Probing embedded topological modes in bulk-like GeTe-Sb2Te3 heterostructures

The interface between topological and normal insulators hosts metallic states that appear due to the change in band topology. While topological states at a surface, i.e., a topological insulator-air/vacuum interface, have been studied intensely, topological states at a solid-solid interface have been less explored. Here we combine experiment and theory to study such embedded topological states (ETSs) in heterostructures of GeTe (normal insulator) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Sb}_2$$\end{document}Sb2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Te}_3$$\end{document}Te3 (topological insulator). We analyse their dependence on the interface and their confinement characteristics. First, to characterise the heterostructures, we evaluate the GeTe-Sb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2Te\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3$$\end{document}3 band offset using X-ray photoemission spectroscopy, and chart the elemental composition using atom probe tomography. We then use first-principles to independently calculate the band offset and also parametrise the band structure within a four-band continuum model. Our analysis reveals, strikingly, that under realistic conditions, the interfacial topological modes are delocalised over many lattice spacings. In addition, the first-principles calculations indicate that the ETSs are relatively robust to disorder and this may have practical ramifications. Our study provides insights into how to manipulate topological modes in heterostructures and also provides a basis for recent experimental findings [Nguyen et al. Sci. Rep. 6, 27716 (2016)] where ETSs were seen to couple over thick layers.


X-ray diffraction analysis
X-ray (XRD) measurements were accomplished with a high-resolution Bruker D8 diffractometer, employing a scintillator detector with a mechanical slit of 3 mm. Figure 1 depicts a 2θ /θ measurement of a Sb 2 Te 3 /GeTe/Sb 2 3Te 3 sample. The scan was carried out in a range from 2θ = 3˘140 • with angular resolution of 0.1 • . Each data point was measured for 10 s. Besides the Si substrate peaks, signatures of the Sb 2 Te 3 and GeTe epilayers are found. The reflections of the epilayers' peaks are always (0,0,n), evidencing that both, Sb 2 Te 3 and GeTe, are single-crystalline films with the z-axis in growth direction.

Atom Probe Tomography
Atom Probe Tomography (APT) was carried out using a LEAP 4000X-HR from Cameca. The system is equipped with a laser-generating picosecond pulses at a wavelength of 355 nm. For the analysis, all samples were cooled down to a temperature of 20 K. The experimental data are collected at a laser pulse rate of 65 -125 kHz using laser powers between 1.75 and 5 pJ per pulse. The data are reconstructed using IVAS 3.6.
In order to create the APT depth profile, the reconstructed volume is sliced into 0.1 nm thick discs along the z-axis. The elemental atoms/ions in each disk are then used to calculate the concentrations of the respective element at the particular position along the z-/depth axis.

First Principles Calculation
In our calculations, we use the SIESTA 1 and Smeagol 2 program packages and adopt the double zeta plus polarization function (DZP) level basis set. Our results are consistent with the experimental findings presented in the previous section. To evaluate the band offset and identify the SG interfacial electronic states, we make use of the non-equilibrium Green's function technique 3 combined with DFT (NEGF-DFT). Since NEGF-DFT satisfies semi-infinite boundary condition, our results are free from artificial size effects, which are often problematic in slab model calculations. NEGF-DFT calculations were carried out using the Smeagol program package 2, 4 . We adopt an exchange correlation (XC) functional of the van der Waals (vW) correction, DF2 5 , for total energy calculations and use the local density approximation (LDA) to determine the band structure in NEGF-DFT calculations, where spin-orbit (SO) interaction are included.
At room temperature, bulk GeTe crystallises in a disordered rock salt (dRk) structure and Sb 2 Te 3 crystallises in a rhombohedral layered (Rh) structure 6 . Hence, the conventional hexagonal cell can be taken as the unit cell for both, where the (111) direction of the rock salt corresponds to the (0001) in the conventional hexagonal cell. We set the c-axis to the (0001) direction and define the z-coordinate along the c-axis. The SG interface plane is then perpendicular to the (0001) direction. In the conventional hexagonal cell, both GeTe (dRk) and

Four-band model calculations
This section discusses the description of the heterostructure band structure using the envelope function formalism that uses effective four-band models for the separate bulk phases, the parameters of which were presented in the main text.
For a slab geometry with an interface in the xy-plane, we loose translational invariance in the z direction. The Hamiltonian of a heterostructure takes the form of Eq. (3) in the main text with z-dependent coefficients A 1 (z), A 2 (z), . . ., i.e., Note that in the heterostructure, the second z-derivative must be written in a manifestly symmetric form to ensure a hermitian Hamiltonian, i.e., we replace 8 Eigenstates that solveĤΨ(z) = EΨ(z) depend on the parallel momentum k as well as the z coordinate. We consider solutions within a layer with the ansatz 8 where λ α is a complex-valued function of E and α, β = ±. The eigenstate equation for Ψ(z, E) reads with The value of λ α is obtained in closed analytical form by solving Eq. (5), which gives 9, 10 where

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For each value of β λ α (E), there are two linearly independent eigenvectors: The full solution in each layer is composed of these eight eigenvectors. Eight boundary conditions then determine the relative magnitude of the eigenvectors (with the overall magnitude fixed by the normalization) and the energy E. The condition on E follows from the determinant of the boundary conditions. It remains to determine boundary conditions. First, we require Ψ = 0 at the two boundaries of the slab (eight boundary conditions). Second, at each interface, we impose the continuity of the wavefunction and the current (which gives four boundary conditions per interface). This can be written as where the flux operator M is obtained by integrating the Hamiltonian across the interface

Continumm model applied to an Sb 2 Te 3 slab
Here we consider a single Sb 2 Te 3 (Rh) film to estimate the strength of the interlayer coupling. The single slab geometry has been solved in Refs. [9][10][11] . Results of this calculation using Sb 2 Te 3 parameters are shown in Fig. 4. Figure 4(a) shows the position of the valence band maximum and the conduction band minimum at the Γ point as a function of the slab thickness L s (orange lines). These states are surface states that are localised at the edge of the slab and gap out due to the intralayer coupling between the surface modes. The plot also includes the first bulk bands as blue lines. As for Bi 2 Se 3 11 , there is an oscillating exponential decay of the gap with increasing slab thickness L s . The inset illustrated the band structure near the Γ point for two values of the thickness L s = 5 nm and 10 nm, where the gapped surface Dirac state is indicated by the orange lines and bulk bands by blue lines. Figure 4(b) shows the gap as a function of slab thickness L s . As is apparent from the figure, surface states on the left and right edges are sufficiently decoupled for L s = 15 nm, at which point the energy gap caused by intralayer coupling between the surface states is negligible (less than 0.01 eV). Furthermore, as is apparent from Fig. 4(a), for thicknesses above 15 nm, the upper and lower energies of the surface states at Γ point coalesce to 0.14 eV below the Fermi level of Sb 2 Te 3 . The Sb 2 Te 3 film is thus "bulk-like" in in this regime.