Noise Thermal Impedance of a Diffusive Wire

The current noise density $S_2$ of a conductor in equilibrium, the Johnson noise, is determined by its temperature $T$: $S_2=4k_{B}TG$, with $G$ the conductance. The sample’s noise temperature $T_N=S_2/(4k_{B}G)$ generalizes $T$ for a system out of equilibrium. We introduce the “noise thermal impedance” of a sample as the ratio $\delta T_N^{\omega }/\delta P_J^{\omega }$ of the amplitude $\delta T_N^{\omega }$ of the oscillation of $T_N$ when heated by an oscillating power $\delta P_J^{\omega }$ at frequency $\omega$. For a macroscopic sample, it is the usual thermal impedance. We show for a diffusive wire how this (complex) frequency-dependent quantity gives access to the electron-phonon interaction time in a long wire and to the diffusion time in a shorter one, and how its real part may also give access to the electron-electron inelastic time. These times are not simply accessible from the frequency dependence of $S_2$ itself.

Figure 1

Real and imaginary parts of $\mathcal{R}(\omega )$ as a function of $\omega {\tau }_D$, in the hot electron, diffusion-cooled regime (ii) (solid lines), and independent-electron regime (iii) (dashed lines). The values of $\mathcal{R}(0)$ differ by $\sim 10\%$ for $e{V}_{\mathrm{dc}}\ll k_{B}T$. (Inset) The geometry considered: a wire of length $L$ between two thick normal metal contacts.

Figure 2

Magnitude (thick lines) and real parts (thin lines) of $\mathcal{R}(\omega )$ as a function of $\omega {\tau }_D$, in the hot electrons, diffusion-cooled regime (ii) (solid lines, and independent electrons regime (iii) (dashed lines). Inset: Current noise amplitude modulated by the time-dependent bias $V(t)$.