Colloquium: Graphene spectroscopy

Spectroscopic studies of electronic phenomena in graphene are reviewed. A variety of methods and techniques are surveyed, from quasiparticle spectroscopies (tunneling, photoemission) to methods probing density and current response (infrared optics, Raman) to scanning probe nanoscopy and ultrafast pump-probe experiments. Vast complimentary information derived from these investigations is shown to highlight unusual properties of Dirac quasiparticles and many-body interaction effects in the physics of graphene.

Figure 1

(a) Schematic of the $\pi$-band dispersion of SLG showing Dirac cones at $K$ and $K^{\prime }$ points. From [287]. (b) Definitions of the intralayer (${\gamma }_0$) and interlayer (${\gamma }_1$–${\gamma }_4$) hopping parameters of Bernal-stacked graphene materials. [For their experimental values, see, e.g., [412].] (c) The energy scales of electronic phenomena in graphene along with the corresponding frequency ranges and spectroscopic methods. The asterisk denotes compatibility of a method with high magnetic fields.

Figure 2

Schematic dispersion of electron density excitations in SLG (lines). The horizontal axis corresponds to the $\mathrm{\Gamma }$-$K$ cut through the Brillouin zone. All excitations experience Landau damping inside the electron-hole pair continuum (shaded).

Figure 3

(a) The ARPES schematics. (b), (c) The ARPES intensity in the energy-momentum space for a potassium-doped epitaxial graphene on SiC (0001) ([30]). In panel (c) he interval of momenta $k_y$ is indicated by the line segment in the inset. (b) The momentum $k_x$ varies along the orthogonal path through the $K$ point. (d) The ARPES maps for a similar sample taken at the energies (top to bottom) $E=E_F\approx E_D+0.4\mathrm{eV}$, $E_D$, and $E_F-0.8\mathrm{eV}$ ([165]). (e) Solid lines, left to right: the ARPES dispersions along the $\mathrm{\Gamma }\text{−}\mathrm{K}$ direction for graphene on $\mathrm{SiC}(000\overline{1})$, hBN, and quartz. Dotted lines: results of $GW$ calculations for $\kappa =\infty$ and $\kappa =1$. Adapted from [166].

Figure 4

Spectroscopic determination of the Dirac dispersion in SLG. (a) The STS tunneling spectra $dI/dV$ taken at the same spatial point and different gate voltages $V_g$. The curves are vertically displaced for clarity. The arrows indicate the positions of the $dI/dV$ minima $V_D$. (b) Optical conductivity of SLG at different gate voltages with respect to the neutrality point. (c) The distance $|E_D|$ between the Dirac point and the Fermi level as a function of $V_g$ obtained from the data in (a). The line is the fit to $E_D\propto |V_g{|}^{1/2}$. The insets are cartoons showing the electron occupation of the Dirac cones. From [416]. (d) The gate voltage dependence of the interband absorption threshold $2E_F$ obtained from the data in (b). From [211].

Figure 5

(a) The carrier density dependence of $v_F$ in suspended SLG extracted from magnetoresistance oscillations (circles) and a fit to a theory (solid curve). Adapted from [96]. (b) Renormalized velocity determined from the gate capacitance of SLG on hBN (symbols) and a fit to Eq. (1.7) (solid curve). From [407]. (c) Renormalized velocity of SLG on hBN from the STS of Landau levels (symbols). The line is a fit to Eq. (1.11). From [52]. (d) Fermi velocity for SLG as a function of the dielectric constant of the substrate. The filled symbols are the data points obtained from the ARPES spectra. The open symbols and the line are from theoretical modeling. From [337].

Figure 6

(a) The schematics of LLs in SLG. Each LL is fourfold degenerate due to spin and valley degrees of freedom. The neutrality point corresponds to the half filling of the $n=0$ LL. (b) The STS spectra of graphene on graphite at different magnetic fields ([208]). (c) A high-resolution STS revealing fourfold states that make up $n=1$ LL of epitaxial graphene on SiC at different magnetic fields. The energy separations $\mathrm{\Delta }{E}_v$ and $\mathrm{\Delta }{E}_s$ due to lifting of the valley and spin degeneracies are enhanced when the Fermi level falls between the spin-split levels at filling factor $\nu =5$. Additional stable states appear at $\nu =11/2$ and $9/2$ ([345]). (d) The IR transmission of a $p$-doped graphene at $\nu =-2$ normalized to that at $\nu =10$ at three different magnetic fields. Two LL resonances, $T_1$ and $T_2$, are observed. The inset shows the allowed LL transitions ([174]).

Figure 7

Gating-induced change $\mathrm{\Delta }{\sigma }^{\prime }(\omega )={\sigma }^{\prime }(\omega )-{\sigma }_{\mathrm{CNP}}^{\prime }(\omega )$ in the optical conductivity of SLG. Solid lines are the fits assuming the Drude model for both $\sigma (\omega )$ and ${\sigma }_{\mathrm{CNP}}(\omega )$. The latter is the conductivity at the charge-neutrality point (CNP). Its Drude form is chosen to account for inhomogeneous local doping; cf. Sec. 4a. From [159].

Figure 8

Collective modes of graphene on polar substrates originate from hybridization of substrate surface phonons with graphene plasmons. Both modes show up as resonances in the near-field amplitude spectrum $s(\omega )$. The main left panel shows the phonon mode measured for ${\mathrm{SiO}}_2$ alone (squares) and the phonon-plasmon hybrid mode of ${\mathrm{SiO}}_2$ covered with SLG (circles). The modeling results are shown by the lines, with the SLG trace revealing the lower hybrid mode of predominantly plasmon character at $\omega \sim 500{\mathrm{cm}}^{-1}$. From [109]. Direct observation of this plasmonlike mode is achieved by real-space imaging of $s(\omega )$ at a fixed frequency $\omega =892{\mathrm{cm}}^{-1}$ (inset). The oscillations seen in the image result from interference of plasmon waves ([106]). The bright lines in the right inset depict the calculated mode dispersions for SLG with the chemical potential $\mu /hc=1600{\mathrm{cm}}^{-1}$ on ${\mathrm{SiO}}_2$. The experimentally relevant momenta are situated near the vertical dashed line. The diagonal dashed line is the border of the electron-hole continuum (cf. Sec. 1d). The main right panel depicts collective modes of epitaxial graphene on SiC measured with electron energy loss spectroscopy at 300 K (circles) and 80 K (squares). The solid and dash-dotted lines are different theoretical fits. The dotted lines indicate the boundaries of the electron-hole continuum. From [364].

Figure 9

Phonon dispersion and Raman spectroscopy of graphene. (a) Calculated phonon dispersion of SLG ([303]) (symbols) compared with the experimental data for graphite ([256]) (lines). (b) Raman spectra of graphene and graphite measured at 514 nm laser excitation showing the $G$ and the $2D$ Raman peaks ([111]). (c) The evolution of the $2D$ Raman peak with the number of graphene layers ([111]). (d) Schematics of the $G$-mode and the $2D$-mode Raman scattering processes.

Figure 10

The ARPES dispersion along the $\mathrm{\Gamma }$-$K$ direction for (a) lightly doped (solid line) and (b) heavily doped (solid line) graphene ([340]). The dashed lines are guides to the eye. The arrows indicate the kinks in the curves due to the el-ph interaction. (c) Zero field STS tunneling spectra ([208]). The spectra are approximately linear except around the phonon energy. (d) The ARPES electron-phonon coupling constant as a function of the carrier density for graphene grown on different substrates: SiC ([30]; [246]; [423]; [247]), Ir (111) ([140]; [23]), and Cu ([337]). The results of ab initio calculations of [42] without (upper dashed line) and with (lower dashed line) el-ph interactions included are shown for comparison.

Figure 11

The ARPES dispersion of doped ($N=8\times 10^{10}{\mathrm{cm}}^{-2}$) graphene (a) perpendicular and (b) parallel to the $\mathrm{\Gamma }$-$K$ direction ([31]). The dashed lines are guides to the eye for the dispersion of the hole and plasmaron bands; the vertical solid line goes through the Dirac point. The inset shows a schematic of the renormalized spectrum in the presence of plasmarons.

Figure 12

The ARPES intensity along (a)–(c) the direction orthogonal to $\mathrm{\Gamma }$-$K$, (d) the path $\mathrm{\Gamma }$-$M$-$K$-$\mathrm{\Gamma }$. (a) Adapted from [100]. (b), (c) Adapted from [317]. (d) Adapted from [189].

Figure 13

(a) Moiré superlattice wavelength (left axis) and rotation (right axis) as a function of the angle between the SLG and hBN lattices. Inset: STM topography images showing (top to bottom) 2.4, 6.0, and 11.5 nm moiré patterns. The scale bars in all the images are 5 nm tall. (b) The tunneling $dI/dV$ curves for samples with moiré wavelengths 9.0 nm (right curve) and 13.4 nm (left curve). The dips in the $dI/dV$ curves marked by arrows occur at secondary Dirac points. (c) Longitudinal resistivity ${\rho }_{xx}$ of SLG on hBN as a function of carrier density. Inset: One possible reconstruction of SLG spectrum. (d) The Hall resistivity ${\rho }_{xy}$ changes sign at high electron and hole doping, revealing well-isolated secondary Dirac points. Inset: Conductive atomic force microscope image of the moiré pattern. The separation between the white spots is 11 nm. (a), (b) Adapted from [406]. (c), (d) Adapted from [309].

Figure 14

STM images and STS spectra taken at 7.5 K. (a) Graphene monolayer patch on Pt(111) with four nanobubbles at the graphene-Pt border and one in the patch interior. Residual ethylene molecules and a small hexagonal graphene patch can be seen in the lower right (3D $z$ scale enhanced $4.6\times$). (Inset) High-resolution image of a graphene nanobubble showing distorted honeycomb lattice resulting from strain in the bubble ($\max z=1.6\mathrm{nm}$, 3D $z$ scale enhanced $2\times$). (b) STS spectra of bare Pt(111), flat graphene on Pt(111) (shifted upward by $3\times 10^{-11}{\mathrm{\Omega }}^{-1}$), and the center of a graphene bubble (shifted upward by $9\times 10^{-11}{\mathrm{\Omega }}^{-1}$). The peaks in the graphene bubble spectrum indicate the formation of pseudo-Landau levels. (c) Normalized peak energy vs $\mathrm{sgn}(n)\sqrt{|n|}$ for peaks observed on five different nanobubbles follow expected scaling behavior (see text). Adapted from [206].

Figure 15

Ultrafast dynamics of excited electrons in graphene. (a) Spectrally integrated transmission change as a function of pump-probe delay (open circles). The solid line is the numerical fit, and the dash-dotted line is the cross correlation of pump and probe pulses. The inset shows linear dependence of the maximum transmission change on the absorbed pump fluence. The transmission increase is due to photoinduced Pauli blocking of interband transitions. The ultrafast decay is due to thermalization between the electrons and holes and optical phonons. (b) Transient transmission changes at probe photon energies of 1.24 eV (solid), 1.55 eV (dash-dotted), and 1.77 eV (dashed) for short and long delays. The slower decay at the picosecond time scale is due to equilibration with acoustic phonons. From [34].

Figure 16

(a) IR conductivity spectra of MLG with layer number $L=1,2,\dots ,8$. (b) A contour plot of the IR conductivity per layer as a function of photon energy and $L$. The dots identify the position of the peaks in experimental IR conductivity. These transition energies follow well-defined energy-scaling relations predicted by the zone-folding model (solid curves). From [228]. (c) An illustration of the Bernal-stacked BLG with electrical field above and below it. (d) Electronic structure of a pristine and a gated BLG. Arrows indicate allowed optical transitions. (e)–(g) ARPES data showing the evolution of an induced band gap in electronic structure of epitaxial BLG on SiC with chemical adsorbed Ca atoms. From [283]. (h) Electrical field dependence of the induced band gap in BLG measured through IR absorption spectroscopy (symbols). From [417].