Theory of reflectivity properties of graphene-coated material plates

The theoretical description for the reflectivity properties of dielectric, metal, and semiconductor plates coated with graphene is developed in the framework of the Dirac model. Graphene is described by the polarization tensor allowing the analytic continuation to the real frequency axis. The plate materials are described by the frequency-dependent dielectric permittivities. The general formulas for the reflection coefficients and reflectivities of the graphene-coated plates, as well as their asymptotic expressions at high and low frequencies, are derived. The developed theory is applied to the graphene-coated dielectric (fused silica), metal (Au and Ni), and semiconductor (Si with various charge carrier concentrations) plates. In all these cases, the impact of graphene coating on the plate reflectivity properties is calculated over a wide frequency range. The obtained results can be used in many applications exploiting the graphene coatings, such as the optical detectors, transparent conductors, antireflection surfaces, etc.

Figure 1

The reflectivities of the graphene-coated and uncoated silica plates (the lines 2 and 1, respectively) at the normal incidence are shown as functions of frequency at high frequencies.

Figure 2

(a) The TM and TE reflectivities of the graphene-coated silica plate at $\omega =3\times {10}^{15}$ rad/s are shown as functions of the incidence angle by the lower and upper solid lines, respectively. The dashed lines show similar results for an uncoated silica plate. (b) The TM reflectivities of graphene-coated and uncoated silica plates at $\omega =3\times {10}^{15}$ rad/s are shown by the solid and dashed lines, respectively, in the vicinity of the Brewster angle.

Figure 3

The reflectivities of the graphene-coated and uncoated silica plates (the lines 2, 3 and 1, respectively) at the normal incidence are shown as functions of frequency at low frequencies. The lines 2 and 3 are plotted for the graphene-coated silica at $T=75$ and 300 K, respectively.

Figure 4

(a) The reflectivities of the uncoated Au plate and (b) the relative change of reflectivity of the graphene-coated Au plate at the normal incidence are shown as functions of frequency.

Figure 5

The TM and TE reflectivities of the graphene-coated Au plate at $\omega =3.65\times {10}^{15}$ rad/s are shown as functions of the incidence angle by the lower and upper solid lines, respectively. The dashed lines show similar results for an uncoated Au plate.

Figure 6

(a) The reflectivity of the uncoated Ni plate and (b) the relative change of reflectivity of the graphene-coated Ni plate at the normal incidence are shown as functions of frequency.

Figure 7

(a) The reflectivity of the uncoated high-resistivity Si plate, (b) the relative change of reflectivity of the high-resistivity graphene-coated Si plate, and (c) the reflectivities of the high-resistivity graphene-coated Si plate at $T=300$ and 0 K (the lines 2 and 1, respectively) at the normal incidence are shown as functions of frequency.

Figure 8

The reflectivities of the low-resistivity graphene-coated Si plates are shown at $T=300$ K by the solid lines 1, 2, and 3 as functions of frequency for the concentrations of free charge carriers equal to $5\times {10}^{14}, 5\times {10}^{16}$, and $5\times {10}^{17}{\text{cm}}^{-3}$, respectively. The respective quantities for the uncoated plates are shown by the dashed lines 1, 2, and 3.

Figure 9

The TM and TE reflectivities of the graphene-coated low-resistivity Si plate at $\omega =1.5\times {10}^{12}$ rad/s are shown at $T=300$ K as functions of the incidence angle by the lower and upper solid lines, respectively, for the doping concentrations (a) $N=5\times {10}^{14}{\text{cm}}^{-3}$ and (b) $N=5\times {10}^{17}{\text{cm}}^{-3}$. The dashed lines show similar results for an uncoated Si.