Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state

Journal name:
Nature
Volume:
505,
Pages:
528–532
Date published:
DOI:
doi:10.1038/nature12800
Received
Accepted
Published online

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.

At a glance

Figures

  1. QSH state in monolayer graphene in extreme tilted magnetic fields.
    Figure 1: QSH state in monolayer graphene in extreme tilted magnetic fields.

    a, Conductance of deviceA at B = 1.4T for different values of BT. As BT increases, the insulating state at ν = 0 is gradually replaced by a high-conductance state, with an accompanying inversion of the sign of ∂Gcnp/∂T (additional data in Extended Data Figs 2 and 3). Inset, Gcnp as a function of BT for deviceA: B = 0.75, 1.0, 1.4, 1.6, 2.0, 2.5, 3.0 and 4.0T (left to right). b, Capacitance (opaque lines) and dissipation (semi-opaque lines) of deviceB at B = 2.5T. The low dissipation confirms that the measurements are in the low-frequency limit, such that the dips in capacitance can be interpreted as corresponding to incompressible states. c, Conductance under the same conditions. The absence of a detectable change in capacitance, even as the two-terminal conductance undergoes a transition from an insulating state to a metallic state (Extended Data Fig. 6), suggests that the conductance transition is due to the emergence of gapless edge states.

  2. Nonlocal two-terminal transport in the QSH regime.
    Figure 2: Nonlocal two-terminal transport in the QSH regime.

    a, Schematic diagram of four distinct two-terminal measurement topologies available in a four-terminal device. Open circles indicate floating contacts whereas filled, coloured circles indicate measurement contacts. Each variation probes two parallel conductance paths between the measurement contacts with a variable number of segments on each path, indicated by black edges. b, Two-terminal conductance measurements of deviceA for B = 1.4T, colour-coded to match the four different measurement configurations. Dashed curves correspond to BT = 1.4T; solid curves correspond to BT = 34.5T (QSH regime). In the QSH regime, Gcnp depends strongly on the number of floating contacts (see Extended Data Fig. 4 for similar data for deviceC). Inset, atomic force microscope (AFM) phase micrograph of deviceA; scale bar, 1μm. c, Gcnp for eighteen different contact configurations based on cyclic permutations of the topologies shown in a. Data are plotted against two model fits. In a numerical simulation based on a diffusive model (black circles), the graphene flake was assumed to be a bulk conductor with the conductivity left as a fitting parameter (σ = 3.25e2/h for the best fit). The QSH model (red circles) is equation(1) and has no fitting parameters. The dashed line indicates a perfect fit of data to model. We note that the measured Gcnp never reaches the value predicted by the QSH model, indicating either contact resistance or finite backscattering between the helical edge states. d, Schematic diagram of bulk order and edge-state spin texture in the fully polarized QSH regime. Arrows indicate the projection of the electron spin on a particular sublattice, with the two sublattices indicated by open and filled circles. The edge-state wavefunctions are evenly distributed on the two sublattices and have opposite spin polarizations, at least for an idealized armchair edge14.

  3. Symmetry-driven quantum phase transition.
    Figure 3: Symmetry-driven quantum phase transition.

    a, Capacitance (top) and conductance (bottom) of deviceA at B = 1.1T. The central dip in capacitance does not change with BT at any point during the transition, implying that the bulk gap does not close. b, Bulk spin order in the three transition regimes. The balls and arrows are respectively schematic representations of the spin and sublattice textures of the ground-state wavefunctions and do not represent individual electrons; the electron density within the zLL at ν = 0 is two electrons per cyclotron guiding centre. Insets, details of the relative alignment of the electron spins on the two sublattices. At large BT, the bulk electron spins are aligned with the field (top panel), resulting in an emergent U(1) spin-rotation symmetry in the plane perpendicular to BT. As the total magnetic field is reduced below some critical value (with B held constant), the spins on opposite sublattices cant with respect to each other while maintaining a net polarization in the direction of BT (middle panel). This state spontaneously breaks the U(1) symmetry, rendering local rotations of the electron spins energetically costly. For pure perpendicular fields (bottom panel), the valley isospin anisotropy energy overwhelms the Zeeman energy and the canting angle, θ, is close to 90°, defining a state with antiferromagnetic order. c, Low-energy band structure in the three phases14. ε is the energy and x is the in-plane coordinate perpendicular to the physical edge of the sample. The intermediate CAF phase smoothly interpolates between the gapless edge states of the QSH phase (top panel; FM, ferromagnetic) and the gapped edge of the perpendicular-field phase (bottom panel; AF, antiferromagnetic) without closing the bulk gap. Colour indicates the spin texture of the bands projected onto the magnetic field direction: red, aligned; blue, antialigned; black, zero net spin along the field direction.

  4. Spin-textured edge states of the CAF phase.
    Figure 4: Spin-textured edge states of the CAF phase.

    a, Temperature dependence in the intermediate-field regime for deviceC at B = 5.9T and BT = 45.0T. The conductance peaks shows a metallic temperature coefficient, whereas the state at charge neutrality remains insulating. b, Non-local two-terminal conductance of deviceA at B = 1.6T and BT = 26.1T. Colour-coding indicates contact geometry following the scheme in Fig. 2a. The height of the conductance peaks depends strongly on the configuration of floating contacts, indicating their origin in the gapped, counterpropagating edge states of the CAF phase. c, Schematic band diagram, including spin order, of the CAF edge states. For the electron and hole bands nearest to zero energy, the canting angle inverts near the sample edge, leading to counterpropagating edge states with inverted CAF spin texture. The dashed grey line indicates the Fermi energy, εF, in the regime corresponding to one of the conductance peaks. d, Schematic of bulk order and edge-state spin texture in the CAF regime, following the convention of Fig. 2d. e, Differential conductance, dI/dVSD, of deviceC in the CAF regime (B = 5.9T, BT = 45.0T) in units of e2/h. A constant source–drain voltage, VSD, along with a 100-μV, 313-Hz excitation voltage, are applied to one contact and the a.c. current is measured through the second, grounded contact. f, dI/dVSD of deviceC in the QSH regime (B = 2.7T, BT = 45.0T) in units of e2/h. In both e and f, a symmetry is observed on reversing both VSD and carrier polarity.

  5. Images of measured devices.
    Extended Data Fig. 1: Images of measured devices.

    False-colour AFM images of the devices enumerated in Extended Data Table 1. Dashed lines outline the graphene boundary. Black scale bars correspond to 1μm.

  6. Conductance as a function of B[perp], BT and gate voltage for devices[thinsp]A, B and C.
    Extended Data Fig. 2: Conductance as a function of B, BT and gate voltage for devicesA, B and C.

    Colouring of lines from blue to red indicates increasing BT, with B as indicated at the top of each panel.

  7. Gcnp as a function of B[perp] and BT for devices[thinsp]A, B and E.
    Extended Data Fig. 3: Gcnp as a function of B and BT for devicesA, B and E.

    Correspondingly higher values of BT are required to induce the transition for higher values of B. For deviceA, the curves correspond to B = 0.75, 1, 1.2, 1.4, 1.6, 2, 2.5, 3, 3.5 and 4T (blue to red). For deviceB, the curves correspond to B = 1, 1.5, 2, 2.5, 3 and 3.5T (blue to red). For deviceE, the curves correspond to B = 0.9, 1, 1.1, 1.2, 1.3, 1.4 and 1.5T (blue to red).

  8. Nonlocal measurements for device[thinsp]C in the QSH and CAF regimes.
    Extended Data Fig. 4: Nonlocal measurements for deviceC in the QSH and CAF regimes.

    In the main text, evidence for conduction via edge states in the QSH and CAF regimes is provided by non-local transport measurements in deviceA (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 deviceC. a, Schematic of distinct two-terminal measurement topologies with different numbers of floating contacts (open circles). b, QSH regime, B = 2.7T and BT = 45T. c, CAF regime, B = 5.9T and BT = 45T. 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.

  9. Double conductance peaks in seven different devices.
    Extended Data Fig. 5: Double conductance peaks in seven different devices.

    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.

  10. Temperature dependence of the charge-neutrality point conductivity for device[thinsp]B.
    Extended Data Fig. 6: Temperature dependence of the charge-neutrality point conductivity for deviceB.

    ac, Gate sweeps for deviceB at constant B = 2.5T and BT = 2.5 (a), 26.5 (b) and 34.5T (c). d, Conductance at the charge-neutrality point as a function of temperature for the data in ac. 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.

  11. Schematic diagram of the capacitance bridge-on-a-chip in a tilted magnetic field.
    Extended Data Fig. 7: Schematic diagram of the capacitance bridge-on-a-chip in a tilted magnetic field.

    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 (~3pF), this sets the low-frequency cut-off for the measurement at ~1kHz. 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 VsVG. 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.

  12. Tilted-field magnetotransport in zero-field insulating monolayer graphene.
    Extended Data Fig. 8: Tilted-field magnetotransport in zero-field insulating monolayer graphene.

    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 825kΩ at the charge-neutrality point in zero magnetic field and at T = 0.3K. 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 3T and BT = B. Dashed lines are for the corresponding sequence with B = 1, 2 and 3T but BT = 45T. Data taken at 0.3K. Semiconducting graphene samples do not show any sign of QSH-type physics, at least up to 45T. Even for B = 1T and BT = 45T, 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 = 45T. This is understandable, because, even neglecting interaction effects, closing a moiré-induced band gap of Δ = 10meV requires a Zeeman field of nearly Δ/gμB85T. We note that in these samples, the ground state at BT = B may not be an antiferromagnet.

Tables

  1. Physical parameters of measured devices
    Extended Data Table 1: Physical parameters of measured devices

References

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Author information

  1. These authors contributed equally to this work.

    • A. F. Young,
    • J. D. Sanchez-Yamagishi &
    • B. Hunt

Affiliations

  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • A. F. Young,
    • J. D. Sanchez-Yamagishi,
    • B. Hunt,
    • S. H. Choi,
    • R. C. Ashoori &
    • P. Jarillo-Herrero
  2. Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

    • K. Watanabe &
    • T. Taniguchi

Contributions

A.F.Y. and J.D.S.-Y. had the idea for the experiment. J.D.S.-Y. and S.H.C. fabricated the samples. A.F.Y., J.D.S.-Y. and B.H. performed the experiments, analysed the data and wrote the paper. T.T. and K.W. grew the crystals of hexagonal boron nitride. R.C.A. and P.J.-H. advised on experiments, data analysis and writing the paper.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

Author details

Extended data figures and tables

Extended Data Figures

  1. 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 1μm.

  2. Extended Data Figure 2: Conductance as a function of B, BT and gate voltage for devicesA, 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.

  3. Extended Data Figure 3: Gcnp as a function of B and BT for devicesA, B and E. (733 KB)

    Correspondingly higher values of BT are required to induce the transition for higher values of B. For deviceA, the curves correspond to B = 0.75, 1, 1.2, 1.4, 1.6, 2, 2.5, 3, 3.5 and 4T (blue to red). For deviceB, the curves correspond to B = 1, 1.5, 2, 2.5, 3 and 3.5T (blue to red). For deviceE, the curves correspond to B = 0.9, 1, 1.1, 1.2, 1.3, 1.4 and 1.5T (blue to red).

  4. Extended Data Figure 4: Nonlocal measurements for deviceC 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 deviceA (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 deviceC. a, Schematic of distinct two-terminal measurement topologies with different numbers of floating contacts (open circles). b, QSH regime, B = 2.7T and BT = 45T. c, CAF regime, B = 5.9T and BT = 45T. 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.

  5. 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.

  6. Extended Data Figure 6: Temperature dependence of the charge-neutrality point conductivity for deviceB. (153 KB)

    ac, Gate sweeps for deviceB at constant B = 2.5T and BT = 2.5 (a), 26.5 (b) and 34.5T (c). d, Conductance at the charge-neutrality point as a function of temperature for the data in ac. 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.

  7. 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 (~3pF), this sets the low-frequency cut-off for the measurement at ~1kHz. 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 VsVG. 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.

  8. 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 825kΩ at the charge-neutrality point in zero magnetic field and at T = 0.3K. 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 3T and BT = B. Dashed lines are for the corresponding sequence with B = 1, 2 and 3T but BT = 45T. Data taken at 0.3K. Semiconducting graphene samples do not show any sign of QSH-type physics, at least up to 45T. Even for B = 1T and BT = 45T, 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 = 45T. This is understandable, because, even neglecting interaction effects, closing a moiré-induced band gap of Δ = 10meV requires a Zeeman field of nearly Δ/gμB85T. We note that in these samples, the ground state at BT = B may not be an antiferromagnet.

Extended Data Tables

  1. Extended Data Table 1: Physical parameters of measured devices (92 KB)

Additional data