Quantum field theoretical description for the reflectivity of graphene

We derive the polarization tensor of graphene at nonzero temperature in $(2+1)$-dimensional space-time. The obtained tensor coincides with the previously known result at all Matsubara frequencies, but, in contrast, it admits analytic continuation to the real frequency axis satisfying all physical requirements. Using the obtained representation for the polarization tensor, we develop a quantum field theoretical description for the reflectivity of graphene. The analytic asymptotic expressions for the reflection coefficients and reflectivities at low and high frequencies are derived for both independent polarizations of the electromagnetic field. The dependencies of the reflectivities on the frequency and angle of incidence are investigated. Numerical computations using the exact expressions for the polarization tensor are performed and application regions for the analytic asymptotic results are determined.

Figure 1

The normalized values of the thermal correction to the 00-component of the polarization tensor of graphene computed as functions of imaginary frequency at $k_{\perp }=10{\xi }_1$ (a) using the representation obtained in this paper and (b) using the representation of Ref. [10].

Figure 2

The normalized values of the thermal correction (47) to the combination of the components of the polarization tensor of graphene computed as functions of imaginary frequency at $k_{\perp }=10{\xi }_1$ (a) using the representation obtained in this paper and (b) using the representation of Ref. [10].

Figure 3

Reflectivities of the TM and TE polarized light of high frequencies on graphene are shown as functions of the incidence angle by the lower and upper lines, respectively.

Figure 4

Reflectivities of graphene at the normal incidence at different temperatures ($T=10$, 100, and 300 K) are shown as functions of frequency measured in K by the solid lines from bottom to top, respectively.

Figure 5

Reflectivities of the TM and TE polarized light of low frequencies ($\omega /T=0.01$ and $\omega /T=0.03$ for the upper and lower pairs of lines, respectively) on graphene are shown as functions of the incidence angle.

Figure 6

Reflectivities of (a) TM and (b) TE polarized light on graphene are shown as functions of $\omega /T$ at the incidence angle ${\theta }_i=30^{\circ }$. The solid, short-dashed, and long-dashed lines are computed by the exact formulas, and by the asymptotic expressions for low and high frequencies, respectively.

Figure 7

The complex $q_4$ plane containing the integration pathes ${\gamma }_1$ and ${\gamma }_2$. The dots indicate the poles at the fermionic Matsubara frequencies and the crosses indicate four additional poles (see text for further discussion).