High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices

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Science  14 Jul 2017:
Vol. 357, Issue 6347, pp. 181-184
DOI: 10.1126/science.aal3357

Heat-loving quantum oscillations

The shape of the Fermi surface in a conductor can be gleaned through quantum oscillations—periodic changes in transport properties as an external magnetic field is varied. Like most quantum properties, the phenomenon can usually be observed only at very low temperatures. Krishna Kumar et al. report quantum oscillations in graphene that do not go away even at the temperature of boiling water. Although “ordinary,” low-temperature quantum oscillations die away, another oscillatory behavior sets in that is extremely robust to heating. These resilient oscillations appear only in samples in which graphene is nearly aligned with its hexagonal boron nitride substrate, indicating that they are caused by the potential of the moiré superlattice that forms in such circumstances.

Science, this issue p. 181


Cyclotron motion of charge carriers in metals and semiconductors leads to Landau quantization and magneto-oscillatory behavior in their properties. Cryogenic temperatures are usually required to observe these oscillations. We show that graphene superlattices support a different type of quantum oscillation that does not rely on Landau quantization. The oscillations are extremely robust and persist well above room temperature in magnetic fields of only a few tesla. We attribute this phenomenon to repetitive changes in the electronic structure of superlattices such that charge carriers experience effectively no magnetic field at simple fractions of the flux quantum per superlattice unit cell. Our work hints at unexplored physics in Hofstadter butterfly systems at high temperatures.

Oscillations of physical properties of materials with magnetic field are a well known and important phenomenon in condensed matter physics. Despite having a variety of experimental manifestations, there are only a few basic types of oscillations: those of either quantum or semiclassical origin (15). Semiclassical size effects, such as Gantmakher and Weiss oscillations, appear owing to commensurability between the cyclotron orbit and a certain length in an experimental system (14). Quantum magneto-oscillations are different in that they arise from periodic changes in the interference along closed electron trajectories (15). Most commonly, quantum oscillations involve cyclotron trajectories. This leads to Landau quantization and, consequently, Shubnikov–de Haas (SdH) oscillations in magnetoresistance and the associated oscillatory behavior in many other properties (13). In addition, quantum oscillations may arise from interference on trajectories imposed by sample geometry, leading to Aharonov-Bohm oscillations in mesoscopic rings, for instance (3, 5). Whatever their exact origin, the observation of such oscillatory effects normally requires low temperatures (T), and this requirement is particularly severe in the case of quantum oscillations that rely on the monochromaticity of interfering electron waves. Even in graphene, with its massless Dirac spectrum and exceptionally large cyclotron gaps, SdH oscillations rarely survive above 100 K. At room T, high magnetic fields (B) of ~30 T are needed to observe the last two SdH oscillations arising from the maximal gaps between the zeroth and first Landau levels (LLs) of graphene (6). In all other materials, quantum oscillations disappear at much lower T.

Electronic systems with superlattices can also exhibit magneto-oscillations. In this case, the interference of electrons diffracting at a superlattice potential in a magnetic field results in fractal, self-similar spectra that are often referred to as Hofstadter butterflies (HBs) (712). Their fractal structure reflects the fact that charge carriers effectively experience no magnetic field if magnetic flux Embedded Image through the superlattice unit cell is commensurate with the magnetic flux quantum, Embedded Image (79). This topic has attracted interest for decades (1116) but received a particular boost thanks to the recent observation of clear self-similar features in transport characteristics and in the density of states (DOS) of graphene superlattices (1725). Because the HB depicts quantum states developed from partial admixing of graphene’s original LLs (12), superlattice-related gaps already become smeared at relatively low T, well below those at which signatures of quantization in the main spectrum disappear. Therefore, it is perhaps not surprising that investigations of Hofstadter systems were confined mostly to low T. As shown below, this has resulted in a failure to notice some unusual physics: Superlattices exhibit robust high-T oscillations in their transport characteristics, which are different in origin from the known oscillatory effects.

We used multiterminal Hall bar devices (Fig. 1A, inset, and fig. S1) made from graphene superlattices (26) to carry out our transport measurements. Monolayer graphene was placed on top of a hexagonal boron nitride (hBN) crystal, and their crystallographic axes were aligned with an accuracy of better than 2° (17, 24). The resulting moiré pattern gives rise to a periodic potential that is known to affect the electronic spectrum of graphene (2325). To ensure that the charge carriers have high mobility, the graphene was encapsulated using a second hBN crystal, which was intentionally misaligned by ~15° with respect to graphene’s axes. Although the second hBN layer also leads to a moiré pattern, it has a short periodicity and, accordingly, any superlattice effects may appear only at high carrier concentrations n or ultrahigh B, beyond those accessible in transport experiments (1725). Therefore, the second hBN effectively serves as an inert, atomically-flat cover protecting graphene from the environment. Six superlattice devices were investigated and showed consistent behavior, which is described below. As a reference, we also studied devices made according to the same procedures but with the graphene misaligned with respect to both top and bottom hBN layers.

Fig. 1 High-temperature oscillations in graphene superlattices.

(A) ρxx at relatively small B for a superlattice device with a moiré periodicity of ≈13.6 nm. The electron density is n ≈ 1.7 × 1012 cm−2 and is induced by a back-gate voltage. (Inset) Optical micrograph of one of our Hall bar devices. Scale bar, 5 μm. Ω, ohms. (B) Same as (A) but using the reference device at the same n. (C) Magnetoresistance curves for the devices in (A) and (B) at 100°C (solid curves). ρxx for the reference device is multiplied by a factor of 4. Dotted curves: Oscillatory behavior is emphasized by subtracting a smooth background, using fourth-order polynomials as best fits to the original ρxx curves (26). At 100°C, graphene superlattices exhibit oscillations starting below 4 T, whereas no sign of oscillatory behavior could be discerned at this T in our reference devices at any B. (Inset) Scanning tunneling image illustrates typical moiré patterns found in our devices [for details, see (30)].

Figure 1A shows typical behavior of the longitudinal resistivity ρxx for graphene superlattices as a function of B at various T. For comparison, Fig. 1B plots similar measurements for the reference device. In the latter case, ρxx exhibits pronounced SdH oscillations at liquid-helium T, which develop into the quantum Hall effect above a few tesla. The SdH oscillations are rapidly suppressed with increasing T and completely vanish above liquid-nitrogen T, the standard behavior for graphene in these relatively weak fields (27, 28). In stark contrast, graphene superlattices exhibit prominent oscillations over the entire T range (Fig. 1A and fig. S2). At both high and low T, the oscillations are periodic in 1/B (figs. S3 and S4). The oscillations in Fig. 1A change their frequency at ~50 K. This is the same T range in which SdH oscillations disappear in the reference device of Fig. 1B. For certain ranges of n and B, we observed that SdH oscillations vanished first, before new oscillations emerged at higher T. An example of such nonmonotonic T dependence is shown in fig. S2. To emphasize the robustness of the high-T oscillations, we show that they remain well developed even at boiling-water T in moderate B (Fig. 1C). The oscillations were observed even at higher T, but above 400 K our devices (both superlattice and reference devices) showed rapid deterioration in quality and strong hysteresis as a function of gate voltage.

The high-T and SdH oscillations differ not only in their periodicity and thermal stability but also because they have distinctly different n dependences. Figure 2, A and B, shows Landau fan diagrams for the longitudinal conductivity σxx of graphene superlattices as a function of B and n (we plot σxx rather than ρxx to facilitate the explanation given below for the origin of the high-T oscillations). At low T (Fig. 2A), we observe the same behavior as reported previously (1722): Numerous LLs fan out from the main (n = 0) and second-generation neutrality points (NPs) that are found at n = ±n0, where n0 = 4/S corresponds to four charge carriers per superlattice unit cell with the area Embedded Image and the superlattice period a (1722). The LL intersections result in third-generation NPs at finite B (19). Minima in σxx evolve linearly in B and originate from first-, second-, and third-generation NPs (1722). This reflects the fact that the DOS for all LLs (including those caused by fractal gaps) is the same and proportional to B (11). At T > 100 K, the Landau quantization dominating the low-T diagrams wanes and, instead, oscillations with a periodicity independent of n emerge (Fig. 2, B and C). This independence of n clearly distinguishes the high-T oscillations from all of the known magneto-oscillatory effects arising from either Landau quantization or commensurability (15). For the reasons that become clear below, we refer to the observed high-T phenomenon as Brown-Zak (BZ) oscillations (7, 8).

Fig. 2 Concentration and field dependence of Brown-Zak (BZ) oscillations.

(A) Low-T fan diagram σxx(n, B) for a superlattice device with a ≈ 13.9 nm. The gray scale is logarithmic: white, 0.015 mS; black, 15 mS. (B) Same device as (A) but at 150 K. Logarithmic gray scale: white, 0.1 mS; black, 10 mS. The dotted lines denote Embedded Image. (C) Same as (B) but for Δσxx obtained by subtracting a smooth best-fit background (26). Linear gray scale: ±0.3 mS. (Inset) Fundamental frequency B0 of BZ oscillations found in our different devices as a function of Embedded Image. (D) Near Embedded Image (dashed lines are for q = 3 to 6), local changes in σxx and σxy resemble magnetotransport in metals near zero field, as illustrated by the green inset curves. (E) Part of (C) near the second-generation NP for electron doping is magnified and plotted as a function of Embedded Image. The main maxima in Δσxx occur at Embedded Image. A few extra maxima for p = 2 and 3 are indicated by black and green arrows, respectively (see fig. S6 for details). (F) Corresponding behavior of Δσxy (smooth background subtracted). Its zeros align with the red maxima in Δσxx.

The BZ oscillations become stronger with increased doping (Fig. 2C), in agreement with the fact that the superlattice spectrum is modified more strongly at energies away from the main Dirac point (1722). The maxima in σxx are found at Embedded Image, which corresponds to unit fractions of Embedded Image piercing a superlattice unit cell (q is an integer). The relation between the superlattice period and the periodicity of the high-T oscillations holds accurately for all of our devices (Fig. 2C, inset; for details, see fig. S3). This is the same periodicity that underlines the Hofstadter spectrum and describes the recurrence of third-generation NPs (17, 18). However, BZ oscillations emerge most profoundly at high T, in the absence of any remaining signs of the Hofstadter spectrum or Landau quantization (fig. S5). Our capacitance measurements (fig. S5A) reveal no sign of behavior similar to BZ oscillations in the DOS, even at liquid-helium T that allow the clearest view of the HB (20, 25). These observations prove that BZ oscillations are a transport phenomenon, unrelated to the spectral gaps that make up the Hofstadter spectrum. BZ oscillations do not disappear at low T and, retrospectively, can be recognized as horizontal streaks connecting third-generation NPs on the transport Landau fan diagrams (1719) (Fig. 2A). However, the streaks are heavily crisscrossed by LLs, which makes them easy to overlook or wrongly associate with the quantized Hofstadter spectrum (17).

The HB spectrum is expected to exhibit a fractal periodicity associated with not only unit fractions but all of the simple fractions, p/q, corresponding to p flux quanta per q cells. No signatures of such higher-order states were found in the previous experiments (1722) nor can they be resolved in our present fan diagrams at low T. However, the fractions with p = 2 and 3 become evident in BZ oscillations (Fig. 2E) and are most prominent at high n (fig. S6). This again indicates that the BZ oscillations are governed by the same underlying periodicity as is the HB spectrum. We also find that BZ oscillations are stronger for electrons than for holes (Fig. 2B and fig. S7). This is in contrast to the relative strengths of all other features reported previously for graphene-on-hBN superlattices (1725). The electron-hole asymmetry is probably connected to the observed stronger electron-phonon scattering for hole doping (fig. S7).

To explain BZ oscillations, we recall that at Embedded Image, the electronic spectrum of superlattices can be reduced to the case of zero magnetic field by introducing new Bloch states and the associated magnetic minibands, different for each p/q. This concept was put forward by Brown (7) and Zak (8) and predates the work by Hofstadter (10). Examples of BZ minibands for several unit fractions of Embedded Image are shown in Fig. 3 and fig. S8, using a generic graphene-on-hBN potential (26). Each miniband can be viewed as a superlattice-broadened LL, such that its energy dispersion Embedded Image disappears in the limit of vanishing superlattice modulation (12). If the Fermi energy εF lies within these superlattice-broadened LLs, the system should exhibit a metallic behavior (25). The Hofstadter spectrum can then be understood as Landau quantization of BZ minibands in the effective field Embedded Image (20, 29). With this concept in mind, let us take a closer look at the experimental behavior of σxx and the Hall conductivity σxy at high T and small Beff—that is, in the absence of Landau quantization in BZ minibands (Fig. 2, D to F, and fig. S6). One can see that every time BZ minibands are formed, σxx exhibits a local maximum and σxy shows a Beff/(1 + Beff2)–like feature on top of a smoothly varying background. This local behavior resembles changes in σxx and σxy expected for any metallic system near zero B and approximated by the functional forms 1/(1 + B2) and B/(1 + B2), respectively (13). The latter are sketched in the insets of Fig. 2D and match well the shape of local changes in σxx and σxy near fractional fluxes Embedded Image, which correspond to Beff = 0 (Figs. 2D and 3 and fig. S6).

Fig. 3 BZ oscillations as recurring Bloch states in small effective fields.

Solid curves: σxx at 100 K for electron and hole doping (n/n0 = ±1.6) (top and bottom panels, respectively) in a superlattice device with a ≈ 13.6 nm. Black dots and curves: σxx calculated in the constant-τ approximation for different p and q. Inset image: BZ minibands Embedded Image inside the first Brillouin zones indicated by the gray hexagons (their size decreases with increasing q). The minibands were calculated for a generic graphene-on-hBN superlattice (29) and correspond to broadened LLs (for example, LLs are 2 and 3 for q = 2 and range from 3 to 8 for q = 5). Only those minibands at energies relevant to the doping level on the experimental curves are shown.

The described analogy between magnetotransport in normal metals and in BZ minibands can be elaborated using the approximation of a constant scattering time τ (13). We assume τ to be the same for all minibands and magnetic fields. In this approximation, Embedded Image and is determined by the group velocity of charge carriers, v (26). Each BZ miniband effectively represents a different two-dimensional system with a different k-dependent velocity. If T is larger than the cyclotron gaps, as in our case, the Fermi step becomes smeared over several minibands, which all contribute to σxx. In this regime, one can define Embedded Image averaged over an interval of ±T around εF. We calculate Embedded Image using a representative miniband spectrum for a graphene-on-hBN superlattice, which was computed with the model developed in (29). The resulting conductivity is evaluated as (26)Embedded Imagewhere e is the electron charge, vF is the Fermi-Dirac velocity, and h is Planck’s constant (26). The only fitting parameter is τ, which we choose so that σxx fits the experimental values for Embedded Image (Fig. 3). For other p/q, the calculated σxx are shown by black dots. Furthermore, according to the classical magnetotransport theory (13), σxx near zero Beff should vary as σxx(Beff) = σxx(0)/(1 + αBeff2), where α is a p- and q-dependent coefficient. It can be evaluated (26) without extra fitting parameters (narrow black parabolas in Fig. 3). One can see that the theory provides qualitative agreement for the observed experimental peaks. The derived values of τ yield σxx(B = 0) ≈ 20 mS, again in qualitative agreement with experiment. It would be unreasonable to expect any better agreement because of the limited knowledge about the graphene-on-hBN superlattice potential (20, 29) and the used τ approximation. The observed exponential T dependence of BZ oscillations (detailed in fig. S4) can also be understood qualitatively as arising from scattering on acoustic phonons such that the scattering length Embedded Image becomes shorter than the characteristic size, aq, of supercells responsible for the q-peak in conductivity (26).

To conclude, graphene superlattices exhibit a distinct quantum oscillatory phenomenon that can be understood as repetitive formation of different metallic systems, the BZ minibands. At simple fractions of Embedded Image, charge carriers effectively experience zero magnetic field, which results in straight rather than curved (cyclotron) trajectories. Straighter trajectories lead to weaker Hall effect and higher conductivity. The smooth background (varying over many q) is attributed to trajectories that involve transitions between different minibands and effectively become curved. The reported oscillations do not require monochromaticity, which allows them to persist up to exceptionally high T, beyond the existence of LLs. The extrapolation of the observed T dependences (fig. S4B) suggests that the quantum oscillations may be observable even at 1000 K. Further theory is required to understand details of temperature, field, and concentration dependences of BZ oscillations; the origin of the electron-hole asymmetry of phonon scattering; the behavior of higher-order fractions; and the effect of inter-miniband scattering, which is responsible for the non-oscillating background.

Supplementary Materials

Supplementary Text

Figs. S1 to S10

References (3145)

References and Notes

  1. See supplementary materials.
  2. Acknowledgments: This work was supported by the European Research Council, Lloyd’s Register Foundation, the Graphene Flagship, and the Royal Society. R.K.K. and E.K. acknowledge support from the Engineering and Physical Sciences Research Council, D.A.B. and I.V.G. from the Marie Curie program SPINOGRAPH, and S.V.M. from the Russian Science Foundation and National University of Science and Technology (MISiS).
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