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Evidence for a fractional fractal quantum Hall effect in graphene superlattices

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Science  04 Dec 2015:
Vol. 350, Issue 6265, pp. 1231-1234
DOI: 10.1126/science.aad2102

Mixing interactions and superlattices

Under the influence of an external magnetic field, the energies of electrons in two-dimensional systems group into the so-called Landau levels. In the cleanest samples, interactions among electrons lead to fractional quantum Hall (FQH) states. If such a system is then subjected to a superlattice potential, it is unclear whether the fragile FQH states will survive. To address this question, Wang et al. sandwiched graphene between two layers of hexagonal boron nitride. Transport measurements on the superlattice showed that some FQH states did survive. Furthermore, the interplay between interactions and the superlattice potential produced additional, anomalous states.

Science, this issue p. 1231

Abstract

The Hofstadter energy spectrum provides a uniquely tunable system to study emergent topological order in the regime of strong interactions. Previous experiments, however, have been limited to low Bloch band fillings where only the Landau level index plays a role. We report measurements of high-mobility graphene superlattices where the complete unit cell of the Hofstadter spectrum is accessible. We observed coexistence of conventional fractional quantum Hall effect (QHE) states together with the integer QHE states associated with the fractal Hofstadter spectrum. At large magnetic field, we observed signatures of another series of states, which appeared at fractional Bloch filling index. These fractional Bloch band QHE states are not anticipated by existing theoretical pictures and point toward a distinct type of many-body state.

In a two-dimensional electron gas (2DEG) subjected to a magnetic field, the Hall conductivity is generically quantized whenever the Fermi energy lies in a gap (1). The integer quantum Hall effect (IQHE) results from the cyclotron gap that separates the Landau energy levels (LLs). The longitudinal resistance drops to zero, and the Hall conductivity develops plateaus quantized to σXY = ve2/h, where v, the Landau level filling fraction, is an integer; h is Planck’s constant; and e is the electron charge. When the 2DEG is modified by a spatially periodic potential, the LLs develop additional subbands separated by minigaps, resulting in the fractal energy diagram known as the Hofstadter butterfly (2). When plotted against normalized magnetic flux ϕ/ϕ0 and normalized density n/n0 (representing the magnetic flux quanta and electron density per unit cell of the superlattice, respectively), the fractal minigaps follow linear trajectories (3) according to a Diophantine equation, n/n0 = tϕ/ϕ0 + s, where s and t are integers; s is the Bloch band-filling index associated with the superlattice, and t is a similar index related to the gap structure along the field axis (4) (in the absence of a superlattice, t reduces to the LL filling fraction). The fractal minigaps give rise to QHE features at partial Landau level filling, but in this case t, rather than the filling fraction, determines the quantization value (1, 5) and the Hall plateaus remain integer-valued.

In very-high-mobility 2DEGs, strong Coulomb interactions can give rise to many-body gapped states also appearing at partial Landau fillings (68). Again, the Hall conductivity exhibits a plateau, but in this case quantized to fractions of e2/h. This effect, termed the fractional quantum Hall effect (FQHE), represents an example of emergent behavior in which electron interactions give rise to collective excitations with properties fundamentally distinct from the fractal IQHE states. A natural theoretical question arises regarding how interactions manifest in a patterned 2DEG (912). In particular, because both the FQHE many-body gaps and the single-particle fractal minigaps can appear at the same filling fraction, it remains unclear whether the FQHE is even possible within the fractal Hofstadter spectrum (1315). Experimental effort to address this question has been limited, owing to the requirement of imposing a well-ordered superlattice potential while preserving a high carrier mobility (1618).

Here, we report a low-temperature magnetotransport study of fully encapsulated hexagonal boron nitride (h-BN)/graphene/h-BN heterostructures, fabricated by van der Waals assembly with edge contact (19, 20). A key requirement to observe the Hofstadter butterfly is the capability to reach the commensurability condition in which the magnetic length Embedded Image (where ħ is Planck’s constant divided by 2π, e is the electron charge, and B is the magnetic field) is comparable to the wavelength of the spatially periodic potential, λ. For experimentally accessible magnetic fields, this requires a superlattice potential with wavelength on the order of tens of nanometers. In this regard, graphene/h-BN heterostructures provide an ideal system, because at near-zero angle mismatch, the slight difference in lattice constants between the graphene and BN crystal structures gives rise to a moiré superlattice with a period of ~14 nm (2124). Moreover, we find that in our van der Waals assembly technique, in which the graphene-BN interface remains pristine (19), alignment between the graphene and BN can be achieved by simple application of heat. Figure 1A shows an example of a heterostructure that was assembled with random (and unknown) orientation of each material. After heating the sample to ~350°C, the graphene flake translates and rotates through several micrometers, despite being fully encapsulated between two BN sheets. This behavior has been observed in several devices (20), in each case resulting in a moiré wavelength of 10 to 14 nm [indicating less than 2° angle mismatch (21)]. We speculate that the thermally induced motion proceeds until the macroscale graphene flake finds a local energy minimum, corresponding to crystallographic alignment to one of the BN surfaces (25).

Fig. 1 Transport measurements in a BN-encapsulated graphene device with a ~10-nm moiré superlattice.

(A) Zero-field resistance versus gate bias. Left inset: False-colored optical image showing macroscopic motion of the graphene (G) after heating; scale bar, 15 μm. Right inset: Gap measured by thermal activation at the CNP and hole satellite peak positions, across four different devices (20). (Note that the satellite peak does not exhibit activated behavior, within the resolution of our measurement, for superlattice wavelengths less than ~10 nm.) Both gaps are observed to vary continuously with rotation angle. Error bars indicate the uncertainty in the gap deduced from a linear fit to the activated temperature response (20). (B) Hall conductivity plotted versus magnetic field and gate bias (top) and longitudinal resistance versus normalized field and density (bottom) for the same device as in (A). (C) Simplified Wannier diagram labeling the QHE states identified in (B). Light gray lines indicate all possible gap trajectories according to the Diophantine equation, where we have assumed that both spin and valley degeneracy may be lifted such that s and t are allowed to take any integer value (for clarity, the range is restricted to |s, t| = 0 … 10). Families of states are identified by color as follows: Black lines indicate fractal IQHE states within the conventional Hofstadter spectrum, including complete lifting of spin and valley degrees of freedom. Blue lines indicate conventional FQHE states. Red lines indicate anomalous QHE states that do not fit either of these descriptions, exhibiting integer Hall quantization but corresponding to a fractional Bloch index (see text).

Figure 1A shows the resistance versus gate voltage at zero applied magnetic field for a device with moiré wavelength of ~10 nm. In addition to the usual peak in resistance at the zero-density charge neutrality point (CNP), two additional satellite peaks appear at equidistant positive and negative gate bias—characteristic signatures of electronic coupling to a moiré superlattice (2123). The CNP peak resistance exhibits thermally activated behavior and exceeds 100 kilohms at low temperature, indicating a moiré coupling–induced band gap (2426). The gap varies continuously with rotation angle, consistent with previous studies of nonencapsulated graphene (20, 24). At zero angle, the energy gap measured by transport is equivalent to the optical gap (27), indicating low disorder broadening of the energy band in our devices, and consistent with the high electron mobility achievable by the van der Waals assembly technique (19, 20) Unlike previous studies of encapsulated devices (23, 25), we find that the gap remains robust despite the graphene being covered with a top BN layer. The precise origin of the gap in h-BN/graphene heterostructures remains uncertain (28), and further experimental and theoretical studies will be required to resolve the differences in the gap magnitude and correlation with twist angle that have been reported so far.

Figure 1B shows the longitudinal resistance and transverse Hall conductivity for the same device as in Fig. 1A. The low disorder in our samples allows smaller energy gap states to be resolved than previously possible, resulting in a rich complexity of observable transport features. A sequence of repeated minifans, resembling a repeated butterfly in the Hall conductivity map, shows evidence of the fractal nesting expected from the Hofstadter spectrum. In Fig. 1C, a simplified Wannier diagram is shown in which the positions of the most prominent QHE states are plotted against normalized flux and normalized density axes. We focus our discussion on the FQHE and anomalous states.

FQHE states are characterized by a longitudinal resistance minimum occurring at a fractional Landau filling index, with the corresponding Hall conductivity plateau quantized to the same fractional value, and with the gap trajectory in the Wannier diagram projecting to n/n0 = 0. The FQHE states are observed at Embedded Image filling fractions in the lowest and first excited Landau level, where m is an integer. The observation of a well-developed Embedded Image state is consistent with previous studies of monolayer graphene in which a zero-field band gap was reported (24, 26), and is presumably due to the lifting of the valley degeneracy that results from coupling to the moiré pattern (11). In the second Landau level, fractional states beyond Embedded Image are absent, apparently obscured by the appearance of the minigap states (Fig. 1B).

As can be seen in Fig. 1C, the visible FQHE states span only a finite range of the perpendicular magnetic field. This is shown in more detail in Fig. 2A, where a selected region of the longitudinal resistance from Fig. 1B is replotted against magnetic field on the vertical axis and Landau filling fraction along the horizontal axis. Select line traces from this diagram, corresponding to varying filling fraction at constant magnetic field, are shown in Fig. 2B. For clarity, we plot the Hall conductivity calculated from the measured longitudinal and Hall resistances (20). In the following, we focus on the Embedded Image state as a representative example of the general behavior.

Fig. 2 Fractional quantum Hall effect in the Hofstadter spectrum.

(A) Longitudinal resistance versus Landau filling fraction corresponding to a high-field region of data from Fig. 1B. (B) Hall conductivity (top) and longitudinal resistance (bottom) corresponding to horizontal line cuts within the dashed region in (A). Conductivity plateaus identified at 30 T and 40 T are labeled blue and red, respectively. (C) Hall conductivity versus magnetic field at fixed filling fraction, showing evolution from FQHE plateaus to integer-valued plateaus.

At B = 30 T, the Hall conductivity at filling fraction Embedded Image is well quantized to σXY = Embedded Image(e2/h). Upon increasing to B = 34 T, this FQHE state has completely disappeared; by B = 40 T, a Hall plateau has reemerged, but it is quantized to the integer-valued σXY = 1(e2/h) and with the plateau no longer precisely coincident with Embedded Image filling (Fig. 2B). We interpret the apparent phase transition to be the result of a competition with a fractal minigap state. This is supported by examining the relative strength of the QHE features on either side of the transition as a qualitative measure of relative gap size; the high-field integer-valued state exhibits a substantially better-developed longitudinal resistance minimum and wider Hall plateau (indicative of a larger gap) than the lower-field FQHE state. Taken together, these observations provide experimental evidence supporting two theoretical predictions (1315): (i) The fractal Hofstadter spectrum can support Laughlin-like FQHE states even at field strengths approaching the commensurability condition, and (ii) at filling fractions in which a conventional FQHE and a Hofstadter minigap state coexist, the state with the larger associated band gap is the one that is observed.

We next discuss the anomalous QHE features associated with the red lines in Fig. 1C. In Fig. 3A, a reduced Wannier plot is shown in which only these anomalous features are replotted (solid red lines), together with a dashed line showing the projection to the n/n0 axis. Linear fits to the RXX minimum position versus magnetic field (20) indicate that these states follow a trajectory with an integer-valued slope, t, but project to non–integer-valued intercepts, s. Figure 3B demonstrates that these features correspond to QHE features with well-quantized Hall plateaus, and further that the quantization value corresponds to the t value, as expected from the Diophantine equation. Determining the fractional s number from the n/n0 intercept of the Wannier plot is imprecise because this depends on calculating the density. Nonetheless, within experimental uncertainty (fig. S8), the fractional intercepts appear to cluster around values of Embedded Image and Embedded Image (Fig. 3A). We observe similar features in a second device (20) fabricated in the same way but with zero orientation angle between the graphene and BN lattices (moiré wavelength ~14 nm). Again, several anomalous features are present, characterized by integer Hall quantization and slope in the Wannier diagram but exhibiting fractional intercept (s number). In the second device, which exhibits in general a more symmetric electron and hole response, these features appear on both the electron and hole side, and with both positive and negative Hall quantization. We note, however, that the fractional s numbers remain limited to multiples of Embedded Image and Embedded Image in both samples.

Fig. 3 The FBQHE states.

(A) Wannier diagram showing only the anomalous features from Fig. 1. Brackets label the corresponding Bloch band (s) and Landau band (t) numbers; dashed lines show linear projections to the n/n0 axis. (B) Hall conductivity and longitudinal resistance versus normalized density, measured at fixed magnetic field B = 40 T, showing transport signatures of selected gap states from (A). The s and t numbers, determined respectively from the n/n0 intercept and the Hall quantization value, are labeled for each QHE plateau. (C) Summary illustration of the energy band structure evolution with magnetic field in the case of no superlattice (left) and with a superlattice (right), and in the limit of no interactions (top) and strong interactions (bottom): (i) With no superlattice, the density of states is continuous at zero magnetic field. A cyclotron gap develops with finite magnetic field, indicated by white against a colored background. (ii) In the presence of a large-wavelength superlattice, the Bloch band edge is accessible by field effect gating. Hofstadter minigaps evolve from this band edge, intersecting the conventional Landau levels at large magnetic field. (iii) With no superlattice present, interactions give rise to the fractional quantum Hall effect, appearing also as subgaps within the Landau level but projecting to zero energy. (iv) In the regime of both strong interactions and large-wavelength superlattice, we observe a set of gaps that do not correspond to either the IQHE of single-particle gaps or the conventional many-body FQHE gaps. These are defined by integer-valued Hall quantization, but they project to fractional Bloch band-filling indices.

Figure 3C shows a cartoon summary of this result. In the regime of very large magnetic field, in addition to the conventional fractional quantum Hall effect, we observe a new series of states outside of the single-particle band structure as described by the Diophantine equation, and coinciding with a fractional Bloch band index. This so-called fractional Bloch band QHE (FBQHE), described by integer t numbers but fractional s numbers, may have several possible origins. We note that at B = 30 T, the Coulomb energy is ~80 meV (ECoulomb = e2/εlB, where we assume the dielectric constant ε to be 4). This is similar in magnitude to the superlattice potential (29), which suggests that interactions play a comparable role. Recent theoretical consideration of graphene superlattices indeed showed (15) that electron interactions may open a gap consistent with a fractional s number. However, the nature of the associated ground state was not identified. Previously, it was predicted that electron interactions may drive a charge density wave (CDW)–type modulation of the electron density, commensurate to the superlattice but with a larger period (30). A superstructure with 3 times the moiré unit cell area (such as a Embedded Image Kekule distortion) could explain n/n0 intercepts of Embedded Image, whereas a doubling of the superlattice cell could explain Embedded Image intercepts. In this regard, our observation may resemble the reentrant QHE seen in high-mobility GaAs (31), also believed to result from a CDW phase. Alternatively, one interpretation of the Wannier diagram is to consider the minigaps as a sequence of mini–Landau fans, residing in a local reduced magnetic field Embedded Image, where ϕ/ϕ0 = 1/m (or equivalently 1 – 1/m by symmetry) labels regions of high density of minigap crossing. Recent band structure calculation of moiré-patterned graphene (11) indicates that the minifans are not exact replicas, but instead can exhibit a local degeneracy with additional Dirac points emerging near ϕ/ϕ0 = 1/m. The FBQHE states may therefore result from an interaction-driven breaking of this degeneracy, similar to quantum Hall ferromagnetism.

Finally, we consider that within the minifan picture, the FBQHE states resemble the FQHE effect in that they follow trajectories that evolve along fractional filling of the minifan LLs, projecting to the B′ = 0 center of the minifan (20). However, both the slope and the corresponding Hall conductivity plateaus are integer-valued. A complete understanding of our findings will require a theory that accounts for both the observed fractional Bloch band numbers and the associated Hall conductivity value. Experimentally, possible ground states could be distinguished by a local probe of the density of states, because (for example) a CDW phase exhibits broken translation symmetry, unlike the Laughlin FQHE state.

Supplementary Materials

www.sciencemag.org/content/350/6265/1231/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S13

References (3234)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank A. MacDonald, T. Chakraborty, I. Aleiner, V. Falko, A. F. Young, and B. Hunt for helpful discussions. Supported by Office of Naval Research grant N000141310662 (L.W., Y.G., and J.H.), NSF grant DMR-1463465 (C.R.D.), and JSPS grant-in-aid for scientific research 25107005 (M.K.). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-0654118, the State of Florida and the U.S. Department of Energy.
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