Correlation-driven electron-hole asymmetry in graphene field effect devices

Electron-hole asymmetry is a fundamental property in solids that can determine the nature of quantum phase transitions and the regime of operation for devices. The observation of electron-hole asymmetry in graphene and recently in twisted graphene and moiré heterostructures has spurred interest into whether it stems from single-particle effects or from correlations, which are core to the emergence of intriguing phases in moiré systems. Here, we report an effective way to access electron-hole asymmetry in 2D materials by directly measuring the quasiparticle self-energy in graphene/Boron Nitride field-effect devices. As the chemical potential moves from the hole to the electron-doped side, we see an increased strength of electronic correlations manifested by an increase in the band velocity and inverse quasiparticle lifetime. These results suggest that electronic correlations intrinsically drive the electron-hole asymmetry in graphene and by leveraging this asymmetry can provide alternative avenues to generate exotic phases in twisted moiré heterostructures.


INTRODUCTION
Electron-hole asymmetry, or the difference in a material's electronic properties upon doping with electrons versus holes, profoundly impacts the character of phase transitions [1][2][3][4][5] , and the choice of doping for devices 6,7 . While it typically arises from differing structures of bands containing electrons and holes 8,9 , in some cases this asymmetry manifests from external sources such as impurities 10,11 , strain 9,12,13 , or simply from intrinsic many-body interactions [14][15][16] . Graphene is an interesting case in this light because its K point band structure is expected to be perfectly electron-hole symmetric 17 , but the combination of its dimensionality and dispersion relation renders it highly susceptible to symmetry-breaking perturbations 18 . Most experimental realizations of the monolayer 16,19 and bilayer 20,21 exhibit electron-hole asymmetry, even after vast improvements in sample preparation, which reduce the effective strain and impurity concentration 16,21,22 . Whether external sources or intrinsic interactions, such as correlations 5,[14][15][16]23,24 drive asymmetry remains to be verified. Understanding how electron-hole symmetry breaks and its relationship to exotic ground states 5,14,15,25 has become even more important with the recent discovery of Mott-like physics and superconductivity in twisted bilayer graphene 26,27 and other moiré heterostructures [28][29][30] . These systems exhibit phase diagrams reminiscent of the cuprates 31 , in which the role of the strong electron-hole asymmetry in driving the remarkable phases remains elusive.
The difficulty in addressing the origin of electron-hole asymmetry in graphene today is the requirement of a probe that has complete access to the material self-energy in both energy and momentum spanning over a large range of electron and hole dopings. Some probes, including transport 19 and quantum capacitance 16 , can easily cover the broad doping range via electrostatic gating, but are only sensitive to the electronic states at the Fermi energy (E F ) and do not provide any momentum information. In contrast, Angle-Resolved Photoemission Spectroscopy (ARPES) can provide access to the full quasiparticle spectral function A(k, ω), but so far has resorted to methods of doping that modify the fundamental properties of the system, including screening 32,33 and impurity concentration 34,35 . The very recent introduction of electrostatic gating into ARPES experiments 36,37 enables studies of the doping-dependent self-energy with full energy and momentum resolution while leaving the sample in pristine condition. Here, we directly reveal a significant electronhole asymmetry in the quasiparticle self-energy of graphene. The doping and momentum resolution of our measurement enables us to provide strong evidence for correlations as the intrinsic driving force. Figure 1 a presents an illustration of the sample geometry used for the ARPES experiment and gating configuration, while panel b shows the optical micrograph of the overall sample S1 (two samples were measured in this report, see Methods section for details). The dashed contours identify regions of monolayer graphene (black), hBN (blue), and graphite (purple) while the yellow thick lines indicate the electrical contacts. The size of the sample is smaller than 1200 μm 2 . The adopted beam size was 1 μm to allow measurements of each individual part of the sample and disentangle different contributions. The equilibrium spectra for the sample in Fig. 1d clearly depicts the characteristic linear bands of graphene's Dirac fermions along the K-K' direction populated up to near the charge neutrality point. A positive (negative) voltage established between the graphite back gate and the graphene sample results in the addition, panel e (subtraction, panel c) of electrons to (from) the sample. Since the Fermi energy E F is held at the ground, the additional negative (positive) charges shift the Dirac spectrum downward (upward). The doping change can be estimated by the peak separation at E F from momentum distribution curves (MDCs), spectra at constant energy as a function of momentum, shown in panel f for different gating values. At V g = 0V the Fermi surface is a point and the momentum separation between MDC peaks is negligibly small. As electrons (holes) are added to the system, two peaks emerge and the momentum separation increases, with a maximum at V g = − 5V(8V) corresponding to a p (n) doping of 2.2 ± 0.3 ⋅ 10 12 cm −2 (0.5 ± 0.3 ⋅ 10 12 cm −2 ) (details on calculation of the carrier density can be found in Supplementary Note 1). The position of the Fermi energy E F − E D , displayed in Fig. 1g, is estimated by the intersection point of linear fits to the Dirac spectra (blue dashed lines in Fig. 1c-e). Assuming a linear dispersion relation, i.e., E F − E D = v F k F , the Fermi velocity v F can be estimated here from a linear fit to the data in panel g. Interestingly, we find distinct behaviors on opposite sides of the charge neutrality point: at hole dopings, the estimated v F (orange/red line) is less steep than v F at neutrality and electron dopings (navy/blue lines). Although the results in Fig.  1g might appear in contrast to a previous report 38 where no asymmetry in the density of states was observed, we note that a true estimate of the density of states can only be reached by comparing E F with quantities such as k F that are unaffected by quantum capacitance contributions near charge neutrality (see Supplementary Note 5). Figure 2 reports the detailed evolution of the K point electronic structure near E F for different doping (gating) values. Figure 2a-c display raw image plots near the K point for dopings of − 0.9 ⋅ 10 12 cm −2 , 0.0 ⋅ 10 12 cm −2 , and 1.1 ⋅ 10 12 cm −2 . Already from the raw data one can see that the spectrum in Fig. 2a is linear, and at the neutrality point ( Fig. 2b) the dispersion looks noticeably steeper near E F (= E D ) than at higher binding energies, in agreement with previous reports 32, 33 . The electron-doped spectrum ( Fig. 2c) presents different structure for the valence band than does the spectrum at similar hole doping: the valence band near the Dirac point (black dashed line) is steeper than the valence band in Fig. 2a (red dashed line).

Electron-hole asymmetry in graphene dispersions
These differences are better visualized by plotting the energy dispersion vs momentum (Fig. 2d), extracted by fitting the momentum distribution curves with standard Lorentzianlike functions in the proximity of the Dirac point. A clear departure from linearity is observed in the data starting at the neutrality point, where the dispersion is steepest, and still observed in the electron-doped side. Band velocities can be directly extracted from these data, being proportional to the slope of the ARPES dispersions. Because the dispersions for hole dopings remain linear, the band velocity at the Dirac point v D (which is above E F at these dopings) can be approximated by the Fermi velocity v F . In contrast, the dispersions at neutrality (purple) and electron dopings (blue) show a large deviation from linearity, with v D nearly twice as large as velocities at E D − 0.5 eV. These results clearly indicate the presence of a distinct electron-hole asymmetry in the electronic response and are summarized in panel e, where the band velocities at the Dirac point (v D ), extracted from the slope of ARPES dispersions, are plotted as a function of doping. Although a divergence of v D is observed in the proximity of the charge neutrality point, as previously reported 35 for the as a function of k F , extracted from linear fits to the graphene spectra. Data for two different samples are shown (S1 and S2). Error bars indicate 1σ deviation of best fit to the linear intersection of the dispersions. Orange (Red) line is a linear fit to the hole-doped E F data for S1 (S2), Navy (Light blue) line is a fit to the electron-doped E F data for S1 (S2). electron-doped side, a clear asymmetry is revealed over the entire doping range, with v D~3 0% higher for electron dopings than for hole dopings. The large renormalization of the Dirac spectra was previously reported at the neutrality point and assigned to electron-electron interactions 35,39 leading to a logarithmic correction of the band velocity via the coefficient α, which represents the long-range Coulomb coupling strength. Using a similar model 35 we determine the coefficient α = e 2 /ϵℏv 0 (panel f), which is the primary contributor to the velocity enhancement. The dielectric strength ϵ = ϵ 0 (1 + a|n e | 1/2 ) is allowed to effectively increase as a function of doping 40,41 to account for charge-carrier screening in graphene 35 , and v 0 = 1.0 ⋅ 10 6 m/s is the local density approximation of the bare band velocity. The long-range coupling strength α shows a strong asymmetry between the electron and hole side, which is the driver for the asymmetry in the band dispersion discussed in panel d. We note that the correlation strength α is a direct measurement of charge-carrier screening and is independent from the substrate-induced screening, which could also modify v D 42 but is doping independent 35 . Though this result is in apparent contrast with some reports using E F sensitive probes 40,41 , we note that the real Coulomb interaction strength α can be isolated more reliably from energy states at the Dirac point 35 rather than from states at E F . Indeed, at E F , the band velocity in graphene is modified by several interactions: notably it is enhanced by the long-range electron-electron interactions 33,35 , and reduced by electron-phonon coupling 33,35,43,44 . In contrast, at the Dirac point the electron-phonon interaction becomes negligible due to the diminished density of states [45][46][47] , and the band velocity is solely enhanced by the electron-electron interaction 39,48 .
Electron-hole asymmetry in graphene imaginary self-energy Figure 3 reports the imaginary part of the self-energy for holes and electrons at several doping values. The momentum distribution curve (MDC)'s FWHM Δk, the energy distribution curve (EDC)'s FWHM ΔE and the imaginary part of the self-energy Im Σ(ω) are related by 2 Im Σ(ω) = ℏv F Δk = ΔE 49 . A clear asymmetry between electrons and holes is already apparent in the raw spectra, EDC (panel a) and MDC (panel b). The full doping dependence of Im Σ is plotted in Fig. 3c for both the MDCs at E F (grey) and the EDCs at E − E D = − 1.5 eV (black), each showing a strong electron-hole asymmetry. Im Σ at E F scales as a 0 ffiffiffiffiffiffiffi ffi jn e j p away from neutrality, with the amplitude a 0 = 0.30 ± 0.05 for electron dopings and a 0 = 0.11 ± 0.04 for hole dopings. This doping dependence is in contrast to alkali-doped graphene samples, which develop a 1= ffiffiffi n p dependence from the added long-range impurities 35 , and the ffiffiffi n p scaling of the self-energy at E F observed in gate-tunable graphene samples 50 can be attributed to the Matthiessen combination of acoustic and optical phonons 47,51,52 and short-range impurities 53,54 . The electronelectron interaction has been observed to modify the E F self-energy contribution from phonons 43,44,47,55 and impurities 53,54,56 through a positive scaling with the renormalization parameter α = e 2 /ϵℏv F which denotes the strength of correlations. The asymmetry in α, as reported in Fig. 2, therefore drives the electron-hole asymmetry in the self-energy at E F . We note that the choice of substrate is important to be able to discern the effect here described. Indeed, a competing electron-hole asymmetry such as the ones generated by moiré potential or breaking of inversion symmetry 13 as for example in the case of an aligned hBN substrate, might induce an overall reduction of the observed asymmetry.
Whereas techniques that are only sensitive to the low energy physics are often marred by impurities 19,34 , the ability of ARPES to access the entire energy range allows us to extract the intrinsic behavior of materials. Figure 3d presents the energy dependence of the imaginary self-energy scaled by the position of the Fermi energy E F for different doping values. That ImΣ=E F collapses to two distinct curves for electron and hole dopings provides further evidence for electron-hole asymmetry in the material. The reported energy dependence is qualitatively similar to numerical calculations of the inverse quasiparticle lifetime from dynamically screened electron-electron correlations 57,58 . From these calculations we can approximate the scattering rate to an empirical form: where c 1 and c 2 are fit parameters. The fit shows an overall good agreement with the data and gives c 1h = 2.5 ± 0.2 for hole dopings, c 1e = 4.6 ± 0.3 for electron dopings, and c 2 = 0.11 ± 0.03 for both dopings. Such differences are another manifestation of the electron correlation strength, as discussed in Ref. 58 .

DISCUSSION
The data reported here provide evidence of a strong electron-hole asymmetry in graphene that is driven, as we will argue below, by strong electronic correlations. We now discuss the possible sources of such asymmetry and show that it is an intrinsic property rather than driven by disorder or other extrinsic effects. As mentioned above, there are several mechanisms that break electron-hole symmetry in graphene, and include intrinsic asymmetries in the band structure, charged impurities, and electronic correlations. The asymmetries in the band structure are induced by the next-nearest-neighbor-hopping 17,59 , which can be effectively enhanced by strain 12 , induced for example from alignment to a substrate with a different lattice constant, and easily modeled by tight binding calculations 13 . When applying the latter to our experimental data, it becomes clear that to account for the 30% asymmetry between conduction and valence band velocity an unrealistic value of jt 0 j $ 3 eV is needed. This is an order of magnitude larger than values reported in the literature (t 0 $ 0:3 eV) 16,19,60 even when graphene is aligned to an hBN substrate 13 , and opposite in sign to the asymmetry produced in graphene strained via wrinkles 12 . Moreover, we note that our samples are aligned at large twist angles to the hBN substrate, where lattice reconstruction is negligibly small 61 (see Supplementary Note 3 for more details), and therefore the effect on t 0 is negligible.
Another possible source of electron-hole asymmetry is the presence of charged impurities leading, in the case of very close (< 5 nm) proximity, to changes in the LDOS as large as 30% 12,22 . However, impurities produce an inverse quasiparticle lifetime that scales inversely with E − E D 53 , in contrast to the empirical function in equation 2 used to fit our data. Additionally, the impurity density required to produce this effect throughout a mesoscopic sample (~10 13 cm −2 ) is large enough to produce signatures in the spectral function in the form of resonance states 22,62,63 or impurity bands 64,65 , which are not observed in our data.
These observations make electronic correlations the primary driver of electron-hole asymmetry observed in our study. Indeed, this interaction can consistently explain the asymmetric logarithmic renormalization of the dispersions across charge neutrality 39,48 , the nonlinear behavior of self-energy at high binding energies 57 , and likely the asymmetry in the self-energy at E F . Finally, we note that though numerical calculations for Σ el-el 58 are much smaller than values found in our experiment, reaching quantitative agreement between experimental and theoretical results often requires additional scaling factors 45,66 .
In conclusion, we have demonstrated the power of electrostatic gated ARPES to study the interplay of interactions and electronhole symmetry in 2D materials. Our results point to electronic correlations as the driving force for an intrinsic electron-hole asymmetry in graphene, manifested in the dispersion and inverse quasiparticle lifetime. These findings open the intriguing possibility that electron-electron interactions might also be responsible for the asymmetries found in the phase diagrams of more strongly correlated materials, including twisted bilayer graphene 26,27 , similar correlated 2D moiré systems 28,30,67 , and in hightemperature cuprate superconductors 15,24,31,68 . Given that the moiré potential can be leveraged to enhance the strength of correlations in 2D heterostructures 30,69-71 , we speculate that breaking electron-hole asymmetry via moiré-enhanced correlations can be an exciting alternative pathway to realize exotic quantum phases in twisted 2D heterostructures 5,14,15,25 .

METHODS
Two devices were made for this experiment-for S1 refer to Fig. 1

Sample preparation
Flakes of single-layer Graphene and hexagonal Boron Nitride were exfoliated onto Silicon Wafers with 90nm-thick oxide. S1 was constructed using a method similar to that used in 72 . A stamp comprised of Polypropylene carbonate (PPC), and Polydimethylsiloxane (PDMS), and the transparent tape was used to pick up Graphite, hBN, and Graphene in sequential order. The PPC stamp holding the stack was flipped onto a 90 nm oxidized Si wafer with the Graphene facing up, and the polymer was subsequently removed by annealing in a vacuum furnace at 350 C for 10 h. S2 was constructed using a technique similar to that outlined in Zomer et. al. 73 . A stamp comprised of Polycarbonate (PC) and Polydimethylsiloxane (PDMS) was used to pick up the Graphene, hBN, and Graphite to form a graphene/hBN/graphite heterostructure, which was then placed onto a fresh 90 nm-oxidized Si wafer. PC polymer residue was removed by placing the stack-on-chip in Chloroform for > 60 min at room temperature. Contacts were patterned onto each sample surface using electron-beam lithography followed by evaporation of 5 nm Cr and 50 nm Au.

ARPES measurements and analysis
Sample 1 was measured using a Scienta R4000 Hemispherical Analyzer at the nanoARPES branch of beamline 7.0.2 (MAESTRO) at the Advanced Light Source using a photon energy of 74 eV, a temperature of 300 K, and a pressure better than 1e-10 Torr. The beam was capillary refocused 74 to a spot size of~1 μm × 1 μm. The overall energy and momentum resolution was 30 meV and 0.014 Å −1 , respectively. The sample was doped electrostatically using a Keithley 2450 Source Meter. Sample 2 was measured using a Scienta R8000 Hemispherical Analyzer at Beamline 4.0.3 (MERLIN) 75 at the Advanced Light Source using a photon energy of 94 eV, a temperature of 20 K, and a pressure better than 5e-11 Torr. The beamspot was~100 μm × 50 μm. The overall energy and momentum resolution was 25 meV and 0.017 Å −1 , respectively. The sample was doped electrostatically using a Keithley 2200 programmable power supply electrically connected to the cryostat.
All ARPES data in this paper were analyzed using pyARPES, an opensource python-based analysis framework 76 . Spectra presented in the figures have had a background (estimated by the mean value of detector counts ≃ 0. 5Å −1 away from the K point) removed, and are smoothed by a gaussian filter with windows in momentum and energy smaller than the experimental resolution.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.