 1.72 eV for Rb2AgCrBr6, respectively, and are 1.55 and 1.74 eV for K2AgCrBr6, respectively. Similarly, the low frequency limit of the isotropically averaged value of ε1 (ω = 0) = ε∞ (called optical dielectric constant or high-frequency dielectric constant) is found to be 6.1 for Cs2AgCrBr6, 5.4 for K2AgCrBr6, and 5.8 for K2AgCrBr6. This result indicates that the contribution of electrons to the static dielectric constant is appreciable and that the studied systems contain ionic bonds. Zakutayev et al. [88], as well as others [84], have previously demonstrated that perovskites with appreciable dielectric constants are defect tolerant. For comparison, MAPbI3 (X = Cl, Br, I), and other perovskite systems [73] were reported to have virtually similar optical dielectric constants (viz.  The magnitude of the bandgap is the determinant of the onset of first optical absorption, which can be approximated using the spectra of ε2. As such, the onset of absorption is approximately lying between 1.15 and 1.40 eV for A2AgCrBr6 ( Figure 5), showing an appreciable absorption mostly in the infrared spectral region. Note that the bandgaps of A2AgCrBr6 evaluated using SCAN + rVV10 were lying between 1.27 and 1.29 eV, and the dielectric functions were calculated using PBEsol in conjunction with DFPT, yet there is a close match between bandgap and onset of optical absorption evaluated using the two different theoretical approaches employed. The magnitude of the bandgap is the determinant of the onset of first optical absorption, which can be approximated using the spectra of ε 2 . As such, the onset of absorption is approximately lying between 1.15 and 1.40 eV for A 2 AgCrBr 6 ( Figure 5), showing an appreciable absorption mostly in the infrared spectral region. Note that the bandgaps of A 2 AgCrBr 6 evaluated using SCAN + rVV10 were lying between 1.27 and 1.29 eV, and the dielectric functions were calculated using PBEsol in conjunction with DFPT, yet there is a close match between bandgap and onset of optical absorption evaluated using the two different theoretical approaches employed. Figure 6a,b compares the energy dependence of ε 1 and ε 2 for Cs 2 AgCrX 6 (X = Cl, Br, I), respectively. From either of the two plots, it is apparent that halogen replacement has a significant effect not only on the magnitude of ε ∞ , but also on the onset of optical absorption. For instance, the value of ε ∞ inferred from Figure 6a is approximately around 4.5, 6.1, and 8.9 for Cs 2 AgCrCl 6 , Cs 2 AgCrBr 6 , and Cs 2 AgCrI 6 , respectively. These indicate that Cs 2 AgCrI 6 has a greater ability to screen charged defects compared to Cs 2 AgCrBr 6 and Cs 2 AgCrCl 6 [89,90]. The observed trend in the increase of ε ∞ caused by halogen substitution at the X-site in Cs 2 AgCrX 6 is consistent with that reported for the MAPbX 3 (X = Cl, Br, I) series [73].
effect not only on the magnitude of ε∞, but also on the onset of optical absorption. For instance, the value of ε∞ inferred from Figure 6a is approximately around 4.5, 6.1, and 8.9 for Cs2AgCrCl6, Cs2AgCrBr6, and Cs2AgCrI6, respectively. These indicate that Cs2AgCrI6 has a greater ability to screen charged defects compared to Cs2AgCrBr6 and Cs2AgCrCl6 [89,90]. The observed trend in the increase of ε∞ caused by halogen substitution at the X-site in Cs2AgCrX6 is consistent with that reported for the MAPbX3 (X = Cl, Br, I) series [73].
The onset of optical absorption corresponding to the ε2 spectra is positioned around 1.55 eV for Cs2AgCrCl6. This is significantly shifted from the near-infrared region to the low energy infrared region of the electromagnetic spectrum caused by the halogen substitution at the X-site, thus showing up at energies around 1.15 eV and 0.45 eV for Cs2AgCrBr6 and Cs2AgCrI6, respectively. These values are close to their corresponding bandgaps predicted using the SCAN + rVV10 functional. It is worth noting that we did not calculate the contribution of dielectric constant due to ions to the static dielectric constant, thus it is not possible to provide views on the polarity of the chemical bonds and the softness of the vibrations [91], as well as the nature and importance of (picosecond) response of lattice vibrations (phonon modes) that explain the ionic and lattice polarizations, and the extent to which the material would control the photovoltaic performance [74,89,90,92]. In any case, the major peaks in the ε2 curve of Cs2AgCrCl6 are positioned at energies of 2.0 and 2.3 eV, whereas the onset of the optical absorption was located at an energy of 1.55 eV (Figure 6b). These peaks in the UV/Vis/NIR diffuse reflectance spectrum were experimentally observed at energies of 1.6 and 2.2 eV for hexagonal Cs2AgCrCl6, respectively, and were assigned to the d-d transitions of 4 A2 → 4 T2 and 4 T2 → 4 T1, respectively. This is not surprising given that the ground state The onset of optical absorption corresponding to the ε 2 spectra is positioned around 1.55 eV for Cs 2 AgCrCl 6 . This is significantly shifted from the near-infrared region to the low energy infrared region of the electromagnetic spectrum caused by the halogen substitution at the X-site, thus showing up at energies around 1.15 eV and 0.45 eV for Cs 2 AgCrBr 6 and Cs 2 AgCrI 6 , respectively. These values are close to their corresponding bandgaps predicted using the SCAN + rVV10 functional. It is worth noting that we did not calculate the contribution of dielectric constant due to ions to the static dielectric constant, thus it is not possible to provide views on the polarity of the chemical bonds and the softness of the vibrations [91], as well as the nature and importance of (picosecond) response of lattice vibrations (phonon modes) that explain the ionic and lattice polarizations, and the extent to which the material would control the photovoltaic performance [74,89,90,92].
In any case, the major peaks in the ε 2 curve of Cs 2 AgCrCl 6 are positioned at energies of 2.0 and 2.3 eV, whereas the onset of the optical absorption was located at an energy of 1.55 eV (Figure 6b). These peaks in the UV/Vis/NIR diffuse reflectance spectrum were experimentally observed at energies of 1.6 and 2.2 eV for hexagonal Cs 2 AgCrCl 6 , respectively, and were assigned to the d-d transitions of 4 A 2 → 4 T 2 and 4 T 2 → 4 T 1 , respectively. This is not surprising given that the ground state of the Cr 3+ (t 2g 3 e g 0 ) cation in an octahedral O h symmetry is described by 4 A 2g (F), and is accompanied with spin-allowed transitions from the 4 A 2g (F) ground state to the excited 4 T 2 and 4 T 1 states (that is, 4 T 2 → 4 T 1 and 4 A 2 → 4 T 1 ). Such d-d transitions were reported to occur at energies of 3.06 eV (405 nm) and 2.25 eV (550 nm) in the UV-vis spectrum of isolated Cr 3+ complex of aspartic acid, respectively [93]. The intense and broad bands observed at energies of 2.93 eV and 2.13 eV for strontium formate dehydrate crystal of Cr 3+ were attributed to the corresponding transitions, respectively [94]. Also, zinc-tellurite glasses doped with Cr 3+ ion feature two intense broad bands with maxima at 454 nm (2.73 eV) and 650 nm (1.91 eV), which were assigned to the transitions 4 A 2 → 4 T 1 and 4 A 2 → 4 T 2 , respectively [95]. In Cs 2 AgInCl 6 :Cr 3+ halide double perovskite, the absorptions peaks at 1.55 eV (800 nm) and 2.19 eV (565 nm) were assigned to the same spin-allowed transitions 4 A 2 → 4 T 2 and 4 T 2 → 4 T 1 , respectively. Because the VBM is dominated with Br(4p) and I(5p) energy states for Cs 2 AgCrBr 6 and Cs 2 AgCrI 6 , respectively, and the contribution of Cr(3d) orbitals states to the VBM is reasonably small, the two electronic transitions appear in the ε 2 spectra in the region 0.3-2.2 eV were shifted toward the low energy regions (Figure 6b). Figure 7 displays the plot of the absorption coefficient, photoconductivity, and complex refractive index for A 2 AgCrBr 6 . Equations (4)-(7) were used, respectively. Figure 7b shows the Tauc plot. These plots replicate the oscillator peaks of the dielectric function ε 2 in the region 0.0-3.0 eV ( Figure 6). From Figure 7a, it is obvious that α for A 2 AgCrBr 6 , which determines the extent to which light of a particular energy can penetrate a material before it is absorbed [96], is non-zero (positive) both in the infrared and visible regions. As can be seen, different regions have different absorption coefficients. This may not be unusual given that the absorption coefficients of MAPbI 3 as low as 10 -14 cm -1 were detected at room temperature for long wavelengths, and were 14 orders of magnitude lower than those observed at shorter wavelengths [97]. Nevertheless, α increases as the light energy increases and becomes a maximum at the strongest peak occurred at an energy around 1.7 eV for Cs 2 AgCrBr 6 . The trend in α in the series A 2 AgCrBr 6 follows the order: Cs 2 AgCrBr 6 (3.3 × 10 5 cm −1 ) < Rb 2 AgCrBr 6 (3.6 × 10 5 cm −1 ) ≈ K 2 AgCrBr 6 (3.6 × 10 5 cm −1 ). These are somehow larger than those of 0.5 × 10 4 cm −1 and 1.5 × 10 4 cm −1 observed at 1.77 eV (700 nm) and 2.25 eV (550 nm) for MAPbI 3 , respectively [98,99].
The above trend both in α and σ is not exactly similar to that found for the static refractive index n (ω = 0), with the latter were around 2.5, 2.3, and 2.4 for Cs 2 AgCrBr 6 , Rb 2 AgCrBr 6 , and K 2 AgCrBr 6 , respectively (Figure 7d). The n(ω) values increase with respect to the increase of photon energy in the region 0.0-3.0 eV. The maximum of n at the highest peak in the region 0.0-3.0 eV varies between 3.0 and 3.2 for A 2 AgCrBr 6 .
Cl and I substitutions at the X-site in Cs 2 AgCrX 6 has resulted in a relatively smaller α of 2.4 × 10 5 cm −1 for Cs 2 AgCrI 6 and a relatively larger α of 4.0 × 10 5 cm −1 for Cs 2 AgCrCl 6 ; all within the range 0.0-2.5 eV (Figure 8a). Although the peaks in the α spectrum of Cs 2 AgCrCl 6 are relatively sharper than those of Cs 2 AgCrBr 6 and Cs 2 AgCrI 6 , the different absorption edges found for the three systems are consistent with the corresponding bandgap properties. Also, the trend in α is concordant with the conductivity spectra shown in Figure 8b, in which, σ decreases in the series in this order: Cs 2 AgCrI 6 < Cs 2 AgCrBr 6 < Cs 2 AgCrCl 6 . On the other hand, the n (ω = 0) values were 2.1, 2.5, and 3.0 for Cs 2 AgCrCl 6 , Cs 2 AgCrBr 6 , and Cs 2 AgCrI 6 , respectively (Figure 9a). The maximum values of n were approximately 2.9, 3.2, and 3.4 for the corresponding systems, respectively, a feature which is consistent with the trend in the optical dielectric constants (see above). Regardless of the nature of the A 2 AgCrX 6 systems examined, the extinction coefficient (imaginary part of the refractive index) was found to be very small (Figures 7d and 9b), and is expected for semiconducting materials [101]. These results lead to a meaning that the studied Cs 2 AgCrX 6 (X = Cl, Br, I) perovskites may result in relatively larger reflection at the perovskite/electrode interface. For comparison, the refractive indices of CH 3 NH 3 PbI 3-x Cl x perovskite thin films were reported approximately to be 2.4 and 2.6 in the visible to near-infrared wavelength region [101]. He et al. have examined bulk single crystals of CH 3 NH 3 PbX 3 (CH 3 NH 3 = MA, X = Cl, Br, I) and have measured refractive indices that rank in this order: MAPbI 3 > MAPbBr 3 > MAPbCl 3 at the same wavelength [102]. These results indicate that n may play an important role in determining the optical response properties of halide perovskites, which may serve as a useful metric in the parameter space for use in optoelectronic device design. Figure 7 displays the plot of the absorption coefficient, photoconductivity, and complex refractive index for A2AgCrBr6. Equations (4-7) were used, respectively. Figure 7b shows the Tauc plot. These plots replicate the oscillator peaks of the dielectric function ε2 in the region 0.0-3.0 eV ( Figure 6). From Figure 7a, it is obvious that α for A2AgCrBr6, which determines the extent to which light of a particular energy can penetrate a material before it is absorbed [96], is non-zero (positive) both in the infrared and visible regions. As can be seen, different regions have different absorption coefficients. This may not be unusual given that the absorption coefficients of MAPbI3 as low as 10 -14 cm -1 were detected at room temperature for long wavelengths, and were 14 orders of magnitude lower than those observed at shorter wavelengths [97]. Nevertheless, α increases as the light energy increases and becomes a maximum at the strongest peak occurred at an energy around 1.7 eV for Cs2AgCrBr6. The trend in α in the series A2AgCrBr6 follows the order: Cs2AgCrBr6 (3.3 × 10 5 cm −1 ) < Rb2AgCrBr6 (3.6 × 10 5 cm −1 ) ≈ K2AgCrBr6 (3.6 × 10 5 cm −1 ). These are somehow larger than those of 0.5 × 10 4 cm −1 and 1.5 × 10 4 cm −1 observed at 1.77 eV (700 nm) and 2.25 eV (550 nm) for MAPbI3, respectively [98,99]. The nature of the curves of α is similar to that found for σ at the strongest peak positions in the region 0.0-2.5 eV (see Figure 7a vs. Figure 7c). The value of σ at these peaks is 2.64 × 10 9 Sm −1 for Cs2AgCrBr6, 2.76 × 10 9 Sm −1 for Rb2AgCrBr6, and 2.84 × 10 9 Sm −1 for K2AgCrBr6. For comparison, the average value of σ for MAPbI3 thin films was reported to be 6 × 10 5 Sm −1 [100].
The above trend both in α and σ is not exactly similar to that found for the static refractive index n (ω = 0), with the latter were around 2.5, 2.3, and 2.4 for Cs2AgCrBr6, Rb2AgCrBr6, and K2AgCrBr6, respectively (Figure 7d). The n(ω) values increase with respect to the increase of photon energy in the region 0.0-3.0 eV. The maximum of n at the highest peak in the region 0.0-3.0 eV varies between 3.0 and 3.2 for A2AgCrBr6.
Cl and I substitutions at the X-site in Cs2AgCrX6 has resulted in a relatively smaller α of 2.4 × 10 5 cm −1 for Cs2AgCrI6 and a relatively larger α of 4.0 × 10 5 cm −1 for Cs2AgCrCl6; all within the range 0.0-2.5 eV (Figure 8a). Although the peaks in the α spectrum of Cs2AgCrCl6 are relatively sharper than those of Cs2AgCrBr6 and Cs2AgCrI6, the different absorption edges found for the three systems are consistent with the corresponding bandgap properties. Also, the trend in α is concordant with the conductivity spectra shown in Figure 8b, in which, σ decreases in the series in this order: Cs2AgCrI6 < Cs2AgCrBr6 < Cs2AgCrCl6. On the other hand, the n (ω = 0) values were 2.1, 2.5, and 3.0 for Cs2AgCrCl6, Cs2AgCrBr6, and Cs2AgCrI6, respectively (Figure 9a). The maximum values of n were approximately 2.9, 3.2, and 3.4 for the corresponding systems, respectively, a feature which is consistent with the trend in the optical dielectric constants (see above). Regardless of the nature of the A2AgCrX6 systems examined, the extinction coefficient (imaginary part of the refractive index) was found to be very small (Figures 7d and 9b), and is expected for semiconducting materials [101]. These results lead to a meaning that the studied Cs2AgCrX6 (X = Cl, Br, I) perovskites may result in relatively larger reflection at the perovskite/electrode interface. For comparison, the refractive indices of CH3NH3PbI3-xClx perovskite thin films were reported approximately to be 2.4 and 2.6 in the visible to near-infrared wavelength region [101]. He et al. have examined bulk single crystals of CH3NH3PbX3 (CH3NH3 = MA, X = Cl, Br, I) and have measured refractive indices that rank in this order: MAPbI3 > MAPbBr3 > MAPbCl3 at the same wavelength [102]. These results indicate that n may play an important role in determining the optical response properties of halide perovskites, which may serve as a useful metric in the parameter space for use in optoelectronic device design.

Conclusions
In this study, a promising van der Waals functional called SCAN + rVV10 was used to theoretically investigate the electronic structures, stability, electronic, transport, and optical properties of A2AgCrBr6 (A = K, Rb, Cs). To shed some light on the effect of halogen substitution (X = Cl, I) that replaces Br in A2AgCrBr6 and for comparison, the same properties of Cs2AgCrX6 (X = Cl, I) obtained with the same functional were included. As such, A2AgCrBr6 (A = K, Rb, Cs) and Cs2AgCrX6 (X = Cl, I) were shown to exhibit a perovskite structure, emerged using the combined application of the octahedral factor, the Goldschmidt tolerance factor, and the global instability index. However, the application of the newly proposed tolerance factor τ has rejected Cs2AgCrI6 and K2AgCrBr6 as stable perovskites, showing its high-level predictability over the traditional ones. Despite the aforementioned rejection, each member of the series A2AgCrBr6 was shown to feature very similar bandgap properties with others, including the nature of the effective masses of charge carriers, and optical constants. Small variation in either of these properties in a given series was due to the small change in the lattice constants, cell volumes, and bonding interactions induced by alkali substitution at the A-site. The positional occurrence of the first onset of absorption which appeared in the spectra of the dielectric function, absorption coefficient, refractive index, and photoconductivity, as well as that in the Tauc plot, has demonstrated that the geometrically stable A2AgCrBr6 double perovskites emanated from this study could be interesting for their eventual experimental characterizations. We have also shown that the onset of the first absorption can be significantly shifted to the high-and low-energy infrared regions via halogen substitution at the X site with X = Cl and X = I in Cs2AgCrX6, respectively. Moreover, the computed electronic transitions of Cs2AgCrCl6 were shown in decent agreement with those reported experimentally for hexagonal Cs2AgCrCl6 and Cr-doped Cs2AgInCl6, meaning that the chosen computational model utilized in this study should be applicable to similar other halide double perovskites to explore similar properties. Because A2AgCrBr6 (A = Rb, Cs) and Cs2AgCrCl6 possess impressive electronic and optical properties, they are expected to find application in optoelectronics.

Conclusions
In this study, a promising van der Waals functional called SCAN + rVV10 was used to theoretically investigate the electronic structures, stability, electronic, transport, and optical properties of A 2 AgCrBr 6 (A = K, Rb, Cs). To shed some light on the effect of halogen substitution (X = Cl, I) that replaces Br in A 2 AgCrBr 6 and for comparison, the same properties of Cs 2 AgCrX 6 (X = Cl, I) obtained with the same functional were included. As such, A 2 AgCrBr 6 (A = K, Rb, Cs) and Cs 2 AgCrX 6 (X = Cl, I) were shown to exhibit a perovskite structure, emerged using the combined application of the octahedral factor, the Goldschmidt tolerance factor, and the global instability index. However, the application of the newly proposed tolerance factor τ has rejected Cs 2 AgCrI 6 and K 2 AgCrBr 6 as stable perovskites, showing its high-level predictability over the traditional ones. Despite the aforementioned rejection, each member of the series A 2 AgCrBr 6 was shown to feature very similar bandgap properties with others, including the nature of the effective masses of charge carriers, and optical constants. Small variation in either of these properties in a given series was due to the small change in the lattice constants, cell volumes, and bonding interactions induced by alkali substitution at the A-site. The positional occurrence of the first onset of absorption which appeared in the spectra of the dielectric function, absorption coefficient, refractive index, and photoconductivity, as well as that in the Tauc plot, has demonstrated that the geometrically stable A 2 AgCrBr 6 double perovskites emanated from this study could be interesting for their eventual experimental characterizations. We have also shown that the onset of the first absorption can be significantly shifted to the high-and low-energy infrared regions via halogen substitution at the X site with X = Cl and X = I in Cs 2 AgCrX 6 , respectively. Moreover, the computed electronic transitions of Cs 2 AgCrCl 6 were shown in decent agreement with those reported experimentally for hexagonal Cs 2 AgCrCl 6 and Cr-doped Cs 2 AgInCl 6 , meaning that the chosen computational model utilized in this study should be applicable to similar other halide double perovskites to explore similar properties. Because A 2 AgCrBr 6 (A = Rb, Cs) and Cs 2 AgCrCl 6 possess impressive electronic and optical properties, they are expected to find application in optoelectronics.
Author Contributions: Conceptualization, problem design, and investigation: P.R.V.; Literature survey and writing-original draft: P.R.V.; Writing-review and editing: P.R.V. The author has read and agreed to the published version of the manuscript.

Funding:
The funders have no role in the conceptualization, design and publication of this work and that this research received no external funding to declare.