Low-dimensional electronic systems have traditionally been obtained by electrostatically confining electrons, either in heterostructures or in intrinsically nanoscale materials such as single molecules, nanowires and graphene. Recently, a new method has emerged with the recognition that symmetry-protected topological (SPT) phases1, 2, which occur in systems with an energy gap to quasiparticle excitations (such as insulators or superconductors), can host robust surface states that remain gapless as long as the relevant global symmetry remains unbroken. The nature of the charge carriers in SPT surface states is intimately tied to the symmetry of the bulk, resulting in one- and two-dimensional electronic systems with novel properties. For example, time reversal symmetry endows the massless charge carriers on the surface of a three-dimensional topological insulator with helicity, fixing the orientation of their spin relative to their momentum3, 4. Weakly breaking this symmetry generates a gap on the surface5, resulting in charge carriers with finite effective mass and exotic spin textures6. Analogous manipulations have yet to be demonstrated in two-dimensional topological insulators, where the primary example of a SPT phase is the quantum spin Hall state7, 8. Here we demonstrate experimentally that charge-neutral monolayer graphene has a quantum spin Hall state9, 10 when it is subjected to a very large magnetic field angled with respect to the graphene plane. In contrast to time-reversal-symmetric systems7, this state is protected by a symmetry of planar spin rotations that emerges as electron spins in a half-filled Landau level are polarized by the large magnetic field. The properties of the resulting helical edge states can be modulated by balancing the applied field against an intrinsic antiferromagnetic instability11, 12, 13, which tends to spontaneously break the spin-rotation symmetry. In the resulting canted antiferromagnetic state, we observe transport signatures of gapped edge states, which constitute a new kind of one-dimensional electronic system with a tunable bandgap and an associated spin texture14.
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Extended data figures and tables
Extended Data Figures
- Extended Data Figure 1: Images of measured devices. (419 KB)
False-colour AFM images of the devices enumerated in Extended Data Table 1. Dashed lines outline the graphene boundary. Black scale bars correspond to 1 μm.
- Extended Data Figure 2: Conductance as a function of B⊥, BT and gate voltage for devices A, B and C. (498 KB)
Colouring of lines from blue to red indicates increasing BT, with B⊥ as indicated at the top of each panel.
- Extended Data Figure 3: Gcnp as a function of B⊥ and BT for devices A, B and E. (733 KB)
Correspondingly higher values of BT are required to induce the transition for higher values of B⊥. For device A, the curves correspond to B⊥ = 0.75, 1, 1.2, 1.4, 1.6, 2, 2.5, 3, 3.5 and 4 T (blue to red). For device B, the curves correspond to B⊥ = 1, 1.5, 2, 2.5, 3 and 3.5 T (blue to red). For device E, the curves correspond to B⊥ = 0.9, 1, 1.1, 1.2, 1.3, 1.4 and 1.5 T (blue to red).
- Extended Data Figure 4: Nonlocal measurements for device C in the QSH and CAF regimes. (230 KB)
In the main text, evidence for conduction via edge states in the QSH and CAF regimes is provided by non-local transport measurements in device A (Figs 2b and 4b). Owing to conduction through counterpropagating edge states, interrupting an edge with a floating contact decreases the two-terminal conductance much more than would be expected in a diffusive transport model. Here we provide an additional example of this behaviour for device C. a, Schematic of distinct two-terminal measurement topologies with different numbers of floating contacts (open circles). b, QSH regime, B⊥ = 2.7 T and BT = 45 T. c, CAF regime, B⊥ = 5.9 T and BT = 45 T. Curves are colour-coded according to the measurement schematics, as in the main text. Owing to a small gate leak in one of the contacts, these specific non-local measurements underestimate the conductance by a scale factor that was adjusted for by fitting the ν = −1 plateau to a conductance of e2/h.
- Extended Data Figure 5: Double conductance peaks in seven different devices. (197 KB)
A generic feature of the intermediate regime between the insulating and metallic QSH regimes is the appearance of double conductance peaks close to ν = 0. The figure shows two-terminal conductance versus back-gate voltage, VG. Purely perpendicular magnetic field only (BT = B⊥, black lines) results in an insulating state at ν = 0. Increasing the total magnetic field while keeping the perpendicular component constant (BT > B⊥, red lines), induces a transition to the CAF with an associated double conductance peak feature. Devices are ordered from left to right by descending aspect ratio.
- Extended Data Figure 6: Temperature dependence of the charge-neutrality point conductivity for device B. (153 KB)
a–c, Gate sweeps for device B at constant B⊥ = 2.5 T and BT = 2.5 (a), 26.5 (b) and 34.5 T (c). d, Conductance at the charge-neutrality point as a function of temperature for the data in a–c. A clear insulating dependence (∂G/∂T > 0) is observed for B⊥ = BT. With increased BT, in the intermediate regime, the double conductance peaks between ν = 0 and ν = ±1 have a weakly metallic temperature dependence (∂G/∂T < 0) whereas Gcnp is very weakly insulating. In the QSH regime (BT B⊥), where the conduction is along edge channels, the temperature dependence at ν = 0 is metallic.
- Extended Data Figure 7: Schematic diagram of the capacitance bridge-on-a-chip in a tilted magnetic field. (209 KB)
The magnetic field points up in the diagram. Beige: sample stage, showing axis of rotation (red arrows). Purple: graphene sample mount. Blue: transistor mount with 90° bend. The HEMT is mounted on the face angled at 90° to the graphene sample mount and with the plane of its 2D conduction channel perpendicular to the sample-stage axis of rotation. A single wire bond connects the two mounts, from the graphite back gate to the balance point of the capacitance bridge. The transistor is gated by applying VG to the balance point/graphite back gate through a 100-MΩ chip resistor. Combined with the total capacitance of the balance point to ground (~3 pF), this sets the low-frequency cut-off for the measurement at ~1 kHz. The density of electrons in the graphene sample is determined by the d.c. voltage difference between the graphene sample and the graphite back gate, namely by Vs − VG. In the main text, this is compensated for and all capacitance measurements are shown as functions of the graphite gate voltage relative to grounded graphene. All components shown in black are at room temperature.
- Extended Data Figure 8: Tilted-field magnetotransport in zero-field insulating monolayer graphene. (223 KB)
In a fraction of devices with the same geometry as those discussed in the main text, we find that, rather than a conductivity of ~e2/h at charge neutrality, these devices instead have insulating behaviour at the charge-neutrality point at zero applied magnetic field. We ascribe this insulating behaviour to the opening of a bandgap at the charge-neutrality point owing to the effect of an aligned hBN substrate31. The top panel shows the resistance of the device in zero magnetic field. This device has a resistance of 825 kΩ at the charge-neutrality point in zero magnetic field and at T = 0.3 K. As with the devices described in the main text, the insulating state becomes stronger in a perpendicular magnetic field. In the bottom panel, solid lines are gate sweeps at constant B⊥ = 1, 2 and 3 T and BT = B⊥. Dashed lines are for the corresponding sequence with B⊥ = 1, 2 and 3 T but BT = 45 T. Data taken at 0.3 K. Semiconducting graphene samples do not show any sign of QSH-type physics, at least up to 45 T. Even for B⊥ = 1 T and BT = 45 T, the conductance at the charge-neutrality point increases only slightly, from 0.02e2/h with zero in-plane field to 0.14e2/h with BT = 45 T. This is understandable, because, even neglecting interaction effects, closing a moiré-induced band gap of Δ = 10 meV requires a Zeeman field of nearly Δ/gμB ≈ 85 T. We note that in these samples, the ground state at BT = B⊥ may not be an antiferromagnet.