A high-mobility electronic system at an electrolyte-gated oxide surface

Electrolyte gating is a powerful technique for accumulating large carrier densities at a surface. Yet this approach suffers from significant sources of disorder: electrochemical reactions can damage or alter the sample, and the ions of the electrolyte and various dissolved contaminants sit Angstroms from the electron system. Accordingly, electrolyte gating is well suited to studies of superconductivity and other phenomena robust to disorder, but of limited use when reactions or disorder must be avoided. Here we demonstrate that these limitations can be overcome by protecting the sample with a chemically inert, atomically smooth sheet of hexagonal boron nitride. We illustrate our technique with electrolyte-gated strontium titanate, whose mobility when protected with boron nitride improves more than 10-fold while achieving carrier densities nearing 1014 cm−2. Our technique is portable to other materials, and should enable future studies where high carrier density modulation is required but electrochemical reactions and surface disorder must be minimized.

Fits (black dashed curves) are performed simultaneously on the ρ xx and R H data for a given cooldown. c, Densities n 1 and n 2 of the two bands; n tot = n 1 + n 2 . d, Mobilities µ 1 and µ 2 ; µ avg = (n 1 µ 1 + n 2 µ 2 )/n tot . sheet resistance ρ xx and Hall coefficient R H versus applied magnetic field for the 10-band system whose parameters are given in Supplementary   simultaneously on the ρ xx and R H data for a given cooldown. c, Densities n 1 and n 2 of the two bands; n tot = n 1 + n 2 . d, Mobilities µ 1 and µ 2 ; µ avg = (n 1 µ 1 + n 2 µ 2 )/n tot . e, Linear magnetoresistance coefficients k i , for i = 1, 2, normalized to the zero field resistivities ρ i of the corresponding bands.  The leakage becomes nonzero as the gate reaches 3 V for the first time (black trace); ρ xx begins to change significantly after the gate reaches -3 V.    Table 1: Sample details. L tot is the total length of the Hall bar channel, and W is the channel width. L c is the center-to-center distance between R xx voltage probes. D coplanar is the distance from the Hall bar to the nearest point on the coplanar gate. Supplementary Table 2: Parameters for ten bands used to generate data in Supplementary Figure 3. These parameters are chosen to be plausible given the densities extracted from quantum oscillations and the average mobility and total density that we extract from the Hall analysis, but are otherwise arbitrary.

Supplementary Note 1: Sample properties Lateral dimensions
The samples studied in this work have different lateral dimensions because we are constrained by the dimensions of the chosen BN barrier flake. We are also typically forced to position our coplanar gate and ohmic contacts away from thick, randomly-positioned flakes that are unintentionally transferred alongside the flake of interest, leading to further geometry variations. Nonetheless, the approximate geometry in all cases is as shown in Figure 1a in the main text, with a small Hall bar about 100 µm from a coplanar gate.

Flake thickness
We determined the flake thicknesses reported in Supplementary Table 1 by atomic force microscopy (AFM) after transferring the flakes onto the STO crystal and annealing at 500 • C. We found that after annealing, a low density of small, mobile particles remained on the sample surface. While these particles were too small and too sparse to affect transport measurements, they could easily be collected by the AFM tip and redeposited, particularly at the step edge whose height we wanted to measure. This introduced sizable errors in our tapping mode AFM (TAFM) measurements. The thickness uncertainties quoted in Supplementary Table 1 reflect the measured thickness variations between linescans at different positions on the same flake.
Sample B was particularly difficult to image with TAFM because of mobile particles, and therefore has a large error bar.
For our later generation of samples (Samples A and C), we used contact AFM (CAFM), which is more suitable than TAFM for studying a surface with mobile particles, and generally leads to a more accurate step height measurement between substrate and flake. 1, 2 However, the difference in friction between STO and few-layer h-BN causes twisting of the CAFM cantilever and thus a different measured flake thickness for left-moving and right-moving scans. 1 Because the offset between left-moving and right-moving scans is small (severalÅ), to first order the correct step height can be found by averaging the step heights measured in each direction. This average thickness is the thickness quoted in Supplementary Table 1 for Samples A and C, and the quoted uncertainties reflect the variation in average thickness between linescans at different positions on the same flake. As a sanity check, we also performed TAFM on Samples A and C, and found thicknesses consistent with the average thickness measurements from CAFM.

Determination of ρ xx and ρ xy
In this study, the quantities that we directly measure as a function of magnetic field are the longitudinal resistance R xx (H) and the transverse resistance R xy (H) of our six-terminal devices. To extract the sheet resistance ρ xx (H) of the 2DES from the measured signal R xx (H), R xx (H) must be multiplied by a factor that accounts for geometrical details such as the number of squares between the voltage probes, the nonzero width of the voltage probes, and the distance between the voltage probes and the current leads. Similarly, to determine the Hall resistance ρ xy (H), the measured transverse resistance R xy (H) must also be corrected for a geometrical factor, since in some geometries the Hall voltage can be shorted out by the current leads. 3 In general, these geometrical correction factors may depend on the Hall angle θ H = tan −1 (ρ xy (H)/ρ xx (H)).
We have used the conformal mapping technique 4 to numerically determine the geometrical correction factors C xx ≡ R xx /ρ xx and C xy ≡ R xy /ρ xy for the geometry of each sample. where f xx and f H are the two-band predictions for ρ xx and R H , and C is a constant that adjusts the relative importance of R H over ρ xx in the fitting. Typically we set C = 1, but for some datasets, in particular those with strong ρ xx oscillations, we increase C to get appropriate fits. The constraints on the fit parameters are very loose: parameters are allowed to range over several orders of magnitude. Our best fits are reproducible over a large range of initial conditions provided to the fitting algorithm.

Linear magnetoresistance model
Our fits presented in the main text use a two-band model in which the resistivity tensor of each band is described by where ρ i ≡ 1/n i eµ i is the zero-field resistivity of band i, n i is the density of band i, k i ≥ 0, and µ 0 is the magnetic constant (in the main text we absorbed µ 0 into k i for simplicity). The conductivity tensor of the complete system is the sum of the conductivity tensors of the individual bands.

Fitting more than two bands with a two-band model
It is clear from the presence of quantum oscillations with many frequencies that more than two bands contribute to transport. This is also the case for the highest-mobility LAO/STO 2DES, in which several bands contribute to quantum oscillations even for total carrier densities as low as 10 13 cm −2 (Ref. 5,6). Despite the large number of bands in our 2DES, the two-band model with linear magnetoresistance presented in the text captures the essential magnetotransport behavior. In fact, fits to the data in Figure   2 in the main text are also possible with the four-parameter, two-band model, 7 although the fit quality is lower (Supplementary Figure 2a The two-band model can give a reasonable fit to the many-band data by averaging the set of high mobility bands into one band, and the set of low-mobility bands into another band. For instance, if we assume the existence of ten bands with the arbitrarily-chosen (but plausible) densities and mobilities shown in Supplementary Table 2, the simulated ρ xx and R H are acceptably fit by the four-parameter, two-band model ( Supplementary Figure 3a,b). The resulting fit parameters are n 1 = 1.1 × 10 13 cm −2 , n 2 = 1.4 × 10 13 cm −2 , µ 1 = 11267 cm 2 V −1 s −1 , and µ 2 = 2467 cm 2 V −1 s −1 . These values yield a total density n tot ≡ i n i = 2.5×10 13 cm −2 and an average mobility is µ avg ≡ i n i µ i /n tot = 6395 cm 2 V −1 s −1 , which agree well with the total density 2.5 ×10 13 cm −2 and average mobility 6320 cm 2 V −1 s −1 calculated from Supplementary The six-parameter fit with linear magnetoresistance yields better agreement to the ρ xx and R H curves, even though no linear magnetoresistance is used to produce the ten-band curves (Supplementary Figure   3a,b). The fit parameters are n 1 = 9.2 × 10 12 cm −2 , n 2 = 1.5 × 10 13 cm −2 , µ 1 = 12382 cm 2 V −1 s −1 , µ 2 = 3039 cm 2 V −1 s −1 , k 1 = 0, and k 2 = 1.6 Ω/T, which give n tot = 2.4×10 13 cm −2 and µ avg = 6587 cm 2 V −1 s −1 , also in good agreement with the correct values.
On the other hand, if we assume that there are in fact two higher-density bands in our system, the addition of several low-density oscillating bands does not strongly affect the shapes of the magnetotransport curves. For a specific example, consider adding four bands, each of density 10 12 cm −2 , with mobilities 5000, 6000, 8000, and 12000 cm 2 V −1 s −1 to the two bands with linear magnetoresistance extracted in the main text ( Figure 2) for Sample A, Cooldown 1. The four additional bands have linear magnetoresistance coefficients of zero, for simplicity. Other than globally shifting R H because of the increased density, the addition of these bands has a modest effect on ρ xx and R H (Supplementary Figure 3c,d).
We conclude that regardless of the actual number of bands, the two-band model with or without linear magnetoresistance is a reliable method for extracting total density and average mobility.

Justification of two-band fits with linear magnetoresistance
The fact that the six-parameter fit could more closely approximate the ten-band ρ xx and R H (Supplementary we ascribe to subpar cleanliness of our procedures away from our home lab, the qualitative magnetoresistance behavior is similar to that of our other samples measured at lower fields. Nonsaturating linear magnetoresistance with an apparently saturating R H has also been observed in LAO/STO up to 31 T (Ref.

8).
We have attempted to fit our data from all samples with a three-band model (parameters n 1 , n 2 , n 3 , µ 1 , µ 2 , µ 3 ), since a high-density, low-mobility band could in principle give rise to a nearly linear growth of ρ xx with a nearly constant R H for the experimentally-accessible field range. However, the three-band model does not fit all of our datasets, while the two-band model with linear magnetoresistance can generally fit the data very well. Even when the three-band fits do capture the data, the resulting densities are surprisingly high, ranging from 10 14 to 10 16 cm −2 , with mobilities ∼ 1 cm 2 V −1 s −1 . The two-band model with linear magnetoresistance is our best known method to accurately fit our data.

Interpretation of k i
In our six-parameter, two-band fits with linear magnetoresistance coefficients k i , we typically find that the high-mobility band has a smaller linear coefficient than does the low-mobility band, even when k i is normalized to the zero-field resistivity ρ i (Supplementary Figure 4e). In fact, our fits in Figure 2 in the main text all yielded exactly zero for k 1 , the linear magnetoresistance coefficient of the high-mobility band (Supplementary Figure 5). One possible interpretation of this result, assuming that the linear magnetoresistance arises from spatial fluctuations in the mobility, 9,10 is that the low-mobility band is nearer the STO surface, while the high-mobility band is buried deeper. The inhomogeneities responsible for the mobility fluctuations plausibly reside on the surface, thereby more strongly affecting the low-mobility band.
However, we caution that the linear term is only the first-order correction to the band resistivities ρ xx,i . The linear magnetoresistance due to mobility fluctuations exhibits a crossover field, below which the magnetoresistance is quadratic. 9, 10 Giving the linear magnetoresistance mainly to the low-mobility band may simply help approximate this shape: if the densities of the two bands are similar, the low-mobility band becomes relevant to the magnetoresistance only at higher fields.

Low-field data
All of the fits shown in the main text and in the Supplementary Information neglect the low-field region, where magnetic effects are likely important. 11 We find that the behavior of ρ xx and R H in this region cannot be simultaneously captured by few-band physics, even with linear magnetoresistance terms (Supplementary

Supplementary Note 3: Chemical resistance of BN flakes
An electrolyte-gating study of gold films has reported carrier density modulation of up to 3 × 10 15 cm −2 and correspondingly large changes in sheet resistance. 12 The capacitance required for these changes exceeds by an order of magnitude the expected capacitance of the electrolyte gate, 13 suggesting that chemical modification of the gold film is responsible for the modulation. Indeed, we have shown that electrolyte gating readily oxidizes the gold film at negative gate voltages, and that this oxide growth accounts for at least 90% of the observed changes in transport properties. 14 To determine whether our BN films can prevent this sort of electrochemical reaction, we transferred a BN flake 6 nm thick onto a gold Hall bar 30 nm thick. The topography scan of the resulting structure shows that the flake mostly conforms to the Hall bar (Supplementary Figure 7a). However, because the Hall bar protrudes 30 nm from the oxide substrate, the flake relaxes strain by forming several wrinkles. The flake is also not perfectly flush with the gold where top two voltage probes meet the channel. Prior to electrolyte gating, all unprotected gold areas were masked by a layer of crosslinked PMMA (Supplementary Figure 7b).
The protected Hall bar and the on-chip coplanar gate were then covered in the ionic liquid N,N-diethyl-Nmethyl-N-2-methoxyethyl ammonium tetrafluoroborate (DEME-BF4) and inserted in a vacuum chamber at room temperature.
First, we swept the gate voltage between -2 and 2 V. The total change in ρ xx over these 4 V is ∼ 5×10 −4 Ω (Supplementary Figure 7c). For 4 V dropped across a BN flake 6 nm thick, electrostatics gives a density change ∆n = 4 0 × 4 V/6 nm = 1.5 × 10 13 cm −2 (we neglect the double layer capacitance, which is at least ten times the capacitance from the 6 nm BN dielectric). The total 2D carrier density in a gold film 30 nm thick is n = 1.8 × 10 17 cm −2 . If the sheet resistance is proportional to 1/n, electrostatics predicts a fractional change in sheet resistance ∼ 10 −4 , close to the fractional change ∼ 3 × 10 −4 that we observe.
Considering that we have neglected changes in electron mobility and the inhomogeneous distribution of the induced carrier density within the gold film, this approximate agreement is consistent with purely electrostatic carrier accumulation. The gate leakage current through the electrolyte and into the sample (Supplementary Figure 7d) is well below the 10 pA resolution of our measurement electronics, further implying the absence of electrochemical reactions, which would involve charge transfer from the electrolyte to the sample.
In contrast, electrolyte gating of an unprotected gold Hall bar with similar film thickness shows a very strong modulation of ρ xx and a large leakage current even for very small gate voltages (Supplementary Figure   7e,f). The fractional change in ρ xx over the -2 to 2 V window is ∼ 130 times larger for the uncovered sample than it is for the covered sample. This cannot be fully accounted for by the added thickness of the BN spacer, which should decrease the parallel plate capacitance from the double layer value of 7 µFcm −2 (Ref. 14) by a factor of 12. Thus, the fractional change in ρ xx for a given electric field is 10 times smaller in the BN-protected sample than it is in the unprotected sample, implying that the BN very strongly suppresses, if not entirely prevents, the oxidation of gold by the electrolyte.
We next proceeded to higher gate voltages on our BN-covered sample. In our first sweep up from 0 to 5 V, the leakage current started to noticeably increase at around 3 V (Supplementary Figure 7h), but the change in ρ xx remains small (Supplementary Figure 7g). However, upon sweeping from 5 to -5 V, we find a sudden ρ xx increase around -3 V, coincident with a leakage current spike. The next sweep, from -5 to 5 V, shows a significant ρ xx modulation even for positive gate voltages. The fractional magnitude of this modulation per volt is comparable to that of the unprotected gold Hall bar (Supplementary Figure 7e).
These data strongly suggest that our BN flake effectively blocks the oxidation reaction at the gold surface. At a sufficiently high voltage (approximately ± 3 V), the BN barrier itself starts to degrade, allowing reactions with the gold surface. The damage to the BN flake and/or the underlying gold is visible in an optical micrograph of the device after measurement (Supplementary Figure 7b), which shows shows dark spots in the upper left corner of the channel where prior to measurement the BN flake was wrinkled and poorly conformed to the gold surface. We expect that BN flakes without such mechanical defects will withstand a larger voltage before degrading.

Supplementary Note 4: Wrinkles in BN flakes
The BN flakes in this study are transferred using a PMMA membrane. 15 This process leaves a layer of polymer residue all over the sample, even after cleaning in standard solvents. We remove this residue by annealing in oxygen at 500 • C. One consequence of this is that the flakes tend to wrinkle, perhaps because of the different thermal expansion coefficients of STO and BN, or because small pockets of trapped contaminants escape from beneath the BN. If these wrinkles are only ∼ 1 nm tall (Supplementary Figure 8a,b), they present no obvious problem to our measurements: our highest mobility sample (Sample A) has small wrinkles, while Sample C has no wrinkles in the channel but still a lower mobility. However, some flakes have wrinkles several nm high (Supplementary Figure 8c,d); wrinkles this tall appear to prevent the underlying STO from being metallic at low temperature, just as one would expect based on electrostatic carrier accumulation.
The flakes as exfoliated are typically not wrinkled. The wrinkling is a byproduct of our particular transfer method and annealing process. Other methods, such as the dry transfer process based on polypropylene carbonate (PPC), are capable of wrinkle-free transfers. 16 We have found in our limited experience with the PPC-based process that PPC leaves less residue than does PMMA. More importantly, this PPC residue can be removed with a remote oxygen plasma, avoiding the need for annealing.

Supplementary Note 5: Superconducting properties of BN-protected STO samples
Samples A and C were measured under appropriate density and temperature conditions to observe superconductivity. Although the critical temperature for superconductivity T C in our samples does appear to be maximized near a total density n tot = 6 × 10 13 cm −2 (Supplementary Figure 9a), we emphasize that other factors such as carrier mobility 17 may strongly affect to T C . The magnitude of the critical current varies with sample size and T C . For Sample C, which is 4 microns wide, at optimal doping (T C = 195 mK) the critical current is 200 nA.
When sweeping a perpendicular magnetic field, we typically observe minimal hysteresis (Supplementary Figure 9b). On some cooldowns of Sample C, we find a resistance peak near the critical field (Supplementary Figure 9c) and critical temperature at zero dc current. Similar behavior has been observed in mesoscopic aluminum samples, and has been ascribed to various mechanisms such as an inhomogeneous distribution of normal metal-superconductor interfaces 18 or vortex dynamics that depend on the sample geometry. 19 While these resistance peaks in our sample show significant hysteresis (Supplementary Figure 9c), we hesitate to ascribe this hysteresis to magnetism or other physical effects in the STO: the hysteresis could simply reflect demagnetization cooling in our probe, which we have not carefully tested.