Effects of Charge Dopants in Quantum Spin Hall Materials

Semiconductors’ sensitivity to electrostatic gating and doping accounts for their widespread use in information communication and new energy technologies. It is demonstrated quantitatively and with no adjustable parameters that the presence of paramagnetic acceptor dopants elucidates a variety of hitherto puzzling properties of two-dimensional topological semiconductors at the topological phase transition and in the regime of the quantum spin Hall effect. The concepts of resonant states, charge correlation, Coulomb gap, exchange interaction between conducting electrons and holes localized on acceptors, strong coupling limit of the Kondo effect, and bound magnetic polaron explain a short topological protection length, high hole mobilities compared with electron mobilities, and different temperature dependence of the spin Hall resistance in HgTe and (Hg,Mn)Te quantum wells.

Figure 1

Band structure and positions of acceptor levels in HgTe QWs of different thicknesses computed with band structure parameters given in Ref. [32]. (a),(b) Band energies $E$ vs wave vector $k$ for unstrained QW thickness of 6 and 8 nm sandwiched between ${\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}$ barriers. Red rectangles depict the band region displayed in (c). (c) Band edges and acceptor levels (symbols connected by dashed lines vs QW thickness $d_{\mathrm{QW}}$). Except for the orange circles computed for the doubly ionized acceptor ($Z=-2$), other symbols represent the single acceptor ($Z=-1$). The orange symbols ($E_{3/2}$) correspond to acceptors associated with the valence band around $k=0$; the blue symbols ($E_{1/2}$) with valence band side maxima visible in (a) and (b). Full symbols represent the acceptors residing in the QW center; the open symbols represent acceptors at the distances $d_{\mathrm{QW}}/4$, $d_{\mathrm{QW}}/2$, and $3d_{\mathrm{QW}}/2$ of the QW center. Colors represent the participation of the $p_{\pm 3/2}$ Kohn-Luttinger amplitude in the wave functions. The discontinuity in the orbital content occurring at $k=0$ and $E_g\to 0$ [see (a)], is blurred in (c) by contributions with $k\ne 0$.

Figure 2

Schematic picture of carrier and acceptor bands at the topological phase transition ($E_g=0$). (a) Bulk 3D case with the Fermi energy pinned in the conduction band (CB) by acceptors negatively charged below the Fermi level. Coulomb gap at $E_F$ is also shown. (b),(c) The same for the 2D case and positions of the Fermi level in the conduction band and the valence band (VB). The acceptor band is wide, as the binding energy depends on the acceptor location with respect to the QW center.

Figure 3

Destructive role of charge dopants in the quantum spin Hall effect. If axial symmetry is maintained (${\mathcal{J}}_x={\mathcal{J}}_y$, ${\mathcal{J}}_{yz}=0$, $D_x=0$) only spin-flop ($\Uparrow \downarrow \leftrightarrows \Downarrow \uparrow$) transitions occur (case 1 in the figure), so that edge current is conserved in the spin-momentum locking situation. Note that arrows up and down refer to time reversal partners rather than to spin up and down. However, if exchange interaction is anisotropic, $\Uparrow \uparrow \leftrightarrows \Downarrow \downarrow$ transitions that violate pseudospin conservation are also allowed (case 2), leading to net backscattering after a chain of spin-dependent interactions of electrons with an acceptor. For realistic concentrations of charge dopants, backscattering is efficient in the strong coupling limit of the Kondo effect.