Superfluid flow in disordered superconductors with Dynes pair-breaking scattering: depairing current, kinetic inductance, and superheating field

We investigate the effects of Dynes pair-breaking scattering rate $\Gamma$ on the superfluid flow in a narrow thin-film superconductor and a semi-infinite superconductor by self-consistently solving the coupled Maxwell and Usadel equations for the BCS theory in the diffusive limit for all temperature $T$, all $\Gamma$, and all superfluid momentum. We obtain the depairing current density $j_d(\Gamma, T)$ and the current-dependent nonlinear kinetic inductance $L_k(j_s, \Gamma, T)$ in a narrow thin-film and the superheating field $H_{sh}(\Gamma, T)$ and the current distribution in a semi-infinite superconductor, taking the nonlinear Meissner effect into account. The analytical expressions for $j_d(\Gamma,T)|_{T=0}$, $L_k(j_s, \Gamma, T)|_{T=0}$, and $H_{sh}(\Gamma, T)|_{T=0}$ are also derived. The theory suggests $j_d$ and $H_{sh}$ can be ameliorated by reducing $\Gamma$, and $L_k$ can be tuned by a combination of the bias current and $\Gamma$. Tunneling spectroscopy can test the theory and also give insight into how to engineer $\Gamma$ via materials processing. Implications of the theory would be useful to improve performances of various superconducting quantum devices.


I. INTRODUCTION
The physics of the superfluid flow in s-wave superconductors is closely tied with the operating principles and performances of various superconducting quantum devices such as superconducting nanowire single-photon detectors (SNSPDs) [1,2], resonators for microwave kinetic inductance detectors (MKIDs) [3,4] and quantum computers [5][6][7], and superconducting radio-frequency (SRF) resonant cavities for particle accelerators [8][9][10][11].The supercurrent density j s is proportional to the superfluid momentum hq and the superfluid density n s .When |q| is such a small value that n s is not significantly suppressed, j s linearly increases with |q|.However, as |q| increases, the reduction of n s becomes significant, and j s ceases to increase [12][13][14].The maximum value of j s is called the depairing current density j d and determines the stability limit of the superfluid flow, above which finite electrical resistance necessarily appears.In SNSPDs, a superconducting nanowire is biased with a dc current close to j d .An incident photon absorbed by the strip heats electrons and reduces the critical current below the bias current, resulting in measurable finite electrical resistance.The reset time after a detection event is often limited by the kinetic inductance [15].In MKIDs, the kinetic inductance plays an essential role in its operating mechanism.Incoming photons with a frequency higher than the superconducting gap break Cooper pairs and lead to an increase of the kinetic inductivity L k ∝ 1/n s .A resultant shift of resonant frequency δf ∝ −δL k can be detected.Besides, the bias current also reduces n s and increase L k and can be utilized to tune a resonator frequency [16] and to observe the nonlinear Meissner Effect [17].In SRF resonant cavities, charged particles are accelerated by the electric component of the microwave, which is proportional to the rf magnetic field at the sur- * kubotaka@post.kek.jpface.Vortex-free cavities [18][19][20] exhibit huge quality factor Q ∼ 10 10 -10 12 at T < 2 K [21][22][23] even under the strong rf magnetic field [24][25][26][27] such that the nonlinear Meissner effect manifest itself.Here the achievable rf field is limited by the induced screening current at the surface, which cannot exceed j d .The surface magnetic field that induces j d is coincident with the superheating field H sh , which is the stability limit of the Meissner state.H sh is thought to define the upper limit of the accelerating field [9][10][11] and is one of the main interests in fundamental SRF studies [28,29].
Microscopic calculations of physical quantities relevant to these devices would provide us with a deeper understanding of experimental results and clues to improving device performances.Those for disordered materials are especially important because these devices are often made from high-resistance films or impurity-doped bulk materials [30][31][32][33][34].Some 60 years ago, Maki calculated j d at the temperature T → 0 [13,14].Kupriyanov and Lukichev obtained j d for an arbitrary T [35].By using the Maki's results, L k for the current-carrying state was calculated afterwards [36].Then those for all T and all j s up to j d were investigated [37].Calculations of H sh also have a long history, starting from those for a clean-limit superconductor [38,39].Effects of homogeneous [40] and inhomogeneous [41] impurities on H sh were recently investigated.Yet, theories including realistic materials features which can limit device performances have been studied lesser extent.Such theories would be useful to pin down causes of performance limitations, e. g., critical current below the ideal j d in nanowires and quenches below the ideal H sh in SRF cavities, etc.
In this work, we focus on effects of Γ on the superfluid flow in disordered superconductors.We consider the geometries shown in Fig. 2: a thin and narrow superconducting film (relevant to, e.g., SNSPD, MKID) and a semi-infinite superconductor (relevant to, e.g., SRF cavities made from bulk materials or thick film).We evaluate the depairing current density j d (Γ, T ), the currentdependent nonlinear kinetic inductance L k (j s , Γ, T ), and the superheating field H sh (Γ, T ) for all T , all Γ, and all current.The results of this work would provide with clues to finding out causes of performance limitations and those used to improve performances of superconducting quantum devices.
The paper is organized as follows.In Section II, we briefly review the Eilenberger-Usadel formalism [61][62][63][64] of the BCS theory and express physical quantities with the Matsubara Green's functions.In Sec.III, we solve the Usadel equation for all T , all Γ, and all q.Some useful formulas of ∆, n s , λ, and j s at T = 0 and ≃ T c are also obtained.In Sec.IV, we consider a thin and narrow superconducting film [Fig. 2 (a)].We evaluate the depairing current density j d (Γ, T ) and the currentdependent nonlinear kinetic inductance L k (j s , Γ, T ).In addition to numerical results, we present the analytical formulas for j d (Γ, T ) and L k (j s , Γ, T ) at T = 0 and ≃ T c .In Sec.V, we consider a semi-infinite superconductor [Fig. 2 (b)].We calculate the current distribution taking the nonlinear Meissner effect into account and evaluate the superheating field H sh (Γ, T ).Analytical formula for T → 0 and T ≃ T c are also derived.In Sec.VI, we discuss the implications of our results.

II. THEORY
We apply the well-established Eilenberger-Usadel formalism [61][62][63][64] of the BCS theory in the diffusive limit to the geometries shown in Fig. 2. The normal and anomalous quasiclassical Matsubara Green's functions G ωn = cos θ and F ωn = sin θ obey the Usadel equation: Here D is the electron diffusivity, the prime denotes differentiation with respect to x, s = (q/q ξ ) 2 ∆ 0 is the superfluid flow parameter, ∆ 0 = ∆(s, Γ, T )| s=Γ=T =0 is the BCS pair potential at T = 0, hq = 2mv s is the superfluid momentum, v s is the superfluid velocity, m is the electron mass, q ξ = 2∆ 0 /hD is the inverse of the coherence length, and hω n = 2πk B T (n + 1/2) is the Matsubara frequency.In Figs. 2 (a) and 2 (b), the current distributes uniformly and varies slowly over the coherence length, respectively.In either cases, the θ ′′ term is negligible and Eq.(1) reduces to Note that, in Fig. 2 (b), θ and ∆ depend on x via s = s(x).The pair potential ∆(s, Γ, T ) satisfies the selfconsistency equation ln where k B T c0 = ∆ 0 exp(γ E )/π ≃ ∆ 0 /1.76 is the BCS critical temperature, and γ E = 0.577 is the Euler constant.
Now consider the current-carrying state (s > 0).In addition to the Dynes pair-breaking scattering rate Γ, the superfluid momentum q give rise to pair-breaking effects, suppressing ∆ and n s and resulting in nonlinearity of j s respect with q.
For T ≃ T c , the GL regime, ∆ in the current-carrying state is calculated from Eq. ( 13).Then, we obtain [52] ∆(s, Γ, for For 0 < T < T c , we need to numerically solve Eqs. ( 2)-( 7) for a finite s.Shown in Figs. 6 (a) and 6 (b) are ∆(s, Γ, T ), n s (s, Γ, T ), and j s (s, Γ, T ) as functions of |q/q ξ | = √ s at T /T c0 = 0.5 for different Γ.The simi-lar calculations for other T are straightforward (see also Ref. [52]).The q dependences of ∆, n s , and j s resemble those for T = 0 (see Fig. 5).As T increases, all these values monotonically decrease, and the depairing current densities (colored blobs) also shift to lower values.

IV. THIN AND NARROW FILM
In this section, we consider the geometry shown in Fig. 2 (a): a narrow thin-film.We calculate the depairing current density and the kinetic inductance using the results obtained in Sec.III.

A. Depairing current density
For T = 0, the depairing current density j d is already obtained for some Γ values (colored blobs in Fig. 5).That for an arbitrary Γ can also be calculated from the formulas, Eqs. ( 21)-( 23), by finding the maximum values of j s .Shown as the solid curves in Fig. 7 are j d , q d and s d = (q d /q ξ ) 2 as functions of Γ.An increase of Γ leads to decreases of n s and j s , resulting in a monotonic decrease of j d .

B. Kinetic inductance
The kinetic inductance of a narrow thin-film is given by L film = (ℓ/W d)L k for the length ℓ, width W , and thickness d.The kinetic inductivity L k is defined by L k js = − Ȧ [37,68], the dot denotes differentiation with respect to the time t, and A = hq/2|e| is the vector potential.Hence, Here hq ξ /2 √ π|e|j s0 = µ 0 λ 2 0 is used.In the following, we investigate L k for the zero-current limit and the currentcarrying state in the fast-and the slow-measurement regimes [37,68] for all T , all Γ, and all j s up to j d (Γ).

Current carrying state (s > 0): Fast measurement
Now consider a current-carrying superconductor in the fast measurement regime [37,68], in which the timedependent current changes rapidly about its time average on a time scale much shorter than the relaxation time of n s .We assume n s cannot follow the time dependence of the current and take ṅs → 0.Then, Eq. ( 43) reduces to where s is the time average of s(t).Experiments in this regime is found in e.g., Ref. [69].We consider a superconductor under a dc biased rf current: q(t) = q bias +q rf (t), s(t) = [q(t)/q ξ ] 2 = s bias +s rf (t), and j s = j s (s bias + s rf (t)) = j bias + j rf (t).In this case, we have s = s bias = (q bias /q ξ ) 2 and j bias = j s (s bias ).Note that, when s bias = 0, Eq. ( 49) reduces to the zero-current kinetic inductance, Eq. ( 44).
For T = 0, we can use Eqs.( 21)-( 23) to evaluate Eq. (49).Shown in Fig. 10 (a) are L k (j bias , Γ, T )| T =0 as functions of the dc bias current j bias for different Γ.As j bias increases, L k monotonically increases and reaches the maximum at the depairing current density j d (colored blob).Shown in Fig. 10 (b) are L k (j bias , Γ, 0) as FIG.10.
functions of the normalized current j bias /j d .The effects of Γ are significant rather than those of j bias .Shown in Fig. 10 (c) are L k (j bias , Γ, 0) as functions of Γ for j bias = 0 (dashed curve) and j bias = j d (solid curve).While both the curves quickly increase with Γ and diverge at Γ = 1/2, the difference between the solid and dashed curves is always smaller than factor 1.5.Shown in Fig. 10 (d) are L k (j bias , Γ, 0)/L k (0, Γ, 0) as functions of (j bias /j d ) 2 for different Γ.The blue (Γ = 0) and red (Γ = 0.4) curves almost overlap, and the effects of j bias is less than 1.5 independent of Γ.It should be noted that the blue curves in Figs. 10 (a), 10 (b), and 10 (d), which represent the ideal BCS superconductor with Γ = 0, are coincident with the results in the previous study [37].
To understand the nonlinear L k for small current regions, we use the approximate formulas, Eqs. ( 27)- (29).After some calculations, we find (see Appendix D): for T = 0 and (j bias /j d ) 2 ≪ 1.Here ∆ d0 and s d0 are given by Eqs. ( 37) and (38).Shown as the dashed gray line in Fig. 10 (d) is calculated from Eq. ( 50), which agrees well with the exact results (solid curves) at (j bias /j d ) 2 ≪ 1.For T ≃ T c , Eq. ( 49) can be calculated from Eqs. ( 30)- (32).Hence, for T ≃ T c .Here and Eq. ( 48) are used.When s bias = s d , we have The bias momentum parameter s bias can be converted to j bias /j d by using Eqs.(32) and (40).For small current regions, we have s bias /3s d = (4/27)(j bias /j d ) 2 and obtain for T ≃ T c and (j bias /j d ) 2 ≪ 1. Eq. ( 53) has the same form as that obtained in the previous study [68] except T c and j d depend on Γ.Note the value of C GL fm differs from that of C fm at T = 0.
For 0 < T < T c , we use the numerical solutions of Eqs. ( 2)- (7).Shown in Fig. 11 (a) are L k (j bias , Γ, T ) as functions of j bias for different Γ and T .The blobs represent the depairing points.Shown in Fig. 11 (b) are L k (j bias , Γ, T )/L k (0, Γ, T ) for different Γ and T , which are not sensitive neither to Γ nor T /T c [see also Fig. 10 (d) for T = 0].The similar curves for Γ = 0 are found in Ref. [37].

Current carrying state (s > 0): Slow measurement
Consider the other limit, the slow measurement regime [37,68], in which the time-dependent current changes on a time scale much longer than the relaxation time of n s .In this case, we can assume n s instantly follows the time dependence of the current: n s = n s (q(t)).Substituting ṅs = q∂ q n s into Eq.( 43), we find where Eq. ( 56) corresponds with the expression given in Ref. [37].
For small-current regions, we have an useful formula to calculate L k (see Appendix D): for T = 0 and (j s /j d ) 2 ≪ 1.Hence, we have a relation C sm = 4C fm [see Eqs. ( 53) and ( 54)].Shown as the dashed gray line in Fig. 12 (b) is the normalized L k calculated from Eq. ( 57), which agrees well with the exact results (solid curves) at (j s /j d ) 2 ≪ 1.
For T ≃ T c , we use the GL results.Substituting Eq. ( 32) into Eq.( 56), we obtain When s ≪ s d , we have s/s d = (4/9)(j s /j d ) 2 and obtain for T ≃ T c and (j s /j d ) 2 ≪ 1.We find C sm for T ≃ T c is smaller than that at T = 0. Eq. ( 60) has the same form as the well-known result [68] except T and j d depend on Γ.
For 0 < T < T c , we use the numerical solutions of Eqs. ( 2)- (7).Shown in Fig. 13 (a) are L k (j s , Γ, T ) as functions of j s for different Γ and T , which increase with j s and diverge at j s = j d (Γ), Γ = 1/2, and T = T c (Γ). Shown in Fig. 13 (b) are L k (j s , Γ, T )/L k (0, Γ, T ) as functions of (j s /j d ) 2 for different Γ and T , sensitive neither to Γ nor T .This insensitivity resembles that for the fast measurement case [see also Fig. 11 (b)].See also Ref. [37] for Γ = 0. Slow-measurement kinetic inductance L k (js, Γ, T )|T =0 calculated from the formulas given by Eqs. ( 21)-( 23) and ( 56).(a) L k as functions of js.(b) L k normalized with L k (js = 0) as functions of (js/j d ) 2 .The dashed gray line is calculated from the approximate formulas given by Eq. ( 57).

V. SEMI-INFINITE SUPERCONDUCTOR
In this section, we consider the geometry shown in Fig. 2 (b): a semi-infinite superconductor occupying x ≥ 0. We calculate the current distribution and the superheating field.

A. Current distribution
In the Meissner state, the current distributes within the depth ∼ λ from the surface.When the superfluid flow is small (|q| ≪ q d ), the pair-breaking effect due to a finite q is negligible, and the distributions of the current j s (x) and the magnetic field H(x) obey the London equation.As |q| increases, the London equation ceases to be valid due to the current-induced pair-breaking effect (nonlinear Meissner effect [17,70]).To obtain the current and field distributions, we need the self-consistent solutions of the coupled Maxwell and Usadel equations, Eqs. ( 2)- (10).
gives H(x)/H c0 = j s (x)/j s0 = H 0 /H c0 exp(−x/λ 0 ): the curves for H(x)/H c0 and j s (x)/j s0 completely overlap.Shown in Fig. 14 (a) are H(x) and j s (x) as functions of x for different H 0 calculated from the self-consistent solutions of Eqs. ( 2)- (10).For small H 0 regions, in fact, H(x)/H c0 (solid curves) and j s (x)/j s0 (dashed curves) almost overlap, expected from the London equation.However, as H 0 increases, the dashed curves deviate from the solid curves in the vicinity of the surface: the nonlinear Meissner effect manifests itself.Shown in Fig. 14 (b) are ∆(x) and λ(x), which differ from the zero-current values (∆ 0 and λ 0 ) at x < ∼ λ but approach ∆ 0 and λ 0 as x increases.

B. Superheating field
The superheating field H sh is given by the value of H 0 which induces j s (x 0 ) = j d .Here x 0 is the depth at which the distribution j s (x) takes the maximum.Note here it is not necessarily the case that we have x 0 = 0.For instance, in the multilayer structure [11,51,[71][72][73] or a superconductor including inhomogeneous impurities in the vicinity of the surface [11,41,51], the surface current can be suppressed, and j s (x) takes the maximum at the inside (x 0 > 0).
In our semi-infinite superconductor, the current is a monotonically decreasing function of x as shown in Fig. 14 (a).Hence, x 0 = 0.In this case, we can derive a simple formula of H sh .Integrating both the sides of Eq. ( 8) from x = 0 to ∞, we obtain q ′ (0 Then, Using Eqs. ( 9) and (10), we find the relation between the applied magnetic field H 0 and the superfluid flow at the surface s(0): H sh can be calculated by substituting s(0) = s d : which is the general formula of the superheating field for a homogeneous dirty superconductor, valid for arbitrary Γ and T .For T = 0, we can evaluate Eq. ( 63) by using Eqs.( 21) and (22) [see also Figs. 5 (b) and 7 (b)].Shown as the solid red curve in Fig. 15 (a) is H sh (Γ, T )| T =0 as a function of Γ.As Γ increases, H sh decreases and vanishes at Γ = 1/2.Shown in Fig. 15 (b) is H sh (Γ, 0) as a function of ∆(0, Γ, 0)/∆(0, 0, 0).
For T = 0 and Γ ≪ 1 (such that Γ ≪ ∆ − s, Eq. ( 63) reduces to a formula (see Appendix E), where ζ d = s d /∆ d and s d are given by Eqs.(34) and (35).For the ideal dirty BCS superconductor (Γ = 0), we have This is slightly smaller than the clean-limit value, H clean sh (0, 0) = 0.84H c0 [38,39], and is consistent with the previous study [40], in which H sh takes the maximum at the mean free path (mfp) = 5.32ξ 0 and decreases with mfp.
For T ≃ T c , we use the GL results.Substituting Eqs.(30) and (31) into Eq.( 63) and using the depairing value of the s parameter in the GL regime, s d (Γ, T ) = s m /3 = (4T c /3π)(1 − T /T c ), we obtain The coefficient is independent of Γ and coincident with the well-known GL result obtained for Γ = 0 [74,75].For 0 < T < T c , we use the numerical solutions of n s (s, Γ, T ) and s d (Γ, T ) [see Fig. 6 (b)].Shown as the solid curves in Fig. 16 (a) is H sh (Γ, T ) as functions of T /T c .We find H sh is a monotonically decreasing function of Γ and T .Shown as the dashed curves in Fig. 16 (a) is H sh (Γ, T ) normalized by H c (Γ, T ).The curves merge at T → T c and reproduce the GL coefficient √ 5/3.It is often useful to plot H sh as functions of (T /T c ) 2 (see, e.g., Ref. [28]).Shown in Fig. 16 (b) is H sh as functions of (T /T c ) 2 .The slope of H sh is steeper than [1 − (T /T c ) 2 ] decreases as Γ increases.

VI. DISCUSSIONS
In Sec.III, we have investigated a disordered superconductor with a finite Dynes Γ parameter by solving the Usadel equation.We have calculated ∆(s, Γ, T ), n s (s, Γ, T ), λ(s, Γ, T ), and j s (s, Γ, T ) for all T , all Γ, and all superfulid flow parameter s ∝ q 2 .Besides, we have derived the formulas of ∆, n s , λ, and j s at T = 0, taking the effects of Γ into account [Eqs.( 21)- (29)].The formulas for T ≃ T c have also been obtained, which have the similar forms as the usual GL results except that T c depends on Γ.Using these results, we have investigated a narrow thin-film in Sec.IV and a semi-infinite superconductor in Sec.V.In the following, we summarize the results and discuss their implications.

A. Depairing current density
In Sec.IV A, we have calculated the depairing current density j d (Γ, T ) for all T and all Γ (see Fig. 8).Also, we have derived the analytical formulas for j d valid for T = 0 and Γ ≪ 1 [see Eqs. ( 33)- (39)].The formulas for T ≃ T c has the similar form as the well-known GL depairing current except that T c depends on Γ [see Eqs. ( 40)-( 42)].
Our results show that j d is given by the Kupriyanov-Lukichev-Maki (KLM) theory for Γ = 0 and decreases as Γ increases.Hence, we can expect that real materials, which usually have Γ > 0, exhibit smaller j d than the ideal KLM value.The previous measurements do not contradict this expectation, but the correlation between j d and Γ is still unclear.Simultaneous measurements of j d and Γ can provide with a deeper insight into observed values of j d .
While materials mechanisms behind Γ are not wellunderstood, it would be possible to engineer Γ by combining tunnel measurements and various materials processing.Finding a better materials processing method which can reduce Γ, we can ameliorate j d .
In SNSPD, the detection efficiency (DE) depends on the bias current j bias /j d .Then, an increase or decrease of j d via a Γ engineering (see Fig. 7) would result in a shift of the necessary dc bias.For instance, when Γ = 0.1, we have more than 10% degradation of T c and 30% degradation of j d compared with the ideal BCS superconductor, which would reduce the necessary dc bias by 30%.Simultaneous measurements of Γ, j d and DE before and after materials treatments (e.g., ion irradiation [76]) can test the theory.

B. Kinetic inductance
In Sec.IV B 1, we have derived the zero-current kinetic inductance formula, Eq. (44).By using this formula, we have calculated L k (0, Γ, T ) for all T and all Γ.Our results show that Γ affects the T dependence of L k (0, Γ, T )/L k (0, Γ, 0) as shown in Fig. 9 (b).Simultaneous measurements of Γ and the zero-current kinetic inductance for different T can confirm the theoretical prediction.
In Sec.IV B 2 and Sec.IV B 3, we have numerically calculated the current dependent nonlinear kinetic inductance in the fast-and the slow-measurement regimes for all T , all Γ, and all current up to j d (see Figs. 11 and 13).In the fast measurement regime, the current-induced increase of L k (j bias , Γ, T )/L k (0, Γ, T ) is at most ∼ 1.5 even at j d .The coefficient of the quadratic expansion at T = 0 is C fm = 0.136 [see Eqs.(50)].On the other hand, in the slow measurement regime, L k diverges at j d , and the coefficient of the quadratic expansion at T = 0 is given by C sm = 4C fm = 0.544 [see Eqs.(57)].The difference between the fast-and the slow-measurement results would be detectable in experiments.The effects of Γ on C fm and C sm are not significant.To test the theory, measurements of the current-dependent nonlinear kinetic inductance should be combined with a measurement of j d and tunneling spectroscopy to extract Γ.Also, our theory suggests that it would be possible to tune L k by engineering Γ as well as by the dc or ac current and controlling T .While L k always increases with the bias current and Γ, the dissipative conductivity σ 1 can be smaller than that of the ideal BCS superconductor by tuning the dc bias and Γ (see Fig. 7 in Ref. [52]).Then, we can simultaneously increase L k and reduce σ 1 (e.g., Γ = 0.05 and j s ≪ j d lead to 10% increase of L k and 20% reduction of σ 1 [9,50,52]).These results might be useful for developing superconducting circuit elements (e.g., superinductor [77]).

C. Superheating field
In Sec.V, we have derived the general formula of the superheating field for a disordered superconductor H sh , taking the effects of Γ into account [see Eq. ( 63)].Using this formula, we have calculated H sh (Γ, T ) for all T and all Γ.A simple analytical formula for T = 0 and Γ ≪ 1 is also derived, which is given by Eq. ( 64).For the ideal dirty BCS superconductor with Γ = 0, we have obtained H sh (0, 0) = 0.79H c0 at T = 0, which is slightly smaller than H clean sh = 0.84H c0 for a clean limit superconductor with a large λ/ξ [38,39] and consistent with the previous study [40] in which H sh decreases with mfp when mfp < 5.32ξ 0 .
According to our results, we can expect that the maximum operating field of an SRF cavity made from a homogeneous dirty BCS superconductor with Γ = 0 is given by H sh (0, 0) = 0.79H c0 .Taking dirty Nb materials for example, µ 0 H c0 = 200 mT yields µ 0 H sh (0, 0) = 160 mT, which translates into the accelerating field E acc = 37 MV/m for the Tesla-shape SRF cavity.This can be tested by measuring quench fields of impuritydoped dirty Nb cavities with mfp ≪ ξ.It should be noted that H sh can increase as impurities decreases [40], then materials with mfp ∼ ξ [30][31][32][33][34] can have slightly higher H sh than 0.79H c0 .
Strong local-heating (and resultant quenches) of SRF cavities are often attributed to geometrical defects on the surface [78][79][80][81][82][83][84][85][86].Our theory suggest there can be another source of local heating.Since real materials have a finite Γ [43,44] and H sh (Γ, 0) < H sh (0, 0), an area with a large Γ on the inner surface can be a hot spot even when H 0 ≪ H sh (0, 0).For instance, H sh at an area with Γ = 0.3 on the surface of Nb 3 Sn can be estimated as µ 0 H sh = 160 mT at T = 0 from Fig. 15, which is much smaller than the ideal value µ 0 H sh (0, 0) = 430 mT.Here µ 0 H c0 = 540 mT is used.Taking another example, H sh at an area with Γ = 0.15 on the surface of disordered Nb materials can be estimated as µ 0 H sh = 110 mT at T = 0, which is smaller than the ideal dirty Nb by 30%.Such an area with Γ > 0 can be a source of local heating and cause quenches.Simultaneous measurements of an onset field of local heating and Γ at the hot spot can test the theory.Materials processing that can reduce Γ would improve the accelerating field of SRF cavities.

FIG. 2 .
FIG. 2. (a) Thin and narrow superconducting film carrying a uniform current js.We assume a thickness d ≪ λ and a width W ≪ λ 2 /d.(b) Semi-infinite superconductor carrying the Meissner current js(x).We assume λ is much larger than the coherence length.

FIG. 15 .
FIG.15.Superheating field H sh (Γ, T ) at T = 0. (a) H sh as a function of Γ.The dashed curve is calculated from the approximate formula given by Eqs.(64)-(66).(b) H sh as a function of ∆ in the zero-current state.