## Breaking down graphene degeneracy

Bilayer graphene has two layers of hexagonally arranged carbon atoms stacked on top of each other in a staggered configuration. This spatial arrangement results in degenerate electronic states: distinct states that have the same energy. Interaction between electrons can cause the states to separate in energy, and so can external fields (see the Perspective by LeRoy and Yankowitz). Kou *et al.*, Lee *et al.*, and Maher *et al.* used three distinct experimental setups that clarify different parameter regimes of bilayer graphene.

## Abstract

Symmetry-breaking in a quantum system often leads to complex emergent behavior. In bilayer graphene (BLG), an electric field applied perpendicular to the basal plane breaks the inversion symmetry of the lattice, opening a band gap at the charge neutrality point. In a quantizing magnetic field, electron interactions can cause spontaneous symmetry-breaking within the spin and valley degrees of freedom, resulting in quantum Hall effect (QHE) states with complex order. Here, we report fractional QHE states in BLG that show phase transitions that can be tuned by a transverse electric field. This result provides a model platform with which to study the role of symmetry-breaking in emergent states with topological order.

The fractional quantum Hall effect (FQHE) (*1*) represents a notable example of emergent behavior in which strong Coulomb interactions drive the existence of a correlated many-body state. In conventional III-V heterostructures, the Laughlin wave function (*2*) together with the composite fermion picture (*3*) provides a complete description of the series of FQHE states that have been observed within the lowest Landau level (LL). In higher LLs, many-body states that are not amenable to this description appear, such as the even-denominator 5/2 state (*4*) [presumed to have non-abelian quantum statistics (*5*)] and a variety of charge-density wave states (*6*, *7*).

In graphene (*8*–*14*), the combined spin and valley degrees of freedom are conjectured to yield FQHE states within an approximate SU(4) symmetry space (neglecting relatively weak spin Zeeman and short-range interaction energies). Although phase transitions in FQHE states have been explored for typical semiconductor systems (including valley degenerate systems) (*15*–*20*), the wide gate tunability of graphene systems coupled with large cyclotron energies allows for the exploration of multiple different SU(4) order parameters for a large range of filling fractions. In bilayer graphene (BLG), coupling to electric and magnetic fields can force transitions between different spin and valley orderings, including different FQHE phases (*21*–*28*). Thus, BLG provides a model system particularly well suited to experimentally study phase transitions between different topologically ordered states.

Whereas the FQHE in monolayer graphene has now been observed in several studies (*8*, *9*, *11*, *12*), including evidence of magnetic-field–induced phase transitions (*29*), achieving the necessary sample quality in BLG has proven challenging (*10*, *13*, *14*). We fabricated BLG devices encapsulated in hexagonal boron nitride using the recently developed van der Waals transfer technique (*30*) [(*31*), section 1.1]. The device geometry includes both a local graphite bottom gate and an aligned metal top gate (Fig. 1A), which allows us to independently control the carrier density in the channel [*n* = (*C*_{TG}*V*_{TG} + *C*_{BG}*V*_{BG})/*e* – *n*_{0}, where *C*_{TG} is the top gate capacitance per area, *C*_{BG} is the bottom gate capacitance per area, *V*_{TG} is the top gate voltage, *V*_{BG} is the bottom gate voltage, *e* is the electron charge, and *n*_{0} is residual doping] and the applied average electric displacement field [*D* = (*C*_{TG}*V*_{TG} – *C*_{BG}*V*_{BG})/2ε_{0} – *D*_{0}, where *D*_{0} is a residual displacement field due to doping]. Crucially, portions of the graphene leads in these devices extend outside of the dual-gated channel. We used the silicon substrate as a third gate in order to set the carrier density of the leads independently (Fig. 1A). Tuning the carrier density in the graphene leads has a dramatic effect on the quality of magnetotransport data [(*31*), section 2.2], especially at large applied magnetic fields. Because of a slight systematic n-doping of our contacts during fabrication, our highest-quality data are obtained for an n-doped channel. We thus restricted our study to the electron side of the band structure.

Under application of low magnetic fields, transport measurements show a sequence of QHE plateaus in *R _{xy}* appearing at

*h*/4

*me*

^{2}, where

*m*is a nonzero integer and

*h*is Planck’s constant, together with resistance minima in

*R*, which is consistent with the single-particle LL spectrum expected for bilayer graphene (

_{xx}*32*). At fields larger than ~5 T, we observed complete symmetry breaking, with QHE states appearing at all integer filling fractions (fig. S8A), which is indicative of the high quality of our sample (Fig. 1B). By cooling the sample to sub-kelvin temperatures (20 to 300 mK) and applying higher magnetic fields (up to 31 T), clearly developed FQHE states appear at partial LL filling, with vanishing

*R*and unambiguous plateaus in

_{xx}*R*(Fig. 1C). By changing

_{xy}*V*

_{TG}while sweeping

*V*

_{BG}, we can observe the effect of different electric displacement fields on these FQHE states. In Fig. 1D,

*V*

_{TG}= 0.2, the

*v*= 2/3 and

*v*= 5/3 FQHE states are clearly visible as minima in σ

*, whereas for*

_{xx}*V*

_{TG}= 1.2 V, the

*v*= 2/3 state is completely absent, the 5/3 state appears weakened, and a new state at

*v*= 4/3 becomes visible. This indicates that both the existence of the FQHE in BLG, and the sequence of the observed states, depend critically on the applied electric displacement field and that a complete study of the fractional hierarchy in this material requires the ability to independently vary the carrier density and displacement field.

To characterize the effect of displacement field in more detail, it is illuminating to remap the conductivity data versus displacement field and LL filling fraction *v* (Fig. 2A). Replotted in this way, a distinct sequence of transitions, marked by compressible regions with increased conductivity, is observed for each LL. For example, at *v* = 1 there is evidently a phase transition exactly at *D* = 0 and then a second transition at large finite *D*. In contrast, at *v* = 2 (Fig. 2B) there is no apparent transition at *D* = 0, and a finite *D* transition appears at a much smaller displacement field than at *v* = 1. Last, at *v* = 3 there is a single transition only observed at *D* = 0. This pattern is in agreement with other recent experiments (*33*). At large displacement fields, it is expected that it is energetically favorable to maximize layer polarization, indicating that low-displacement-field states that undergo a transition into another state at finite displacement field (such as *v* = 1,2) likely exhibit an ordering different from full layer polarization. Following predictions that polarization in the 0-1 orbital degeneracy space is energetically unfavorable (*34*), this could be a spin ordering, like ferromagnetism or antiferromagnetism, or a layer-coherent phase (*34*–*37*). This interpretation is consistent with several previous experimental studies that reported transitions within the symmetry-broken integer QHE states to a layer-polarized phase under a finite displacement field (*21*–*25*, *33*, *38*).

At higher magnetic field and lower temperature (Fig. 2C), the integer states remain robust at all displacement fields, indicating that the observed transitions points in Fig. 2A exhibit insulating temperature dependence. This could be due to a small persistent bulk gap or disorder effects. At these fields and temperatures, FQHE states within each LL become evident, exhibiting transitions of their own with displacement field. Of particular interest are the states at *v* = 2/3 and *v* = 5/3, which are the most well developed. A strong *v* = 2/3 state is consistent with recent theory (*39*), which predicts this state to be fully polarized in orbital index in the 0 direction. At *v* = 2/3, there is a transition at *D* = 0, as well as two more at mV/nm (shown in Fig. 2C by the increased conductivity at the top and bottom boundaries of the plot. Behavior at larger is shown in fig. S8B). These transitions are qualitatively and quantitatively similar to the transitions in the *v* = 1 state seen in Fig. 2C. For the 5/3 state, we present high-resolution scans at fields from 20 to 30 T in Fig. 2D. We again observed a transition at *D* = 0 as well as at finite *D*. The finite *D* transitions, however, occur at much smaller values than for either *v* = 2/3 or 1. Indeed, they are much closer in *D* value to the transitions taking place at *v* = 2, which are a factor of ~8 smaller than that of *v* = 1.

In Fig. 3, A and B, we plot the resistance minima of 2/3 and 5/3 state, respectively, as a function of *D* (corresponding to vertical line cuts through the two-dimensional map in Fig. 2C). For the 2/3 state, we observed broad transitions, whose positions we define by estimating the middle points. Where the transition cannot be fully resolved because of limited gate range, this is done by the identification of a local minimum in conductance within the transition. The 5/3 state exhibits much narrower transitions, which we mark by the local maximum of resistance. We observed some small asymmetry (<5 mV/nm) between the negative and positive *D* transitions, the source of which is not clear to us and which we include in our error associated with transition *D* values. The location of transitions in *D* is plotted in Fig. 3C as a function of magnetic field *B* for both integer and fractional filling fractions. Our main observation is that the transitions in the fractional quantum Hall states and the transitions in the parent integer state (the smallest integer larger than the fraction) fall along the same line in *D* versus *B*. More specifically, the finite *D* transitions for *v* = 2/3 and *v* = 1 fall along a single line of slope 7 mV/nm · T, and the transitions for *v* = 5/3 and *v* = 2 fall along a single line of slope 0.9 mV/nm · T. This difference of a factor of ~8 in the slopes of the transition lines suggests a difference in the nature of the competing broken symmetry states at *v* = 1 and *v* = 2. The quantitative agreement observed between multiple devices indicates that these values are largely sample-independent.

We now turn to a discussion of the nature of these transitions. For the lowest LL of BLG, the possible internal quantum state of electrons comprises an octet described by the SU(4) spin-valley space and the 0-1 LL orbital degeneracy. Because *v* = 1 corresponds to filling five of the eight degenerate states in the lowest LL, the system will necessarily be polarized in some direction of the SU(4) spin-valley space. The presence of a *D* = 0 transition for *v* = 1 indicates that even at low displacement field, the ground state exhibits a layer polarization that changes as *D* goes through zero. At the same time, we expect that at large *D* the system is maximally layer polarized. We therefore propose that the transition at finite *D* is between a 1/5 layer-polarized state (for example, 3 top layer and 2 bottom layer levels filled) and a 3/5 layer-polarized state (for example, 4 top layer and 1 bottom layer levels filled) (*35*–*37*). The quantitative agreement of the transitions for *v* = 2/3 and *v* = 1, shown in Fig. 3C, strongly suggest that the composite fermions undergo a similar transition in layer polarization.

Whereas the observed transition at *v* = 1 and its associated FQHE (*v* = 2/3) can be explained by a partial-to-full layer polarization transition, the nature of the transition at *v* = 2 is less clear: The high *D* state presumably also exhibits layer polarization, but we do not have any experimental insight as to the ordering of the low *D* state. Additionally, although the finite *D* transitions in the 5/3 state seem to follow the *v* = 2 transitions quantitatively (Fig. 3C), the 5/3 state also has a clear transition at *D* = 0, suggesting that there may be a different ground state ordering of the 5/3 and 2 states very near *D* = 0. In particular, the transition at *D* = 0 indicates that the *v* = 5/3 state exhibits layer polarization even at low displacement field, in a state separate from the high-displacement-field layer-polarized state, whereas this does not appear to be true at *v* = 2. One possible explanation for this result could be the formation of a layer-coherent phase that forms at *v* = 2 filling (*37*) but that is not stabilized at partial filling of the LL.

We also briefly mention other observed FQHE states. At the highest fields, we see evidence of weaker states at *v* = 1/3 and *v* = 4/3 (Fig. 2D) [(*31*), section 2.3], with both exhibiting phase transitions that appear to follow those of *v* = 2/3 and *v* = 5/3, respectively. We have also observed transitions in the *v* = 8/3 state with displacement field (Fig. 2C), though we do not have a systematic study of its dependence on magnetic field because of our limited gate range. Qualitatively, there is a region at low *D* where no minimum is apparent that gives way to a plateau and minimum at finite *D*. The *v* = 3 state also exhibits a transition at *D* = 0 (Fig. 2A), indicating that there may also be a correspondence between the integer and fractional states in the *v* = 3 LL. These observations are summarized in Fig. 3C, inset; the detailed data are available in (*31*), section 2.6. Last, we have indications that the fractional hierarchy breaks electron-hole symmetry (*14*) [(*31*), section 2.3] because the clearest fractional states we observed can be described as *v* = *m* – 1/3, where *m* is an integer.

The electric-field–driven phase transitions observed in BLG’s FQHE indicate that ordering in the SU(4) degeneracy space is critical to the stability of the FQHE. In particular, quantitative agreement between transitions in FQHE states and those in parent integer QHE states suggests that generally, the composite fermions in BLG inherit the SU(4) polarization of the integer state and couple to symmetry-breaking terms with the same strength. However, an apparent disagreement in the transition structure at *v* = 5/3 and *v* = 2 indicates that there may be subtle differences in the ground-state ordering for the integer and fractional quantum Hall states.

## Supplementary Materials

www.sciencemag.org/content/345/6192/61/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S9

Reference (*40*)

## References and Notes

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**Acknowledgments:**We acknowledge M. Shinn and G. Myers for assistance with measurements. We thank A. Yacoby for helpful discussions. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by U.S. National Science Foundation cooperative agreement DMR-0654118, the State of Florida, and the U.S. Department of Energy (DOE). This work is supported by the National Science Foundation (DMR-1124894) and FAME under STARnet. J.H. acknowledges support from Office of Naval Research N000141310662. P.K. acknowledges support from DOE (DE-FG02-05ER46215).