Charge-Carrier Screening in Single-Layer Graphene

The effect of charge-carrier screening on the transport properties of a neutral graphene sheet is studied by directly probing its electronic structure. We find that the Fermi velocity, Dirac point velocity, and overall distortion of the Dirac cone are renormalized due to the screening of the electron-electron interaction in an unusual way. We also observe an increase of the electron mean free path due to the screening of charged impurities. These observations help us to understand the basis for the transport properties of graphene, as well as the fundamental physics of these interesting electron-electron interactions at the Dirac point crossing.

Figure 1

ARPES spectra of graphene on $h\mathrm{\text{−}}\mathrm{BN}$. (a) ARPES spectrum of single-layer graphene along the $\Gamma \mathrm{\text{−}}K$ direction. This is the direction along which all of our data are analyzed in subsequent figures. (b) ARPES spectrum along the $K\mathrm{\text{−}}{K}^{\prime }$ direction (perpendicular to $\Gamma \mathrm{\text{−}}K$) shows that the Dirac point is at the Fermi level. The spectra in panels (a), (b) have been normalized by the area under the MDCs. (c) The pointlike Fermi surface of graphene. (d)–(h) Doping dependence along the $\Gamma \mathrm{\text{−}}K$ direction, with Fermi $k$ vectors corresponding to $k_F=-0.0035$, 0.0037, 0.0126, 0.0206, $0.0282{\AA }^{-1}$, respectively.

Figure 2

The screening of charged impurities can lower the quasiparticle scattering rate. (a) The imaginary self-energy ($\mathrm{Im}\Sigma$), proportional to the width of ARPES spectra and inversely proportional to the quantum lifetime of the photohole, is shown for two different dopings ($n$ is proportional to $k_F^2$). In addition to the roughly linear binding energy dependence for each spectrum is an overall offset between them: The spectra with added potassium have sharper spectral features than the as-grown sample, a naively counterintuitive finding. (b) A comparison of the MDC widths at the Fermi level shows that increasing the charge density sharpens spectral features by screening the interaction between quasiparticles and impurities. The dashed line is a $1/k_F$ fit to the data, which allows us to extract an impurity density of $n_{\mathrm{imp}}\sim 1.6\times {10}^{11}{\mathrm{cm}}^{-2}$. The inset shows the raw MDCs at the Fermi level, confirming that the blue (slightly darker) curve is sharper than the red (slightly brighter) one.

Figure 3

Velocity renormalization by charge-carrier screening. (a) An electronic energy-momentum dispersion extracted from the peak positions of the MDCs from undoped $\mathrm{\text{graphene}}/h\mathrm{\text{−}}\mathrm{BN}$ is shown as the solid black curve. The curvature is due to the long-range electron-electron interaction [23] and can be compared to the (arbitrary) straight dashed line in the same figure. (b) Extracted dispersions show the Fermi velocity as a function of doping. For ease of viewing, the momenta of these dispersions have been aligned so that the Fermi $k$ vectors coincide (the $x$ axis is given as $k\mathrm{\text{−}}{k}_F$). (c) Extracted dispersions show the Dirac point velocity as a function of doping. For ease of viewing, the energies of these dispersions have been aligned so that the Dirac points coincide (the $y$ axis is given as $E\mathrm{\text{−}}{E}_D$). (d) The Fermi velocities and Dirac point velocities are given as a function of doping ($n$ is proportional to $k_F^2$). Both the Dirac point and Fermi level show velocity enhancements as the Dirac point approaches the Fermi energy. The dashed line is a logarithmic fit to the Dirac point velocities; see the text for further details. (e) These two Dirac cones give a schematic of where the data are taken in (b) and (c). Panel (b) shows dispersions near the Fermi level [gray planes in (e)], which change position with respect to the Dirac point as the Fermi level is adjusted. Panel (c) shows dispersions near the Dirac point, which remain at the same place on the Dirac cone even as the Fermi energy is altered. See Supplemental Material [42] for further discussion of data analysis techniques.

Figure 4

Dirac cone renormalization by charge-carrier screening. (a) Electronic dispersions near the Dirac point show higher binding energy behavior than Fig. 3c. The inset shows the difference between the experimental bands and an arbitrary straight line. Panel (b) illustrates the difference between charge-carrier screening and dielectric screening. Three dispersions are shown: (i) as-grown $\mathrm{\text{graphene}}/h\mathrm{\text{−}}\mathrm{BN}$, (ii) as-grown $\mathrm{\text{graphene}}/\mathrm{SiC}(000\overline{1})$, and (iii) doped $\mathrm{\text{graphene}}/h\mathrm{\text{−}}\mathrm{BN}$. The doping for (iii) was chosen so that the band velocity near the Dirac point would match that of (ii). The cartoons in (c) and (d) illustrate the renormalization effect on the Dirac cones for charge-carrier screening and dielectric screening, respectively. In (c), the renormalization is primarily restricted to low momenta. In (d), the renormalization extends to all momenta within our range of measurement and becomes larger in magnitude at higher momenta. See Supplemental Material [42] for further discussion of data analysis techniques.