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A critical study of the elastic properties and stability of Heusler compounds: Cubic Co$_{2}YZ$ compounds with $L2_{1}$ structure

Elastic constants and their derived properties of various cubic Heusler compounds were calculated using first-principles density functional theory. To begin with, Cu$_2$MnAl is used as a case study to explain the interpretation of the basic quantities and compare them with experiments. The main part of the work focuses on Co$_2$-based compounds that are Co$_2$Mn$M$ with the main group elements $M=$~Al, Ga, In, Si, Ge, Sn, Pb, Sb, Bi, and Co$_2TM$ with the main group elements Si or Ge, and the $3d$ transition metals $T=$~Sc, Ti, V, Cr, Mn, and Fe. It is found that many properties of Heusler compounds correlate to the mass or nuclear charge $Z$ of the main group element. Blackman's and Every's diagrams are used to compare the elastic properties of the materials, whereas Pugh's and Poisson's ratios are used to analyze the relationship between interatomic bonding and physical properties. It is found that the {\it Pugh's criterion} on brittleness needs to be revised whereas {\it Christensen's criterion} describes the ductile--brittle transition of Heusler compounds very well. The calculated elastic properties give hint on a metallic bonding with an intermediate brittleness for the studied Heusler compounds. The universal anisotropy of the stable compounds has values in the range of $0.57 <A_U <2.73$. The compounds with higher $A_U$ values are found close to the middle of the transition metal series. In particular, Co$_2$ScAl with $A_U=0.01$ is predicted to be an isotropic material that comes closest to an ideal Cauchy solid as compared to the remaining Co$_2$-based compounds. Apart from the elastic constants and moduli, the sound velocities, Debye temperatures, and hardness are predicted and discussed for the studied systems. The calculated slowness surfaces for sound waves reflect the degree of anisotropy of the compounds.