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 transitions1,2,3,4,5, and the choice of doping for devices6,7. While it typically arises from differing structures of bands containing electrons and holes8,9, in some cases this asymmetry manifests from external sources such as impurities10,11, strain9,12,13, or simply from intrinsic many-body interactions14,15,16. Graphene is an interesting case in this light because its K point band structure is expected to be perfectly electron-hole symmetric17, but the combination of its dimensionality and dispersion relation renders it highly susceptible to symmetry-breaking perturbations18. Most experimental realizations of the monolayer16,19 and bilayer20,21 exhibit electron-hole asymmetry, even after vast improvements in sample preparation, which reduce the effective strain and impurity concentration16,21,22. Whether external sources or intrinsic interactions, such as correlations5,14,15,16,23,24 drive asymmetry remains to be verified. Understanding how electron-hole symmetry breaks and its relationship to exotic ground states5,14,15,25 has become even more important with the recent discovery of Mott-like physics and superconductivity in twisted bilayer graphene26,27 and other moiré heterostructures28,29,30. These systems exhibit phase diagrams reminiscent of the cuprates31, 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 transport19 and quantum capacitance16, can easily cover the broad doping range via electrostatic gating, but are only sensitive to the electronic states at the Fermi energy (EF) 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 screening32,33 and impurity concentration34,35. The very recent introduction of electrostatic gating into ARPES experiments36,37 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 electron-hole 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.

Results

ARPES in-operando

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 μm2. 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 EF 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 EF from momentum distribution curves (MDCs), spectra at constant energy as a function of momentum, shown in panel f for different gating values. At Vg = 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 Vg = − 5V(8V) corresponding to a p (n) doping of 2.2 ± 0.3 ⋅ 1012cm−2 (0.5 ± 0.3 ⋅ 1012cm−2) (details on calculation of the carrier density can be found in Supplementary Note 1). The position of the Fermi energy EF − ED, 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., EF − ED = vFkF, the Fermi velocity vF 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 vF (orange/red line) is less steep than vF at neutrality and electron dopings (navy/blue lines). Although the results in Fig. 1g might appear in contrast to a previous report38 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 EF with quantities such as kF that are unaffected by quantum capacitance contributions near charge neutrality (see Supplementary Note 5).

Fig. 1: ARPES in-operando.
figure 1

a, b. Schematic of experimental setup (a-) and optical micrograph of the graphene/hBN sample (b-). Dashed lines outline regions of graphene (black), hBN (blue), and graphite (purple). Scale bar: 20 μm (c.-e.) ARPES spectra for S1 along the \(K-K^{\prime}\) direction (perpendicular Γ − K) at (c-) p doping (-5V), (d-) equilibrium (0V), and (e-) n doping (8V). Blue dashed lines indicate quasiparticle peak positions extracted from Lorentzian fits. (f.) doping-dependent MDCs spectra and quasiparticle peak positions (black arrows) at EF, indicated by the white dashed line in (b-d). (g.) EF − ED as a function of kF, 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 EF data for S1 (S2), Navy (Light blue) line is a fit to the electron-doped EF data for S1 (S2).

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Electron-hole asymmetry in graphene dispersions

Figure 2 reports the detailed evolution of the K point electronic structure near EF for different doping (gating) values. Figure 2a–c display raw image plots near the K point for dopings of − 0.9 ⋅ 1012cm−2, 0.0 ⋅ 1012cm−2, and 1.1 ⋅ 1012cm−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 EF (= ED) than at higher binding energies, in agreement with previous reports32,33. 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).

Fig. 2: Electron-hole asymmetry in graphene dispersions.
figure 2

(a.-c.) graphene spectra at three representative dopings: (a.) − 0.9 ⋅ 1012cm−2, (b.) 0.0 ⋅ 1012cm−2, (c.) 1.1 ⋅ 1012cm−2. Black (red) dashed lines indicate linear fits to dispersions near ED (EF in (a)). Inset cartoons illustrate the deviations from Dirac cone dispersions at respective dopings. (d.) ED dispersions near the charge neutrality point indicate asymmetry in band velocity for electron and hole dopings. (e.) Extracted band velocities as a function of doping. Error bars indicate 1σ deviation of best fit to graphene dispersions. (f.) Graphene fine structure constant (α) as a function of doping, is extracted from band velocity fits. Teal shaded regions indicate 1σ deviation of best fit to the vD data using the logarithm-based lineshape described in the text.

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These differences are better visualized by plotting the energy dispersion vs momentum (Fig. 2d), extracted by fitting the momentum distribution curves with standard Lorentzian-like 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 vD (which is above EF at these dopings) can be approximated by the Fermi velocity vF. In contrast, the dispersions at neutrality (purple) and electron dopings (blue) show a large deviation from linearity, with vD nearly twice as large as velocities at ED − 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 (vD), extracted from the slope of ARPES dispersions, are plotted as a function of doping. Although a divergence of vD is observed in the proximity of the charge neutrality point, as previously reported35 for the electron-doped side, a clear asymmetry is revealed over the entire doping range, with vD ~ 30% 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 interactions35,39 leading to a logarithmic correction of the band velocity via the coefficient α, which represents the long-range Coulomb coupling strength. Using a similar model35

$$\frac{v}{{v}_{0}}=1+\frac{\alpha }{8}{{\mathrm{log}}}\,\left(\frac{{n}_{0}}{{n}_{{{{\rm{e}}}}}}\right)$$
(1)

we determine the coefficient α = e2/ϵℏv0 (panel f), which is the primary contributor to the velocity enhancement. The dielectric strength ϵ = ϵ0(1 + a∣ne∣1/2) is allowed to effectively increase as a function of doping40,41 to account for charge-carrier screening in graphene35, and v0 = 1.0 ⋅ 106 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 vD42 but is doping independent35. Though this result is in apparent contrast with some reports using EF sensitive probes40,41, we note that the real Coulomb interaction strength α can be isolated more reliably from energy states at the Dirac point35 rather than from states at EF. Indeed, at EF, the band velocity in graphene is modified by several interactions: notably it is enhanced by the long-range electron-electron interactions33,35, and reduced by electron-phonon coupling33,35,43,44. In contrast, at the Dirac point the electron-phonon interaction becomes negligible due to the diminished density of states45,46,47, and the band velocity is solely enhanced by the electron-electron interaction39,48.

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) = ℏvFΔk = ΔE49. 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 EF (grey) and the EDCs at E − ED = − 1.5 eV (black), each showing a strong electron-hole asymmetry. Im Σ at EF scales as \({a}_{0}\sqrt{| {n}_{{{{\rm{e}}}}}| }\) away from neutrality, with the amplitude a0 = 0.30 ± 0.05 for electron dopings and a0 = 0.11 ± 0.04 for hole dopings. This doping dependence is in contrast to alkali-doped graphene samples, which develop a \(1/\sqrt{n}\) dependence from the added long-range impurities35, and the \(\sqrt{n}\) scaling of the self-energy at EF observed in gate-tunable graphene samples50 can be attributed to the Matthiessen combination of acoustic and optical phonons47,51,52 and short-range impurities53,54. The electron-electron interaction has been observed to modify the EF self-energy contribution from phonons43,44,47,55 and impurities53,54,56 through a positive scaling with the renormalization parameter α = e2/ϵℏvF 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 EF. 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 symmetry13 as for example in the case of an aligned hBN substrate, might induce an overall reduction of the observed asymmetry.

Fig. 3: Electron-hole asymmetry in graphene imaginary self-energy.
figure 3

(a., b.) Normalized EDCs at E − ED = −1.5 eV (a-), and MDCs at EF (b-) for hole (red) and electron (blue) doped graphene. (c.) The imaginary part of the self-energy as a function of doping for EDCs at E − ED = −1.5 eV (black) and MDCs at EF (grey). Error bars indicate 1σ deviation obtained from peak fit to graphene spectral function. The shaded grey region indicates 1σ deviation of best fit to the square-root-based function described in the text. (d.) The imaginary part of the self-energy normalized by the Fermi energy. Blue (red) solid lines indicate fits to the data, as described in the text.

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Whereas techniques that are only sensitive to the low energy physics are often marred by impurities19,34, 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 EF for different doping values. That \({{{\rm{Im}}}}{{\Sigma }}/{E}_{{{{\rm{F}}}}}\) 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 correlations57,58. From these calculations we can approximate the scattering rate to an empirical form:

$$\frac{{{{\rm{Im}}}}{{{\Sigma }}}_{{{{\rm{ee}}}}}}{{E}_{{{{\rm{F}}}}}}={c}_{1}\tanh \left[{c}_{2}\cdot \frac{(E-{E}_{{{{\rm{F}}}}})}{{E}_{{{{\rm{F}}}}}}\right]$$
(2)

where c1 and c2 are fit parameters. The fit shows an overall good agreement with the data and gives c1h = 2.5 ± 0.2 for hole dopings, c1e = 4.6 ± 0.3 for electron dopings, and c2 = 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-hopping17,59, which can be effectively enhanced by strain12, induced for example from alignment to a substrate with a different lattice constant, and easily modeled by tight binding calculations13. 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 \(| t^{\prime} | \sim 3\) eV is needed. This is an order of magnitude larger than values reported in the literature (\(t^{\prime} \sim 0.3\) eV)16,19,60 even when graphene is aligned to an hBN substrate13, and opposite in sign to the asymmetry produced in graphene strained via wrinkles12. Moreover, we note that our samples are aligned at large twist angles to the hBN substrate, where lattice reconstruction is negligibly small61 (see Supplementary Note 3 for more details), and therefore the effect on \(t^{\prime}\) 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 − ED53, 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 ( ~ 1013 cm−2) is large enough to produce signatures in the spectral function in the form of resonance states22,62,63 or impurity bands64,65, 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 neutrality39,48, the nonlinear behavior of self-energy at high binding energies57, and likely the asymmetry in the self-energy at EF. Finally, we note that though numerical calculations for Σel-el58 are much smaller than values found in our experiment, reaching quantitative agreement between experimental and theoretical results often requires additional scaling factors45,66.

In conclusion, we have demonstrated the power of electrostatic gated ARPES to study the interplay of interactions and electron-hole 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 graphene26,27, similar correlated 2D moiré systems28,30,67, and in high-temperature cuprate superconductors15,24,31,68. Given that the moiré potential can be leveraged to enhance the strength of correlations in 2D heterostructures30,69,70,71, 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 heterostructures5,14,15,25.

Methods

Two devices were made for this experiment—for S1 refer to Fig. 1 and for S2 refer to Figs. 2 and 3

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 in72. 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 refocused74 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 open-source python-based analysis framework76. 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.