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

Tomasz Dietl
Phys. Rev. B 107, 085421 – Published 21 February 2023
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  • INTRODUCTION
  • THEORETICAL RESULTS
  • COMPARISON TO EXPERIMENTAL RESULTS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    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−p and sp−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.

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    • Received 9 September 2022
    • Revised 28 January 2023
    • Accepted 30 January 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.085421

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Electrical conductivityElectrical propertiesKondo effectQuantum spin Hall effect
    1. Physical Systems
    Quantum wells
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Tomasz Dietl*

    • International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland and WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

    • *dietl@MagTop.ifpan.edu.pl

    See Also

    Effects of Charge Dopants in Quantum Spin Hall Materials

    Tomasz Dietl
    Phys. Rev. Lett. 130, 086202 (2023)

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    Vol. 107, Iss. 8 — 15 February 2023

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    • Figure 1
      Figure 1

      Envelope functions f1(z), f3(z), f4(z), and f7(z) at k=0 for E1 and H1 subbands in HgTe quantum well of 8-nm width. The envelopes f3 and f6=f3 correspond to the Kramer's pair for the H1 subband, whereas f1, f4, f7, and f2=−f1, f5=−f4, f8=f7 for the E1 subband with a relative weight 53.6, 45.9, and 0.5%, respectively. As seen, the values of fj are either real and symmetric or imaginary and antisymmetric in respect to the QW center at z=0.

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    • Figure 2
      Figure 2

      Verification of axial approximation. Subband dispersions for a HgTe QW of the thickness dQW=6 nm (a) and 8 nm (b) computed for k⃗∥〈11〉 (dashed lines), k⃗∥〈10〉 (dotted lines), and within the axial approximation (solid lines). Character of particular subbands at k=0 is also marked.

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    • Figure 3
      Figure 3

      Effects of biaxial strain on band dispersion in HgTe QWs. (a) εxx=0.31% (CdTe substrate); (b) εxx=0; (c) εxx=−0.31%. Colors describe the participation of the p3/2,±3/2 Kohn-Luttinger amplitude in the electronic wave function; HgTe QW thickness dQW=6 nm.

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    • Figure 4
      Figure 4

      Example of the identification of acceptor levels E3/2(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 lmax in Eq. (5) and for nmax=25 [(Eq. (3)]. (c) E as a function of the distance of acceptor to QW center z0 for lmax=15 and nmax=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 E3/2 acceptor level; dotted lines represent examples of acceptor excited states. HgTe QW thickness dQW=6 nm; no strain.

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    • Figure 5
      Figure 5

      Computed values (points) of exchange integrals Jα/A, the Dzyaloshinskii-Moriya energy Dx/A, and the spin-independent term Δeh for coupling between edge electrons and acceptor holes in topological HgTe QW, plotted as a function of the distance ym between the acceptor wave function maximum and the edge. (a) The electron wavevector k=0.05 nm−1 and the penetration of helical state into the QW, b=7 nm; (b) k=0.1 nm−1b=5 nm. The material constant A is defined in Eq. (20). Lines are linear fits to the computed points.

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    • Figure 6
      Figure 6

      Computed magnitudes of exchange anisotropy ratio rxy=[(2(Jx−Jy)/(Jx+Jy)]2 and rDx=[(3Dx/(Jx+Jy+Jz)]2 for edge electrons in topological HgTe QWs as a function of the distance ym between the acceptor wave function maximum and the edge, obtained for Jα and Dx values display in Fig. 5; the results are for two electron wavevectors k=0.05 and 0.1 nm−1, for which the penetration of helical state into the QW b=7 and 5 nm, respectively. In the case of rDx, the distance between the QW center and the wave function maximum zm=2 nm (full points) or varies between 0.5 and 2.5 nm with 0.5-nm step (empty triangles); k=0.1 nm−1; rDx(zm=0)=0. Lines are linear fits to the computed points.

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    • Figure 7
      Figure 7

      Calculated magnitudes (points) of Kondo temperatures, obtained for Jx and Jy values display in Fig. 5, for edge electrons in HgTe QWs containing holes bound to acceptors as a function of the distance ym between the acceptor wave function maximum and the edge calculated for various values of the electron-hole energy interval Eeh; full points: electron wavevector k=0.05 nm−1 and the penetration of helical state into the QW, b=7 nm; open points: k=0.1 nm−1b=5 nm. Lines connect calculated points.

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    • Figure 8
      Figure 8

      Configurationally averaged two-terminal conductance G in the units 2e2/h vs temperature for a HgTe QW of the thickness dQW=8 nm and length Lx=10 µm computed for various energy distance of the Fermi level to the acceptor state Eeh; k=0.1 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.

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