First-principles study on electronic, optic, elastic, dynamic and thermodynamic properties of RbH compound

– We performed ﬁrst-principles calculations to obtain the electronic, optical, elastic, lattice-dynamical and thermodynamic properties of RbH compound with rock salt structure. The ground-state properties, i.e., the lattice constant and the band gap were investigated using a plane wave pseudopotential method within density functional theory. The calculated lattice constant, bulk modulus, energy band gap and elastic constants are reported and compared with previous theoretical and experimental results. Our calculated results and the previous results which are obtained from literature are in a good agreement. Moreover, real and imaginary parts of complex dielectric function, reﬂectivity spectrum, absorption, extinction coefﬁcient and loss function as a function of photon energy and refractive index with respect to photon wavelength were calculated. In addition, temperature dependent thermodynamic properties such as Helmholtz free energy, internal energy, entropy and speciﬁc heat have been studied.


Introduction
Alkali hydrides (LiH, NaH, KH, RbH and CsH) have possible applications in nuclear and chemical industries [1]. Thus, this materials have been investigated both experimentally and theoretically for many years. Like alkali metal halides, alkali metal hydrides crystallize with the rock-salt structure. The X-ray experimental study [2] showed that RbH crystallize with the rock-salt (B1) structure at room temperature. Gaydon and Pearse [3] studied the band structure of RbH experimentally and theoretically. A structural phase transition from NaCl to CsCl phase was observed in RbH at high pressure [4]. Jain [5] investigated ground state of RbH using the Rydberg-Klein-Ress method. Stwalley et al. [6] reported structure of the alkali hydride diatomic molecules. Hooper and Zurek [7] investigated the structures of compressed rubidium polyhydrides, RbH n with n > 1. Zrafi et al. [8] calculated the evaluation of the first and second derivative of nonadiabatic coupling between the several I R þ adiabatic states of LiH, RbH and CsH molecules. Ahuja [9] investigated the structure of RbH under high compressions by means of first principles self consistent total-energy calculations within local-density approximation. Fatmi et al. [10] showed structural, electronic, optical and thermodynamic properties of Na x Rb 1Àx H and Na x K 1Àx H alloys. Hooper et al. [11] discussed three new structures of high-pressure alkali metal hydrides (P6 3 / mmc, I4 I /amd, and Pnma) and found structural phase transitions via evolutionary methodologies. Sudha Priyonga et al. [12] calculated structural, electronic and elastic properties of alkali hydrides (MH: M = Li, Na, K, Rb, Cs).

Computational methods
The first-principles calculations are performed with the ABINIT [13] and the WIEN2K [14] code programs. The structural, electronic and optical properties of RbH crystal was investigated using WIEN2K package within the Generalized Gradient Approximation (GGA) based on the density functional theory (DFT). 286 k points were applied in the Brillouin zone for the structural, electronic and optical properties of RbH crystal.
In addition, the dynamic and thermodynamic properties of RbH crystal was investigated using ABINIT code. The calculated results were obtained using GGA included the exchange-correlation effects [15] of DFT. The self-consistent norm-conserving pseudopotentials of FHI-type with a Troullier-Martins scheme [16] for all atoms of RbH crystal were used. The Kohn-Sham equations [17] were solved the conjugate gradient minimization method [18]. Plane waves were used for the electronic wave functions. For the true valence electron of Rubidium atom and for Hydrogen atom were deliberated the 5s electron and the 1s electron, respectively. Firstly, in all calculations total energy of this material was accommodated with respect to cutoff energy and Monkhorst-Pack mesh grid. The calculated total energy calculations were done to a good convergence at 30 hartree of cutoff energy. Besides, 60k points using 6 · 6 · 6 Monkhorst-Pack mesh grid [19] in RbH crystal for structural optimization and the calculations of dynamic and thermodynamic properties.

Structural and electronic properties
RbH crystal contains two atoms per face centered cubic unit cell and crystallizes in the Rock salt-type structure. Rb atom is centered at (0, 0, 0) and H atom is located at (0.5, 0.5, 0.5) in the reduced coordinates. The volume of unit cell is optimized using WIEN2K code. The total energy of RbAu compound has been investigated as a function of the unit cell volume and fitted to the Murnaghan equation. Volume parameter was calculated as 376.39 (a.u 3 ) of cubic atomic. In this value, the total force is equal to zero at minimum total energy. The calculated values of lattice parameters for RbH compound is 11.461 bohr (6.064 Å), while experimental value is 6.037 Å [2]. As seen, there is a good agreement between calculated and experimental result for lattice parameters. Then, calculated result was used for calculation of all studied physical properties of this compound ( Table 1).
The pseudopotential method based on density functional theory within general gradient approximation was used for electronic structure calculations of RbH. According to the band structure calculations, RbH crystal has a direct band gap at high symmetry point L with the value of 2.943 eV and experimental values are 2.96 eV. We found that RbH is a semiconductor. Electronic band structure was plotted as in Figure 1.

Optical properties
RbH crystal is a cubic crystal and space group number is 216. This structure is requested to investigate one dielectric component. Our calculation were performed 0-40 eV photon energy range. The calculated real (e 1 ) and imaginary (e 2 ) parts of dielectric function of this compound are illustrated in Figure 2. As seen from this figure, the real part of dielectric function, e 1 , goes to its maximum value at 4.8 eV of photon energy for RbH. There is no absorption through 2.93 eV. After this value, optic transitions from valance band to conduction band can be detected. Furthermore, real part goes to zero at 20.3133, 20.69429, 20.80311, 21.94599 and 21.9732 eV. The incident electromagnetic waves are completely reflected at the negative values of real part. The reflectivity (R) and absorption (a) active regions are from 12 eV to 24 eV and from 2 eV to 18 eV, respectively. The extinction coefficient (k) varies similar to the absorption coefficient (a). The region of the active damping is between 2 eV and 18 eV. Refractive index (n) is 1.75 for low and very low energy electromagnetic waves.   The range of the optical loss in surface and volume is maximum in the ultraviolet region of the electromagnetic spectrum. The calculated extinction coefficient, absorption coefficient, reflectivity and refractive index (n) for RbH are shown in Figures 3 and 4, respectively. The spectral dependence of calculated refractive index is in the wavelength range of 0-0.6 lm, and the calculated value of refractive indices is 1.75 for RbH. The energy loss functions (L), for both surface and volume, are illustrated in Figure 5. As seen in Figure 5, sharp peaks are incorporated with plasma oscillations concerned to the photon energy values.

Elastic properties
To obtain elastic constants of this compound, we used the method developed by Morteza Jamal integrated in WIEN2 K code [24]. Here, to analyze the elastic properties, we calculated the elastic constants (C ij ), Young module, shear modulus, bulk modulus, and Poisson's coefficients with three different approximations that are namely Voight, Reuss and Hill using the mentioned method in reference [23]. The elastic constants are calculated to be C 11 = 32.835 GPa, C 12 = 3.815 GPa, and C 44 = 9.657 GPa for RbH. In this case, RbH has a stable structure. The calculated elastic constants of RbH are listed in Table 2. The Born mechanical stability criteria for cubic structure is given below: C 11 > 0; C 12 > 0; C 44 > 0; C 12 > C 44 ; C 11 þ 2 C 12 > 0; C 11 À C 12 > 0 The Young's modulus E, Zener anisotropy factor A and Poisson's ratio m, which have the plenty of exciting elastic properties for applications, are also computed in terms of the reckoned data using the following relations: Voigt's shear modulus concerned to the upstairs bound of G values, and G R (5/ G R = 4/(C 11 C 12 ) + 3/C 44 ) the Reuss's shear modulus concerned to the lower bound of G values. The calculated Zener anisotropy factor, Poisson's ratio, Young's modulus, and shear modulus for RbH are given in Table 3. The calculated and other results are in a good agreement.

Dynamic properties
Phonon dispersion curves and phonon density of states of RbH along the high-symmetry directions are calculated by using the ABINIT. In RbH crystal, unit cell involves two atoms at zero pressure. The Rb atom and H atom are positioned at the  Table 3. Calculated single crystal bulk modulus (B 0 ), Young modulus (E), shear modulus (G), anisotropy factor (A), Poisson's ratio (m) obtained for RbH.    Figure 6. RbH has two atoms per unit cell so that six phonon branches occur. Three of them are acoustic and three of them are optical branches. Some of the branches are degenerate along more symmetrical directions (C-X and C-L) inside Brillouin zone of RbH crystal. The calculated phonon frequencies resulting from the high symmetry points (C, X and L) of the Brillouin zone are listed in Table 4. There is a gap between the acoustic and the optic phonon branches which shows that RbH is a phononic crystal. Optical branches are almost horizontal around U symmetry point. Optical and acoustic branches degenerate at left and right side of U point. All branches are non-degenerate between X-W, W-K, K-U points. Slopes of acoustic branches around U point are steep, which shows large elastic constant for RbH. High phonon frequencies are attributed to H atoms, because its mass is small and can vibrate very rapidly.

Thermodynamic properties
After calculating the phonon band structure, we have inquired the thermodynamic properties of RbH using these phonon band structure calculations. In terms of thermodynamic properties, we have examined the entropy, constant-volume specific heat and the phonon contributions to the Helmholtz free energy and internal energy. We have plotted all these thermodynamic quantities as functions of temperature as can be seen in Figure 7. As seen from this figure, the values of free energy and internal energy are different from zero at zero temperature which shows zero point oscillations. The values of free energy and internal energy at zero temperature are very close to each other that is 9.92 kJ/mol for free energy and 10.47 kJ/mol for internal energy. The constant-volume specific heat of RbH approaches a limit value of 49.34 J/mol K at 1466.30 K.

Conclusion
We have investigated the dynamic and thermodynamic properties of Rubidium Hydride (RbH) by an ab-initio pseudopotential method applying density functional theory within the generalized gradient approximation. RbH show semiconducting property and the value of band gap is 2.943 eV. We have declared real and imaginary parts of dielectric function, reflectivity spectrum, absorption coefficient, extinction coefficient and loss function as a function of photon energy and refractive index as a function of photon wavelength. During the investigation of  dynamical properties of RbH, we have calculated and plotted phonon density of states function graphs. While examining the thermodynamic properties of RbH, we have calculated the phonon contribution to the Helmholtz free energy and to the internal energy, the entropy and constant-volume specific heat as functions of temperature.