Charge imbalance in superconductors in the low-temperature limit

We explore charge imbalance in mesoscopic normal-metal/superconductor multiterminal structures at very low temperatures. The investigated samples, fabricated by e-beam lithography and shadow evaporation, consist of a superconducting aluminum bar with several copper wires forming tunnel contacts at different distances from each other. We have measured in detail the local and nonlocal conductance of these structures as a function of the applied bias voltage $V$, the applied magnetic field $B$, the temperature $T$, and the contact distance $d$. From these data the charge-imbalance relaxation length ${\lambda }_{Q^{\ast }}$ is derived. The bias-resolved measurements show a transition from dominant elastic scattering close to the energy gap to an inelastic two-stage relaxation at higher bias. We observe a strong suppression of charge imbalance with magnetic field, which can be directly linked to the pair-breaking parameter. In contrast, practically no temperature dependence of the charge-imbalance signal was observed below 0.5 K. These results are relevant for the investigation of other nonlocal effects such as crossed Andreev reflection and spin diffusion.

Figure 1

(Color online) Scanning electron microscopy image of sample A illustrating the experimental scheme. Five copper (Cu) fingers are connected by tunnel contacts to an aluminum bar (Al). For one of the possible injector-detector pairs the bias and measurement scheme used for charge imbalance detection is shown.

Figure 2

(Color online) Local differential conductance $g=dI/dV$ of one contact of sample B as a function of bias voltage $V$ for (a) different temperatures $T$ and (b) different applied magnetic fields $B$. Symbols represent measured data, lines are fits to the model described in the text.

Figure 3

(Color online) (a) Pair potential $\Delta$ as a function of the applied magnetic field $B$. $\Delta$ is normalized to its zero-field value ${\Delta }_0$ and $B$ is normalized to the critical field $B_{\text{c}}$. (b) Normalized pair-breaking parameter $\Gamma /{\Delta }_0$ as a function of ${\left(B/B_{\text{c}}\right)}^2$. The lines are predictions from pair-breaking theory.

Figure 4

(Color online) (a) Differential conductance $g_{\text{det}}$ of the left-most contact of sample A as a function of bias voltage $V_{\text{det}}$ with additional voltage bias $V_{\text{inj}}$ applied to the neighbor contact at $1\mu \text{m}$ distance. (b) Normalized pair potential $\Delta /{\Delta }_0$ as a function of injector bias $V_{\text{inj}}$. The line is a guide to the eye.

Figure 5

(Color online) Nonlocal differential conductance $g_{\text{nl}}=d{I}_{\text{det}}/d{V}_{\text{inj}}$ for an injector/detector pair of sample A as a function of bias voltage $V_{\text{inj}}$ (a) for different temperatures $T$ and (b) for different applied magnetic fields $B$.

Figure 6

(Color online) (a) Normalized nonlocal differential conductance $g_{\text{nl}}/G_{\text{inj}}{G}_{\text{det}}$ as a function of injector bias voltage $V_{\text{inj}}$ for different contact distances $d$. (b) Semilogarithmic plot of $g_{\text{nl}}/G_{\text{inj}}{G}_{\text{det}}$ as a function of contact distance $d$ for different injector bias $V_{\text{inj}}$. The solid lines are fits to Eq. (9).

Figure 7

(Color online) Charge imbalance relaxation length ${\lambda }_{Q^{\ast }}$ as a function of injector bias voltage $V_{\text{inj}}$ for (a) different temperatures $T$ and (b) different applied magnetic fields $B$. The line is the result of a numerical simulation described in Sec. 5.

Figure 8

(Color online) (a) Normalized charge-imbalance relaxation rate $\hslash /{\Delta }_0{\tau }_{Q^{\ast }}$ as a function of the normalized pair-breaking parameter $\Gamma /{\Delta }_0$. The line is a guide to the eye. (b) Normalized charge-imbalance relaxation rate $\hslash /\Gamma {\tau }_{Q^{\ast }}$ as a function of normalized injector bias $e{V}_{\text{inj}}/{\Delta }_0$. The line is a joint fit to the data of all three samples.

Figure 9

(Color online) $g^{\ast }$ as a function of normalized injector bias $e{V}_{\text{inj}}/{\Delta }_0$ (a) for all samples at $B=0$ and (b) for sample A at different magnetic fields $B$.