Magnetic exchange interaction in topological insulators

The effective Hamiltonian that describes magnetic exchange interaction in a topological insulator is calculated, projecting bulk contact $s$-$d$ interaction onto surface electron states. This model consistently describes the indirect exchange between magnetic atoms mediated either by massless Dirac fermions in thick films or by massive excitations in a thin slab with tunneling between opposite surfaces. It is shown that an effective $s$-$d$ exchange is sensitive to the symmetry of surface wave functions and depends on the position of an impurity spin relative to slab surfaces. Under a vertical bias the indirect exchange between localized spins is switchable between opposite surfaces, and the oscillating range function reveals zero magnetic field beating as a signature of Rashba spin splitting.

Figure 1

$s$-$d$ exchange parameters vs $z$ position of a magnetic impurity.

Figure 2

Spin-flip $s$-$d$ parameter vs $z$ position of a magnetic impurity.

Figure 3

Interband $s$-$d$ parameter ${\tilde{J}}_{12}$ vs $z$ position of a magnetic impurity in the slab.

Figure 4

Factor ${\tilde{V}}_{as}{\tilde{J}}_{12}+{\tilde{J}}_z$ under applied voltage. Solid line: $V=10\mathrm{mV}$; dotted line: $V=-10\mathrm{mV}$.