Calculated Resistances of Single Grain Boundaries in Copper

The resistance of copper grain boundaries (GBs) is calculated systematically through a full atomistic quantum approach. A set of twin GBs, including the coherent twin GB, is generated by density functional theory (DFT) total energy relaxation starting from the coincidence site lattice (CSL) model. The atomic structure of the GBs is used to construct two-probe transport junctions for quantum-transport analysis by carrying out DFT within the Green’s function formalism. The specific resistivity calculated for the coherent twin GB is found to be quantitatively consistent with the available experimental and theoretical data. The specific resistivity and reflection coefficient of other more complex GBs are predicted. The interfacial energy density and specific resistivity are both found to inversely relate with the planar density of coincidence sites. Comparison of our calculated specific resistivities and reflection coefficients with the corresponding GB-averaged experimental quantities shines light on the microstructure of the samples.

Figure 1

Calculated total energy $E_0$ (blue diamonds) and corresponding Birch-Murnaghan fit (red line) as a function of the lattice constant $a$.

Figure 2

Periodic unit cell of the $\mathrm{\Sigma }3$ CSL boundary in a 2D projection (a) and in a 3D perspective (b).

Figure 3

$\mathrm{\Sigma }13\mathrm{a}$ CSL boundary before (a) and after (b) merging the pair of closest-neighbor atoms. Blue atoms represent the pair of closest-neighbor atoms (a) and the new atom resulting from the merging process (b).

Figure 4

$\mathrm{\Sigma }11$ CSL boundary incremental relaxation at steps $NL=0$ (a), $NL=1$ (b), $NL=3$ (c), and $NL=9$ (d). Blue atoms are fully relaxed, while the remaining ones are kept fixed.

Figure 5

Total energy as a function of the number of relaxed layers $NL$ for the $\mathrm{\Sigma }3$ (a), $\mathrm{\Sigma }5$ (c), and $\mathrm{\Sigma }9$ (e) boundaries. Atomic forces in the direction perpendicular to the GB plane as a function of distance in the $\mathrm{\Sigma }3$ (b), $\mathrm{\Sigma }5$ (d), and $\mathrm{\Sigma }9$ (f) GBs.

Figure 6

Bond-orientational order parameter $S_6(i)$ plotted on a color scale inside the $\mathrm{\Sigma }3$ (a), $\mathrm{\Sigma }5$ (b), $\mathrm{\Sigma }9$ (c), $\mathrm{\Sigma }13\mathrm{a}$ (d), $\mathrm{\Sigma }11$ (e), and $\mathrm{\Sigma }17\mathrm{a}$ (f) GBs.

Figure 7

Area per coincidence site $\sigma$ (a), interfacial energy density ${\gamma }_E$ (b), and specific resistivity ${\gamma }_R$ (c) of the investigated GBs.