Absence of helical surface states in bulk semimetals with broken inversion symmetry
Carmine Ortix, Jörn W. F. Venderbos, Roland Hayn, Jeroen van den Brink
AAbsence of helical surface states in bulk semimetals with broken inversion symmetry
Carmine Ortix, J¨orn W.F. Venderbos, Roland Hayn, and Jeroen van den Brink Institute for Theoretical Solid State Physics, IFW Dresden, D01171 Dresden, Germany Aix-Marseille Univ., CNRS, IM2NP-UMR 7334, 13397 Marseille Cedex 20, France (Dated: September 24, 2018)Whereas the concept of topological band-structures was developed originally for insulators witha bulk bandgap, it has become increasingly clear that the prime consequences of a non-trivialtopology – spin-momentum locking of surface states – can also be encountered in gapless systems.Concentrating on the paradigmatic example of mercury chalcogenides HgX (X = Te, Se, S), we showthat the existence of helical semimetals , i.e. semimetals with topological surface states, criticallydepends on the presence of crystal inversion symmetry. An infinitesimally small broken inversionsymmetry (BIS) renders the helical semimetallic state unstable. The BIS is also very importantin the fully gapped regime, renormalizing the surface Dirac cones in an anisotropic manner. As aconsequence the handedness of the Dirac cones can be flipped by a biaxial stress field.
PACS numbers: 73.20.At,71.55.Gs, 72.80.Sk
Introduction –
The discovery of two- and three-dimensional (3D) topological insulators (TIs) [1–15] hasbrought to light a new state of quantum matter. Thishas had a tremendous impact in the field of fundamen-tal condensed matter physics as well as for potential ap-plications in spintronics and quantum computation [16].The TIs are insulating in the bulk but have topologi-cally protected surface states [2, 4, 14] and the topol-ogy dictates that the metallic surface states are spin-momentum locked: surface electrons with opposite spincounter-propagate at the sample boundaries [3, 4, 7, 9].Materials with a TI band structure such as Sb [13],Bi Se [17] and Bi Rh I [15] often show the presenceof a finite bulk carrier density. In such materials, thebulk Fermi surface does not simply swallow up the topo-logical surface states. They survive and coexist with abulk Fermi surface [13], leading to the notion of a heli-cal metal . The coexistence of a bulk Fermi surface andtopological surface states can be understood as a dopedTI being made out of a bulk TI with a non-topologicalmetallic band inside the gap. The hybridization betweenthe topological surface states and the additional metal-lic band pushes the topological surface states away fromoverlapping with the bulk states in energy and momen-tum. This preserves them in a slightly modified format those points in the Brillouin zone (BZ) where thesurface and the bulk bands do not cross [18]. Also inbulk semimetals with a topological non-trivial band or-dering surface Dirac-like states are expected to coexistwith metallic states [19, 20], suggesting the analogouspresence of a helical semimetallic state.Using the paradigmatic example of the series of cubicmercury chalcogenides HgX (X = Te, Se, S), we showhowever that the existence of a helical semimetallic statecritically relies on the presence of crystal inversion sym-metry even in the absence of disorder. An infinitesimallysmall broken inversion symmetry (BIS) is detrimental forthe topological surface states of a helical semimetal, in- FIG. 1: Schematic band-structure close to the Brillouin zonecentre of the cubic mercury chalcogenides inverted semicon-ductors HgX as compared to the topologically trivial CdTesemiconductor. dependent of any overlap in energy and momentum ofbulk and topological surface states. We show further-more that in the fully gapped TI regime, a BIS does notendanger the existence of topological surface states. Inthis case the BIS rather renormalizes the Fermi velocityof the surface Dirac fermions in an anisotropic manner,similarly to the effect envisioned in anisotropic graphenesuperlattices [21, 22]. This in principle allows an exter-nally applied biaxial stress field to flip the surface statechirality in a material with BIS.
HgX compounds – Pristine HgTe is a semimetal whichis charge neutral when the Fermi energy is at the touch-ing point between the light-hole (LH) and the heavy-hole(HH) Γ bands at the BZ center [20, 23]. The topologi- a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l cal nature of the electronic states in this material cannotbe inferred from these p / atomic levels but rather fol-lows from the inverted band ordering at the zone centerof the LH Γ band which is particle-like and the Γ s -band which is hole-like. In normal semiconductors, suchas CdTe [see Fig.1], Γ forms the conduction band andΓ is one of the valence bands. The consequence of thisband inversion can be understood from the simple crite-rion derived by Fu and Kane [6] to distinguish normaland non-trivial topological classes. This criterion, whichrelies on the presence of inversion symmetry, establishesa material to be in a topologically non-trivial class if twobands of opposite parity have level-crossed with respectto the normal band ordering. The zinc-blende crystalstructures of HgTe lacks inversion symmetry, but it isnormally considered that the BIS acts as a small pertur-bation and, by invoking the principle of adiabatic conti-nuity, does not hinder the topological nature of the levelcrossing. As the HH bands do not participate in the topo-logical level crossing, it can be assumed that they act asinserted ”parasitic” bulk bands closing the full band gapand preventing the system to be a strong 3D TI. It is ex-pected [19, 20] that the existence of topological surfacestates resulting from the LH-Γ TI bulk is not under-mined by the presence of the HH bulk bands suggestingHgTe to be a helical semimetal.Similar arguments apply to HgSe which has the sameband ordering as HgTe but with the difference that thespin-orbit (SO) split-off Γ bands are above the Γ bands[24] – the SO splitting ∆ = E (Γ ) − E (Γ ) is smallerthan the gap − E = E (Γ ) − E (Γ ) [c.f. Fig. 1]. In thiscase the SO split-off bands, the LH and the Γ bandsrealise a bulk TI with the HH bands playing as in HgTethe role of parasitic bands which close the full band gap.Yet another material of the same family – metacinnabar– was proposed to be in a topologically non-trivial class:a recent fully-relativistic electronic structure calculation[25] finds the required band-ordering, although in thisparticular case the Γ bands and Γ have switched placeswith respect to the normal ordering [c.f. Fig. 1] and thus∆ <
0. This reversed order originates from a small butsignificant contribution of Hg 5 d orbitals whose spin-orbit(SO) coupling dominates over the sulfur 3 p states andreverses its sign [26] by an amount sufficient enough tocreate a small gap thus rendering β -HgS a stoichiometricstrong 3D TI . Inversion invariant effective Hamiltonian –
For ananalysis of the topological surface states one can rely oneffective low-energy theories that are based upon a k · p expansion of the lowest energy bands around the Γ point.In the simplest case, as for instance Bi Se , the k · p theory yields a Dirac-like Hamiltonian [9, 27] with theDirac-mass representing the energy gap and a change ofits sign corresponding to a nontrivial level crossing. Thesame approach applies to the mercury chalcogenides andhas correctly predicted the Quantum Spin Hall effect in -4 -2 0 2 402468101214 -4 -2 0 2 400.10.20.30.40.50.6 -4 -2 0 2 4-0.6-0.4-0.200.2 ∆ / |E | l c | E | / P ∆ / |E | ∆ / |E | ∆ / |E | E Γ / | E | v F / P S x , S y ( a . u . ) (a) (b)(d)(c) -4 -2 0 2 4-1-0.500.511.52 FIG. 2: (a) Behavior of the energy of the surface Dirac pointmeasured in units of the gap | E | as a function of the ratioamong the spin-orbit splitting ∆ and | E | in the absenceof broken inversion symmetry. The gray lines correspond tothe conduction and valence bulk band-edges by artificially re-moving the parasitic HH bands. (b) Behavior of the decaylength for the surface states at the centre of the surface Bril-louin zone as a function of ∆ / | E | . (c) Same for the Fermivelocity of the surface Dirac cones v F measured in units ofthe band structure parameter P . (d) Behavior of the spinconstants S x,y as a function of the spin-splitting energy ∆ . HgTe/CdTe quantum wells [3]. The generic form of alow-energy k · p expansion at the BZ center Γ for semi-conductors with a zinc-blende crystal structure is givenby the Kane model Hamiltonian [28]. Previous work hasfocused predominantly on the bands responsible for thelevel crossing, the Γ and Γ bands [20, 23]. While suchan analysis is capable of correctly describing the topo-logical characteristics of HgTe and its consequences forsurface excitations, here we consider the full eight-bandKane model Hamiltonian which takes into account theΓ , Γ and Γ bands and correctly describes the bandordering near the Brillouin zone centre of the series ofmercury chalcogenides HgX once the spin-orbit splittingenergy ∆ is varied. This allows us to smoothly connectfrom the intrinsic, fully gapped, TI regime realised in β -HgS to the putative helical semimetal regime for ∆ > not only provides a convenienttuning parameter, its variation represents the physicaleffect of biaxial strain fields. Simultaneous application oftwo stress fields directed along the [100] ([010]) and [001]directions will under specific conditions [see the Supple-mental Material] preserve the degeneracy at the Γ pointamong the LH and HH bands, but renormalize the SOenergy ∆ .To establish the helical semimetal state in the cubicmercury chalcogenides when inversion symmetry is pre-served, we explicitly calculate the [001] surface states ofthe eight-band Kane model Hamiltonian by neglectingBIS effects on the half-space z > T = 0 K [29]. At the Γ point of the surface BZ, theKane model Hamiltonian predicts the HH bands to becompletely decoupled from the other bands. This decou-pling guarantees the absence of any mixing among theparasitic HH bands and the topological surface states re-sulting from the TI bulk. The corresponding part of theHamiltonian is block diagonal with the two blocks forthe chalcogen p -type (mercury s -type) states of total an-gular momentum J z = 1 / J z = − / ↑ ( z ) = ( ψ ↑ , ) T and Ψ ↓ ( z ) = ( , ψ ↓ ) T where ψ ↑ , ↓ is a three-dimensionalspinor and is a five component zero vector. For the sur-face states, the wavefunction ψ ↑ , ↓ ( z ) is localized at the[001] surface in which case Ψ ↑ , ↓ play the role of a spinone-half surface Kramer’s doublet [see the SupplementalMaterial].Fig.2(a) shows the energy of the surface Kramer’s dou-blet as a function of the ratio among the SO splitting en-ergy ∆ and the Γ − Γ gap − E . In the intrinsic, fullygapped, TI regime, ∆ <
0, the surface state’s energyresides in the direct bulk insulating gap at the Γ point.In the ∆ < , -LH bands. Fig.2(b) shows the behavior of the de-cay length of the surface states. We find that preciselyat ∆ ≡ z >
0. By projecting thebulk Hamiltonian onto the subspace of these two surfacestates [9] , we obtain an effective surface Hamiltonian tothe leading order of k x,y H surf ( k x , k y ) = E Γ I + v F ( σ x k y − σ y k x ) , (1)with the Fermi velocity v F whose behavior as a func-tion of the spin-orbit splitting is shown in Fig.2(c). Thatthe σ matrices in the effective surface model Hamilto-nian are proportional to the real spin can be shown byprojecting the total angular momentum operators J x,y,z onto the surface state subspace. Independent of the spin-orbit splitting energy, we do find that (cid:104) Ψ | J x,y,z | Ψ (cid:105) ≡ S x,y,z σ x,y,z where S z ≡ / S x ≡ S y with a fi-nite value whose behavior as a function of ∆ is shownin Fig.2(d). As a result, the surface states show a lineardispersion with helical spin-textures left-handed for thesurface conduction band and right-handed for the surfacevalence band proving the spin-momentum locking of thesurface state solutions. Broken inversion symmetry –
Having established thatin presence of inversion symmetry, the series of cubic mer-
FIG. 3: (a) Phase diagram for the existence of renormaliz-able surface state solutions in the [001] surface of HgX mer-cury chalcogenide compounds as obtained from the eight-band Kane model Hamiltonian. The gray area correspondsto regions with topological surface states at the BZ centrewhere the opening of the indirect bulk band gap cannot becorrectly captured by the Kane model . (b) Behavior of theenergy of the surface Dirac point measured in units of thegap | E | as a function of the ratio among the spin-orbit split-ting ∆ and | E | for different strengths of the linear in k BISterms. The Dirac point energy always lies in the gap at theΓ point among the Γ and the SO split-off bands Γ whoseband-edges are represent by the gray lines. (c) Same for thedecay length of the surface states. cury chalcogenides will either be in the strong 3D TI or inthe helical semimetal state, we now take into account theintrinsic BIS of the zinc-blende crystal structure. Froma k . p perspective, the BIS allows for additional termsin the bulk Hamiltonian once the point group symme-try is reduced to D d [23, 28]. In the valence band block H , of the Kane model Hamiltonian [see the Supplemen-tal Material] the BIS indeed yields an additional term[30] H , BIS = c (cid:2)(cid:8) J x , J y − J z (cid:9) k x + c.p. (cid:3) / √ H , block H , BIS = − i √ c (cid:2) T † yz k x + c.p. (cid:3) .The presence of this linear in k additional terms stemsfrom bilinear terms consisting of k . p and SO interac-tion with the uppermost d core levels [31]. As a result,the parameter c is an elementary parameter of the Kanemodel that unlike the higher order spin splitting termsinduced by BIS cannot be expressed in terms of the ex-tended Kane model [28]. Because of the smallness of theelementary parameter c (cid:39)
80 meV ˚A[31] as comparedto the linear parameter coupling P (cid:39) c value, indeed, the BIS-induced linear in k terms couplethe HH with the TI bulk at the Γ point of the surface BZ. FIG. 4: (a),(b) Behavior of the two non-equivalent Fermi ve-locities of the surface Dirac cones as a function of the spin-orbit splitting ∆ for different values of the elementary pa-rameter c of the eight-band Kane model Hamiltonian. (c),(d)Same for the behavior of the spin constants S , ,z . As a result, the BIS leads to an effective hybridizationamong the topological surface states and the parasiticHH bands. One would then expect that whenever thetopological surface states overlap in momentum and en-ergy with the parasitic HH bands they should be pushedaway. And indeed we find that for positive values of thespin-orbit splitting ∆ , in which case the energy of thesurface Kramer doublet lies below the zero-energy HHband-edge, localized surface state wavefunctions Ψ ↑ , ↓ atthe BZ centre do not exist. In the ∆ < c parameterand the spin-orbit splitting. On the contrary we find thatthe existence of topological surface states is intrinsicallyrelated to the strength of the linear in k BIS terms andleads to the phase diagram shown in Fig.3(a). Remark-ably for small values of the BIS parameter c , renormal-izable surface states appear only whenever the spin-orbitsplitting is negative by an amount sufficient to create afull indirect band gap. Thus, even in the absence of anoverlap in momentum and energy with the parasitic HHbands, the topological surface states are prevented in theabsence of a full bulk band-gap proving that the helicalsemimetal state is completely suppressed by the BIS.Fig.3(b) shows the behavior of the surface Kramer’sdoublet energy at the surface Brillouin zone centre fordifferent values of the BIS elementary parameter c whenthe intrinsic, fully gapped, TI regime is reached. It al-ways lyes in the bulk gap at the zone centre among theΓ and the SO split-off Γ bands. We also show [c.f.Fig.3(c)] the behaviour of the penetration depth of thesurface states which, increasing the value of the spin- orbit splitting ∆ , increases and eventually diverges atthe ”topological phase transition” of the phase diagramin Fig. 3(a). By projecting again the bulk Hamiltonianonto the subspace of the surface BZ surface states, weobtain that in the presence of BIS terms the effectivesurface Hamiltonian to the leading order of k x,y reads H surf ( k x , k y ) = E Γ I + v F k σ − v F k σ , (2)where k , = ( k x ± k y ) / √ σ , are the correspond-ing rotated Pauli matrices σ , = ( σ x ∓ σ y ) / √
2. As aresult the surface Dirac cones are anisotropic ( v F (cid:54) = v F )along the diagonal directions of the surface BZ as can beshown by a two-dimensional k . p analysis [see the Sup-plemental Material] and in perfect agreement with thedensity functional electronic structure calculations in β -HgS [25]. In addition, the spin-momentum locking of thesurface states is guaranteed by the fact that by projectingthe π/ J , ,z onto the subspace of the surface states at the BZ centrewe find (cid:104) Ψ | J , ,z | Ψ (cid:105) ≡ S , ,z σ , ,z with S , ,z some con-stants the behaviour of which, as function of the spin-orbit splitting ∆ is shown in Fig.4(c),(d). Fig.4(a),(b)show the behaviour of the two inequivalent Fermi veloci-ties for different values of the BIS parameter c . It is evi-dent that for ∆ (cid:28) E , the surface Dirac cone is stronglyanisotropic with a large dispersion along the diagonal k and a nearly flat band along the perpendicular direction.By varying the strength of the spin-orbit splitting en-ergy, we find a critical value of the spin-orbit splitting ∆ c where the degree of anisotropy v F /v F diverges and thesurface states will be completely one-dimensional. Evenmore, the fact that only one of the two non-equivalentFermi velocities changes sign, implies a change in thehandedness of the surface Dirac cone – left-handed for∆ < ∆ c and right-handed for ∆ > ∆ c in the sur-face conduction band. Therefore, a suitable applicationof anisotropic biaxial stresses can induce a flip of chi-rality which would immediately manifest itself as a signchange of the quantized Hall conductance in the presenceof a time-reversal symmetry breaking perturbation at thesurface. Conclusions –
Coexistence of bulk metallic states withtopological surface states can be encountered in a largeclass of materials. Inverted zero-gap semiconductors fallinto this class and provide a prominent example of anhelical semimetal. We have shown here that while such atopological state of matter can be established in crystalswith inversion symmetry, a breaking of the bulk inver-sion symmetry initiates a bulk-surface state struggle inwhich the topological surface states, and thereby the heli-cal semimetallic state as a whole, perish. In the intrinsic,fully gapped, TI regime, the broken inversion symmetrystrongly renormalizes the anisotropy of the group veloc-ity of the surface Dirac fermions similarly to graphenesuperlattices [21, 22]. This might be a relevant featurefor spin conduction experiments where a large anisotropyof the surface Dirac cone has been predicted to lead tovery large spin lifetimes [32].The authors thank M. Richter and F. Virot for veryfruitful discussions. [1] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005).[2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005).[3] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science , 1757 (2006).[4] C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. , 106401 (2006).[5] M. K¨onig, S. Wiedmann, C. Br?ne, A. Roth, H. Buh-mann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang,Science , 766 (2007).[6] L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007).[7] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. ,106803 (2007).[8] J. E. Moore and L. Balents, Phys. Rev. B , 121306(2007).[9] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.Zhang, Nat Phys , 438 (2009).[10] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J.Cava, and M. Z. Hasan, Nature , 970 (2008).[11] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A.Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z.Hasan, Nat Phys , 398 (2009).[12] Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K.Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang,S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen,Science , 178 (2009).[13] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J.Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S.Hor, R. J. Cava, and M. Z. Hasan, Science , 919(2009).[14] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[15] B. Rasche, A. Isaeva, M. Ruck, S. Borisenko, V. Zabolot-nyy, B. B¨uchner, K. Koepernik, C. Ortix, M. Richter, andJ. van den Brink, Nat Mater , 422 (2013).[16] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker,Phys. Rev. Lett. , 216404 (2009).[17] K. Eto, Z. Ren, A. A. Taskin, K. Segawa, and Y. Ando,Phys. Rev. B , 195309 (2010).[18] D. L. Bergman and G. Refael, Phys. Rev. B , 195417(2010).[19] C. Br¨une, C. X. Liu, E. G. Novik, E. M. Hankiewicz,H. Buhmann, Y. L. Chen, X. L. Qi, Z. X. Shen, S. C.Zhang, and L. W. Molenkamp, Phys. Rev. Lett. ,126803 (2011).[20] R.-L. Chu, W.-Y. Shan, J. Lu, and S.-Q. Shen, Phys.Rev. B , 075110 (2011).[21] C.-H. Park, L. Yang, Y.-W. Son, M. L. Cohen, and S. G.Louie, Nat. Phys , 213 (2008).[22] C.-H. Park, Y.-W. Son, L. Yang, M. L. Cohen, and S. G.Louie, Nano Letters , 2920 (2008).[23] X. Dai, T. L. Hughes, X.-L. Qi, Z. Fang, and S.-C. Zhang,Phys. Rev. B , 125319 (2008).[24] A. Svane, N. E. Christensen, M. Cardona, A. N. Chantis, M. van Schilfgaarde, and T. Kotani, Phys. Rev. B ,205205 (2011).[25] F. Virot, R. Hayn, M. Richter, and J. van den Brink,Phys. Rev. Lett. , 236806 (2011).[26] A. Delin, Phys. Rev. B , 153205 (2002).[27] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C.Zhang, Phys. Rev. B , 045122 (2010).[28] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer,Berlin, 2003).[29] E. G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. La-tussek, C. R. Becker, G. Landwehr, H. Buhmann, andL. W. Molenkamp, Phys. Rev. B , 035321 (2005).[30] G. Dresselhaus, Phys. Rev. , 580 (1955).[31] M. Cardona, N. E. Christensen, and G. Fasol, Phys. Rev.Lett. , 2831 (1986).[32] V. E. Sacksteder, S. Kettemann, Q. Wu, X. Dai, and Z.Fang, Phys. Rev. B , 205303 (2012).[33] W.-Y. Shan, H.-Z. Lu, and S.-Q. Shen, New Journal ofPhysics , 043048 (2010).[34] L. Fu, Phys. Rev. Lett. , 266801 (2009). Appendix A: Kanel model Hamiltonian
As long as the intrinsic bulk-inversion asymmetry ofthe zinc-blende crystal structure is not taken into ac-count, the eight-band Kane model Hamiltonian reads H = H , H , H , · H , H , · · H , (3)with the expression of the Hamiltonian subblocks H α,β listed in Table I. They are expressed in terms of the usualPauli matrices σ x,y,z , the J = 3 / J x = √ / I ⊗ σ x + ( σ x ⊗ σ x + σ y ⊗ σ y ) / J y = √ / I ⊗ σ y +( σ y ⊗ σ x − σ x ⊗ σ y ) / J z = σ z ⊗I + I ⊗ σ z / I the identity matrix and the following T i matrices: T x = 13 √ (cid:18) −√ − √ (cid:19) T y = − i √ (cid:18) √ √ (cid:19) T z = √ (cid:18) (cid:19) T xx = 13 √ (cid:18) − √ −√ (cid:19) T yy = 13 √ (cid:18) − −√ √ (cid:19) T zz = √ (cid:18) − (cid:19) T yz = i √ (cid:18) − −√ √ (cid:19) T zx = 12 √ (cid:18) − √ √ − (cid:19) T xy = i √ (cid:18) − − (cid:19) The parameters
F, γ , γ , γ describe the coupling toremote bands and are, as well as P, E , material-specificparameters. We have considered for simplicity the axialapproximation γ = ( γ + γ ) / µ = ( γ − γ ) / ≡ k x,y plane [29].Effects of strain can be taken into consideration by ap-plying the formalism of Bir and Pikus. They lead to addi-tional terms in the eight-band Kane model Hamiltonianproportional to the strain tensor (cid:15) and expressed in termsof the valence band deformation potentials C , D u , D (cid:48) u [see TableII] The strain-induced interactions then lead toa change in the band-edges with the degeneracy at the Γpoint among the LH and HH bands that is preserved fora strain field with (cid:15) xx + (cid:15) yy = 2 (cid:15) zz . This condition canbe achieved, for instance by a simultaneous application TABLE I: Expressions of the Kane model in the axial ap-proximation. Here { A, B } denotes the anticommutator forthe A, B operators, c.p. cyclic permutations of the precedingterm and we defined B = (cid:126) / (2 m ) with m the free electronmass. We also list the band structure parameters for HgTe ofRef.[29] .Hamiltonian blocks k · p interactions H , E + B (2 F + 1) k H , √ P T · k H , − √ P σ · k − Bγ k + 2 Bγ (cid:104)(cid:16) J x − J (cid:17) k x + c.p. (cid:105) H , + Bγ [ { J x , J y } { k x , k y } + c.p. ]6 Bγ (cid:2)(cid:0) T † xx k x + c.p. (cid:1) + H , (cid:0) T † xy { k x , k y } + c.p. (cid:1)(cid:3) H , − ∆ − Bγ k E F P /B γ γ γ -0.3 eV 0 18.8 eV 4.1 0.5 1.3TABLE II: Strain-induced terms in the eight-band Kanemodel HamiltonianHamiltonian blocks strain-induced interactions H , C T r(cid:15)D d T r(cid:15) + D u (cid:2)(cid:0) J x − J (cid:1) (cid:15) xx + c.p. (cid:3) H , D (cid:48) u [ { J x , J y } (cid:15) xy + c.p. ] H , D d T r(cid:15) of two uniaxial stresses along the [100] ([010]) and [001]direction of magnitudes X ( Y ) and Z respectively with X, Y = 2 Z in which case T r(cid:15) = 3 ( S + 2 S ) Z where S , S are the elastic compliance constants. Appendix B: Surface state solutions
We solve the eight-band Kane model Hamiltonianon the half-infinite space z > k x,y ≡
0) where the degeneracy of the surface bandsis protected by time-reversal symmetry. We therefore re-place k z → − i∂ z and obtain the Schr¨odinger equation H k x,y ≡ ( k z → − i∂ z ) Ψ( z ) ≡ E Γ Ψ( z ) In the absence ofBIS, the eigenstates take the formΨ ↑ ( z ) = (cid:32) ψ ↑ (cid:33) Ψ ↓ ( z ) = (cid:32) ψ ↓ (cid:33) . (4)where ψ ↑ , ↓ is a three-dimensional spinor and is a fivecomponent zero vector. Obviously to obtain the surfacestates, the wavefunction ψ ↑ , ↓ ( z ) should be localised at the[001] surface. We therefore put a trial spinorial wavefunc-tion of the form ψ ↑ , ↓ ( z ) = ψ ↑ , ↓ λ e λz into the Schr¨odingerequation thereby obtaining two eigenvalue equations witha unique secular equation the solution of which yields thegeneral wavefunction solution ψ ↑ , ↓ ( z ) = (cid:88) α =1 (cid:88) β = ± C αβ ψ ↑ , ↓ αβ e βλ α ( E Γ ) z . The normalizability of the wavefunction in the z > β negative and immediately yields theexistence condition of the surface states R ( λ α ) (cid:54) = 0 pre-venting the surface states to penetrate into the bulkand defines the decay length of the surface states l c = max { / R ( λ α ) } . Furthermore, applying the boundarycondition ψ ↑ , ↓ ( z = 0) ≡
0, we obtain a secular equation[33] of the non-trivial solution for the coefficients C αβ that determines the energy of the two degenerate surfacestates at the surface Brillouin zone centre.When the intrinsic BIS of the zinc-blende crystal struc-ture is taken into account, the eigenstates of the Hamil-tonian H k x,y ≡ have the form Ψ ↑ ( z ) = ( ψ ↑ , ) T andΨ ↓ ( z ) = ( , ψ ↓ ) T with ψ ↑ , ↓ now a four-dimensionalspinor. With this, the general solution can be written as ψ ↑ , ↓ ( z ) = (cid:80) α =1 (cid:80) β = ± C αβ ψ ↑ , ↓ αβ e βλ α ( E Γ ) z with the con-dition of renormalizability of the wavefunction implying R ( λ α ) (cid:54) = 0 and β negative. For positive spin-splittingenergy ∆ the condition for the existence of the surfacestates is never verified since one of the λ α is purely imag-inary independent of the c value. For negative spin-orbitsplittings ∆ instead, a renormalizable surface state so-lution can exist provided a non-trivial solution for thecoefficients C αβ can be found. Appendix C: Two-dimensional k . p theory The point group of cubic mercury chalcogenides, is thegroup T d , which does not contain the inversion opera-tion. If one of these materials is terminated in the (001)direction, leading to a two-dimensional (001)-surface, thepoint group of that surface is reduced to C v , consistingof a twofold rotation symmetry along the z axis, and twomirror symmetries M x : x → − x and M y : y → − y along the two diagonal Γ − M directions of the surface BZ. Bychoosing as a natural basis for the Kramer doublet at thecentre of the surface Brillouin zone, the total angular mo-mentum J = ± /
2, we have that the twofold rotationalsymmetry can be represented as C = − iσ z . Similarly wecan represent the two mirror operations as M x = − iσ x and M y = − iσ y by choosing the phases of | ψ ↑ , ↓ (cid:105) appro-priately. Finally the anti unitary time-reversal operatoris represented as usual as T = iσ y K . The Kramers dou-blet is split away from the centre of the surface BZ andthe corresponding surface band structure can be stud-ied within the k . p framework. The form of the effectivesurface Hamiltonian H ( k ) is highly constrained by time-reversal and crystal symmetries. Indeed under C and M x,y , spin and momentum transform as follows: C : k x,y → − k x,y , σ x,y → − σ x,y , σ z → σ z M x : k x → − k x , k y → k y , σ x → σ x , σ y,z → − σ y,z M y = M x ( x → y ) (5)The Hamiltonian H ( k ) must be invariant under Eq.5. Inaddition, time-reversal symmetry gives the constraint H ( k ) = σ y H (cid:63) ( − k ) σ y (6)As a result, we find that the effective Hamiltonian musttake the following form up to second order in k : H ( k ) v = E v ( k ) I + v xF k x σ y − v yF k y σ x (7)where E v ( k ) = (cid:126) k x / (2 m x ) + (cid:126) k y / (2 m y ) generatesparticle-hole asymmetry. The Hamiltonian for the sur-face states at the (001) surface of a material with dia-mond crystal structure which instead possesses inversionsymmetry can be easily calculated from Eq.7 by addingthe additional constraint due to the four-fold rotationalsymmetry along the z axis. As C can be represented ase iπσ z / we have that spin and momentum transform as C : k x → − k y , k y → k x , σ x → − σ y , σ y → σ x . (8)As a result, the Hamiltonian for the corresponding bulkinversion symmetric material would read as H ( k ) v = E v ( k ) I + v F ( k x σ y − k y σ x ) (9)where E v ( k ) = (cid:126) k / (2 m ). Therefore systems with in-version symmetry display the same emerging U (1) rota-tional symmetry as is Bi Se3