Switchable friction enabled by nanoscale self-assembly on graphene

Graphene monolayers are known to display domains of anisotropic friction with twofold symmetry and anisotropy exceeding 200%. This anisotropy has been thought to originate from periodic nanoscale ripples in the graphene sheet, which enhance puckering around a sliding asperity to a degree determined by the sliding direction. Here we demonstrate that these frictional domains derive not from structural features in the graphene but from self-assembly of environmental adsorbates into a highly regular superlattice of stripes with period 4–6 nm. The stripes and resulting frictional domains appear on monolayer and multilayer graphene on a variety of substrates, as well as on exfoliated flakes of hexagonal boron nitride. We show that the stripe-superlattices can be reproducibly and reversibly manipulated with submicrometre precision using a scanning probe microscope, allowing us to create arbitrary arrangements of frictional domains within a single flake. Our results suggest a revised understanding of the anisotropic friction observed on graphene and bulk graphite in terms of adsorbates.

Suspended graphene monolayers display an intrinsic roughness of order 1 nm as a consequence of their twodimensionality [1].An early scanning tunneling microscopy (STM) study of graphene supported by an oxidized silicon wafer did not observe this intrinsic roughness, suggesting that supported graphene instead inherits most of its observed roughness from the substrate [2].Subsequent scanned probe studies of graphene on atomically flat substrates such as mica [3] and multilayer hBN [4] indeed found graphene surface roughnesses comparable to substrate roughness; the clear resolution of moiré patterns further suggested the highly conformal behavior of graphene on hBN [4].Recent STM studies have shown that graphene tends to form nanoscale periodic ripples in the vicinity of grain boundaries [5] or substrate step-edges [6], or when suspended over a nanoscale trench [7], but in none of these cases does the periodic rippling extend more than tens of nanometers away from the nucleating defect.
We demonstrate that nanoscale periodic ripples spontaneously form throughout the topmost graphene or hBN layer of exfoliated flakes and graphene/hBN heterostructures prepared by mechanical assembly or chemical growth.These ripple superlattices are not in general nucleated by defects, but rather appear in response to biaxial compressive stress from the substrate supporting the flake or heterostructure.Here we present data for flakes and heterostructures on thermally oxidized silicon wafers, the most widely used substrate, but we have observed equivalent ripple superlattices on other substrates, suggesting that periodic rippling is a generic behavior of graphene and hBN sheets (Fig. S1 of Supplemental Material [8]).
In nearly all of the as-deposited graphene flakes that we prepared by standard techniques, we could directly resolve one-dimensional superlattices of ripples at room temperature in air with an atomic force microscope (AFM) operated in tapping mode [8].The topmost graphene layer of a typical flake forms multiple domains of superlattices with distinct ripple axes [Fig.1(a)], but the period (usually 3 to 6 nm), amplitude (10 to 100 pm, peak to trough), and angular orientation of the ripple superlattice within a given domain do not measurably vary (Fig. S2, [8]).The ripple axes obey crystal symmetry: within a pristine single layer, we observe at most three distinct ripple axes, which are rotationally separated by 60 • .This phenomenology is reminiscent of a study of monolayer graphene by Choi et al. [9], which reported domains of anisotropic friction that obeyed a 180 • rotational symmetry, and found at most three distinct lowfriction axes per flake, rotationally separated by 60 • .A comparison of the high-resolution tapping-mode topography signal [Fig.1(a)] and the friction signal in contactmode AFM [Fig.1(b)] reveals that the ripple superlattices are microscopically responsible for the anisotropic friction in graphene per the puckering mechanism suggested by Choi et al. [9]: the ripples pucker most as the tip scans perpendicular to a ripple axis, creating the most friction.
After thermal cycling to liquid nitrogen temperatures or below, domains of periodic ripples appeared at room temperature on the exposed layers of all graphene flakes, hBN flakes, and graphene/hBN heterostructures that we tested.We performed these thermal cycles by immersion in liquid nitrogen, as well as by more gentle cooldown in three different cryostats with vastly different cooling rates and atmospheric conditions, on both unprocessed samples and etched graphene/hBN samples with pat- terned electrodes.Conversely, we found that thermal cycling to above 200 • C in oxygen or hydrogen environments would often remove the ripples [8].We examined flakes and heterostructures up to 200 nm thick, and found no clear dependence of the superlattice period, amplitude, or phenomenology on sample thickness, or on whether the sample was graphene or hBN-apart from the fact that the top layer of almost all graphene flakes, but very few hBN flakes, was periodically rippled as deposited, prior to thermal cycling.Although we have not characterized the frictional properties of hBN, in many studies with tapping-mode AFM we have never observed more than three distinct ripple axes within a single hBN crystal.A comparison of the ripple axes to the moiré lattice vectors in perfectly-aligned graphene/hBN heterostructures prepared by van der Waals epitaxy matches the ripple axes in both graphene and hBN to the armchair axes to within 3 • (Fig. S3, [8]).
To characterize the evolution of periodic ripples with temperature, we studied the surface topography of an asdeposited hBN flake with a variable-temperature AFM operated in non-contact mode in ultrahigh vacuum [8].Prior to any thermal cycling, the as-deposited hBN flake at 300 K was flat [Fig.2(a)]: the measured rms roughness was 10 pm at a lateral resolution of order 1 nm (limited by tip sharpness).After cooling to 250 K, we observed periodic ripples on the exposed hBN layer [Fig.

2(b)]
, which persisted to 225 K [Fig.2(c)], but could no longer be resolved by 200 K [Fig.2(d)] or below.We carried out a similar experiment on an hBN flake that was initially periodically rippled at 300 K; again the ripples disappeared between 250 K and 200 K as the temperature decreased.The periodic ripples are also sup-pressed at low temperature in graphene: no hint of these ripples has ever been reported in the many atomicallyresolved, low-temperature STM studies of graphene on various substrates [2,4,10,11].Notably, the periodic ripples did not reappear as we stepped the temperature back up to 300 K in any of our variable-temperature AFM experiments-a sharp contrast with our many thermal cycles in other cryostats, which always produced periodic ripples at 300 K.This discrepancy may be related to the relatively high minimum temperature accessible in the variable-temperature AFM (110 K), the rates of temperature change, or the ultrahigh vacuum environment itself.
The creation and subsequent suppression of the periodic ripples with decreasing temperature can be understood in the context of substrate-induced compressive stress [Fig.2(e)], following similar explanations for various structural distortions in graphene [7,[12][13][14].At room temperature, the unstrained (or minimally strained) stack of graphene and/or hBN layers rests on an oxidized silicon wafer.Because the thermal expansion coefficient of the substrate is positive while the in-plane thermal expansion coefficients of graphene and hBN are negative near room temperature [15][16][17][18], lowering the temperature applies biaxial compressive stress to the flake or heterostructure.This stress is transmitted through the stacked layers to the top layer, which relieves some of its strain by rippling; while additional layers near the top could perhaps form small ripples [not shown in Fig. 2(e) for simplicity], deeper layers would pay a high energy cost to ripple, so they instead remain flat and compressively strained.As the temperature is further lowered, the pinning force between the topmost layer and the layer beneath it dominates over thermal fluctuations, suppressing the ripples, and returning the topmost layer to a fully strained configuration.Upon warming, thermal fluctuations overcome the pinning force, and the top layer springs out of its strained configuration to form domains of periodic ripples.The suppression and reappearance of ripples under this model is inherently hysteretic: the strained or rippled configuration at a given temperature may be metastable, and layers may slip relative to one another during the thermal cycle.An issue of great practical relevance is how the onedimensional ripple superlattice interacts with the moiré superlattice in assembled heterostructures of graphene on hBN.The moiré superlattice, which relies on close contact and rotational alignment between the graphene and the underlying hBN, opens a gap around the Dirac point, as well as high-energy minigaps in the graphene spectrum [19][20][21].In all five rotationally-aligned, assembled graphene/hBN heterostructures that we studied, we found that instead of competing with the moiré superlattice, the periodic ripples are simply superimposed on the moiré pattern at room temperature [Fig.3(a)].At larger length scales, most samples segregate into domains of periodic ripples [Fig.3(b,c)]; domain sizes range from tens of nanometers to several microns.The fast Fourier transform [Fig.3(d)] of the topography signal contains a set of six peaks from the moiré superlattice, as well as two peaks per distinct ripple axis contained in the scan win-dow.The ripple superlattice wavevector is here approximately parallel to the moiré wavevector, but in other samples these wavevectors are not aligned (Fig. S4, [8]).Rather than adjusting for commensuracy, the ripple superlattice and moiré superlattice appear not to influence each other: their relative orientation in each sample is consistent with the ripple axes being locked to the armchair axes (Fig. S5, [8]).The absence of any measurable interaction between ripple and moiré superlattices is especially striking in light of recent evidence that even at room temperature, the graphene lattice within the moiré hexagons is strained to match the hBN lattice [11].The ripple superlattice may eliminate any such atomic-scale adjustment.
Our transport measurements revealed that any effect of the periodic rippling on the low-energy, low-field transport properties of graphene/hBN heterostructures is subtle (Fig. S6, [8]), despite theoretical predictions of the formation of midgap states at zero energy in periodically rippled graphene [22,23].Stronger effects should be observed in transport as the Fermi wavelength approaches the wavelength of the ripples; the required carrier densities (near 10 13 cm −2 ) are within reach using suitable dielectrics or electrolyte gating.The nanoscale one-dimensional superlattice could be harnessed to control graphene's band structure: under applied fields, periodically rippled graphene should experience periodic scalar and vector potentials, which are predicted to open gaps [22,24,25] and renormalize the Fermi velocity [26].The ripple superlattice should also impact the mechanical properties of graphene and hBN, and may be particularly relevant for explaining discrepancies in the measured thermal expansion coefficient of graphene [14,16].We gratefully acknowledge Byong-man Kim and Ryan Yoo of Park Systems for verifying the presence of periodic ripples in our samples using their Park NX10 AFM.
Flakes of graphene and hBN were prepared on silicon wafers with 90 nm or 300 nm of thermal oxide by mechanical exfoliation (3M Scotch 600 Transparent Tape or 3M Scotch 810 Magic Tape) from bulk crystals.The substrates were not exposed to any chemical processing following thermal oxidation.For graphene deposition, we used both Kish graphite and highly-oriented pyrolitic graphite (HOPG), and observed no relevant differences in superlattice phenomena between these two graphite sources.We also prepared graphene flakes on 200 nm of Au(111) on mica (Phasis, Switzerland) by mechanical exfoliation from Kish and HOPG crystals.
Assembled heterostructures of graphene on hBN were prepared using both wet [1] and dry transfer methods.Because the dry method cannot easily produce heterostructures with an exposed layer of graphene, we extended the technique pioneered by Wang et al. [2] by introducing the following procedure to invert the heterostructures.First, monolayer graphene atop multilayer hBN was prepared on a polypropylene carbonate/polydimethylsiloxane (PPC/PDMS) stamp as described by Wang et al.This heterostructure was then released from the PPC/PDMS stamp (graphene side down) onto a silicon-supported polymer stack consisting of 300 nm of polymethyl methacrylate (PMMA) atop 1 µm of PPC.The release was performed at 80 • C so that the heterostructure cleanly transferred from the PPC/PDMS stamp onto the PMMA surface without releasing the PPC itself from the PDMS stamp.The PMMA/PPC membrane was then mechanically peeled from its supporting silicon wafer and deposited (PPC side down) onto PDMS.Finally, the stack of hBN/graphene/PMMA/PPC on PDMS was brought into contact (hBN side down) with the desired substrate and subsequently released from the PDMS by heating to 120 • C. The majority of the polymer residue was cleaned off in acetone at 80 • C. Remaining residues were removed by annealing the sample in a tube furnace for 4 hours at 500 • C under continuous flow of oxygen and argon.
All samples that we studied with transport measurements were assembled heterostructures of monolayer graphene on hBN with multilayer graphene back gates; some of these samples were produced by aligning the hBN/graphene/PMMA/PPC/PDMS stack to a target multilayer graphene flake on silicon dioxide, while others were produced by first assembling a graphene/hBN/multilayer-graphene/hBN heterostructure on PPC/PDMS and then inverting this fourcomponent heterostructure by the procedure described above.To make electrical contact to the samples, we used electron-beam lithography to define the desired pattern in PMMA, and deposited 1-3 nm Cr/100 nm Au by electron-beam evaporation.We next etched the samples into a Hall bar geometry in a remote oxygen plasma using a PMMA mask defined by electron-beam lithography.After fabrication, we removed resist residues by annealing in a tube furnace for 4 hours at 325 • C under continuous flow of hydrogen and argon.
We prepared epitaxial graphene heterostructures on oxidized silicon substrates by mechanical exfoliation of hBN followed by graphene growth at 500 • C by a remote plasma-enhanced chemical vapor deposition process described previously [3].

THERMAL CYCLING
Thermal cycling to 77 K or below was performed using four different instruments.(1) Liquid nitrogen bath.Samples were glued with PMMA to a ceramic chip carrier and then rapidly immersed in a bath of liquid nitrogen.We waited for the liquid nitrogen to stop boiling around the sample, and then waited about 60 additional seconds before rapidly removing the sample to room temperature.Condensation was blown off the sample with dry air.(2) Quantum Design PPMS.Samples were glued with PMMA to a printed circuit board with an integrated thermometer and loaded into the PPMS.The sample chamber was evacuated with a turbo pump and then sealed for the duration of the experiment.We used various rates of temperature change between 0.5 K/min and 10 K/min, and typically reached a base temperature of 10 K. (3) Variable-temperature insert.Samples were glued with PMMA to a ceramic chip carrier and placed in a load lock, which was evacuated with a turbo pump.The pump was then removed and the samples were exposed to continuous flow of boiloff-grade helium vapor during measurement.The typical rate of temperature change was 20 K/min, with a base temperature of 1.5 K. (4) Dilution refrigerator.Samples were glued with PMMA to a ceramic chip carrier and placed in a load lock, which was evacuated with a turbo pump.The pump was then removed and the sample holder was brought into contact with a metal plate cooled to 4 K by a pulse tube cooler.The sample cooled down to 4 K over several hours in a low pressure of helium gas.The base temperature was 20 mK.Samples were warmed to room temperature over two hours by rapidly removing the sample holder from base temperature to the evacuated (but not continuously pumped) load lock outside the inner vacuum chamber of the refrigerator.
Thermal cycling to above room temperature was performed routinely for assembled heterostructures as part of the cleaning procedures described above.Our standard anneals to 325 • C or 500 • C would result in periodically rippled flakes and heterostructures at room temperature about half of the time; samples were cooled down to room temperature at an estimated 5 to 10 K/min.All of the epitaxial heterostructures that we studied were periodically rippled following 500 • C growth.For the purposes of some transport measurements, we successfully suppressed the ripple superlattice to below our experimental resolution by placing the sample on a hot plate at 200 • C for several minutes and then immediately removing the sample from the hotplate.

AMBIENT AFM MEASUREMENTS
For AFM under ambient conditions, we used a Park XE-100 AFM.To resolve the periodic ripples in tapping mode, we used sharp silicon probes (MikroMasch Hi'Res-C15/Cr-Au) with a nominal 1 nm spike radius, a typical resonant frequency of 265 kHz, and a typical cantilever Q of 400.Because even a slightly blunt tip will obscure the periodic ripples, we frequently checked the sharpness of our tips by scanning on a sample with a ripple superlattice that we had previously characterized.

VARIABLE-TEMPERATURE AFM MEASUREMENTS
To measure the behavior of the ripple superlattice below room temperature, we used an Omicron varibletemperature AFM/STM operating in ultrahigh vacuum (8 × 10 −11 mbar).The sample stage was cooled by a copper braid attached to a cold sink held at low temperature by continuous flow of liquid nitrogen; by this method we achieved a base temperature of 110 K.We used the same sharp probes as for ambient AFM (MikroMasch Hi'Res-C15/Cr-Au).In ultrahigh vacuum, the cantilever Q reached 5000, which significantly restricted scan speed for tapping mode; we therefore used on-resonance frequencymodulation mode, imaging at a typical frequency shift of -30 Hz.All images were collected by biasing the tip with respect to the sample to nullify the contact potential difference.
We cooled down three different oxidized silicon pieces with hBN flakes; two of these pieces had graphene flakes and assembled graphene/hBN heterostructures as well, although below room temperature we only scanned on hBN.To distinguish periodic ripples from noise, we verified that the ripples behaved as expected under large changes in scan speed, scan size, and scan axis.The absence of a moiré pattern in hBN prevented us from correcting our images for thermal drift, which creates an approximately affine distortion of the images.These distortions can easily change the apparent angle of periodic ripples by tens of degrees, and in turn change the apparent ripple period by tens of percent.Furthermore, because of uncontrolled lateral movement (∼ 1 µm) of the tip relative to the sample between temperature setpoints, we were not able to return at every temperature to precisely the same location on the relatively featureless hBN flakes that we studied.For these reasons, we hesitate to make any claims about the apparent change of period and ripple axis between 250 K and 225 K in Fig. 2.
While warming up to 300 K from a base temperature between 110 K and 175 K, we could not resolve a ripple superlattice at any of our several temperature steps (including 300 K) on all three hBN samples that we studied.We later verified by ambient AFM that no ripple superlattices could be resolved on any of the layered materials on these silicon pieces, including graphene and graphene/hBN heterostructures.

ERROR BARS AND LATERAL CALIBRATION
All values quoted for moiré wavelength and angular orientation are extracted from the FFT of the AFM images.AFM images of all heterostructures described in this study are corrected for thermal drift by performing an affine transformation to produce regular moiré hexagons (we used the free software Gwyddion, available at gwyddion.net).All error bars reflect the full width at half maximum (FWHM) of the peaks in the FFT; for instance, 12.0 ± 0.5 nm means that the FWHM of the peak maps to 1 nm in real space.The lateral scale of the Park XE-100 ambient AFM was calibrated by measuring the moiré wavelength of perfectly-aligned graphene/hBN heterostructures grown by van der Waals epitaxy and defining this wavelength to be 13.6 nm.This definition corresponds to the assumption made for Fig. S5 that the lattice constants for hBN and graphene are a hBN = 0.25 nm and a graphene = a hBN /1.018.The lateral scale of FIG.S1.Ripple superlattice on a gold/mica substrate.Tapping-mode topography signal, differentiated along the scan axis (horizontal), at the edge of an as-deposited, fewlayer graphene flake on gold on mica.A ripple superlattice is visible on the graphene (upper left half of image) with period 3.8 nm; peak-to-trough amplitude in the topography signal is 20 pm.Scale bar is 50 nm.
the Omicron variable-temperature AFM was calibrated to the lateral scale of the Park XE-100 by measuring the moiré pattern of the same sample in both systems.

CONDITIONS FOR DIRECT OBSERVATION OF THE RIPPLE SUPERLATTICE
Graphene has been heavily studied by a variety of scanned probes, many with atomic resolution.The fact that the ripple superlattice has not previously been imaged may be surprising, but is understandable given that to observe the nanoscale ripples requires extremely fine lateral and vertical resolution.In AFM, the ripples cannot be resolved without a very sharp tip (these are expensive and fragile, and are therefore not commonly used) and excellent vibration isolation.While atomic resolution is routinely available in STM, these experiments are usually performed in ultrahigh vacuum, for which standard procedure is to first bake the sample to ∼ 300 • C, which may suppress the superlattice in subsequent measurements at room temperature.Most atomic-resolution STM studies have moreover focused on low temperature [4][5][6][7], where based on our variable-temperture AFM measurements we expect the ripple superlattice to be suppressed.Furthermore, even under appropriate imaging conditions, the periodic ripples can easily be confused for noise given their high periodicity and small amplitude.Unless one is specifically looking for a nanoscale structural superlattice, the periodic ripples are challenging to identify.

RIPPLE SUPERLATTICE IN GRAPHENE AND HBN ON SUBSTRATES OTHER THAN SILICON
To determine the relevance of the specific choice of substrate to the ripple superlattice phenomenon that we have observed on oxidized silicon wafers, we exfoliated graphene flakes onto 200 nm of Au(111) on mica.We observed the ripple superlattice on two flakes out of approximately ten that we checked (Fig. S1).After thermal cycling by immersion in liquid nitrogen, we found the same two flakes to display the ripple superlattice, while none of the other flakes formed a ripple superlattice.Evidently the formation of ripple superlattices is heavily favored by certain substrates, consistent with our explanation based on substrate-induced biaxial compressive stress.Nonetheless, when a superlattice forms it is clearly a property of the exfoliated crystal itself, rather than reflecting structure in the particular substrate.

RIPPLE SUPERLATTICE IN OTHER LAYERED COMPOUNDS
The appearance of superlattices on both graphene and hBN sheets suggests that such periodic rippling might be observed in other layered compounds.We exfoliated flakes of molybdenum disulfide onto silicon dioxide by the same procedure that we used for graphene and hBN, but found no evidence of a ripple superlattice on flakes of varying thicknesses, both before and after thermal cycling by immersion in liquid nitrogen.

UNIFORMITY OF RIPPLE SUPERLATTICE WITHIN A SINGLE DOMAIN
Single domains of ripple superlattices may span many microns.Here we illustrate the extreme uniformity of the ripple superlattice by analyzing a representative large area scan (scan window approximately 600 nm) over a single domain in hBN [Fig.S2(a)].To illustrate potential defects in the superlattice, we have deliberately chosen an image that contains a phase slip-but we emphasize that phase slips are uncommon away from domain boundaries or structural defects, and that phase slips could be induced by the moving tip rather than being a feature of an unperturbed sample.The peak-totrough amplitude of the ripples is 30 ± 10 pm throughout the scan window [Fig.S2(b)]; making precise quantitative statements about the homogeneity of the ripple amplitude over large areas is difficult because the tip often picks up and drops nanoscale particles, which temporarily affect the tip radius and therefore the apparent ripple amplitude.Our ability to resolve the angle of the ripple axis is limited by slowly-varying scanner drifts, which manifest as smearing of the superlattice peak in the fast Fourier transform (FFT) along the direction perpendicular to the scan axis [Fig.S2(c)].Here this smearing permitted an angular resolution of 4 • [Fig.S2(d)]; while scan parameters can be optimized for slightly improved angular resolution (for instance, by increasing the scan speed to mitigate thermal drifts), dramatic improvement is difficult.To this few-degree precision, we observe no changes in ripple axis within a domain.The period over a large area can be measured with very high precision: a radial cut through the superlattice peak yields a period 6.43 ± 0.04 nm [Fig.S2(e)].However, extremely slowly varing thermal drifts will cause an affine distortion of the image, which strongly impacts the accuracy of the measured period.

CRYSTALLOGRAPHIC ORIENTATION OF RIPPLES
In perfectly-aligned heterostructures of graphene grown on hBN by van der Waals epitaxy, the moiré lattice vectors are aligned with the lattice vectors of graphene and hBN.We can therefore determine the crystallographic orientation of the ripple axes by comparison to the moiré lattice vectors (Fig. S3).In all 25 epitaxial heterostructures that we studied, we found (with typi- cal precision ± 3 • ) only armchair ripple axes for both graphene and hBN.
In our assembled heterostructures of graphene and hBN with less than 1 • misalignment, we again found only three distinct ripple axes (to within our typical ± 3 • precision) on the exposed layers of a given heterostructure.We have not carefully studied the number of distinct ripple axes that appear on the exposed layers of misaligned heterostructures, in part because we require a clear moiré pattern to correct our images for affine distortion, but also because a surprisingly large fraction of our samples are nearly aligned despite no attempt by us to do so.
In nearly-aligned heterostructures, we find that the ripple axes and moiré lattice vectors need not be rotationally aligned (Fig. S4), suggesting that the moiré pattern and the ripple superlattice do not strongly influence each other.We extract the crystallographic orientation of the ripple axes using the calculated angular misalignment [8] between the moiré lattice vectors and the graphene lattice vectors.Our data rule out the possibility that the ripple axes are always parallel to the graphene lattice vectors (zigzag axes), since the measured angle between the ripple axes and the nearest moiré lattice vector does not follow the expected trend as a function of moiré superlattice wavelength [Fig.S5(a)].In contrast, our angular misalignment data conform to expectations if we assume that the ripple axes are always armchair [Fig.S5(b)].One datapoint slightly better fits the expected curve if its ripple axes are zigzag [Fig.S5(c)], but because the ripple axes are always armchair in epitaxial heterostruc-tures, we find it most likely that armchair is also preferred in nearly-aligned, assembled heterostructures.
While we cannot exclude the possibility that the underlying hBN influences the chosen ripple axes of graphene, for instance by forming small armchair ripples directly beneath the graphene, our data suggest that graphene intrinsically prefers armchair axes: in regions of epitaxial bilayer and trilayer graphene, we always observe armchair ripple axes on the topmost graphene layer, which we expect to be minimally influenced by the buried hBN.While a photoemission experiment [9] found evidence for zigzag low-friction axes for monolayer graphene on silicon dioxide, we note that the angular resolution in that experiment was 30 degrees, and that only a single flake was studied.

SIGNATURES OF RIPPLE SUPERLATTICE IN TRANSPORT MEASUREMENTS
We studied the temperature dependence of the longitudinal resistance in ten samples of monolayer graphene on hBN, all with multilayer graphene back gates.All transport measurements shown in Fig. S6 were performed in a Quantum Design PPMS, which provided very accurate thermometry and temperature control using a thermometer mounted on the same small circuit board as the sample.Longitudinal resistance data were collected using a lock-in amplifier in a four-terminal Hall bar geometry with approximately one square between the voltage probes and about 3 µm channel width.In one experiment, we loaded into our PPMS a sample on which we initially could not resolve periodic ripples by AFM, and measured the temperature dependence of the peak resistance at charge neutrality as a function of temperature for two consecutive thermal cycles between 300 K and 10 K [Fig.S6(a)].After the first thermal cycle, the sample presumably formed a ripple superlattice at room temperature.While there was a slight overall change in room temperature resistance before and after the first cooldown [Fig.S6(b)], and a slightly different temperature dependence of the peak resistance on the second cooldown [Fig.S6(a)], we cannot rule out the adsorption or desorption of surface impurities as the cause of the observed changes.More generally, we have never observed a significant difference between the room temperature resistance of a sample before and after a thermal cycle, despite studying several samples on which we could not resolve periodic ripples prior to thermal cycling.The only feature in our transport measurements that we ascribe to the ripple superlattice is a brief but sharp deviation from an otherwise smooth temperature dependence while warming up [Fig.S6(c)].We observed this sort of resistance glitch while warming between 200 K and 230 K on multiple thermal cycles of both samples whose resistance during warmup we carefully studied (four glitches on one sample, two on the other, over about ten thermal cycles for each sample).The glitch is not isolated to the charge neutrality point; we also observed it while repeatedly sweeping the gate voltage over the secondary Dirac point [Fig.S6(d)], and found that the glitch approximately uniformly increased the resistance throughout the window of carrier density that we explored (∼ 5 × 10 11 cm −2 around the secondary Dirac point).
Because the temperature of the glitch (200 K to 230 K) coincides with the temperature at which periodic ripples disappeared in our variable-temperature AFM studies, we speculate that the glitch arises as the strained graphene sheet suddenly overcomes the substrate pinning force to form domains of periodic ripples.We expect this unpinning process upon warmup to be much more violent than the disappearance of the periodic ripples upon cooldown, which would explain why we never observe a glitch while cooling.While we again cannot rule out effects such as adsorption of contaminants, the responsible effect seems more likely to be structural given the sharpness of the glitch and the fact that the sample resistance appears to quickly reassume its original temperature dependence at higher temperatures, consistent with the apparently minimal impact of the ripple superlattice on transport properties at low Fermi energies.

FIG. 1 .
FIG. 1. One-dimensional ripple superlattices at room temperature in bilayer graphene on silicon dioxide.(a) Tapping-mode AFM topography signal, differentiated along the scan axis (vertical) for clarity.The upper left corner of the image shows the silicon dioxide, which displays uncorrelated roughness.The bilayer flake shows two domains of periodic ripples with distinct ripple axes (black dashed lines in each domain are a guide to the eye).Scale bar is 100 nm.(b) Friction signal in contact-mode AFM over a larger area (scale bar is 1 µm).White dashed square indicates the scan region in (a).Two frictional domains appear on the bilayer graphene flake, which correspond to the two domains of distinct ripple axes; insets show cartoons of the ripple topography in each domain.Scan direction is right to left, and darker regions have higher friction.

FIG. 2 .
FIG.2.Formation and suppression of ripple superlattice with decreasing temperature.(a-d) Non-contact AFM topography of an hBN crystal 60 nm thick, as deposited on silicon dioxide, as the temperature is successively lowered.All window sizes are 50 nm, and all grayscales are the same.At 300 K (a), the sample is very flat, with rms roughness 10 pm.At 250 K (b), periodic ripples have formed, with peak-to-trough amplitude 15 pm.The periodic ripples are still resolved at 225 K (c) with similar amplitude.Changes in the apparent period and ripple axis between 250 K and 225 K could entirely result from thermal scanner drift, or from slightly differing scan locations[8].At 200 K (d), the ripples can no longer be resolved.Horizontal features faintly visible in all scans, but especially in (a) and (d), are artifacts of the horizontal scan axis.(e) Model for ripple formation on the topmost graphene or hBN layer of a flake or heterostructure on silicon dioxide.The stacked layers start out in a relaxed configuration, and as the temperature is lowered, the contracting silicon wafer applies biaxial compressive stress to the layers, whose in-plane thermal expansion coefficient is negative.The topmost layer relieves some strain by forming periodic ripples.At sufficiently low temperature, the substrate interaction overcomes the thermal fluctuations driving the out-of-plane distortion, forcing the top layer into a strained state.

FIG. 3 .
FIG. 3. Ripple superlattice at room temperature in nearly-aligned graphene/hBN heterostructures.(a) Tapping-mode AFM topography showing an 11.8 ± 1.2 nm moiré pattern with a one-dimensional ripple superlattice superimposed (ripple axis runs lower left to upper right).Scale bar is 25 nm.(b) Tapping-mode topography on a different device showing domains with all three ripple axes.A 12.7 ± 0.5 nm moiré pattern is present but difficult to pick out by eye.Scale bar is 100 nm.(c) Cantilever phase shift measured simultaneously with the data in (b).The ripples are easier to see in the phase image, while the moiré pattern is nearly invisible.Scale bar is 100 nm.Both (b) and (c) are corrected for thermal scanner drift by applying an affine transformation [8].(d) Magnitude of the fast Fourier transform of (b).The hexagonal arrangement of six peaks of wavevector magnitude 79 µm −1 arises from the moiré pattern.The hexagonal arrangement of six peaks of wavevector magnitude 170 µm −1 results from the three different ripple axes.Scale bar is 100 µm −1 .
FIG. S2.Uniformity of ripple superlattice within a single domain on hBN.(a) Tapping-mode topography signal, differentiated along the scan axis (horizontal) to suppress the gentle topographic background (the broad feature running bottom left to top right is a smooth depression of depth ∼ 100 pm).The ripple axis runs upper left to lower right.Black circle is centered around a rare phase slip: three parallel ripple lines merge into two.Scale bar is 100 nm.(b) Height along black dashed line in (a).Each point is averaged over 16 nm transverse to the black dashed line.(c) FFT of the topography signal used to produce (a).One of the two superlattice peaks is circled in red.Scanner drift smears the peaks along the vertical axis.Scale bar is 100 µm −1 .(d) Vertical cut through the superlattice peak in (c); k vertical = 0 corresponds to the vertical center of panel (c).FWHM is 10 µm −1 , which corresponds to a 4 • error in estimation of the ripple axis.(e) Radial cut through the superlattice peak in (c); k radial = 0 corresponds to the center of panel (c).The ripple period is 6.43 ± 0.04 nm, where the error represents the FWHM of the peak (2 µm −1 ).
FIG.S3.Determining crystallographic orientation of ripple axes using perfectly-aligned heterostructures.(a) Tapping-mode topography signal (corrected for affine distortion, and differentiated for clarity) from a perfectly-aligned heterostructure of graphene grown on hBN by van der Waals epitaxy.Three islands of monolayer graphene are surrounded by hBN (graphene can be distinguished from hBN by the presence of a moiré pattern).Each graphene island has a ripple superlattice with a distinct ripple axis along an armchair direction of the moire superlattice.The hBN shows two ripple superlattice domains, also with armchair ripple axes.Scale bar is 100 nm.(b) FFT of the topography signal used to produce (a).The angles between the moire wavevectors (parallel to red dotted lines) and ripple superlattice wavevectors (parallel to black dotted lines) are 28 ± 4 • , 29 ± 3 • , and 29 ± 3 • .Scale bar is 200 µm −1 .
FIG.S4.Incommensuracy of ripple superlattice with moiré pattern.(a) Tapping-mode topography signal (corrected for affine distortion) from a rotationally-aligned heterostructure of graphene on hBN displaying a 12.5 ± 0.7 nm moiré pattern.Three distinct ripple axes can be seen.The domain wall in the upper right is disturbed by the AFM tip, and is therefore blurred along the scan axis.Scale bar is 50 nm.(b) FFT of the signal in (a).Red dotted lines extend from the origin; one passes through a ripple superlattice peak, while the other passes through the nearest moiré superlattice peak.The angle between the lines is 18 ± 3 • , which agrees with expectations for the measured moiré wavelength under the assumption that graphene ripple axes are armchair (or zigzag; Fig.S5).Scale bar is 200 µm −1 .
FIG.S6.Effect of ripple superlattice on transport properties of graphene/hBN heterostructures.(a-c) Transport measurements as a function of temperature for two consecutive thermal cycles of a single Hall bar device (graphene on hBN with a multilayer graphene back gate).No ripple superlattice could be resolved prior to the first cooldown.(a) Peak four-terminal resistance Rxx at the charge neutrality point as the temperature was decreased from 300 K to 10 K for each cooldown.Data were collected by measuring Rxx while repeatedly sweeping the gate voltage over the charge neutrality point at various temperatures.(b) Rxx as a function of back gate voltage at 300 K prior to each cooldown.(c) Rxx as a function of increasing temperature (ramp rate 10 K/min) for each warmup from 10 K to 300 K.Here Rxx is measured at a fixed back gate voltage near the charge neutrality point.Between 200 and 220 K, Rxx sharply departs from the otherwise smooth curve.(d) Peak Rxx near the secondary Dirac peak as a function of temperature for a different device of similar structure, but with close rotational alignment between the graphene and hBN.Data were collected by repeatedly sweeping the gate voltage over the secondary Dirac peak as the temperature was cycled from 390 K to 10 K to 390 K at 10 K/min.The peak Rxx sharply departs from the otherwise smooth curve near 225 K during warmup.