Due its toxicity, lead has caused serious problems in human health and environmental pollution; thus, the use of lead-containing solder in electronic device packaging has been restricted by legislation . In the last few decades, aiming to replace lead-containing solders, great efforts have been dedicated to develop lead-free solder with respect to cost-effectiveness, wettability, melting point, corrosion resistance, and mechanical and electrical properties . A series of binary alloy solders, like Sn-Zn alloys , Sn-Cu alloys , Sn-Ag alloys , and Sn-Bi alloys , have been extensively investigated. Moreover, ternary and quaternary alloys have recently received considerable attention, and the Sn-Ag-Cu ternary alloy is considered as a promising candidate to substitute conventional lead-containing solder alloys .

Driven by the miniaturization of electronic devices and their widespread application in harsh environments such as high temperature and high humidity, the reliability of solder joints has become a major issue in practice . As far as lead-containing solders are concerned, the high quality of solder joints can be attributed to the formation of a continuous Pb layer, serving as a barrier layer to separate the intermetallics in solders from the substrate . In contrast to the barrier layer formed with lead-containing solder, the lead-free solder forms a compact interface between the intermetallics and the substrate without a barrier layer . The compact interface could result in poor resistance to high temperatures and thermal shocks. Therefore, high-lead solder remains the preferred choice for high-temperature applications at present .

In view of the fact that Ni and Pb belong to the same group in the periodic table of elements and have similar chemical and electronic properties , researchers have been intrigued to enhance the reliability of lead-free solder joints by Ni alloying. For instance, Zhang et al. observed that the corrosion resistance of Sn-Zn solder can be enhanced by alloying with Ni, Cr, Cu, or Ag; the higher corrosion resistance follows the order Ag < Cu < Cr < Ni . El-Taher et al. demonstrated that the ductility and the strength of Sn–3.0Ag–0.5Cu lead-free solders could be enhanced by Ni alloying . Although the beneficial effects of Ni alloying on the properties of lead-free solder have been demonstrated by these investigations, little attention has been paid to the effects of Ni alloying on the interfacial mechanical properties of lead-free solder joints.

Considering the important role the interface between the intermetallics and the substrate plays in the strength and reliability of a solder joint, Gan et al. investigated the formation of Cu3Sn and Cu6Sn5 on a Cu substrate and determined the orientation relationship of ε-Cu3Sn/Cu interfaces as (001)ε//(111)Cu and [100]ε//[−110]Cu, that is, the interface was constructed by attaching the (001) facet of ε-Cu3Sn to the (111) facet of the Cu substrate and making the [100] axis of ε-Cu3Sn parallel to the [−110] axis of Cu substrate. Based on the orientation relationship of the ε-Cu3Sn/Cu interface, an attempt to reveal the effects of Ni alloying on the strength and toughness of (Cu1−xNix)3Sn/Cu interface has been made in this study. At first, the elastic properties of (Cu1−xNix)3Sn and Cu were calculated, followed by evaluation of the intrinsic ductility in terms of elastic moduli. Subsequently, the orientation-dependent Young’s moduli of Cu and (Cu1−xNix)3Sn were calculated. Finally, tensile modulus, ultimate tensile stress, work of adhesion, and interfacial toughness of (Cu1−xNix)3Sn/Cu were calculated based the interface model with the orientation relationship of (001)ε//(111)Cu and [100]ε//[−110]Cu; the underlying mechanisms responsible for the influence of Ni alloying on the work of adhesion and interfacial toughness are demonstrated.

In this study, first-principles calculations within the framework of density functional theory were implemented by the ABINIT package . The norm-conserving pseudopotentials and Perdew–Burke–Ernzerhof generalized gradient approximation (GGA) of the exchange–correlation functional were adopted for the calculation. Regarding the calculations on the intermetallics (CuxNi1−x)3Sn, virtual crystal approximation (VCA) was used to construct the virtual atoms standing for the mixture of Cu and Ni atoms, namely, the pseudopotentials of the virtual atoms were constructed by :

As demonstrated in previous studies , the VCA could significantly enhance the calculation efficiency without losing the accuracy by reducing the model size of the alloy systems. Considering the phase stability of (CuxNi1−x)3Sn , the content of Ni was set within the range from 0 to 30 atom %. As far as the calculations of the structure optimizations and elastic properties are concerned, a kinetic energy cutoff of 30 Hartree, a k-point mesh of 8 × 8 × 8 and a potential residual V(r) of less than 10−8 Hartree were used to achieve self-consistent convergence.

Based on the optimized crystal structures, the elastic constants of FCC Cu and orthorhombic (CuxNi1−x)3Sn were calculated by finite strain methods, where three and nine deformations were built for Cu and (CuxNi1−x)3Sn, respectively, related to the three and nine independent elastic constants corresponding to cubic and orthorhombic crystals, respectively . Strain magnitudes of −0.02, −0.01, 0.0, 0.01, and 0.02 were used to calculated the energy increments of the deformed cells. Via quadratic fits of the relation between the energy increments and the strains, the elastic constants C11, C12, and C44 for Cu and C11, C22, C33, C12, C13, C23, C44, C55, and C66 for (CuxNi1−x)3Sn were extracted. Based on the calculated elastic constants, bulk modulus, shear modulus, Young’s modulus, anisotropy, and Poisson’s ratio of Cu and (CuxNi1−x)3Sn were calculated according to Voight–Reuss–Hill bounds . Furthermore, from the calculated elastic constants, the orientation-dependent Young’s moduli of Cu and (CuxNi1−x)3Sn were calculated.

Based on the orientation relationship of Cu3Sn/Cu interfaces , namely, (001)ε//(111)Cu and [100]ε//[−110]Cu, the interface was constructed by adhering Cu and (CuxNi1−x)3Sn slabs. The Cu slab consisted of four atomic layers, the (CuxNi1−x)3Sn slab consisted of three atomic layers, and the thickness of the vacuum layer was 1 nm. Interfacial modulus, ultimate tensile stress, work of adhesion, and the interfacial toughness of (CuxNi1−x)3Sn /Cu interfaces were determined by a tensile test along the direction normal to the interface plane, that is, along the z-axis. During the tensile deformation, the strain along the z-axis was fixed; at the same time, the stresses along the x-axis and the y-axis were relaxed to less than 0.5 GPa. For the calculations on the interface structure, a kinetic energy cutoff of 30 Hartree, a k-point mesh of 4 × 4 × 1 and a potential residual V(r) of less than 10−8 Hartree were used to achieve self-consistent convergence.

Figure 1: (a) Crystal structures of Cu and (CuxNi1−x)3Sn, where the virtual crystal approximation is adopted for (CuxNi1−x)3Sn, and the atomic models were plotted using VESTA . This content is not subject to CC BY 4.0. (b) The independent elastic constants, C11, C22, C33, C12, C13, C23, C44, C55, and C66 of (CuxNi1−x)3Sn, and the independent elastic constants, C11, C12, and C44 of Cu.

Table 1: Space groups, lattice constants, and independent elastic constants of Cu and (CuxNi1−x)3Sn; Voigt-type bulk modulus BV and Reuss-type bulk modulus BR; Voigt-type shear modulus GV and Reuss-type shear modulus GR. The data presented in parentheses are given for comparison.

aThe experimental lattice constants and elastic constants of Cu are cited from . bThe calculated lattice constants and elastic constants of Cu3Sn are cited from .

Using the calculated elastic constants of Cu and (CuxNi1−x)3Sn, the average bulk modulus, shear modulus, Young’s modulus, universal anisotropy, and Poisson’s ratio were calculated according to the Voigt–Reuss–Hill approximations . For FCC Cu, the Voigt-type bulk modulus BV and shear modulus GV, and the Reuss-type bulk modulus BR and shear modulus GR, can be calculated from C11, C12, and C44 as :

For orthorhombic (CuxNi1−x)3Sn, the Voigt-type bulk modulus BV and shear modulus GV, and the Reuss-type bulk modulus BR and shear modulus GR, can be calculated from C11, C22, C33, C12, C13, C23, C44, C55, and C66 as :

where

The average bulk modulus B was calculated as the arithmetic average of BV and BR, that is, B = (1/2)(BV + BR). Likewise, the average shear modulus G was calculated by G = (1/2)(GV + GR). From the bulk modulus B and the shear modulus G, the Young’s modulus E and the Poisson’s ratio ν can be calculated as :

Figure 2: (a) Bulk moduli, shear moduli, and Young’s moduli of Cu and (CuxNi1−x)3Sn; (b) universal anisotropies AU of Cu and (CuxNi1−x)3Sn; (c) Poisson’s ratios of Cu and (CuxNi1−x)3Sn; (d) ratios of bulk modulus to shear modulus, B/G, of Cu and (CuxNi1−x)3Sn. The green horizontal lines in (c) and (d) correspond to the boundaries of the ductile-to-brittle transition.

Table 2: Bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν; universal anisotropy AU; ductility index B/G, minimum Young’s modulus, Emin and the corresponding orientation [l1, l2, l3]min; as well as maximum Young’s modulus Emax and the corresponding orientation [l1, l2, l3]max. l1, l2, l3 are the direction cosines of the orientation axis. The data presented in parentheses are given for comparison.

aThe experimental elastic moduli and Poisson’s ratios of Cu are cited from . bThe experimental elastic moduli and Poisson’s ratios of Cu3Sn are cited from .

Moreover, using the calculated values of BV, BR, GV, and GR, the universal anisotropy index AU, developed by Ostoja-Starzewski et al., can be expressed as :

Considering the critical roles of orientation-dependent elastic properties in the mechanical properties of interfaces , the orientation-dependent Young’s moduli of Cu and (CuxNi1−x)3Sn were investigated. Regarding FCC Cu, the orientation-dependent Young’s moduli,, along the directions ⟨hkl⟩ were calculated as :

For orthorhombic (CuxNi1−x)3Sn, the orientation-dependent Young’s moduli, , along the directions ⟨hkl⟩ were calculated as :

Figure 3: 3D surfaces of oriented Young’s moduli for (a) Cu, (b) Cu3Sn, (c) (Cu0.9Ni0.1)3Sn, (d) (Cu0.8Ni0.2)3Sn, and (e) (Cu0.7Ni0.3)3Sn. 2D profiles on the (110) plane of oriented Young’s moduli for (f) Cu, (g) Cu3Sn, (h) (Cu0.9Ni0.1)3Sn, (i) (Cu0.8Ni0.2)3Sn, and (j) (Cu0.7Ni0.3)3Sn; the orientations of the maximum Young’s moduli in the 2D profiles are denoted by the red lines.

Figure 4: (a) Orientation relationship and structure of (CuxNi1−x)3Sn/Cu interfaces; the atomic models were plotted using VESTA . This content is not subject to CC BY 4.0; (b) tensile stress vs tensile strain curves of (CuxNi1−x)3Sn/Cu interfaces; (c) tensile moduli Etensile of (CuxNi1−x)3Sn/Cu interfaces; (d) ultimate tensile strengths UTS and tensile strains at UTS of (CuxNi1−x)3Sn/Cu interfaces; (e) work of adhesion Wad of (CuxNi1−x)3Sn/Cu interfaces; (f) interfacial toughness of (CuxNi1−x)3Sn/Cu interfaces.

From the tensile stress vs tensile strain cur