Quantitative theory of backscattering in topological HgTe and (Hg,Mn)Te quantum wells: Acceptor states, Kondo effect, precessional dephasing, and bound magnetic polaron

We present the theory and numerical evaluations of the backscattering rate determined by acceptor holes or Mn spins in HgTe and (Hg,Mn)Te quantum wells in the quantum spin Hall regime. The role of anisotropic $s\text{−}p$ and $sp\text{−}d$ exchange interactions, Kondo coupling, Luttinger liquid effects, precessional dephasing, and bound magnetic polarons is quantified. The determined magnitude and temperature dependence of conductance are in accord with experimental results for HgTe and (Hg,Mn)Te quantum wells.

Figure 1

Envelope functions $f_1\left(z\right), f_3\left(z\right), f_4\left(z\right)$, and $f_7\left(z\right)$ at $k=0$ for E1 and H1 subbands in HgTe quantum well of 8-nm width. The envelopes $f_3$ and $f_6=f_3$ correspond to the Kramer's pair for the H1 subband, whereas $f_1, f_4, f_7$, and $f_2=-f_1, f_5=-f_4, f_8=f_7$ for the E1 subband with a relative weight 53.6, 45.9, and 0.5%, respectively. As seen, the values of $f_j$ are either real and symmetric or imaginary and antisymmetric in respect to the QW center at $z=0$.

Figure 2

Verification of axial approximation. Subband dispersions for a HgTe QW of the thickness $d_{\text{QW}}=6$ nm (a) and 8 nm (b) computed for $\vec{k}\parallel \langle 11\rangle$ (dashed lines), $\vec{k}\parallel \langle 10\rangle$ (dotted lines), and within the axial approximation (solid lines). Character of particular subbands at $k=0$ is also marked.

Figure 3

Effects of biaxial strain on band dispersion in HgTe QWs. (a) ${\epsilon }_{xx}=0.31$% (CdTe substrate); (b) ${\epsilon }_{xx}=0$; (c) ${\epsilon }_{xx}=-0.31$%. Colors describe the participation of the $p_{3/2,\pm 3/2}$ Kohn-Luttinger amplitude in the electronic wave function; HgTe QW thickness $d_{\text{QW}}=6$ nm.

Figure 4

Example of the identification of acceptor levels $E_{3/2}(m=1)$ degenerate with the QW band states. (a) Energy levels $E$ as a function of the center charge $Z$. (b) $E$ as a function of the number of exponential functions $l_{\text{max}}$ in Eq. (5) and for $n_{\text{max}}=25$ [(Eq. (3)]. (c) $E$ as a function of the distance of acceptor to QW center $z_0$ for $l_{\text{max}}=15$ and $n_{\text{max}}=50$. Color scale depicts the magnitude of an effective in-plane Bohr radius $a^{*}$ in the logarithmic scale. Red-solid lines show the ground state $E_{3/2}$ acceptor level; dotted lines represent examples of acceptor excited states. HgTe QW thickness $d_{\text{QW}}=6$ nm; no strain.

Figure 5

Computed values (points) of exchange integrals ${\mathcal{J}}_{\alpha }/A$, the Dzyaloshinskii-Moriya energy $D_x/A$, and the spin-independent term ${\mathrm{\Delta }}_{eh}$ for coupling between edge electrons and acceptor holes in topological HgTe QW, plotted as a function of the distance $y_{\text{m}}$ between the acceptor wave function maximum and the edge. (a) The electron wavevector $k=0.05$ ${\mathrm{nm}}^{-1}$ and the penetration of helical state into the QW, $b=7$ nm; (b) $k=0.1$ ${\mathrm{nm}}^{-1}b=5$ nm. The material constant $A$ is defined in Eq. (20). Lines are linear fits to the computed points.

Figure 6

Computed magnitudes of exchange anisotropy ratio $r_{xy}=[{(2({\mathcal{J}}_x-{\mathcal{J}}_y)/({\mathcal{J}}_x+{\mathcal{J}}_y)]}^2$ and $r_{Dx}=[{(3D_x/({\mathcal{J}}_x+{\mathcal{J}}_y+{\mathcal{J}}_z)]}^2$ for edge electrons in topological HgTe QWs as a function of the distance $y_{\text{m}}$ between the acceptor wave function maximum and the edge, obtained for ${\mathcal{J}}_{\alpha }$ and $D_x$ values display in Fig. 5; the results are for two electron wavevectors $k=0.05$ and 0.1 ${\mathrm{nm}}^{-1}$, for which the penetration of helical state into the QW $b=7$ and 5 nm, respectively. In the case of $r_{Dx}$, the distance between the QW center and the wave function maximum $z_{\text{m}}=2$ nm (full points) or varies between 0.5 and 2.5 nm with 0.5-nm step (empty triangles); $k=0.1$ ${\mathrm{nm}}^{-1}$; $r_{Dx}(z_{\text{m}}=0)=0$. Lines are linear fits to the computed points.

Figure 7

Calculated magnitudes (points) of Kondo temperatures, obtained for ${\mathcal{J}}_x$ and ${\mathcal{J}}_y$ values display in Fig. 5, for edge electrons in HgTe QWs containing holes bound to acceptors as a function of the distance $y_{\text{m}}$ between the acceptor wave function maximum and the edge calculated for various values of the electron-hole energy interval $E_{eh}$; full points: electron wavevector $k=0.05$ ${\mathrm{nm}}^{-1}$ and the penetration of helical state into the QW, $b=7$ nm; open points: $k=0.1$ ${\mathrm{nm}}^{-1}b=5$ nm. Lines connect calculated points.

Figure 8

Configurationally averaged two-terminal conductance $G$ in the units $2e^2/h$ vs temperature for a HgTe QW of the thickness $d_{\text{QW}}=8$ nm and length $L_x=10$ $\mu \mathrm{m}$ computed for various energy distance of the Fermi level to the acceptor state $E_{eh}$; $k=0.1$ ${\mathrm{nm}}^{-1},b=5$ nm; full points: no Luttinger liquid effects ($K=1$); empty points: Luttinger liquid effects taken into account ($K=0.88$). Lines connect calculated points.