Coulomb drag between massless and massive fermions

We theoretically investigate the frictional drag induced by the Coulomb interaction between spatially separated massless and massive fermions in the Boltzmann regime and at low temperatures. As a model system, we use a double-layer structure composed of a two-dimensional electron gas (2DEG) and an $n$-doped graphene layer. We analyze this system numerically and also present analytical formulas for the drag resistivity in the limit of large and small interlayer separation. Both, the temperature and density dependence are investigated and compared to 2DEG-2DEG and graphene-graphene double-layer structures. Whereas the density dependence of the transresistivity for small interlayer separation differs already in the leading order for each of those three structures, we find the leading order contribution of the density dependence in the large interlayer separation limit to exhibit the same density dependence in each case. In order to distinguish between the different systems in the large interlayer separation limit, we also investigate the subleading contribution to the transresistivity. Furthermore, we study the Coulomb drag in a double-layer structure consisting of $n$-doped bilayer and monolayer graphene, which we find to possess the same qualitative behavior as the 2DEG-graphene system.

Figure 1

Schematic illustration of the geometry considered. The 2DEG is located within a quantum well of width $w$ and its localization along the $z$ direction is described by ${\chi }_{2\mathrm{D}}\left(z\right)$, whereas the location of the graphene sheet is given by ${\chi }_g\left(z\right)$. The relative dielectric constants of the structure are given by ${\kappa }_1$, ${\kappa }_2$, ${\kappa }_3$, and ${\kappa }_{2\mathrm{D}}$.

Figure 2

Dependence of the transresistivity on the temperature $T$ for air/graphene/Al${}_2$O${}_3$/GaAs/AlGaAs and air/graphene/SiO${}_2$/GaAs/AlGaAs structures with an interlayer distance of $d=20$ nm for different widths of the GaAs quantum well ($w=0,5,10$ nm). The electronic densities of graphene and GaAs have been set to the values $n_{\mathrm{g}}=1.0\times {10}^{12}$ cm${}^{-2}$ and $n_{\mathrm{GaAs}}=1.0\times {10}^{11}$ cm${}^{-2}$, respectively.

Figure 3

Dependence of the transresistivity on the electronic density in graphene, $n_{\mathrm{g}}$, at $T=100$ K for air/graphene/Al${}_2$O${}_3$/GaAs/AlGaAs and air/graphene/SiO${}_2$/GaAs/AlGaAs structures with an interlayer distance of $d=20$ nm and for different widths of the GaAs quantum well ($w=0,5,10$ nm). The electronic density of GaAs has been set to the value $n_{\mathrm{GaAs}}=1.0\times {10}^{11}$ cm${}^{-2}$.

Figure 4

(a) Dependence of the transresistivity of an air/graphene/Al${}_2$O${}_3$/bilayer graphene/SiO${}_2$ structure on the electronic density of graphene ($n_{\mathrm{g}}$), for different electronic densities in bilayer graphene ($n_{\mathrm{bg}}$) at $T=100$ K. (b) Temperature dependence of the transresistivity for the same structure as in (a) at fixed graphene electronic density, $n_{\mathrm{g}}=5\times {10}^{11}$ cm${}^{-2}$. In both cases, (a) and (b), the interlayer distance is $d=20$ nm.