Spin splitting of surface states in HgTe quantum wells
A. A. Dobretsova, Z. D. Kvon, S. S. Krishtopenko, N. N. Mikhailov, S. A. Dvoretsky
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Spin splitting of surface states in HgTe quantum wells
A.A. Dobretsova , , Z.D. Kvon , , S.S. Krishtopenko , , N.N. Mikhailov , S.A. Dvoretsky Rzhanov Institute of Semiconductor Physics, Novosibirsk 630090, Russia Novosibirsk State University, Novosibirsk 630090, Russia Institute for Physics of Microstructures RAS, GSP-105, 603950, Nizhni Novgorod, Russia and Laboratoire Charles Coulomb, UMR CNRS 5221,University of Montpellier, 34095 Montpellier, France. (Dated: October 17, 2018)We report on beating appearance in Shubnikov–de Haas oscillations in conduction band of 18–22 nm HgTe quantum wells under applied top-gate voltage. Analysis of the beatings reveals twoelectron concentrations at the Fermi level arising due to Rashba-like spin splitting of the first con-duction subband H
1. The difference ∆ N s in two concentrations as a function of the gate voltage isqualitatively explained by a proposed toy electrostatic model involving the surface states localizedat quantum well interfaces. Experimental values of ∆ N s are also in a good quantitative agreementwith self-consistent calculations of Poisson and Schr¨odinger equations with eight-band k · p Hamil-tonian. Our results clearly demonstrate that the large spin splitting of the first conduction subbandis caused by surface nature of H PACS numbers: key words: spin splitting, Rashba effect, surface states, Shubnikov–de Haas oscillations, quantum wells
INTRODUCTION
Thin films based on HgTe are known by a number of itsunusual properties originating from inverted band struc-ture of HgTe [1–4]. The latter particularly results in ex-istence of topologically protected gapless states, arisingat HgTe boundaries with vacuum or materials with con-ventional band structure. Although these states weretheoretically predicted more than 30 years ago [5–7],clear experimental confirmation was not possible at thattime due to lack of growth technology of high qualityHgTe-based films. Experimental investigations of wide(the width d >
70 nm) strained HgTe quantum wells(QWs), which started only in 2011, confirmed existenceof the predicted surface states and revealed their two-dimensional (2D) nature [4, 8, 9].In comparison with other materials with the invertedband structure, in which the surface states are knownbeing Dirac-like [10–12], HgTe spectrum involves heavy-hole band | Γ , ± / i modifying the surface state disper-sion. Although strain opens a bulk band-gap and re-sults thus in three dimensional (3D) topological insula-tor state of wide HgTe quantum wells [4, 8, 9], it doesnot cancel strong hybridization of the surface states withthe | Γ , ± / i band. As a result, the surface states instrained HgTe films can be resolved only at large ener-gies, while at the low ones they are indistinguishable fromconventional heavy-hole states [13, 14].In thin films of 3D topological insulator the surfacestates from the opposite boundaries may be coupled byquantum tunneling, so that small thickness-dependentgap is opened up [15–17]. In strained HgTe thin films,the latter arises deeply inside the heavy-hole band at theenergies significantly lower than the top of the valenceband [4]. In the ultrathin limit, the HgTe quantum well transforms into semimetal [2, 18] and then to 2D topo-logical insulator [1, 19] with both gapped surface andquantized bulk states.On the other hand, the electronic states in HgTe QWsare classified as hole-like H n , electron-like E n or light-hole-like LH n levels according to the dominant contribu-tion from the bulk | Γ , ± / i , | Γ , ± / i or | Γ , ± / i bands at zero quasimomentum k = 0 [19]. The stronghybridization in inverted HgTe QWs results in the upperbranch of the gapped surface states being representedby the H subband [4]. At large quasimomentum k thewave-functions of H subband are localized at the QWinterfaces, while at Γ point of the Brillouin zone they arelocalized in the QW center and are thus indistinguishablefrom other 2D states.The gapped surface states in the films of 3D topologi-cal insulators exhibit sizable Rashba-type spin splitting,arising due to electrical potential difference between thetwo surfaces [20]. Such spin splitting was first observed inQWs of Bi Se [21], which is a conventional 3D topolog-ical insulator with Dirac-like surface states [10–12, 21].The spin splitting of the gapped surface states also ex-ists in HgTe QWs and should be naturally connectedwith the splitting of the H subband. Previous experi-mental studies of 12–21 nm wide HgTe QWs [22–24] haveattributed large spin splitting of the H subband to theRashba mechanism in 2D systems [25, 26], enhanced bynarrow gap, large spin-orbit gap between the | Γ , ± / i and | Γ , ± / i bands, and the heavy-hole character ofthe H subband. The latter however contradicts the factthat the splitting of other subbands H , H , H etc. withthe heavy-hole character is significantly lower.In this work, we investigate spin splitting of conduc-tion band in 18–22 nm HgTe QWs with asymmetrical po-tential profile tuned by applied top gate voltage. Thebeating pattern of Shubnikov–de Haas (ShdH) oscilla-tions, observed in all the samples at the applied top gatevoltage, reveals two electron concentrations at the Fermilevel due to the spin splitting of the H subband. Ex-perimental difference in the concentrations as a functionof the gate voltage is qualitatively explained by a pro-posed toy electrostatic model involving the surface statesat the QW interfaces. Self-consistent Hartree calcula-tions based on eight-band k · p Hamiltonian [27], being ingood quantitative agreement with the experimental data,clearly show that the large Rashba-like spin splitting ofthe H H EXPERIMENT
Our experiments were carried out on undoped 22 nm( n -doped 18 nm ( µ m and the bar width 50 µ m. Electronconcentration of n -doped sample N s = 7 . × cm − . The experiments wereperformed at temperatures from 2 to 0.2 K and magneticfields up to 8 T. For magnetotransport measurements thestandard lock-in technique was used with the excitationcurrent 100 nA and frequencies 6 - 12 Hz. In this study wewere interested in electron transport when only the firstconduction subband is occupied. Electron concentrationwas thus in the range 1 − × cm − . The electronmobility in this region was rather high (see Fig. 1 (b))within 10 −
60 m /V s for undoped and 8 −
20 m /V s for (a) (b) µ t r , m / V s N s , 10 cm -2 undopedn-doped FIG. 1. (a) The cross section of the structures studied.(b) Transport mobility dependence on electron concentra-tion for undoped ( n -doped( ρ xx dependences on magneticfield B at top gate voltages V g = 0 − doped samples.Let us consider our results obtained for the undopedstructures first. In Fig. 2 longitudinal resistivity ρ xx as afunction of magnetic field B is shown for top gate voltages V g from 0 to 7 V. Due to good sample quality Shubnikov–de Haas oscillations are already seen at 0.4 T. The key ex-perimental result is an appearance of oscillation beatingsat gate voltage V g > V g = 0 V resistivityoscillations are homogeneous. The oscillation beatingsgive an evidence of presence of two carrier types in thesystem with close concentrations. Fourier analysis of re-sistivity dependence on inverse magnetic field ρ xx ( B − )with monotone background removed indeed shows twonearby peaks (see Fig. 3 (a)). From the Fourier ana-lyzes two electron concentrations N s and N s can bestraight calculated by N si = ef i /h , where we denote by f and f the lower and upper frequency positions of theFourier peaks correspondingly. Note the above expres-sion is written for spin non-degenerate electrons, this isjustified since at considering gate voltage range only thefirst conduction subband is occupied.Although the Fourier analysis enables finding electronconcentrations reasonably precisely, we found more accu-rate getting the frequencies from fitting of Shubnikov–deHaas oscillations by Lifshits - Kosevich formula [30–32]:∆ ρ xx ρ = X i =1 , A i D ( X )exp (cid:18) − πµ qi B (cid:19) cos (cid:18) πf i B + φ i (cid:19) , (1)where ρ is the monotone resistivity part and ∆ ρ xx =( ρ xx − ρ ) is the oscillatory part; D ( X ) = X/ sinh( X ) isthe thermal damping factor with X = 2 π k B T / ~ ω c , k B being Boltzmann constant and ω c being cyclotron fre- (a) (b)(c) (d)FIG. 3. Results obtained for undoped 22 nm HgTe quantumwell ρ xx ( B − )at gate voltage V g = 7 V. (b) Electron concentrations N s (red circles) and N s (blue triangular) and their sum (greensquares) obtained from Shubnikov–de Haas oscillations andtotal electron concentration N s obtained from Hall measure-ments (pink line) versus gate voltage. (c) The oscillatoryresistivity part ∆ ρ xx normalized to the monotone resistivitypart ρ versus inverse magnetic field. Black line shows the re-sult obtained experimentally at V g = 7 V while red line is thefitting curve calculated by Exp. (1). (d) Quantum mobilities µ q and µ q versus total electron concentration. quency; µ qi are the quantum mobilities; A i and φ i aresome constants.Before fitting the experimental curves we first removedany residual background, which we extracted from theinitial curves by Fourier filtering. A i , φ i , µ i and f i wereused as fitting parameters. We used frequencies achievedfrom Fourier analysis (see Fig. 3 (a)) as starting frequencyvalues. To increase sensitivity to the low-field data weused the weight of 10 for data points at magnetic fieldless than ∼ . V g = 7 Vis shown in Fig. 3 (c). Concentrations N s and N s ob-tained from the fitting process described above as func-tions of gate voltage are shown in Fig. 3 (b). The sumof two concentrations N s + N s matches very well withthe total concentration N s obtained from Hall measure-ments.An additional advantage of oscillation fitting is obtain- FIG. 4. Longitudinal resistivity ρ xx dependences on magneticfield B at top gate voltages V g from 0 to -4 V obtained forsymmetrically n -doped 18 nm HgTe quantum well ing quantum mobilities µ qi , which are shown in Fig. 3 (d)as functions of the total electron concentration N s . µ q and µ q are almost the same and do not change in a fullconcentration range from 5 to 9 × cm − , also they aremore than one order smaller than the transport mobilityshown in Fig. 1 (b). The difference between transport andquantum mobilities implies presence of long-range scat-tering, which might be electron density inhomogeneities.The experimental results for symmetrically n -dopedquantum well ρ xx ( B ) measured at top gate voltages V g from 0 to-4 V. Here oscillations are also homogeneous at zero gatevoltage while at V g < − ρ xx (1 /B ) (see Fig. 5 (a)), ∆ ρ xx is again the oscillatorypart of ρ xx . Since electron mobility in these structuresis smaller than in the undoped ones (see Fig. 1 (b)), os-cillations arise only at B ∼ B by Zeeman splitting it enables onlyone beating being resolved. Since the beating shifts tolarger fields with decreasing gate voltage at V g < − . − (a) (b)(c) (d)FIG. 5. Results obtained for symmetrically n -doped 18 nmHgTe quantum well ρ xx ( B − ) at gate voltage V g = − . N s (red circles) and N s (blue triangular) and theirsum (green squares) obtained from Shubnikov–de Haas oscil-lations and total electron concentration N s obtained from Hallmeasurements (pink line) versus gate voltage. (c) The oscilla-tory resistivity part ∆ ρ xx normalized to the monotone resis-tivity part ρ versus inverse magnetic field. Black line showsthe result obtained experimentally at V g = − . µ q and µ q versus total electron concentration. in the undoped structure almost the same, do not changein a presented concentration range and one order smallerthan the transport mobility. DISCUSSION
Beating pattern of Shubnikov–de Haas oscillations athigh gate voltages, while at V g = 0 the oscillations arehomogeneous, in both symmetrically doped and undopedQWs, indicates the origin of the spin splitting beingasymmetry of the QW profile, changing with V g . Letus first demonstrate that the difference in the electronconcentrations extracted from the ShdH oscillations canbe qualitatively explained by a toy electrostatic model in-volving the surface states at QW interfaces. This modelwas previously proposed for wide HgTe quantum wells [9],and here we briefly repeat its derivation. FIG. 6. Simplified band diagram and electron distributionover surface states for gate voltages V g = 0 (the left panel)and V g > As for the relative changes in the concentrations, theinitial conditions are not important, therefore, for sim-plicity, we assume electron concentrations on the topand bottom surfaces being the same at zero V g . Fig. 6schematically shows simplified band diagrams and elec-tron distribution over the surface states for a structurewith metallic top gate at zero and positive gate voltages.In the absence of gate voltage, the Fermi level remains thesame across the structure. When gate voltage is applied,the Fermi level differs in the metallic gate and QW layerby eV g , where e is the elementary charge. Since the leftsurface is closer to the gate, it partially screens the gatepotential from the right surface. The change of electronconcentration ∆ N s at the left surface exceeds thus itschanging ∆ N s at the right one. In their turn, the differ-ence in the concentrations induces an additional electricalpotential growth eφ HgT e between left and right surfaces,while the Fermi level over the QW layer remains constant.The difference in the concentrations can be written as∆ N si = ∆ E fi × D i , where D i ( i = 1 ,
2) is the density ofstates and ∆ E fi is the local change of the Fermi energyfor the right (1) and left (2) surface states. ∆ E f and∆ E f are connected thus as ∆ E f = ∆ E f + eφ HgTe . Thepotential difference between the two surface states canbe evaluated from the charge neutrality and the Gauss’slaw as φ HgTe = E HgTe d eff = e ∆ N s d eff /ǫ HgTe ǫ , where d eff is the effective distance between the opposite sur-face states and E HgTe is electric field in the well. Here,we neglect a distortion of the QW profile from the lineardependence caused by distribution of charge carriers inthe bulk of QW layer. Finally, we find∆ N s / ∆ N s = D /D + e d eff D /ǫ HgTe ǫ . (2)The effective distance between the surface states d eff can differ from the QW width due to localization ofthe surface states wave-functions not exactly on theboundaries of HgTe layer. In addition, the QW widthin our samples is comparable with the scale of sur-face states localization [33] to exclude the interactionbetween electrons at different boundaries. Parameter d eff can be evaluated by fitting experimental value of (a) (b)(c) (d)FIG. 7. (a) and (b) shows the difference between electronconcentrations ∆ N s = N s − N s as a function of total con-centration N s obtained experimentally (red circles) for sam-ples N s = 9 × cm − . All electrons are assumedcoming to the well due to top gate voltage. (c) shows theenergy spectrum, where black dashed lines correspond to sur-face states without hybridization with heavy holes. (d) showsHgTe quantum well potential profile (blue and red lines are Γ and Γ bands correspondingly) and squared absolute valuesof wave functions of electron states at the Fermi level (greenlines). ∆ N s / ∆ N s ≃ ( dN s /dV g ) / ( dN s /dV g ) with Eq. (2).It gives d eff = 9 nm for the sample dN s /dV g ) / ( dN s /dV g ) = 1 .
43 (see Fig. 3), which looksvery reasonable for given QW width.Let us obtain the expression for the difference in elec-tron concentrations at two different surfaces ∆ N s = N s − N s as a function of the total concentration N s .Now, the initial distribution of electrons over the struc-ture becomes important. For simplicity, we assume that N s = 0 for symmetric QW profile at V g = 0, and allelectrons at non-zero V g come to the HgTe layer dueto the top gate voltage. Thus, from N si = ∆ N si and N s = N s + N s , we get linear dependence of ∆ N s on N s : ∆ N s = ∆ N s / ∆ N s − N s / ∆ N s + 1 N s . (3)Fig. 7 (a) provides a comparison between experimen- tal data and estimation within our toy electrostaticmodel (presented by green curve) for the undoped sam-ple ǫ = 20, d eff = 9 nm and D = D = m ∗ / π ~ valid for parabolic dispersion ofthe surface states. The latter holds since hybridizationwith heavy holes modifies the band dispersion of the sur-face states, making it close to parabolic. From cyclotronresonance measurements [34] the effective mass of thesurface states was obtained equal to m ∗ ≈ . m , with m being free electron mass.Our toy electrostatic model is seen perfectly reproduc-ing the slope of the experimental behavior of ∆ N s ( N s ).Moreover, it can fit experimental data if one assumes theresidual concentration of 4 × cm − in the absence ofgate voltage. Note that this value is twice higher thanit was measured for the sample V g = 0 (seeFig. 3). The difference between theoretical estimationand experimental values gives the evidence of the impor-tance of microscopic details of the surface states, whichwere completely ignored within our toy model.Therefore, we also perform self-consistent calculationsof Poisson and Schr¨odinger equations with 8-band k · p Hamiltonian [27]. These calculations take into accountall microscopic details of the surface states and thus al-low obtaining a realistic QW profile. As it is done for atoy electrostatic model, here we also assume that all elec-trons at non-zero V g come to the HgTe layer due to thetop gate. At the final iteration of solving self-consistentlyPoisson and Schr¨odinger equations, we obtain energy dis-persions E ( k ) ( k is a quasimomentum in the QW plane).Then, for a given value of N s , we find the position ofFermi level and obtain the values of Fermi wave-vectors k and k . Finally, we find electron concentrations by N si = k i / π . Theoretical values of ∆ N s ( N s ) found fromself-consistent calculations are shown in Fig. 7 (a) by bluecurve and are in a good agreement with the experimentaldata.Fig. 7 (c) provides an energy dispersion of the surfacestates at N s = 9 × cm − , where they are representedby H subband due to hybridization with the states ofheavy-hole band. Surface state connection with the H subband is also supported by Fig. 7 (d). The figure showstheoretical QW profile and wave-functions of the states atthe Fermi level (see green curves). Spin-split states cor-responding to k and k wave-vectors are clearly seen tolocalize at the opposite boundaries of HgTe QW. Largeoverlapping between the surface states in our samplesalso explains only qualitative agreement of the experi-mental data with our toy electrostatic model. We notethat hybridization of the surface states with the heavy-hole band is partially included in the toy model by us-ing expression for the density of states D = m ∗ / π ~ ,which is inherent for parabolic spectrum. The dashedblack curves show dispersion of the surface states ne-glecting hybridization with the heavy holes. The surfacestates mixing with the | Γ , ± / i band is indeed seentransforming the linear dispersion of surface states intoparabolic. Interestingly, the spin splitting of the surfacestates is significantly suppressed if the hybridization isincluded.∆ N s ( N s ) obtained experimentally for the n -dopedstructure CONCLUSION
To sum up we have investigated Rashba-like spin split-ting of the conduction H n -doped structures, providestwo close electron concentrations. We have qualitativelydescribed the evolution of the difference between theseconcentrations with gate voltage by a toy electrostaticmodel involving electron states localization at the wellinterfaces. The quantitative agreement between the ex-perimental data and theoretical calculations was achievedby self-consistent solving Poisson and Schr¨odinger equa-tions with eight-band k · p Hamiltonian, which takes intoaccount microscopic details of the surface states omit-ted in our toy model. Comparison of the toy electro-static model with the rigorous self-consistent calculationsclearly shows large spin-splitting of H ACKNOWLEDGMENTS
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