Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2013 / Paper 1 / 2 / a / Solution

The spacer electrons mediate interlayer exchange coupling. Spin-dependent reflection at the two magnetic interfaces changes their electronic energy as the spacer thickness varies. In the oscillatory spacer-mediated exchange picture, Fermi-surface spanning wavevectors produce terms such as $J(t_2)\propto \cos (Q{t}_2+\phi )$ with a decaying envelope; real spacers can have several periods.

Define the coupling energy per area by $E_{\mathrm{ex}}=-J(t_2){\hat{\mathbf{m}}}_1\cdot {\hat{\mathbf{m}}}_2$. $J>0$ favours parallel moments and $J<0$ antiparallel moments. The oscillation period and phase depend on spacer band structure and interface reflection; there is no universal monotone rule or numerical crossover thickness from $t_2$ alone. Roughness and nonuniform thickness average and weaken the coupling. This is conduction-electron-mediated exchange, not a requirement that the nonmagnetic spacer itself be ferromagnetic.