Tunable Dirac interface states in topological superlattices

G. Krizman, B. A. Assaf, T. Phuphachong, G. Bauer, G. Springholz, G. Bastard, R. Ferreira, L. A. de Vaulchier, and Y. Guldner

Phys. Rev. B 98, 075303 – Published 2 August 2018

Abstract

Relativistic Dirac fermions are ubiquitous in condensed-matter physics. Their mass is proportional to the material energy gap and the ability to control and tune the mass has become an essential tool to engineer quantum phenomena that mimic high-energy particles and provide novel device functionalities. In topological insulator thin films, new states of matter can be generated by hybridizing the massless Dirac states that occur at material surfaces. In this paper, we experimentally and theoretically introduce a platform where this hybridization can be continuously tuned: the $P b_{1 - x} S n_{x} Se$ topological superlattice. In this system, topological Dirac states occur at the interfaces between a topological crystalline insulator $P b_{1 - x} S n_{x} Se$ and a trivial insulator, realized in the form of topological quantum wells (TQWs) epitaxially stacked on top of each other. Using magnetooptical transmission spectroscopy on high-quality molecular-beam epitaxy grown $P b_{1 - x} S n_{x} Se$ superlattices, we show that the penetration depth of the TQW interface states and therefore their Dirac mass are continuously tunable with temperature. This presents a pathway to engineer the Dirac mass of topological systems and paves the way towards the realization of emergent quantum states of matter using $P b_{1 - x} S n_{x} Se$ topological superlattices.

Received 24 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.075303

Physics Subject Headings (PhySH)

Topological insulators Topological phase transition

Superlattices

Infrared spectroscopy Molecular beam epitaxy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

G. Krizman^1,2, B. A. Assaf¹, T. Phuphachong², G. Bauer³, G. Springholz³, G. Bastard², R. Ferreira², L. A. de Vaulchier², and Y. Guldner²

¹Département de Physique, Ecole Normale Supérieure, Paris Sciences et Lettres Research University, Centre National de la Recherche Scientifique, 24 rue Lhomond, 75005 Paris, France
²Laboratoire Pierre Aigrain, Département de Physique, Ecole Normale Supérieure, Paris Sciences et Lettres Research University, Sorbonne Université, Centre National de la Recherche Scientifique, 24 rue Lhomond, 75005 Paris, France
³Institut für Halbleiter und Festkörperphysik, Johannes Kepler Universität, Altenberger Straβe 69, 4040 Linz, Austria

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Issue

Vol. 98, Iss. 7 — 15 August 2018

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Images

Figure 1
(a) Identical Dirac cones on the (111) surfaces of bulk $P b_{0.75} S n_{0.25} Se$ . (b) Topological quantum-well $P b_{0.75} S n_{0.25} Se$ / $P b_{1 - y} E u_{y} Se$ superlattices of alternating TCI/NI layers studied in this paper. The band alignment and band profiles are shown in (c). ${L_{6}}^{\pm}$ denote the conduction- and valence-band extrema at the L points in the rocksalt Brillouin zone of $P b_{0.75} S n_{0.25} Se$ and $P b_{1 - y} E u_{y} Se$ . $ɛ_{A}$ (<0 at 4.2 K) is the ${L_{6}}^{\pm}$ energy separation in the band-inverted $P b_{0.75} S n_{0.25} Se$ quantum wells (A) and $ɛ_{B}$ (>0) is that of the normal insulator $P b_{1 - y} E u_{y} Se$ barriers (B). V is the conduction band offset and $d$ is the well thickness. (d) Sketch of the evolution of the wave-function probability density and Dirac cones of the topological interface state as a function of temperature across the topological phase transition T*, illustrating the tunability of the penetration depth and hybridization gap of the TIS.
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Figure 2
(a) Relative transmission spectra [T(B)/T(0)] vs photon energy at 4.2 K for magnetic fields between 2 and 15 T. The curves are shifted for clarity. (b) Landau fan chart derived from the experiments and $k \cdot p$ envelope function calculations (solid lines). Red data points and lines are the TIS (E1-H1) interband transitions, black points and lines are the H2-E2 transitions, and the purple points and lines denote hybrid transitions between the TIS and E3. The green rectangle indicates the reststrahlen region of the $Ba F_{2}$ substrate that blocks the infrared transmission. (c) Calculated $k \cdot p$ sub-bands and dispersion of TQW-36. The inset shows the probability density (χ) of the E1 state, i.e., the TIS at 4.2 K. (d) Computed Landau levels vs magnetic field. The lowest LL is denoted by $N = - 1$ for each sub-band. The dashed lines are computed by taking into account an anticrossing (see Appendix pp4) of 5 meV between $- 1^{E 2 / H 2}$ and $0^{E 1 / H 1}$ levels. The red arrows in (a) and (d) indicate the allowed magneto-optical transitions between the TIS (E1-H1), and the black and purple arrows indicate the transition between the second QW sub-bands (E2-H2) and between the TIS and E3, respectively.
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Figure 3
(a) Relative transmission spectra [T(B)/T(0)] for TQW-36 recorded at $T = 60 – 200 K$ for four magnetic fields from 6 to 15 T. The curves are shifted vertically for clarity. (b)–(e) Resulting Landau level fan charts for $T = 60$ , 80, 120, and 200 K. The experimental data are shown as red dots, black circles, and purple circles for the E1-H1, E2-H2, and hybrid H1-E3 and E1-E3 transitions, respectively, and the fit by $k \cdot p$ theory is represented by the solid lines. The black arrows mark the E1-H1 Dirac gap Δ. The green rectangle indicates the reststrahlen region of the $Ba F_{2}$ substrate that blocks the infrared transmission. (f) Derived hybridization gap Δ (left axis) and Dirac mass $m_{D}$ (right axis) of the TIS vs temperature for TQW-36 (red dots) and TQW-24 (red open squares). The variation of the bulk gap $| ɛ_{A} |$ vs temperature for $P b_{1 - x} S n_{x} Se$ with the given composition ( $x = 0.25$ ) is shown by blue dots and solid line. The orange shaded region represents the topological regime where $Δ < | ɛ_{A} |$ for TQW-36.
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Figure 4
Results of $k \cdot p$ modeling of the data for TQW-36 at 4.2 K (a), 60 K (b), 80 K (c), and 200 K (d), showing for each temperature the quantum-well band alignment with the position of the TQW ground states (red line), the probability density χ of the wave function of the TIS (E1) (red line), and the second QW sub-band E2 (dashed line), and their corresponding band dispersions (from left to right). (e) Penetration depth of the TIS vs temperature for the two investigated samples. (f) Phase diagram of the topological $P b_{1 - x} S n_{x} Se$ quantum wells with $x = 0.25$ vs TQW thickness and temperature, respectively. The solid blue line represents the phase boundary between regime i and ii. The experimental data points for the TQW-36 and TQW-24 are shown as full circles and empty squares, respectively. The orange shading for (a), (b), (e), and (f) denotes the cases where $Δ < | ɛ_{A} |$ (regime i) and the Dirac states are either massless or massive but interfacial. In the other cases (c), (d), $Δ > | ɛ_{A} |$ and the massive Dirac states are no longer pinned to the interface.
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Figure 5
High-resolution x-ray-diffraction data for the superlattice samples TQW24 (a)–(c) and TQW 36(d)–(f) taken using Cu-Kα1 radiation. (a), (d) Symmetric scans along the [111] growth direction. SL0–SL±10/18, respectively, denote superlattice satellite peaks stemming from the PbSnSe/PbEuSe stack, and the peak labeled $Ba F_{2}$ corresponds to the (222) Bragg reflection from the substrate. (b), (e) Reciprocal space maps depicting the scattered intensity distribution around the symmetric (222) reciprocal lattice point. (c), (f) Same as (b), (e) but around the asymmetric (153) reflection. Reciprocal lattice point $q_{z}$ denotes the reciprocal space coordinate along [111], and $q_{x}$ denotes that along an in-plane direction.
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