Competing ordered states with filling factor two in bilayer graphene

Abstract

The quantum Hall effect, in which a two-dimensional sample’s Hall conductivities become quantized, is a remarkable transport anomaly commonly observed at strong magnetic fields. However, it may also appear at zero magnetic field if time-reversal symmetry is broken. Charge-neutral bilayer graphene is unstable to a variety of competing and closely related broken symmetry states, some of which have non-zero quantized Hall conductivities. Here we explore those states by stabilizing them with external fields. Transport spectroscopy measurements reveal two distinct states that have two quantum units of Hall conductivity, stabilized by large magnetic and electric fields, respectively. The majority spins of both phases form a quantum anomalous Hall state, and the minority spins constitute a Kekulé state with spontaneous valley coherence for phase I and a quantum valley Hall state for phase II. Our results shed light on the rich set of competing ordered states in bilayer graphene.

Introduction

Because of its large density-of-states and the 2π Berry phase near its low-energy band-contact points^1,2,3,4,5,6, neutral bilayer graphene (BLG) at zero magnetic field (B) is susceptible to chiral-symmetry breaking, leading to several gapped spontaneous quantum Hall (QH) states distinguished by valley- and spin-dependent quantized Hall conductivities^{7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34}. In these states, each spin-valley flavour spontaneously transfers charge between layers^10,11. When spin is ignored, only two classes of gapped states can be distinguished: (i) ones in which electrons in opposite valleys are polarized to opposite layers, producing a quantum anomalous Hall (QAH) state with broken time-reversal symmetry and orbital magnetization^7,8,9, and (ii) states in which electrons in opposite valleys are polarized to the same layer, yielding a quantum valley Hall (QVH) state with broken inversion symmetry and Hall conductivity, σ_H=0 (refs 7, 8, 9). For spinful electrons, symmetry-broken states further include the layer antiferromagnetic state (LAF) and the quantum spin Hall state with two-terminal conductivity 0 and 4e²/h, respectively. An additional competing state is the Kekulé order with spontaneous coherence between valleys. Electronic configurations of these states are summarized in Table 1.

Table 1 Schematic diagram of electronic configurations of the states stabilized by electric field and magnetic field.

So far, only the LAF state, which has QVH states with opposite layer polarization signs for opposite spins⁸ and appears to be the thermodynamic ground state^35,36,37, and (at finite interlayer electric field E_⊥) the layer polarized state, which has QVH states with spin-independent layer polarization, have been observed experimentally^35,36,37,38. Other spontaneous QH states, and competing states with Kekulé order, have eluded experimental observation (although prior work on singly gated devices has shown that the QH states at filling factors ν=4 and ν=0 can persist to very small B³⁹). The experiments we describe are motivated by the sensitivity of the competing orders in BLG to both E_⊥ and B fields.

Spontaneous Hall states can be explored by taking advantage of their adiabatic connection to QH states at integer filling factors within BLG’s eightfold degenerate lowest Landau levels (LLs). Our prior work⁴⁰ has shown that the LL gaps in dual-gated devices with controlled layer polarization are different from those in singly gated devices. Here we use transport spectroscopy^35,36,40 to study high-quality dual-gated suspended BLG devices at ν=±2 (refs 36, 38, 39, 41, 42, 43, 44, 45, 46) as a function of B and E_⊥. Our results reveal two distinct states: (1) phase I is fully resolved only near E_⊥=0 and large B, and has a relatively small gap that extrapolates to 0 at B=0. This phase is expected⁸ to be spontaneously spin-polarized and to have coherence between valleys (layers), that is, with Kekulé valley order^47,48,49, (2) phase II appears only at finite E_⊥, but can be stabilized at a much smaller B and has a much larger gap with a finite B=0 intercept, suggesting that this state survives to anomalously weak B. Because finite B couples to the spin and orbital moments, and finite E_⊥ favours layer (valley) polarization, this state is expected⁸ to form a majority-spin QAH state but a minority-spin QVH state, where majority (minority) spins refer to those that occupy all four (half) of the orbital-degenerated LLs. It follows that its charge, spin and valley Hall conductivities are all quantized to 2e²/h. The anomalous stability of phase II at small B is consistent with the stability of QVH (QAH) state at large E_⊥ (finite B). Our results represent the first spectroscopic mapping of the exotic behaviour hosted in BLG at filling factor ν=2. By shedding light on the competition between ordered states in BLG, we provide a road map for studying the spontaneous symmetry breaking in BLG and its thicker chiral cousins^8,50,51,52.

Results

Realization of two distinct ν=±2 QH states

Suspended dual-gated devices (Fig. 1) with mobilities as high as 150,000 cm² V⁻¹ s⁻¹ were measured in He-3 refrigerators. Here we present data from two different devices (devices 1 and 2) with field effect mobilities 80,000 and 40,000 cm² V⁻¹ s⁻¹, respectively. (Similar data are observed in two additional devices). All measurements were taken at temperature T=260 mK.

Figure 1: SEM image of a dual-gated device.

False colored SEM image of a device with suspended bilayer graphene (deep pink), clamped on by source-drain electrodes (cobalt blue) and straddled from above by a suspended top gate (bright pink).

Figure 2a plots the two-terminal differential conductance G of device 1 in units of e²/h at B=3.5 T as a function of charge density n and out-of-plane electric field E_⊥. The QH plateaus at ν=0, ±1, ±2 and ±4 appear as dark blue, blue, white and red colour bands, respectively. Most interestingly, the resolution of ν=±2 QH states depends on E_⊥. This can be seen in the G(ν) traces in Fig. 2b, near E_⊥=0, only the QH plateaus at ν=0 and ±4 are fully resolved; in contrast, at larger E_⊥=−21 mV nm⁻¹, the ν=±2 plateaus are clearly visible. The resolution of the ν=±2 states is abrupt: at ν=2, G( E_⊥) stays at 4e²/h for small E_⊥, but drops sharply to a 2e²/h plateau at a well-defined critical value E_⊥c. This critical E_⊥c is ~10 mV nm⁻¹ at 8 T, and is only weakly dependent on B, with a slope ~0.72 mV nm⁻¹ T⁻¹ (Fig. 2c).

Figure 2: Magnetotransport data.

(a) G( E_⊥, n) of device 1 at B=3.5T. (b) Line traces G(ν) at B=3.5 T and E_⊥=0 (brown dotted line) and E_⊥=−21 mV nm⁻¹ (purple solid line), respectively. (c) Line trace G(E_⊥) at B=3.5 T and filling factor ν=2 (density n=1.7 × 10¹¹ cm⁻²). Data in a–c are from device 1. Inset: critical electric field E_c versus B and linear fit to the data points with slope ~0.72 mV nm⁻¹ T⁻¹. (d) G(ν) of device 2 at E_⊥=0, and B=10 and 24 T, respectively.

The above observations reflect unprecedented sample quality, but agree with previous studies in which the ν=2 state is fully resolved only when E_⊥ is non-zero either in a controlled fashion in dual-gated devices^36,38, or inadvertently in singly gated devices^35,37,43,45. This suggests that the ν=2 state observed at B=3.5 T is layer polarized. The intriguing possibility of a ν=2 state at E_⊥=0 in higher quality samples or in stronger fields has not been demonstrated previously. Figure 2d displays G(ν) for device 2. As in the case of device 1, for B<20 T, the ν=2 state is resolved only in the presence of finite E_⊥. At E_⊥=0, the state is fully resolved for B>24 T.

Characterization of the two distinct ν=±2 QH states

The experimental observation presented above allows us to identify the nature of these two distinct ν=2 QH states. Phase I is only fully resolved at E_⊥=0 and large B, and phase II is resolved at large E_⊥ and relatively small B. To further explore these two distinct ν=2 QH states, we perform transport spectroscopy by using the source-drain bias V as a spectroscopic tool^35,40. The details of this measurement technique and its application to QH states can be found in refs 36, 40. Briefly, at each conductance plateau, BLG’s Fermi level is pinned between the highest filled and the next unfilled LLs. Charges are carried by edge states that are separated from the gapped bulk by a gap on the order of the LL spacing. Increasing bias aligns the source electrode’s Fermi level with the next unfilled LL and allows additional charge transport through the bulk. Consequently, near V=0, the device displays a conductance valley, whose half-width yields the LL gap Δ. Figure 3a,b displays G(V,ν) from device 1 at B=3.5 T and E_⊥=0 and −14.4 mV nm⁻¹, respectively. At E_⊥=−14.4 mV nm⁻¹, the diamond at ν=−2 is significantly larger than that at E_⊥=0, demonstrating an enlarged LL gap. Such difference in gap sizes can be seen more clearly in line traces in Fig. 3c, which are taken from Fig. 3a,b at ν=−2, respectively. Similar traces at B=6 T are shown in Fig. 3d. At E_⊥=−14.4 mV nm⁻¹, prominent zero-bias valleys appear in G(V) traces (Fig. 3c,d, red lines), indicating that the ν=−2 state (phase II) is fully resolved. In contrast, at E_⊥=0, the G(V) traces display only small conductance dips superimposed on a peak at zero bias (Fig. 3c,d, blue traces), suggesting that phase I is only partially resolved.

Figure 3: Transport spectroscopy data from device 1.

(a,b) G(V,n) at E_⊥=0 and −14.4 mV nm⁻¹, respectively. Data are taken at B=3.5 T. (c) Line traces G(V) at ν=−2 and B=3.5 T. Blue and red traces are taken at E_⊥=0 and −14.4 mV nm⁻¹, respectively. (d) Similar data taken at B=6 T. (e) Measured LL gap Δ( B) at ν=2 and E_⊥=0 and −14.4 mV nm⁻¹, respectively. Error bars are estimated from bias resolution and uncertainties in the valley width measurement. (f) Schematics of transitions between phases I and II at ν=2. (T/B, top/bottom layer; S, symmetric state |S>:=|T>+|B>; |AS>: antisymmetric state |T>−|B>).

The measured values of Δ_ν=−2 are shown in Fig. 3e. At E_⊥=0, the LL gap for phase I is Δ_I ~0.17 meV T⁻¹ with a zero intercept at B=0; in contrast, at E_⊥=−14.4 mV nm⁻¹, the gap for phase II is larger by more than a factor of 5, Δ_II ~1.0 meV T⁻¹, with a non-vanishing B=0 intercept.

Discussion

We now discuss the nature of these two distinct ν=2 QH states. Phase I is only fully resolved at E_⊥=0 and large B, with a vanishing LL gap at B=0. The fact that a large B is required to stabilize this phase has been anticipated in BLG QH ferromagnetism theory^5,8,22. In the absence of interactions, the zero-energy LLs are eightfold degenerate and QH effects only exhibit at ν=±4; thus interactions are necessary to induce phase I. Since the valley K and K′ states in the N=0 subspace are localized at the top and bottom layers, respectively, the valley pseudospin and layer pseudospin coincide. At E_⊥=0, there is an electrostatic energy cost of valley (or equivalently layer) polarization, and it is energetically favourable to achieve a gap at ν=2 by establishing coherence between the valleys/layers. This is what is observed experimentally: as phase I is only observed in the absence of E_⊥, it is evidently not layer polarized but a symmetric linear combination of the top and bottom layers. Such a state is also a valley symmetric state, that is, a valley-Kekulé order in the presence of B due to electron–electron interactions. This Kekulé order appears at large B, even though it is absent at B=0. Accordingly, our experiments demonstrate that the stability of the Kekulé state, as measured by its gaps, disappears as B goes to zero.

The valley-Kekulé order that we have observed is reminiscent of the Kekulé lattice distortion proposed for carbon nanotubes and graphene^47,48,49, albeit here it only exists in the presence of a magnetic field. As B approaches 0, a spontaneous QH state⁸ emerges as the ground state, as observed previously³⁶. In the absence of B fields, the intervalley interactions in BLG are too weak to induce any valley coherence. At finite B, however, owing to the equivalency of the layer and valley pseudospins, any interlayer coherence naturally leads to intervalley coherence.

In contrast to phase I, phase II is observed at anomalously small B and large E_⊥, with a LL gap that extrapolates to a finite B=0 intercept. Its appearance at much smaller B than phase I is reminiscent of the spontaneous QH states at B=0 (refs 8, 9). This phase is adiabatically connected to the B=0 collinear ferromagnet ordered state with a majority-spin QAH state and a minority-spin QVH state^6,8,22. Indeed phase II is only metastable^6,8 at B=0, most likely because it loses the ordering competition to the LAF state at E_⊥=0 and to the QVH state at E_⊥≠0. As observed here, however, phase II can be preferred in the presence of finite B and E_⊥, since states with different total Hall conductivity are most stable at different carrier densities; moreover, the energy of this phase is lowered by the orbital and spin coupling to B and by the compensation of the Hartree energy cost of its layer polarization by E_⊥. It is partially polarized in spin, valley and layer, and is consistent with a QH ferromagnet state^5,8,22,43.

In terms of their layer characteristics, phase I has XY valley (layer) coherence order, whereas phase II has layer polarization Ising order. Schematic representations of these two phases are provided in Fig. 3f. Here S and AS indicate symmetric and antisymmetric combination of the K and K′ valleys, respectively. The energy differences between these two phases arise mainly from (i) the Hartree energy of layer polarization, and (ii) the difference between interlayer and intralayer exchange for the occupied N=0 LLs. Both differences scale as (d/l_β) × (e²/l_β) ~ B, in agreement with the weak linear B dependence of E_⊥c in our BLG measurements, where the layer separation d is much smaller than the magnetic length l_β.

We also note that data from singly gated devices are similar to those at finite E_⊥: owing to the presence of an inadvertently induced electric field ne/2ε₀ (here ε₀ is the permittivity of vacuum), the ν=±2 states in singly gated devices are almost always in phase II and stabilized by E_⊥. Thus, care must be taken when interpreting QH data from singly gated devices.

Finally, using transport spectroscopy, we explore the dependence of the LL gap on E_⊥. Figure 4a plots G(V, E_⊥) at ν=2 from device 2. The most striking feature is the red region at the centre of the plot, that is, at small E_⊥, where G~4e²/h, surrounded by blue–white regions, where G~2e²/h. The abrupt transition between the two regions (Fig. 4b) is the same as that observed in Fig. 2c. At E_⊥=0, G(V) displays a narrow conductance dip at V=0, with half-width ~1.6 mV, corresponding to Δ_I (Fig. 4c). At large E_⊥=−35 mV nm⁻¹, the G(V) trace is significantly different, with much wider conductance valley and half-width of 15 mV. We thus take the half-width of the wider valley to be Δ_II. The dependence of Δ_II on E_⊥ is shown in Fig. 4d. It increases with E_⊥ of both polarities, although with slight asymmetry. Interestingly, the wider conductance valley, which appears as the white curves superimposed on top of the blue background, extend into |E_⊥|<E_⊥c, that is, the narrow dip co-exists with wider valley, suggesting co-existence of both states near E_⊥c.

Figure 4: Spectroscopy data from device 2 at B=10 T and ν=2.

(a) G(V, E_⊥) data. (b) Line traces G(V) at E_⊥=0 (blue), −35 (red) and 15 mV nm⁻¹ (green). (c) G(E_⊥) at V=0. (d) Gap of layer polarized ν=2 state versus E_⊥.

In conclusion, we utilized high-quality dual-gated suspended BLG devices to explore the spontaneous symmetry breaking physics near charge neutrality. We resolved two distinct ν=2 QH states: phase I is fully resolved only at E_⊥=0 and large B, and is likely a Kekulé state with interlayer and intervalley coherence, whereas phase II is observed at small B and large E_⊥. Our measurements demonstrate that phase II is metastable at B=0. Our data represent the first spectroscopic mapping of the exotic competing orders at ν=±2 in BLG, and pave the way for studies of other symmetry-broken states at different filling factors in BLG or in the much less-explored TLG^8,53. Our study also motivates future in-plane B measurements^9,54,55 to realize more complete control of spin, layer and orbital degrees of freedom.

Methods

BLG sheets are exfoliated from Kish bulk graphite onto Si/SiO₂ substrates, and identified via colour contrast in an optical microscope and through Raman spectroscopy⁵⁶. They are coupled to Cr (10 nm)/Au (150 nm) electrodes and Cr-suspended top gates (~300 nm above substrate) using a multi-layer lithography technique^57,58, then etched in hydrofluoric acid to remove ~150–250 nm of SiO₂. Devices are current annealed at 260 mK in He-3 refrigerators⁵⁹, and measured using standard lock-in techniques.