Indirect exchange interaction between magnetic impurities in the two-dimensional topological insulator based on CdTe/HgTe/CdTe quantum wells

We study indirect exchange interaction between magnetic impurities in the (001) CdTe/HgTe/CdTe symmetric quantum well. We consider low temperatures and the case of the chemical potential placed in the energy gap of the two-dimensional quasiparticle spectrum. We find that the indirect exchange interaction is suppressed exponentially with the distance between magnetic impurities. The presence of inversion asymmetry results in oscillations of the indirect exchange interaction with the distance and generates additional terms which are noninvariant under rotations in the (001) plane. The indirect exchange interaction matrix has complicated structure with some terms proportional to the sign of the energy gap.

Figure 1

The energy levels $E_{S_z^A,S_z^B}$ for the Hamiltonian $H_{\mathrm{anis}}^A+H_{\mathrm{anis}}^B+H_{\text{IEI}}$ of the two-impurity problem. Splitting of nine quartets for $M<0$ $\left(M>0\right)$ is shown on the left (right) figure. On each figure, the left four lines correspond to the energy levels without IEI while the right four lines correspond to the energy levels when IEI is taken into account. The parameters are chosen as follows: $V_{zz}^A=300,V_{xy}^A=50,V_{zz}^B=75,V_{xy}^B=15,J_0^A=1,J_1^A=1,J_2^A=1,J_0^B=4,J_1^B=4,J_2^B=4,n_x=0,n_y=1,F=1$, and $F_c=F_s=1/\sqrt{2}$. The energy levels are normalized on $max\{E_{|S_z^A|}^A+E_{|S_z^B|}^B\}$.

Figure 2

The contour of integration in the $\kappa$ plane. The branch cuts of ${\left|\kappa \right|}_q$ are shown by thick gray lines.

Figure 3

Solutions of the equation $\epsilon =\sqrt{f\left(\kappa \right)}$ for $\epsilon >{\epsilon }_0$.

Figure 4

The contours of integration in the $\epsilon$ plane before $\left(C\right)$ and after $\left(\tilde{C}\right)$ deformation. Nonzero ${\varphi }_f$ [see Eq. (A16)] implies that the saddle point is situated away from the real axis.

Figure 5

The contour of integration in $\kappa$ plane is depicted schematically by red curves. The branch cuts for ${\left|\kappa \right|}_q$ and $\sqrt{f\left(\kappa \right)}$ are shown by gray lines.

Figure 6

Energy levels for Hamiltonian $H_{\mathrm{anis}}$ as a function of $V_{xy}/V_{zz}$. It is assumed that $V_{zz}>0$.