The temperature dependence of the elastic properties of Ni${}_{2+x}$Mn${}_{1-x}$Ga and Ni${}_2$Mn(Ga${}_{1-x}$Al${}_x$) ($x=0.0$, 0.1, and 0.2) random alloys are investigated by using the first-principles exact muffin-tin orbitals method. At 0 K, the calculated equilibrium parameters in both cubic $L{2}_1$ and nonmodulated tetragonal ${\beta }^{\prime \prime \prime }$-Ni${}_2$MnGa are in good agreement with the available experimental data and other theoretical results. Separating the thermal effects into single electron excitation, volume expansion, phonon smearing, and magnetic terms, we find that phonon smearing gives the dominating positive tetragonal elastic constant ($C^{\prime }$) versus temperature ($T$) slope for the cubic phase. For Ni${}_{2+x}$Mn${}_{1-x}$Ga, the competition between the negative alloying effect ($\partial C^{\prime }/\partial x<0$) and the positive temperature effect ($\partial C^{\prime }/\partial T>0$) leads to nearly constant $C^{\prime }[x,T_M\left(x\right)]$ at the martensitic transition temperature $T_M\left(x\right)$. For Ni${}_2$Mn(Ga${}_{1-x}$Al${}_x$), where both $\partial C^{\prime }/\partial x$ and $\partial C^{\prime }/\partial T$ are positive, however, due to the significantly decrease of $T_M\left(x\right)$, the critical $C^{\prime }[x,T_M\left(x\right)]$ slightly decreases with Al doping. Furthermore, it is demonstrated that both the composition and the temperature dependence of $C^{\prime }$ are indispensable to get a reasonable theoretical $T_M\left(x\right)$.

• Received 22 August 2011

DOI:https://doi.org/10.1103/PhysRevB.84.174117