Nonlocal transport through multiterminal diffusive superconducting nanostructures

Motivated by recent experiments on nonlocal transport through multiterminal superconducting hybrid structures, we present self-consistent calculations based on quasiclassical Green’s functions for the order parameter, currents, and voltages in a system consisting of a diffusive superconductor connected to two normal and one superconducting electrodes. We investigate nonequilibrium effects for different biasing conditions corresponding to measurements of the nonlocal conductance and of the nonlocal resistance. It is shown that while the nonlocal conductance does not change its sign, this change might be observed in a nonlocal resistance measurement for certain parameter range. The change in sign of the nonlocal signal takes places at a voltage of the order of the self-consistent gap of the superconducting region. We show that this is not related to the nonlocal Andreev processes but rather to nonequilibrium effects. We finally discuss the case of four terminal measurements and demonstrate that a change in sign in the nonlocal resistance appears when the current injected into the superconductor exceeds a critical value. The connection to the existing experiments is discussed.

Figure 1

(Color online) Sketch of a three-terminal device for (a) measurement of the nonlocal conductance and (b) measurement of the nonlocal resistance. (c) Typical experimental setup for measurement of the nonlocal resistance.

Figure 2

(Color online) The amplitude of the self-consistent order parameter, the current $I_L$ injected from the left normal electrode and the current $I_R$ measured at the right electrode, as a function of the voltage $V_L$. Panels (a)–(c) for ${ϵ}_S=0.2$, ${ϵ}_L=0.1$, and $T=0.01{\Delta }_0$ and different values of ${ϵ}_R$. Panels (d)–(f) for ${ϵ}_S=0.2$, ${ϵ}_R=0.1$, and $T=0.01{\Delta }_0$ and different values of ${ϵ}_L$. We have defined $R=R_N{ϵ}_{\text{Th}}/{\Delta }_0$.

Figure 3

(Color online) The amplitude of the self-consistent order parameter as a function of the voltage $V_L$ for different temperatures. Notice that in the range of temperatures shown and for large values of $V_L$ the gap is enhanced by increasing $T$.

Figure 4

(Color online) The voltage dependence of the (negative) nonlocal conductance $G_{\text{nl}}$ normalized with respect to $R=R_N{ϵ}_{\text{Th}}/{\Delta }_0$. We have chosen $T=0.01{\Delta }_0$, ${ϵ}_S=0.2$, ${ϵ}_L=0.1$, and $T=0.01{\Delta }_0$ and different values of ${ϵ}_R$.

Figure 5

(Color online) (a) The amplitude of the self-consistent order parameter, (b) the current $I_L$ injected from the left normal electrode, and (c) the induced voltage $V_R$ measured at the right electrode as a function of the voltage $V_L$ for ${ϵ}_S=0.2$, ${ϵ}_L=0.1$, and $T=0.01{\Delta }_0$ and different values of ${ϵ}_R$.

Figure 6

(Color online) The voltage dependence of the nonlocal resistance calculated from Eq. (13) for ${ϵ}_S=0.2{\Delta }_0$, ${ϵ}_L=0.1{\Delta }_0$, and $T=0.01{\Delta }_0$, and two values of ${ϵ}_R$.

Figure 7

(Color online) The amplitude of the self-consistent order parameter, the current $I_L$ injected from the left normal electrode and the induced voltage $V_R$ measured at the right electrode, as a function of the voltage $V_L$, for ${ϵ}_L={ϵ}_R=0.1$, $T=0.01{\Delta }_0$, and different values of ${ϵ}_S$.

Figure 8

(Color online) The voltage dependence of nonlocal resistance for the same parameters as in Fig. 7.

Figure 9

(Color online) The temperature dependence of the zero bias nonlocal resistance measure in a three-terminal device for different values of ${ϵ}_S$. We have chosen ${ϵ}_L={ϵ}_R=0.1{\Delta }_0$.

Figure 10

(Color online) Sketch of the four-terminal structure under consideration.

Figure 11

(Color online) The nonlocal resistance measured in a four terminal structure as a function of the bias voltage $V_L$ for ${ϵ}_L={ϵ}_R=0.1{\Delta }_0$, ${ϵ}_S=0.2$, and different values of the temperature. Inset: the nonlocal resistance as a function of the temperature at zero bias.