From many valleys to many topological phases - quantum anomalous Hall effect in IV-VI semiconductor quantum wells

TL;DR

This work models the quantum anomalous Hall effect in Pb$_{1-x}$Sn$_x$Se/(PbSe)$_{1-y}$(EuS)$_y$ quantum wells using a generalized four-band $\bm{\mathrm{k}}\cdot\bm{\mathrm{p}}$ framework that incorporates confinement, spin-orbit coupling, Zeeman splitting, and exchange with magnetic barriers. A basis-transformation approach enables accurate treatment of wells grown along $\langle 111\rangle$, $\langle 110\rangle$, and $\langle 001\rangle$ directions, capturing L-valley anisotropy and projecting valleys onto the 2D Brillouin zone to compute valley-resolved Chern numbers via the Fukui plaquette method. The results show tunable Chern numbers $\mathcal{C}$ from $1$ to $4$ across orientations, with $\mathcal{C}=3$ possible in $[111]$ wells due to three equivalent M valleys, and emphasize the essential role of strain compensation and barrier magnetization to realize robust QAHE at practical fields and temperatures. The work provides Chern-phase diagrams and detailed guidance for experimental efforts, suggesting that tensile strain, piezoelectric tuning, and careful valley-edge alignment can enable higher-$\mathcal{C}$ QAHE in IV–VI quantum wells and potential integration with superconductors for new functionalities.

Abstract

Consistent with prior qualitative expectations for group IV-VI topological crystalline insulators, this work demonstrates, based on band structure and Chern number calculations, that Pb$_{1-x}$Sn$_x$Se/(PbSe)$_{1-y}$(EuS)$_y$ quantum wells constitute a promising and viable platform for realizing a variety of quantum anomalous Hall phases. The proposed basis transformation procedure for the multiband $\mathit{k} \cdot \mathit{p}$ Hamiltonian enables the treatment of wells grown along arbitrary crystallographic directions while explicitly accounting for the anisotropy of the material's isoenergetic surfaces. Numerical studies of $\langle 111\rangle$-, $\langle 110\rangle$- and $\langle 001\rangle$-oriented quantum wells predict attainable Chern numbers with magnitudes ranging from $1$ to $4$, depending on the quantum well width, Sn content, and relative orientation of the four projected $\mathrm{L}$ valleys with respect to the growth direction. The results further indicate that appropriate strain compensation is required to achieve high-quality quantization of the Hall conductance.

Figures (10)

• Figure 1: (a) Rock-salt crystal structure of IV--VI compounds with representative $\{001\}$, $\{110\}$ and $\{111\}$ planes; the symmetries of these planes ensure topological protection of gapless surface states Hsieh:2012_Nat_Commun. (b) Projections of the three-dimensional FBZ of IV--VI semiconductors onto the quantum well growth directions considered in this work. Schematic constant-energy surfaces near the L-point band edges --- approximately spheroidal Nimtz:1983 --- are also shown; their anisotropy and reciprocal-space orientations induce lifting of the L-valley degeneracy upon FBZ projection onto $\langle 111 \rangle$ and $\langle 110 \rangle$ directions.
• Figure 2: Schematic of the modeled quantum wells. The conduction and valence band edges, $E_{c}$ and $E_{v}$, are sketched along the $z$-axis, and the electron and hole states, $E^{e}_n$ and $E^{h}_n$, are indicated schematically. The energy gap in this system is $E_{\text{g}} = E^{e}_1 - E^{h}_1$.
• Figure 3: Geometry of the basis (coordinate-system) transformation in the $\bm{\mathit{k}}$$\cdot$$\bm{\mathit{p}}$ model. Rotation angles about the respective axes are: $\alpha$$=$$\arccos(1/\sqrt{3})$$\approx$$54.7$°, $\beta$$=$$\arccos(\sqrt{2/3})$$\approx$$35.3$°, $\gamma$$=$$\arccos(1/3)$$\approx$$70.5$° and $\delta$$=$$90$°. Solid arrows indicate the coordinate axes directions in which the Hamiltonian (\ref{['eq:kp_Ham']}) is written. The orientation of the constant-energy ellipsoids with respect to the unit vector $\hat{\bm{\mathit{e}}}_z$ determines the quantization conditions of the energy levels derived from a given valley when electrons are confined along that direction.
• Figure 4: Schematic diagrams of the discrete FBZ meshes for the two-dimensional systems considered in this study (cf. Fig. \ref{['subfig:IV-VIs_b']}). The spanning vectors $\bm{\mathit{\kappa}}_{\mu}$ are given by (a) $\bm{\mathit{\kappa}}_{1}=\frac{2\pi}{a_0}\left[\sqrt{2},-1\right]{}^{\mathrm{T}}$, $\bm{\mathit{\kappa}}_{2}=\frac{2\pi}{a_0}[0, 1]^{\mathrm{T}}$, (b) $\bm{\mathit{\kappa}}_{1}=\frac{2\pi}{a_0}\sqrt{\frac{2}{3}}\left[\sqrt{3}, 1\right]{}^{\mathrm{T}}$, $\bm{\mathit{\kappa}}_{2}=\frac{2\pi}{a_0}\sqrt{\frac{2}{3}}\left[-\sqrt{3}, 1\right]{}^{\mathrm{T}}$, (c) $\bm{\mathit{\kappa}}_{1}=\frac{2\sqrt{2}\pi}{a_0}[1, 0]^{\mathrm{T}}$, $\bm{\mathit{\kappa}}_{2}=\frac{2\sqrt{2}\pi}{a_0}[0, 1]^{\mathrm{T}}$, where $a_0$ is the fcc lattice constant.
• Figure 5: Band gap diagrams as functions of tin concentration $x_{\text{Sn}}$ and the quantum well thickness $d_{\text{QW}}$ for the individual valleys of structures grown along $[hkl]$. The value $\vartheta$ beneath the symbol for a given $\mathrm{L}$-valley projection indicates the three-dimensional angular deviation of the valley axis from the well growth direction (cf. Fig. \ref{['fig:TR']}). Other system parameters are $B = 0.0\, \mathrm{T}$, $T = 1.5\,\mathrm{K}$, $d_{\text{BR}} = 15\,\mathrm{nm}$, $y_{\text{EuS}} = 0.25$. The lower-right panel displays cross sections of the maps $E_{\text{g}}(x_{\text{Sn}}, d_{\text{QW}})$ taken along the dashed lines at $d_{\text{QW}} = 11.25\, \mathrm{nm}$.