Miniband engineering and topological phase transitions in topological-insulator--normal-insulator superlattices

Miniband engineering and topological phase transitions in topological-insulator–normal-insulator superlattices

G. Krizman, B. A. Assaf, G. Bauer, G. Springholz, L. A. de Vaulchier, and Y. Guldner

Phys. Rev. B 103, 235302 – Published 3 June 2021

Abstract

Periodic stacking of topologically trivial and nontrivial layers with opposite symmetry of the valence and conduction bands induces topological interface states that, in the strong coupling limit, hybridize both across the topological and normal insulator layers. Using band structure engineering, such superlattices (SLs) can be effectively realized using the IV–VI lead tin chalcogenides. This leads to emergent minibands with a tunable topology, as demonstrated both by theory and experiments. The topological minibands are proven by magneto-optical spectroscopy, revealing Landau level transitions both at the center and edges of the artificial SL mini-Brillouin zone. Their topological character is identified by the topological phase transitions within the minibands observed as a function of temperature. The critical temperature of this transition as well as the miniband gap and miniband width can be precisely controlled by the layer thicknesses and compositions. This witnesses the generation of a fully tunable quasi-three-dimensional topological state that provides a template for realization of magnetic Weyl semimetals and other strongly interacting topological phases.

Received 24 December 2020
Accepted 18 May 2021

DOI:https://doi.org/10.1103/PhysRevB.103.235302

Physics Subject Headings (PhySH)

Superlattices Topological insulators

Infrared spectroscopy k dot p method

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

G. Krizman ^1,2,*, B. A. Assaf³, G. Bauer², G. Springholz ², L. A. de Vaulchier¹, and Y. Guldner¹

¹Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, 24 rue Lhomond, 75005 Paris, France
²Institut für Halbleiter und Festkörperphysik, Johannes Kepler Universität, Altenberger Strasse 69, 4040 Linz, Austria
³Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA

^*gauthier.krizman@jku.at

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Vol. 103, Iss. 23 — 15 June 2021

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Images

Figure 1
High-resolution x-ray characterization of the topological crystalline insulator (TCI)/normal insulator (NI) superlattice (SL) structures. (a) and (b) Reciprocal space maps of ${Pb}_{1 - x} {Sn}_{x} Se$ / ${Pb}_{1 - y - x} {Eu}_{y} {Sn}_{x} Se$ SLs around the symmetric (222) and asymmetric (153) Bragg reflection evidencing perfect pseudomorphic growth. (c) Radial diffraction scans along $Q_{[111]}$ normal to the surface for samples SL27-1.5 and SL9-3.5. The satellite peaks are labeled as $SL x$ , and the diffraction peaks of the ${BaF}_{2}$ substrate and ${Pb}_{1 - y} {Eu}_{y} Se$ capping layer are also indicated. The sample parameters obtained are listed in Table 1.
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Figure 2
Topological miniband (TMB) formation in topological crystalline insulator (TCI)/normal insulator (NI) superlattices (SLs). (a) Modulation of the conduction and valence band edges along the SL structure in the growth direction $z$ . The envelope wave function of the TMB concentrated at the interface is illustrated by the red curve; the black arrows indicate the intrawell and interwell tunnel coupling $τ_{QW}$ and $τ_{B}$ , respectively. (b) Miniband dispersion $E (q_{z})$ for the longitudinal (solid lines) and oblique valleys (dashed lines) derived by $k \cdot p$ theory at $k_{x} = k_{y} = 0$ and 4.2 K for the SL structure SL9-3.5 listed in Table 1. The color scale represents the symmetry ( $L_{6}^{+}$ vs $L_{6}^{-}$ ) of the minibands [see scale bar in (c)]. Label numbers denote the $L_{6}^{+}$ proportion. (c) Evolution of the minibands and their symmetry (color scale) in the conduction and valence bands $(TMB, respectively, {TMB}^{'}$ ) as a function of ${Pb}_{1 - x} {Sn}_{x} Se$ thickness $d_{QW}$ . The barrier width is fixed to $d_{B} = 3.5$ nm and the ${Pb}_{1 - x} {Sn}_{x} Se$ composition to $x_{Sn} = 0.27$ . The corresponding bulk band gaps are $2 Δ_{QW} = - 72.5$ meV (dashed horizontal lines) and $2 Δ_{B} = + 150$ meV. (d) Probability density of the TMB envelope wave function across the SL structure for different miniband topologies. Green line: normal SL (NSL) with $2 δ_{0}$ $= + 20$ meV and $τ_{B} < | τ_{QW} |$ , blue line: zero gap SL with $2 δ_{0}$ $= 0$ and $τ_{B} = | τ_{QW} |$ , red line: topological SL (TSL) with $2 δ_{0}$ $= - 47.5$ meV and $τ_{B} > | τ_{QW} |$ . The different character is set by changing the ${Pb}_{1 - x} {Sn}_{x} Se$ band gap from $2 Δ_{QW} = 10$ , $- 10$ , $- 60$ meV.
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Figure 3
Magneto-optical spectroscopy. (a) and (b) Normalized transmission spectra of the superlattice (SL) samples SL9-3.5 and SL15-3.5 at 4.2 and 160 K at different magnetic fields of up to $B = 15 T$ . The minima are due to Landau level transitions between the minibands at $q_{z} = 0$ and $q_{z} = π / L$ , marked by red and blue arrows, respectively. (c) and (d) Magneto-optical fan charts derived from the experiments (red/blue dots) compared with the calculations by the $k \cdot p$ model for the longitudinal and oblique valleys (solid and dashed lines, respectively). The extrapolated transition energies $2 δ_{0}$ and $2 δ_{π / L}$ at $B = 0$ of the two transition sets are indicated by the arrows and in the insert. The green shaded regions indicate the experimentally nonaccessible energy range blocked by the reststrahlen band of the substrate and window cutoffs.
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Figure 4
Temperature dependence of the miniband gap. (a) Magneto-optical fan chart of superlattice (SL) SL27-1.5 at 4.2 K, showing the ground transitions between the minibands at $q_{z} = 0$ (red) and $| q_{z} | = π / L$ (blue) on an enlarged scale. (b) Temperature dependence of the far-infrared transmission spectra at $B = 15$ T in which the lowest energy transition is indicated by the red dots. The energy position of this transition is shown in (c) as a function of temperature together with the theoretical fit (solid line) obtained by the $k \cdot p$ model. The critical temperature $T_{c} ≅$ 130 K, indicated by the blue arrow, separates the topological SL (TSL) from the normal SL (NSL) phases and where $\partial | δ_{0} | / \partial T$ changes sign. Note that transitions <70 meV are masked by the reststrahlen band of the substrate.
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Figure 5
Demonstration of topological phase transitions. (a)–(c) Temperature dependence of the superlattice (SL) miniband gaps $2 δ_{0}$ (red) and $2 δ_{π / L}$ (blue) at $q_{z} = 0$ and $\pm π / L$ obtained by experiments (dots) and $k \cdot p$ model (solid lines). The region of the miniband band inversion is highlighted in yellow. The vertical dashed lines indicate the critical temperature $T_{c}$ below which the SLs are nontrivial. Above $T_{c}$ , they are trivial. This transition is due to a symmetry inversion which changes the sign of the miniband gaps. The blue shaded area represents twice the miniband width $2 Δ_{MB}$ . Also shown is the temperature dependence of the band gaps of the quantum well [QW; $2 Δ_{QW} (x, T)$ , black circles] and the barriers $[2 Δ_{B} (y, T)$ , green circles] obtained from the fits that nicely agree with our previous work (dashed lines) [30, 33]. (d) and (e) Topological phase diagrams of the SL structures as a function of temperature and ${Pb}_{1 - x} {Sn}_{x} Se$ composition for fixed barrier thickness of (d) $d_{B} = 3.5$ nm and (e) $1.5$ nm. The solid lines represent the phase boundaries for different QW thicknesses $d_{QW}$ , and the shaded regions indicate topological SL (TSL) phases. The black dots in the phase diagrams mark the experimental phase transitions observed for our samples. (f) Temperature dependence of the miniband width $Δ_{MB}$ derived from experiments (symbols) and the $k \cdot p$ model (solid lines). The cusps mark the topological-to-normal insulator SL phase transition, as indicated by the arrows.
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Figure 6
Landau levels of the topological minibands (TMBs). Calculated Landau levels of $TMB$ and ${TMB}^{'}$ at $q_{z} = 0$ (red) and $q_{z} = \pm π / L$ (blue). Calculations have been performed with the parameters $d_{QW} = 9$ nm, $d_{B} = 3.5$ nm, $2 Δ_{QW} = - 72.5$ meV, $2 Δ_{B} = + 150$ meV, and $v_{∥} = v_{z} = 4.40 \times 10^{5}$ m/s. Some Landau level indexes $n$ are written at the right.
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