Temperature and voltage probes far from equilibrium

We consider an open system of noninteracting electrons consisting of a small sample connected to several reservoirs and temperature or voltage probes. We study the nonlinear system of equations that determines the probe parameters. We show that it has a unique solution, which can be computed with a fast converging iterative algorithm. We illustrate our method with two well-known models: the three-terminal system and the open Aharonov-Bohm interferometer.

Figure 1

A multiterminal system: The sample is connected through leads to $N=N_{\mathrm{p}}+N_{\mathrm{t}}$ particle reservoirs. The reservoirs $1,...,N_{\mathrm{p}}$ are probes and the reservoirs $N_{\mathrm{p}}+1,...,N_{\mathrm{p}}+N_{\mathrm{t}}$ are thermostats driving the system out of equilibrium.

Figure 2

A one-channel system with two thermostats (2 and 3) and one probe (1).

Figure 3

The temperature dependence of $\mathcal{N}(T,{\mu }_2=0,{\mu }_3=100)$.

Figure 4

The potential dependence of $\mathcal{N}(T=1,{\mu }_2=0,{\mu }_3)$.

Figure 5

The open AB interferometer. A magnetic flux $\phi$ crosses the ring and a QD is placed in the upper branch of the ring. The terminals 1 to 12 are voltage probes and the terminals $L$ and $R$ are thermostats. $k_T$, $k_P$, and $k_D$ are the hopping constants coupling the ring to the thermostats, the probes, and the QD. The energy levels of the dot are $V$, $V+U$, and $V+2U$.

Figure 6

The “intrinsic” phase ${\alpha }_{\mathrm{QD}}$ (blue dashed line) and the “experimental” phase $\beta$ (red solid line) as a function of the gate voltage $V$ applied on the QD. This corresponds to the setup of Fig. 5 with a three-level QD, all temperatures equal to zero, the chemical potential of the thermostats are ${\mu }_L=0$ and ${\mu }_R=0.2$. The couplings to the thermostats, probes, and QD are $k_T=0.5$, $k_P=0.5k_T$, and $k_D=0.01k_T$, respectively (the parameter $k_P$ plays a role similar to $J_x$ in Ref. 20). All remaining parameters are set as in Ref. 20.