Ballistic thermal conductance of electrons in graphene ribbons

We investigate the ballistic thermal conductance of electrons in gated graphene ribbons with width above 20 nm and clarify both the temperature and the Fermi-level dependences. In the intrinsic graphene ribbons, the normalized thermal conductance by the quantum conductance, ${\kappa }_0$, increases monotonically with temperature. In the gated graphene ribbons, the normalized thermal conductance increases steplikely as the Fermi level increases but it has nonmonotonic temperature dependence when the Fermi level is a little larger than the bottom of the subband. The value of the step height changes from $4{\kappa }_0$ to $2{\kappa }_0$ with increasing temperature. The ballistic electron thermal conductance per unit width of graphene ribbons is smaller than those of corresponding single-walled nanotubes and a graphene sheet and it approaches at 100 K that of a graphene sheet above the ribbon width of about 80 and 50 nm for zigzag ribbons and armchair ribbons, respectively.

Figure 1

Schematic figures of (a) a zigzag ribbon with a width of $N_z=4$ and (b) an armchair ribbon with $N_a=7$.

Figure 2

Energy dispersion relations in the tight-binding model of (a) a zigzag ribbon with a width of $N_z=8$ and (b) an armchair ribbon with $N_a=8$.

Figure 3

Electron-derived thermal conductance ${\kappa }_{\text{el}}$ in the tight-binding model of (a) zigzag ribbons at $N_z=8,101,200,302,401,602,800$ and (b) armchair ribbons at $N_a=11,101,200,302,401,602,800$ as a function of temperature. ${\kappa }_{\text{el}}$ for (400,400), (300,300), (200,200),(100,100), (50,50), and (4,4) armchair nanotubes are also plotted by broken lines in (a) and those for (399,0), (300,0), (201,0), (99,0), (51,0), and (6,0) zigzag nanotubes are plotted by broken lines in (b). No temperature dependence is noticed for two narrowest zigzag nanotubes in (b).

Figure 4

Electron-derived thermal conductance ${\kappa }_{\text{el}}$ of (a) a zigzag ribbon with $N_z=200$ and (b) an armchair ribbon with $N_a=200$ at $\mu =0,20,40,60,80,100\text{meV}$ as a function of temperature.

Figure 5

Electron-derived thermal conductance ${\kappa }_{\text{el}}$ of zigzag ribbons with $N_z=101,152,200,251$, as a function of the Fermi level, $\mu$, at (a) $T=1\text{K}$, (b)$T=10\text{K}$, (c)$T=100\text{K}$, and (d) $T=300\text{K}$.

Figure 6

Electron-derived thermal conductance ${\kappa }_{\text{el}}$ of armchair ribbons with $N_a=101,152,200,251$, as a function of the Fermi level, $\mu$, at (a) $T=1\text{K}$, (b) $T=10\text{K}$, (c) $T=100\text{K}$ and (d) $T=300\text{K}$.