Proximity effect model of ultranarrow NbN strips

We show that narrow superconducting strips in superconducting (S) and normal (N) states are universally described by the model presenting them as lateral NSN proximity systems in which the superconducting central band is sandwiched between damaged edge bands with suppressed superconductivity. The width of the superconducting band was experimentally determined from the value of magnetic field at which the band transits from the Meissner state to the static vortex state. Systematic experimental study of 4.9-nm-thick NbN strips with widths in the interval from 50 nm to 20 μm, which are all smaller than the Pearl's length, demonstrates gradual evolution of the temperature dependence of the critical current with the change of the strip width.

Figure 1

SEM images of stripes, which were prepared from NbN film by positive (a) and negative (b) lithography over PMMA resist. White arrows indicate the exposed areas of the resist.

Figure 2

Temperature dependence of the square resistance of NbN film (symbols). The solid line is the best fit by the 2D Aslamazov-Larkin fluctuation model [Eq. (1)]. (Inset) Temperature dependence of the second critical magnetic field. Red line represents a fitting of linear dependence in the vicinity of critical temperature.

Figure 3

Dependence of the critical temperature on the nominal width for PT (black squares) and NT (red circles) strips. The dashed lines are to guide the eyes.

Figure 4

Dependence of the nominal density of the critical current at 4.2 K on the nominal width for PT (black squares) and for NT (red circles) strips. The dashed lines are to guide the eyes.

Figure 5

Dependence of the nominal density of the retrapping current at 4.2 K on the nominal width for PT (black squares) and NT (red circles) strips. The dashed lines are to guide the eyes. The solid line is the fit by Eq. (11) with $j_{\mathrm{r}}^{\mathrm{eff}}=3.2\mathrm{MA}/\mathrm{c}{\mathrm{m}}^2$ and $\mathrm{\Delta }{W}_{\mathrm{N}}=25\mathrm{nm}$ as fitting parameters.

Figure 6

Dependence of the residual resistivity ${\rho }_0$ on the nominal width for PT (black squares) and for NT strips. The solid lines are best fits of the experimental data by Eq. (8).

Figure 7

(a) Dependence of the critical current (red circles) for the 80-nm-wide strip (nominal width) on the normalized temperature $t={(1-T/T_{\mathrm{C}})}^{3/2},I_{\mathrm{C}}\left(t\right)$. Thin blue solid line shows the temperature dependence of the extracted current $I_{\mathrm{c}}^{\mathrm{extr}}\left(t\right)=j_{\mathrm{C}}^{\mathrm{extr}}Wdt$ [the linear fit of the $I_{\mathrm{C}}\left(t\right)$ curve at $T\to T_{\mathrm{C}}]$. The dashed green line $\left[I_{\mathrm{C}}1\left(t\right)\right]$ is the extracted current with the temperature dependent correction of Kuprijanov and Lukichev [Eq. (15)] $I_{\mathrm{c}}^{\mathrm{extr}}\left(t\right)KL\left(t\right)$. Thick black solid line $\left[I_{\mathrm{C}}2\left(t\right)\right]$ is the calculated critical current of the superconducting core ($W_{\mathrm{S}}=67\mathrm{nm}$) of the strip $j_{\mathrm{C}}^{\mathrm{extr}}KL\left(t\right)W_{\mathrm{S}}d$. The up arrow marks the splitting temperature $t_0$. The down arrow marks the end of the transition region. (b) Dependence of relative splitting temperature on the nominal width of PT (black squares) and NT (red circles) strips.

Figure 8

Dependencies of the nominal density of critical current on the normalized temperature for three NT strips with different widths indicated in the legend. The blue dashed line represents the linear fit in vicinity of $T_{\mathrm{C}}$.

Figure 9

Sketch of the lateral proximity system (N-S-N) in the thin superconducting NbN strip with the thickness $d$, nominal width $W$, the width of each damaged edge band $\mathrm{\Delta }{W}_{\mathrm{N}}/2$, and the effective width $W_{\mathrm{S}}$.

Figure 10

Dependence of the critical current density on the magnetic field for the strip with the width 1 μm at 6.2 K. Solid lines are fits of the data points corresponding to the Meissner state by Eq. (7).

Figure 11

Dependence of the effective width $W_{\mathrm{S}}$ on the nominal width $W$ calculated with Eq. (5) for PT (black squares) and NT (red circles) strips. Solid line corresponds to $W_{\mathrm{S}}=W$. The inset zooms in the interval $W<100\mathrm{nm}$. Dashed curves are fits by $W_{\mathrm{S}}=W-\mathrm{\Delta }{W}_{\mathrm{N}}$: the red curve for $\mathrm{\Delta }{W}_{\mathrm{N}}=10\mathrm{nm}$ and the black curve for $\mathrm{\Delta }{W}_{\mathrm{N}}=25\mathrm{nm}$.

Figure 12

Dependencies of the critical temperature $T_{\mathrm{c}}$ (a), the effective densities of the critical current ${j_{\mathrm{c}}}^{\mathrm{eff}}=I_{\mathrm{c}}/d{W}_{\mathrm{SC}}$ (b), the effective density of the retrapping current $j_{\mathrm{r}}^{\mathrm{eff}}=I_{\mathrm{r}}/d{W}_{\mathrm{SC}}$ (c) currents, and the relative temperature $T_0/T_{\mathrm{C}}$ of splitting (d) on the effective width of strips. (a) Solid line is the $T_{\mathrm{C}}$($W_{\mathrm{S}}$) dependence calculated by Eq. (12); green triangles show the same dependence computed in the framework of the Usadel's formalism for the resistivity of edge bands twice as large as the resistivity of the central core. (b) Thick dashed horizontal line corresponds to the value of the depairing current density calculated with superconducting and normal state parameters of the superconducting core by Eq. (13). Thin dashed curves in (b) and (c) are to guide the eyes. (d) Solid curve corresponds to the inverse dependence for $W_{\mathrm{S}}=4{\xi }_{\mathrm{GL}}(T_0/T_{\mathrm{C}})$ calculated by Eq. (6).