Polarization dependence of x-ray absorption spectra in graphene

We calculate the x-ray absorption spectra (XAS) for $1s$-${\pi }^{*}$ and $1s$-${\sigma }^{*}$ transitions in single layer graphene using the dipole approximation and we compare with experimental results. It is found that the in-plane and out-of-plane orientations of the dipole vectors, which correspond to the $1s$-${\pi }^{*}$ and $1s$-${\sigma }^{*}$ transitions, respectively, are responsible for the polarization dependence of the x-ray absorption intensity in graphene. Using the atomic matrix elements, the low-energy XAS peaks can be assigned.

Figure 1

(a) The calculated energy dispersion relations of the $\pi$ and $\sigma$ bands of 2D graphene along various high-symmetry directions. (b) The calculated energy dispersion relations of the $1s$ bands of 2D graphene along various high-symmetry directions.

Figure 2

The joint density of states (JDOS) of graphene as a function of energy. The JDOS is calculated from the occupied $1s$ orbitals to unoccupied states.

Figure 3

(a) The oscillator strength for the $1s$ to ${\pi }^{*}$ transition as a function of the final wave vector ${\vec{k}}_f$ in the 2D Brillouin zone. The bright (dark) area shows the strong (weak) oscillator strength. The numbers in the color bar are given in the units of $m_{\mathrm{opt}}^{\mathrm{on}}$. (b) The x-ray absorption intensity [the bright (dark) area shows strong (weak) x-ray absorption] of the $1s$ to ${\pi }^{*}$ transition as a function of the final wave vector ${\vec{k}}_f$ in the 2D Brillouin zone of graphene. Here we use the x-ray energy $E=286.4\mathit{eV}$ and $\alpha ={30}^{\circ }$. The units of the x-ray absorption intensity are ${(\mathrm{a}.\mathrm{u}.)}^{-2}$ as shown in the color bars. (c) The polarization angle $\alpha$ is defined by the angle between the propagating direction of the x-ray and the dipole vector $D$, which is perpendicular to the graphene basal plane. $P$ is the polarization direction of the x-ray light.

Figure 4

On-site normalized dipole vectors for (a) the $1s$ to ${\sigma }_1^{*}$ transition and (b) the $1s$ to ${\sigma }_2^{*}$ transition as a function of the final wave vector ${\vec{k}}_f$ in the 2D Brillouin zone.

Figure 5

The oscillator strength (a) for the $1s$ to ${\sigma }_1^{*}$ on-site transition and (b) for $1s$ to ${\sigma }_2^{*}$ on-site transition. The bright (dark) area shows as strong (weak) oscillator strength. The numbers in the color bar are given in units of $m_{\mathrm{opt}}^{\mathrm{on}}$. Here we use the value of the on-site atomic matrix element as $m_{\mathrm{opt}}^{\mathrm{on}}=0.30{(\mathrm{a}.\mathrm{u}.)}^{-1}$. The x-ray absorption intensity [the bright (dark) area shows strong (weak) x-ray absorption] of (c) the $1s$ to ${\sigma }_1^{*}$ transition and (d) the $1s$ to ${\sigma }_2^{*}$ transition as a function of the final wave vector ${\vec{k}}_f$ in the 2D Brillouin zone of graphene for the x-ray energy $E=293.4\mathit{eV}$ and $\alpha ={30}^{\circ }$. The direction of the polarization vector $P$ and the direction of the dipole vector $D$ for the $1s$ to ${\sigma }^{*}$ transition, which acts along the $xy$ graphene basal plane, are shown. The units of the x-ray absorption intensity shown in the color bars are ${(\mathrm{a}.\mathrm{u}.)}^{-2}$.

Figure 6

(a) The C($K$)-edge absorption spectra of single-crystal graphite at various polarization angles $\alpha$ between the surface normal and the Poynting vector of the light. Short lines with labels from A to J at the bottom of the figure are lines that denote the peak energies of various spectral features: dashed lines represent the states of ${\pi }^{*}$ symmetry, while solid lines represent the states of ${\sigma }^{*}$ symmetry. States whose symmetry could not be determined are represented by dashed dotted lines. The monochromatic photon energy calibration is estimated to be accurate to $\pm 0.5\mathit{eV}$ (reproduced from Fig. 1 of Ref. 7); (b) the calculated XAS spectra of graphene. The dashed (solid) lines at the bottom show the contribution from the ${\pi }^{*}$ (${\sigma }^{*}$) orbitals.

Figure 7

(a) Relative peak intensities for the $1s$ to ${\pi }^{*}$ transitions as a function of ${\sin }^2\alpha$. (b) Relative peak intensities for the $1s$ to ${\sigma }^{*}$ transitions as a function of ${\cos }^2\alpha$. The circles are calculated results the lines are a guide to the eye.