Observation of Volkov-Pankratov states in topological HgTe heterojunctions using high-frequency compressibility

TL;DR

This study experimentally observes massive surface states in topological HgTe heterojunctions, revealing their influence on electronic properties and confirming theoretical models, which enhances understanding of topological phases for quantum technology applications.

Contribution

First experimental detection of massive surface states in HgTe topological insulators using high-frequency compressibility measurements, aligning with theoretical predictions.

Findings

Identification of conductance peaks and compressibility bumps at high energies

Observation of high mobility Dirac surface states up to large energies

Detection of charge metastability and Hall resistance anomalies

Abstract

It is well established that topological insulators sustain Dirac fermion surface states as a consequence of band inversion in the bulk. These states have a helical spin polarization and a linear dispersion with large Fermi velocity. In this article we report on a set of experimental observations indicating the existence of massive surface states. These states are confined at the interface and dominate equilibrium and transport properties at high energy and/or high electric field. By monitoring the AC admittance of HgTe topological insulator field-effect capacitors, we access the compressibility and conductivity of surface states in a broad range of energy and electric fields. The Dirac surface states are characterized by a compressibility minimum, a linear energy dependence and a high mobility persisting up to energies much larger than the transport bandgap of the bulk. New features are revealed at high energies with signatures such as conductance peaks, compressibility bumps, a strong charge metastability and a Hall resistance anomaly. These features point to the existence of excited massive surface states, responsible for a strong intersubband scattering with the Dirac states and the nucleation of metastable bulk carriers. The spectrum of excited states agrees with predictions of a phenomenological model of the topological-trivial semiconductor interface. The model accounts for the finite interface depth and the effect of electric fields. The existence of excited topological states is essential for the understanding of topological phases and opens a route for engineering and exploiting topological resources in quantum technology.

Equations

$$\hat{H}_{0}({\bf k},\Delta)=\left[\begin{array}[]{cccc}\Delta&\hbar v_{F}k_{y}&0&\hbar v_{F}(k_{z}-ik_{x})\\ \hbar v_{F}k_{y}&-\Delta&\hbar v_{F}(k_{z}-ik_{x})&0\\ 0&\hbar v_{F}(k_{z}+ik_{x})&\Delta&-\hbar v_{F}k_{y}\\ \hbar v_{F}(k_{z}+ik_{x})&0&-\hbar v_{F}k_{y}&-\Delta\end{array}\right],$$

$${\mathcal{E}}_T=\frac{{\Delta }_1+{\Delta }_2}{e\xi }.$$

$$\hat{H}_{s0}^{\prime}=\hat{U}^{\dagger}\hat{H}_{s0}\hat{U}=\left[\begin{array}[]{cccc}\hbar v_{F}k_{x}&\hbar v_{F}k_{y}&0&\sqrt{2\hbar v_{F}e\mathcal{E}_{T}}\hat{a}^{\dagger}\\ \hbar v_{F}k_{y}&-\hbar v_{F}k_{x}&\sqrt{2\hbar v_{F}e\mathcal{E}_{T}}\hat{a}^{\dagger}&0\\ 0&\sqrt{2\hbar v_{F}e\mathcal{E}_{T}}\hat{a}&\hbar v_{F}k_{x}&-\hbar v_{F}k_{y}\\ \sqrt{2\hbar v_{F}e\mathcal{E}_{T}}\hat{a}&0&-\hbar v_{F}k_{y}&-\hbar v_{F}k_{x}\end{array}\right],$$

$${\epsilon }_{m,k_x,k_y}^{(\pm )}=\pm \sqrt{{\hslash }^2{v}_F^2(k_x^2+k_y^2)+2\hslash v_{F}e{\mathcal{E}}_{T}m},$$

$$E_{\pm m}={\epsilon }_{m,k_x=k_y=0}^{(\pm )}=\pm \sqrt{2\hslash v_{F}e{\mathcal{E}}_{T}m}=\pm \sqrt{2\hslash v_F({\Delta }_1+{\Delta }_2)m/\xi }$$

$$E_{\pm m}(\mathcal{E})=\pm \sqrt{m}\sqrt{2\hslash v_{F}e{\mathcal{E}}_T}(1-{\mathcal{E}}^2/{\mathcal{E}}_T^2{)}^{3/4},$$

$$n_m(\mathcal{E})=\frac{m(m+1)}{2}\frac{e{\mathcal{E}}_T}{2\pi \hslash v_F}(1-{\mathcal{E}}^2/{\mathcal{E}}_T^2{)}^{1/2}.$$

$$v_F(\mathcal{E})=v_F(1-{\mathcal{E}}^2/{\mathcal{E}}_T^2{)}^{1/2}.$$

$$\rho (\epsilon )=\frac{|\epsilon |}{2\pi {\hslash }^2{v}_F^2(1-{\beta }^2)}\sum _l\Theta \left[|\epsilon |-(1-{\beta }^2{)}^{3/4}\sqrt{2\hslash v_{F}e{\mathcal{E}}_{T}m}\right],$$

$$\hat{H}_{s}^{\prime}=\left[\begin{array}[]{cccc}\hbar v^{\prime}_{F}k_{x}&\hbar v^{\prime}_{F}k_{y}&0&\sqrt{2\hbar v^{\prime}_{F}e\mathcal{E}^{\prime}_{T}}\hat{b}\\ \hbar v^{\prime}_{F}k_{y}&-\hbar v^{\prime}_{F}k_{x}&\sqrt{2\hbar v^{\prime}_{F}e\mathcal{E}^{\prime}_{T}}\hat{b}&0\\ 0&\sqrt{2\hbar v^{\prime}_{F}e\mathcal{E}^{\prime}_{T}}\hat{b}^{\dagger}&\hbar v^{\prime}_{F}k_{x}&-\hbar v^{\prime}_{F}k_{y}\\ \sqrt{2\hbar v^{\prime}_{F}e\mathcal{E}^{\prime}_{T}}\hat{b}^{\dagger}&0&-\hbar v^{\prime}_{F}k_{y}&-\hbar v^{\prime}_{F}k_{x}\end{array}\right],$$

$$\hat{b}=\frac{1}{(1-{\beta }^2{)}^{1/4}\sqrt{2\hslash v_F^{\prime }e{\mathcal{E}}_T^{\prime }}}\left\{\hslash v_F{k}_x-i\sqrt{1-{\beta }^2}\left[e{\mathcal{E}}_{T}z-\left({\Delta }_1+\frac{\beta }{{\beta }^2-1}\epsilon \right)\right]\right\},$$

$${\hat{b}}^{†}=\frac{1}{(1-{\beta }^2{)}^{1/4}\sqrt{2\hslash v_F^{\prime }e{\mathcal{E}}_T^{\prime }}}\left\{\hslash v_F{k}_x+i\sqrt{1-{\beta }^2}\left[e{\mathcal{E}}_{T}z-\left({\Delta }_1+\frac{\beta }{{\beta }^2-1}\epsilon \right)\right]\right\}.$$

Full text

I Introduction

Topological materials have emerged [qizhang2011review, ] as a new class of matter with properties arising from peculiar crystal properties. These include topological insulators (TIs) with band inversion due to e.g. spin-orbit interaction [Bernevig2006science, ; Fu2007prl, ], induced topological superconductors [qizhang2011review, ], and topological semi-metals [Hosur1013crp, ; Xu2015science, ]. They have attracted much attention as hosts of exotic modes such as Majorana, Dirac or Weyl fermions, depending on symmetry and dimensionality. Topology also leaves macroscopic foot-prints such as the Quantum Spin Hall Effect in 2D-TIs [Konig2007science, ], quantum Hall and Dirac screening in 3D-TIs [Brune2014prx, ], the fractional Josephson effect in TI-SC junctions [Wiedemann2016ncomm, ; Bocquillon2016nnat, ; Deacon2016arxiv, ], and more recently the quantized Faraday rotation in 3D-TIs [Wu2016science, ; Dziom2017ncomm, ].

In spite of intensive studies some important questions remain. One concerns the robustness of the topological states in practical implementations for quantum or classical topological electronics. Another is the possibility of new members in the family of topological states and the related question of higher energy excitations. Indeed, as predicted in Ref.[Karzig2013prx, ], a limiting factor for the dynamics of Majorana braiding is the existence of surface states at higher energies. One may wonder if similar high-energy states accompany Dirac states in 3D-TIs. Remarkably this question has been anticipated theoretically by Volkov and Pankratov in Ref.[Volkov1985jetp, ] who considered topological hetero-junctions (THJ) between inverted and non inverted PbxSn1-xTe semiconductors. They predicted the existence of both the helical Dirac ground state, and of a series of massive surface states, thereafter called Volkov-Pankratov states (VPS). VPS theory is sketched below and detailed in Ref.[Tchoumakov2017arXiv, ] with the important addition of field effects. Some of the properties of a THJ will certainly depend on the existence of VPS; let us mention the anomalous screening properties in 3D TIs reported in Ref.[Brune2014prx, ]. The present work aims at unveiling VP states and highlighting their role in transport by a high-frequency (RF) measurement of compressibility in high-mobility strained HgTe 3D-TIs using a RF capacitor geometry which is a building block of topological electronics.

The RF compressibility approach complements the ARPES spectroscopy which has proven very efficient in identifying Dirac states (see e.g. Refs.[Hsieh2008nature, ; Hsieh2009nature, ; Zhang2009nphys, ; Liu2015prb, ]). Indeed ARPES is not fully versatile as it is limited to vacuum-TI interfaces. It therefore excludes TI capping and gating which turn out to be essential to detect VPS as well as for a future topological electronics. While massive surface states were reported for oxidized Bi2Se3 and Bi2Te3 samples [Bianchi2010ncomm, ; Bahramy2012ncomm, ], they were extrinsically induced by on-purpose surface doping. On the other hand, the transport technique is sensitive to buried interface states, even in the presence of capping and top gating and is therefore well suited to map Dirac states and the VPS in THJs. In fact a signature of VPS lies in their anomalous electric field $\mathcal{E}$ response, coupled with the Dirac nature of states in a THJ, such an electric field susceptibility can be regarded as a pseudo magnetic response with an effective magnetic field $\mathcal{E}/v_{F}$, where $v_{F}$ is the velocity of Dirac carriers. With $\mathcal{E}\sim 10^{8}\;\mathrm{Vm^{-1}}$ and $v_{F}\sim 10^{6}\;\mathrm{ms^{-1}}$, one has access to very large fields $\mathcal{E}/v_{F}\sim 100\;\mathrm{T}$. Our technique is based on high-frequency compressibility measurements introduced in Ref.[Pallecchi2011prb, ]. The device is a metal-insulator-TI field-effect capacitor (MITI-cap), with a metallic plate acting as a DC and RF gate and the THJ acting as a RF drain. The broad-band admittance measures the differential capacitance per unit area $C(V_{g})$ and the TI sheet conductivity $\sigma(V_{g})$ as function of the DC gate voltage $V_{g}$. From $C(V_{g})$ we extract the carrier density $n(V_{g})$, the quantum capacitance $C_{Q}(V_{g})$, analog to the compressibility $\chi=\partial n/\partial\mu=C_{Q}/e^{2}$, from which we deduce the surface Fermi energy $\mu(V_{g})$ that is essential for VPS spectroscopy. The joint measurement of $\sigma$ and $C_{Q}$ gives access to the diffusion constant $\mathcal{D}(\mu)=\sigma/C_{Q}$ used here as a spectroscopic tool of the surface states. Below the first VPS gap ($\mu\lesssim E_{1}$), we find $\mathcal{D}\propto\mu$ and a large mobility $\mu_{e}=2\mathcal{D}/\mu\simeq 12\;\mathrm{m^{2}V^{-1}s^{-1}}$, characteristic of the Dirac state. As chemical potential crosses the VPS1 band edge ($\mu\simeq E_{1}$), an efficient inter-subband scattering sets in [Bastard1996, ], that is responsible for a drop of $\mathcal{D}$. The scattering spectroscopy turns out to have a better resolution than compressibility itself which is eventually blurred by residual heavy-hole bulk contributions in cubic HgTe.

A fundamental concern is the distinction between VPS and classical massive surface states such as reported in Refs.[Bianchi2010ncomm, ; Bahramy2012ncomm, ]. Indeed both are massive, doubly degenerate, and occur at similar energies. The main distinction lies in the nature of confinement: relativistic VPS are confined in an energy gradient, whereas classical states are trapped in a potential well involving electrostatic band bending. Experimentally they can be distinguished by their electric field sensitivity as discussed below (Section IV) and in Ref.[Tchoumakov2017arXiv, ]: VPS are red-shifted whereas trivial states are naturally blue-shifted as a positive electric field enhances quantum confinement whereas a negative one destroys the well altogether. This anomalous field-effect is linked to the existence of a critical field $\mathcal{E}_{T}=(\Delta_{2}+\Delta_{1})/e\xi$ at which VPS collapse. As discussed in Ref.[Tchoumakov2017arXiv, ] it corresponds to the situation where the gap alignment between the trivial and topological regions closes. Here $2\Delta_{2}$ (resp. $-2\Delta_{1}$) are the positive (resp. negative) band gaps of the direct (resp. inverted) semiconductors, and $\xi$ is the penetration depth of the Dirac state. $\mathcal{E}_{T}$ sets both the energy scale of zero-field VPS subband series, $E_{m}(0)=\sqrt{2m\hbar v_{F}e\mathcal{E}_{T}}$ with $m$ a positive integer, and the electric field dependence $E_{m}(\mathcal{E})=E_{m}(0)(1-\mathcal{E}^{2}/\mathcal{E}_{T}^{2})^{3/4}$. In particular the $E_{1}(\mathcal{E})$ curve defines the phase boundary for Dirac screening. Taking $\xi\gtrsim\hbar v_{F}/\Delta_{1}\simeq 5\;\mathrm{nm}$ and $\Delta_{1}\simeq\Delta_{2}/3\simeq 0.15\;\mathrm{eV}$ [Baum2014prb, ] for the CdHgTe/HgTe smooth THJs used in this work, the predicted critical field and VPS1 energy are $\mathcal{E}_{T}\lesssim 1.2\;10^{8}\;\mathrm{Vm^{-1}}$ and $E_{1}\lesssim\Delta_{1}\sqrt{2(1+\Delta_{2}/\Delta_{1})}\simeq 0.4\;\mathrm{eV}$. Five signatures of the VPS are reported here : the observation of a Dirac-state scattering peak in undoped samples at an energy $E_{1}\simeq 0.35\;\mathrm{eV}$ close to the predicted value, the onset of a secondary type of carrier and Dirac screening breakdown at the same energy, the observation of three scattering peaks obeying the predicted VPS series $E_{m}=\sqrt{m}E_{1}$, and their electric-field red-shift in oxide capped doped samples.

The paper is organized as follows. In Section II we describe the experimental principles including the fabrication of high-mobility CdHgTe/HgTe THJs. Section III is a report of RF surface compressibility measured over a broad range of electric field in undoped and doped HgTe samples. Special attention is devoted to charge metastability effects and their relationship to VPS. Observations are confirmed by control experiments performed in Hall-bars with similar gate stacks. Section IV present a heuristic model of the VPS and their electric field dependence, complemented by $k\cdot P$ numerical calculations. The comparison between theory and experiment in Section V supports the existence of VPS in CdHgTe/HgTe topological hetero-junctions. We conclude in Section VI with perspectives offered by the VPS for a future topological electronics.

II Experimental principles

The MITI-Caps are based on high mobility HgTe/CdHgTe hetero-structures (Figure1-a) grown by molecular beam epitaxy [Brune2011prl, ; Brune2014prx, ; Kozlov2014prl, ] where the HgTe layer, of thickness $t_{HgTe}=68\;\mathrm{nm}$, is strained by the CdTe insulating substrate so as to open a small bandgap $\simeq 0.02\;\mathrm{eV}$ in the topological HgTe layer between the light electron and heavy hole $\Gamma_{8}$-bands. The gap is smaller than the light electron- light hole ($\Gamma_{8}$-$\Gamma_{6}$) inverted bandgap, $-2\Delta_{1}=\Delta_{\Gamma_{6}-\Gamma_{8}}\simeq-300\;\mathrm{meV}$ [Guldner1973prb, ], responsible for the Dirac surface states. We use a wet etching technique [Bendias2017arXiv, ] to design a mesa and optical lithography to deposit contacts and define the gated area of the capacitor. A gold gate electrode is evaporated on top of a $10\;\mathrm{nm}$-thin HfO2 insulating layer, grown by low temperature atomic layer deposition (ALD) techniques. An ohmic contact, with a resistance $R_{c}\simeq 50\;\mathrm{Ohms}$, is deposited by Ge/Au evaporation. A false color scanning electron microscopy (SEM) image of the gate-contact alignment is shown in Figure1-b. Two series of samples have been fabricated. Type $A$ samples ($S_{A}$) are covered by a $5\;\mathrm{nm}$ Cd0.7Hg0.3Te capping layer protecting the HgTe during the process and providing a well-defined trivial insulator boundary of band gap $2\Delta_{2}^{A}\simeq 0.9\;\mathrm{eV}$ [Baars1972ssc, ; Capper2010, ] (Figure1-a). The total insulator thickness is $t_{ins}=15\;\mathrm{nm}$. Type B samples ($S_{B}$) are devoid of capping layer and HfO2 is directly deposited on HgTe (not shown in the figure). This entails an unintentional electron doping of the HgTe bulk (density $n_{0}\simeq 2.6\;10^{12}\;\mathrm{cm^{-2}}$ see below). The HgTe-HfO2 interface involves an oxidized HgTe insulating layer of unknown, and presumably small, bandgap ($\Delta_{2}^{B}\ll\Delta_{2}^{A}$). We have characterized in-situ the permittivity, \varepsilon_{\text{HfO{}{2}}}\simeq 3.6, of the thin oxide layer using a metal-oxide-metal (MOM) control capacitor. This value is process dependent and deviates here from to the accepted bulk value \varepsilon_{\text{HfO{}{2}}}\simeq 11.7 [Neumaier2015kaprun, ]. Taking $\varepsilon_{CdHgTe}\simeq 8.5$ for the capping layer [Baars1972ssc, ; Capper2010, ] we estimate the capacitance of our insulating stack $C_{ins}=2.65\;\mathrm{mF/m^{2}}$ in agreement with our measurement below. The MITI-Caps are embedded in coplanar wave guides (Fig.1-d), designed for [math]–$40\;\mathrm{GHz}$ measurements. Five similar MITI-Caps of varied dimensions are fabricated per chip complemented by MOM, dummy and thru-line structures used for calibration purpose (see below). For definiteness, experiments reported here refer to two capacitors of dimensions $L\times W=44\times 20\;\mathrm{\mu m}$, one of type $A$ and one of type $B$.

Compressibility measurements have been considered so far mainly for the characterization of Bi2Se3 thin crystals [Xu2015nl, ], and for the purpose of Landau-level spectroscopy in HgTe films [Kozlov2016prl, ]. Here we subtract the series combination of insulator capacitance $C_{ins}$ to the total capacitance $C_{tot}$ to obtain the quantum capacitance correction, $C_{Q}=e^{2}\chi$. $C_{tot}$ can be tuned by a DC gate voltage $V_{g}$ controlling altogether the charge density $n$ and the applied electric field $\mathcal{E}$. High frequency measurements ($\lesssim 10\;\mathrm{GHz}$) give access to the dissipative regime, governed by the conductivity $\sigma$, and yield the diffusion constant $\mathcal{D}=\sigma/C_{Q}$. Thin insulators allow for high fields eventually exceeding the critical field $\mathcal{E}_{T}\simeq 1.2\;10^{8}\;\mathrm{Vm^{-1}}$ of our CdHgTe/HgTe THJs. Compressibility measurement plays an important role here. First, it provides an absolute chemical potential scale with a zero given by the Dirac dip, and secondly it gives the precise behavior of the diffusion constant, which helps identifying the scattering mechanisms.

Following earlier work on graphene [Pallecchi2011prb, ] we primarily measure the electronic compressibility by vectorial network analyzer (VNA) techniques (frequency range $\omega/2\pi=50\;\mathrm{kHz}$–$8\;\mathrm{GHz}$) in a cryogenic RF probe station at $10\;\mathrm{K}$. Standard in-situ calibration techniques enable to de-embed the circuitry and the contact resistance contributions from the MITI-Cap admittance $Y(\omega)$ of interest. The admittance spectrum $Y(\omega)$ is accurately described by a distributed resistance-capacitance model, $Y(\omega)/W=+\jmath C_{tot}\omega L\times\tanh(\sqrt{\jmath\sigma^{-1}C_{tot}\omega L^{2}})/\sqrt{\jmath\sigma^{-1}C_{tot}\omega L^{2}}$. With $1/C_{tot}=1/C_{ins}+1/C_{Q}$, the measured quantum capacitance per unit area $C_{Q}=e^{2}\partial n/\partial\mu$ is a non-local compressibility where $n$ stands for the total TI charge and $\mu$ the value of the chemical potential at the TI surface. When two fluids are in mutual equilibrium, e.g. Dirac and/or VPS with bulk states (BSs), they share the same surface chemical potential and their respective densities simply add in the total compressibility. The large thickness of our HgTe-TI samples allows us to model them as semi-infinite, and the compressibility is the sum of the upper surface and a semi-infinite bulk contribution. In the doped sample $S_{B}$, the sheet conductance includes a frequency-independent BS contribution.

Typical complex admittance spectra of $S_{A}$ are shown in Figs.2-(a-c) for $V_{g}=0$ (Dirac point), $V_{g}=0.5$, $3\;\mathrm{V}$ (electron doped regime). We observe three frequency domains: a quasi-stationary domain $Y/W=\jmath C_{tot}L\omega+(C_{tot}L\omega)^{2}\sigma^{-1}L/3$ (green shading), an intermediate domain (unshaded) and finally the evanescent wave domain (grey shading), where $\Re(Y)\simeq\Im(Y)\simeq W\sqrt{C_{tot}\omega\sigma/2}$. The agreement between the experimental spectra and the distributed RC line model (solid lines in Figs.2-(a-c)) is excellent and warrants the accuracy of the RF MITI-Cap approach of compressibility. The RC line model allows extracting reliable value of the frequency independent total capacitance $C_{tot}$, insulator capacitance $C_{ins}=C_{tot}(V_{g}\rightarrow-\infty)$, quantum capacitance $C_{Q}(V_{g})=C_{ins}C_{tot}/(C_{ins}-C_{tot})$ (Fig.2-d), and conductivity $\sigma(V_{g})$. Accounting for the non-linear charge voltage characteristic we calculate the total electron density $n=\frac{1}{e}\int C_{tot}(V_{g})dV_{g}$, the applied electric field $\mathcal{E}_{ins}=ne/(C_{ins}t_{ins})$ and the surface chemical potential, $\mu(V_{g})=e\int C_{tot}/C_{Q}dV_{g}$. This allows plotting $C_{Q}(\mu)$ and $\sigma(\mu)$ in Fig.2-d as well as the diffusion constant $\mathcal{D}(\mu)=\sigma/C_{Q}$ used for scattering spectroscopy in Fig.2-e.

A similar chip with the same HgTe/CdHgTe hetero-structure is produced, hosting $600\times 200\;\mathrm{\mu m}$ Hall-bar structures, with the same optical lithography wet-etching technique equipped with a gold electrode as gate on top of $10\,\mathrm{nm}$ thick HfO2 insulator. Standard low frequency AC techniques in a four point measurement configuration are used at a temperature of $2\,\text{K}$ in magnetic fields up to $2\,\mathrm{T}$ to measure the magnetic field and gate voltage $V_{g}$ dependent longitudinal sheet resistance $R_{xx}$ and Hall resistance $R_{xy}$. The carrier density $n$ can be accessed in the classical Drude transport regime by the magnetic field dependence of the Hall resistance. Two samples of types $A$ and $B$ analog to the MITI-Cap samples with and without capping layer (not shown) were measured.

III Experiment

Let us first give an overview of the compressibility measurements in samples $S_{A}$ and $S_{B}$. As shown in the $10\;\mathrm{kHz}$ lock-in measurements of Fig.3, a striking feature is the existence of a fully reversible compressibility in Figs.3-a (and a quasi reversible one in Figs.3-b) below a maximum gate voltage $V_{g}^{m}=3\;\mathrm{V}$. The absence of metastability warrants the absence of a bulk contribution consistently with the magneto-transport measurements of Ref.[Brune2014prx, ]. By contrast we observe above $V_{g}^{m}=3\;\mathrm{V}$ a prominent charge metastability up to the maximum amplitude, $V_{g}^{m}\gtrsim 10\;\mathrm{V}$, sustainable by the insulating dielectric stack. Metastability signals the onset of a second type of carrier and characterizes the breakdown of Dirac screening. The hysteresis loops have a butterfly shape and a trend toward saturation at large negative $V_{g}$. We use this asymptotic value (horizontal lines in Figs.3-a,b) to estimate the insulator capacitance $C_{ins}$. Note that the reversible curves are eventually history dependent themselves, the displayed ones corresponding to the first charging cycles after cool-down to $T=10K$. Reversible cycles subsequent to a high field sweep are qualitatively similar but with a dip that is shifted in gate voltage and amplitude signaling the existence of trapped bulk carriers. We have checked that the MOM capacitance is linear and reversible in the same conditions, confirming that hysteresis is indeed associated to metastability in HgTe charging.

III.1 Compressibility of Dirac surface states

In both samples the reversible charge response shows a capacitance dip attributed to the Dirac surface states near charge neutrality. In sample $S_{A}$ the dip is sharp and located at $V_{g}=0\;\mathrm{V}$ as expected for an intrinsic TI response. In sample $S_{B}$ the dip is shifted to $V_{g}=-1.2\;\mathrm{V}$, corresponding to an n-type doping density $n_{0}=-2.6\;10^{12}\;\mathrm{cm^{-2}}$ (bulk donor density $N_{D}=4\;10^{17}\;\mathrm{cm^{-3}}$), and smeared by this bulk carrier contribution. On subtracting the insulator contribution $1/C_{ins}$ and integrating the relevant quantities, we deduce the surface chemical potential $\mu$ in the reversible state which is plotted in Figure 3-c (sample $S_{A}$) as a function of $\mathcal{E}_{ins}$. The S-shape curve represents the charging path of the MITI-Cap in the $\mu(\mathcal{E}_{ins})$ representation. Similar curves are obtained for sample $S_{B}$ (not shown). Finally, Fig.3-d shows the reversible $C_{Q}(\mu)$ plots of $S_{A}$ (blue line) and $S_{B}$ (green line). As expected from the additivity of compressibilities in a two-fluid system, one has $C_{Q}^{B}>C_{Q}^{A}$. In order to quantify the effect of bulk carriers we have added in the figure the theoretical expectation (green dashed line) for the additional contribution of a trivial semiconductor to quantum capacitance with the above estimated dopant density $N_{D}$. From the good experimental agreement we conclude that the reversible $C_{Q}(\mu)$ plot of sample $S_{A}$ is a close estimate of the intrinsic TI response. The linear energy dependence precludes an interpretation of compressibility in terms of conventional massive surface states. In fact, its energy dependence can be mapped to a Dirac fermion density of states, $C_{Q}=e^{2}\mu/(2\pi\hbar^{2}v_{F}^{2})$, with $v_{F}=1.6\;10^{6}\;\mathrm{ms^{-1}}$ in the electron regime and $v_{F}=0.5\;10^{6}\;\mathrm{ms^{-1}}$ in the hole regime. These values are larger than the accepted $v_{F}\simeq 1\pm 0.2\;10^{6}\;\mathrm{ms^{-1}}$ [Liu2015prb, ]. Remarkably this Dirac-like response extends over a broad energy range ($\mu=-0.05\rightarrow+0.30\;\mathrm{eV}$) widely exceeding the bulk transport bandgap, consistent with the Dirac screening reported in Ref.[Brune2014prx, ]. In the data reduction we have included a constant background $\sim 10\;\mathrm{mFm^{-2}}$ whose origin is not fully clarified. A possible explanation would be the nesting of the Dirac point in the heavy-hole $\Gamma_{8}$ branch, meaning that the observed compressibility minimum results from a superposition of a Dirac contribution with that of a heavy-hole $\Gamma_{8}$ band. 111The resurgence of a bulk contribution in the vicinity of a Dirac point is expected however, due to the vanishing Dirac screening at neutrality, illustrated by the divergence of the effective screening length $\lambda=\varepsilon/C_{Q}$. Such effects have been recently demonstrated in gated graphene-on-metal contacts [Wilmart2016scirep, ]. The analog of TIs would be the case of graphene-on-semiconductor, where the finite compressibility of the small gap semiconductor is taken into account. This case, which has not been investigated yet, is beyond the scope of our paper that focusses on the high field regime. In the following we shall therefore focus on the electron-doped regime and subtract a background compressibility $\sim 0.01\;\mathrm{Fm^{-2}}$.

III.2 Scattering spectroscopy evidence of the first VPS

We focus here on the sample $S_{A}$ which is the closest realization of an intrinsic CdHgTe/HgTe smooth THJ. Fig.2-d shows $C_{tot}(\mu)$ (blue dots) and $\sigma^{-1}(\mu)$ (red dots) deduced from admittance spectra in the reversible regime. The DC capacitance (grey dots) has been reproduced for comparison. The good agreement, beside a small amplitude shift due to calibration, shows that compressibility is frequency-independent, confirming our analysis of admittance spectra that dissipative effects mainly stem from finite conductivity and not from an intrinsic $\chi(f)$ dependence. The resistivity exhibits a Dirac-like peak (DP) which is slightly shifted with respect to the capacitance dip, supporting our conjecture that the DP is nested in the $\Gamma_{8}$ band. The shape of the resistance reflects the electron-hole asymmetry of HgTe. The diffusion constant $\mathcal{D}(\mu)=\sigma/C_{Q}$ in Fig.2-e shows a minimum at $\mu\simeq-0.05\;\mathrm{eV}$ and a linear increase in the electron regime up to $\mu\simeq 0.3\;\mathrm{eV}$, corresponding to a constant mobility $\mu_{e}=2e\mathcal{D}/\mu\simeq 12\;\mathrm{m^{2}V^{-1}s^{-1}}$. This large value is characteristic of massless Dirac fermions and is comparable with values deduced from magneto-transport experiments [Brune2014prx, ]. The diffusion constant $\mathcal{D}(\mu)$ is a marker of the scattering mechanism, its linear dependence is indicative of a screened charge disorder [Nomura2006prl, ; Culcer2010prb, ]. The most remarkable feature is a drop of $\mathcal{D}(\mu)$ at $\mu\simeq 0.3\;\mathrm{eV}$. We attribute this scattering peak to the onset of a new scattering channel for Dirac fermions when the surface Fermi energy $E_{1}=\mu-E_{DP}\simeq 0.35\;\mathrm{eV}$ crosses the bottom of a massive subband [Bastard1996, ].

Sample $S_{A}$ being intrinsic we can rule out the possibility that the scattering peak corresponds to a doping induced massive surface state as reported in [Bianchi2010ncomm, ; Bahramy2012ncomm, ]. The scattering peak energy $E_{1}\simeq 0.35\;\mathrm{eV}$ is very close to the theoretical estimate for the VPS $\Delta_{1}\sqrt{2(1+\Delta_{2}/\Delta_{1})}\simeq 0.4\;\mathrm{eV}$. Identifying this energy with that of the first VPS band edge, $E_{1}=\sqrt{2\hbar v_{F}(\Delta_{1}+\Delta_{2})/\xi}$ (see Eq. (5) in the theory section IV), gives $\xi\simeq 6\;\mathrm{nm}$ (with $v_{F}=1.10^{6}\;\mathrm{m/s}$) close to the estimate $\xi\simeq 5\;\mathrm{nm}$ deduced from numerical studies in Ref.[Baum2014prb, ]. From the agreement between theoretical and experimental determinations of the peak energy we conclude that the scattering peak observed in Fig.2-d is a signature of the topological VPS subband [Volkov1985jetp, ,Tchoumakov2017arXiv, ] described in Sec. IV. A theoretical modeling of the peak shape, which would involve a detailed analysis of scattering and screening mechanisms in HgTe, remains beyond the scope of our work. $C_{Q}$ being monotonic in the VPS1 energy range, the peak in $\mathcal{D}(\mu)$ translates into a peak in $\sigma(\mu)$. Fig.5-a is an extension of Fig.2-d at the maximum gate voltage span, taking the upward voltage sweep of the hysteresis loops. It shows that the conductance peak evolves into a resistance plateau at large electric field, and that no additional conductance peak is observed in this sample.

III.3 Additional signatures of VPS carriers

VPS, charge metastability and Dirac screening breakdown. In Fig.4 we analyze two complementary aspects of the TI charging phenomenology. Fig.4-a shows a set of hysteretic charging characteristics $C_{tot}(V_{g})$ for increasing gate voltage amplitudes $V_{g}^{m}$ in sample $S_{A}$. They have a butterfly shape with two capacitance minima that are shifted upward or backward depending on the direction of the voltage sweep. We attribute this metastability to the breakdown of Dirac screening and the nucleation of bulk carriers at high electric field, either electrons or holes depending on the field history. The loop width, $\Delta V_{g}(V_{g}^{m})$ (inset) sets-in at $V_{g}^{m}\simeq 3\;\mathrm{V}$ and gradually increases up to $\Delta V_{g}\simeq 3\;\mathrm{V}$ at $V_{g}^{m}\simeq 10\;\mathrm{V}$, corresponding to a bulk charge variation $\Delta n_{B}=\pm 2\;10^{12}\;\mathrm{cm^{-2}}$. Note that the capacitance minima are shifted upward at large $V_{g}^{m}$ supporting the contribution of additional bulk carriers in the compressibility. The presence of charge metastability signals the Dirac screening breakdown.

Hall signature of VPS carriers. Figs.4-b,c show the $R_{xy}$, $R_{xx}$ magneto-transport measurements performed in a type-A Hall bar. The capped sample shows a clear maximum of the sheet resistance accompanied by a change in the sign of the Hall resistance (not shown) indicating the state with lowest carrier density at $V_{g}\approx-1.0\;\mathrm{V}$. Increasing the gate voltage (electron side) decreases the sheet resistivity $\sigma^{-1}$ up to $V_{g}\approx 1.0\;\mathrm{V}$. For higher gate voltages $\sigma^{-1}$ increases monotonically with $V_{g}$ (Fig.4-c) and a non-linearity in the Hall resistance develops for low magnetic fields (inset of Fig.4-b). The uncapped sample consistently shows additional peaks in the longitudinal resistance (not shown), with the only difference, that the p-conducting regime could not be reached in the available gate voltage range due to the high (unintentional) n-doping. Beside the Dirac peak (at $V_{g}=-1.1\;\mathrm{V}$), $R_{xx}(V_{g})$ exhibits a secondary peak (at $V_{g}\simeq 3\;\mathrm{V}$) reminiscent of the one observed in the capacitor measurement of Fig5-a. The Hall resistance $R_{xy}\propto B$ develops an anomaly at $V_{g}\gtrsim 1\;\mathrm{V}$ (inset of Fig.4-b) signaling the advent of a secondary carrier type. The anomaly is best depicted by plotting the derivative $\partial R_{xy}/\partial B$ in Fig.4-c, where the base line is shifted according to the applied gate voltage. Data are fitted using an empirical function $\alpha/\cosh^{2}(B/B_{0})$ (solid lines) used to extract the magnetic field range $B_{0}$ of the anomaly. We find $B_{0}\propto(V_{g}-V_{g1})$ with $V_{g1}=1.0\;\mathrm{V}$ corresponding to the resistance minimum in that sample. This shows that the scattering peak is indeed accompanied by the nucleation of a second carrier type, consistently with the VPS-subband interpretation. A similar analysis has been carried out in type-B samples where we observe multiple $R_{xx}$ peaks (not shown), also reminiscent of the capacitor measurement in Fig.5-b.

III.4 The VPS phase diagram

The heuristic model below predicts a VP state series, $E_{m}=\sqrt{m}E_{1}$. In sample $S_{A}$ the second VPS, at $E_{2}\simeq 0.5\;\mathrm{eV}$, is at the limit of experimental reach and deep in the metastable regime. This difficulty is circumvented in sample $S_{B}$ which has a finite n-type chemical doping and smaller $E_{m}$’s due to a smaller capping bandgap $\Delta_{2}$ (see below). As shown in Fig.5-b the scattering spectroscopy is even richer: besides the Dirac peak identified by the capacitance dip, we observe two resistance peaks in the electron side and one faint resistance peak in the hole side. Similarly to sample $S_{A}$ (Fig.5-a), the conductance peaks (resistance dips) are not accompanied by capacitance features.

For direct comparison with theory we provide in Figs.6 a summary of our experimental observations. Figs.6-a,c show the quantum capacitance and conductivity as a function of the TI charge density for both samples. We identify the Dirac peak position by the coinciding capacitance/conductance minima, and the VP states by the conductance maxima. These peaks are reported in the density-electric field $n(\mathcal{E}_{ins})$ diagrams of Figs.6-b,d which facilitates contact with more standard measurements where the surface chemical potential is not accessible experimentally. Note that $\mathcal{E}_{ins}$ is the surface electric field at the insulator side, which differs from the HgTe-TI side by the ratio $\varepsilon_{ins}/\varepsilon_{HgTe}$ of permittivities. The green lines in Figs.6-b,d are theoretical fits with the model below. As mentioned before, an advantage of this $n(\mathcal{E}_{ins})$ diagram lies in the fact that the capacitor charging law reduces to a straight line. Actually, this is strictly true for the total TI charge, including bulk carriers. Later on when comparing with theory, one should keep in mind that the theoretical phase diagram holds only for the surface state density.

IV Theory

In this section, we describe the topological-normal junction giving rise to surface states over a penetration depth $\xi$ within a simplified effective four-band model and a gradual interface of length $\xi$ between the topological (inverted-gap) and the trivial (normal-gap) insulator. Beside the Dirac fermion we find a set of degenerate massive surface states and we study the effect of an applied electric field on the surface-states spectra. Our theoretical modeling is corroborated with numerical $k\cdot P$ calculations of a gradual junction using six-band models particularly designed for HgTe.

IV.1 Effective model of surface states

In order to model the insulating phases, we use the simplified linear four-band $k\cdot P$-Hamiltonian describing the bands around the $\Gamma$-point of an inverted band structure [qizhang2011review, ]

[TABLE]

for which the spectrum consists of two doubly degenerate bands $\varepsilon_{k}^{(\pm)}=\pm\sqrt{\Delta^{2}+\hbar^{2}v_{F}^{2}k^{2}}$. This spectrum is independent of the sign of the gap parameter $\Delta$. In the following we model HgTe as an inverted insulator of gap $-\Delta_{1}<0$ that is in contact with a normal insulator of gap $\Delta_{2}>0$. The corresponding bulk Hamiltonians are $\hat{H}_{0}({\bf k},-\Delta_{1})$ and $\hat{H}_{0}({\bf k},\Delta_{2})$ as represented on the left- and right-hand sides of panel (a) of Fig. (7).

We consider that HgTe is located in a region $z<0$ and that the normal insulator (CdHgTe or HfO2) is located at $z>\xi$. We model the interface $0<z<\xi$ between the two semi-conductors using an interpolating Hamiltonian $\hat{H}_{s0}=\hat{H}_{0}\left[{\bf k},-\Delta_{1}+e\mathcal{E}_{T}z\right]$ with the characteristic field

[TABLE]

This field plays the role of a confinement or gap field that we have chosen to have the same physical dimension as an electric field. The evolution of the gap is sketched as a green line in Fig. 7-a and can be viewed as a three-dimensional generalization of the procedure described in Ref.[Karzig2013prx, ] for a one-dimensional system and in Ref. [tchoumakovarxiv, ] for Weyl semimetals. The Hamiltonian contains a pair of non-commuting variables $[z,k_{z}]\neq 0$ that can be merged into the same matrix elements with the help of a $k-$independent unitary transformation $|\Psi\rangle=\hat{U}|\Psi^{\prime}\rangle$. One finds in this new basis

[TABLE]

where we have introduced the ladder operators $\hat{a}=\left[\hbar v_{F}k_{z}-i\left(e\mathcal{E}_{T}z-\Delta_{1}\right)\right]\left/\sqrt{2\hbar v_{F}e\mathcal{E}_{T}}\right.$ and $\hat{a}^{\dagger}=\left[\hbar v_{F}k_{z}+i\left(e\mathcal{E}_{T}z-\Delta_{1}\right)\right]\left/\sqrt{2\hbar v_{F}e\mathcal{E}_{T}}\right.$ such that $\left[\hat{a},\hat{a}^{\dagger}\right]=1$. This Hamiltonian is similar to that in a magnetic field [markreview, ] and we introduce the number states $|m\rangle$ associated to the number operator $\hat{m}=\hat{a}^{\dagger}\hat{a}$, with $\hat{m}|m\rangle=m|m\rangle$. These states are localized at the interface between two insulator with a mean position $\langle z\rangle=\Delta_{1}/e\mathcal{E}_{T}=\Delta_{1}\xi/(\Delta_{1}+\Delta_{2})\in[0,\xi]$.

For $m\geq 1$ one finds an infinite number of states of the form $|\Psi_{m,\bf{k}}\rangle=\left(\alpha_{1}|m\rangle,\alpha_{2}|m\rangle,\alpha_{3}|m-1\rangle,\alpha_{4}|m-1\rangle\right).$ Their band dispersion is doubled in that one finds, for each value of $m$, a band at positive and negative energy,

[TABLE]

which yields the relevant ${\bf k}=0$ separation

[TABLE]

between the VPS, as already mentioned in Sec.III.B in the estimation of the surface width $\xi$. In addition, these states, which were first identified by Volkov and Pankratov in 1985 [Volkov1985jetp, ], are doubly degenerate and depend explicitly on the characteristic field (2) and thus on the parameters characterizing the interface. In the limit of a sharp surface, i.e. $\xi\rightarrow 0$ so that $\mathcal{E}_{T}\rightarrow\infty$, these states are shifted to high energies and do not play any physical role.

In contrast to theses bands, the $m=0$ surface states are not degenerate. They are of the form $|\Psi_{0,{\bf k}}\rangle=\left(\alpha_{1}|0\rangle,\alpha_{2}|0\rangle,0,0\right)$ and their energy dispersion is that of a two-dimensional Dirac cone $\varepsilon_{0,k_{x},k_{y}}^{(\pm)}=\pm\hbar v_{F}\sqrt{k_{x}^{2}+k_{y}^{2}}$. It is independent of $\mathcal{E}_{T}$ and one can show [tchoumakovarxiv, ] that this state is of topological nature. This topological surface state survives in the limit of an infinitely sharp interface, as expected, and shows a dispersion that only depends on the bulk parameter $v_{F}$, in agreement with previous studies [qizhang2011review, ]. We represent the spectra of the VPS*±m* states and the Dirac state in panel (a) of Fig. 7.

Most saliently, the VPS can be modified to great extent by an electric field applied perpendicular to the surface. As an effect of charge screening, the associated electrostatic potential drops in the interface and generates and electric field $\boldsymbol{\mathcal{E}}\equiv-\mathcal{E}{\bf e}_{z}\approx-V_{0}/\xi{\bf e}_{z}$, where $V_{0}=\mu/e$ is the surface potential. The interface Hamiltonian, for $0<z<\xi$, is now $\hat{H}_{s}=\hat{H}_{s0}^{\prime}+(V_{0}-e\mathcal{E}z)\mathbbm{1}$. Notice that the added term remains invariant under the above-mentioned rotation that leads to the form of the Hamiltonian given by Eq. (3). As detailed in App.VII, the spectrum now depends on the ratio $\beta=\mathcal{E}/\mathcal{E}_{T}$ between the applied and the characteristic electric fields. The spectrum for the $m\geq 1$ states is still doubly degenerate with the dispersion relations $\varepsilon_{m,k_{x},k_{y}}(\beta)=\sqrt{(1-\beta^{2})\hbar^{2}v_{F}^{2}(k_{x}^{2}+k_{y}^{2})+2m(1-\beta^{2})^{3/2}\hbar v_{F}e\mathcal{E}_{T}}$. One observes that the gap (5) is reduced by the applied electric field according to

[TABLE]

and that the critical surface density obeys

[TABLE]

At the same time the dispersion relation flattens out because of the reduced Fermi velocity that vanishes at $\mathcal{E}_{T}$ according to

[TABLE]

The special $m=0$ surface state is also flattened and one finds $\varepsilon_{0,k_{x},k_{y}}^{(\pm)}=\pm(1-\beta^{2})^{1/2}\hbar v_{F}\sqrt{k_{x}^{2}+k_{y}^{2}}$.

In panel (b) of Fig. 7 we represent the extrema of the VPS*±m* (in green) and of the Dirac state at $k_{x}=k_{y}=0$ (in red) as a function of $\beta=\mathcal{E}/\mathcal{E}_{T}$. We observe that VPS merge for an electric field close to the critical field $\mathcal{E}_{T}$. Beyond this limit our model (10) has no bound state so that the interface behaves as a conventional semiconducting heterojunction. This shows that $\mathcal{E}_{T}$ is not only a characteristic field governing the massive surface state spectrum but actually a genuine critical field for the topological nature of the interface. The applied electric potential also influences the chemical potential [blue line in Fig. (7)] that will eventually cross the VPS. This leads to the experimentally observed kinks in the compressibility and features in the conductivity that we discuss in detail in the following section.

We finish this theoretical section with a discussion of the density of states $\rho(\epsilon)$, which is directly proportional to the quantum capacitance (at $T=0$). The density of states associated with the surface states per unit area reads

[TABLE]

where $\Theta(x)$ is the Heaviside function. The corresponding behavior of the quantum capacitance is represented in panel (c) of Fig. 7 for typical values of $\beta=\mathcal{E}/\mathcal{E}_{T}$. One observes that the gaps are smaller and that the density of states becomes enhanced for larger electric fields (in red). This is a direct consequence of the reduced Fermi velocity in the presence of an electric field, as pointed out above. Notice, however, that in the experimental setup the chemical potential depends itself on the applied electric potential and therefore jumps from one curve to the other. Moreover, we expect this density of state to be smeared in the presence of disorder.

IV.2 Numerical $k\cdot P$ treatment of surface states

To complement the previous analytical study, we have performed a numerical study of the band structure of an HgTe/CdTe interface, based on a $k.P$ model. Our numerical approach amounts to discretizing a standard Kane model for the $6$ bands $\Gamma_{6,\pm 1/2},\Gamma_{8,\pm 1/2},\Gamma_{8,\pm 3/2}$ hamiltonian with parameters for HgTe and CdTe known from the literature [Novik2005prb, ]. The parameters of the model is interpolate between their values in both materials over a distance $\xi$, corresponding to the size of the interface. This description incorporates the stress induced by the lattice mismatch through a Bir-Pikus term. The resulting band structure is shown in Fig.8, where the color encodes the eigenstate’s density around the HgTe/CdTe interface. The band structure is calculated for an HgTe thickness of $70\;\mathrm{nm}$ and $\xi=5\;\mathrm{nm}$. Note that this band structure is obtained at zero electric field and using CdTe ($2\Delta_{2}\simeq 1.5\;\mathrm{eV}$) as a capping layer boundary. We find a massive surface subband at $1\;\mathrm{eV}$ which is accompanied by a strong depletion of the bulk state amplitude in the surface layer that confirms the above picture of high energy surface states (VPS). The excitation energy is quite large (still smaller than $2\Delta_{2}$) and found to be sensitive to the detail of the shape of the smooth interface. This is consistent with the analytical approach of section IV.1 which focussed on a linear interpolation between the two materials and neglects asymmetry between the two materials beyond the gap inversion. We stress that these VPS are predicted at zero electric field, and assume that they follow the electric field dependence predicted by the approach of section IV.1, as corroborated by Ref. [Tchoumakov2017arXiv, ].

V Comparison with experiment

The main evidence of VPS is the scattering peak observed in the intrinsic sample $S_{A}$ (Fig2-e and Fig6-a), where the absence of classical massive surface state and the Dirac screening of bulk states have been previously warranted experimentally [Brune2014prx, ]. The measured VPS1 gap $E_{1}=0.35\;\mathrm{eV}$ is close to the theoretical value $E_{1}(0)\simeq 0.4\;\mathrm{eV}$, deduced from Eq.(6), and is accompanied by a second carrier type observed in the Hall bar measurements. In addition we report that the VPS1 triggers the onset of charge metastability signaling the breakdown of Dirac screening [Brune2014prx, ].

The second evidence is the observation of a series of scattering peaks in the uncapped sample $S_{B}$ (Fig5-b and Fig6-c). The capping of sample $S_{B}$ being ill-defined, we cannot make a direct comparison of VPS energy with theory so that $\Delta_{2}$ becomes a fitting parameter. Still we can adjust the peaks series with the predicted sequence $n_{m}\propto m(m+1)/2$ of Eq.(7).

The third evidence relates to the electric field red-shift. As seen in Fig6-d, the agreement of VPS spectroscopy with theory involves a $\sim 10\%$ electric field red-shift for the VPS2 state of sample $S_{B}$ consistent with Eq.(6). As for the VPS-1 state, its observation fully relies on a strong field renormalization of the gaps, meaning that VPS-1 peak position signals the vicinity of the critical field $\mathcal{E}_{T}$. The fact that VPS-1 is smeared is also consistent with theory since the Fermi velocity vanishes at $\mathcal{E}_{T}$ according to Eq.(8). Moreover the conductivity becomes featureless above $\mathcal{E}_{T}$ also in agreement with theory. Our scattering spectroscopy measurements thus support the field-effect induced red-shift and suppression of VPS predicted in Section IV. Although a deeper insight into the field suppression is desirable, it remains very challenging as a full mapping of the $n(\mathcal{E})$ diagram would require an in-situ tuning of the insulator permittivity and/or TI chemical doping. An extension of this work can be envisioned using a series of similar HgTe THJs with varied capping bandgap $\Delta_{2}$ and/or insulator permittivity $\varepsilon_{ins}$.

Before concluding, let us detail our analysis of the $n(\mathcal{E})$ phase diagrams in Figs.6-b,d. Eq.(7) predicts sub-band minima for a surface state density $n_{m}=n_{1}\times\frac{m(m+1)}{2}\sqrt{1-\beta^{2}}$ where $n_{1}=e\mathcal{E}_{T}/2\pi\hbar v_{F}$ and $\beta=\mathcal{E}_{HgTe}/\mathcal{E}_{T}=\varepsilon_{ins}\mathcal{E}_{ins}/\varepsilon_{HgTe}\mathcal{E}_{T}$. Taking $\mathcal{E}_{T}=1.2\;10^{8}\;\mathrm{V/m}$, $\varepsilon_{ins}^{A}=C_{ins}t_{ins}/\varepsilon_{0}\simeq 4.5$ and $\varepsilon_{HgTe}\simeq 20$ [Baars1972ssc, ; Capper2010, ], we estimate for sample $S_{A}$ the applied critical field $\mathcal{E}_{c}^{A}=\varepsilon_{HgTe}\mathcal{E}_{T}/\varepsilon_{ins}^{A}\simeq 5\;10^{8}\;\mathrm{Vm^{-1}}$ and a surface critical density $n_{1}^{A}=2.9\;10^{12}\;\mathrm{cm^{-2}}$ (with $v_{F}=10^{6}\;\mathrm{m/s}$). Direct total charge measurement gives a larger $n_{1}\simeq 4\;10^{12}\;\mathrm{cm^{-2}}$ due to the presence of bulk carriers. The same analysis can be carried out for the doped sample $S_{B}$ where we obtain a quantitative agreement for the position of three VPS peaks (VPS-1, VPS2 and VPS3) with a single parameter $\mathcal{E}^{B}_{T}=(0.4\pm 0.1)\;10^{8}\;\mathrm{Vm^{-1}}$, yielding $n_{1}^{B}=1.\;10^{12}\;\mathrm{cm^{-2}}$. The fact that $\mathcal{E}^{B}_{T}\simeq\mathcal{E}^{A}_{T}/3$ highlights the importance of a capping layer in strengthening the robustness of surface topological states.

Finally we have plotted in Figs.6-(a,c) (solid black lines) the theoretical prediction for the quantum capacitance calculated from the density of states of surface states in Eq.(9). While we see the linear increase of a secondary carrier density in the Hall bar measurements, we do not resolve the compressibility steps signaling the onset of massive subbands. A possible reason for this discrepancy is the presence of residual bulk carriers, obscuring the compressibility resolution and explaining the compressibility background observed in Fig.3-d. These carriers can be disorder-induced small scale charge puddles or more likely, in our high mobility samples, large-scale puddles induced by the non-uniformity of the applied electrostatic surface potential. Considering the small transport bandgap of HgTe, the potential uniformity requirements are especially stringent.

VI Conclusion

Our comprehensive set of measurement, complemented by a heuristic model, support the existence of intrinsic massive surface states, the Volkov-Pankratov states, accompanying Dirac states at the interface between TI HgTe and the insulator CdHgTe. Such high-energy topological states, that are intrinsic to topological matter, have long been predicted [Volkov1985jetp, ; Karzig2013prx, ] but remained elusive until now. Our work is the first report and systematic study of these states, and we hope it will trigger further investigations into different materials such as the Bi-based TI [Inhofer2017arXiv, ] or different topological phases such as massive Majorana states, helical edge states or Fermi arcs. As pointed out in Ref.[Karzig2013prx, ] and demonstrated in this experiment, massive surface states play an important role in restricting the phase diagram where topological protection is robust such as the Dirac screening [Karzig2013prx, ]. This phase diagram can be enriched by applying an external magnetic field, or simply by increasing the temperature, highlighting the possible role of electron-electron interactions.

VII Appendix : Lorentz boost of surface states in an electric field

In section IV.1 we have introduced the Hamiltonian $\hat{H}_{s}=\hat{H}_{s0}^{\prime}+(V_{0}-e\mathcal{E}z)\mathbbm{1}$. In order to deal with this new $z-$dependence, we perform a Lorentz boost on the time-independent Schrödinger equation $e^{\frac{\eta}{2}\hat{J}}(\hat{H}_{s}^{\prime}-\varepsilon\mathbbm{1})e^{\frac{\eta}{2}\hat{J}}|\bar{\Psi}\rangle=0$, where the Lorentz boost [lukose2007prl, ] is realized by the hyperbolic transformation $|\Psi^{\prime}\rangle=\mathcal{N}e^{-\frac{\eta}{2}\hat{J}}|\Psi\rangle$ generated by $\hat{J}=\sigma_{y}\otimes\sigma_{x}$, in terms of the Pauli matrices $\sigma_{x}$ and $\sigma_{y}$.

In the case $\tanh(\eta)\equiv\beta=\mathcal{E}/\mathcal{E}_{T}\in[-1,1]$ and $V_{0}=-\beta\Delta_{1}$, one can thus make the $(V_{0}-e\mathcal{E}z)\mathbbm{1}$-term vanish in the co-moving frame of reference. If we define the pseudo-magnetic field $B=\mathcal{E}_{T}/v_{F}$, this transformation can be understood from the viewpoint of special relativity as a boost to a frame of reference where the drift velocity $v_{D}=\mathcal{E}/B$ vanishes [lukose2007prl, ]. The condition $\mathcal{E}/\mathcal{E}_{T}=v_{D}/v_{F}\in[-1,1]$ is similar to the existence of a limiting velocity, the Fermi velocity $v_{F}$ plays the role of the speed of light in special relativity. The Schrödinger equation becomes $\hat{H}_{s}^{\prime}|\bar{\Psi}\rangle=\varepsilon|\bar{\Psi}\rangle$ with

[TABLE]

with $v_{F}^{\prime}=\sqrt{1-\beta^{2}}v_{F}$, $\mathcal{E}_{T}^{\prime}=(1-\beta^{2})\mathcal{E}_{T}$ and the ladder operators $\hat{b},\hat{b}^{\dagger}$ where

[TABLE]

As a consequence of the Lorentz boost, the ladder operators are now explicitly energy-dependent as well as the mean position of the number states $\langle z\rangle=\left.\left(\Delta_{1}+\frac{\beta}{{\beta^{2}-1}}\varepsilon\right)\right/e\mathcal{E}_{T}$. For our theory to describe a surface state, this position must be within $\langle z\rangle\in[0,\xi]$, and one notices that for $\beta\rightarrow 1$ this condition is not fulfilled for $\varepsilon\neq 0$.

The spectrum is found using states of the form $|\Psi_{m,{\bf k}}\rangle=\left(\alpha_{1}|m\rangle,\alpha_{2}|l\rangle,\alpha_{3}|m-1\rangle,\alpha_{4}|m-1\rangle\right)$ for $m\geq 1$ and $|\Psi_{0,{\bf k}}\rangle=\left(\alpha_{1}|0\rangle,\alpha_{2}|0\rangle,0,0\right)$ for $m=0$, in the comoving frame of reference. The $|m\rangle$ states are the eigenstates of the number operator $\hat{m}=\hat{b}^{\dagger}\hat{b}$.

Acknowledgements.

We gratefully acknowledge B.A. Bernevig, N. Regnault, R. Ferreira, Y. Guldner, G. Bastard and M. Civelli for fruitful discussions.