Electromagnetic properties of a double-layer graphene system with electron-hole pairing

We study electromagnetic properties of a double-layer graphene system in which electrons from one layer are coupled with holes from the other layer. The gauge invariant linear response functions are obtained. The frequency dependences of the transmission, reflection, and absorption coefficients are computed. We predict a peak in the reflection and absorption at the frequency equal to the gap in the quasiparticle spectrum. It is shown that the electron-hole pairing results in an essential modification of the spectrum of surface TM plasmons. We find that the optical TM mode splits into a low frequency undamped branch and a high frequency damped branch. At zero temperature the lower branch disappears. It is established that the pairing does not influence the acoustic TM mode. It is also shown that the pairing opens the frequency window in the subgap range for the surface TE wave.

Figure 1

Frequency dependence of the parallel current conductivity, the real (a) and imaginary (b) parts, in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid line) and $\mathrm{\Delta }=0.2\mu$ (dash-dotted line), and in the normal state (dashed line). The conductivity is given in $e^2/4\hslash$ units. The temperature $T=0.1\mu$.

Figure 2

Frequency dependence of the transmission ($T$), reflection ($R$) and absorption ($A$) coefficients for the normal incidence. Left panel, the double-layer graphene system in the normal state; right panel, the same system in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid curves) and $\mathrm{\Delta }=0.2\mu$ (dashed curves). The temperature $T=0.1\mu$.

Figure 3

The continuum of electron-hole excitations (shaded area) for the double-layer graphene system in the normal state (a) and in the paired state with $\mathrm{\Delta }=0.5\mu$ (b).

Figure 4

The spectrum (upper panel) and damping rate (lower panel) for two branches of the symmetric TM mode in the system with electron-hole pairing at $\mathrm{\Delta }=0.5\mu$ and two different temperatures $T=0.1\mu$ and $T=0.2\mu$. In the upper panel the spectrum of the symmetric TM mode in the normal state is shown by dash-dotted line and the lower boundary for the continuum of electron-hole excitations is shown by dotted line.

Figure 5

The same as in Fig. 4 at $\mathrm{\Delta }=0.2\mu$ and $T=0.1\mu$.

Figure 6

Frequency dependence of the loss function $L_{+}(q,\omega )$ for $q=0.1k_F$ at $T=0.1\mu$ (upper panel) and $T=0$ (lower panel) in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid line) and $\mathrm{\Delta }=0.2\mu$ (dashed line), and in the normal state (dotted line).

Figure 7

The imaginary part of the counterflow conductivity (in units of $e^2/4\hslash$) at $T=0.1\mu$ in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid line) and in the normal state (dashed line).

Figure 8

The spectrum of the antisymmetric TM wave at $T=0.1\mu$ in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid line) and in the normal state (dashed line).

Figure 9

Frequency dependence of the loss function $L_{-}(q,\omega )$ for $q=0.1k_F$ and $q=0.3k_F$ at $T=0.1\mu$ in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid line) and in the normal state (dotted line).

Figure 10

The imaginary part of the parallel current conductivity (in $e^2/4\hslash$ units) at $T=0$ in the paired state with $\mathrm{\Delta }=0.5\mu$ (solid line) and $\mathrm{\Delta }=0.2\mu$ (dash-dotted line), and in the normal state (dashed line).