Threshold fluctuations in a superconducting current-carrying bridge

TL;DR

This paper calculates the energy barrier for switching from superconducting to resistive state in a current-carrying bridge, showing the dependence on current and defect presence across different regimes and limits.

Contribution

It extends the understanding of threshold fluctuations in superconducting bridges, demonstrating the applicability of long-bridge results to shorter or defected bridges and across temperature regimes.

Findings

The threshold fluctuation energy follows a specific current dependence in long bridges.

Weak defects significantly alter the fluctuation energy dependence, resembling short bridges or Josephson junctions.

The results are valid in both dirty and clean limits far below the critical temperature.

Abstract

We calculate the energy of threshold fluctuation $\delta F_{thr}$ which triggers the transition of superconducting current-carrying bridge to resistive state. We show that the dependence $\delta F_{thr}(I)\propto I_{dep}\hbar(1-I/I_{dep})^{5/4}/e$, found by Langer and Ambegaokar for a long bridge with length $L \gg \xi$, holds far below the critical temperature both in dirty and clean limits (here $I_{dep}$ is the depairing current of the bridge and $\xi$ is a coherence length). We also find that even 'weak' local defect (leading to the small suppression of the critical current of the bridge $I_c \lesssim I_{dep}$) provides $\delta F_{thr}\propto I_c\hbar(1-I/I_c)^{3/2}/e$, typical for a short bridge with $L \ll \xi$ or a Josephson junction.

Equations

$$\delta F_{LA}=\frac{4\sqrt{2}}{3}{F}_0{\left(1-\frac{I}{I_{dep}}\right)}^{5/4}$$

$$=\frac{\sqrt{6}}{2}\frac{I_{dep}\hslash }{e}{\left(1-\frac{I}{I_{dep}}\right)}^{5/4},$$

$${\xi }_{GL}^2(0){\nabla }^2\Delta +(1-T/T_c-|\Delta {|}^2/{\Delta }_{GL}^2(0))\Delta =0,$$

$$\frac{d^{2}f}{d{x}^2}-\frac{j^2}{f^3}+f-f^3=0,$$

$$f\big{|}_{-\frac{L}{2}}=f\big{|}_{\frac{L}{2}}=1,$$

$$\delta F_{thr}=F_{saddle}-F_{ground}-\frac{\hslash }{2e}I\delta \phi ,$$

$$F_{saddle,ground}=-\frac{F_0}{2}\int f^{4}dx.$$

$$\frac{d^{2}f}{d{x}^2}-\frac{j^2}{f^3}+\alpha f-f^3=0,$$

$$\displaystyle\frac{df^{L}}{dx}\bigg{|}_{-\frac{l}{2}}=\frac{S}{S_{d}}\frac{df^{C}}{dx}\bigg{|}_{-\frac{l}{2}},\frac{S}{S_{d}}\frac{df^{C}}{dx}\bigg{|}_{\frac{l}{2}}=\frac{df^{R}}{dx}\bigg{|}_{\frac{l}{2}},$$

$$\displaystyle f^{L}\big{|}_{-\frac{l}{2}}=f^{C}\big{|}_{-\frac{l}{2}}=f^{C}\big{|}_{\frac{l}{2}}=f^{R}\big{|}_{\frac{l}{2}},$$

$$\hslash v_F\frac{dg}{dx}+\Delta f^{+}-{\Delta }^{\ast }f=0,$$

$$-\hslash v_F\frac{df}{dx}-2{\omega }_{n}f+2\Delta g=0,$$

$$\hslash v_F\frac{d{f}^{+}}{dx}-2{\omega }_n{f}^{+}+2{\Delta }^{\ast }g=0,$$

$$\frac{\Delta (x)}{\lambda }=\pi N_0{k}_{B}T\sum _{{\omega }_n}\frac{1}{2}[f(x,{\omega }_n,v_F)+$$

$$+f(x,{\omega }_n,-v_F)],$$

$$j=-2\pi ıe{N}_0{k}_{B}T\sum _{{\omega }_n}\frac{v_F}{2}[g(x,{\omega }_n,v_F)-$$

$$-g(x,{\omega }_n,-v_F)].$$

$$\delta F_{thr}=S{N}_0\pi k_{B}T\hslash v_{F}Re\sum _{{\omega }_n}\left(\ln \frac{a_{+}}{a_{-}}-\frac{2{\Delta }_{R0}}{\sqrt{{{\omega }_n^{\prime }}^2+{\Delta }_0^2}}\right)$$

$$-\frac{\hslash }{e}I\arctan \frac{{\Delta }_{R0}}{|{\Delta }_I|}.$$

$$\pi k_{B}T\sum _{{\omega }_n}\left(Re\frac{1}{\sqrt{{{\omega }_n^{\prime }}^2+{\Delta }_0^2}}-\frac{1}{|{\omega }_n|}\right)=\ln T,$$

$$\sum _{{\omega }_n}Im\left[\frac{1}{{\omega }_n^{\prime }+i{\Delta }_I}\frac{1}{\sqrt{{{\omega }_n^{\prime }}^2+{\Delta }_0^2}}\right]=0.$$

$$g({\omega }_n,x)=\cos \theta ({\omega }_n,x),$$

$$f({\omega }_n,x)=\sin \theta ({\omega }_n,x)e^{ı\chi (x)},$$

$$\frac{\hslash D}{2}\frac{d^2\theta }{d{x}^2}-\left({\omega }_n+\frac{D}{2\hslash }{q}_s^2\cos \theta \right)\sin \theta +\Delta \cos \theta =0,$$

$$\Delta \ln \frac{T}{T_c}=2\pi k_{B}T\sum _{{\omega }_n>0}\left(\sin \theta -\frac{\Delta }{{\omega }_n}\right),$$

$$j=4e{N}_{0}D\pi T\frac{q_s}{\hslash }\sum _{{\omega }_n>0}{\sin }^2\theta .$$

$$F=2\pi N_0{k}_{B}TS\sum _{{\omega }_n>0}\int dx\{\frac{\hslash D}{2}[{\left(\frac{d\theta }{dx}\right)}^2$$

$$+{\left(\frac{q_s\sin \theta }{\hslash }\right)}^2]-2{\omega }_n(\cos \theta -1)-2\Delta \sin \theta$$

$$+\frac{{\Delta }^2}{{\omega }_n}\}+N_{0}S\int dx{\Delta }^2\ln \frac{T}{T_c}.$$

$$-\left({\omega }_n+\frac{D}{2\hslash }{q}_s^2\cos {\theta }_{\infty }\right)\sin {\theta }_{\infty }+{\Delta }_{\infty }\cos {\theta }_{\infty }=0.$$

$$\frac{\hslash D}{2}\frac{d^2{\theta }^C}{d{x}^2}-\frac{D}{2\hslash }{q}_s^2\cos {\theta }^C\sin \theta =0,$$

$$I(\varphi )=\frac{4\pi k_{B}T}{e{R}_N}\sum _{{\omega }_n>0}\frac{{\Delta }_{\infty }\cos \frac{\varphi }{2}}{\delta }\arctan \frac{{\Delta }_{\infty }\sin \frac{\varphi }{2}}{\delta },$$

$$\displaystyle\frac{d\theta^{L,R}}{dx}\bigg{|}_{\pm\frac{L}{2}}=\frac{S}{S_{pad}}\frac{d\theta^{C}}{dx}\bigg{|}_{\pm\frac{L}{2}}$$

$$=\frac{S}{S_{pad}}\frac{2}{L}\frac{{\omega }_n\sin \frac{\varphi }{2}\arctan \frac{{\Delta }_{\infty }\sin \frac{\varphi }{2}}{\delta }}{\delta },$$

$$\displaystyle\theta^{L}\big{|}_{-\frac{L}{2}}=\theta^{C}\big{|}_{-\frac{L}{2}}=\theta^{C}\big{|}_{\frac{L}{2}}=\theta^{R}\big{|}_{\frac{L}{2}},$$

$$\displaystyle\theta^{L}\big{|}_{-\frac{L_{sys}}{2}}=\theta^{R}\big{|}_{\frac{L_{sys}}{2}}=\theta_{\infty},$$

Full text

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1 Introduction

Superconducting state of bridge or wire with current is stable with respect of infinitesimally small perturbations of superconducting order parameter $\Delta$ up to currents close to critical (depairing) current. But, if fluctuation induced change of $\Delta$ is sufficiently large, instability is developed in the superconductor even at $I<I_{c}$, leading to the appearance of a finite resistance and dissipation. Theoretically, fluctuation–induced switching first was studied in the work of Langer and Ambegaokar (LA) [1]. They considered long (length $L\gg\xi$, $\xi$ — is the coherence length) quasi-one-dimensional (transverse dimensions smaller than $\xi$) superconducting bridge. To calculate threshold fluctuation LA proposed to find a saddle-point state in the system nearest in energy to the ground state. In their work authors obtained that threshold fluctuation corresponds to a partial suppression of the superconducting order parameter in a finite segment of the bridge with size of about $\xi$ and derived the dependence of the energy of threshold fluctuation on the applied current. Their result is described well by following approximate expression [2]

[TABLE]

where $F_{0}=\Phi_{0}^{2}S/32\pi^{3}\lambda^{2}\xi$, $\Phi_{0}$ is the magnetic flux quantum, $S=wd$ is the area of the cross section of the bridge with the width $w$ and thickness $d$, $\lambda$ is the London penetration depth of the magnetic field, and $I_{dep}=2I_{0}/3\sqrt{3}$ ($I_{0}=c\Phi_{0}S/8\pi^{2}\lambda^{2}\xi$) is the depairing current in the Ginzburg -Landau model, which coincides with the expected critical current of the long ($L\gg\xi$) bridge. In the work [3] LA approach was generalized for superconducting bridges with arbitrary length and it was shown that dependence $\delta F_{thr}(I)$ tends to the expression $\delta F_{thr}=\hbar I_{c}(1-I/I_{c})^{3/2}/e$ for short bridges ($L\ll\xi$, $I_{c}\propto 1/L$ is a critical current of the bridge). This dependence is typical for the energy of threshold fluctuation for Josephson junctions with a sinusoidal current–phase relation [4].

The energy of threshold fluctuation also was calculated for long bridge using microscopic approach [5, 6]. In the work  [5] temperature and current dependencies of $\delta F_{thr}$ were calculated on the basis of the Eilenberger equations [7] for clean long superconducting bridge with only one conducting channel. However, $\delta F_{thr}$ has been significantly overestimated at finite current since the contribution to $\delta F_{thr}$ owing to the work performed by the current source was not taken into account. In the present work on the basis of the Eilenberger equations we recalculate the dependence $\delta F_{thr}(I)$ at different temperatures and find the agreement with power–$5/4$ law up to $T=0.5T_{c}$ which coincides with the result found in work [6] with help of Usadel equations [8] for long dirty bridge. We argue that the relation $\delta F_{thr}(0)\sim I_{dep}\hbar/e$ found in framework of GL model (see Eq. (1)) approximately holds in a broad temperature range below $T_{c}$ not only for long bridges (dirty or clean ones) but for short bridges too, with the replacement of $I_{dep}(T)$ by actual critical current of the bridge $I_{c}(T)$.

Our interest to the role of defects on $\delta F_{thr}(I)$ is motivated by recent experimental works  [9, 10, 11]. In the experiment one usually measures many times the switching current $I_{sw}$ (which has random value due to fluctuations) to find the average value $\langle I_{sw}\rangle$ and the dispersion $\sigma$ which are directly related to $\delta F_{thr}(I)$ (for explicit relation between $\langle I_{sw}\rangle$, $\sigma$ and $\delta F_{thr}(I)$ see for example (2,3) in  [11]). Note, that alternatively $F_{thr}(0)$ could be found from temperature dependence of resistivity near $T_{c}$, because $R(T)\propto\exp(-\delta F_{thr}(0)/kT)$ [12]. Although in  [10, 11] experiments were done for long superconducting bridges in wide temperature interval below $T_{c}$ the good agreement with power–$3/2$ law was found. To explain this result Khlebnikov [13] recently has developed the model which considered the bridge as a discrete set of nodes connected by superconducting links and in his model he neglected local suppression of the superconducting order parameter. Below we show that power–$3/2$ law can be obtained for a long bridge in the framework of LA model, if one takes into consideration presence of defects in the bridge such as constrictions or local variation of the critical temperature or mean path length. We argue that dependence $\delta F_{thr}(I)$ can deviate from power–$5/4$ law even in case of relatively ”weak” defects, when the critical current $I_{c}$ of the bridge with defect is not far from the depairing current $I_{dep}$.

2 Effect of defects on $\delta F_{thr}(I)$

Here we consider a model system consisting of the superconducting bridge with cross section $S$ and length $L$, which connects two superconducting banks whose cross section has the area $S_{pad}\gg S$. Assuming that the maximal characteristic transverse size $d\sim\sqrt{S}\ll\xi$, the problem can be considered as one-dimensional and only the dependence on the longitudinal coordinate x is taken into account.

To consider the effect of defects on the dependence $\delta F_{thr}(I)$, we use the Ginzburg-Landau theory. To determine the energy of threshold fluctuation it is necessary to find the saddle state of the system corresponding to the local maximum of the free energy in presence of external current source. Since it is stationary state (albeit unstable), it is described by the Ginzburg–Landau equation

[TABLE]

where $\xi_{GL}(0)$ and $\Delta_{GL}(0)$ are the coherence length and the superconducting order parameter in the GL model at zero temperature respectively [12].

We seek the solution in the form $\Delta(x)/\Delta_{GL}=f(x)exp(i\varphi(x))$. Then the dimensionless Ginzburg–Landau equation has the form

[TABLE]

where the condition of the constant current in the system, $I=const$, is used (here $j=f^{2}d\varphi/dx=I/S$ is the current density in the bridge). In  (3) the magnitude of the superconducting order parameter $f$, length, and current density are measured in units of $\Delta_{GL}=\Delta_{GL}(0)\sqrt{1-t}$, $\xi=\xi_{GL}(0)/\sqrt{1-t}$ and $j_{0}=I_{0}/S$ ($t=T/T_{c}$ is the dimensionless temperature). (3) should be supplemented with boundary conditions at the ends of the bridge

[TABLE]

which follow from the assumption about nearly zero current density at banks, and thus the order parameter reaches its equilibrium value $f=1$.

The energy of threshold fluctuation can be found using the expression

[TABLE]

where $\delta\varphi$ is the additional phase difference between the ends of the bridge appearing in the saddle-point state and $F_{saddle}$ and $F_{ground}$ are the free energies of the saddle-point and ground states, respectively. In our units these energies take the form

[TABLE]

Equation (3) with boundary conditions  (4) is solved numerically for bridge with length $L=30\xi$. In the numerical solution, we use the relaxation method: the time derivative $\partial f/\partial t$ is added to GL equation  (3) and iterations are performed until the time derivative become zero within a specified accuracy. To find the saddle- point state, we use the numerical method proposed in [14]: at a given current, we fix the magnitude of the order parameter $f(0)$ at the center of the bridge and allow $f$ to change at all other points. The state with the minimum fixed $f(0)$ value for which a steady-state solution exists is a saddle-point state.

We consider three types of defects. The first type corresponds to variation of critical temperature $T_{c}$ along the bridge. To describe such defect in the model, we write GL equation at the defect region (placed in the center of the bridge) in the form

[TABLE]

where the parameter $\alpha=(1-t^{\ast})/(1-t)$ characterizes the deviation from the critical temperature of the rest of the bridge (here $t^{\ast}=T/T^{\ast}_{c}$). Absence of a defect corresponds to the case $\alpha=1$, and decrease of local critical temperature $T^{\ast}_{c}<T_{c}$ corresponds to $\alpha<1$.

We consider defects with lengths $l=0.5\xi$, $\xi$ and $2\xi$ and calculate dependencies $\delta F_{thr}(I)$ at different $\alpha$. Results of our calculations for length $l=0.5\xi$ are shown in figure 1 where we also present fitting expression $\delta F_{thr}=\delta F_{thr}(0)(1-I/I_{c})^{b}$. For bridge with critical current $I_{c}=0.95I_{dep}$ ($\alpha=0.6$) we have $b\approx 1.36$, for bridge with $I_{c}=0.74I_{dep}$ ($\alpha=-0.55$) $b\approx 1.45$, and bridge with $I_{c}=0.66I_{dep}$ ($\alpha=-1.05$) is well fitted by $b\approx 1.5=3/2$ typical for short bridge [3] and Josephson junction [4]. Besides we find that in all cases $\delta F_{thr}(0)\simeq\hbar I_{c}/e$ (see inset in figure 1) which is typical for a Josephson junction and resembles result found in framework of GL model both in limiting cases of long $L\gg\xi$ and short $L\ll\xi$ bridges.

The second type of defect models the inhomogeneity of cross-section area of the bridge. We assume that there is region with the cross-section area $S_{d}<S$ and length $l$ in the center of the bridge (see figure 2. To describe such a constriction, the boundary condition  (4) is supplemented by conditions, similar to the conditions from [3]

[TABLE]

where $f^{L},f^{C},f^{R}$ are the magnitudes of the order parameter to the left of the defect, in the defect and to the right of the defect, respectively. The condition (8a) appears from the variation of the Ginzburg–Landau functional for the superconductor with the cross-section depending on $x$ (which is responsible for the appearance of the derivative $d/dx(S(x)df/dx)$). Here $S/S_{d}$ is not the actual ratio of areas of cross-sections but it is a reference parameter characterizing a change in the derivative of the function $f$ in $x$ direction at the transition through the bridge–defect interface.

Calculated dependencies $\delta F_{thr}(I)$ for the constriction with $l=\xi$ and different cross-sections $S_{d}$ are shown in the figure 3 together with fitting expressions $\delta F_{thr}=\delta F_{thr}(0)(1-I/I_{c})^{b}$. In case of bridge with $S_{d}=0.9S$ critical current $I_{c}=0.987I_{dep}$ and $b\approx 1.36$, for bridge with $S_{d}=0.76S$ $I_{c}=0.945I_{dep}$ ($b\approx 1.42$) and for bridge with $S_{d}=0.5S$ $I_{c}=0.795I_{dep}$ and $b\approx 1.48$. This result demonstrate that even small variation of cross-section area can significantly change dependence $\delta F_{thr}(I)$ and both power–$5/4$ law and power–$3/2$ law are not suitable to fit the current dependence of $\delta F_{thr}$. As in case of local variation $T_{c}$ even relatively ’weak’ constriction ’provides’ power law $3/2$ and $\delta F_{thr}(0)\simeq\hbar I_{c}/e$ (see inset in figure 3).

Very similar results could be obtained if in the bridge there is local variation of mean path length $\ell$ (third type of defect). In principle, to calculate $\delta F_{thr}(I)$ one can use analytical results for distribution of $f$ and phase along the superconducting bridge from [15] but we use numerical procedure because dependence $f$ on coordinate is expressed via special functions. We find that when $\ell$ is five times smaller in the region with length $l=0.5\xi$ dependence $\delta F_{thr}(I)\simeq 1.06\delta F_{J}(1-I/I_{c})^{3/2}$ with $I_{c}\simeq 0.73I_{dep}$.

Change of the exponent for considered types of defect from $5/4$ to $3/2$ could be understood in the following way. In a long defectless bridge length of the critical nucleus (the region with suppressed $\Delta$) diverges as $I\rightarrow I_{dep}$  [1], while in a bridge with defect its length is restricted by length of defect plus $\sim 2\xi$ (when $I_{c}\lesssim 0.7I_{dep}$). It resembles the situation with a short superconducting bridge, which behaves like a Josephson junction and has $\delta F_{thr}\propto(1-I/I_{c})^{3/2}$.

3 The energy of threshold fluctuation at arbitrary temperature

Obtained in section II results are valid near critical temperature $T_{c}$, since they are based on the Ginzburg–Landau model. Below we show on the example of defectless bridges that the dependencies $\delta F_{thr}=\delta F_{thr}(0)(1-I/I_{c})^{b}$ ($b=5/4,3/2$) and $\delta F_{thr}(0)\propto I_{c}$ are valid at $T\ll T_{c}$ too.

At first, we consider, similar to Zharov et al. [5], the case of a long clean one-dimensional superconducting bridge ($\ell\gg\xi_{0}$, with $\xi_{0}=\hbar v_{F}/\pi\Delta_{0}$ is the coherence length in clean limit at $T=0$) containing only one conduction channel. To find the saddle state in that case we use the one-dimensional Eilenberger equations for the normal and anomalous Green’s functions, $g(x,\omega_{n},v_{F})$ and $f(x,\omega_{n},v_{F})$ respectively

[TABLE]

where $v_{F}$ is the Fermi velocity, $\omega_{n}=2\pi k_{B}T(n+1/2)$ is the Matsubara frequency. The Green’s functions obey the normalization condition $g^{2}+ff^{+}=1$. These equations are completed with the self–consistency equation for the order parameter $\Delta$

[TABLE]

and the expression for the supercurrent density

[TABLE]

Here $\lambda$ is the coupling constant, $N_{0}$ is the density of states on the Fermi level. The summation is going over all Matsubara frequencies.

Following [5], we seek the solution in the form of plane waves $\Delta,f\propto e^{\imath kx}$ with complex amplitudes and solve (8i). To calculate the energy of threshold fluctuation, we use the expression  (5) derived by Eilenberger in his work  [7]. Using this expression and saddle-point solution of (8i), one can calculate the energy of threshold fluctuation

[TABLE]

Here $a_{\pm}=\Delta^{2}_{0}-i\Delta_{I}\omega^{\prime}_{n}\pm\Delta_{R0}\sqrt{{\omega^{\prime}_{n}}^{2}+\Delta^{2}_{0}}$, ${\omega^{\prime}}_{n}=\omega_{n}+i\hbar k/2$, $\Delta_{0}$ is the absolute value of the complex amplitude of the order parameter, $\Delta_{R0}$ and $\Delta_{I}$ are the real and imaginary parts of the complex amplitude, which is determined by the equations

[TABLE]

In the work [5] the expression (8l) does not contain last term, which includes the work performed by the current source on the system during the transition of the system from the ground state into the saddle–point state. The comparison of our results with the results of [5] and the LA theory is shown in the figure 4. It is seen that accounting of this term significantly changes the dependence $\delta F_{thr}(I)$ and brings it to the form that is similar to (1) in wide temperature range below the critical temperature (only at $T/T_{c}=0.05$ there is noticeable deviation from the power–$5/4$).

Than we consider the case of dirty superconducting bridge ($l\ll\xi_{0}$). To calculate the energy of saddle–point state, we use the Usadel equation [8] for the normal $g(\omega_{n},x)$ and anomalous $f(\omega_{n},x)$ Green’s functions in standard parametrization  [16]

[TABLE]

where $\theta$ and $\chi$ are real functions. With that parametrization the Usadel equation reads as

[TABLE]

while the self–consistency equation and the expression for the supercurrent density takes the form

[TABLE]

Here $D$ is the diffusion coefficient, $q_{s}=\hbar(d\chi/dx)$ is the superfluid momentum. The free energy in (5) can be written as

[TABLE]

The equations (8p – 8r) are numerically solved for a long bridge using the Newton’s method with the boundary conditions $\theta=\theta_{\infty}$ at $x=\pm 15\xi_{T_{c}}$ ($\xi_{T_{c}}=\sqrt{\hbar D/k_{B}T_{c}}$), where $\theta_{\infty}$ is the solution of the uniform Usadel equation

[TABLE]

Search of the saddle state is performed in a similar way as we do on the basis of the GL theory with the only difference that we fix the ratio $\sin\theta(0)/\sin\theta_{\infty}$ at $x=0$ instead of the magnitude of the order parameter. The dependence $\delta F_{thr}(I)$ is shown in figure 4. It can be seen that for dirty long bridge the current dependence of $\delta F_{thr}$ remains close to the dependence described by Eq. (1). Besides $\delta F_{thr}(0)\simeq\delta F_{LA}(0)$ (see figure 5) in broad range of temperatures below $T_{c}$ and if one uses for $I_{dep}(T)$ result following from microscopic calculations and not the Ginzburg-Landau depairing current. In the clean limit the deviation is stronger, reaching about $15\%$ for $\delta F_{thr}(0)$ as $T\rightarrow 0$ [5].

Alongside the case of long bridges, we also study short bridges ($L\ll\xi(T)$) in dirty limit. In this case we can neglect nongradient terms in (8p) inside the bridge like it was done by Kulik and Omelyanchuk [17] and we obtain the following equation

[TABLE]

where $\theta^{C}$ defines $\theta$ inside the bridge. In the work [17] the solution of this equation was found together with current-phase relation $I(\phi)$

[TABLE]

where $\delta=\sqrt{(\Delta_{\infty}\cos\phi/2)^{2}+{\omega_{n}}^{2}}$ and $\phi$ is the phase difference across the bridge. In (8v) for each current there are two values of $\phi$ corresponding to two different states - the smaller $\phi$ corresponds to the ground state, and the larger $\phi$ corresponds to the saddle state. The strategy to find $\delta F_{thr}$ is following - for fixed current we find two values of $\phi$, than with these $\phi$ we use analytical solution from [17] for $\theta^{C}$ while for $\theta$ outside the bridge we numerically solve equations (8p) and  (8q), neglecting by the pair breaking effect of the current/supervelocity in the banks (which is applicable when cross-section of banks $S_{pad}\gg S$). Solutions in the bridge and in the banks are matched by using the boundary conditions

[TABLE]

where $\theta^{L},\theta^{R}$ are the functions $\theta$ in the left bank and right bank, respectively. Here $L_{sys}=40\xi_{T_{c}}+L$ is length of modelled system, including the bridge (with length L) and the banks with cross-section $S_{pad}$ and length $(L_{sys}-L)/2$ which are contacted with much wider banks where $\theta$ is equal to its value at given temperature and zero current. Above conditions appear from the conservation law for spectral currents [18] and is similar to the boundary conditions (8).

Calculated $\delta F_{thr}(I)$ are shown in figure 6. For $L\ll\xi(T)$ ($L=0.2\xi_{T_{c}}$) the power–$3/2$ law is approximately valid at all temperatures (note noticeable difference at $I\gtrsim 0.8I_{c}$ for $T=0.5T_{c}$ and $T=0.05T_{c}$) while for bridge with $L=0.6\xi_{T_{c}}$ the condition $L\ll\xi(T)$ is not applicable at low temperatures, which leads to stronger deviation from the power–$3/2$ law in wide range of currents near $I_{c}$. Note, that $\delta F_{thr}(0)\simeq\hbar I_{c}/e$ (see inset in  6) with the largest deviation at low temperatures.

And finally, in dirty limit we find how $\delta F_{thr}(0)$ depends on the length of the bridge. Earlier, in work  [3] we claimed that dependence $\delta F_{thr}(0,L)$ may have a minimum at $L\simeq 2-3\xi(T)$ at proper choice of widths of banks and bridge. We carried calculations (to determine the saddle–state, the condition $\theta(x=0,y)=0$ is added) using two-dimensional Usadel equation in the the same geometry as in  [3] (see figure 4 there) and the same geometrical parameters but we did not find a minimum (see figure 7). Instead $\delta F_{thr}(0)$ monotonically increases when $L$ decreases following increase of $I_{c}$. This result force us to check our calculations made in framework of GL model  [3] and we find that this result is an artefact of used grid approximation. With proper grid we confirm absence of minimum in dependence $\delta F_{thr}(L)$ in GL model too.

4 Discussion

We demonstrate, that functional dependence of energy of threshold fluctuation (perturbation) on current following from Ginzburg-Landau model stays valid at temperatures well below $T_{c}$ both in dirty and clean limits if one uses actual critical (depairing) current but not the Ginzburg-Landau depairing current. This result gives us the hope that the strong effect of even relatively ’weak’ defect (which does not strongly suppress critical current of the bridge and provide $I_{c}\simeq I_{dep}$) on dependence $\delta F_{thr}(I)$ that was found at $T\sim T_{c}$ is temperature independent and could be applicable at low temperatures, too.

Our results could be used for qualitative explanation of the dependence $\delta F_{thr}(I)\sim(1-I/I_{c})^{3/2}$ found in works [10, 11] for long bridges/wires by presence of intrinsic defects in their samples. Unfortunately we are not able to make quantitative comparison due to lack of important parameters (resistivity and diffusion coefficient of the bridges/wires, their width and thickness) which are needed to see how far the actual critical current of the bridge is from the depairing current. Alternative explanation of that experiments is based on the model of the bridge/wire as chain of weakly connected, via Josephson coupling, granules [13] which naturally leads to power –$3/2$ but it is not clear how this model could be applicable to works  [10, 11].

5 Conclusion

We calculate the energy of threshold fluctuation which switches the current-carrying superconducting bridge to resistive state. We make calculations at arbitrary temperature, different length of the bridge and in presence of defects connected with local variation $T_{c}$, mean path length $\ell$ or cross-section of the superconductor. It is found that the presence of defect has strong influence on the form of current dependence of the energy of threshold fluctuation, changing it from $\delta F(I)\simeq(1-I/I_{dep})^{5/4}$ valid for long defectless bridge to $\delta F(I)\simeq(1-I/I_{c})^{3/2}$ which is typical for short bridge and Josephson junction. Additionally, using microscopic theory we show that the results, obtained on the basis of Ginzburg–Landau theory, stay valid at temperatures significantly below $T_{c}$, if one uses proper temperature dependent critical (depairing) current.

The study is supported by the Russian Foundation for Basic Research (grant No 15-42-02365).