Main

Under a strong perpendicular magnetic field, bilayer graphene (BLG) develops a ν = 0 quantum Hall state at the charge neutrality point (CNP) that displays anomalous insulating behaviour8,9,10,11,12,13,14. Transport studies in a dual-gated geometry10,11,12 indicate that this gapped state results from an interaction-driven spontaneous symmetry breaking in the valley-spin space15. However, the exact order of the resulting ground state remains controversial. In high-mobility suspended BLG devices, a broken-symmetry state at charge neutrality has also been observed at zero magnetic field whose nature has been under intense theoretical16,17,18,19,20,21,22 and experimental10,12,13,14,23 investigation. Some experiments indicate there may be a connection between these two insulating states12,14.

The ν = 0 state in BLG occurs at half-filling of the lowest Landau level. Although the BLG zero-energy Landau level has an additional orbital degeneracy stemming from two accidentally degenerate magnetic oscillator wavefunctions24, minimization of exchange energy favours singlet pairs in the orbital space at all even filling factors25. This leaves an approximate SU(4) spin and pseudospin symmetry for the remaining ordering of the ground state. Comparable ground-state competition has also been studied in double quantum wells in GaAs two-dimensional electron systems26. Quantum phase transitions, in particular from canted antiferromagnetic to ferromagnetic ordering, can occur in those systems as an in-plane magnetic field is applied (Zeeman energy is increased)27,28.

Our experiment is carried out in BLG samples with top and bottom gates in which thin single-crystal hexagonal boron nitride (hBN) serves as a high-quality dielectric on both sides (Fig. 1 bottom inset). By controlling the top gate voltage (V TG) and the bottom gate voltage (V BG) we can adjust the carrier density n and the perpendicular electric displacement field D independently. Furthermore, tilting the sample in the magnetic field allows us to independently control the Coulomb energy in a Landau level, e2/ℓB, where the magnetic length ℓB is determined by the perpendicular magnetic field , and Zeeman energy (determined by the total magnetic field Btot). In BLG, Btot and D separately couple to spin and pseudospin, respectively, so our experimental set-up allows us to tune anisotropies and characterize the broken-symmetry quantum Hall states in the approximate SU(4) spin-pseudospin space (see Supplementary Information).

Figure 1: Device summary.
figure 1

Plot of longitudinal resistance (Rx x, colour scale) versus the top gate voltage V TG and the bottom gate voltage V BG at . The data were taken at 1.5 K. The dashed line indicates the CNP. The top right inset shows four-terminal conductance at the CNP plotted against and D. Labels (i) and (ii) indicate the two different insulating ν = 0 phases. The dashed line in the inset corresponds to the dashed line in the main plot. The bottom left inset shows an optical image of the device used in this experiment with false colouring of the graphene (pink).

Full size image

Figure 1 shows the resistance at measured as a function of the two gate voltages. At zero magnetic field, the resistance at the CNP depends monotonically on |D| owing to the opening of a single particle energy gap24,29. With increasing applied magnetic field (Fig. 1 top inset) our device shows the same behaviour reported elsewhere10,11,12, with a layer-polarized insulator (which we denote phase (i)) observed at large applied displacement field, separated from a low displacement field and high magnetic field quantum Hall insulator (phase (ii)) by a well-defined boundary of finite conductance. Previous studies have reported slopes for this phase boundary in the range of 11–18 mV nm−1 T−1 (refs 10, 11, 12). We obtain values of 7 and 10 mV nm−1 T−1 for the electron and hole sides, respectively, in reasonable agreement with these previous results. For the remainder of the paper we focus on the behaviour of phase (ii) as we vary the relative Zeeman versus Coulomb energy scales in a rotated field geometry.

Figure 2a shows a series of conductance maps in the vicinity of the CNP. Each pane corresponds to a fixed perpendicular applied magnetic field of 1.75 T, but with varying total field. At a total field of 1.75 T (that is, ), the distinct insulating regimes at low and high |D| are clearly visible, as discussed in Fig. 1. At low |D| the four-terminal conductance increases markedly with increasing total magnetic field Btot, reaching a value of around 3–4e2/h, after which it becomes relatively insensitive to either displacement field or magnetic field (Fig. 2b). This rapidly increasing behaviour of conductance at fixed is consistent with the gradual closing of a gap in phase (ii) as Btot increases. At fixed Coulomb energy, this decreasing gap with increasing Zeeman energy suggests that phase (ii) corresponds to a spin-unpolarized ground state7,8,9,13. In addition, the saturation of conductance at a metallic value around ∼4e2/h at Btot≳15 T strongly suggests a transition into a new phase with a fully closed transport gap, which we label phase (iii).

Figure 2: Tilted-field transport.
figure 2

a, 1/Rx x plotted against carrier density, displacement field and total magnetic field with fixed at 1.75 T. Labels (i) and (ii) mark the same phases identified in Fig. 1, and (iii) marks the new conducting phase that appears at large Btot. The data were taken at 350 mK. b, Four-terminal conductance at the CNP (G = 1/Rx x) plotted against displacement field D for a variety of total magnetic fields, fixed at 1.75 T. The dashed line indicates the zero displacement field points plotted in c. The solid line marks the theoretically expected 4e2/h conductance for the ferromagnetic phase. The data were taken at 350 mK. c,d, Temperature-dependent four-terminal conductance at the CNP plotted against displacement field, with , Btot = 2.5 T (c), 13.1 T (d). The coloured curves were taken at 3 K, black curves at 18 K, and intermediate traces are spaced by 1.5 K. The dashed (solid) vertical lines at a fixed displacement field correspond to the dashed (solid) lines in e. e, Four-terminal conductance at the CNP for different total magnetic fields plotted against temperature, all at . The solid lines demonstrate how the behaviour changes as the total field is increased for zero displacement field. The dashed lines demonstrate similar behaviour for all total magnetic fields at D = 0.085 V nm−1.

Full size image

The transition between insulating and metallic behaviours can further be confirmed by investigating the temperature dependence of the conductance. Figure 2c,d plots four-terminal conductance G at the CNP against D at fixed with Btot = 2.5 Tand 13.1 T, respectively. Temperature T varies between 3 and 18 K. At small |D|, G exhibits weakly insulating behaviour for (see Supplementary Information), whereas at , a metallic temperature dependence (increasing G with decreasing T) is observed. Figure 2e further contrasts the metallic versus insulating classification in different phases by plotting conductance at the CNP against temperature for both large D and D = 0. G(T) shows insulating behaviour for phase (ii) () and metallic behaviour for phase (iii) (). At intermediate values of Btot the conductance is relatively temperature insensitive, suggesting a gradual crossover between the two phases. Interestingly, the cusps of high conductance that mark the transition between the layer-polarized and non-layer-polarized phase show a metallic temperature dependence even in a fully perpendicular field. At high D (that is in phase (i)), the conductance remains relatively insensitive to the applied in-plane magnetic field. This is expected for a layer-polarized ν = 0 quantum Hall state, where the energy gap is determined by the applied displacement field.

A general theoretical analysis7 proposes four possible ground states for ν = 0 in BLG: partially layer-polarized (PLP), fully layer-polarized (FLP), canted antiferromagnetic (CAF) or ferromagnetic. Considering that layer pseudospin is synonymous with sublattice and valley pseudospin in the lowest Landau level24, these four different scenarios can be visualized in terms of spin and sublattice of BLG (Fig. 3a). Which state the system favours depends on both anisotropies in the electron–electron and electron–phonon interactions, and external symmetry-breaking terms such as Zeeman splitting εZ and interlayer potential energy εV. The PLP/FLP phases are favoured in the large- εV limit, and the ferromagnetic phase is favoured in the large- εZ limit. Unlike in GaAs double wells, these ground states may be accompanied by markedly different transport signatures. The BLG ground-state configurations discussed above are all predicted to be insulators except for the ferromagnetic phase. For the ferromagnetic phase, the highest filled band acquires an electron-like dispersion at the edge whereas the lowest unoccupied band acquires a hole-like dispersion (Fig. 3b; ref. 6). This forces a level crossing at the edge that should result in metallic conductance of 4e2/h due to spin-polarized counter-propagating edge states4,5,6. Interpreting our results in this framework, we can identify phase (i) with either the PLP or FLP state, phase (ii) with the CAF state, and phase (iii) with the ferromagnetic state.

Figure 3: Candidate states.
figure 3

a, Spin and pseudospin configurations in the BLG ν = 0 state. The illustrations represent the layer-polarized (i), CAF (ii) and ferromagnetic (iii) phases. b, Diagram of predicted bulk and edge energy levels for the CAF (ii) and ferromagnetic (iii) phases.

Full size image

We now focus on the phase diagram of the different ground states classified above. The phase transitions between the layer-polarized, CAF, and ferromagnetic phases can be mapped out by locating the boundary between the metallic (G≳e2/h) and insulating (G≪e2/h) regimes as a function of D and Btot with fixed . For a given Btot, we can define the critical displacement field for transition Dc as the D value that gives maximum change of ∂ G/∂ D in G(D,Btot) at fixed . Figure 4a shows the values of Dc and Btot extracted from Fig. 2b ( T). In this D–Btot map three different phases can be identified, corresponding to the layer-polarized, CAF and ferromagnetic phases. Note that the transition between the CAF and layer-polarized phases is sharp, as the conductance changes by an order of magnitude over a D variation ΔD≲0.02 V nm−1. In contrast the transition from CAF to ferromagnetic is gradual. This observation is consistent with the theoretical expectation that the layer-polarized to CAF or ferromagnetic transition is first order whereas the CAF to ferromagnetic transition is second order7. Further experiments on hysteresis behaviour across the phase boundaries would help to clarify the nature of these phase transitions.

Figure 4: Phase transitions.
figure 4

a, Different phases of the ν = 0state mapped as a function of D and Btot at . At low Btot, the phase boundary between layer-polarized and CAF is determined by the conductance spike separating the two insulating phases. At large Btot, the phase boundary between layer-polarized and ferromagnetic is determined by the point where conductance begins to exponentially decrease with displacement field. The gradual transition between CAF and ferromagnetic with increasing Btot is marked by the gradual colour change, with ferromagnetic stabilizing at high Btot. Error bars are due to the spacing between adjacent points in D. b, Four-terminal conductance at the CNP with D = 0, plotted against Btot for three different values of . Dashed lines indicate regions of exponential increase of conductance and the dotted line marks saturated conductance. The critical magnetic field Btotc for the phase transition is marked by arrows. The inset shows Btotc as a function of separating the CAF and ferromagnetic phases in a Btot versus parameter space. Error bars are due to uncertainties in fitting guide lines.

Full size image

By identifying characteristic energy scales for the external symmetry-breaking terms, we can comment quantitatively on the phase boundaries for a fixed . We identify Zeeman energy εZ = g μBBtot, where g and μB are the electron spin g-factor and the Bohr magneton respectively, and the interlayer potential energy εV = D d e/2, where d is the interlayer distance in BLG. For , at low Btot the CAF–layer-polarized transition occurs at εV≈5 meV, and at εV = 0 the CAF–ferromagnet transition occurs at εZ≈0.8 meV. The ferromagnet–layer-polarized transition occurs along a line whose slope εV/εZ≈2. The fact that εV/εZ at the ferromagnet–layer-polarized phase boundary is of order unity is consistent with the increasing stability of a spin-polarized state with increasing magnetic field, providing further confirmation that the conducting ν = 0 phase is a ferromagnet. This observation is in qualitative agreement with theoretical predictions7.

At the CNP and D = 0, is the only physical quantity to control the strength of electron–electron interactions, so the phase boundary between the CAF and ferromagnetic phases can be further investigated by observing G as a function of Btot with different values of fixed (see Supplementary Information). Figure 4b shows G(Btot) for three different fixed values of at D = 0. Generally, the behaviour of G can be characterized by two different regimes. For (that is, the CAF phase), G increases exponentially as Btot increases. For (that is, the ferromagnetic phase), G is insensitive to Btot, saturating close to 4e2/h. We define the critical magnetic field Btotc that separates these two regimes by the intersection of guide lines for the exponential and saturated regimes, as marked by the arrows in Fig. 3b. The identification of Btotc at different allows us to map the phase boundary between the CAF and ferromagnetic as a function of . The inset of Fig. 4b shows Btotc as a function of . Although there are only a few data points available to indicate the phase boundary, the general trend of the CAF–ferromagnet phase transition can be inferred from them.

Finally, we remark on the similarity between the ferromagnetic phase of BLG at ν = 0 and the quantum spin Hall effect in HgTe (refs 2, 3). In the quantum spin Hall phase, quantized conduction is due to spin-polarized counter-propagating edge states, whereas the bulk remains incompressible. An analogous scenario is expected for BLG in the ferromagnetic quantum Hall state4,5,6, except that BLG carries an additional orbital degeneracy. The ferromagnetic state in BLG is therefore expected to exhibit a four-terminal conductance of 4e2/h, compared with 2e2/h for HgTe. Consistent with reported measurements of HgTe (refs 3, 30), our devices show imperfect quantization with measured conductance values less than 4e2/h in the ferromagnetic phase. The reduced conductance may be due to backscattering in the edge state or through remaining impurity states in the bulk. Unambiguous experimental support for the existence of spin-polarized counter-propagating edge states should be provided by well-quantized nonlocal measurements30 (see Supplementary Information).

We have measured dual-gated graphene bilayers at the ν = 0 state. In a perpendicular magnetic field, we observe a transition between two incompressible states at a finite displacement field, indicating that the D = 0 state is not layer polarized. At low displacement fields, we observe the four-terminal conductance increase and then saturate as the in-plane magnetic field increases, showing that the D = 0, state is also not spin polarized. This is consistent with a quantum phase transition driven by in-plane magnetic field between the CAF quantum Hall state and the ferromagnet. The near quantization of the four-terminal conductance to 4e2/h agrees with predictions that the ferromagnetic quantum Hall ground state carries spin-polarized counter-propagating edge states.

Methods

The device we report data from was fabricated with mechanically exfoliated graphene and hBN using a polymer membrane transfer process described in previous work31. The graphene was etched into a Hall bar pattern before deposition of the top hBN layer. The bottom gate is a global silicon gate, and both the contacts and the top gate are Cr/Pd/Au stacks of thickness 1/10/50 nm. Three devices fabricated in a similar way exhibited similar data trends, although the phase boundaries may be quantitatively sample- and disorder-dependent (see Supplementary Information). We present data sets obtained from the highest quality device.

These devices were measured either in a sample-in-4He-vapour variable temperature cryostat with a 14 T magnet in-house or in a sample-in-3He-vapour cryostat, mounted in the bore of a 31 T resistive magnet at the National High Magnetic Field Laboratory in Tallahassee, Florida. Measurements were performed using a 1 mV voltage bias. The numerous features present in a gate voltage trace allowed precise angle calibration, with determined to better than 1% accuracy. The longitudinal resistance Rx x and Hall resistance Rx y are measured in a four-terminal geometry, so that contact resistances can be excluded. Four-terminal conductance at the CNP is obtained by taking G = 1/Rx x.