An all electron topological insulator in InAs double wells
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b All electron topological insulator in InAs double wells
Sigurdur I. Erlingsson ∗ and J. Carlos Egues School of Science and Engineering, Reykjavik University,Menntavegi 1, IS-101 Reykjavik Reykjavik, Iceland Instituto de F´ısica de S˜ao Carlos, University of S˜ao Paulo, 13560-970 S˜ao Carlos, SP, Brazil
We show that electrons in ordinary III-V semiconductor double wells with an in-plane modulatingperiodic potential and inter well spin-orbit interaction are tunable Topological Insulators (TIs). Herethe essential TI ingredients, namely, band inversion and the opening of an overall bulk gap in thespectrum arise, respectively, from (i) the combined effect of the double well even-odd state splitting∆
SAS together with the superlattice potential and (ii) the interband Rashba spin-orbit coupling η .We corroborate our exact diagonalization results by an analytical nearly-free electron descriptionthat allows us to derive an effective Bernevig-Hughes-Zhang (BHZ) model. Interestingly, the gate-tunable mass gap M drives a topological phase transition featuring a discontinuous Chern number at∆ SAS ∼ . PACS numbers: 73.63.Hs,71.70.Ej,73.40.-c
I. INTRODUCTION
Topological Insulators (TI) have been theoreticallypredicted in graphene and in negative-gap or inverted-band HgTe-based quantum wells , being experimentallyrealized in the latter shortly after . These are ex-otic solids being bulk insulators with metallic edges orsurfaces . An essential ingredient for a system to ex-hibit TI phases is the existence of a tunable bulk bandgap that can not only be tuned to zero but also invertits sign. Defining as “positive gap” the case in whichit can be mapped without closing onto the m c > . Time reversal symmetry and spin orbitinteraction inextricably lock the spin and momentum ofthese states making them helical .More recently, interesting works have proposed TIswith ordinary bulk materials , as naturally occurringinverted-band or negative-gap materials are usually un-conventional narrow band-gap systems. These propos-als rely on externally inducing a band inversion of theelectron and hole states, e.g., electrically in double wellsystems . Hexagonal patterns fabricated in p -dopedGaAs well to resemble the physics of Dirac carriers ingraphene offer another means to attain TI phases . An-other appealing idea is the use of built-in polarizationfields to induce band inversion and TI phases in Ge sand-wiched between GaAs layers . All of these works relyon electrons and holes or holes only Here we propose a TI based on ordinary III-V semicon-ductor nanostructures with only electrons. We considera bilayer quantum well with two confined electron sub-bands and intersubband spin-orbit coupling (ISOC) , ∆ SAS a ) L L y xb ) πL k x ǫ ( k ) ǫ ( k ) ∆ SAS V V c ) FIG. 1: a) Proposed double well structure and its potentialprofile showing the two lowest states split by ∆
SAS . The in-plane superlattice is denoted by the black dots on top of thestructure. b) Unit cell of the periodic “pits” in a). The rel-ative depth of the pits, indicated by different shadings, de-termine the values V and V that parametrize the periodicpotential . c) Free and nearly-free electron bands (dashedand solid, respectively) in units of ~ Q m along the k x axis.The interband spin-orbit coupling η will open up a gap at thecrossing (circle) of the inverted bands, see Fig. 2. Fig. 1a. Such even/odd two band systems have been re-alized in wide quantum well . By fabricating a periodicpattern on top of the structure (e.g. via etching) anddepositing metal gates gives rise to an in-plane modulat-ing superlattice potential within the QW , Fig. 1a, weare able to obtain the necessary ingredients for a TI: (i)tunable inverted subbands controlled by the ‘mass gap’2 M = ∆ SAS − ∆ V that depends on both the double-well even-odd state splitting ∆ SAS and the quantity ∆ V that is determined by the gate controllable parametersof the periodic potential, and (ii) a bulk overall gap con-trolled via the ISOC η that gives rise to anti crossings,see bands around the Γ-point in Fig. 2. In principle theFermi energy can be tuned so as to lie in the bulk gap.The sign of M can be tuned either through ∆ SAS (dif-ferent quantum well structures) or the gate-controllable∆ V , see Appendix A.We solve the problem within the physically appealingnearly-free electron description. In this approach we an-alytically derive an effective BHZ model for our system .We also solve the problem numerically via exact diagonal-ization thus determining the full energy spectrum withinthe Brillouin zone. In the appropriate parameter rangethe two descriptions agree very well. We also calculatethe topological invariant from our bulk band structureand show that the system undergoes a topological phasetransition when the mass M changes sign, indicated by adiscontinuity in the topological invariant as a function of∆ SAS or ∆ V , see Fig. 4. We then find the solutions fora strip configuration to verify the bulk-edge correspon-dence explicitly: in the non-topological phase ( M >
M <
0) it features, in addi-tion, gapless edge states with Dirac-like bands, see Fig.5. Interestingly, we find that the edge states display os-cillations as they spatially decay away from the borderinto bulk, see Fig. 6. These oscillations can in principlebe mapped via scanning gate microscopy recently usedto probe edge states in HgTe-based TI wells . II. MODEL SYSTEM
Our effective 4 × {| p , e ↑i , | p , o ↓i , | p , e ↓i , | p , o ↑i} of the (e)ven and(o)dd eigenstates we have H w ( p )= p m ∗ − i η ~ p − i η ~ p + p m ∗ + ∆ SAS p m ∗ i η ~ p + − i η ~ p − p m ∗ + ∆ SAS , (1)where p denotes the electron momentum and p ± = p x ± ip y . Here η is the interband spin-orbit coupling and m ∗ the electron effective mass. The Hamiltonian in (1)can be put into the standard BHZ form by the unitarytransformation U = e − πσ y / acting on each 2 × × H w, × ( p ) = p m ∗ + ∆ SAS η ~ p + η ~ p − p m ∗ ! . (2)The lower diagonal block is the time reversed version ofEq. (2): H ∗ w, × ( − p ). Equation (2) has a form reminis-cent of the 2 × . In the BHZ model theinverted band structure arises from the peculiar ordering -3-2-10123456 Γ X M Γ E [ m e V ] δ η ∆ SAS ∆ V M = ∆ SAS − ∆ V < M = 0 . M = − . M = − . FIG. 2: In-plane superlattice bandstructure (red curves)for V = 3 . V = 12 meV, ∆ SAS = 4 . η = 20 meV nm. The gray (dashed) curves are the en-ergy bands for η = 0. A finite interband spin-orbit cou-pling η gives rise to anti crossing of the inverted bands, thusgenerating an overall gap (shaded rectangular gray area).The shift δ η is due to ISOC corrections to the band ener-gies. The inset shows the bandstrucure for different values of M (∆ SAS ) = − . . , − . . . .
6) meV. of bands of HgTe combined with the tunability of theelectron and hole levels in a well geometry. Here we will engineer an inverted-band system from the ordinary dou-ble well with normal ordering of bands by superimposinga two dimensional superlattice on top of it, Fig. 1a andb. We choose the periodic potential (period L ) as V ( r ) = V (cos( Qx ) + cos( Qy )) + V cos( Qx ) cos( Qy ) , (3)where Q = πL . This potential gives rise to parabolicdispersions around the Γ-point with positive curvature(mass) for the first and second subbands and negativecurvature for the third band, schematically shown in Fig.1c. More specifically, the superlattice Hamiltonian is H SL = H w, × ( − i ~ ∂ x , − i ~ ∂ y ) + V ( x, y ) I × . (4)where we have used p = − i ~ ∇ . The correspond-ing eigensolutions are Bloch wave functions ψ k ,n ( r ) = e i k · r u k ,n ( r ) with energies ε n ( k ), u k ,n ( r ) has the sameperiodicity as V ( r ) . It is convenient to define the en-ergy scale E Q = ~ Q m ∗ , which for InAs ( m ∗ = 0 . L = 80 nm, yields E Q ≈
10 meV.
III. GAPPED BULK SPECTRUM: NUMERICS
Figure 2 shows the band structure (red curves) ob-tained via exact diagonalization using the parameters: V = 3 . V = 12 . SAS = 4 . η =20 meVnm . The interband coupling η can be furtherincreased by optimizing the quantum well structure .The V term opens up a gap at the Γ-point, givingrise to a negative curvature band, and V facilitates thecoupling between the second and third states for finite k values, see Eqs. (18) and (19). When V ≈ E Q the gapopens up over the full Brillouin zone. The gray dashedcurves show the bands in the absence of the spin-orbitcoupling η . The 2nd and 3rd bands clearly show inversionand crossings for η = 0, while a non-zero η opens upgaps at the crossing (red curves). The energy splittingof the inverted bands is given by the tunable mass gap2 M = ∆ SAS − ∆ V . The parameter ∆ V is defined as theenergy difference between the 1st and 3rd energy bandsat the Γ-point, see Fig. 2. The value of the even-oddenergy splitting ∆ SAS is controlled by the structure ofthe quantum well confining potential. The inset in Fig.2 is a blowup of the band crossing for ∆ V = 5 . SAS , going from an invertedordering of bands for M = − .
45 meV and − . M = 0 . IV. NEARLY-FREE ELECTRON DESCRIPTION
Here we focus on the 2nd and 3rd bands for η = 0 (seegray curves in Fig. 2), which comprise the two invertedcrossing bands required by the BHZ model. To obtainanalytical results and a better qualitative understand-ing of our system we now follow a perturbative approachbased on the nearly free electron model (NFEM). A. Energy bands and controlled band inversion
We start by looking at the single quantum well in thepresence of a periodic potential. Using Bloch’s theorem,the eigenvalue problem corresponding to the Hamiltonianin Eq. (4) reduces to (cid:18) p m ∗ + ~ m ∗ k · p + V ( x, y ) (cid:19) u n, k ( x, y )= (cid:18) ε n, k − ~ k m ∗ (cid:19) u n, k ( x, y ) . (5)In the k · p spirit, we are interested in finding the en-ergy spectrum at the Γ-point ( k = 0) and then use thesestates to calculate the energy bands away from the Γ-point, using pertubation theory to obtain corrections dueto ~ m ∗ k · p . For the free electron model, the energy lev-els at the Γ-point occur at ε = 0 (1 state), ε = E Q (4 degerate states), ε = 2 E Q (4 degerate states), etc. ,where E Q = ~ Q / m ∗ . To describe the inverted bandswe need to consider the four normalized states that crossat ε = E Q (cid:26) e iQx L , e iQy
L , e − iQx L , e − iQy L (cid:27) , (6)and the ground state at ε , i.e. (cid:8) L (cid:9) . The 4 states inEq. (6) are coupled by the V term and using degenerate perturbation theory the new states and correspondingeigenenergies are u ,A ( x, y ) = 12 L ( e iQx + e − iQx − e iQy − e − iQy ) , (7) u ,B ( x, y ) = 1 √ L ( e iQx − e − iQx ) , (8) u ,C ( x, y ) = 1 √ L ( e iQy − e − iQy ) , (9) u ,D ( x, y ) = 12 L ( e iQx + e − iQx + e iQy + e − iQy ) , (10)and the corresponding eigenergies are ε ,A = E Q − V / ε ,B = E Q , ε ,C = E Q , and ε ,D = E Q + V /
2. The state u ,A ( x, y ), that gets lowered in energy by V /
2, alongwith the ground state that we denote by u ( x, y ) = 1 /L (eigenenergy ε = 0) will form the inverted bands whenthe even/odd state energy separation in the bilayer sys-tem is considered.Anticipating the energy spectrum for the bilayer sys-tem we simplify the notation and use the same labelingscheme as in Fig. 1c) and denote state 2A in Eq. (7) by u and corresponding eigenenergy ε . The second order per-turbation theory correction to states u and u at k = 0are ε ( k = 0) = 0 − V E Q − V E Q + V (11) ε ( k = 0) = E Q − V − V E Q + V − V E Q + V + 1 E Q + V ! . (12)The quantity related to the superlattice potential thatenters into the mass gap is the bandwidth ∆ V , which isdefined as the difference between the top of the secondband and the bottom of the first band:∆ V = ε ( k = 0) − ε ( k = 0) . (13)The NFEM calculation, along with full numerics, of thebandwidth ∆ V are plotted as a function of V , for threevalues of V in Figure 3a). In the range correspondingto the values used in the paper the behavior of ∆ V ispredominantly linear in V with second order correctionscontributing to higher values of V and V . An intu-itive way, although not mathematically rigorous, to un-derstand why the nearly-free electron model gives qual-itatively good results, even for V > E Q , is to write theperiodic potential in terms of Fourier components V ( x, y ) = V e iQx + e iQy ) + V e iQ ( x + y ) + e iQ ( x − y ) ) + c . c . , which shows that the coupling constants entering thepertubative calculations are effectively V and V , i.e.smaller than E Q , even for V = 1 . E Q as we use in themanuscript. ∆ V [ m e V ℄ V [meV℄a) -1-0.500.5110 12 14 16 M ( V ) [ m e V ℄ V [meV℄ ∆ SAS = 5 . meV ∆ SAS = 5 . meV V = 0 . meV V = 2 . meV V = 3 . meV E [ m e V ] k x /Q b) ε ( k x ), NFEM ε ( k x ), NFEM ε ( k x ), numerical ε ( k x ), numerical FIG. 3: a) The bandwidth ∆ V as a function of V , for threevalues of V . The dashed curves are the pertubative resultobtained using Eqs. (11), (12) and (13). The inset shows themass gap 2 M = ∆ SAS − ∆ V ( V ) as a function of V aroundthe value of V = 12 meV, showing that the bands can beinverted via the gate that defines the superlattice potential.b) Numerical and pertubative results, Eqs. (14) and (15), forbands 2 and 3 and η = 0. A nonzero η opens up a gap at thecrossing point, see Fig. 2. For simplicity we exhibit the bands along the k x di-rection and suppress the k y varible for clarity. Next weshow that the curvature of ε ( k x ) is negative. Lowestorder perturbation in ~ m ∗ k x p x results in ε ( k x ) = ε ( k x = 0) + ~ k x m ∗ + (cid:12)(cid:12)(cid:12) ~ m ∗ k x Q √ (cid:12)(cid:12)(cid:12) E Q − V − E Q = ε ( k x = 0) + E Q (cid:18) − E Q V (cid:19) k x Q , (14)which shows that the curvature is indeed negative forvalues of V < E Q . In our calculations we use V = 1 . E Q yielding (cid:16) − E Q V (cid:17) ≈ − .
33, which com-pares well to the numerical value of − .
29, extracted byfitting the full numerics with a parabolic dispersion. The other band that will form the inverted band structure is ε ( k x ), which is the same as ε ( k x ) apart from a shift inenergy of ∆ SAS . The curvature of ε ( k x ) is simply de-termined by the free electron dispersion since the k · p term does not couple the lowest band to any higher-lyingbands, resulting in ε ( k x ) = ∆ SAS + ε ( k x = 0) + ~ k x m ∗ . (15)The two relevant inverted energy bands are shown in Fig-ure 3b). The NFEM result and the numerics show quali-tative agreement, i.e. an inverted band ordering and neg-ative curvature of the hole-like band.When the bilayer quantum well is added together withthe superlattice spectrum we get a spectrum similar tothe one shown in Fig. 1c), i.e. the lowest superlattice bandof the odd quantum well state is shifted by ∆ SAS , leadingto the band ε in Eq. (15). The interband spin-orbitcoupling η opens up a gap at the crossing point, as shownin the inset of Figure 2. The size of the anti-crossing gap in terms of the BHZ parameters (see App. B) is given by∆ η ≡ A q MB , or term of NFEM parameters∆ η ≈ ηL " √ π V V s V E Q (∆ V − ∆ SAS ) ≈ . − . . (16)The size of the gap is predominantly determined by theratio ηL since the quantity in the square brackets is typi-cally of order one. The the quantity in the square brack-ets is ≈ . V since ε ( k x = 0) aquires a spin-orbit correc-tion. This shift can be estimated using second order per-tubation theory in η ( p x ± ip y ). Denoting the bandwidthin the presence of spin-orbit coupling by ∆ V,η , we candefine the spin-orbit induced change in bandwidth as δ η ≡ ∆ V − ∆ V,η = η Q ∆ SAS + V , (17)which gives a calculated spin-orbit induced shift ofaround δ η = ∆ V,η − ∆ V ≈ .
23 meV for the parame-ters used in the paper ( η = 20 meV nm, L = 80 nm,∆ SAS = 4 . V = 12 meV). This correspondsquite well to the numerically obtained value of δ η =∆ V,η − ∆ V ≈ .
19 meV, see Fig. 2.
B. Wavefunction away from the Γ -point The two inverted bands are formed by ε and ε aregiven in Eqs. (15) and (14). Due to the square symmetryof the periodic potential the k x dependence in the bandsis the same as for k y , which is also true for the wavefunc-tions. The wavefunctions corresponding to ε and ε (for η = 0) can be found by doing lowest order perturbationtheory in V , V , and k x,y (see solid lines in Fig. 1c) u k , ( r ) = 1 − m ∗ V ~ Q (cos( Qx ) + cos( Qy )) (18) u k , ( r ) = cos( Qx ) − cos( Qy )+ i ~ Q √ mV (cid:16) k x sin( Qx ) + k y sin( Qy ) (cid:17) . (19)The zeroth order form of u k , ( r ) is a linear combinationof cos( Qx ) and cos( Qy ) since V splits the four degeneratefree-electron bands at the Γ-point. C. Effective BHZ model
We now use the η = 0 states u k , ( r ), u k , ( r ) to con-struct an effective Hamiltonian for η = 0 by projectingthe off-diagonal part of Eq. (4) onto this subspace. Theseoff-diagonal terms are the components of d -vector of theBHZ model given by d x,y ( k ) = − iη Z d r u ∗ k , ( r ) ∂ x,y u k , ( r ) (20)The approximate solutions in Eq. (18) and (19) result in d x,y ( k ) ≈ Ak x,y , A ≡ η V √ V . (21)The properties of the BHZ model, along with the relevantparameters M , A , etc. , are discussed in detail in Refs. 2and 5, and summarized in App. B. The z -component isdefined as d z ( k ) = ( ε ( k ) − ε ( k )) /
2, which correspondsto the separation of ε ( k ) and ε ( k ) Fig. 1c. In Fig.4a) we plot d z ( k ) / | d ( k ) | (color plot) and d x ( k ) , d y ( k )(arrows) using the exact solutions for the energy bands 2and 3 in Fig. 1c) and 2. D. Topological index
From the d -vector we calculate numerically the topo-logical invariant C = 14 π Z d k ˆ d ( k ) · ( ∂ ˆ d ( k ) × ∂ ˆ d ( k )); ˆ d = d | d | (22)for ( i ) fixed superlattice parameters V = 3 . V = 12 meV and varying ∆ SAS and ( ii ) fixed ∆ SAS =5 . V , keeping V = 3 . C shows a clear jump when M changes sign,either by varying ∆ SAS or ∆ V via V , as can be seen inFig. 4b). a ) ∆ SAS [meV℄ b ) V [meV℄ C -0.5 -0.25 0 0.25 0.5 k x -0.5-0.2500.250.5 k y -1-0.500.51 FIG. 4: Vector field plot of d (a) and the topological index C (b). In a) the arrows denote the xy components and thecolor scale the z component of d , respectively. Note that d x ( d y ) is linear in k x ( k y ) only close to the Γ-point, in contrastto the BHZ model for which d x ( d y ) is linear irrespective ofthe value of k . C shows a jump as ∆ SAS is varied (for fixed V = 3 . V = 12 meV) and when V is varied (fora fixed value of ∆ SAS = 5 . V. STRIP CONFIGURATION: EDGE STATES.
Here we verify the bulk-edge correspondence by ex-plicitly finding the edge states of the system in a fi-nite geometry. We also determine the bulk and edgespectrum for both the topological and non-topologicalphases, see Fig. 5. We solve the Hamiltonian in Eq. (4)for a strip of width L x , using hard wall boundary con-ditions ψ (0 , y ) = ψ ( L x , y ) = 0. Bloch’s theorem stillapplies in the longitudinal y -direction. We expand thetransverse part of the wavefunction in a normalized sinebasis. The number of transverse states is truncated at M max = 5 N per where N per is the number of superlatticeperiods that fit within a strip width. This correspondsto including, roughly, 5 Q -vectors in the x -direction inthe bulk model. Solving Eq. (4) for a given value of k y yields 2 × N per eigenvalues. Focusing on the eigenvaluesin the energy interval corresponding to the BHZ bands,one can plot the relevant set of eigenvalues as function of k y . A. Gapless edge dispersion
In Fig. 5a) the value of ∆
SAS = 4 . SAS = 5 . SAS controlsthe magnitude and the sign of the gap. Using the BHZnotation, the edge state in Fig. 5a) corresponds to a neg-ative gap M = d z ( ) <
0, Fig. 5b) where M is positiveand no gap states appear. Note that for ∆ SAS = 4 . E [ m e V ] k y [ Q ] a ) -0.2 -0.1 0 0.1 0.2 0.3 2.82.933.13.23.33.43.53.63.73.8 k y [ Q ] b ) FIG. 5: Energy spectra resulting from the exact diagonal-ization of Eq. 4. a) ∆
SAS = 4 . SAS = 5 . B. Oscillatory decaying edges.
In order to compare our results to the known analyticalsolution of the BHZ model, we focus on ∆
SAS = 5 . . From thesevalues we can calculate the properties of the edge statesusing the ansatz ψ ( x ) ∝ e λx . Solving for λ in themiddle of the gap ( k y = 0 . E = − DM/B ) resultsin λ = ± A √ B − D ± i s MB − A B − D ) ! . (23)Note that for the parameters extracted from the bulk spectrum MB − A B − D ) >
0, which yields a λ with anon-zero imaginary part in addition to a real part. Thisgives rise to a localized edge state that oscillate spa-tially. Indeed we see from the numerical diagonalizationof Eq. (4) that the edge state decays into the bulk withslower oscillations due to the imaginary part of λ andrapid oscillations due to the period of the superlattice.The probability densities of the edge states correspond-ing to ∆ SAS = 5 . L x = 80 L and L x = 40 L are shown in Figs. 6a) and b), respectively. For∆ SAS = 5 . / Re { λ } = 11 . L and π Im { λ } = 11 . L ,respectively. For a given energy in the gap there aretwo edge states localized at opposite edges, ψ + k y , ↑ ( x )localized around x = 0 and ψ − k y , ↑ ( x ) localized around x = L x . The density of the BHZ edge state localizedaround x = 0, using parameters extracted from Fig. 2and k y = 0, | ψ BHZ ( x ) | ∝ exp( − x Re { λ } ) sin (Im { λ } x ) , (24)agrees well with the numerical results (blue curves inFig. 6). Note that the edge state density is symmetric ρ ( x ) x [ L ] | ψ BHZ ( x ) | a ) b ) 3.053.13.153.2-0.2 0 0.2 E [ m e V ] k x [ Q ] | ψ BHZ ( x ) | a ) b ) c ) c ) FIG. 6: The edge probability density ρ ( x ) for ∆ SAS =5 . L x = 80 L and b) 40 L . Thecurve for 80 L is shifted up by 0.25 for clarity. The inset ina) shows the edge states along with the BHZ dispersions ob-tained from the bulk spectrum. c) Same as in b) but for∆ SAS = 4 . for k y = 0 . . The helical character of the edge-statescomes from the time reversed part of the 4 × SAS = 5 . M = − . SAS = 4 . p M/B , see Eq. (23). Note thatexperiments based on detecting the edge transport andscanning gate microscopy that can image the modulationof the edge charge density profile are feasible.A potential drawback of our proposal is the relativesmall size of the gap 2 A p M/B ≈ . − . x Ga − x As materials, as compared to HgTe based sys-tems. Optimizing the quantum wells can yield a 3-foldincrease in η , tuning the superlattice parameters willgive a factor 2, and pushing down the period to 40 nm,yields a gap ∼ − V and V ), Fermi energy, chargedensity and ∆ SAS – are easily controllable.This work was supported by the Icelandic ResearchFund, the Brazilian funding agencies CNPq and FAPESPand PRP/USP within the Research Support Center Ini-tiative (NAP Q-NANO). SIE would like to acknowl-edge helpful discussions with H.G. Svavarsson and G.Thorgilsson for assistance with graphics.
Appendix A: The role of ∆ SAS and ∆ V The mass gap 2 M = ∆ SAS − ∆ V is controlled by twoparameters: ( i ) the energy splitting ∆ SAS of the two low-est even and odd double quantum well states and ( ii ) the’bandwidth’ of the energy bands ∆ V introduced in theprevious section.For large enough barriers, which is the case in our dou-ble barrier, the width of the central barrier is the majorfactor in controlling ∆ SAS . By varying the barrier thick-ness the value of ∆
SAS can be controlled, but it will befixed for a given sample. The even-odd splitting ∆
SAS is predominantly determined by the quantum well struc-ture, i.e. barrier thickness, quantum well width etc. andit is almost unaffected by the presence of the periodicpotential. In the absence of the lateral superlattice, andafter projecting the full double quantum well Hamilto-nian onto the, even/odd subspace results in H DQW = − ~ m ( ∂ x + ∂ y ) I + ∆ SAS τ z , (A1)where τ z is the Pauli matrix for the double quantum welleven-odd subspace. We now assume that the electrostaticpotential can be written as a periodic function in x and y , see Eq. (3) in manuscript. The constants V and V are replaced by functions ˜ V ( z ) and ˜ V ( z ) V per ( x, y, z ) = ˜ V ( z ) (cos( Qx ) + cos( Qy ))+ ˜ V ( z ) cos( Qx ) cos( Qy ) , (A2)and when inserted into the Laplace equation ∇ V per =0 , results in the following equations ∂ z ˜ V ( z ) − Q ˜ V ( z ) = 0 , ˜ V (0) = V , ˜ V ( ∞ ) = 0(A3) ∂ z ˜ V ( z ) − Q ˜ V ( z ) = 0 , ˜ V (0) = V , ˜ V ( ∞ ) = 0 . (A4)The solution to these equations are˜ V ( z ) = V e − Qz ≈ V e − Qd (1 − Q ( z − d )) (A5)˜ V ( z ) = V e −√ Qz ≈ V e −√ Qd (1 − √ Q ( z − d )) , (A6)where d is the distance from the surface to the quantumwell. Here we have also assumed that Qw ≪
1, where w is the QW width. When the Hamiltonian with thefull electrostatic potential is projected onto the even-oddsubstace we get˜ H SL = − ~ m ( ∂ x + ∂ y ) + V ( x, y ) + ∆ SAS τ z + wa eo L h V ( x, y ) + ( √ − V cos( Qx ) cos( Qy ) i τ x , (A7)where V ( x, y ) is given in Eq. (3) and a eo = 2 πw Z dzχ ∗ e ( z ) zχ o ( z ) , (A8) is a dimensionless constant of order one coming from thematrix element of Eqs. (A5) and (A6) in the the evenand odd basis. We can disregard the mixing of the evenand odd states due to lateral periodic potential (the τ x term) as long as w L max { V , V } ∆ ≪ . (A9)Note that ∆ SAS can be controlled by the quantum wellstructure, e.g. it can be made larger by a thinner bar-rier, so this condition can always be satisfied. For typicalquantum wells the barrier and well thicknesses, are oforder w ≈
10 nm and the lower limit of periodic poten-tial is L ≈
40 nm. So, even for relatively thick barriersand short period superlattice we have w L ∼ . τ x and τ z in Eq. (A7). Since thecontribution of the different Pauli matrices are added assquares, the condition for discarding the τ x comes fromcomparing the squares of the two contributions, whichleads to Eq. (A9). Appendix B: Connection to BHZ model
In the BHZ model the two bands that comprise theinverted bands are written as ε e , ↑ ( k ) = C + M + ( D − B ) k (B1) ε h , ↓ ( k ) = C − M + ( D + B ) k . (B2)where k = k x + k y . The bands will only cross when M >
B > D >
M <
B 0. The parameter C is simply a trivial energy shiftwhose value has not impact on the physical properties ofthe system. These two bands are then coupled, with acoupling strength A , resulting in a 2 × H BHZ = (cid:18) M + ( D − B ) k Ak + Ak − − M + ( D + B ) k (cid:19) (B3)= − Dk I + d ( k ) · σ , (B4)where k ± = k x + ik y and the vector d is given by d x ( k ) = Ak x , (B5) d y ( k ) = Ak y , (B6) d z ( k ) = M − Bk . (B7)Here we put the trivial constant C = 0. The energydifference between the two bands, in the absence of thespin-orbit coupling, is given by 2 d z ( k ). Note that thefull 4 × A was introduced inEq. (8) in the manuscript and the other parameters canbe related to band parameters in our proposal as follows:2 M ≡ ∆ SAS − ∆ V (B8) B = − E Q Q E Q V < D = E Q Q (cid:18) − E Q V (cid:19) < . (B10) ∗ Electronic address: [email protected] C.L. Kane and E.J. Mele, Phys. Rev. Lett. , 226801(2005). A. Bernevig, T.L. Hughes, and S.-C. Zhang, Science ,1757 (2006). M. Konig, S. Wiedmann, C. Br¨une, A. Roth, H. Buhmann,L. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science ,766 (2007). See also G. M. Gusev et al. Phys. Rev. Lett. , 076805 (2013). M. Z. Hasan, C. L. Kane, Rev. Mod. Phys. , 30453067(2010) X. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057 (2011). O. Pankratov, S. Pakhomov, and B. Volkov, Solid StateCommun. , 93 (1987). M. Konig et al. Phys. Rev. X , 021003 (2013). C. Liu, T.L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang,Phys. Rev. Lett. , 236601 (2008). P. Michetti, J.C. Budich, E.G. Novik, and P. Recher, Phys.Rev. B , 125309 (2012) O.P Sushkov and A.H. Castro Neto, Phys. Rev. Lett. ,186601 (2013). D. Zhang, W. Lou, M. Miao, S. C. Zhang, and K. Chang,Phys. Rev. Lett. , 156402 (2013). Some experimental evidence for the proposal in Ref. isreported in I. Knez, R.R. Du, and G. Sullivan, Phys. Rev.Lett. , 136603 (2011). R.S. Calsaverini, E.S. Bernardes, J.C. Egues, and D. Loss,Phys. Rev. B , 155313 (2008). E. Bernardes et al. Phys. Rev. Lett. , 076603 (2007);S.I. Erlingsson, J.C. Egues and D. Loss, Physica E F.G.G. Hernandez et al. Phys. Rev. , 161305(R) (2013). C. Kittel, Introduction to Solid State Physics (Wiley Pub-lishing, 1995), see chapter 7. T. Schlosser, K. Ensslin, J.P. Kotthaus, and M. Holland.Europhys. Lett. , 683 (1996) Different magnitudes of V and V in Eq. (3) are achievedby making the central holes deeper or shallower than thoseat the unit cell corners. This pattern is then inherited bythe metal deposited on top thus leading to a modulatedelectric potential within the QW upon the application ofa proper gate potential. J. Li, K. Chang, G. Hai, and K. Chen, Appl. Phys. Lett. , 152107 (2008). B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu,Phys. Rev. Lett. , 246807 (2008). P. Michetti, P.H. Penteado, J.C. Egues, and P. Recher,Semicond. Sci. Technol. The BHZ parameter values corresponding to ∆ SAS = 5 . C = 3 . D = 1 . × meVnm , M = 0 . B = 2 . × meVnm and A = 5 . D. Grundler, Phys. Rev. Lett. , 6074 (2000). C. Brune, A. Roth, H. Buhmann, E. Hankiewicz,L. Molenkamp, J. Maceijko, X.-L. Qi, and S.-C. Zhang,Nature Physics , 485 (2012). N.W. Ashcroft and N.D. Mermin, Solid state physics,Saunders, 1976 [see figure 9.4]26