Optical absorption and conductivity in quasi-two-dimensional crystals from first principles: Application to graphene

This paper gives a theoretical formulation of the electromagnetic response of the quasi-two-dimensional crystals suitable for investigation of optical activity and polariton modes. The response to external electromagnetic field is described by current-current response tensor ${\mathrm{\Pi }}_{\mu \nu }$ calculated by solving the Dyson equation in the random phase approximation, where current-current interaction is mediated by the photon propagator $D_{\mu \nu }$. The irreducible current-current response tensor ${\mathrm{\Pi }}_{\mu \nu }^0$ is calculated from the ab initio Kohn-Sham orbitals. The accuracy of ${\mathrm{\Pi }}_{\mu \nu }^0$ is tested in the long-wavelength limit where it gives correct Drude dielectric function and conductivity. The theory is applied to the calculation of optical absorption and conductivity in pristine and doped single-layer graphene and successfully compared with previous calculations and measurements.

Figure 1

Diagrammatic representation of the noninteracting current-current response tensor. The first term of the right-hand side represents the paramagnetic term (10) and the second term represents the diamagnetic term (11). Thin solid line represents one particle (hole) propagator and the wavy line represents the free-photon propagator ${\mathbf{D}}^0$.

Figure 2

Diagrammatic expansion of the induced current within RPA (16). ${\mathrm{\Pi }}^0$ represents noninteracting current-current response tensor (9), $\mathrm{\Pi }$ interacting current-current response tensor within RPA (13), and $D^0$ free-photon propagator (14).

Figure 3

Geometry of the system. (a) The orientation of the crystal layers and incident electromagnetic field. (b) Evanescent character of the incident electromagnetic field for $\omega <\mathrm{Qc}$. (c) Radiative character of the incident electromagnetic field for $\omega >\mathrm{Qc}$. Layered crystal electronic density is restricted in the region $-\frac{L}{2}<z<\frac{L}{2}$.

Figure 4

(a) Absorption spectrum in pristine graphene $\left(E_F=0\right)$ for normal incidence $\left(\mathrm{Q}=0\right)$ and $T=0\mathrm{K}$. Inset: details of optical absorption in IR, visible, and UV regions; black solid lines: this work; blue dashed lines: theoretical results taken from Ref. [15]; and red dotted lines: experimental results taken from Ref. [16]. The damping parameters are ${\eta }_{\mathrm{intra}}=10\mathrm{meV}$ and ${\eta }_{\mathrm{inter}}=50\mathrm{meV}$. (b) Noninteracting current-current correlation tensor (51) (green line) and the interacting one (23) (yellow line) for $\mu =\nu =z$ and $\mathrm{Q}=0$.

Figure 5

Calculated absorption spectrum in slightly doped graphene $\left(E_F=0.1\mathrm{eV}\right)$ for normal incidence $\left(\mathrm{Q}=0\right)$ and $T=300\mathrm{K}$ obtained with Eqs. (58) (yellow line) and (59) (blue line). Experimental absorption spectrum for graphene sample taken from Ref. [18] is represented with red line. Horizontal dashed line denotes the universal absorption constant $\pi \alpha$. The interband damping parameter used in this calculations is ${\eta }_{\mathrm{inter}}=50\mathrm{meV}$.

Figure 6

Absorption spectrum for $\mathbf{s}$ (blue lines) and $\mathbf{p}$ (red lines) mode in doped graphene. Results are presented for (a) $E_F=0.5\mathrm{eV}$ and (b) $E_F=1\mathrm{eV}$. Incident wave vectors are in the $\mathrm{\Gamma }M$ direction with the values ${\mathbf{Q}}_n=n\mathrm{\Delta }\mathrm{Q}\widehat{\mathbf{y}},n=0–8$, where $\mathrm{\Delta }\mathrm{Q}=1.35\times {10}^{-7}\mathrm{a}.\mathrm{u}.$ Vertical solid lines represent positions of the energy of light $\left({\omega }_{\mathrm{light}}=\mathrm{Qc}\right)$ for each of the wave vector Q. Insets: dispersion of the longitudinal 2D plasmon (TM mode) for each of the dopings, calculated when retardation effects are included $(c$ is speed of light in the vacuum) (green line) and when retardation effects are not included $c\to \infty$ (yellow line).

Figure 7

(a) Absorption spectrum for $\mathbf{s}$ (blue lines) modes in doped graphene $\left(E_F=0.5\mathrm{eV}\right)$ where incident wave vectors are ${\mathbf{Q}}_n={\mathrm{Q}}_0+n\mathrm{\Delta }\mathrm{Q}\widehat{\mathbf{y}},n=0–3$, with ${\mathrm{Q}}_0=2.6\times {10}^{-4}\mathrm{a}.\mathrm{u}.$ and $\mathrm{\Delta }\mathrm{Q}=6.76\times {10}^{-7}\mathrm{a}.\mathrm{u}.$ (b) Same as in (a) but with $E_F=1\mathrm{eV},{\mathrm{Q}}_0=4.6\times {10}^{-4}\mathrm{a}.\mathrm{u}.$, and $\mathrm{\Delta }\mathrm{Q}=6.76\times {10}^{-7}\mathrm{a}.\mathrm{u}.$ Vertical solid lines are positions of the energy of light $\left({\omega }_{\mathrm{light}}=\mathrm{Qc}\right)$ for the corresponding wave vectors Q. Insets: dispersion of the TE mode for each of the dopings compared with the dispersion of light. (c) Absorption intensities for $\mathbf{s}$ mode around the energies of $\pi \to {\pi }^{*}$ transitions for very small $\text{Q}$ wave vectors $\left(\mathrm{Q}\sim {\mathrm{Q}}_{\mathrm{light}}\right)$.

Figure 8

Optical absorption spectrum for $\mathbf{s}$ and $\mathbf{p}$ polarized incident electromagnetic field in graphene with different doping: (a), (d) $E_F=0$; (b), (e) $E_F=0.5\mathrm{eV}$; and (c), (f) $E_F=1\mathrm{eV}$. In each of the graphs, the five absorption spectra are shown for five incident wave vectors along the $\mathrm{\Gamma }\mathrm{M}$ direction ${\mathbf{Q}}_n=n\mathrm{\Delta }\mathrm{Q}\mathbf{y},n=0,1,2,3,4$, where $\mathrm{\Delta }\mathrm{Q}=0.0039\mathrm{a}.\mathrm{u}.$ The brown arrows indicate the direction of increasing wave vector. Insets in (a), (b), and (c) show the dispersion relations of low-energy $\pi \to \pi$ intraband peaks, while insets in (e) and (f) show the dispersion relations of 2DPP. The dashed lines represent the upper edge $\left(v_{F}Q\right)$ of the $\pi \to \pi$ intraband electron-hole continuum where the Fermi velocity is $v_F=\sqrt{3}ta/2$ and the hopping parameter is $t=2.7\mathrm{eV}$. Red solid lines represent the long-wavelength approximation of the 2DPP dispersion relation $\sqrt{2E_F\mathrm{Q}}$.

Figure 9

(a) Real part of interband optical conductivity in pristine graphene $\left(E_F=0\right)$ calculated using Eq. (58) where the damping constants are ${\eta }_{\mathrm{inter}}=60\mathrm{meV}$ (blue), ${\eta }_{\mathrm{inter}}=90\mathrm{meV}$ (red), and ${\eta }_{\mathrm{inter}}=120\mathrm{meV}$ (yellow). (b) Same as in (a) but using Eq. (59). (c) Real part of intraband optical conductivity in pristine graphene $\left(E_F=0\right)$ calculated using Eq. (57) with ${\eta }_{\mathrm{intra}}=5\mathrm{meV}$ (blue), ${\eta }_{\mathrm{intra}}=25\mathrm{meV}$ (red), and ${\eta }_{\mathrm{intra}}=50\mathrm{meV}$ (yellow), and $T=300\mathrm{K}$ for all three cases. (d) Total optical conductivity as a result of combining Eq. (57) with (58) (black) and (59) (green), where ${\eta }_{\mathrm{intra}}=5\mathrm{meV}$ and ${\eta }_{\mathrm{inter}}=60\mathrm{meV}$. The wave vector is $\mathrm{Q}=0$ which corresponds to the electromagnetic field of normal incidence and polarization $\mu =x$ or $y$.