Quantum Transport Thermometry for Electrons in Graphene

We propose a method of measuring the electron temperature $T_e$ in mesoscopic conductors and demonstrate experimentally its applicability to micron-size graphene devices in the linear-response regime ($T_e\approx T$, the bath temperature). The method can be especially useful in case of overheating, $T_e>T$. It is based on analysis of the correlation function of mesoscopic conductance fluctuations. Although the fluctuation amplitude strongly depends on the details of electron scattering in graphene, we show that $T_e$ extracted from the correlation function is insensitive to these details.

Figure 1

The electronic temperature $T_e$, determined via Eq. (2), as a function of the bath temperature, $T$, for four graphene flakes (F1: open circles, F2: open squares, B1: closed circles, B2: closed squares; see Table ). The inset shows ${\Delta }_c/2.7$ for sample D with $L_T\sim L_x$, i.e., not in the regime (3); the dashed horizontal line shows the Thouless energy, $h/{\tau }_D$.

Figure 2

The diagrams which contribute to the main order in the diagrammatic expansion of the conductivity-conductivity correlation function. The wavy lines stand for the propagators of the diffusion modes describing motion at length scales $\gg \ell$; shaded blocks stand for Hikami boxes, describing motion at length scales $\sim \ell$.

Figure 3

The normalized correlation function, $F_n(\Delta )$, for a wire under conditions (3). The solid line corresponds to the asymptotic Eq. (7). The dotted, dashed, and dash-dotted lines correspond to the numerical integration of Eqs. (4) with $\alpha =0.01$, $\beta =0.3$; $\alpha =\beta =0.1$; $\alpha =0.1$, $\beta =0.3$, respectively ($\alpha \equiv L_T/L_x$, $\beta \equiv L_T/L_{\phi }$). The width at half maximum is ${\Delta }_c\approx (2.8\pm 0.1)k_B{T}_e$. The inset shows $F_n(\Delta )$ for a square sample where each line has the same values of $\alpha$ and $\beta$.

Figure 4

(a) Typical fingerprint of $\delta G$ (normalized by $e^2/h$) for sample B1 at $T=0.25\mathrm{K}$ and $B=90\mathrm{mT}$ as a function of Fermi energy (controlled by the gate voltage [21].) (b) Correlation function for sample B1 at $T=0.25\mathrm{K}$. The inset sketches the circuit used in the experiment. The resistor has resistance much greater than that of the graphene flake to maintain a fixed ac current of 1 nA.