Self-Consistent Model for Gate Control of Narrow-, Broken-, and Inverted-Gap (Topological) Heterostructures

TL;DR

This work tackles the challenge of accurately predicting band structures in narrow-, broken-, and inverted-gap materials under electrostatic gating. It advocates a full-band envelope-function approach within a self-consistent Hartree framework and implements it in kdotpy to model HgTe quantum wells. The results show quantitative agreement between calculated subband densities and Shubnikov–de Haas oscillations across thick, inverted HgTe wells, validating the method and boundary conditions. The open-source toolset enables broader study of gate-controlled topological heterostructures and related devices.

Abstract

Even small electrostatic potentials can dramatically influence the band structure of narrow-, broken-, and inverted-gap materials. A quantitative understanding often necessitates a self-consistent Hartree approach. The valence and conduction band states strongly hybridize and/or cross in these systems. This makes distinguishing between electrons and holes impossible and the assumption of a flat charge carrier distribution at the charge neutrality point hard to justify. Consequently the wide-gap approach often fails in these systems. An alternative is the full-band envelope-function approach by Andlauer and Vogl, which has been implemented into the open-source software package kdotpy (arXiv:2407.12651). We show that this approach and implementation gives numerically stable and quantitatively accurate results where the conventional method fails by modeling the experimental subband density evolution with top-gate voltage in thick (26 nm - 110 nm), topologically inverted HgTe quantum wells. We expect our openly-available implementation to greatly benefit the investigation of narrow-, broken-, and inverted-gap materials.

Figures (9)

• Figure 1: Illustration of the conceptual problems of the conventional approach. (a) In a heterostructure of broken-gap type, the valence (blue) and conduction (red) band edges overlap and hybridize strongly, and charge transfer can occur. (b) In a narrow-gap heterostructure, electric potentials can strongly bend the valence and conduction band edges, also leading to mixing and hybridization. In both cases, an unambiguous separation into electron- and hole-type states is not possible. (c) In a thin HgTe quantum well with normal band ordering (left), the H1 subband is occupied if the chemical potential $\mu$ lies at the CNP, and the E1 subband is empty (wave functions $|\psi(z)|^2$ shown as red and blue curves, respectively). For an inverted quantum well (right), the subband order is reversed. It follows that the carrier distribution $\rho_0(z)$ at the CNP differs by $\Delta\rho_0(z)\propto |\psi_{\mathrm{E}1}(z)|^2 - |\psi_{\mathrm{H}1}(z)|^2$, the functional dependence of which is shown in the center of the figure.
• Figure 2: Self-consistently calculated Hartree potentials of a 45nm thick HgTe quantum well for several total carrier densities $n_\text{tot}$. The potentials $U$ are plotted along the growth direction $z$ of the quantum well. The HgTe well region is indicated by two dashed lines.
• Figure 3: Energy dispersions $E$ of a $d=45nm$ thick HgTe quantum well along the in" plane $k$" directions $[\overline{1} 1]$ and $[1 1]$ from three self-consistent Hartree $\mathbf{k}\cdot\mathbf{p}$ calculations shown in Figure \ref{['fig:potential_45nm']}. (a) For a total carrier density of $n_\text{tot}=-2.0e11cm^{-2}$, (b) $n_\text{tot}=2.0e11cm^{-2}$, resulting in a near" symmetric Hartree potential for the chosen boundary conditions (cf. Figure \ref{['fig:potential_45nm']}), and (c) for $n_\text{tot}=6.0e11cm^{-2}$. The dashed lines give the position of the chemical potential. Labels in (a) indicate the topological surface states (TSS) and the first bulk valence (VB) and conduction (CB) subbands. The color coding shows the spatial expectation value $\langle z \rangle$ of the wave functions along the quantum well growth direction $z$, normalized by the quantum well thickness $d$.
• Figure 4: (a) Experimental longitudinal resistance curves $R_{xx}$ in an out" of" plane magnetic field $B$ at different gate voltages of a 45nm thick HgTe quantum well. The absolute scale applies to the -0.8V curve. For clarity, the other curves are offset vertically and the 0.0V and 0.4V curves are scaled as indicated. (b) Extracted subband densities. Colored symbols show the individual subband densities $n_\text{part}$ of both topological surface states (TSS), the first bulk conduction bands (CB), and the hole pockets at the top of the valence band (VB) from self" consistent Hartree $\mathbf{k}\cdot\mathbf{p}$ calculations (cf. Figs. \ref{['fig:potential_45nm']} and \ref{['fig:dispersion_45nm']}). The background shows the power spectral density (PSD) at density $n$ obtained from FFTs of the experimental low-field Shubnikov" de Haas oscillations at different gate voltages $V_g$ on a logarithmic scale. The black line corresponds to the experimental gate action for the total carrier density obtained from Hall measurements. The black dots give the total carrier density from the calculations. Red" dashed lines divide the gate voltage range into four transport regimes I-IV, see discussion and Ref. Fuchs2025.
• Figure 5: For Table of Contents Only.