Recovery of SINIS turnstile accuracy in a strongly nonequilibrium regime

We perform a theoretical study of nonequilibrium effects in charge transport through a hybrid single-electron transistor based on a small normal metal (N) island with the gate-controlled number of electrons, tunnel-coupled to voltage-biased superconducting (S) electrodes (SINIS). Focusing on the turnstile mode of the transistor operation with the gate voltage driven periodically, and electrons on the island being out of equilibrium, we find that the current quantization accuracy is a nonmonotonic function of the relaxation rate ${\mathrm{\Gamma }}_{\mathcal{F}}$ of the distribution function $\mathcal{F}\left(\epsilon \right)$ on the island due to tunneling, as compared to the drive frequency $f$, electron-electron $1/{\tau }_{ee}$, and electron-phonon $1/{\tau }_{eph}$ relaxation rates. Surprisingly, in the strongly nonequilibrium regime, $f\gg {\mathrm{\Gamma }}_{\mathcal{F}}\gg {\tau }_{ee}^{-1},{\tau }_{eph}^{-1}$, the turnstile current plateau is recovered, similarly to the ideal equilibrium regime, ${\tau }_{eph}^{-1}\gg {\mathrm{\Gamma }}_{\mathcal{F}}$. The plateau is destroyed in the quasiequilibrium regime when the electron-electron relaxation is faster than tunneling.

Figure 1

Schematics of a hybrid SINIS SET used in turnstile experiments.

Figure 2

An illustration of the SINIS turnstile operation in the equilibrium regime. Panels show energy diagrams of quasiparticle distributions in superconducting electrodes with a gap $\mathrm{\Delta }$ and in the normal island for the two stages of the turnstile operation. The two positions $\mu =\mp A_g$ of the island chemical potential, shown by solid black lines, correspond to (a) injection and (b) ejection stages, respectively. The filled (empty) states are shown by blue (white) color, while the electrodes are numbered by 1 and 2 in the circles. Both in the island and in electrodes, the electronic distributions are zero-temperature Fermi distributions. The electrode chemical potentials are shifted by a bias voltage $V$ and are shown by horizontal dashed lines. The main tunneling process of injection (ejection) is shown by the green arrow.

Figure 3

The energy diagrams of a SINIS SET turnstile in the fast drive regimes mentioned in the text, when the drive frequency $f\gg {\mathrm{\Gamma }}_{\mathcal{F}}$, the distribution function relaxation rate ${\mathrm{\Gamma }}_{\mathcal{F}}$ due to tunneling. (a) Equilibrium regime ${\mathrm{\Gamma }}_{\mathcal{F}}\ll {\tau }_{eph}^{-1}$ at low bath temperature; (b) Quasiequilibrium regime ${\tau }_{eph}^{-1}\ll {\mathrm{\Gamma }}_{\mathcal{F}}\ll {\tau }_{ee}^{-1}$ with the long thermal tails of the distribution function $\mathcal{F}\left(\epsilon \right)$; (c) and (d) Nonequilibrium regime ${\mathrm{\Gamma }}_{\mathcal{F}}\gg {\tau }_{ee}^{-1},{\tau }_{eph}^{-1}$ for (c) small $A_g<\mathrm{\Delta }$ and (d) large $A_g>\mathrm{\Delta }$ amplitudes. In all panels the distribution functions in the island shown by blue color correspond to the first half-period ($\mu$ is shown by black solid lines). The solid green, dashed red, and dotted purple arrows show the main forward tunneling, back tunneling, and leakage contributions, respectively.

Figure 4

A schematic diagram of different regimes of a SINIS SET turnstile out of equilibrium, ${\tau }_{eph}^{-1}\ll {\mathrm{\Gamma }}_{\mathcal{F}}$, showing the three regimes discussed in the text.

Figure 5

The period-averaged current $\langle I\rangle$ through the SINIS SET versus the drive amplitude $A_g/\mathrm{\Delta }$ for the equilibrium (blue solid line), quasiequilibrium (green dashed-dotted line), and fully nonequilibrium (red dashed line) regimes discussed in the text, and $eV=1.75\mathrm{\Delta }$. The threshold values, $A_g^{\mathrm{ft}}=0.125\mathrm{\Delta }$ and $A_g^{\mathrm{bt}}=1.875\mathrm{\Delta }$, are shown explicitly on the horizontal axis. The calculation method and parameters' values are discussed below in Sec. 4d and in Fig. 7. The finite widths of the current plateau onsets at small $(A_g-A_g^{\mathrm{ft}})/\mathrm{\Delta }\lesssim 0.2$ are related to the missed tunneling events due to the regime $\mathrm{\Gamma }/f\lesssim 1$.

Figure 6

A sketch of the dependence ${\overline{\mathcal{F}}}_{\mathrm{neq}}\left(\epsilon \right)$ in the turnstile regime $A_g^{\mathrm{ft}}<A_g<A_g^{\mathrm{bt}}$ for (a) $A_g<\mathrm{\Delta }$ and (b) $A_g>\mathrm{\Delta }$.

Figure 7

The period-averaged current $\langle I\rangle$ through the SINIS SET versus the drive amplitude $(A_g-A_g^{\mathrm{ft}})/\mathrm{\Delta }$ relative to the threshold value for the equilibrium (Eq), ${\mathrm{\Gamma }}_{\mathcal{F}}\ll {\tau }_{eph}^{-1}$, quasiequilibrium (QEq), ${\tau }_{eph}^{-1}\ll {\mathrm{\Gamma }}_{\mathcal{F}}\ll {\tau }_{ee}^{-1}$, and fully nonequilibrium (NEq), ${\tau }_{eph}^{-1},{\tau }_{ee}^{-1}\ll {\mathrm{\Gamma }}_{\mathcal{F}}$, regimes and crossovers between them. In (a), (c), and (e) we represent zooms of the regions shown by dashed rectangles in (b), (d), and (f) near the plateau $\langle I\rangle =ef$. (a), (b) NEq – QEq crossover governed by the decrease of $l_{\mathrm{mfp}}$ from 1 (red thin solid) to $2.4\times {10}^{-2}$ (violet thick dashed-dotted) by a factor $\sqrt{10}$ at each step. This corresponds to the change of the Thouless energy $E_{\mathrm{Th}}/\mathrm{\Delta }$ from 75 (red thin solid) to 1.8 (violet thick dashed-dotted). We take $\delta /\mathrm{\Delta }={10}^{-2}$ keeping ${\mathrm{\Gamma }}_{\mathcal{F}}$ constant. (c), (d) QEq-Eq crossover governed by the decrease of $\delta /\mathrm{\Delta }$ from ${10}^{-2}$ (red thin solid) to ${10}^{-7}$ (violet thick dashed-dotted curve) by a factor 10 at each step. We choose $l_{\mathrm{mfp}}\propto {(\delta /\mathrm{\Delta })}^{-1/3}$ and change it from $2.4\times {10}^{-2}$ up to 1 to keep the Thouless energy constant, $E_{\mathrm{Th}}/\mathrm{\Delta }=1.8$. This effectively changes only ${\mathrm{\Gamma }}_{\mathcal{F}}\simeq f\delta /\mathrm{\Delta }$. In all curves ${\tau }_{ee}^{-1}\gg {\mathrm{\Gamma }}_{\mathcal{F}}$. (e), (f) NEq – Eq crossover governed by the decrease of $\delta /\mathrm{\Delta }$ from ${10}^{-2}$ (red thin solid) to ${10}^{-7}$ (violet thick dashed-dotted curve) by a factor 10 at each step. We consider the maximal possible Thouless energy taken at $l_{\mathrm{mfp}}=1$, however, due to $\delta$ dependence of $E_{\mathrm{Th}}/\mathrm{\Delta }$, it also varies from 75 (red thin solid) to 1.8 (violet thick dashed-dotted). In all panels we considered the island made of AlMn ($T_{eph}=250$ K, $E_F=11.7\mathrm{eV}$, and ${\nu }_{\mathcal{N}}\left(0\right)=1.45\times {10}^{47}{\mathrm{J}}^{-1}{\mathrm{m}}^{-3}$) and take $eV=1.75\mathrm{\Delta },f/\mathrm{\Delta }=2\times {10}^{-3}, \gamma /\delta =4\times {10}^{-2}\gg f/\mathrm{\Delta }$. In (b), (d), and (f) the vertical dashed-dotted line shows the amplitude threshold value $A_g=\mathrm{\Delta }$ for the nonequilibrium case. The finite widths of the current plateau onsets at small $(A_g-A_g^{\mathrm{ft}})/\mathrm{\Delta }\lesssim 0.2$ are related to the missed tunneling events due to the regime $\mathrm{\Gamma }/f\lesssim 1$.

Figure 8

A sketch of ${\mathcal{F}}_{\mathrm{neq}}\left(\epsilon \right)$ in the turnstile regime $A_g^{\mathrm{ft}}<A_g<A_g^{\mathrm{bt}}$ for a slow drive out of equilibrium $f,{\tau }_{eph}^{-1},{\tau }_{ee}^{-1}\ll {\mathrm{\Gamma }}_{\mathcal{F}}$ at (a) the first and (b) second halves of the period. In (a) the approximate values of the distribution function relaxation rate ${\mathrm{\Gamma }}_{\mathcal{F}}$ due to tunneling are shown for the energy intervals (i)–(iii) mentioned in the text.

Figure 9

The energy diagram of a SINIS SET turnstile in the regimes of slow drive out of equilibrium $f,{\tau }_{eph}^{-1}\ll {\mathrm{\Gamma }}_{\mathcal{F}}$. The distribution function in the island shown by blue color corresponds to the first half-period. The corresponding chemical potential is shown by the black solid line. The shifts of the effective chemical potentials of the Fermi functions relative to the edges of the superconducting DOSes of the leads are shown explicitly.