## Abstract

Nonlocal transport through edge channels holds great promise for low-power information processing. However, edge channels have so far only been demonstrated to occur in the quantum Hall regime, at high magnetic fields. We found that mercury telluride quantum wells in the quantum spin Hall regime exhibit nonlocal edge channel transport at zero external magnetic field. The data confirm that the quantum transport through the (helical) edge channels is dissipationless and that the contacts lead to equilibration between the counterpropagating spin states at the edge. The experimental data agree quantitatively with the theory of the quantum spin Hall effect. The edge channel transport paves the way for a new generation of spintronic devices for low-power information processing.

The search for topological states of quantum matter has become an important goal in condensed matter physics. Inside a topological insulator, the conventional laws of electrodynamics are substantially altered (*1*), which may have applications in constructing novel devices for the processing of (quantum) information. The quantum spin Hall (QSH) state (*2*, *3*) is a topologically nontrivial state of matter that exists in the absence of any external magnetic field. It has a bulk energy gap but gapless helical edge states protected by time reversal symmetry. In the QSH regime, opposite spin states forming a Kramers doublet counterpropagate at the edge (*4*, *5*). Recently, the QSH state was theoretically predicted in HgTe quantum wells (*6*). There is a topological quantum phase transition at a critical thickness *d*_{c} of the quantum well, separating the trivial insulator state for *d < d*_{c} from the QSH insulator state for *d > d*_{c}. Soon after the theoretical prediction, evidence for the QSH state was observed in transport measurements (*7*). In the QSH regime, experiments measured a conductance *G* close to twice the quantum unit of conductance *G* = 2*e*^{2}*/h* (where *e* is the charge on the electron and *h* is Planck’s constant); this value is consistent with quantum transport due to helical edge states. However, such a conductance quantization in small Hall bar geometries does not allow us to distinguish experimentally between ballistic and edge channel transport in a convincing manner. Thus, it is important to be able to prove experimentally in an unambiguous manner the existence of edge channels in HgTe quantum wells.

**Ohm’s law versus nonlocal transport.** In conventional diffusive electronics, bulk transport satisfies Ohm’s law. The resistance is proportional to the length and inversely proportional to the cross-sectional area, implying the existence of a local resistivity or conductivity tensor. However, the existence of edge states necessarily leads to nonlocal transport, which invalidates the concept of local resistivity. Such nonlocal transport has been experimentally observed in the quantum Hall (QH) regime in the presence of a large magnetic field (*8*), and the nonlocal transport is well described by a quantum transport theory based on the Landauer-Büttiker formalism (*9*). These measurements constitute definitive experimental evidence for the existence of edge states in the QH regime.

We report nonlocal transport measurements in HgTe quantum wells that demonstrate the existence of the predicted extended edge channels. We have fabricated structures more complicated than a standard Hall bar that allow a detailed investigation of the transport mechanism. In addition, we present the theory of quantum transport in the QSH regime, and uncover the effects of macroscopic time irreversibility on the helical edge states.

**Device structure.** We present experimental results on four different devices. The behavior in these structures is exemplary for the ~50 devices we studied. The devices were fabricated from HgTe/(Hg,Cd)Te quantum well structures with well thicknesses of *d* = 7.5 nm (samples D1, D2, and D3) and 9.0 nm (sample D4). Note that all wells have a thickness *d > d*_{c}* *≈ 6.3 nm, and thus exhibit the topologically nontrivial inverted band structure. At zero gate voltage, the samples are n-type and have a carrier density of *n*_{s} ≈ 3 × 10^{11} cm^{–2} and a mobility of 1.5 × 10^{5} cm^{2} V^{–1} s^{–1}, with small variations between the different wafers. The devices are lithographically patterned using electron-beam lithography and subsequent Ar ion-beam etching. Devices D1 and D2 are micrometer-scale Hall bars with exact dimensions as indicated in the insets of Fig. 1. Devices D3 and D4 are dedicated structures for identifying nonlocal transport, with schematic structure given in Fig. 2. All devices are fitted with a 110-nm-thick Si_{3}N_{4}/SiO_{2} multilayer gate insulator and a Ti (5 nm)–Au (50 nm) gate electrode stack.

By applying a voltage *V*_{g} to the top gate, the electron carrier density of the quantum well can be adjusted, going from an n-type behavior at positive gate voltages through the bulk insulator state into a p-type regime at negative gate voltages. For reasons of comparison, the experimental data in Figs. 1, 3, and 4 are plotted as a function of a normalized gate voltage *V** = *V*_{g} – *V*_{thr} (*V*_{thr} is defined as the voltage for which the resistance is largest). Measurements were performed at a lattice temperature of 10 mK for samples D1, D2, and D3 and at 1.8 K for sample D4, using low-frequency (13 Hz) lock-in techniques under voltage bias. The four-terminal resistance (Fig. 1) shows a maximum at about *h/*2*e*^{2}, in agreement with the results of (*7*). The contact resistance should be insensitive to the gate voltage and can be measured from the resistance deep in the metallic region. By subtracting the contact resistance, we find that the two-terminal resistance has its maximum at about 3*h/*2*e*^{2} (Fig. 1). This value is exactly what is expected from the theory of QSH edge transport obtained from the Landauer-Büttiker formula.

**Transport on the edge.** Within the general Landauer-Büttiker formalism (*10*), the current-voltage relationship is expressed as*I _{i}* is the current flowing out of the

*i*th electrode into the sample region,

*V*is the voltage on the

_{i}*i*th electrode, and

*T*is the transmission probability from the

_{ji}*i*th to the

*j*th electrode. The total current is conserved in the sense that ∑

*= 0. A voltage lead*

_{i}I_{i}*j*is defined by the condition that it draws no net current (i.e.,

*I*= 0). The physical currents are left invariant if the voltages on all electrodes are shifted by a constant amount μ, implying that ∑

_{j}*= ∑*

_{i}T_{ij}*. In a time reversal–invariant system, the transmission coefficients satisfy the condition*

_{i}T_{ji}*T*=

_{ij}*T*.

_{ji}For a general two-dimensional sample, the number of transmission channels scales with the width of the sample, so that the transmission matrix *T _{ij}* is complicated and nonuniversal. However, a tremendous simplification arises if the quantum transport is entirely dominated by the edge states. In the QH regime, chiral edge states are responsible for the transport. For a standard Hall bar with

*N*current and voltage leads attached (compare the insets of Fig. 1 with

*N*= 6), the transmission matrix elements for the ν = 1 QH state are given by

*T*(QH)

_{ i}_{+1,}

*= 1, for*

_{i}*i*= 1, …,

*N*, and all other matrix elements vanish identically. Here we periodically identify the

*i*=

*N*+ 1 electrode with

*i*= 1. Chiral edge states are protected from backscattering; therefore, the

*i*th electrode transmits perfectly to the neighboring (

*i*+ 1)th electrode on one side only. In the example of current leads on electrodes 1 and 4, and voltage leads on electrodes 2, 3, 5, and 6, one finds that

*I*

_{1}= –

*I*

_{4}≡

*I*

_{14},

*V*

_{2}–

*V*

_{3}= 0, and

*V*

_{1}–

*V*

_{4}= (

*h/e*

^{2})

*I*

_{14}, giving a four-terminal resistance of

*R*

_{14,23}= 0 and a two-terminal resistance of

*R*

_{14,14}=

*h/e*

^{2}.

In the case of helical edge states in the QSH regime, opposite spin states form a Kramers pair, counterpropagating on the same edge. The helical edge states are protected from backscattering due to time reversal symmetry, and the transmission from one electrode to the next is perfect. From this point of view, the helical edge states can be viewed as two copies of chiral edge states related by time reversal symmetry. Therefore, the transmission matrix is given by *T*(QSH) = *T*(QH) + *T*^{†}(QH), implying that the only nonvanishing matrix elements are given by*T*(QSH)_{i}_{+1,}* _{i}* =

*T*(QSH)

_{i}_{,}

_{i}_{+1}= 1 (2)Considering again the example of current leads on electrodes 1 and 4 and voltage leads on electrodes 2, 3, 5, and 6, one finds that

*I*

_{1}= –

*I*

_{4}≡

*I*

_{14},

*V*

_{2}–

*V*

_{3}= (

*h/*2

*e*

^{2})

*I*

_{14}, and

*V*

_{1}–

*V*

_{4}= (3

*h/e*

^{2})

*I*

_{14}, giving a four-terminal resistance of

*R*

_{14,23}=

*h/*2

*e*

^{2}and a two-terminal resistance of

*R*

_{14,14}= 3

*h/*2

*e*

^{2}. The experimental data in Fig. 1 confirm this picture. For both micro–Hall bar structures D1 and D2 that differ only in the dimensions of the area between voltage contacts 3 and 4, we observe exactly the expected resistance values for

*R*

_{14,23}=

*h/*2

*e*

^{2}and

*R*

_{14,14}= 3

*h/*2

*e*

^{2}for gate voltages where the samples are in the QSH regime.

**Dissipationless transport.** Conceptually, one might sense a paradox between the dissipationless nature of the QSH edge states and the finite four-terminal longitudinal resistance *R*_{14,23}, which vanishes for the QH state. We can generally assume that the microscopic Hamiltonian governing the voltage leads is invariant under time reversal symmetry; therefore, one would naturally ask how such leads could cause the dissipation of the helical edge states, which are protected by time reversal symmetry. In nature, the time reversal symmetry can be broken in two ways: at the level of the microscopic Hamiltonian, or at the level of the macroscopic irreversibility in systems whose microscopic Hamiltonian respects the time reversal symmetry. When the helical edge states propagate without dissipation inside the QSH insulator between the electrodes, neither form of time reversal symmetry breaking is present. As a result, the two counterpropagating channels can be maintained at two different quasi–chemical potentials, leading to a net current flow. However, once they enter the voltage leads, they interact with a reservoir containing infinitely many low-energy degrees of freedom, and the time reversal symmetry is effectively broken by the macroscopic irreversibility. As a result, the two counterpropagating channels equilibrate at the same chemical potential, determined by the voltage of the lead. Dissipation occurs with the equilibration process. The transport Eq. 1 breaks the macroscopic time reversal symmetry, even though the microscopic time reversal symmetry is ensured by the relationship *T _{ij}* =

*T*. In contrast to the case of QH state, the absence of dissipation of the QSH helical edge states is protected by Kramers’ theorem, which relies on the quantum phase coherence of wave functions. Thus, dissipation can occur once the phase coherence is destroyed in the metallic leads. By contrast, the robustness of QH chiral edge states does not require phase coherence. The result of a more rigorous and microscopic analysis on the different role played by a metallic lead in QH and QSH states (

_{ji}*11*) agrees with the simple transport Eqs. 1 and 2. These two equations, which correctly describe the dissipationless quantum transport inside the QSH insulator and the dissipation inside the electrodes, can be subjected to more stringent experimental tests than the two- and four-terminal experiments of Fig. 1 by considering devices D3 and D4 (Fig. 2).

**Helical versus chiral.** A further difference between helical and chiral edge channels is evident from our experiments on the six-terminal device D3 (Fig. 3). When the longitudinal resistance of device D3 is measured by passing a current through contacts 1 and 4 and by detecting the voltage between contacts 2 and 3 (*R*_{14,23}), we find, similar to the results of Fig. 1, a resistance value of *h/*2*e*^{2} when the bulk of the device is gated into the insulating regime (Fig. 3A). However, the longitudinal resistance is markedly different in a slightly modified configuration, where the current is passed through contacts 1 and 3 and the voltage is measured between contacts 5 and 4 (*R*_{13,54}) (Fig. 3B). We now find *R*_{13,54} ≈ 8.6 kilohms, which is markedly different from what one would expect for either the QH transport or the purely diffusive transport, where this configuration would be equivalent to the previous one. Application of Eqs. 1 and 2 actually predicts that the observed behavior is indeed what one expects for helical edge channels. This resistance value can again be expressed as an integer fraction of the inverse conductance quanta *e*^{2}*/h*: *R*_{13,54} = *h/*3*e*^{2}. This result shows that the current through the device is influenced by the number of ohmic contacts in the current path. These ohmic contacts lead to the equilibration of the chemical potentials between the two counterpropagating helical edge channels inside the contact. There are also some devices for which the maximal resistance does not match the theoretical value obtained from Eqs. 1 and 2, but still remains an integer fraction of the quantum *h/e*^{2}. This result can be naturally understood as due to inhomogeneities in the gate action (e.g., due to interface trap states) inducing some metallic droplets close to the edge channels while the bulk of the sample is insulating. A metallic droplet can cause dephasing of the electronic wave function, leading to fluctuations in the device resistance. For full dephasing, the droplet plays the role of an additional ohmic contact, just as for the chiral edge channels in the QH regime (*8*). More details on the effects of additional ohmic contacts in the QSH state are given in (*11*).

Another measurement that directly confirms the nonlocal character of the helical edge channel transport in the QSH regime is in Fig. 4, which shows data obtained from device D4, in the shape of the letter H. In this four-terminal device, the current is passed through contacts 1 and 4 and the voltage is measured between contacts 2 and 3. In the metallic n-type regime (low gate voltage), the voltage signal tends to zero. In the insulating regime, however, the nonlocal resistance signal increases to ~6.5 kilohms, which again fits perfectly to the result of Laudauer-Büttiker considerations: *R*_{14,23} = *h/*4*e*^{2} ≈ 6.45 kilohms. Classically, one would expect only a minimal signal in this configuration (from Poisson’s equation, assuming diffusive transport, one estimates a signal of about 40 ohms), and certainly not one that increases so strongly when the bulk of the sample is depleted. This signal measured here is fully nonlocal and can be taken (as was done 20 years ago for the QH regime) as definite evidence of the existence of edge channel transport in the QSH regime. A similar nonlocal voltage has been studied in a metallic spin Hall system with the same H-bar geometry (*12*), in which case the nonlocal voltage can be understood as a combination of the spin Hall effect and the inverse spin Hall effect (*13*). The quantized nonlocal resistance *h/*4*e*^{2} we find here is the quantum counterpart of the metallic case. For example, if we assume that the chemical potential in contact 1 is higher than that in contact 4 (compare to the layout of D4 in Fig. 2B), more electrons will be injected into the upper edge state in the horizontal segment of the H-bar than into the lower edge state. Because on opposite edges the right-propagating edge states have opposite spin, this implies that a spin-polarized current is generated by an applied bias *V*_{1} – *V*_{4}, comparable to a spin Hall effect. When this spin-polarized current is injected into the right leg of the device, the inverse effect occurs. Electrons in the upper edge flow to contact 2 while those in the lower edge flow to contact 3, establishing a voltage difference between those two contacts due to the charge imbalance between the edges. The right leg of the device thus acts as a detector for the injected spin-polarized current, which corresponds to the inverse spin Hall effect.

**Concluding remarks.** The multiterminal and nonlocal transport experiments on HgTe microstructures in the QSH regime demonstrate that charge transport occurs through extended helical edge channels. We have extended the Landauer-Büttiker model for multiterminal transport in the QH regime to the case of helical QSH edge channels and have shown that this model convincingly explains the observations. Logic devices based on the complementary metal oxide semiconductor design generate considerable heating due to the ohmic dissipation within the channel. Our work on conductance quantization demonstrates that electrons can be transported coherently within the edge channel without ohmic dissipation. Such an effect can be used to construct logic devices with improved performance.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/325/5938/294/DC1

SOM Text

Figs. S1 to S9

Table S1

References

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- We thank T. Beringer, N. Eikenberg, M. König, and S. Wiedmann for assistance in some of the experiments. Supported by Deutsche Forschungsgemeinschaft grant SFB 410; German-Israeli Foundation grant I-881-138.7/2005; NSF grant DMR-0342832; the Alexander von Humboldt Foundation; the U.S. Department of Energy, Office of Basic Energy Sciences, under contract DE-AC03-76SF00515; the Focus Center Research Program (FCRP) Center on Functional Engineered Nanoarchitectonics; the National Science and Engineering Research Council of Canada; and the Stanford Graduate Fellowship Program. Computational work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (www.sharcnet.ca).