Anderson-Bogoliubov and Carlson-Goldman modes in counterflow superconductors: Case study of a double monolayer graphene

The impact of electron-hole pairing on the spectrum of plasma excitations in double layer systems is investigated. The theory is developed with reference to a double monolayer graphene. Taking into account the coupling of scalar potential oscillations with oscillations of the order parameter $\mathrm{\Delta }$, we show that the spectrum of antisymmetric (acoustic) plasma excitations contains two modes: a weakly damped mode below the gap $2\mathrm{\Delta }$ and a strongly damped mode above the gap. The lower mode can be interpreted as an analog of the Carlson-Goldman mode. This mode has an acoustic dispersion relation at small wave vectors and it saturates at the level $2\mathrm{\Delta }$ at large wave vectors. Its velocity is larger than the velocity of the Anderson-Bogoliubov mode $v_{AB}=v_F/\sqrt{2}$, and it can be smaller than the Fermi velocity $v_F$. The damping rate of this mode strongly increases under increase of temperature. Out-of-phase oscillations of two order parameters in two spin subsystems are also considered. This part of the spectrum contains two more modes. One of them is interpreted as an analog of the Anderson-Bogoliubov (phase) mode and the other as an analog of the Schmid (amplitude) mode. With minor modifications the theory can be extended to describe collective modes in a double bilayer graphene as well.

Figure 1

Graphene lattice. The red (dark) and yellow (light) circles correspond to the A and B sublattices. The unit cells are shown by dashed lines and two primitive lattice vectors are shown by arrows.

Figure 2

Frequency dependence of the dielectric loss function (52) at $T=0.1\mu ,0.07\mu ,0.05\mu$, $q=0.1k_F$, $\mathrm{\Delta }=0.2\mu$, and $\hslash \gamma ={10}^{-3}\mu$.

Figure 3

Frequency dependence of the dielectric loss function (52) at $T=0.1\mu$, $q=0.02k_F,0.1k_F,0.2k_F,0.3k_F$, $\mathrm{\Delta }=0.2\mu$, and $\hslash \gamma ={10}^{-3}\mu$.

Figure 4

Frequency dependence of the energy loss functions (54), (55) in $\mu /g{k}_F^2$ units at $T=0.1\mu$, $q=0.1k_F,0.2k_F,0.3k_F$, $\mathrm{\Delta }=0.2\mu$, and $\hslash \gamma ={10}^{-3}\mu$.

Figure 5

The same as in Fig. 4 at $T=0$.

Figure 6

The spectra of the Anderson-Bogoliubov (solid curve) and Schmid (dashed curve) modes in the double layer graphene system.

Figure 7

Frequency dependence of the energy loss function (61) at $T=0.1\mu$ (a) and $T=0$ (b). The high-frequency peaks at $T=0$ are shown in the inset in another scale.

Figure 8

The dispersion curves for the Carlson-Goldman mode (solid curve) and for the upper antisymmetric plasmon mode (dashed curve). The spectrum of the antisymmetric (acoustic) plasmon mode in the normal state is shown by the dash-dotted line.