Interplay of Dirac Nodes and Volkov-Pankratov Surface States in Compressively Strained HgTe

Preceded by the discovery of topological insulators, Dirac and Weyl semimetals have become a pivotal direction of research in contemporary condensed matter physics. While easily accessible from a theoretical viewpoint, these topological semimetals pose a serious challenge in terms of experimental synthesis and analysis to allow for their unambiguous identification. In this work, we report on detailed transport experiments on compressively strained HgTe. Because of the superior sample quality in comparison to other topological semimetallic materials, this enables us to resolve the interplay of topological surface states and semimetallic bulk states to an unprecedented degree of precision and complexity. As our gate design allows us to precisely tune the Fermi level at the Weyl and Dirac points, we identify a magnetotransport regime dominated by Weyl/Dirac bulk state conduction for small carrier densities and by topological surface state conduction for larger carrier densities. As such, similar to topological insulators, HgTe provides the archetypical reference for the experimental investigation of topological semimetals.

Popular Summary

Topological materials are a diverse range of materials with exotic electronic properties. The first class discovered was topological insulators, which have insulating interiors but conducting surfaces. Another, more recently discovered class is called topological semimetals. One of their most interesting electronic signatures is that, when subjected to a small magnetic field, the collective low-energy excitations of the electrons in them, called “quasiparticles,” are analogs of the Weyl fermion, a massless particle once proposed to describe neutrinos and long searched for by particle physicists. In this paper, we demonstrate that we can—with great control—turn HgTe, a prototype topological insulator, into a Weyl semimetal and explore the physics of both classes in one system at will.

Our approach exploits the physical insight that appropriate mechanical strain on HgTe turns its electronic structure from that of a topological insulator to that of a Weyl semimetal. We create such mechanical strain through molecular-beam-epitaxy-based sample growth. We also use semiconductor fabrication technology to fit the sample with a gate electrode, which allows us to vary the Fermi energy at will and to measure its electronic transport.

Tuning the Fermi level to a very special value (“Weyl points”), we clearly observe an anomaly specific to Weyl semimetals, called chiral anomaly, which manifests as a strong decrease in resistance when a magnetic field is applied parallel to the current. Tuning the Fermi level to other values, i.e., exploring other parts of the electronic structure, we observe electronic transport that we unambiguously attribute to surface conduction in the Weyl semimetal. This finding points to a deep connection between the electronic structural feature responsible for the original topological insulator behavior (“band inversion”) and that for the Weyl semimetal (“band crossing”).

At the same time, we do not see evidence of other surface states believed to be specific to Weyl semimetals, the so-called Fermi arcs. This is fundamentally intriguing: “Telltales” of Fermi arcs, such as “Fermi-arc electron transport” or superconductivity, were said to have been observed in other, previously studied Weyl semimetal materials. Since their electronic structures are very similar to that of our strained HgTe, our null finding here raises questions about the actual origin of those earlier telltales: Might their origin be simply in those topological surface states that dominate the physics of Weyl semimetals when the Fermi level is not exactly at the Weyl points?

Figure 1

Panel (a) shows a schematic picture of the band structure at an interface of two semiconductor materials with mutually inverted bulk bands (gray) with two types of interface states (named after the authors of Ref. [6] Volkov-Pankratov states), the massless interface states in black and the massive ones in dashed magenta. (b) Strain-imposed growth of HgTe. Tensile strain is realized by a CdTe substrate, compressive strain by a ${\mathrm{Cd}}_{0.9}{\mathrm{Zn}}_{0.1}\mathrm{Te}$ virtual substrate. Plotted as a function of $k_z$, i.e., along the growth direction, the band structure profile changes from quadratic band touching for unstrained HgTe to a topological bulk gap for tensile strain and a pair of linear level crossing for compressive strain. (c) Band structure plot for the compressive strain regime in the $k_z\text{−}{k}_x$ momentum plane. The bulk inversion asymmetry that would split the Dirac points into Weyl nodes is not accounted for in the calculation.

Figure 2

(a) Band structure along the $k_x$ direction of the surface Brillouin zone of a semi-infinite thick slab of compressively strained HgTe ($\approx 0.3\%$) from density functional theory. Spectral features in dark red show prominent surface character, while lighter colours display the continuum of bulk states with relatively weaker projections onto the (001) surface. The sharp dark lines highlight the massless Volkov-Pankratov states and Dirac point stemming from the ${\mathrm{\Gamma }}_6\text{−}{\mathrm{\Gamma }}_8$ band inversion. Because of the sharp interface and the absence of a Hartree potential in this calculations, the massive Pankratov states are shifted to high energies and not visible here. (b) Enlargement of (a) around the Fermi level showing two Weyl points with the same chirality $+2$. At large momenta, i.e., far from the Weyl points low-energy physics, the surface states originating from the Weyl nodes are the massless Volkov-Pankratov states that disperse in the unoccupied part of the spectrum [cf. the surface states above the Fermi level in (a)]. The color code identifies the spectral function $A(k_x,\omega )=(-1/\pi )\mathrm{Im}{G}_{\text{surf}}(k_x,\omega )$, where $G_{\text{surf}}(k_x,\omega )$ is the momentum and energy resolved surface Green’s function.

Figure 3

(a) Longitudinal resistance $R_{xx}$ as a function of the applied gate voltage $U_{\text{gate}}$ for $B=0$. The inset of (a) shows an optical microscope picture of the finished sample. (b) $R_{xx}$ as a function of the applied magnetic field $B$ along the current direction, as indicated by the sketch in the inset, for $U_{\text{gate}}=0\mathrm{V}$.

Figure 4

$R_{xx}$ as a function of the magnetic field applied parallel to the current for different gate voltages. The dashed lines represent $R_{xx}$ as a function of the applied gate voltage for zero (black) and maximum magnetic field ($\pm 14\mathrm{T}$) (gray).

Figure 5

(a) $R_{xx}$ as a function of the magnetic field $B$ rotated along $\varphi$ from perpendicular to the sample plane towards parallel to the current. In (b), $R_{xx}$ as a function of in-plane rotation with angle $\theta$, where $(\theta =0°)$ is parallel to the current and $\theta =85°$ nearly perpendicular to the current (inset) at $T=0.3\mathrm{K}$.

Figure 6

Longitudinal resistance $R_{xx}$ and Hall resistance $R_{xy}$ as a function of the out of plane magnetic field $B$ for high (a) electron and (b) hole densities at $T=0.3\mathrm{K}$.

Figure 7

(a) Calculated $k·p$ band structure for a 66-nm-thick sample including bulk inversion asymmetry terms and a Hartree potential corresponding to an applied gate voltage of $-2\mathrm{V}$ ($n=-4\times {10}^{11}{\mathrm{cm}}^{-2}$). The Hartree potential is presented in the inset of (a). The massless Volkov-Pankratov states are shown in red, while the massive Volkov-Pankratov states are in blue. The bulk subbands are shown in black and the Fermi energy $E_F$ is indicated by the orange line. (b) Derivative of the Hall conductivity with respect to the gate voltage ($\partial {\sigma }_{xy}/\partial U_{\text{gate}}$) as a function of the applied gate voltage $U_{\text{gate}}$ and magnetic field $B$ at $T=0.02\mathrm{K}$. The solid lines assign the experimental data to two types of Volkov-Pankratov surface (VPS) states: the massless ones (black) and massive ones (magenta).