Correlation-Driven Electron-Hole Asymmetry in Graphene Field Effect Devices

Electron-hole asymmetry, or the difference in a material’s electronic properties upon doping with electrons versus holes, profoundly impacts the character of phase transitionsSharpe et al. (2019)Sarkar et al. (2020)Sajadi et al. (2018)Hsu et al. (2017), and the choice of doping for devicesArora et al. (1982)Larbalestier et al. (2001). While it typically arises from differing structures of bands containing electrons and holesAshcroft and Mermin (1976)Jost et al. (2017), in some cases this asymmetry manifests from external sources such as impuritiesYazdani et al. (1997)Novikov (2007), strainBai et al. (2015)Jost et al. (2017)Dasilva et al. (2015), or simply from intrinsic many body interactionsAnderson and Ong (2006)Kretinin et al. (2013). Graphene is an interesting case in this light, because its K point band structure is expected to be perfectly electron-hole symmetricNeto et al. (2007), but the combination of its dimensionality and dispersion relation renders it highly susceptible to symmetry-breaking perturbationsKotov et al. (2012). Most experimental realizations of the monolayerDeacon et al. (2007) Kretinin et al. (2013) and bilayerZou et al. (2011)Cao et al. (2018a)Lu et al. (2019) exhibit electron hole asymmetry, even after vast improvements in sample preparation, which reduce the effective strain and impurity concentrationWang et al. (2013a)Kretinin et al. (2013). Whether external sources or intrinsic interactions such as correlationsZou et al. (2011)Anderson and Ong (2006)Cai et al. (2016)Kretinin et al. (2013) drive asymmetry remains to be verified. This has become even more important with the recent discovery of Mott-like physics and superconductivity in bilayer grapheneCao et al. (2018a)Lu et al. (2019), displaying a phase diagram which is strongly electron-hole asymmetric and reminiscent of the cuprates Gooding et al. (1994).

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 Deacon et al. (2007) and quantum capacitance Kretinin et al. (2013), 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 $A(k,\omega)$, but so far has resorted to methods of doping that modify the fundamental properties of the system, including screeningHwang et al. (2012)Siegel et al. (2011) and impurity concentration Chen et al. (2008)Siegel et al. (2013). The very recent introduction of electrostatic gating into ARPES experiments Nguyen et al. (2019)Joucken et al. (2019) enables studies of the doping dependent self-energy with full energy and momentum resolution while leaving the sample in pristine condition. Here we report the first of such study, revealing significant asymmetries in the quasiparticle self energy. The doping and momentum resolution of our measurement enables us to effectively characterize the relationship between electronic correlations and these asymmetries.

Fig. 1a 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. 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 $\mu\textnormal{m}^{2}$. The adopted beam size was 1 $\mu\textnormal{m}$ 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 $K-K^{\prime}$ 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 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 $2.2\pm 0.3\cdot 10^{12}\textnormal{cm}^{-2}\,(0.6\pm 0.3\cdot 10^{12}\textnormal{cm}^{-2})$. 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 Figs 1c-e). The scaling of $E_{D}$ with gate voltage away from neutrality follows $E_{D}\sim\sqrt{V_{g}}$, suggesting that the charge doping is well controlled, and a result of geometric capacitance Yu et al. (2013).

Fig. 2 reports the detailed evolution of the K point electronic structure near $E_{F}$ for different doping (gating) values. Figs 2a-c display raw image plots near the K point for dopings of $-0.9\cdot 10^{12}\textnormal{cm}^{-2}$, $0.0\cdot 10^{12}\textnormal{cm}^{-2}$, and $1.1\cdot 10^{12}\textnormal{cm}^{-2}$ (details on calculation of the carrier density can be found in Supplementary Note 1). 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 Siegel et al. (2011)Hwang et al. (2012). 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).

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 $v_{D}$ (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 Siegel et al. (2013) for the electron doped side, a clear asymmetry is revealed over the entire doping range, with $v_{D}$ $\sim$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 interactions González et al. (1994)Siegel et al. (2013) leading to a logarithmic correction of the band velocity. Using a similar model Siegel et al. (2013)

we extract the long-range coulomb coupling $\alpha=\frac{e^{2}}{\epsilon\hbar v_{0}}$ (panel f), which is the primary contributor to the velocity enhancement. The dielectric strength $\epsilon=\epsilon_{0}(1+a|n_{e}|^{1/2})$ is allowed to effectively increase as a function of doping Elias et al. (2011)Yu et al. (2013), and $v_{0}=1.0\cdot 10^{6}$ m/s is the local density approximation of the bare band velocity. The long range coupling strength $\alpha$ 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. Though this result is in apparent contrast with some reports using $E_{F}$ sensitive probes Elias et al. (2011)Yu et al. (2013), we note that the real coulomb interaction strength $\alpha$ can be isolated more reliably from energy states at the Dirac pointSiegel et al. (2013) rather than from states at $E_{F}$, which are affected by multiple different interactions Calandra and Mauri (2007).

Figure 3 reports the imaginary part of the self energy for holes and electrons at several doping values. The MDC’s FWHM $\Delta k$, the EDC’s FWHM $\Delta E$ and the imaginary part of the self energy Im $\Sigma(\omega)$ are related by $2\,\textnormal{Im}\Sigma(\omega)=\hbar v_{F}\Delta k=\Delta E$ Valla et al. (1999). 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 $\Sigma$ is plotted in Fig 3c for both the MDCs at $E_{F}$ (grey) and the EDCs at $k-K=-0.19\AA^{-1}$ (black), each showing a strong electron hole asymmetry. Im $\Sigma$ at $E_{F}$ scales as $a_{0}\sqrt{|n_{e}|}$ away from neutrality, with the amplitude $a_{0}=0.45\pm 0.05$ for electron dopings and $a_{0}=0.09\pm 0.02$ 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 impuritiesSiegel et al. (2013), and the $\sqrt{n}$ scaling of the self energy at $E_{F}$ observed in gate-tunable graphene samples Muzzio et al. (2020) can be attributed to either acoustic phononsHwang and Das Sarma (2008)Kaasbjerg et al. (2012) or short range impuritiesAdam et al. (2007). However, since both scattering sources are directly affected by the screening of the Coulomb interactionAdam et al. (2007)Hwang and Das Sarma (2008), we can confidently conclude that the prominent electron-hole asymmetry in the self energy at $E_{F}$ is due to the asymmetry in the correlation strength $\alpha$ (reported in Fig 2). A similar asymmetry has been observed in a graphene sample nearly aligned with the hBN substrate Muzzio et al. (2020), which may be weaker due to the competing asymmetry produced by the moiré potentialDasilva et al. (2015) and/or from the presence of a screening graphite back gateStepanov et al. (2020).

Whereas techniques that are only sensitive to the low energy physics are often marred by impurities Chen et al. (2008)Deacon et al. (2007), the ability of ARPES to access the entire energy range allows us to extract the intrinsic behavior of materials. Fig. 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 $\textnormal{Im}\Sigma/E_{F}$ 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 correlationsHwang et al. (2007)Tse et al. (2008). 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 $c_{1h}=2.5\pm 0.2$ for hole dopings, $c_{1e}=4.6\pm 0.3$ for electron dopings, and $c_{2}=0.11\pm 0.03$ for both dopings. Such differences are another manifestation of the electron correlation strength, as discussed in Ref. Tse et al. (2008).

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 next nearest neighbor hopping Neto et al. (2007)Reich et al. (2002) which can be effectively enhanced by strainBai et al. (2015), induced for example from alignment to a substrate with a different lattice constant, and easily modeled by tight binding calculationsDasilva et al. (2015). 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) Bostwick et al. (2007)Deacon et al. (2007)Kretinin et al. (2013), and opposite in sign to the asymmetries produced in graphene aligned to hBNDasilva et al. (2015), and graphene strained via wrinklesBai et al. (2015). Moreover, we note that our samples are aligned at large twist angles to the hBN substrate, where lattice reconstruction is negligibly small Jung et al. (2015) (see supplementary information 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% Wang et al. (2013b)Bai et al. (2015). However, impurities produce an inverse quasiparticle lifetime that scales inversely with $E\,-\,E_{D}$ Adam et al. (2007), 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 ($\sim 10^{13}\,\textnormal{cm}^{-2}$) is large enough to produce signatures in the spectral function in the form of resonance statesPeres et al. (2006)Skrypnyk and Loktev (2008)Wang et al. (2013b) or impurity bandsAvsar et al. (2014)Hwang et al. (2018), 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 neutralityGonzález et al. (1994)Das Sarma et al. (2007), the nonlinear behavior of self energy at high binding energiesHwang et al. (2007), and likely the asymmetry in the self energy at $E_{F}$. Finally, we note that though numerical calculations for $\Sigma_{el-el}$Tse et al. (2008) are much smaller than values found in our experiment, reaching quantitative agreement between experimental and theoretical results often requires additional scaling factors Calandra and Mauri (2007)Park et al. (2007).

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 twisted bilayer graphene Cao et al. (2018a)Lu et al. (2019) and similar correlated 2D moiré systems Wang et al. (2020)Chen et al. (2020), as is found in high temperature cuprate superconductors Gooding et al. (1994)Phillips (2006)Anderson and Ong (2006)Cai et al. (2016).

Acknowledgements:
We thank Salman Kahn for technical assistance in the sample fabrication setup. This work was primarily supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DEAC02- 05CH11231 (Ultrafast Materials Science Program KC2203). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan ,Grant Number JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and the CREST(JPMJCR15F3), JST.

Competing Interests:
The authors declare that they have no competing financial interests.

Correspondence:
Correspondence and requests for materials should be addressed to A.L. (email: alanzara@ lbl.gov).

Author Contributions:
N.D. and A.L. initiated and directed the research project. T.T. and K.W. synthesized the hBN crystals. N.D., M.I.B.U., S.Z., and K.L. fabricated the graphene samples. N.D., R.M., J.D.D., C.G.F, and A.B. performed the ARPES measurements. N.D. analyzed the ARPES data using software designed by C.S., and inputs from A.L. N.D. and A.L. wrote the manuscript, with inputs from all of the authors.

Data Availability:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

The charge-carrier density can be obtained from the size of the Fermi surface using Luttinger’s theorem. Assuming a conical dispersion for graphene, the Fermi surface is a circle with a radius of the Fermi wavevector $k_{F}$, i.e. $n_{e}=k_{F}^{2}/\pi$ (see Fig S1a, and S1b). We extract the Fermi wavevector from the distance between peak positions at the Fermi level, which are displayed in Fig S1c for S1 and Fig S1d for S2. Whereas data from S1 includes contributions from both sides of the Dirac cone, data from S2 primarily contains one band, and $k_{F}$ can be extracted more accurately as (see Fig S1b) the intersection of the band dispersion (black dashed line) with the Fermi level (grey line). The K point position for S2 is then extracted as value of $k_{F}$ at the doping with the steepest $v_{D}$, which in our case occurs at a gate voltage of 0.5V (see Figure 2). The summary of extracted $k_{F}$ values, and calculated $n_{e}$ values are presented in Fig S1e and Fig S1f, respectively. For S1, error bars for $k_{F}$ are estimated based on the broadness of the bands and the momentum resolution, which was $\sim 0.014\,\AA^{-1}$, whereas for S2, error bars are extracted from the statistical error in the linear fit to the dispersion. Error bars for $n_{e}$ are obtained by propagating errors in $k_{F}$. Away from charge neutrality, both devices appear to have a roughly linear dependence between doping and gate voltage i.e. $n_{e}\simeq CV_{g}$, which is expected when treating the system as a parallel-plate capacitor with geometric capacitance $C$. The dip in geometric capacitance near the neutrality point indicates contributions of the quantum capacitance, which scale with the strength of electron-electron interactionsYu et al. (2013).

Supplementary Note 2: MDC and EDC Analysis

Graphene scattering rates described in the main text are extracted using the following procedure. The quasiparticle scattering rate $\Gamma$ is proportional to the width of MDCs $\Delta k$, which usually have a Lorentzian lineshape Damascelli (2003). Fig S2a presents a typical graphene spectra for S2, which has matrix elements strongly favoring the $k-K<0$ side of the Dirac cone. The faint presence of the other side of the Dirac cone for $k-K>0$ requires MDCs at $E_{F}$ to be fit to two Lorentzians with an affine background. We restrict the two peaks to have the same width and be placed symmetrically about the Dirac point, (see Fig S2b). EDCs at higher binding energies, for example at $K-0.19\AA^{-1}$ can simply be fit to a single Voigt lineshape with an affine background (see Fig S2c). Figures S2d and S2e show the MDCs (blue) and EDCs (red) as well as the fits (black) used to extract the quasiparticle scattering rate in the main text. MDCs have been smoothed using a gaussian filter with a momentum window sized smaller than the expermental resolution, and EDCs have been smoothed using a window (60 meV) that is smaller than the smallest recorded value of Im $\Sigma$ for sample S2. The statistical error of the fit is used to make the error bars in Fig 3c. The presence of hBN spectra can be seen in the MDCs at hole dopings $n_{e}<-0.5\cdot 10^{12}\textnormal{cm}^{-2}$, and in the EDCs at hole dopings $n_{e}<0.0\cdot 10^{12}\textnormal{cm}^{-2}$, though they do not strongly affect the extracted MDC or EDC widths due to the large discrepancy in broadness of hBN and graphene bands.

Supplementary Note 3: Alignment to the hBN Substrate

Alignment angle between the graphene sample and hBN substrate is extracted by overlaying their respective Brillouin Zones atop of photoemission data in Figs S3a, and S3b. The graphene K point is characterized by the radial center of the Dirac cone dispersion near $E_{F}$ (Fig S3a) and the hBN K point is characterized as the peak of a parabolloid band at a binding energy of $\sim$ 2.2 eV as seen from Fig S3c. The angular orientation of the hBN can also be determined using the equation $\theta\simeq 2\arcsin(\frac{\Delta k}{2K})$ Cao et al. (2018b). Using the momentum separation $\Delta k\simeq 0.18\AA^{-1}$ extracted from Fig S3c, we obtain an relative alignment of $\sim 6$ degrees, which agrees with the alignment of the overlaid Brillouin Zones in Figs S3a and S3b. Using the same technique, we obtain a relative twist of $\sim 13$ degrees for sample S2.

Supplementary Note 4: Shift of K point upon Application of Gate Voltage

The analysis presented here is very sensitive to the exact identification of the K point. Indeed, given the linear dispersion of graphene, a small misalignment within the sample can cause a large error in the doping value. Previous gated ARPES experiments have suffered from stray fields that drive a shift of the measured spectra with gating Muzzio et al. (2020). To minimize the effects of experimental stray fields, in our experiment we have adopted a sample geometry with a sizeable sample grounding plane Nguyen et al. (2019). The resulting constant energy maps for our sample S2 (see panels a-f) show that the K point does not shift significantly for negative (-2V) and positive (+3V) gate voltages, allowing a precise identification of the K point from the peak in the spectral weight at $E_{F}$ (see panel g). With this geometry we find that the K point shift is negligible and much less than our momentum resolution ($\sim 0.017\AA^{-1}$).