Figure 8: Calculated projected density of states using PBE functional
Mechanical properties
For LiBH4 with the orthorhombic crystal structure, there are nine independent-elastic-constants namely C11, C12, C13, C22, C23, C33, C44, C55, and C66. They should satisfy the following Born’s stability criterions [33] for mechanical stability which is found to be satisfied rigorously according to our outcomes as given in table-3.
\(C_{11}>0\), \(C_{22}>0\), \(C_{33}>0\), \(C_{44}>0\),\(C_{55}>0\), \(C_{66}>0\) (1)
\(\left[C_{11}+\ C_{22}+C_{33}+{2C}_{12}{+2C}_{13}+{2C}_{23}\right]>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\)
\(C_{11}+C_{22}-{2C}_{12}>0\),\(C_{11}+C_{33}-{2C}_{13}>0\),\(C_{22}+C_{33}-{2C}_{23}>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\)
Moreover, other mechanical parameters are calculated using Voigt-Reuss-Hill’s approximation [34-35]. It is well-known that elastic properties are closely related to the basic solid state properties such as Bulk, Young and Shear moduli, thermal expansion, Debye temperature etc. The values of elastic parameters are summarized under table-3. It is worth mentioning that neither theoretical nor former experimental data is available to compare with our outcomes. Thus, mechanical behavior of the studied compound is being explored for the first time with varying pressure prior to report its fitness for hydrogen storage devices.
\begin{equation} B_{V}=\ \frac{[C_{11}+\ C_{22}+C_{33}+{2(C}_{12}{+C}_{13}+C_{23})]}{9}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\ \end{equation}\begin{equation} B_{R}=\ \begin{matrix}[C_{11}\left(C_{22}+C_{33}-{2C}_{23}\right){+C}_{22}\left(C_{33}-2C_{13}\right)-2C_{33}C_{12}\\ +C_{12}\left(2C_{33}-C_{12}\right)+C_{13}\left(2C_{12}-C_{13}\right)+C_{23}\left(2C_{13}-C_{23}\right)]^{-1}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\ \end{equation}
Where B is Hill’s bulk modulus which is average of Voigt-Reuss bulk modulus and
\begin{equation} B=\ {[B}_{V}+B_{R}]/2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\nonumber \\ \end{equation}
And\(=C_{13}\left(C_{12}C_{23}-{C_{13}C}_{22}\right)+C_{23}\left(C_{12}C_{13}-C_{23}C_{11}\right){+C}_{33}\left(C_{11}C_{22}-C_{12\ }^{2}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)\)
\begin{equation} G_{V}=\ \frac{\left[C_{11}+\ C_{22}+C_{33}+{3(C}_{44}{+C}_{55}+C_{66}\right)-{(C}_{12}{+C}_{13}+C_{23}]}{15}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)\nonumber \\ \end{equation}\begin{equation} G_{R}=\ \begin{matrix}15\{4[C_{11}\left(C_{22}+C_{33}+C_{23}\right){+C}_{22}\left(C_{33}+C_{13}\right)+C_{33}C_{12}\\ -C_{12}\left(C_{23}+C_{12}\right)-C_{13}\left(C_{12}+C_{13}\right)-C_{23}\left(C_{13}+C_{23}\right)]/\\ +3[\left(\frac{1}{C_{44}}\right)+\left(\frac{1}{C_{55}}\right)+\left(\frac{1}{C_{66}}\right)\}^{-1}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)\nonumber \\ \end{equation}\begin{equation} G=\ {[G}_{V}+G_{R}]/2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)\nonumber \\ \end{equation}
Hill’s Shear modulus G (GPa) is an average of Voigt-Reuss shear moduli. While the ratio of Hill’s bulk and shear moduli be recognized as Pugh’s ratio which is an essential parameter to predict whether the studied material associates to brittle or ductile nature [36].
Table 3: Values of the elastic constants C11, C12, C13, C22, C23, C33, C44, C55, and C66 in \((GPa)\), Bulk, Young and Shear moduli in \((GPa)\), Poisson’s coefficient \(\nu\), Anisotropy factor A and Pugh’s ratio \(\frac{B}{G}\).