Temperature-driven transition from a semiconductor to a topological insulator
Steffen Wiedmann, Andreas Jost, Cornelius Thienel, Christoph Brüne, Philipp Leubner, Hartmut Buhmann, Laurens W. Molenkamp, J. C. Maan, Uli Zeitler
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Temperature-driven transition from a semiconductor to a topological insulator
Steffen Wiedmann, ∗ Andreas Jost, Cornelius Thienel, Christoph Br¨une, PhilippLeubner, Hartmut Buhmann, Laurens W. Molenkamp, J. C. Maan, and Uli Zeitler High Field Magnet Laboratory and Institute for Molecules and Materials,Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands. Physikalisches Institut (EP3), Universit¨at W¨urzburg, Am Hubland, D-97074, W¨urzburg, Germany. (Dated: July 12, 2018)We report on a temperature-induced transition from a conventional semiconductor to a two-dimensional topological insulator investigated by means of magneto-transport experiments onHgTe/CdTe quantum well structures. At low temperatures, we are in the regime of the quan-tum spin Hall effect and observe an ambipolar quantized Hall resistance by tuning the Fermi energythrough the bulk band gap. At room temperature, we find electron and hole conduction that canbe described by a classical two-carrier model. Above the onset of quantized magneto-transport atlow temperature, we observe a pronounced linear magneto-resistance that develops from a classicalquadratic low-field magneto-resistance if electrons and holes coexist. Temperature-dependent bulkband structure calculations predict a transition from a conventional semiconductor to a topologicalinsulator in the regime where the linear magneto-resistance occurs.
PACS numbers: 73.25.+i, 73.20.At, 73.43.
INTRODUCTION
Narrow-gap semiconductors possess conduction bandswhich are strongly non-parabolic and spin-orbit splittingsthat can be even larger than the fundamental band gap[1]. A particular interesting system is a type-III het-erostructure composed of the semimetal HgTe and thewide-gap semiconductor HgCdTe with a low Hg con-tent. These quantum well (QW) structures with an en-ergy gap of several meV have been experimentally in-vestigated by means of optics and magneto-transport al-ready in the late 90s in order to obtain information aboutthe band structure (BS) and the Landau level dispersion[2, 3]. In 2006, Bernevig et al. predicted that the quan-tum spin Hall effect (QSHE) can be observed in invertedHgTe/CdTe QW structures if the layer thickness is largerthan a critical value [4]. The system is then referred to asa two-dimensional (2D) topological insulator (TI) [5, 6].The hallmark of this new state of matter is a quantizedlongitudinal conductance when the Fermi energy is inthe bulk band gap and transport is governed by spin-polarized counter-propagating edge states. This quan-tized conductance has been found experimentally first ininverted HgTe QWs [7] and later on in InAs/GaSb het-erostructures [8, 9]. The existence of the helical stateshas been confirmed in inverted HgTe QWs by non-localmeasurements [10] and by verifying their spin polariza-tion [11].Bulk HgTe crystallizes in zincblende structure. Whenthe semimetal HgTe with a negative energy gap of E g =-0.3 eV is combined with the semiconductor HgCdTe, atype-III QW is formed. The band order in HgTe QWs ∗ [email protected] with HgCdTe barriers depends strongly on the quantumconfinement, i.e., the width d of the QW. For d < d c , thesystem is a conventional direct band-gap semiconductorwith a s -type Γ conduction band and p -type Γ valenceband. d c is the critical thickness of the QW where thesystem becomes a zero-gap semiconductor [12]. Calcula-tions within the 8 × k · p Kane model yield d c =6.3 nm fora QW on a Cd . Zn . Te substrate and d c =6.7 nm on aCdTe substrate in the zero-temperature limit. For d > d c the BS is inverted, i.e. the H E T > T c ) to a topologically non-trivial state ofmatter ( T < T c ). This is caused by the strong tem-perature dependence of the E d c at T =0. We show that we are in theregime of the QSHE and observe an ambipolar quan-tized Hall resistance at low temperature when the Fermienergy is tuned through the bulk band gap. In con-trast, at room temperature we find electron and hole con-duction that can be described by a classical two-carriermodel. In an intermediate temperature range (100 K ≤ T ≤
205 K), where Shubnikov-de Haas oscillations andquantum Hall effect are absent, we observe a pronouncedlinear magneto-resistance (LMR) that develops from aclassical quadratic low-field magneto-resistance (MR).Bulk band structure calculations using an eight-band k · p model demonstrate a transition from a conventional semi-conductor to a topological insulator in the regime wherethe LMR is observed. EXPERIMENTAL DETAILS AND SAMPLECHARACTERIZATION AT T =4.2 K We have grown inverted HgTe QWs with (001) surfaceorientation by molecular beam epitaxy on a CdTe (sam-ple S1, and S4) and on a Cd . Zn . Te substrate (sam-ple S2 and S3). Details for the samples S1-S4 are givenin Table T1 in the supplemental material [14]. Litho-graphically defined Hall-bar structures have been pro-duced with the dimension ( L × W ) = (600 × µ m .All samples are equipped with a metallic Au top-gate,separated from the structure by an insulator made of asuper-lattice of Si N and SiO , with a total thickness of110 nm to tune the carrier concentration as a function ofthe applied gate voltage V g . Four-probe measurementsof longitudinal and transverse electrical resistances havebeen carried out using Stanford Research Systems SR830Lock-In amplifiers with low constant voltage excitation.The samples were placed in a flow-cryostat in a 33 TBitter-type magnet.In Fig. 1, we present transport at T =4.2 K for sampleS1 and sample S2 with a well width of d =12 nm. Fig-ures 1(a) and (b) show the longitudinal resistance R xx at T =4.2 K as a function of top-gate voltage V g for samplesS1 and S2. According to band structure calculations,both QWs are inverted at 4.2 K. The H H R xx is an indication for the QSHE [4]. Itsvalue is higher than h/ e which can be explained by in-elastic scattering in large samples [7, 8]. This resistanceis by an order of magnitude smaller than in samples withcomparable size and a 20 meV bulk band gap [7] (seeband structure calculations in Fig. 4(c) and in the sup-plemental material [14]). At V g =0 (ii), both samples are n -conducting and sample S1 (S2) has a carrier concentra-tion of n =3.7 · cm − ( n =4.5 · cm − ) and a mo-bility of µ = 5 . · cm /Vs ( µ = 4 . · cm /Vs). Thefact that we can indeed tune the Fermi energy throughthe bulk band gap is demonstrated by measuring the Hallresistivity ρ xy , shown in Figs. 1(c) and (d). Depending on V g , we find a positive ρ xy caused by negatively chargedelectrons in (i) and (ii), and a negative ρ xy in (iii) in-dicating hole transport. At higher magnetic fields, weobserve the quantum Hall effect for electrons and ρ xy forholes diverges. Quantization in ρ xy for holes occurs atlower temperatures, see Fig. 1(d) at 0.3 K owing to thehigher effective mass for holes [2]. -3 -2 -1 0 1 2 30102030 -3 -2 -1 0 1 2 301020300 5 10 15 20-30-20-1001020300 5 10 15 20-30-20-1001020300.00.51.01.50 5 10 15 20 01230 5 10 15 20(iii) (ii) T=4.2 K;B=0 (d)(c) (b) R xx ( k ) V g (V) (a) T=4.2 K;B=0 (i) H2 (iii) (ii) (i) R xx ( k ) V g (V) S1 H1 =-2 T=0.3 K
T=4.2 K S2 (ii)(i)(iii)=2 =1 xy ( k ) B (T) (ii)T=4.2 K (iii)(ii)(i) =3=3=2 xy ( k ) B (T) =1T=4.2 K (e) xx ( k ) (ii)T=4.2 K (f) xx ( k ) FIG. 1. (color online) Magneto-transport in the QSH regimefor sample S1 and S2: Gate sweeps at B =0 for samples S1 (a)and S2 (b). Panels (c) and (d) show the Hall resistivity ρ xy ( B )for three gate voltage positions indicating the transition fromelectrons, in (i) and (ii), to holes in (iii). The inset in (b)sketches a HgTe type-III QW with inverted bands at T =4.2 K.When the Fermi energy is in the bulk band gap, the H H MAGNETO-TRANSPORT AT ROOMTEMPERATURE
We now present magneto-transport at room tempera-ture. At high temperatures we are limited to apply a high | V g | due to an increase in the leak current through theinsulator. In Fig. 2, we show magneto-transport for sam-ple S1 at T =300 K. Applying a gate voltage V g at B =0,we find that ρ xx increases with decreasing V g , see insetof Fig. 2(a), indicating that we deplete electrons whendecreasing the gate voltage. In Fig. 2(a) and (b), we il-lustrate ρ xx and ρ xy as a function of the magnetic fieldfor various fixed V g . For all gate voltages, ρ xx displaysa pronounced MR and ρ xy shows a strong non-linear be-havior. For positive V g , ρ xx increases quadratically as afunction of B but with decreasing V g , we find that ρ xx deviates from the quadratic behavior and exhibits a small M R seemingly saturating in higher magnetic fields.Both observations point towards a system where elec-trons and holes coexist. Notably, the slope of ρ xy is firstpositive, indicating a dominant contribution of mobileelectrons. With increasing magnetic field, the slope be-comes negative due to holes with a higher concentrationand lower mobility. -0.4 -0.2 0.0 0.2 0.4012 (b) xx ( k ) V g (V) (a) T=300 K;B=0 xx ( k ) B (T) -0.4 V xy ( k ) B (T)
FIG. 2. (color online) Magneto-transport at room tempera-ture. (a) ρ xx (offset for clarity by 2 kΩ except trace at V g =-0.4 V) and (b) ρ xy (offset for clarity by 1 kΩ except trace at V g =-0.4 V) as a function of B for fixed V g : for high positive V g , ρ xx has a quadratic MR accompanied by saturating ρ xy ,whereas ρ xx exhibits a transition from quadratic to almostfield-independent MR for negative V g accompanied by a lin-ear decrease in ρ xy at higher fields. Inset: ρ xx as a functionof V g at B =0 (solid line: continuous gate-sweep; symbols: ρ xx ( B ) for a fixed V g ). We can extract quantitative information on the chargecarrier properties using a semi-classical Drude-modelwith field-independent electron and hole concentrationsand mobilities where we sum up the individual contri-butions of both electrons and holes to the conductivitytensor b σ σ xx = neµ e (1+ µ e B ) + peµ p (1+ µ p B ) ; σ xy = neµ e B (1+ µ e B ) − peµ p B (1+ µ p B ) . (1)and perform a fit to the experimentally measured resis-tivity tensor b ρ = b σ − .The results of the simultaneous fitting of the rela-tive magneto-resistance, defined as M R = [ ρ xx ( B ) − ρ xx (0)] /ρ xx (0), and the Hall resistance ρ xy ( B ) at fourchosen fixed gate voltages are shown as the solid lines inFig. 3(a-d). The gate dependences of the electron andhole concentrations n and p , and the mobilities µ e and µ p extracted from the fits are illustrated in Fig. 3(e) andFig. 3(f). Therefore, transport is governed by an electronband with very mobile carriers, i.e. carriers with a smalleffective mass, coexisting with a hole band with a largeamount of charge carriers with a large effective mass andlow mobility. xy ( k ) V g =0 V (d)(c) M R B (T) -505 V g =-0.2 V B (T) -505 xy ( k ) V g =-0.4 V M R V g =0.2 V (a) T=300 KT=300 K (f)(e) holes n , p ( * c m - ) V g (V)electrons holeselectrons e , p ( c m / V * s ) V g (V) FIG. 3. (color online) Analysis of room-temperaturemagneto-transport with a two-carrier Drude model (symbolsrepresent the experimental data, solid lines fits to the model):(a)-(d) MR and ρ xy up to B =30 T for several chosen gatevoltages. (e) Charge carrier concentrations and (f) chargecarrier mobilities for electrons and holes at different V g as ex-tracted from the fits. With decreasing gate voltage, the elec-tron (hole) concentration and mobility decreases (increases). TEMPERATURE-DEPENDENT BANDSTRUCTURE CALCULATIONS
So far, we have presented that our system is a 2D TI atlow-temperature whereas magneto-transport can be de-scribed within a classical two-carrier model for one elec-tron and one hole band at room temperature as expectedfor a conventional semiconductor if the thermal energy k B T is larger than the band gap E g . This transition canbe elucidated by performing temperature-dependent bulkband structure calculations of the QW structures shownfor sample S1 in Fig. 4. Additional BS calculations forsample S2 are illustrated in the supplemental material[14]. Our calculations are based on an eight-band k · p model in the envelope function approach [15]. The k · p model takes into account the temperature dependenceof all relevant parameters, in particular the change inthe lattice constants of Hg − x Cd x Te and the elastic con-stants C , C (bulk modulus) and C with tempera-ture [16, 17]. The elastic constants C ij increase by a few% with decreasing T but the ratios which enter the calcu-lations remain constant. In Figs. 4(a-d), we plot E ( k ) forthe 12 nm thick QW at different temperatures used in ourexperiment. At 300 K, the gap between the conductionband E H E g ≃
26 meV. Withdecreasing temperature, the band-gap E g considerablydecreases and the 12 nm HgTe QW becomes a zero-gapSC at T =223 K [10], see Fig. 4(b). In this temperaturerange E g ≤ k B T , and transport is still governed by ther-mally activated charge carriers within the E H T =100 K, see Fig. 4(c), transportis still dominated by thermally activated charge carriersbut now in the H H E g ≤ k B T . At low temperatures, see Fig. 4(d) for T =4.2 K, E g > k B T and thermal activation becomesnegligible and we would observe an infinite resistance inthe bulk band gap for a perfectly homogeneous gate po-tential if our system was not a 2D TI.In Fig. 4(e), we plot an overview of the temperature de-pendence of the electron band E
1, the heavy-hole bands H H H L k =0). As can be seen from the calculation, thesystem undergoes a transition from a conventional semi-conductor to a 2D TI due to the decrease of the E E H T >
223 K,our system is a conventional semiconductor with the con-duction band E H k || (1,0) [ k || (1,1)]. REGIME OF LINEAR MAGNETO-RESISTANCE
Let us now draw our attention to the intermediate tem-perature regime where, according to the presented BScalculation, the band order is inverted but transport isstill dominated by thermally activated charge carriers.In Fig. 5(a), we plot ρ xx ( T ) at V g =0 and observe that ρ xx first increases with decreasing temperature, which ischaracteristic for a semiconductor with thermally acti-vated carriers. Around 130 K, ρ xx displays a maximumand then starts to decrease with further decreasing tem-perature before saturating for T ≤
10 K. This behav-ior is characteristic for a metallic system with a constantcarrier concentration and an increasing mobility with de-creasing temperature. In Fig. 5(b), we plot the resistiv-ity ρ xx as a function of V g at 100, 150 and 204 K. For T >
100 K, ρ xx decreases monotonously with increasing V g . For T = 100 K, we observe a maximum in ρ xx which k=0 L1H3 H2H1E1 E ( m e V ) T (K) k=0
TI H1 E ( m e V ) T (K)
E1SC zero-gap T c H1E1 k || (1,0) k || (1,1) E ( m e V ) k (nm -1 ) T=223 K’zero-gap’ k || (1,0) k || (1,1) H1 E ( m e V ) k (nm -1 ) H2 T=100 K H1 k || (1,0) k || (1,1) H2 E ( m e V ) k (nm -1 ) T=4.2 K k || (1,0) k || (1,1) (f)(e) (d)(c) (b) H1 E ( m e V ) k (nm -1 ) E1 T=300 K (a)
FIG. 4. (color online) Band structure calculations for sampleS1 for k || (1,0) and k || (1,1): (a)-(d) E ( k ) for (a) at 300 K witha normal BS ( E > H E ∝ k forsmall k where the E H k =0),(c) at 100 K and (d) at 4.2 K with an inverted BS. We findan indirect bulk band-gap at 4.2 K (Note that the curvatureof the H E
1, the heavy-hole like H H H L k =0. The E L T . (f) Crossing of E H T c : for T >
223 K, the system is a normal semiconductor. For
T <
223 K, we have a QW with an inverted BS and thus, a2D TI. we refer to as the region of charge neutrality.A particular feature of this intermediate-temperatureregime is the emergence of a strong LMR that developsfrom a classical quadratic low-field MR. In Figs. 5(c) and(d) we plot ρ xx and ρ xy at V g =0 as a function of B for sev-eral temperatures. In the temperature range presented inFig. 5(c), E g ≤ k B T at V g =0 and magneto-transport isgoverned by bulk electrons and holes. For high tempera-tures ( T >
200 K), we observe a quadratic MR that can -4 -2 0 2 401231 10 1000123 0 10 20 30-10-505100 10 20 300102030 100 200 3000.00.51.02345 100 200 30001002003004002000040000 100 K150 K204 K xx ( k ) V g (V) V g =0 V (b) xx ( k ) T (K) (a) metalic SC
204 K245 K300 K150 K100 K (d) xy ( k ) B (T)
245 K150 K204 K100 K300 K V g =0 V (c) xx ( k ) B (T) V g =0 VV g =0 V (f)(e) holes n , p ( * c m - ) T (K)electrons V g =0 V holeselectrons e , p ( c m / V * s ) T (K)
FIG. 5. (color online) Temperature-dependent magneto-transport: (a) ρ xx ( T ) at V g =0 shows a maximum around130 K, then decreases and saturates with decreasing temper-ature. (b) ρ xx as a function of V g at different temperatures.For T =100 K, R xx exhibits a maximum at V g =1 V whichwe identify as the CNP. (c) ρ xx and (d) Hall resistivity as afunction of B at V g =0 for 300, 245, 204, 150 and 100 K. For150 and 100 K, ρ xx exhibits LMR in a wide range of magneticfield. (e) Carrier concentrations and mobilities for electronsand holes as a function of temperature. The dashed lines(dashed-dotted lines) mark the border between two-carriertransport and one-carrier n -conduction for k || (1,0) when thethermal energy is smaller than the bulk band gap extractedfrom BS calculations. be perfectly modeled by a two-carrier Drude model (seealso Fig. 3). For lower temperatures, however, we findtwo different regimes in ρ xx ( B ): a quadratic MR at lowmagnetic fields, as expected from the classical two-carrierDrude model, and a LMR at high B . Furthermore, theonset of LMR shifts continuously to lower B with decreas-ing temperature. Interestingly, at T =150 and 100 K, weobserve a wide range of LMR, e.g. from 8 to 30 T at 150 Kand 3 to 30 T at 100 K, respectively. The experimentallyobserved MR can not be described by the classical two-carrier model though the corresponding ρ xy traces indi- cate that both electrons and holes still contribute to thetransport. We estimate the electron concentrations fromthe linear increase of ρ xy at low B and the hole concen-tration from the slope of ρ xy at high B and the results areshown in Fig. 5(e). When we decrease the temperaturefrom 300 to 100 K, n and p decrease by approximatelya factor of two and 1.5, respectively, which can be againbe explained by the decrease of thermally exited carriers.We can also estimate the carrier mobilities, by limitingthe two-carrier model to the classical regime at low mag-netic fields. An example of a two-carrier fit is shown inFig. 6(c) for B ≤ T c whenthe band structure is inverted, we plot ρ xx and ρ xy inFigs. 6(a) and (b) as a function of the magnetic field upto 30 T at different V g , respectively. For V g ≥ ρ xx exhibits a strong positive MR at low fields that evolvesinto a strong linear MR with increasing B . For V g ≤ B and becomes linear asthe field is increased, see also inset of Fig. 6(a), and theonset of LM R shifts to lower magnetic field with decreas-ing V g . The linear dependence of ρ xx and its onset can beclearly illustrated in the first-order derivative dρ xx /dB ,as plotted in Fig. 6(d) and (e) as a function of the mag-netic field. We define a critical field B crit as the magneticfield corresponding to the maximum in dρ xx /dB , whichmarks the deviation from a squared dependence in thetwo-carrier model for low B . For all V g , ρ xy is first pos-itive due to mobile electrons and becomes negative withincreasing B due to the presence of holes. DISCUSSION
From the above bulk BS calculations we see that tem-perature induces a transition from a normal state toa topologically non-trivial state in HgTe QWs. Withdecreasing temperature (
T > T c ), the gap closes, seeFig. 4(e). The conduction band exhibits a significant de-pendence on k , yielding a small effective mass 0.015 m e
T <
223 K impliesthat transport can also take place in helical edge states (e)(d)(c) (b) -1.5 V 0 V 1 V2 V3 V-3.5 V4 V xx ( k ) B (T) (a) xy ( k ) B (T)
T=100 K -3.5 V V g =0 V xx ( k ) B (T) -3-2-101 xy ( k ) V g =4 V xx ( k ) B crit d xx / d B B crit V g =-3.5 V xx ( k ) B (T) d xx / d B -1.5 V0 V1 V2 V3 V -3.5 V4 V xx ( k ) B (T)
FIG. 6. (color online) Magneto-transport at T =100 K: (a) ρ xx (inset: low-field behavior of ρ xx ) and (b) ρ xy as a functionof B for different gate voltages. For all V g , ρ xx exhibits strongLMR. (c) Applied two-carrier fit model (solid lines), measured ρ xx and ρ xy (open symbols and circles) at V g =0. (d), (e) Theonset of LMR shifts to lower B with decreasing V g where both ρ xx and dρ xx /dB are plotted as a function of B for V g =4 Vand V g =-3.5 V, respectively. B crit marks the transition fromquadratic MR to LMR . with a linear dispersion in the bulk band gap [4–7].The observation of LMR has been reported in varioussystems such as bulk narrow-band gap semiconductors[18, 19] and semi-metals [20] as well as recently in TIs[21]. In fact, the occurrence of a strong LMR has beenascribed to surface states in three-dimensional TIs [22].In a 2D TI (HgTe QW), LMR has been also found at lowmagnetic field and low temperature when the chemicalpotential moves through the bulk gap [23]. In contrast,since k B T > E g , LMR in our system is governed bymobile bulk electrons with low density and less mobileholes with high carrier concentration and helical edgestates. From V g -dependent measurements we found thatthe onset B crit of LMR shifts to lower B with increasingcarrier concentration of holes.Theoretical models have also addressed the appear-ance of LMR [24, 25]. The classical percolation model byParish and Littlewood [24] for a non-saturating LMR dueto distorted current paths caused by disorder-induced in- homogeneities in the electron mobility cannot be appliedto our system since our MBE grown samples do not showstrong fluctuations in µ , and, it does not explain the tran-sition from classical MR to LMR with increasing mag-netic field. The quantum model, that has been proposedby Abrikosov [25] is valid for systems with a gapless lineardispersion spectrum when only the lowest Landau level(LL) remains occupied. Moreover, the energy differencebetween the lowest LL E and the first LL E shouldbe much larger than E F and k B T . We reach the quan-tum limit for one type of charge carriers, e.g. at V g =0 for T =100 K, since E − E > E F > k B T , however, the LMRin our 2D system occurs in the presence of two types ofcharge carriers in the bulk and charge carriers in the heli-cal edge states in contrast to the three-dimensional modelfor one type of charge carrier proposed by Abrikosov [25].Recently, MR has been theoretically investigated intwo-component systems such as narrow-band semicon-ductors or semi-metals at high temperatures [26]. Forequal carrier concentrations of electrons and holes, a non-saturating LMR has been predicted to occur in finitesize at charge neutrality due to the interplay betweenbulk and edge contributions. At room temperature, ourdata in Fig. 2 shows qualitatively the expected behaviorfor ρ xx and ρ xy as proposed and illustrated in Ref. [26]for broken electron-hole symmetry. Yet we have shownthat both ρ xx and ρ xy can also be explained within theclassical two-carrier model without any contribution dueto a quasi-particle density that develops near the sam-ple edges. A satisfactory theoretical explanation of theorigin of LMR for T < T c and B > B crit , that also ad-dresses the role of helical edge states at high temperatureremains open and is certainly challenging for theoreticalmodels in the future.
CONCLUSION
We have demonstrated in bulk band structure calcu-lations on HgTe QWs that temperature induces a tran-sition from a semiconductor at room temperature to aTI at low temperature. Experimentally, we can distin-guish between three regimes in magneto-transport: (i)transport of coexisting electrons and holes that can be de-scribed within a classical two-carrier model at room tem-perature, (ii) the appearance of a strong LMR for
B >B crit and
T < T c where still electrons and holes coexistand (iii) the regime of quantized transport ( ~ ω c > k B T )at low temperature where we are also in the regime of theQSHE. We note that apart from inverted HgTe QWs, theonly other system known to be a 2D TI is the InAs/GaSbhybrid system [27] that has been investigated at low tem-perature [8, 9, 28]. Temperature-dependent magneto-transport experiments could demonstrate whether theMR effects are unique in inverted HgTe QWs due to theirbulk band structure or are a fundamental property of 2Dtopological insulators. Acknowledgments
This work has been performed at the HFML-RU/FOMmember of the European Magnetic Field Laboratory(EMFL) and has been supported by EuroMagNETII under EU Contract No. 228043, by the DARPAMESO project through the contract number N66001-11-1-4105, by the German Research Foundation (DFG grantHA5893/4-1 within SPP 1666, the Leibniz Program andDFG-JST joint research project Topological Electronics)and the EU ERC-AG program (Project 3-TOP). S.W. isfinancially supported by a VENI grant of the NederlandseOrganisatie voor Wetenschappelijk Onderzoek (NWO). [1] J. Chu, & A. Sher,
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