Insights on electronic structures, elastic features and optical properties of mixed-valence double perovskites Cs2Au2X6 (X=F, Cl, Br, I)

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Insights on electronic structures, elastic features and optical properties of mixed-valence double perovskites Cs2 Au2 X6 (X=F, Cl, Br, I) Xue Du, Dafang He, Huayue Mei, Yuhan Zhong, Nanpu Cheng ∗ School of Materials and Energy, Southwest University, Chongqing 400715, China

a r t i c l e

i n f o

Article history: Received 2 October 2019 Received in revised form 3 November 2019 Accepted 22 November 2019 Available online xxxx Communicated by R. Wu Keywords: Cs2 Au2 X6 Mixed-valence double perovskites Electronic structures Elastic features Optical properties Density function theory

a b s t r a c t In this work, we investigate the X-dependent crystal, electronic, elastic and optical properties of mixedvalence double perovskites Cs2 Au2 X6 (X=F, Cl, Br, I) based on first principles. The effects of all atomic sites, especially the [AuX2 ]− and [AuX4 ]− clusters with different valences of Au atoms, in the crystal structures of Cs2 Au2 X6 on their electronic and optical properties have been clarified. Meanwhile, Cs2 Au2 X6 double perovskites with direct band gaps have strong absorptions, low loss functions and low reflections, and they promise to be used as photoelectric absorption layers of solar cells. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Metal oxide perovskites (ABO3 ), halide perovskites (ABX3 ) and their double perovskites (A2 BB O6 and A2 BB X6 ) are currently the important areas of novel materials [1–3], and many studies have been carried out on them [4,5]. The Pb-based halide perovskites exhibit excellent optoelectronic properties. However, they are not environmentally friendly and the toxicity of Pb limits their practical applications as absorption layers of solar cells [6]. In recent years, deriving from Pb-based halide perovskites, many researchers have made great efforts to explore non-toxic, stable and efficient perovskite solar cells. A good strategy is to replace bivalent Pb cations by non-toxic monovalent BI and trivalent BIII cations and then to establish A2 BI BIII X6 halide double perovskites [7]. As we know that band gaps are important for photoelectric materials. In A2 BI BIII X6 halide double perovskites, A and BI sites prefer to be occupied by monovalent alkaline metals, Ag+1 , Cu+1 , or Au+1 , and they have little influence on the band gaps [8]. However, BIII site prefers to be occupied by Al3+ , Ga3+ , In3+ , Tl3+ , Ti3+ , Sb3+ , Bi3+ or Au3+ , and it has great influence on the band gaps [8]. It should be pointed out that BI and BIII sites in A2 BI BIII X6 halide double perovskites can be occupied by the same or different elements. For example, from HBr solution containing CsBr, AgBr and BiBr3 , Slavney et al. successfully prepared Cs2 AgBiBr6

*

Corresponding author. E-mail address: [email protected] (N. Cheng).

https://doi.org/10.1016/j.physleta.2019.126169 0375-9601/© 2019 Elsevier B.V. All rights reserved.

double perovskite having an indirect band gap of 1.95 eV [9], with BI and BIII sites occupied by different elements (namely Ag in BI site and Bi in BIII site). Though precipitation from a solution of hydrohalic acid and hypophosphorous acid, McClure et al. synthesized Cs2 AgBiX6 (X=Cl, Br) double perovskites with band gaps of 2.62 and 2.06 eV [10]. Adopting an anti-solvent recrystallization method, Yang et al. obtained Cs2 AgBiX6 (X=Cl, Br, I) double perovskites and found that these semiconductors have characteristic absorption peaks in the visible light range [11]. However, from DFT calculations, Zhang et al. [12] & Han et al. [13] revealed that these double perovskites have poor thermodynamic stability. Volonakis et al. also stated in their works that Cs2 AgBiX6 (X=Cl, Br, I) double perovskites are not suitable for photovoltaic materials due to their poor thermodynamic stability [14]. Then, through substituting Bi by In in Cs2 AgBiX6 (X=Cl, Br, I) double perovskites, Volonakis and coworkers studied the electronic and optical properties of Cs2 AgInX6 (X=Cl, Br, I) double perovskites (BI and BIII sites still occupied by different elements) based on first-principles calculations. Their theoretical results reveal that Cs2 AgInX6 (X=Cl, Br, I) double perovskites are direct band gap semiconductors (with band gaps of ∼2.4, ∼1.7 and ∼1.0 eV for X=Cl, Br and I still lying in the suitable visible light range) and have better optical properties than Cs2 AgBiX6 double perovskites. The Pb-free A2 BI BIII X6 halide double perovskites with BI and BIII sites occupied by the same element in different valences are also important for photoelectric materials. Previously, Wells prepared Cs2 Au2 Cl6 double perovskite though recrystallizing from solution

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of cesium chloride containing aurous chloride at the first time [15]. Then, through Mossbauer spectra study, Stanek found that Au atoms in Cs2 Au2 Cl6 double perovskite have different valences [16]. Kitagawa and coworkers [17,18] prepared Cs2 Au2 X6 (X=Cl, Br) double perovskites from acetonitrile solution by using H-type double test tube of glass and analyzed their P-T phase diagrams in details. They found that pressure can induce phase transition of Cs2 Au2 X6 (X=Cl, Br) double perovskites with the mixed-valence states of Au atoms changing to a single-valence state. Kojima and coworkers theoretically studied the optical properties of Cs2 Au2 X6 (X=Cl, Br, I) double perovskites by analyzing the charge transfer of ions based on group theory [19]. Through the same method used by Kitagawa, Liu et al. experimentally synthesized Cs2 Au2 X6 (X=Cl, Br, I) double perovskites with direct band gaps of 2.04, 1.60 and 1.30 eV [20]. Riggs et al. prepared Cs2 Au2 X6 (X=Br, I) double perovskites by a self-flux method, and they pointed out that Cs2 Au2 X6 (X=Br, I) double perovskites have the distorted perovskite structure in space group of I4/mmm [21]. Debbichi et al. indicated in their work that Cs2 Au2 I6 double perovskite as a direct band gap semiconductor, compared with the MAPbI3 perovskite, has better optical properties [22]. From the experimental and theoretical studies mentioned above, we know that Cs2 Au2 X6 (X=Cl, Br, I) double perovskites possess better band gaps compared with Cs2 AgBiX6 and Cs2 AgInX6 double perovskites. However, there have no comprehensive theoretical studies on Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites, and especially there is no any report on Cs2 Au2 F6 double perovskite. Therefore, we take Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites as examples to investigate their structural, electronic, elastic and optical properties through DFT calculations. We hope that our work can deepen the understanding of physical properties of Cs2 Au2 X6 double perovskites and will be helpful for photoelectric applications of these novel materials.

Fig. 1. Illustration of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

energy is used for integration over the first Brillouin zone. The calculations are finished when the total energy is within 5.0 × 10−6 eV/atom, the maximum force on each atom is less than 0.01 eV/Å, the maximum stress on each atom is below 0.02 GPa, and the maximum displacement of each atom is smaller than 5.0 × 10−4 Å. The adopted valence electron configurations are 6s1 for Cs, 5d10 6s1 for Au, 2s2 2p 5 for F, 3s2 3p 5 for Cl, 4s2 4p 5 for Br and 5s2 5p 5 for I. 3. Results and discussions 3.1. Crystal structures The unit cell of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites with the same space group of I4/mmm (Z=2; No. 139) belongs to the tetragonal system and is shown in Fig. 1. Where Cs(4d) , X(4e) and X(8h) respectively locate at (0, 0.5, 0.25), (0, 0, 0.29) and (0.22, 0.22, 0) sites. The two kinds of gold atoms, namely Au(2a) and Au(2b) , respectively locate at (0, 0, 0) and (0, 0, 0.5) sites. The lattice parameters of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites after fully geometrical relaxation, together with the experimental values, are summarized in Table 1. The obtained lattice constants of Cs2 Au2 X6 double perovskites increase with the increasing atomic number of X and coincide well with the experimental ones, meaning that the current computational method is reliable. Hence, the optimized crystal structures of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are then used to calculate their electronic, elastic and optical properties. In order to determine the crystal stability of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites, we calculate the binding energies (E b ) and formation energies (E f ) through Eqs. (1) and (2).

2. Theoretical methods We perform first principles calculations of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites through the Cambridge Sequential Total Energy Package (CASTEP) in Material Studio software [23]. During the calculations of crystal structures, elastic properties and Hirschfeld charge, the generalized gradient approximation (GGA) proposed by Perdew-Burke-Ernzerhof (PBE) [24] is used to deal with the exchange-correlation term, and the ultra-soft pseudopotential with a plane wave energy cutoff of 360 eV is utilized to treat the electron-ion interactions. Because the GGA-PBE level often underestimates band gaps of semiconductors [25] due to the discontinuity of exchange correlation energy, the hybrid functional HSE06 [26] is adopted to calculate the electronic structures and optical properties of Cs2 Au2 X6 double perovskites. Correspondingly, the norm-conserving pseudopotential [27] with a plane wave energy cutoff of 550 eV is applied to cope with the electron-ion interactions in the Hamilton system at this time. During the calculations, a 6 × 6 × 7 Γ -centered MonkhorstPack [28] k-point grid based on convergence testing of the lattice

E b = ( E Cs2 Au2 X6 − nCs × μCs − nAu × μAu − nX × μX )/10

(1)

E f = ( E Cs2 Au2 X6 − nCs × E Cs − nAu × E Au − nX × E X )/10

(2)

where, E Cs2 Au2 X6 is the total energy of primitive cells, E Cs , E Au and E X are respectively the energies of Cs, Au and X atoms in simple substances, n is the number of atoms, μ is the chemical potential of single atom. From Table 1, both the calculated binding energies

Table 1 Lattice parameters, binding energies (E b ) and formation energies (E f ) of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites. a=b

Cs2 Au2 F6 Cs2 Au2 Cl6 666 Cs2 Au2 Br6 Cs2 Au2 I6 a b c

Ref. [29]. Ref. [30]. Ref. [31].

c

Exp.

This work

Exp.

This work

– 7.4951a 7.7593b 8.2841c

6.6871 7.5962 7.9426 8.4184

– 10.8802a 11.3079b 12.092c

9.8405 11.3089 11.8048 12.0534

Eb

Ef

−3.6176 −2.9909 −2.7067 −2.4473

−2.6709 −1.0520 −1.4110 −0.7353

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Table 2 The calculated elastic constants of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

C 11 C 12 C 13 C 33 C 44 C 66

Cs2 Au2 F6

Cs2 Au2 Cl6

Cs2 Au2 Br6

Cs2 Au2 I6

25.745 9.257 7.088 27.835 5.668 11.734

18.923 10.908 3.075 22.387 2.250 5.175

18.429 11.579 2.255 20.179 2.546 4.765

32.738 18.850 6.264 43.894 4.294 18.745

and formation energies of Cs2 Au2 X6 double perovskites are negative and increase with the atomic number of X increasing. This indicates that these double perovskites can be synthesized and their stabilities become weak in the order of F, Cl, Br, and I. 3.2. Elastic properties Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites belonging to the tetragonal system have six independent elastic constants of C 11 , C 12 , C 13 , C 33 , C 44 and C 66 (Table 2), and their mechanical stabilities can be determined through the criteria [32] in Eqs. (3) and (4).

C ii > 0 (i = 1, 3, 4, 6),

(C 11 + C 33 − 2C 12 ) > 0 (C 11 − C 12 ) > 0,   (2C 11 + C 12 ) + C 33 − 4C 13 > 0

(3) (4)

As is well known, the bulk modulus (B) and shear modulus (G) are important parameters of crystals to depict their abilities of resisting compression and shear deformations. In view of the Voigt and Ruess models [33], the bulk and shear moduli of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites can be obtained by Eqs. (5)-(8) [34]. The Voigt and Ruess models [34] respectively correspond to the upper and lower limits of moduli, while results in Eqs. (9) and (10) from the Voigt-Ruess-Hill model are much closer to experimental values. The calculated B and G values of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are grouped in Table 3. It is clear that B and G values of Cs2 Au2 I6 double perovskite are the largest while those of Cs2 Au2 Br6 double perovskite are the smallest. It means that Cs2 Au2 I6 (Cs2 Au2 Br6 ) double perovskite has the strongest (weakest) capacities of resisting compression and shear deformations.

3

Table 3 The calculated shear moduli (G, N/m), Young’s moduli (E, N/m) and Poisson’s ratios (ν ) of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

BV BR B GV GR G B/G E

ν AU

Cs2 Au2 F6

Cs2 Au2 Cl6

Cs2 Au2 Br6

Cs2 Au2 I6

14.021 14.021 14.021 8.340 7.626 7.983 1.756 15.260 0.261 0.468

10.483 10.415 10.449 4.813 3.500 4.156 2.514 8.920 0.325 1.883

9.913 9.705 9.809 4.701 3.604 4.152 2.362 8.752 0.315 1.543

19.125 19.120 19.123 10.666 6.987 8.827 2.166 18.118 0.300 2.633

The B /G ratio is an important factor for judging the ductility and brittleness of materials [35]. From Table 3, we find that all the B /G values of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are slightly larger than the critical value of 1.75. It means that, except for Cs2 Au2 F6 double perovskite which lies in the critical point between brittleness and ductility, other three double perovskites are ductile and their ductilities increase as the atomic number of X increases. The elastic anisotropy ( A U ) obtained from elastic constants of crystals (see Eq. (13)) is also an important mechanical property factor [36]. A crystal is isotropic when its A U is 0, while it is anisotropic when its A U is larger than 0. All the obtained anisotropic factors of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites in Table 3 are greater than 0.4, indicating that all these crystals have obvious anisotropy. Meanwhile, Cs2 Au2 I6 double perovskite has the strongest anisotropy while Cs2 Au2 F6 double perovskite has the weakest anisotropy. The calculated three dimensional and two dimensional projections of Young’s moduli for Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are shown in Fig. 2. The deviation degree of the three dimensional Young’s moduli from sphere can reflect the anisotropy degree of crystals. Fig. 2 reveals that Cs2 Au2 X6 double perovskites own strong elastic anisotropies, and that Cs2 Au2 I6 double perovskite has the strongest anisotropy, which is coincident with the anisotropic factor ( A U ) of Cs2 Au2 I6 double perovskite. From the two dimensional projections of Young’s moduli for Cs2 Au2 X6 double perovskites in Fig. 2, it can be seen that, different from (001) plane, (010) and (100) planes have the same elastic anisotropy.

B V = (2C 11 + 2C 12 + 4C 13 + C 33 )/9

(5)

3.3. Electronic structures

2 /(C 11 + C 12 − 4C 13 + 2C 33 ) B R = (C 11+ C 12 )C 33 − 2C 33

(6)

G V = (2C 11 + C 33 − 2C 13 − C 12 + 6C 44 + 3C 66 )/15

(7)

Fig. 3 shows the band structures and densities of states of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites, in which the Fermi surface is set at zero. The theoretical band gaps together with the experimental ones are listed in Table 4. The band structures along the high symmetry points in the first Brillouin zone in Fig. 3 indicate that all Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are direct band gap semiconductors. In Table 4, the theoretical band gaps of Cs2 Au2 X6 (X=Cl, Br, I) double perovskites calculated by HSE06 are very close to their experimental ones, and they decrease with the increasing atomic number of X due to enhancement of the nearest-neighbor coulomb repulsion and Jahn-Teller distortion of [AuX4 ]− clusters [20]. So we can infer that the theoretical band gap of Cs2 Au2 F6 double perovskite is also reliable, although there is no experimental result. Optical properties of semiconductors strongly depend on their band gaps. If some a semiconductor is used as visible light absorber, its band gap is expected to lie in the range of ∼1.62-3.10 eV. More specifically, semiconductors with band gaps lying in the range of ∼0.8-2.2 eV will have much wider applications [37]. The band gaps of Cs2 Au2 X6 double perovskites listed in Table 4 are in











2 + 6/(C 11 − C 12 ) G R = 18B V / (C 11 + C 12 )C 33 − 2C 13

+ 6/C 44 + 3/C 66

− 1

(8)

B = ( B V + B G )/2

(9)

G = (G V + G R )/2

(10)

The Young’s moduli (E), Poisson ratios (ν ) and anisotropic factors ( A U ) of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites in Table 3 are respectively obtained from Eqs. (11)-(13). From Table 3, Cs2 Au2 I6 double perovskite has the biggest Young’s modulus (E) and anisotropic factor ( A U ), and Cs2 Au2 Cl6 double perovskite has the biggest Poisson ratio (ν ).

E = 9B G /3( B + G )

(11)

ν = (3B − 2G )/(6B + 2G )

(12)

U

A = 5G V /G R + B V / B R − 6 ≥ 0

(13)

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Fig. 2. The calculated Young’s moduli of Cs2 Au2 X6 double perovskites in order of X=F, Cl, Br and I (a-d) three dimensional images and (a -d ) two dimensional projections.

to the similar crystal structures. The valance bands from −4-0 eV are mainly contributed by X-p and Au-5d states, and the bottom conduction bands are predominately determined by X(8h) -p and Au-5d states. X-s, Au-6s/6p and Cs-6s/6p states only play roles in the higher energy region far away from the Fermi surface. Simultaneously, with the increasing atomic number of X, the increased nearest-neighbor coulomb repulsion and Jahn-Teller distortion of [AuX4 ]− clusters cause X-p and Au-5d/6s/6p states to shift toward the Fermi surface, as explained in the band structures in Fig. 3. 3.4. Hirschfeld charge

Fig. 3. The calculated band structures of Cs2 Au2 X6 double perovskites (a) Cs2 Au2 F6 , (b) Cs2 Au2 Cl6 , (c) Cs2 Au2 Br6 and (d) Cs2 Au2 I6 .

Table 4 The calculated and experimental band gaps of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

HSE06 Exp. [20]

Cs2 Au2 F6

Cs2 Au2 Cl6

Cs2 Au2 Br6

Cs2 Au2 I6

2.410 –

2.008 2.040

1.609 1.600

1.371 1.310

the range of ∼1.3-2.5 eV (also in the visible light range), revealing that Cs2 Au2 X6 double perovskites possibly have wide optoelectronic applications. Here, we should point out that although the band gap of Cs2 Au2 F6 double perovskite is slightly larger than 2.2 eV, the direct band gap promises its potential applications in optical devices. Fig. 4 displays the densities of states of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites in the energy range of −4-12 eV. It is significant that their densities of states are similar to each other due

In order to understand the nature of atomic bonds in Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites, the electron occupancy numbers, bond lengths and Hirshfeld charges are listed in Table 5. From Table 5, one can see that, in Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites, Cs and Au atoms lose electrons, while X atoms get electrons. As the atomic number of X increases, X atoms have gradually weakened ability to get electrons, while Cs and Au atoms own weakened ability to lose electrons. In each of Cs2 Au2 X6 double perovskites, X(4e) atoms possess stronger ability of getting electrons than X(8h) atoms, while Au(2a) atoms have stronger ability of losing electrons than Au(2b) . We also clearly see that the bond lengths of X(4e) -Au(2b) and X(8h) -Au(2a) bonds in each of Cs2 Au2 X6 double perovskites are relatively shorter than those of the rest ones, and that all the bond lengths in Cs2 Au2 X6 double perovskites increase with the atomic number of X increasing. In order to further figure out the charge transfers and bonding rules, we analyze the charge densities in (100) and (112) planes of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites in 2 × 2 × 2 supercells. It can be seen from Fig. 5 that the electronic clouds between Au(2b) and X(4e) ((Au(2a) and X(8h) ) atoms are slightly overlapped, indicating that X(4e) and Au(2b) (Au(2a) and X(8h) ) atoms mainly form X(4e) -Au(2b) (X(8h) -Au(2a) ) ionic bonds with small covalent components. Cs and X atoms completely form ionic bonds. For the halogen atoms, with their atomic numbers increasing, their abilities of getting electrons gradually reduce due to the decreased electronegativities.

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Fig. 4. Total and partial densities of states of Cs2 Au2 X6 double perovskites (a) Cs2 Au2 F6 , (b) Cs2 Au2 Cl6 , (c) Cs2 Au2 Br6 and (d) Cs2 Au2 I6 .

Table 5 The calculated electron occupancy numbers, bond lengths and Hirschfeld charges of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites. Atom

s

p

d

Total

Hirshfeld charge (e)

Bond

L (Å)

Cs2 Au2 F6

Cs(4d) Au(2a) Au(2b) F(4e) F(8h)

2.13 0.72 0.83 1.97 1.97

5.76 0.47 0.37 5.53 5.59

0.00 9.18 9.51 0.00 0.00

7.89 10.38 10.71 7.50 7.56

0.28 0.55 0.38 −0.27 −0.24

F(4e) -Cs(4d) F(4e) -Au(2b) F(8h) -Cs(4d) F(4e) -Au(2a) F(8h) -Au(2b) F(8h) -Au(2a)

3.366 2.072 3.425 2.848 2.659 2.069

Cs2 Au2 Cl6

Cs(4d) Au(2a) Au(2b) Cl(4e) Cl(8h)

2.07 0.86 0.93 1.95 1.85

5.74 0.74 0.58 5.47 5.38

0.00 9.42 9.64 0.00 0.00

7.78 11.02 11.16 7.42 7.34

0.19 0.35 0.18 −0.18 −0.14

Cl(4e) -Cs(4d) Cl(4e) -Au(2b) Cl(8h) -Cs(4d) Cl(4e) -Au(2a) Cl(8h) -Au(2b) Cl(8h) -Au(2a)

3.825 2.340 3.911 3.322 2.980 2.383

Cs2 Au2 Br6

Cs(4d) Au(2a) Au(2b) Br(4e) Br(8h)

2.17 1.01 1.03 1.79 1.70

6.26 0.92 0.73 5.37 5.29

0.00 9.51 9.67 0.00 0.00

8.43 11.44 11.43 7.16 6.99

0.16 0.27 0.16 −0.15 −0.12

Br(4e) -Cs(4d) Br(4e) -Au(2b) Br(8h) -Cs(4d) Br(4e) -Au(2a) Br(8h) -Au(2b) Br(8h) -Au(2a)

3.994 2.476 4.079 3.424 3.075 2.533

Cs2 Au2 I6

Cs(4d) Au(2a) Au(2b) I(4e) I(8h)

2.16 1.11 1.10 1.79 1.70

6.27 0.84 1.04 5.27 5.19

0.00 9.74 9.61 0.00 0.00

8.43 11.69 11.76 7.06 6.89

0.15 0.19 0.11 −0.12 −0.09

I(4e) -Cs(4d) I(4e) -Au(2b) I(8h) -Cs(4d) I(4e) -Au(2a) I(8h) -Au(2b) I(8h) -Au(2a)

4.331 2.723 4.256 3.260 3.588 2.652

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Fig. 5. The calculated charge densities in (100) and (112) planes of Cs2 Au2 X6 double perovskites (a) Cs2 Au2 F6 , (b) Cs2 Au2 Cl6 , (c) Cs2 Au2 Br6 and (d) Cs2 Au2 I6 . (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 6. The calculated (a) real part

ε1 (ω) and (b) imaginary part ε2 (ω) of dielectric functions for Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

3.5. Optical properties Optical properties of solids are dependent on their band gaps and directly described by the complicated dielectric function ε(ω) = ε1 (ω) + iε2 (ω). The imaginary part ε2 (ω) describes the electron transitions from occupied states to unoccupied states while light passing through a medium, and it can be obtained by the momentum matrix of electronic transitions [38,39]. As for the real part ε1 (ω), it usually depicts the polarization degree of medium under an external electric field and can be obtained through the Kramers-Krong relationship [40]. Further, in ε2 (ω), electronic transitions include inter-band transitions and inner-band transitions [41]. Here, inter-band transitions are mainly present in metals while inner-band transitions occur primarily in semiconductors. Inner-band transitions can be further

divided into direct and indirect transitions. Moreover, other optical properties such as the refractive index, absorption coefficient, loss function and reflectivity are derived from the dielectric functions of solids [38]. As regards to Cs2 Au2 X6 double perovskites, they are direct semiconductors (see Fig. 3), and their complicated dielectric functions in the energy scale of 0-9 eV are shown in Fig. 6. From Fig. 6 (a), we can see that the calculated static dielectric constants ε1 (0) of Cs2 Au2 X6 double perovskites are respectively 2.39, 3.17, 4.21 and 5.75 in the order of X=F, Cl, Br and I. There are respectively two main peaks in the energy ranges of ∼0.5-3 eV and ∼4.5-9 eV for each of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites. Macroscopically, the first main peaks are composed of single peaks while the second main peaks consist of several superimposed small peaks. The peak values of the first main peaks

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Fig. 7. The calculated (a) refractive index n(ω) and (b) extinction coefficient k(ω) of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

Fig. 8. The calculated (a) absorption coefficients, (b) loss functions and (c) reflectivities of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites.

of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites lie at around 2.01, 1.76, 1.34 and 0.90 eV, and those of the second main peaks are situated at around 7.42, 6.11, 5.54 and 5.88 eV. Meanwhile, the second main peaks show the same trend as the first main peaks. As the atomic number of X increases, both the first and second main peaks shift toward the low energy region with their intensities becoming stronger. In Fig. 6 (b), macroscopically, there are also two main peaks in ε2 (ω) for each of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites. The first main peaks appear in the energy range lower than ∼4.5 eV with their peak positions respectively locating at 2.71, 2.55, 2.56 and 2.03 eV, and the second main peaks locate in the energy range higher than ∼4.5 eV with their peak positions respectively seated at 8.51, 8.12, 6.42 and 7.42 eV. Similar to those in ε1 (ω), both the first and second main peaks in ε2 (ω) also increase and shift toward the low energy region as the atomic number of X increases. It should be pointed out that the peak positions in ε2 (ω) correspond to the inflection points in ε1 (ω). The first main peaks locating in the visible light range are formed by the electronic transitions from the top valence bands to the bottom conduction bands, while the second main peaks in the higher energy range are composed of electronic transitions from the top valence bands to the conduction bands far away from the Fermi surface. For practical application, we are only concerned with the first main peaks locating in the visible light range. The complex refractive indexes of solids can be described by n˜ (ω) = n(ω) + ik(ω), where n(ω) is refractive index and k(ω) is extinction coefficient. In Fig. 7, compared with Fig. 6, we can see that n(ω) and k(ω) respectively have the similar trends as ε1 (ω) and ε2 (ω). From Fig. 7 (a), the calculated static refractive indexes n(0) are respectively 1.55, 1.78, 2.05 and 2.41 for X=F, Cl, Br and I. It is clear that n(ω) values of Cs2 Au2 X6 perovskites gradually increase in the infrared light region. And, after respectively reaching their maximum values of 1.68, 1.96, 2.23 and 2.49 at the incident photon

energies of 2.04, 1.82, 1.42 and 0.99 eV in the visible light range, they decrease quickly to get their minimum values in the energy range of ∼3.2-5.0 eV. We can clearly see that these characteristic peaks shift toward the low energy region and their strengths become stronger as the atomic number of X increases. Absorption coefficient is an important indicator of photoelectric materials, and its initial value corresponds to the band gap determined by electronic transitions from top valence bands to bottom conduction bands. The photon energy-dependent absorption coefficients of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites shown in Fig. 8 (a) reveal that they have the similar trends due to their similar electronic structures. The first characteristic peaks of absorption coefficients for Cs2 Au2 X6 double perovskites locate in the visible light region, and they are mainly derived from the electron transitions from Au(2b) -5d and X(4e) -p states to Au(2a) -5d states [19]. These peaks all shift toward the low energy region and their strengths increase with the increasing atomic number of X. In the range of ∼4.5-9 eV, the absorption coefficient for each of Cs2 Au2 X6 double perovskites increases rapidly and forms the second main peak derived from intramolecular transitions, namely from Au(2a) -5d (Au(2b) -5d) states to X(8h) -s(X(4e) -s) states in [AuX2 ]− and [AuX4 ]− clusters [22]. However, these electronic transitions far away from the Fermi surface are relatively difficult, and thus the second main peak has no realistic significance. Fortunately, the first main peak is wide enough to cover almost the whole visible light region. That’s to say, Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites can absorb light in a wide energy range. The loss functions L (ω), describing the decay of light passing through a homogeneous medium, of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are shown in Fig. 8 (b). It indicates that loss functions for Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites are very small with their maximum value of ∼0.2 in the visible light range. Fig. 8 (c) displays the reflectivities R (ω) of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites. The calculated static reflection coeffi-

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cients R(0) for Cs2 Au2 X6 double perovskites are respectively 0.046, 0.079, 0.119 and 0.169. Obviously, the phonon energy-dependent reflectivities R (ω) of Cs2 Au2 X6 double perovskites increase with the increasing atomic number of X. However, their values are very small in the observed energy region, and even the maximum values of Cs2 Au2 I6 in the visible and ultraviolet regions are respectively ∼0.191 and ∼0.24. We should point out that the peaks of loss functions correspond to the inflection points of reflectivities. 4. Conclusion In this work, we have built theoretical models of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites based on experimental results, and mainly explored their electronic structures, elastic features and optical properties. In Cs2 Au2 X6 double perovskites, X(4e) and Au(2b) (X(8h) and Au(2a) ) atoms form X(4e) -Au(2b) (X(8h) -Au(2a) ) ionic bonds with small covalent components. Cs and X atoms completely form ionic bonds. The bond lengths and lattice parameters in Cs2 Au2 X6 double perovskites increase with the increasing atomic radios and the decreasing electronegativities of X. Cs2 Au2 I6 (Cs2 Au2 Br6 ) double perovskite has the strongest (weakest) capacities of resisting compressive deformation and shear deformation. Cs2 Au2 F6 double perovskite is between brittleness and ductility, and Cs2 Au2 X6 (X=Cl, Br, I) double perovskites are ductile. Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites have obvious elastic anisotropy. Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites have similar electronic structures because of their similar crystal structures, and they are direct band gap semiconductors with band gaps of 2.410, 2.008, 1.609 and 1.371 eV, which are close to experimental values. The top portions of valance bands are mainly contributed by X-p and Au-5d states; the bottom conduction bands are predominately determined by X(8h) -p and Au-5d states; the conduction bands far away from the Fermi surface are dominated by X-s, Au-6s/6p and Cs-6s/6p states. In the visible light range, the optical properties of Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites decided by electron transitions from Au(2b) -5d and X(4e) -p states to Au(2a) -5d states. With the atomic number of X increasing, all of the absorptions, loss functions and reflectivities of Cs2 Au2 X6 shift toward the low energy region with their intensities becoming stronger. Cs2 Au2 X6 (X=F, Cl, Br, I) double perovskites can be used as potential photoelectric absorption layers of solar cells owning to their strong absorptions, low loss functions and low reflectivities. Declaration of competing interest The authors declare no conflict of interest. Acknowledgements The current work was supported by the National Natural Science Foundation of China (Grant No. 51171156) and Chongqing Science & Technology Commission (Grant No. cstc2018jcyjax0582). References [1] S.A. Dar, V. Srivastava, U. Kumar Sakalle, V. Parey, G. Pagare, A combined DFT, DFT + U and mBJ investigation on electronic structure, magnetic, mechanical and thermodynamics of double perovskite Ba2 ZnOsO6 , Mater. Sci. Eng. B 236–237 (2018) 217–224. [2] S.A. Dar, R. Sharma, V. Srivastava, U.K. Sakalle, Investigation on the electronic structure, optical, elastic, mechanical, thermodynamic and thermoelectric properties of wide band gap semiconductor double perovskite Ba2 InTaO6 , RSC Adv. 9 (2019) 9522–9532. [3] S.A. Dar, V. Srivastava, U.K. Sakalle, V. Parey, Electronic structure, magnetic, mechanical and thermo-physical behavior of double perovskite Ba2 MgOsO6 , Eur. Phys. J. Plus 133 (2018) 64.

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