Structure of surface electronic states in strained mercury telluride

TL;DR

This paper develops a theoretical framework for understanding two distinct types of surface electronic states in strained mercury telluride, revealing their dispersion characteristics and potential topological significance.

Contribution

It introduces a comprehensive model describing the formation and properties of surface states in strained HgTe, highlighting their dispersion types and topological implications.

Findings

Identification of two types of surface states in strained HgTe.

Analytical and numerical analysis of surface state structures.

Revealing the topological nature of certain surface states.

Abstract

We present the theory describing the various surface electronic states arisen from the mixing of conduction and valence bands in a strained mercury telluride (HgTe) bulk material. We demonstrate that the strain-induced band gap in the Brillouin zone center of HgTe results in the surface states of two different kinds. Surface states of the first kind exist in the small region of electron wave vectors near the center of the Brillouin zone and have the Dirac linear electron dispersion characteristic for topological states. The surface states of the second kind exist only far from the center of the Brillouin zone and have the parabolic dispersion for large wave vectors. The structure of these surface electronic states is studied both analytically and numerically in the broad range of their parameters, aiming to develop its systematic understanding for the relevant model Hamiltonian. The results bring attention to the rich surface physics relevant for topological systems.

Equations

$$\hat{\mathcal{H}}={\hat{\mathcal{H}}}_{\mathrm{L}}+{\hat{\mathcal{H}}}_{\mathrm{strain}}+{\hat{\mathcal{H}}}_{\mathrm{BIA}}.$$

$${\hat{\mathcal{H}}}_{\mathrm{L}}=\left({\gamma }_1+\frac{5}{2}{\gamma }_2\right){\mathbf{k}}^2-2{\gamma }_2(J_x^2{k}_x^2+J_y^2{k}_y^2+J_z^2{k}_z^2)-2{\gamma }_3(\{J_x,J_y\}{k}_x{k}_y+\{J_x,J_z\}{k}_x{k}_z+\{J_y,J_z\}{k}_y{k}_z)$$

$${\hat{\mathcal{H}}}_{\mathrm{strain}}=\left(a+\frac{5}{4}b\right)(u_{xx}+u_{yy}+u_{zz})-b(J_x^2{u}_{xx}+J_y^2{u}_{yy}+J_z^2{u}_{zz})-\frac{d}{\sqrt{3}}(\{J_x,J_y\}{u}_{xy}+\{J_x,J_z\}{u}_{xz}+\{J_y,J_z\}{u}_{yz})$$

$${\hat{\mathcal{H}}}_{\mathrm{BIA}}=\alpha [k_x\{J_x,(J_y^2-J_z^2)\}+k_y\{J_y,(J_z^2-J_x^2)\}+k_z\{J_z,(J_x^2-J_y^2)\}]$$

$${\phi }_m={\left[C_{1m},C_{2m},C_{3m}{C}_{4m}\right]}^T{e}^{i\mathbf{kr}},$$

$$\det |\hat{\mathcal{H}}({\mathbf{k}}_s,i\kappa )-\epsilon |=0.$$

$$\Psi =e^{i{\mathbf{k}}_s{\mathbf{r}}_s}\sum _{m=1}^4{\left[C_{1m}({\mathbf{k}}_s,i{\kappa }_m),C_{2m}({\mathbf{k}}_s,i{\kappa }_m),C_{3n}({\mathbf{k}}_s,i{\kappa }_m),C_{4n}({\mathbf{k}}_s,i{\kappa }_m)\right]}^T{A}_m{e}^{-{\kappa }_{m}z},$$

$$\sum _{m=1}^4{A}_m{C}_{nm}({\mathbf{k}}_s,i{\kappa }_m)=0,(n=1,2,3,4).$$

$$\det |C_{nm}({\mathbf{k}}_s,i{\kappa }_m)|=0,(n,m=1,2,3,4),$$

$${\hat{\mathcal{H}}}_{\mathrm{strain}}=a(u_{xx}+u_{yy}+u_{zz})+({\epsilon }_g/2)(J_z^2-5/4),$$

$${\hat{\mathcal{H}}}_0=\left[\begin{array}{cccc}-2{\gamma }_2{\hat{k}}_z^2+{\epsilon }_g/2& \sqrt{3}\alpha {\hat{k}}_z& 0& 0\\ \sqrt{3}\alpha {\hat{k}}_z& 2{\gamma }_2{\hat{k}}_z^2-{\epsilon }_g/2& 0& 0\\ 0& 0& 2{\gamma }_2{\hat{k}}_z^2-{\epsilon }_g/2& -\sqrt{3}\alpha {\hat{k}}_z\\ 0& 0& -\sqrt{3}\alpha {\hat{k}}_z& -2{\gamma }_2{\hat{k}}_z^2+{\epsilon }_g/2\end{array}\right],$$

$${\Psi }_1$$

$${\Psi }_2$$

$${\kappa }_{\pm }=\frac{\sqrt{3}\alpha }{4{\gamma }_2}\pm \sqrt{{\left(\frac{\sqrt{3}\alpha }{4{\gamma }_2}\right)}^2-\frac{{\epsilon }_g}{4{\gamma }_2}}$$

$${\hat{\mathcal{H}}}_{\mathrm{eff}}={\epsilon }_0-\frac{3\alpha }{2}({\sigma }_x{k}_x+{\sigma }_y{k}_y)-\frac{\sqrt{3}\alpha }{2}({\sigma }_x{k}_y+{\sigma }_y{k}_x),$$

$$\epsilon ({\mathbf{k}}_s)={\epsilon }_0\pm \sqrt{3}\alpha \sqrt{k_x^2+k_y^2+\sqrt{3}{k}_x{k}_y}.$$

$$\left[{\lambda }_{-}^{(1)}{\lambda }_{+}^{(2)}-1\right]\left[{\lambda }_{+}^{(1)}{\lambda }_{-}^{(2)}-1\right]=0,$$

$${\lambda }_{\pm }^{(j)}=\frac{\sqrt{3}\gamma (k_s^2\mp 2k_s{\kappa }_j)}{\epsilon -{\gamma }_1(k_s^2-{\kappa }_j^2)+(-1{)}^j[\gamma (k_s^2+2{\kappa }_j^2)+{\epsilon }_g/2]},$$

$${\kappa }_j=\sqrt{\frac{A+(-1{)}^j\sqrt{A^2+4BC}}{2B}},$$

$$A=2{\gamma }_1(\epsilon -{\gamma }_1{k}_s^2)+2\gamma (4\gamma k_s^2-{\epsilon }_g),B=({\gamma }_1+2\gamma )(2\gamma -{\gamma }_1),C=(\epsilon -{\gamma }_1{k}_s^2{)}^2-(\gamma k_s^2+{\epsilon }_g/2{)}^2-3{\gamma }^2{k}_s^4.$$

$$\epsilon (k_s)=\left[1-{\left(\frac{1+\sqrt{3(2\gamma -{\gamma }_1)/(2\gamma +{\gamma }_1)}}{2}\right)}^2\right]({\gamma }_1+2\gamma )k_s^2,$$

$$\epsilon ={\gamma }_1{k}_s^2\pm \sqrt{(\gamma k_s^2+{\epsilon }_g/2{)}^2+3{\gamma }^2{k}_s^4}.$$

$${\psi }_{+3/2}=\frac{1}{\sqrt{2}}(X+iY)\uparrow ,{\psi }_{-1/2}=-\frac{1}{\sqrt{6}}[(X-iY)\uparrow +2Z\downarrow ],{\psi }_{+1/2}=\frac{1}{\sqrt{6}}[(X+iY)\downarrow -2Z\uparrow ],{\psi }_{-3/2}=-\frac{1}{\sqrt{2}}(X-iY)\downarrow ,$$

$$J_x=\left[\begin{array}{cccc}0& 0& \frac{\sqrt{3}}{2}& 0\\ 0& 0& 1& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& 1& 0& 0\\ 0& \frac{\sqrt{3}}{2}& 0& 0\end{array}\right],J_y=\left[\begin{array}{cccc}0& 0& -i\frac{\sqrt{3}}{2}& 0\\ 0& 0& i& -i\frac{\sqrt{3}}{2}\\ i\frac{\sqrt{3}}{2}& -i& 0& 0\\ 0& i\frac{\sqrt{3}}{2}& 0& 0\end{array}\right],J_z=\left[\begin{array}{cccc}\frac{3}{2}& 0& 0& 0\\ 0& -\frac{1}{2}& 0& 0\\ 0& 0& \frac{1}{2}& 0\\ 0& 0& 0& -\frac{3}{2}\end{array}\right].$$

$$\hat{\cal H}=\begin{tabular}[]{|c||c c c c|}\hline\cr$\psi_{j_{z}}\backslash\psi_{j_{z}}$&$\psi_{+3/2}$&$\psi_{-1/2}$&$\psi_{+1/2}$&$\psi_{-3/2}$\\ \hline\cr\hline\cr$\psi_{+3/2}$&$F$&$I+L$&$H+M$&$N$\\ $\psi_{-1/2}$&$I^{*}+L$&$G$&$-N$&$-H+M$\\ $\psi_{+1/2}$&$H^{*}+M^{*}$&$-N^{*}$&$G$&$I-L$\\ $\psi_{-3/2}$&$N^{*}$&$-H^{*}+M^{*}$&$I^{*}-L$&$F$\\ \hline\cr\end{tabular}\,,$$

$$F=({\gamma }_1+{\gamma }_2)(k_x^2+k_y^2)+({\gamma }_1-2{\gamma }_2)k_z^2+(a-b)u_{zz}+(a+b/2)(u_{xx}+u_{yy}),$$

$$G=({\gamma }_1-{\gamma }_2)(k_x^2+k_y^2)+({\gamma }_1+2{\gamma }_2)k_z^2+(a+b)u_{zz}+(a-b/2)(u_{xx}+u_{yy}),$$

$$I=-\sqrt{3}{\gamma }_2(k_x^2-k_y^2)+i2\sqrt{3}{\gamma }_3{k}_x{k}_y-(\sqrt{3}b/2)(u_{xx}-u_{yy})+id{u}_{xy},M=-(\sqrt{3}\alpha /2)(k_x+i{k}_y),$$

$$H=-2\sqrt{3}{\gamma }_3(k_x-i{k}_y)k_z-d(u_{xz}-i{u}_{yz}),L=\sqrt{3}\alpha k_z,N=-(3\alpha /2)(k_x-i{k}_y).$$

Full text

I Introduction

The studies of two-dimensional (2D) electronic modes localized near the surface of a three-dimensional (3D) condensed matter structure (surface electronic states) represent one of the most actively studied directions of modern science of the last decade. The increasing interest of scientific community to this research area is caused by the topologically nontrivial nature of the surface states in structures of certain type known as topological insulators (TIs). Namely, TIs are condensed matter systems which behave like an insulator in their 3D bulk but have 2D gapless conducting electronic states protected by the time-reversal symmetry at their boundaries Hasan_2010 . Up to date, such topologically protected electronic states were intensively studied theoretically and experimentally in various condensed matter structures Kane_2005 ; Fu_2007 ; Fu_2007_1 ; Moore_2007 ; Roy_2009 ; Xia_2009 ; Zhang_2009 ; Chen_2009 ; Yudin_2016 ; Hasan_2017 ; Pervishko_2018 ; Kozin_2018 , and their optical analogs were also revealed Lu2014 ; Mittal2016 ; Whittaker2018 ; Klembt2018 ; Banerjee2018 .

Particularly, it follows from the theoretical analysis based on the $Z_{2}$ topological invariants that the surface electronic states in bulk mercury telluride (HgTe) can be topologically nontrivial Fu_2007 . However, the band structure of natural HgTe is semi-metallic: There is the small overlap of conduction and valence bands originated from the bulk inversion asymmetry of the crystal structure Bir_Pikus . In order to observe the predicted topological surface states in HgTe, one needs to turn this semi-metal into insulator. To solve the problem, an uniaxial strain as a tool to open the band gap between the valence and conduction bands of HgTe was proposed Dai_2008 . Following this methodology, the topological surface states in strained HgTe were recently observed experimentally within the strain-induced band gap Brune_2011 . As a consequence, physical properties of the surface electronic states in strained HgTe-based materials are currently in the focus of attention Maier_2012 ; Yan_2012 ; Chen_2012 ; Kozlov_2014 ; Dantscher_2015 ; Li_2015 ; Ruan_2016 ; Kirtschig_2016 ; Leubner_2016 ; Haas_2017 ; Grendysa_2017 ; Thomas_2017 . However, the theory describing the structure of these surface states is still far from being complete. Particularly, the characteristic feature of HgTe is the coexistence of surface electronic states of different physical nature: Besides of the discussed topological surface states in gapped HgTe, there are the surface states in gapless HgTe analyzed for the first time by D’yakonov and Khaetskii Dyakonov_1981 . Certainly, the consistent theory should be able to describe the dependence of all surface states on the strain. The present article takes a step towards such a consistent theory and provides an intuitive understanding of the system.

The paper is organized as follows. In Sec. II, we formulate a formalism describing the surface electronic states of various kinds in strained HgTe. In Sec. III, we solve the corresponding Schrödinger problem analytically in the simplest particular cases, calculate the dispersion of the surface states numerically, and analyze their energy spectrum. The last two sections contain the conclusion and acknowledgments.

II Model

We consider the surface electronic states which originate from the mixing of conduction and valence bands of HgTe near the center of the Brillouin zone (the electronic term $\Gamma_{8}$). In bulk strained HgTe, the states of this electronic term are described by the Hamiltonian Bir_Pikus

[TABLE]

where

[TABLE]

is the conventional Luttinger Hamiltonian describing the conduction and valence bands of unstrained HgTe,

[TABLE]

is the Bir-Pikus Hamiltonian describing the modification of conduction and valence bands of HgTe under strain, and

[TABLE]

is the term arisen from the bulk inversion asymmetry (BIA) of HgTe crystal. Correspondingly, $\mathbf{k}=(k_{x},k_{y},k_{z})$ is the electron wave vector, $\gamma_{1,2,3}$ are the Luttinger parameters of HgTe, $a$, $b$ and $d$ are the deformation potentials of HgTe, $u_{ij}$ are the components of the deformation tensor of the strained HgTe, $\alpha$ is the BIA parameter, $J_{x,y,z}$ are the $4\times 4$ matrices corresponding to the electron angular momentum $J=3/2$, and the curly brackets $\{A,B\}$ represent anti-commutators of the matrices $A$ and $B$. The eigenstates of the Hamiltonian (1) can be written in the most general form as four independent spinors,

[TABLE]

with indices $m=1,2,3,4$, and the spinor components $C_{nm}(k_{x},k_{y},k_{z})$ are the functions of the electron wave vector, $\mathbf{k}=(k_{x},k_{y},k_{z})$.

For definiteness, let us consider electronic states localized near the $(001)$-surface, assuming that the bulk HgTe fills the half-space for $z>0$. It follows from the conservation laws that the surface $(001)$ mixes different electronic states of the Hamiltonian (1) with the same energy, $\varepsilon(\mathbf{k}_{s},k_{z})$, and the same wave vector in the surface plane, $\mathbf{k}_{s}=(k_{x},k_{y})$, but with different normal components of the wave vector, $k_{z}$. As a consequence, the surface-localized electronic states which arise from the mixing are described by the same spinors (5) with the imaginary $z$-component of electron wave vector, $k_{z}=i\kappa$, where $\kappa>0$ is the localization parameters, corresponding to the inverse of localization length for electronic states near the surface. Generally, the localization parameter, $\kappa$, can be a complex number but its real part must be positive. In bulk HgTe, there are the four different branches of electron energy, $\varepsilon(\mathbf{k})$, corresponding to the four branches of the conduction and valence bands spin-split due to the BIA terms of the Hamiltonian (1). Therefore, there are four different parameters, $\kappa_{1,2,3,4}(\mathbf{k}_{s},\varepsilon)$, which can be found as four solutions of the secular equation,

[TABLE]

Making the replacement, $k_{z}\rightarrow i\kappa_{1,2,3,4}$, in the eigenspinors (5), we can write the surface-localized eigenfunction of the Hamiltonian (1) as a linear combination of the spinors,

[TABLE]

where $\mathbf{r}_{s}=(x,y)$ is the electron radius-vector in the surface plane and $A_{1,2,3,4}$ are the constants to be determined. To do so, we chose the model of a surface potential which can be approximated by the infinitely-high barrier at position $z=0$. This sets the boundary condition for the electron wave function (7) as $\Psi|_{z=0}=0$, and results into the homogeneous system of four algebraic equations defining the constants $A_{j}$,

[TABLE]

The secular equation of the algebraic system (8),

[TABLE]

defines the sought energy spectrum of the surface electronic states, $\varepsilon(\mathbf{k}_{s})$. In the next section of the article, we apply the strategy described above to find the spectrum $\varepsilon(\mathbf{k}_{s})$ in systems with various band parameters. In the case of HgTe, we use the following parameters Adachi_2004 ; Ruan_2016 : $\gamma_{1}=15.6\,\hbar^{2}/2m_{0}$, $\gamma_{2}=9.6\,\hbar^{2}/2m_{0}$, $\gamma_{3}=8.6\,\hbar^{2}/2m_{0}$, $b=-1.22$ eV and $\alpha=0.208$ Å$\cdot$eV.

III Results and Discussion

In what follows, we consider the Hamiltonian (1) to be written as a $4\times 4$ matrix in the conventional Luttinger-Kohn basis, $\psi_{j_{z}}$, corresponding to the different projections of electron angular momentum on the $z$ axis, $j_{z}=\pm 1/2,\pm 3/2$ (see the Appendix for details). In unstrained bulk HgTe, the four wave functions, $\psi_{j_{z}}$, are the eigenfunctions of the Hamiltonian (1): They describe the states of the electronic term $\Gamma_{8}$ at $\mathbf{k}=0$, which are four-fold degenerate. As for strained bulk HgTe, we consider for definiteness the case of an uniaxial stress applied along the $z$ axis. Under the uniaxial stress, the deformation tensor of HgTe, $u_{ij}$, written in the principal crystallographic axes, $x,\,y,\,z$, is diagonal. Particularly, the case of $u_{zz}<0$ and $u_{xx}=u_{yy}>0$ corresponds to the compressive strain, whereas the opposite case of $u_{zz}>0$ and $u_{xx}=u_{yy}<0$ corresponds to the tensile strain. As a result, the strain Hamiltonian (3) can be rewritten as

[TABLE]

where $\varepsilon_{g}=2b(u_{xx}-u_{zz})=2b(u_{yy}-u_{zz})$, and we note that $b<0$ for HgTe. The first term of the Hamiltonian (10) is the strain-induced shift of zero energy, which will be omitted in the following. As for the second term, it describes the strain-induced splitting of the electronic states with $j_{z}=\pm 1/2$ and $j_{z}=\pm 3/2$ at $\mathbf{k}=0$. It follows from the total Hamiltonian (1) with the strain Hamiltonian (10) that $\varepsilon_{g}=\varepsilon_{\pm 3/2}-\varepsilon_{\pm 1/2}$, where $\varepsilon_{j_{z}}$ are the energies of these states at $\mathbf{k}=0$ (see Fig. 1a). It should be noted, particularly, that the compressive strain ($\varepsilon_{g}<0$) and the tensile strain ($\varepsilon_{g}>0$) result in the opposite sequence orders for energies of the basic states with the wave functions $\psi_{\pm 1/2}$ and $\psi_{\pm 3/2}$. Substituting the strain Hamiltonian (3) into the total Hamiltonian (1), we can apply the methodology developed in Sec. II to analyze the evolution of the surface electronic states in HgTe under stress.

First of all, let us consider the electronic states with the zero planar electron wave vector, $\mathbf{k}_{s}=0$, which are localized near the $(001)$-surface of the uniaxially strained HgTe. The simplest case, which can be readily solved, corresponds to the Luttinger parameter $\gamma_{1}=0$. Physically, this model situation describes a semiconductor with the Hamiltonian (1), where the masses of electrons and holes along the $z$ axis, $m_{e}$ and $m_{h}$, are equal to each other, $m_{e}/m_{h}=(2\gamma_{2}-\gamma_{1})/(2\gamma_{2}+\gamma_{1})$. For such a symmetric electron-hole system, the Hamiltonian (1) at $\mathbf{k}_{s}=0$ can be written in the Luttinger-Kohn basis as a block-diagonal matrix (see the Appendix),

[TABLE]

where $\hat{k}_{z}=-i\partial/\partial z$ is the operator of the electron momentum normal to the considered surface. In the particular case of the Hamiltonian (11), the surface-localized eigenspinors (7) correspond to the eigenenergy $\varepsilon_{0}=0$ and can be written as

[TABLE]

where

[TABLE]

are the localization parameters of the surface state, and $D=\sqrt{\sqrt{3}\alpha\varepsilon_{g}/(6\alpha^{2}-8\gamma_{2}\varepsilon_{g})}$ is the normalization constant. The eigenspinors (III) can be easily verified by direct substitution into the Schrödinger equation, $\hat{\cal H}_{0}\Psi_{1,2}=\varepsilon_{0}\Psi_{1,2}$, with the Hamiltonian (11) and the eigenenergy $\varepsilon_{0}=0$. It follows from Eq. (13) that $\kappa_{-}<0$ if $\varepsilon_{g}<0$. Since the real part of both localization parameters, $\kappa_{\pm}$, must be positive, the surface states (III) exist only for $\varepsilon_{g}>0$ (tensile strain). Physically, this means that the existence of the surface states (III), first, arises from the BIA terms ($\alpha\neq 0$) and, second, it strongly depends on the sequence order of their parent bulk states, $\psi_{\pm 1/2}$ and $\psi_{\pm 3/2}$.

To find the dispersion of the surface states (III) for small wave vectors $\mathbf{k}_{s}$, we have to project the total Hamiltonian (1) to the subspace spanned by these two states, $\{\Psi_{1},\Psi_{2}\}$. Keeping the terms linear in $\mathbf{k}_{s}$, we arrive at the effective Hamiltonian,

[TABLE]

where $\sigma_{x,y}$ are the Pauli matrices written in the basis $\{\Psi_{1},\Psi_{2}\}$. Diagonalizing the Hamiltonian (14), we can write the sought energy spectrum of the surface states (III) near $\mathbf{k}_{s}=0$ as

[TABLE]

It follows from Eq. (15) that the two degenerate states (III) form the Dirac point at $\mathbf{k}_{s}=0$ with the energy $\varepsilon_{0}=0$ and anisotropic linear dispersion near the point. Applying the methodology of Sec. II, one can calculate numerically the dispersion of the surface states in the broad range of the wave vectors, $\mathbf{k}_{s}$. To demonstrate the properties of these surface states in more details, we plotted their dispersion for the HgTe band parameters $\gamma_{2,3}$ and $\alpha$ but for different Luttinger parameters $\gamma_{1}$ (see Fig. 1). In the model case of symmetric electron-hole system discussed above ($\gamma_{1}=0$), the Dirac point energy, $\varepsilon_{0}=0$, lies exactly in the middle of the bulk states $\psi_{\pm 1/2}$ and $\psi_{\pm 3/2}$ (see Fig. 1a), whereas the nonzero Luttinger parameter $\gamma_{1}$ shifts the Dirac point energy, $\varepsilon_{0}$, from the middle towards the bulk term $\psi_{\pm 3/2}$ (see Fig. 1b). As a result, the branches of the surface states are localized near the bulk term $\psi_{\pm 3/2}$ in the real case of HgTe (see Fig. 1c), where the electron-hole system is strongly asymmetric ($m_{e}/m_{h}\ll 1$). It should be stressed that the effective Hamiltonian (14) and the dispersion (15), which were derived formally for the particular case of $\gamma_{1}=0$, are applicable to describe the energy spectrum of surface states near the Dirac point for any band parameters and the strain-induced band gap. Namely, the energy of the Dirac point $\varepsilon_{0}$ is proportional to the band gap value, $\varepsilon_{0}\propto\varepsilon_{g}$, where the proportionality constant depends on the bulk band parameters, $\gamma_{1,2,3}$ and $\alpha$, and turns into zero if $\gamma_{1}=0$. Taking this into account, both the analytical expression for the Dirac dispersion (15) and the numerically calculated dispersion in Fig. 1c can be used to analyze the surface states in strained HgTe for any gap, $\varepsilon_{g}>0$. Particularly, the Dirac velocity, $v_{D}=\sqrt{3}\alpha/\hbar$, which can be extracted from the dispersion (15), does not depend on the gap.

It follows from the aforesaid that the branches of the surface states merge into the spectrum of bulk electronic states if the plane electron wave vector, $\mathbf{k}_{s}$, is large enough (see Fig. 1c). Therefore, they exist only near $\mathbf{k}_{s}=0$. However, there are the surface electronic states of other kind, which can exist far from $\mathbf{k}_{s}=0$. In contrast to the states (III), the BIA Hamiltonian (4) is not crucial for their existence and will be omitted in the following analysis. To simplify calculations, we also will neglect the weak anisotropy of electron-hole dispersion in HgTe. Following Ref. Dyakonov_1981, , this neglect corresponds to the replacement of the Luttinger parameters, $\gamma_{2,3}\rightarrow\gamma=(2\gamma_{2}+3\gamma_{3})/5$, in the Hamiltonian (2). Under these assumptions, the secular equation (9) reads

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

Solving the equation (16) together with Eqs. (17) and (18), one can find the sought energy spectrum of the surface states, $\varepsilon({k}_{s})$, under the uniaxial strain. This spectrum is plotted in Fig. 2 for the different strain-induced gaps, $\varepsilon_{g}$. In the absence of the strain ($\varepsilon_{g}=0$), Eq. (16) can be solved analytically and leads to the parabolic branch of the D’yakonov-Khaetskii (DKh) surface states Dyakonov_1981 ,

[TABLE]

which is plotted in Fig. 2a. Solving Eq. (16) numerically for $\varepsilon_{g}\neq 0$, we can plot the spectrum of the DKh surface states in strained HgTe. In contrast to the surface states discussed above, the DKh states exist for both tensile (see Fig. 2b) and compressive (see Fig. 2c) strain. However, their structure is crucially different for these two cases: In the case of tensile strain, the DKh states lie only within the bulk spectrum of conduction band (see Fig. 2b), whereas the case of compressive strain corresponds to the DKh states which lie also within the strain-induced band gap (see Fig. 2c). It should be noted also that the DKh states exist only for large electron wave vectors, $k_{s}$, and vanish near $k_{s}=0$. It follows from Figs. 2b and 2c that the DKh branch merges into the continuum of bulk conduction band in the case of tensile strain (see Fig. 2b) and the continuum of bulk valence band in the case of compressive strain (see Fig. 2c) at a some critical electron wave vector. The value of the critical wave vector, $k_{s}=k_{0}$, is defined by Eqs. (16)-(18), where the energy of the surface electron states, $\varepsilon$, is equal to the energy of one of the two bulk electron branches,

[TABLE]

Solving Eqs. (16)-(18) with Eq. (20), one can find that $k_{0}\propto\sqrt{|\varepsilon_{g}|}$. Thus, the increase of the strain-induced gap $\varepsilon_{g}$ shifts the existence domain of the DKh surface states to the region of large electron wave vectors, $k_{s}$. It follows from Eq. (16) that the spectrum of the DKh surface states in strained HgTe is parabolic and described approximately by Eq. (19) for large wave vectors, $k_{s}$, satisfying the condition $\gamma k_{s}^{2}\gg\varepsilon_{g}$. It should be noted that the spectrum of the DKh surface states in strained materials of HgTe-class crucially depends on the Luttinger parameters. To demonstrate this, we plotted the dispersion of the DKh surface states in Fig. 3 for the HgTe band parameter $\gamma=9.0\,\hbar^{2}/2m_{0}$ but for different Luttinger parameters $\gamma_{1}$. In the case of symmetric electron-hole system ($\gamma_{1}=0$) there are the two branches of the DKh states which behave as surface electrons and holes (see Fig. 3a). The nonzero Luttinger parameter $\gamma_{1}$ merges the upper (electronic) branch into the continuum of bulk electronic states and changes the curvature of the lower (hole) branch (see Fig. 3b). As for the real case of HgTe parameters, only the lower branch survives and turns into the surface states of electronic kind (see Fig. 3c).

Finally, it should be noted that the experimental methodology based on the angle resolved photoemission spectroscopy (ARPES) technique, which is commonly used to study surface electronic states in various condensed-matter structures Chen_2012_1 , can allow to observe the surface states discussed above. It is important from an experimental viewpoint that the structure of all considered surface states crucially depends on the bulk Luttinger parameters, $\gamma_{1,2,3}$. Therefore, they should be chosen carefully to interpret experimental results adequately in HgTe-based materials (e.g., solid solutions CdHgTe and MnHgTe), where the band parameters depend on their stoichiometric composition.

IV Conclusion

We developed the theory describing the structure of various surface electronic states which appear due to the mixing of the conduction and valence bands in strained mercury telluride (HgTe). It predicts the coexistence of surface electronic states of two different kinds. First of them originate from the bulk inversion asymmetry of HgTe, have the linear Dirac dispersion, which is characteristic for topological states, and are localized in very narrow region of electron wave vectors near the Brillouin zone center. The surface states of second kind originate from the D’yakonov-Khaetskii surface states existing in gapless HgTe. Due to the strain-induced band gap, they are shifted far from the Brillouin zone center to the region of large electron wave vectors. Thus, the found surface states of the two kinds exist in different areas of the Brillouin zone and can be detected independently. The energy spectrum of the states and their structure are calculated both analytically and numerically in the broad range of band parameters.

Acknowledgements.

The work was partially supported by Horizon2020 RISE project COEXAN, Russian Foundation for Basic Research (project 17-02-00053), Rannis project 163082-051, Ministry of Education and Science of Russian Federation (projects 3.4573.2017/6.7, 3.2614.2017/4.6, 14.Y26.31.0015), and the Government of the Russian Federation through the ITMO Fellowship and Professorship Program. O.V.K. and O.K. thank the University of Iceland for hospitality.

Appendix A The Hamiltonian in the Luttinger-Kohn basis

The wave functions of the Luttinger-Kohn basis, $\psi_{j_{z}}$, corresponding to the different projections of electron angular momentum on the $z$ axis, $j_{z}=\pm 1/2,\pm 3/2$, can be written as

[TABLE]

where the vertical arrows, $\uparrow$ and $\downarrow$, represent the spinors corresponding to the $\pm 1/2$ spin projections on the $z$ axis, and $X,Y,Z$ are the Bloch functions in the Brillouin zone center, which behave like the Cartesian coordinates, $x,y,z$, under rotation of the coordinate axes Bir_Pikus . In the basis (21), the angular momentum matrices, $J_{x,y,z}$, read

[TABLE]

Substituting the matrices (22) into Eq. (1), we arrive at the Hamiltonian (1) written in the Luttinger-Kohn basis,

[TABLE]

where

[TABLE]

In the particular case of the uniaxial strain along the $z$ axis and $\gamma_{1}=0$, the matrix (23) for $k_{x}=k_{y}=0$ turns into the block-diagonal Hamiltonian (11).