Ni₂-based Heusler alloys have received increasing attention due to their shape memory effects and the relevant application properties. It is interesting to explore new Ni₂-based shape memory alloys with novel properties. In this work, the site preference, electronic structure, elastic parameters and martensitic transformation of new Ni₂Cu-based Heusler alloys Ni₂CuZ (Z = Al, Ga, In, Si, Ge, Sn and Sb) are investigated theoretically. Between the two highly-ordered structures of Heusler alloys, Ni₂CuZ alloy tends to crystallize in the L2₁ structure with Cu atom entering the B site in the cubic lattice. In contrast, the XA structure is higher in energy and lower in stability. This is different from the usual rule that transition metal atoms with more valence electrons tend to occupy the A, C sites at first and can be related to the strong covalent hybridization between Ni and main group elements Z in L2₁ type Ni₂CuZ.

Ni₂CuZ martensites are all lower in energy than the corresponding austenites, which makes them candidates for shape memory alloys. This can be explained by the Jahn-Teller effect characterized by the reduced states near E_F in the DOS structure and the mechanical instability of the cubic austenite lattice. The martensite-austenite energy difference ΔE_M is strongly influenced by main group elements Z. When Z are in the same group, the ΔE_M increases with their atomic number increasing, but when Z are in the same period, an opposite trend is observed. The ΔE_M can be regarded as a driving force for the martensitic transformation: a larger ΔE_M corresponds to a higher martensitic transformation T_M. In Heusler alloys, electron concentration e/a and electron density n are usually used to discuss the variation of T_M. An increase of e/a or n tends to increase T_M. However, this is in discrepancy with the results in Ni₂CuZ, which can be explained by using, the new factors, the negative shear modulus C' and softening of the elastic constant C₄₄ and their variations with Z elements. These results reveal the close relation between the martensitic transformation and mechanical parameters and indicate that they are important factors to predict new shape memory alloys and analyse their properties in Heusler alloys. It is also found that the Young’s modulus and shear modulus increase and Poisson’s ratio decreases after the martensitic transformation. Thus, the Ni₂CuZ martensite has higher stiffness and rigidity but lower ductility than the austenite.