Ballistic miniband conduction in a graphene superlattice

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Science  30 Sep 2016:
Vol. 353, Issue 6307, pp. 1526-1529
DOI: 10.1126/science.aaf1095


Rational design of long-period artificial lattices yields effects unavailable in simple solids. The moiré pattern in highly aligned graphene/hexagonal boron nitride (h-BN) heterostructures is a lateral superlattice with high electron mobility and an unusual electronic dispersion whose miniband edges and saddle points can be reached by electrostatic gating. We investigated the dynamics of electrons in moiré minibands by measuring ballistic transport between adjacent local contacts in a magnetic field, known as the transverse electron focusing effect. At low temperatures, we observed caustics of skipping orbits extending over hundreds of superlattice periods, reversals of the cyclotron revolution for successive minibands, and breakdown of cyclotron motion near van Hove singularities. At high temperatures, electron-electron collisions suppress focusing. Probing such miniband conduction properties is a necessity for engineering novel transport behaviors in superlattice devices.

In solids, the quantum nature of electrons generates band structure, which controls conduction and optical properties. Similarly, longer-period superlattices generate minibands that disperse at a finer energy scale over a reduced Brillouin zone, enabling phenomena such as negative differential conductance and Bloch oscillations (13). However, fabricating long-range periodic patterns that strongly modulate the potential to form well-separated minibands without undermining the material quality and electron coherence remains challenging. Most experiments on laterally patterned semiconductor heterostructures have revealed classical commensurability effects (46), which do not require well-formed and separated minibands. Despite evidence for Fermi surface reconstruction in a patterned superlattice, details of Fermi surfaces were obscured by poor separation between minibands and consequent magnetic breakdown across weakly avoided crossings (7).

The arrival of high-quality graphene/h-BN van der Waals heterostructures with misalignment angle below 1° (8, 9) has drastically changed the situation. In such systems, the periodic potential for electrons in graphene is imposed by the hexagonal moiré pattern generated by the incommensurability and misalignment between the two crystals (1012). Formation of minibands for Dirac electrons has been demonstrated by scanning tunneling (13), capacitance (14), and optical (15) spectroscopies, as well as magnetotransport (1619). These studies have elucidated the electronic structure known as the Hofstadter butterfly, which emerges in a quantizing magnetic field (20). By contrast, a small magnetic field may be treated semiclassically. Then the connection between the miniband dispersion ε(k) and transport properties is established by the equations of motion for an electron in an out-of-plane magnetic field B=Bz^,ħv=εk,   ħk˙=eE+eBz^×v (1)where e is the charge on the electron, ħ is the Planck constant divided by 2π, and the relation between carrier velocity v and momentum ħk is approximately v = vk/k (v ≈ 106 m/s) close to the Dirac point of graphene’s spectrum (10, 11, 13, 14).

The shape of the cyclotron orbit in a two-dimensional (2D) metal is a 90° rotation of the shape of the Fermi surface, and the carrier revolves along it clockwise or counterclockwise. Electron trajectories near the boundary of a metal open into skipping orbits (21), which drift in the direction determined by the effective charge of the carrier. These skipping orbits bunch along caustics (2227), leading to the transverse electron focusing (TEF) effect (22, 23). Experimentally, TEF takes place when the magnetic field is tuned such that caustics of skipping orbits, emanating from an emitter E, end up at a collector C, located at position x = L along the boundary. Then a voltage VC is induced at C, proportional to the current IE injected into E. Figure 1B illustrates skipping orbits and caustics in a material with an isotropic Fermi surface, such as unperturbed graphene near the Dirac point, where TEF occurs for B = Bj ≡ 2jħkFeL (for j = 1, 2, …). An equidistant series of peaks (oscillations) appears in the focusing “spectrum”—the nonlocal magnetoresistance VC/IE(B) (Fig. 1C, lower trace), from which the Fermi momentum ħkF and the sign of effective charge ±e can be inferred. TEF was initially used to study the Fermi surfaces of bulk metals (22, 28) and was later extended to 2D systems (23), including graphene (29).

Fig. 1 Experimental concept.

(A) Schematic of the experiment overlaid on a photo of the device. The h-BN/graphene/h-BN/bilayer graphene heterostructure is shown in green, the SiO2 substrate is in purple, and the dashed line denotes the upper boundary of the graphene flake. In the electrical measurement configuration applied to obtain the data in Fig. 2B, the two leftmost contacts are grounded to act as absorbers. We inject current into the left local contact L and measure the voltage difference between two local contacts, M and R. Arrows depict skipping orbits a hole would take if injected at normal incidence with B = B1 ≡ 2ħkF/eL (red) or B2 ≡ 4ħkF/eL (blue). (B) Ensemble of simulated skipping orbits emanating from an emitter (red star). Electron trajectories bunch along caustics (red dashed curves) and focus onto an equidistant array of points at the boundary. Scale markers show the cyclotron diameter 2RC = 2ħkF/eB. (C) Transverse electron focusing (TEF) spectra collected at a single voltage probe M [VM/IL(B), lower trace] and differentially between voltage probes M and R [(VMVR)/IL(B), upper trace], with n = –1.1 × 1012 cm–2 and T = 1.55 K. The first, third, and sixth focusing peaks are labeled. Taking the differential measurement of the spectrum does not shift peak positions, because the device geometry partially shields R from being reached by skipping orbits from L, such that oscillations of VR are much weaker.

Here, we report the observation of TEF in a moiré superlattice at the interface between graphene and h-BN in a van der Waals heterostructure (from top to bottom) h-BN/graphene/h-BN/bilayer graphene assembled on an SiO2 substrate. One of the h-BN layers (we do not know which) is aligned with graphene to better than 1°, forming a moiré pattern with a period of 14 nm (30). We use the bilayer graphene as an electrostatic gate, tuning electron density in the superlattice by applying voltage Vg to it. The device (Fig. 1A) has three etched local contacts along the linear sample boundary. Two other ohmic contacts are grounded and act as absorbers. We measure the multiterminal, nonlocal resistance (VMVR)/IL at our base temperature T = Tbase = 1.55 K. Figure 2B is the resulting map of (VMVR)/IL as a function of B and Vg, exhibiting TEF spectra and their evolution as a function of electron density n. When the Fermi level in graphene is close to the Dirac point at Vg = –0.4 V, the superlattice spectrum is almost isotropic, and kF=π|n|. Hence, the TEF spectra show TEF oscillations with peaks at Bj=(2j/±eL)π|n| (dashed curves in Fig. 2B) as in unperturbed graphene (29). The observation of TEF confirms that electrons travel ballistically from the emitter to the collector. The visibility of as many as eight focusing peaks (Figs. 1C and 2B) shows that carrier reflection at the boundary is mostly specular: Each peak is lower than the last by a factor of the probability of diffuse scattering (22). Quantum effects are suppressed because the thermal length ħvF/kT is shorter than the emitter-collector separation L.

Fig. 2 TEF spectra at base temperature.

(A) Miniband structure of the graphene/h-BN superlattice, calculated as in (12, 30). Each miniband for which we observe TEF is labeled. This dispersion results from a symmetric moiré perturbation: ε+ = 17 meV and ε = 0 meV; this choice gives the best match between experiment (B) and theory (C) (30). Equipotential contours are shown; the dashed contours are at the energy levels of saddle-point VHSs. (B) TEF spectra as a function of gate voltage Vg. T = 1.55 K. The plotted ratio (VMVR)/IL is measured as depicted in Fig. 1A. Black dashed curves indicate B1, B3, and B6, which are some of the peak positions expected when the Fermi level is close to the Dirac point. Green dashed lines indicate the abrupt termination of TEF caused by the breakdown of cyclotron motion at each VHS. Dashed arrows from (B) to (A) point to the energy levels (dashed contours) of the corresponding VHS; voltage values are labeled by the miniband in which the breakdown occurs (e.g., Vc1 for the breakdown of cyclotron motion in C1). Dotted lines show selected densities (I, II, III, and IV) that place the Fermi level in minibands C2, C1, V1, and V2, respectively. (C) TEF spectra as a function of Vg, calculated from the dispersion in (A) and Eq. 1 (30).

At higher densities approaching four electrons (or holes) per moiré unit cell, the Fermi level is near the first minibands’ outer edges, and TEF spectra reflect the modification of electronic states by the superlattice potential. A candidate miniband structure from the model family proposed in (12) is shown in Fig. 2A with relevant minibands labeled. Carrier dynamics in the form of skipping orbits and caustics are represented using ensembles of simulated electron trajectories in Fig. 3. The map of measured TEF spectra (Fig. 2B) matches the theoretically simulated spectra (Fig. 2C) obtained by applying Eq. 1 to the electrons emitted into the minibands of Fig. 2A from a local emitter at the sample edge (30).

Fig. 3 Simulated skipping orbits.

Representative ensembles of simulated skipping orbits emanating from an emitter (red star) at the boundary of the graphene/h-BN superlattice possessing the miniband dispersion of Fig. 2A, for selected electron densities I, II, III, and IV marked in Fig. 2B. The corresponding Fermi surfaces are in minibands C2, C1, V1, and V2, respectively, and each is drawn as a thick, dashed constant-energy contour on the color map of the dispersion. The magnetic field points out of the page, so electron-like carriers turn counterclockwise and their skipping orbits drift left, and holelike carriers do the opposite. Red dashed curves mark caustics.

In addition to TEF of electrons in C1 and holes in V1, both theory and experiment show focusing of holes in C2 and electrons in V2: The carrier charge is reversed relative to the corresponding (conduction or valence) band of graphene. For Vg > Vc3, where Vc3 is the lower band edge of C3, the electron-like pocket of C3 overlaps in energy with the holelike pocket of C2, leading to TEF oscillations for both signs of B. TEF oscillations abruptly terminate at gate voltage values Vv1, Vc1, Vv2, and Vc2, which coincide with the passing of the Fermi level across the saddle-point van Hove singularities (VHSs) at which the constant energy contour of the miniband dispersion percolates across all repeated Brillouin minizones. At these saddle points, cyclotron orbits experience an extreme variant of magnetic breakdown termed orbital switching (31), opening up into runaway trajectories such that electrons do not follow skipping orbits. In the ranges Vv2 < Vg < Vv1 and Vc1 < Vg < Vc2, the Fermi surface consists of small and highly anisotropic pockets just above or below the secondary Dirac points. Thus, even the theoretically calculated pattern in Fig. 2C is both weak and dense; experimental observation of these pockets is obscured because of smearing by finite emitter and collector widths and suppression by partial diffusivity of reflection from the edge.

The positions of saddle points Vv1, Vc1, Vv2, and Vc2 can be directly compared to miniband models. We tested the observed ratios (Vv1Vv2)/(Vc1Vv1) and (Vc2Vc1)/(Vc1Vv1) against predictions from a family of moiré superlattice models parameterized by strengths of inversion-symmetric (ε+) and antisymmetric (ε) interlayer coupling between graphene carbons and the boron and nitrogen sites of h-BN (30, 32). The best match to experimental data corresponds to an inversion-symmetric moiré perturbation with ε+ ≈ 17 meV, ε ≈ 0 (fig. S2), which we used to calculate the miniband structure, electron dynamics, and TEF maps in Figs. 2 and 3. This value for ε+ is similar to previous estimates from optical spectroscopy (15).

We can learn more about carrier dynamics, in particular the effect of their scattering, by examining the temperature dependence of TEF oscillations (28). Throughout the probed temperatures and densities, the suppression of TEF upon heating (Fig. 4A) is faster than what could be expected from merely thermal broadening of injected electron momenta, as |kkF| ~ kBT/ħvF << kF. For quantitative analysis, we determine the area A1 under the first (j = 1) focusing peak and interpret the ratio A1(T)/A1(Tbase) as the fraction of electrons exp(–πL/2vFτ) that propagated ballistically from the emitter to the collector, along the semicircular cyclotron trajectory of length πL/2 that touches the caustic near the collector, even though the electrons scatter with a characteristic rate τ–1. Figure 4B shows the temperature dependence of this effective scattering rate, extracted from the data according to the formula τ(T)–1 = –2vFL log[A1(T)/A1(Tbase)]. The experimentally observed dependence τ(T)–1T2 points toward an electron-electron (e-e) scattering mechanism for the suppression of TEF oscillations upon heating, the same mechanism responsible for the evolution of electronic transport from the ballistic to the viscous regime (3335). Theoretical analysis of spreading of a narrow beam of electrons due to low-angle scattering processes, governed by the Thomas-Fermi–screened e-e interaction, shows that for TT* (where kBT*=2vFkF/πL), the decay of TEF signal can be described by a scattering rate τe-e1(kBT)22ħvFkFlog(3.6Lw) (2)where w is the width of the emitting and collecting contacts (30). Figure 4B shows the theoretical values [calculated without free parameters (30)] of τe-e1, including the theoretically predicted crossover to a slower scattering rate for T > T*. The rate of scattering by phonons, τph1, can be inferred from temperature-dependent sheet resistivity of similar encapsulated graphene/h-BN heterostructures (35, 36) (Fig. 4B). This inferred τph1 is much lower than both the experimentally measured value of τ(T)–1 and the calculated value of τe-e1, so it appears that low-angle e-e scattering is the dominant mechanism for suppression of TEF.

Fig. 4 Temperature dependence of TEF spectra.

(A) VM/IL(B) minus a smooth background (30), for the electron densities I, II, III, and IV (marked in Fig. 2B) and temperatures up to 150 K. Increasing the temperature suppresses TEF. (B) Circles and triangles denote effective scattering rates τ(T)–1 extracted from the amplitude of TEF oscillations in (A). The heavy green curve shows the theoretical scattering rate τe-e1 related to the electron-electron interaction. Square and diamond symbols denote the rate of scattering by phonons τph1 inferred from temperature-dependent sheet resistivity reported in Bandurin et al. (35) and Wang et al. (36). The detection limit set by noise is shaded.

The direct observation and manipulation of ballistic transport is a powerful probe of the low-energy physics of an electron system. Here, the quasiparticles propagate freely from the emitter to the collector through ballistic trajectories as long as πL/2 = 10 μm, which is 700 in dimensionless units of the superlattice period. Ballistic motion of ultracold atoms has been seen in homogeneous optical lattices as large as 100 unit cells (37), but in the solid state, the mean free path of electrons in semiconductor 1D superlattices has been limited to 10 unit cells (38). Our experiment elucidates the key features of miniband electron dynamics in a moiré superlattice and points toward further explorations of novel transport effects. For instance, the saddle-point VHS could host exotic effects caused by enhanced electron-electron interactions (19, 39), and valley-contrasting physics could be accessed by taking advantage of the severe trigonal warping of minibands (40). For technology, such a clear validation of the miniband conduction properties suggests that graphene/h-BN (and perhaps other moiré superlattices) may be a practical platform for devices based on miniband physics. Efficient photocurrent generation at the edge of a graphene superlattice in a magnetic field (41) may be caused by the skipping orbits we have observed; furthermore, THz devices such as the Bloch oscillator can benefit from the much longer scattering times in this system.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S6

References (4253)

References and Notes

  1. See supplementary materials on Science Online.
Acknowledgments: We thank A. L. Sharpe and G. Pan for technical assistance and W. A. Goddard, M. S. Jang, H. Kim, A. Maharaj, S. Goswami, E. J. Heller, L. S. Levitov, J. C. W. Song, and A. Benyamini for discussions. Supported by a Stanford graduate fellowship and a Samsung scholarship (M.L.) and by ERC Synergy grant Hetero2D, the EU Graphene Flagship Project, and a Lloyd Register Foundation nanotechnology grant (J.R.W. and V.I.F.). Work done at Stanford was funded in part by Air Force Office of Scientific Research award FA9550-16-1-0126 and by Gordon and Betty Moore Foundation grant GBMF3429 and was performed in part in the Stanford Nano Shared Facilities.

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