The study of structural, elastic, electronic and optical properties of CsY_x I_{(1 − x)}(Y = F, Cl, Br) using density functional theory

To obtain the structural parameters for all concentrations (0 ⩽ x ⩽1), optimization of CsY_xI_{1 − x}(Y = F, Cl and Br) alloys at the ground state energy was carried out using Birch Murnaghan’s equation of state [29]. As an example, the optimized structures of CsBr and CsBr_0.25I_0.75 are given in Fig. 1.

The variation in the unit cell volume optimizes the unit cell energy to the ground state energy of the system as shown in Fig. 2.

Lighter halogens (F, Cl and Br) were doped in CsI. The calculated structural properties such as the lattice constant (a_o), bulk modulus (K), and energy at equilibrium state (E_o) for both B₁ and B₂ phases are shown in Table 1 and Table 2.

As can be seen, the lattice constant exhibits inverse relationship with the dopant concentration. The bulk modulus (K) shows a random variation with increasing doping concentration (x) of halogens. However, for each doping element, the bulk modulus is smaller at x = 0 and greater at x = 1. In addition, the lattice constant (a₀) and bulk modulus (K) of the materials are in fair agreement with experimental data. Variations of the lattice constant (a₀) and bulk modulus (K) are also plotted vs. concentration (x). Bulk modulus (K) is proportional to crystal rigidity i.e. the larger its value the more rigid is the crystal.

Elastic properties of a material can better explain the behavior of a solid under an applied stress. The stability of a material and its stiffness purely depend on elastic properties. These fundamental properties of a solid can be calculated provided we have stress tensor parameters. Knowing the elastic constants, one can calculate bulk modulus K, Young’s modulus E, Poisson’s ratio σ, shear modulus G, anisotropy factor A and other physical properties. To describe mechanical properties of cubic compounds CsY (Y = F, Cl, Br and I) we only need to calculate three elastic constants C₁₁, C₁₂ and C₄₄.

The elastic and structural parameters of alkali halides are listed in Table 3. It can be seen that the materials show a mix behavior of properties of cubic compounds in both phases. Binary compounds of Cs halides exist in CsCl structure except CsF which exists in NaCl structure at normal conditions. The available experimental and theoretical data concern only the normal phases, in general. We have also calculated the elastic properties of the compounds in other phases i.e. CsCl, CsBr and CsI in FCC structural phase and CsF in BCC phase. Elastically stable cubic structures must satisfy the following relations [30]: (C11/3+2/3C12)><!-- > -->0(C11−<!-- - -->C12)/2><!-- ? -->0andC44><!-- ? -->0;C12<<!-- ? -->K<<!-- ? -->C11 $$ \begin{array}{} (C_{11}/3+2/3C_{12}) \gt 0(C_{11}-C_{12})/2\gt 0\\ \text{and}\,\, C_{44} \gt 0;\, C_{12} \lt K \lt C_{11} \end{array} $$ (1)Table 3

The Elastic constants and mechanical properties of alkali halides.

The data presented in the Table do not show isotropic behavior which means that the structures are anisotropic [48]. Following equations are used to compute the elastic properties E, G, A, K and σ, etc.: K=C11+2C123 $$ K=\frac{C_{11}+2C_{12}}{3} $$ (2)E=9GK3K+G $$ E=\frac{9GK}{3K+G} $$ (3)σ<!-- s -->=3K−<!-- - -->2G2(3K+G) $$ \sigma=\frac{3K-2G}{2(3K+G)} $$ (4)A=2C44C11−<!-- - -->C12 $$ A=\frac{2C_{44}}{C_{11}-C_{12}} $$ (5)

Young’s modulus (E) tells us about the stiffness of materials. The greater the value of (E), the stiffer is the material. From Table 3, it is evident that CsF in B₁ phase is stiffer (E = 50.149) as compared to other compounds in both phases. Using elastic constants C_ij, Cauchy’s pressure C₁₂–C₄₄, the ratio (K/G) Pugh’s index of ductility and Poisson’s ratio σ we obtain a good prediction of stiffness and ductility. The negative value of Cauchy’s pressure represents brittleness, while the positive means that the material is ductile [31]. From Table 3, only CsF in B₂ phase has positive Cauchy’s value, while it is negative for all other compounds. Similarly, the higher the ratio K/G, the more ductile is the material and vice versa. The critical value that separates ductility from brittleness is 1.75 [30]. CsF in B₂ phase and CsBr in B₁ phase have this ratio greater than the critical value while for other compounds it is smaller. Poisson’s ratio σ is another method to calculate brittleness and ductility of a compound. For a compound to be ductile, the value of its Poisson’s ratio must be greater than 0.26, while the one smaller than this value represents the brittle nature [48]. The CsBr in B₁ phase, and CsF in B₂ phase have σ value larger than 0.26 which shows their ductile nature.

The energies difference between valence and conduction band determines the electronic properties of materials. Band structures of all binary and their ternary alloys, calculated using two different methods for the compounds in our study; Wu-Cohen GGA and Engel-Vasco (EV) GGA using 1000 k-points grid during DFT simulation, are presented in Fig. 4 to Fig. 7.

Fig. 4 and Fig. 5 represent the energy band structures of binary CsBr in B₁ and B₂ phases, while that of ternary alloy CsBr_0.25I_0.75 are shown in Fig. 6 and Fig. 7.

The calculated band gap values for binary compounds and their ternary alloys in B₁ phase are tabulated in Table 4, while for B₂ phase are presented in Table 5 along with experimental and other theoretically calculated values. Most of the band structures of the binary and ternary alloys are direct band gap. The results obtained using Wu-Cohen GGA and EV-GGA for binary compounds reveal that the band gap (E_g) is smaller for rock salt phase of cesium halides than B₂ phase, while for CsCl band gap in B₁ phase is larger than B₂ only in EV-GGA scheme. Similarly, the band gap calculated for binary cesium halides using Wu-Cohen GGA is smaller than EV-GGA except for CsI (B₁ phase). Band gap is larger as EV-GGA method has been used for B₂ phase. This increase is also observed in the case of all ternary alloys in both phases calculated using EV-GGA method. Furthermore, the increase in band gap is proportional to the dopant concentration in all cases and phases.

Density of states (DOS) is an important tool to determine different electronic states, dielectric function, reflectivity and band structure of solids.

In Fig. 8, total and partial density of states are given at all concentrations (0 ⩽ x ⩽ 1) and in both phases of CsY_x I _{(1 − x)} (Y = F, Cl, Br) using Wu-Cohen GGA scheme. In the figure of binary compounds and ternary alloys of B₁ and B₂ phases, mixed d-states of all the elements contribute in a very small proportion in conduction band, whereas far region of conduction band is upheld by Cs f-state with a very minute contribution. Fluorine p-state gives a separate peak in all compounds and in both phases (binary as well as ternary) in the valence band near Fermi level. The second sharp peak in the valence band region of CsCl and CsBr corresponds to the halogen p-state, excluding fluorine. The third peak at the lower energy side of valence band is purely the contribution of Cs p-state. In the density of states plot, overall behavior of all compounds is the same for both phases shown in Fig. 8. This behavior is attributed to the overlapping of energy states to form a ternary alloy.

To assess the probability of identifying the specific location of electrons and to obtain information about the nature of bonds among the atoms of a molecule, electron density calculations are important. In the present work, the electron density has been calculated in CsY_xI_{1 − x}(Y = Br, F and Cl) in rock salt phase as well as CsCl structure using the generalized gradient approximation (GGA) method. The bonding nature is ionic in which charge transfer between the anion and cation occurs.

The electron density plots for CsY_xI_{1 − x} at different concentrations are plotted in Fig. 9 in (1 1 0) plane for B₁ phase and Fig. 10 in (0 1 1) plane for B₂ phase.

The first column in both figures represents CsI with increasing concentration (x = 0.25, 0.50, 0.75 and 1) of halogens in the order of F, Cl and Br, respectively. From the plots of these compounds it is revealed that the boding nature is mostly ionic since there is a strong localization of the charge around the anion side. The contour plots are expressed in the units of electrons per unit cell. It is observed that doping of lighter halogens produced slightly covalent character in binary compounds which are pure ionic in nature. In B₁ phase of CsF, the covalent character is strong as compared to other binary Cs halides because fluorine is the smallest halogen atom while the ternary alloys, CsF_.75I_.25 and CsCl_.75I_.25 show the most covalent character out of all in B₁ phase. The rest of all binary compounds and ternary alloys are ionic in both phases.

The dielectric functions for CsY_xI_{1 − x} in the energy range of 0 eV to 40 eV are presented in Fig. 11 and Fig. 12 showing the real ε₁(ω) and imaginary ε₂(ω) parts for all concentrations in B₁ and B₂ structure.

In the left panel of Fig. 11 and Fig. 12, ε₁ (ω) is the real part while in the right panel of the figures, imaginary part ε₂(ω) is shown. The positive value of the real part at low frequency is the static part of the dielectric constant. After reaching the maximum value, the dielectric constant decreases with frequency and some peaks with negative values appear. This corresponds to the metallic region of materials that exhibit negative values of ε₁(ω). From Fig. 12 it is noted that different peaks appear in the spectra of the compounds and the alloys. The transition of electrons from the occupied states of the valence band to the unoccupied states of the conduction band causes the appearance of the peaks.

In CsF_xI_{1 − x} with B₁ phase at x = 0 the real part ε₁(ω) is positive for all energy ranges but is negative in the energy ranges of 7.52 eV to 8.79 eV and 14.12 eV to 17.47 eV which gives maximum peak 5.31 at energy 4.74 eV. The imaginary part ε₂(ω) indicates two peaks: the maximum peak 5.30 at energy 7.28 eV and second 4.28 at 5.97 eV. As the concentration of lighter halogens was increased, the maximum peak decreases in both real and imaginary parts of dielectric function. This reflects that the behavior of the material has shifted from lower to higher energy values as the dopant concentration is increased. This is observed in both real and imaginary parts of dielectric function. Reflectivity is the ratio of the reflected power to the incident power of the material. The reflectivity (R) is plotted for CsY_xI_{1 − x} for concentrations x (0 ⩽ x ⩽ 1) as shown in the Fig. 13.

Binary CsI (B₁) reflectivity is maximum in energy range from 5 eV to 10 eV (0.29) and from 15 eV to 20 eV (0.38) and it decreases at high energy. Fluorine doped CsI peak diminishes with increasing concentration x = from 0 to 1. Furthermore, no significant peak changes are observed in the range of 15 eV to 20 eV. By chlorine doping into CsI, the peak reduces in a similar way with increasing concentration while slight increase is observed in the range of 15 eV to 20 eV. The reflectivity at x = 0.25, 0.50, 0.75 and 1 is given as 0.33, 0.34, 0.37 and 0.47, respectively. Slight peaks change from 0.28 to 0.23 is observed in bromine doping in the range of 5 eV to 10 eV. The observed changes are very clear in the second set of peaks from 0.29 to 0.37 with bromine doping for x equal to 0, 0.25, 0.5, 0.75, and 1.0.

In B₂ structure of CsI, maximum reflectivity (0.38) is found in the energy range from 5 eV to 10 eV and the second set of two peaks with reflectivity of 0.35 is located in 15 eV to 20 eV energy range. Fluorine doping with x = 0.25, 0.50, 0.75 and 1 in the range of 15 eV to 20 eV changes the peaks to 0.39, 0.41, 0.45 and 0.48, respectively, whereas the peak in the range of 5 eV to 10 eV decreases from 0.37 to 0.17. For chlorine doping, two peaks in the energy range 5 eV to 10 eV are evident which shift from 0.28 and 0.30 at x = 0.25 to 0.24 and 0.25 at x = 1 (CsCl). Two more intense peaks in the energy range of 15 eV to 20 eV change from 0.39 and 0.35 at x = 0.25 to 0.49 and 0.39 at x = 1. These clearly indicate that increasing dopant concentration results in larger peaks but at higher photon energies. Finally, in case of bromine, two peaks in the range of 5 eV to 10 eV decrease with increasing dopant concentration from 0.29 and 0.32 at x = 0.25 to 0.25 and 0.29 at x = 1. The peaks observed in the range of 15 eV to 20 eV increase from 0.39 and 0.36 at x = 0.25 to 0.44 and 0.39 at x = 1.

The electron transportation in a material can be observed by optical conductivity (σ). In Fig. 14, optical conductivity of CsY_xI_{1 − x} at different concentrations for both B₁ and B2 structures is presented.

The conductivity changes in the range of 4 eV to 40 eV for all compounds with different concentrations. It is observed that for BCC phase optical conductivity shifts to higher energy with increasing concentration, whereas in FCC structure, one of the two peaks gets smaller.

In binary CsI (B₁), two peaks of 5080 Ω^–1⋅cm^–1 and 5577 Ω⁻¹⋅cm⁻¹ are located, respectively, at 7.26 eV and 13.89 eV. Finally, the conductivity decreases and approaches to zero at 39 eV. The decreasing behavior in the peak values of optical conductivity is observed when the halogens are doped at concentrations x = 0.25, 0.5, 0.75 and 1.

The optical conductivity for B₂ phase of CsY_xI_{1 − x} is also shown Fig. 14. The highest peak is found for pure CsI (6222 Ω⁻¹⋅cm⁻¹) at 7.7 eV, while there is a set of three peaks (4860 Ω⁻¹⋅cm⁻¹) lying in energy range of 12 eV to 14.5 eV. By fluorine doping in B₂ phase CsI, the first peak shifts from 6222 Ω⁻¹⋅cm⁻¹ to 3459 Ω⁻¹⋅cm⁻¹, when the concentration varies from x = 0 to 1, while the second peak increases as concentration increases, i.e. from 4860 Ω⁻¹⋅cm⁻¹ to 9099 Ω⁻¹⋅cm⁻¹ at x changing from 0 to 1. Introduction of chlorine (Cl) caused that the plot of optical conductivity shows three major peaks, two in the energy range of 6 eV to 10 eV, while the third one lies between 12 eV and 14 eV. The very first peak keeps decreasing with increasing Cl content, while the second one decreases till x = 0.75 and then starts to increase for x = 1 (CsCl). The third peak increases with Cl concentration and has the highest value of 7803 Ω⁻¹⋅cm⁻¹ at x = 1 (CsCl) at energy of 13.6 eV. After addition of bromine (Br), CsI shows three major peaks. Two of them are in the energy range of 6 eV to 9 eV, while the third one is in the range of 11.5 eV to 12.5 eV. The first peak decreases with an increase in Br concentration. The second peak is of the same height as that of the first one but it gradually increases from 5521 Ω⁻¹⋅cm⁻¹to 5821 Ω⁻¹⋅cm⁻¹ (CsBr). The third peak keeps increasing till it attains a maximum value of optical conductivity (7346 Ω⁻¹⋅cm⁻¹) at 12.2 eV for CsBr (B₂) phase.