Entanglement Creation in Electron-Electron Collisions at Solid Surfaces

For spin-polarized low-energy electrons impinging on a crystalline surface, an important reaction channel is the collision with a bound valence electron of opposite spin, followed by the emission of a correlated electron pair with antiparallel spins. While primary and valence electrons are not entangled, the screened Coulomb interaction generates spin entanglement between the two outgoing electrons. As a quantitative measure of this entanglement, we calculated a modified von Neumann entropy in terms of direct and exchange transition matrix elements. For coplanar symmetric setups with equal energies of antiparallel-spin electrons, maximal entanglement is analytically shown to occur quite universally, irrespective of the choice of the primary electron energy, the outgoing electron energy, and polar emission angle, and even of the choice of the surface system. Numerical results for Fe(110) and Cu(111) demonstrate first that strong entanglement can persist for unequal energies and second that an overall entanglement reduction due to nonentangled parallel-spin electrons can be avoided for ferromagnetic and even for nonmagnetic surfaces.