Local densities, distribution functions, and wave-function correlations for spatially resolved shot noise at nanocontacts

We consider a current-carrying, phase-coherent multiprobe conductor to which a small tunneling contact is attached. We treat the conductor and the tunneling contact as a phase-coherent entity and use a Green’s function formulation of the scattering approach. We show that the average current and the current fluctuations at the tunneling contact are determined by an effective local nonequilibrium distribution function. This function characterizes the distribution of charge carriers (or quasiparticles) inside the conductor. It is an exact quantum-mechanical expression and contains the phase coherence of the particles via local partial densities of states, called injectivities. The distribution function is analyzed for different systems in the zero-temperature limit as well as at finite temperature. Furthermore, we investigate in detail the correlations of the currents measured at two different contacts of a four-probe sample, where two of the probes are only weakly coupled contacts. In particular, we show that the correlations of the currents are at zero temperature given by spatially nondiagonal injectivities and emissivities. These nondiagonal densities are sensitive to correlations of wave functions and the phase of the wave functions. We consider ballistic conductors and metallic diffusive conductors. We also analyze the Aharonov-Bohm oscillations in the shot-noise correlations of a conductor which in the absence of the nanocontacts exhibits no flux sensitivity in the conductance.