RESEARCH ARTICLES
Quantum Spin Hall Insulator State in HgTe Quantum Wells
Markus König,1 Steffen Wiedmann,1 Christoph Brüne,1 Andreas Roth,1 Hartmut Buhmann,1 Laurens W. Molenkamp,1* Xiao-Liang Qi,2 Shou-Cheng Zhang2
AUTHORS’ SUMMARY
he discovery more than 25 Conductance theoretically that the electronic years ago of the quantum channel with structure of inverted HgTe quan- THall effect (1), in which the up-spin charge tum wells exhibits the properties “Hall,” or “transverse electrical” con- carriers that should enable an observation ductance of a material is quantized, of the quantum spin Hall insula- came as a total surprise to the physics tor state. Our experimental obser- community. This effect occurs in vations confirm this. layered metals at high magnetic These experiments only be- fields and results from the forma- came possible after the devel- tion of conducting one-dimensional opment of quantum wells of
channels that develop at the edges sufficiently high carrier mobility, on April 6, 2015 of the sample. Each of these edge combined with the lithographic channels, in which the current moves techniques needed to pattern the only in one direction, exhibits a quan- sample. The patterning is espe- tized conductance that is character- Conductance cially difficult because of the very istic of one-dimensional transport. The channel with high volatility of Hg. Moreover, number of edge channels in the sam- Quantum down-spin we have developed a special low– ple is directly related to the value of well charge carriers deposition temperature Si-O-N the quantum Hall conductance. More- Schematic of the spin-polarized edge channels in a quantum spin Hall gate insulator (7), which allows
over, the charge carriers in these chan- insulator. us to control the Fermi level (the www.sciencemag.org nels are very resistant to scattering. energy level up to which all Not only can the quantum Hall effect be observed in macroscopic samples electronics states are filled) in the quantum well from the conduction band, for this reason, but within the channels, charge carriers can be transported through the insulating gap, and into the valence band. Using both electron without energy dissipation. Therefore, quantum Hall edge channels may be beam and optical lithography, we have fabricated simple rectangular useful for applications in integrated circuit technology, where power dis- structures in various sizes from quantum wells of varying width and sipation is becoming more and more of a problem as devices become smaller. measured the conductance as a function of gate voltage. Of course, there are some formidable obstacles to overcome—the quantum We observe that samples made from narrow quantum wells with a
Hall effect only occurs at low temperatures and high magnetic fields. “normal” electronic structure basically show zero conductance when the Downloaded from In the past few years, theoretical physicists have suggested that Fermi level is inside the gap. Quantum wells with an inverted electronic edge channel transport of current might be possible in the absence of a structure, by contrast, show a conductance close to what is expected for the magnetic field. They predicted (2–4) that in insulators with suitable edge channel transport in a quantum spin Hall insulator. This interpretation electronic structure, edge states would develop where—and this is is further corroborated by magnetoresistance data. For example, high– different from the quantum Hall effect—the carriers with opposite magnetic field data on samples with an inverted electronic structure show a spins move in opposite directions on a given edge, as shown sche- very unusual insulator-metal-insulator transition as a function of field, matically in the figure. This is the quantum spin Hall effect, and its which we demonstrate is a direct consequence of the electronic structure. observation has been hotly pursued in the field. The spin-polarized character of the edge channels still needs to be Although there are many insulators in nature, most of them do not have unequivocably demonstrated. For applications of the effect in actual the right structural properties to allow the quantum spin Hall effect to be microelectronic technology, this low-temperature effect (we observe it observed. This is where HgTe comes in. Bulk HgTe is a II-VI semi- below 10 K) will have to be demonstrated at room temperature, which may conductor, but has a peculiar electronic structure: In most such materials, be possible in wells with wider gaps. the conduction band usually derives from s-states located on the group II atoms, and the valence band from p-states at the VI atoms. In HgTe this Summary References order is inverted, however (5). Using molecular beam epitaxy, we can 1. K. v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980). grow thin HgTe quantum wells, sandwiched between (Hg,Cd)Te barriers, 2. S. Murakami, N. Nagaosa, S.-C. Zhang, Phys. Rev. Lett. 93, 156804 (2004). Phys. Rev. Lett.
3. C. L. Kane, E. J. Mele, 95, 146802 (2005). SCIENCE
that offer a unique way to tune the electronic structure of the material: When = 4. B. A. Bernevig, S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006). the quantum well is wide, the electronic structure in the well remains 5. A. Novik et al., Phys. Rev. B 72, 035321 (2005).
“ ” BICKEL inverted. However, for narrow wells, it is possible to obtain a normal 6. B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006). : alignment of the quantum well states. Recently, Bernevig et al.(6) predicted 7. J. Hinz et al., Semicond. Sci. Technol. 21, 501 (2006). CREDIT: C
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quantum phase transition has also been inves- FULL-LENGTH ARTICLE tigated in more realistic models beyond the simple four-band model presented here, reach- Recent theory predicted that the quantum spin Hall effect, a fundamentally new quantum state of ing the same conclusion (11). matter that exists at zero external magnetic field, may be realized in HgTe/(Hg,Cd)Te quantum wells. The QSH phase occurs in the inverted re- We fabricated such sample structures with low density and high mobility in which we could tune, gime where M < 0, i.e., when d > dc. The through an external gate voltage, the carrier conduction from n-type to p-type, passing through an sample edge can be viewed as a domain wall insulating regime. For thin quantum wells with well width d < 6.3 nanometers, the insulating regime of the mass parameter M, separating the topo- showed the conventional behavior of vanishingly small conductance at low temperature. However, logically nontrivial phase with M < 0 from for thicker quantum wells (d > 6.3 nanometers), the nominally insulating regime showed a the topologically trivial phase with M >0, plateau of residual conductance close to 2e2/h, where e is the electron charge and h is Planck’s which is adiabatically connected to the vac- constant. The residual conductance was independent of the sample width, indicating that it is caused uum (12). Massless helical states are confined by edge states. Furthermore, the residual conductance was destroyed by a small external magnetic on the sample edge. The sample has a finite field. The quantum phase transition at the critical thickness, d = 6.3 nanometers, was also conductance even when the Fermi level lies independently determined from the magnetic field–induced insulator-to-metal transition. These inside the bulk insulating gap. Therefore, as observations provide experimental evidence of the quantum spin Hall effect. suggested in (6), the QSH state can be ex- perimentally detected by measuring a residual he theoretical prediction of the intrinsic alization of the quantum spin Hall effect (6). conductance plateau as one varies the gate spin Hall effect in metals and insulators In zincblende-type semiconductor QWs, there voltage in the nominally insulating regime. T(1–3) has generated great interest in the are four relevant bands close to the Fermi Furthermore, because the current is carried by field of spintronics, because this effect allows level. The E1 band consists of the two spin the edge states, the conductance should be direct electric manipulation of the spin de- states of the s orbital, whereas the HH1 band independent of sample width. Protected by grees of freedom without a magnetic field, consists of the |px + ipy, ↑〉 and | −(px − ipy), ↓〉 the time-reversal symmetry, nonmagnetic im- and the resulting spin current can flow with- orbitals. The effective Hamiltonian near the G purities or any other time-reversal invariant out dissipation. These properties could lead to point, the center of the Brillouin zone, is local perturbations cannot cause elastic back- promising spintronic devices with low power given by scattering of the helical edge states, which war- dissipation. rants the topological robustness of the edge state However, beyond the potential technolog- ð HðkÞ 0 Þ conductance. However, the presence of a mag- H ðk ,k Þ¼ ∗ , ical applications, the intrinsic spin Hall effect eff x y 0 H ð−kÞ netic field breaks time-reversal symmetry and has guided us in the search for new and topo- therefore can open up a gap in the energy spec- H ¼ eðkÞþd ðkÞs ð1Þ logically nontrivial states of matter. The quan- i i trum of the edge states and remove the residual tum Hall state is the first, and so far the only conductance due to the edge states. example, of a topologically nontrivial state of where si are the Pauli matrices, and Experimental details. We set out to test these theoretical predictions by measuring matter, where the quantization of the Hall con- þ ¼ ð þ Þ ≡ ductance is protected by a topological invar- d1 id2 A kx iky Akþ the transport properties of HgTe/(Hg,Cd)Te iant. The quantum spin Hall (QSH) insulators d ¼ M − Bðk2 þ k2Þ, QWs as a function of the sample thickness, (4–6) have a similar, but distinct, nontrivial 3 x y the gate voltage, and the external magnetic e ¼ − ð 2 þ 2Þ: ð Þ topological property. The QSH insulators are k C D kx ky 2 field. We used modulation-doped type III (13) invariant under time reversal, have a charge HgTe/Hg0.3Cd0.7Te QW structures fabricated excitation gap in the bulk, but have topologi- Here, kx and ky are momenta in the plane by molecular beam epitaxy (14), with widths cally protected gapless edge states that lie of the two-dimensional electron gas (2DEG), (15) varying from 5.5 nm (d < dc)to12nm(d > inside the bulk insulating gap. This type of and A, B, C,andD are material specific con- dc). Dedicated low–thermal budget optical and insulator is typically realized in spin-orbit stants. Spin-orbit coupling is naturally built- e-beam lithography was used to structure de- coupled systems; the corresponding edge states in in this Hamiltonian through the spin-orbit vices in Hall bar geometry with dimensions have a distinct helical property: two states with coupled p orbitals |px + ipy, ↑〉 and | −(px − ipy), ↓〉. (L × W ) of (600 × 200), (20.0 × 13.3), (1.0 × opposite spin-polarization counterpropagate at Two-dimensional materials can be grouped into 1.0), and (1.0 × 0.5) mm2 (Fig. 1A, inset). All a given edge (4, 7, 8). The edge states come in three types according to the sign of the Dirac devices had a 110-nm-thick Si3N4/SiO2 mul- Kramers’ doublets, and time-reversal symmetry mass parameter M. In conventional semicon- tilayer gate insulator (16) and a 5/30 nm Ti/Au ensures the crossing of their energy levels at ductors such as GaAs and CdTe, the s-like E1 gate electrode stack. Transport measurements special points in the Brillouin zone. Because band lies above the p-like HH1 band, and the were done in a 3He/4He-dilution refrigerator of this energy-level crossing, the spectrum of a mass parameter M is positive. Semi-metals (base temperature T < 30 mK, uniaxial fields QSH insulator cannot be adiabatically de- such as grapheme (9, 10) are described by a up to 18 T) and in a 4He cryostat fitted with a formed into that of a topologically trivial in- massless Dirac model with M = 0, although the vector magnet (T = 1.4 K, and fields up to sulator without helical edge states; therefore, in bands have a different physical interpretation. 300 mT in variable direction), using lock-in this precise sense, the QSH insulators represent a In so-called inverted semiconductors such as techniques. At zero gate voltage, the samples topologically distinct new state of matter. HgTe, the s-like orbital lies below the p-like were n-type, exhibiting carrier densities be- − Theoretical background. It has been pro- orbitals; therefore, the Dirac mass parameter tween 1.3 × 1011 and 3.5 × 1011 cm 2 and mob- posed theoretically that HgTe/(Hg,Cd)Te M in the HgTe/(Hg,Cd)Te QWs can be con- ilities up to 1.5 × 105 cm2 V−1 s−1. The carrier quantum wells (QWs) provide a natural re- tinuously tuned from a positive value M >0 density could be reduced continuously by ap- for thin QWs with thickness d < d to a neg- plying a negative gate voltage to the Au elec- 1 c Physikalisches Institut (EP III), Universität Würzburg, ative value M <0forthickQWswithd > d .A trode with respect to carriers in the QW. The D-97074 Würzburg, Germany. 2Department of Physics, c McCullough Building, Stanford University, Stanford, CA topological quantum phase transition occurs at Si-O-N gate insulator stack allowed for quite 94305–4045, USA. a critical thickness d = dc, where the system is large gate voltages, enabling us to gate the *To whom correspondence should be addressed. E-mail: effectively described by a massless Dirac samples through the gap from n-type to p-type [email protected] theory as for graphene. The nature of this conductance.
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Field-induced phase transition. The change which can be directly obtained from the symmetry, they are not robustly protected. Start- in carrier type can be monitored from Hall minimal coupling of the simple Dirac model ing from this case, increasing the magnetic experiments, as shown in Fig. 1A for a large (Eq. 1) to a perpendicular magnetic field B⊥. field shifts the red Landau level toward higher [(L × W) = (600 × 200) mm2] Hall bar with a If we only consider the orbital effects of the and the blue one toward lower energies. When well width of 6.5 nm, at 30 mK. The change magnetic field, two series of Landau levels one of the bulk Landau level crosses the in carrier type is directly reflected in a sign are obtained from the upper and lower 2 × 2 Fermi level, there remains a single edge state change of the slope of the Hall resistance Rxy, blocks of the Hamiltonian, Eq. 1, in which on each edge, and one obtains a net QH effect 2 and we can directly infer that the carrier density the two levels closest to the Fermi energy with either Gxy = e /h (Fig. 2E) or with Gxy = 11 −2 2 varies from an electron density n = 1.2 × 10 are given by E+ = C + M − (D + B)lcpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiand E– = −e /h (Fig. 2F). When the magnetic field is −2 −2 cm at gate voltage Vg = –1.0 V to a hole density C − M − (D − B)lc ,with lc ≡ ℏ=ðeB⊥Þ. increased further, the second bulk Landau 10 −2 p =1.0×10 cm at Vg = –2.0 V. At modest Thus, the condition for level crossing is given level crosses the Fermi level, and one reaches −2 c magnetic fields, for both n-type and p-type by E+ − E− =2M − 2Blc =0orB ⊥ =(ℏM)/(eB). the conventional insulator case (Fig. 2C) but channels, Rxy exhibits quantum Hall plateaus, Generally, the B parameter is always neg- with the colors of the Landau level interchanged. indicative of the good quality of the material, ative; therefore, we can see that the level cross- In models with bulk inversion asymmetry (BIA) until at fields above ~3 T the last Landau level is ing occurs only in the inverted region with (11), the level crossing between E1 and HH1 c depleted. M <0. Landau levels at B ⊥ can be avoided, and the Notable transport behavior is observed for The Landau levels for the normal (d < dc) phase regions (i) and (ii) in Fig. 2B become –1.9 V ≤ Vg ≤ –1.4 V (see the green and the and the inverted (d > dc) regime are shown connected. Generally, the nonmonotonic de- red traces in Fig. 1), where the sample is in- in Fig. 2, A and B, respectively. Edge states pendence of the Landau-level energies leads to sulating at zero magnetic field (i.e., the Fermi in the presence of an external magnetic field the transition from the insulating state to the level is in the gap). For these gate voltages, can be easily obtained by solving our simple QH state at a constant Fermi energy, when the we observe that the sample undergoes a phase Dirac model in Eq. 1 with open boundary con- magnetic field is varied. transition from an insulating state to a QH state dition along one direction and periodic bound- 2 with a quantized Hall conductance of Gxy =±e /h, ary condition along the other direction. Figure either n- or p-type depending on Vg, at a small 2C shows the bulk and edge states for a con- A B (~1 to 2 T) applied magnetic field. The sam- ventional insulator. With increasing thickness d, iii
ple remains in the QH state for a few more T, the two states closest to the Fermi energy ap- E E i i ii and then becomes once again insulating. We proach and then cross each other. This “band iv observed this phenomenon in a number of inversion” leads to the bulk and edge states B Bc B samples in the inverted regime, with 6.5 nm < (Fig. 2D). The Fermi energy crosses a pair of C D σ σ d <12nm. counterpropagating edge states on each edge, case i, xy=0 case ii, xy=0 E The phase transition from an insulating resulting in no net Hall effect. These counter- E E E state to a QH state is a nontrivial consequence propagating edge states are similar to the F F of the inverted band structure of the QSH in- helical edge states of the QSH insulator. How-
F E E Fig. 1. (A) Hall resistance, Rxy,ofa(L × F B / T
(green), –1.7 V (red), –1.8 V (orange), Ω regime. (B) The bulk Landau levels in the
–1.9 V (brown), and –2.0 V (black, lower / k 0 inverted regime. A pair of low-lying Landau c curve). For decreasing Vg, the n-type carrier xy levels cross at a finite magnetic field B ⊥. The concentration decreases and a transition to a R –10 crossing divides the phase diagram of gate p-type conductor is observed, passing –20 V = –2 V voltage and magnetic field into four regimes, through an insulating regime between g labeled (i) to (iv). (C) The low-lying bulk and –1.4 and –1.9 V at B = 0 T. The inset –30 edge state energies as a function of the centers shows a schematic sample layout with of the Landau orbitals in the normal regime. 20 E (V = –1 V) ohmic contacts labeled 1 to 6. The gray F g (D) The low-lying bulk and edge state energies shaded region indicates the top gate B as a function of the centers of the Landau electrode and defines the part of the orbitals for the inverted regime, where the 0 sample where the carrier concentration Fermi energy lies in-between the two bulk and type can be changed. Red and blue inverted Landau levels. The Fermi energy / meV
arrows indicate the counterpropagating E crosses the Landau levels, giving rise to the –20 spin-polarized edge channels of the QSH pair of counterpropagating edge states. When E (V = –2 V) effect. (B) The Landau-level fan chart of a F g the magnetic field is increased, the two lowest- 6.5-nm QW obtained from an eight-band lying bulk Landau levels approach each other, –40 k·p calculation. Black dashed lines indi- and they cross the Fermi energy in a different cate the energetic position of the Fermi order, depending on the value of the gate energy, E ,forV = −1.0 and −2.0 V. Red 0 2 4 6 8 voltage. The crossing of the Fermi energy by F g B / T and green dashed lines correspond to the one of the Landau levels gives rise to either the position of the Fermi energies of the red and green Hall resistance traces of (A). The Landau-level crossing n- or the p-type QHE for the cases shown in (E) points are marked by arrows of the same color. and (F).
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Although the four-band Dirac model (Eq. 1) starts to decrease in magnitude. This is be- mobilities in the nominally insulating regime]. gives a simple qualitative understanding of cause for these well widths, the band gap no The pertinent data are shown in Fig. 4, which this novel phase transition, we also performed longer occurs between the E1andHH1lev- plots the zero B-field four-terminal resistance more realistic and self-consistent eight-band els, but rather between HH1andHH2—the R14,23 ≡ V23/I14 as a function of normalized gate k·p model calculations (13) for a 6.5-nm quan- second confined hole-like level, as schemat- voltage (Vthr is defined as the voltage for which tum well, with the fan chart of the Landau ically shown in the inset of Fig. 3 [see also the resistance is largest) for several devices that levels displayed in Fig. 1B. The two anoma- (17)]. Also in this regime, a band crossing of are representative of the large number of lous Landau levels cross at a critical magnetic conductance- (HH1) and valence- (HH2) band– structures we investigated. R14,23 is measured c field B ⊥, which evidently depends on well derived Landau levels occurs with increasing while the Fermi level in the device is scanned width. This implies that when a sample has its magnetic field (13, 17, 18). Figure 3 clearly through the gap. In the low-resistance regions at Fermi energy in the gap at zero magnetic illustrates the quantum phase transition that positive Vg − Vthr, the sample is n-type; at field, this energy will always be crossed by occurs as a function of d in the HgTe QWs: negative Vg − Vthr, the sample is p-type. c the two anomalous Landau levels, resulting in Only for d > dc does B ⊥ exist, and at the The black curve labeled I in Fig. 4 was a QH plateau in-between the two crossing same time the energy gap is negative (i.e., obtained from a medium-sized [(20.0 × 13.3) fields. Figure 3 summarizes the dependence the band structure is inverted). The experimen- mm2] device with a 5.5-nm QW and shows the c of B ⊥ on well width d. The open red squares tal data allow for a quite accurate determi- behavior we observe for all devices with a are experimental data points that result from nation of the critical thickness, yielding dc = normal band structure: When the Fermi level fitting the eight-band k·p model to experi- 6.3 ± 0.1 nm. is in the gap, R14,23 increases strongly and is mental data as in Fig. 1, while the filled red Zero-field edge channels and the QSH at least several tens of megohm (this is the de- triangles result solely from the k·p calcula- effect. The actual existence of edge channels tection limit of the lock-in equipment used in tion. For reference, the calculated gap ener- in insulating inverted QWs is only revealed the experiment). This clearly is the expected gies are also plotted in this graph as open when studying smaller Hall bars [the typical behavior for a conventional insulator. How- blue circles. The band inversion is reflected mobility of 105 cm2 V−1 s−1 in n-type material ever, for all devices containing an inverted QW, in the sign change of the gap. For relatively implies an elastic mean free path of lmfp ≈ the resistance in the insulating regime remains wide wells (d > 8.5 nm), the (inverted) gap 1 mm(19, 20)—and one may anticipate lower finite. R14,23 plateaus at well below 100 kilohm 2 (i.e., G14,23 =0.3e /h) for the blue curve labeled II, which is again for a (20.0 × 13.3) m 2 Fig. 3. Crossing field, 100 10 m device fabricated by optical lithography, c B ⊥ (red triangles), and normal inverted but that contains a 7.3-nm-wide QW. For much m energy gap, Eg (blue 80 shorter samples (L =1.0 m, green and red open dots), as a func- 200 8 curves III and IV) fabricated from the same tion of QW width d 60 150 wafer, G14,23 actually reaches the predicted 2 resulting from an eight- 100 E2 value close to 2e /h, demonstrating the exis- band k·p calculation. 40 50 6 tence of the QSH insulator state for inverted E1 B E / meV For well widths larger 0 c HgTe QW structures. / T than 6.3 nm, the QW is –50 HH1 / meV 20 HH4 Figure 4 includes data on two devices with
g HH2 –100 HH3 inverted and a mid-gap E m 4 6 8 10 12 14 4 d = 7.3 nm, L = 1.0 m. The green trace (III) crossing of Landau levels d / nm is from a device with W = 1.0 mm, and the red HH 0 deriving from the 1 trace (IV) corresponds to a device with W = conductance and E1va- –20 2 0.5 mm. Clearly, the residual resistance of the lence band occurs at fi- devices does not depend on the width of the nite magnetic fields. The structure, which indicates that the transport experimentally observed –40 occurs through edge channels (21). The traces crossing points are in- 0 3 4 5 6 7 8 9 10 11 12 for the d = 7.3 nm, L = 1.0 mm devices do not dicated by open red d / nm squares. The inset shows reach all the way into the p-region because the the energetic ordering of the QW subband structure as a function of QW width d.[Seealso(17)]. electron-beam lithography needed to fabricate the devices increases the intrinsic (Vg =0V) carrier concentration. In addition, fluctuations on the conductance plateaus in traces II, III, Fig. 4. The longitudinal four- G = 0.01 e2/h 20 terminal resistance, R ,of T = 0.03 K and IV are reproducible and do not stem from, 14,23 7 various normal (d =5.5nm) 10 15 G = 2 e2/h e.g., electrical noise. Although all R14,23 traces (I) and inverted (d =7.3nm) Ω discussed so far were taken at the base / k 10 (II, III, and IV) QW structures T = 30 mK temperature (30 mK) of our dilution refriger- 6 as a function of the gate volt- 10 14,23 T = 1.8 K ator, the conductance plateaus are not limited R 5 age measured for B =0Tat Ω to this very-low-temperature regime. In the / T =30mK.Thedevicesizes 0 inset of Fig. 4, we reproduce the green 30-mK m 2 5 –1.0 –0.5 0.0 0.5 1.0 are (20.0 × 13.3) m for 14,23 10 (Vg – Vthr) / V trace III on a linear scale and compare it with devices I and II, (1.0 × 1.0) R G = 0.3 e2/h a trace (in black) taken at 1.8 K from another 2 mm for device III, and (1.0 × (L × W) = (1.0 × 1.0) mm2 sample, which was m 2 4 0.5) m for device IV. The 10 fabricated from the same wafer. In the fabrica- R V G = 2 e2/h inset shows 14,23( g)oftwo tion of this sample, we used a lower-illumination samples from the same wafer, dose in the e-beam lithography, resulting in a havingthesamedevicesize 3 10 better (but still not quite complete) coverage of (III) at 30 mK (green) and –1.0 –0.5 0.0 0.5 1.0 1.5 2.0 the n-i-p transition. Clearly, in this further 1.8 K (black) on a linear scale. (Vg – Vthr) / V sample, and at 1.8 K, the 2e2/h conductance
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plateau is again present, including (thermally The peak broadens strongly when the magnetic properties in that regime with that of the smeared) conductance fluctuations. field is tilted into the QW plane. For fully in- recently discovered graphene. In many ways, In the pure two-terminal geometry, with plane fields, the QSH conductance can be ob- theHgTeQWsystemcanbeviewedasa only source and drain contacts (contacts 1 and served over a much wider magnetic field range tunable graphene system, where the Dirac FWHM 4, inset of Fig. 1A), the two counterpropagat- (B|| ≈ 0.7 T). mass term can be tuned continuously to zero ing helical edge states at one given edge con- The robustness of the helical edge states is from either the positive (topologically nect the chemical potential from the source ensured by the time-reversal symmetry. A mag- trivial) or the negative (topologically non- and drain, respectively, and they are not in netic field breaks time-reversal symmetry and trivial) side. equilibrium with each other because the elastic thus turns on a gap between the two otherwise backscattering vanishes between these two chan- degenerate helical edge states. The perpendic- References and Notes nels. In the absence of voltage probes 2, 3, 5, and ular and in-plane magnetic field lead to dif- 1. S. Murakami, N. Nagaosa, S. C. Zhang, Science 301, 6, as indicated in the inset of Fig. 1A, the two- ferent gaps, depending on the matrix elements of 1348 (2003). 2 et al Phys. Rev. Lett. terminal conductance should give 2e /h.Inthe the magnetization operator: Egap⊥ = 〈↑|(z% ⋅ r% 2. J. Sinova ., 92, 126603 (2004). j% þ m ↓〉 〈↑ m ↓〉 3. S. Murakami, N. Nagaosa, S. C. Zhang, Phys. Rev. Lett. presence of the voltage probes, the voltage g⊥ BS⊥)| |B|andEgap|| = |g|| BS||| |B|, in r% j% 93, 156804 (2004). measurement necessarily leads to the equilibra- which , are the position and electric current 4. C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 146802 tion of the two helical channels with the opposite operator, respectively, and z% is the unit-vector (2005). Phys. Rev. Lett. spin orientation, because the voltage probes are perpendicular to the QW plane. S⊥(||) stands for 5. B. A. Bernevig, S. C. Zhang, 96, 106802 not spin sensitive. A simple Landauer-Büttiker the dimensionless part of the Zeeman-coupling (2006). Science type of calculation shows that the four-terminal matrix element in a perpendicular (parallel) 6. B. A. Bernevig, T. L. Hughes, S. C. Zhang, 314, 2 1757 (2006). resistance should be given by R14,23 = h/2e .In magnetic field. We can estimate the magnitude 7. C. Wu, B. A. Bernevig, S. C. Zhang, Phys. Rev. Lett. 96, the presence of the voltage probes, the voltage of these two gaps by noting that 〈↑|z% ⋅ r% j%|↓〉 ~ 106401 (2006). 8. C. Xu, J. Moore, Phys. Rev. B 73, 045322 (2006). drops V12, V23,andV34 add in series to give a evx and 〈↑|S||,⊥|↓〉 ~1,inwhichv and x are the ≡ 2 9. K. S. Novoselov et al., Nature 438, 197 (2005). higher resistance of R14 V14/I14 =3h/2e . Fermi velocity and width of the edge channels, Nature x 10. Y. Zhang, Y. Tan, H. L. Stormer, P. Kim, 438, 201 These results are valid as long as the distance respectively. v and can be obtained from the (2005). between the voltage probes is less than the Dirac parameters as v = A/ℏ and x ≈ ℏv/|M|. The 11. X. Dai, T. L. Hughes, X.-L. Qi, Z. Fang, S.-C. Zhang, inelastic mean free path lin. Although elastic parameters for the d = 7.3-nm QW give the di- http://arxiv.org/abs/0705.1516 (2007). 12. Similar mass domain walls have been proposed to scatterers cannot cause backscattering of the mensionless ratio evx/mB ~ 280, which thus leads 2 occur in three-dimensional in PbTe/(Pb,Sn)Te hetero- helical edge states, inelastic scatterers can. We to Egap⊥/Egap|| ~10. From this estimate, we can L± structures, where the 6 bands change position as estimate the inelastic mean free path to be lin > see that the strong anisotropy observed in the a function of Sn concentration (25, 26). 1 mm at our measurement temperature. There- experiments originates from the high Fermi 13. E. G. Novik et al., Phys. Rev. B 72, 035321 (2005). fore, for the large sample (trace II), where the velocity of the edge states and the small bulk 14. C. Becker et al., Phys. Stat. Sol. (C) 4, 3382 (2007). 15. Well thicknesses have been calibrated by x-ray reflectivity distance between the voltage probes exceeds the gap M, which together make the orbital magne- measurements at the DESY synchrotron in Hamburg, inelastic mean free path lin,weexpectthere- tization dominant. Germany. sidual resistance to be higher, consistent with the Concluding remarks. Up to this point, our 16. J. Hinz et al., Semicond. Sci. Technol. 21, 501 (2006). experimental measurement shown in trace II in experiments only measured the charge-transport 17. A. Pfeuffer-Jeschke, thesis, University of Würzburg, Germany (2000). Fig. 4. properties. Although the QSH effect was man- 18. M. Schultz et al., Phys. Rev. B 57, 14772 (1998). Breaking time-reversal symmetry. Another ifested in the change in transport properties, 19. V. Daumer et al., Appl. Phys. Lett. 83, 1376 (2003). intriguing observation is that the QSH effect is we still want to experimentally confirm the 20. M. König et al., Phys. Rev. Lett. 96, 076804 (2006). destroyed by applying only a small magnetic spin accumulation resulting from the spin Hall 21. We have observed a similar independence of resistance d L m field perpendicular to the 2DEG plane. Figure 5 effect (23, 24) in the topologically nontrivial on sample width in the = 7 nm, = 20.0 m devices, showing that also in these larger structures, shows that the magnetoresistance is strongly insulating regime and compare both electric the conductance is completely dominated by edge anisotropic. (These data were obtained in the and magnetic results with the experiments of channels. vector magnet system at 1.4 K.) A very sharp, the spin Hall effect in the metallic regime. It 22. FWHM of the magnetoresistance peak is about 10 mT at cusplike conductance peak is observed for per- would also be interesting to explore the 30 mK, increasing to 28 mT at 1.4 K. 23. Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom, pendicular field, with the full width at half- regime close to the quantum phase transition Science 306, 1910 (2004). FWHM maximum (FWHM) B⊥ ≈ 28 mT (22). point of d = dc and compare the transport 24. J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). 25. B. A. Volkov, O. A. Pankratov, JETP Lett. 42, 178 Fig. 5. Four-terminal mag- (1985). 0.14 0° 26. E. Fradkin, E. Dagotto, D. Boyanovsky, Phys. Rev. Lett. netoconductance, G14,23,in the QSH regime as a func- 57, 2967 (1986). 27. We thank A. Bernevig, X. Dai, Z. Fang, T. Hughes, tion of tilt angle between 0.12 15° C. X. Liu, and C. J. Wu for insightful discussions; the plane of the 2DEG and C. R. Becker and V. Hock for sample preparation; and applied magnetic field for a 0.10 C. Kumpf for calibrating the well widths of the HgTe d = 7.3-nm QW structure samples. This work is supported by the Deutsche L W 30° Forschungsgemeinschaft (grant SFB 410); the German- with dimensions ( × )= /h 0.08 (20 × 13.3) mm2 measured 2 Israeli Foundation for Scientific Research and 45° Development (grant 881/05); the NSF (grant DMR-
in a vector field cryostat at G / e 0.06 0342832); the U.S. Department of Energy, Office of 1.4 K. Basic Energy Sciences, under contract DE-AC03- 0.04 B 60° 76SF00515; and Focus Center Research Program 75° (FCRP) Center on Functional Engineered α Nanoarchitectonics (FENA). 0.02 90° 19 June 2007; accepted 10 September 2007 0.00 Published online 20 September 2007; –0.05 0.00 0.05 10.1126/science.1148047 B / T Include this information when citing this paper.
770 2 NOVEMBER 2007 VOL 318 SCIENCE www.sciencemag.org Quantum Spin Hall Insulator State in HgTe Quantum Wells Markus König et al. Science 318, 766 (2007); DOI: 10.1126/science.1148047
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