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Copper is the current interconnect metal of choice in integrated circuits. As interconnect dimensions decrease, the resistivity of copper increases dramatically because of electron scattering from surfaces, impurities, and grain boundaries (GBs) and threatens to stymie continued device scaling. Lacking direct measurements of individual scattering sources, understanding of the relative importance of these scattering mechanisms has largely relied on semiempirical modeling. Here we present the first ever attempt to measure and calculate individual GB resistances in copper nanowires with a one-to-one correspondence to the GB structure. Large resistance jumps are directly measured at the random GBs with a value far greater than at coincidence GBs and first-principles calculations. The high resistivity of the random GB appears to be intrinsic, arising from the scaling of electron mean free path with the size of the lattice relaxation region. The striking impact of random GB scattering adds vital information for understanding nanoscale conductors.

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Using the adiabatic approximation, we derive the equations of motion for local spin moments which are valid for itinerant as well as tight-binding spins. Material parameters in the equations of motion can be obtained using standard density functional methods, because they depend only on the energy and Berry phase of the constrained ground state of frozen spin configurations. For the calculation of spin waves in a collinear magnet, it is sufficient to know the quadratic forms of total energy and spin component along the symmetry axis as functions of the spin deviations from the ground state configuration.