The linear response function K(q,ω) for superconductors subjected to excitation by an electromagnetic field is calculated using Keldysh’s formulation of non-equilibrium response in the quasiclassical approximation (Rainer and Sauls, 1994). The current response depends on the frequency and wavevector, ω and q, of the electromagnetic field within the metal and includes the condensate response (supercurrent) and the dissipative response of unbound quasiparticles scattered by the random potential with scattering rate, 1/τ. The other important internal timescale is the pair formation time, τ ₀ ≡ ℏ/2πk _B T _c , and the corresponding ballistic pair correlation length is ξ ₀ = v _f τ ₀ = ℏv _f /2πk _B T _c . For pure Nb, the pair correlation length and London penetration depth are comparable, ξ 0 ≈ λ L 0 ≃ 33 nm . However, disorder leads to a finite mean free path for quasiparticles, ℓ = v _f τ, converting ballistic motion to diffusive motion, and, thus, a reduction in the pair correlation length. Disorder is also pair breaking, particularly in the presence of screening currents, resulting in a reduction of the superfluid density, n _s . Thus, the London penetration depth, λ L ∝ 1 / n s , increases due to scattering by the random potential.

For the analysis reported here, we consider the long-wavelength (“local”) limit, q ≲ 1/λ _L ≪ 1/ξ ₀, for the current response, which is achieved in Nb when the mean scattering time is comparable to the pair formation time, τ ∼ τ ₀. This level of disorder leads to weak suppression of T _c , a reduction in the superconducting coherence length, ξ, and an increase in the London penetration depth such that λ ≫ ξ. In this limit, the screening current is determined by the local value of the electromagnetic field, and we can set q = 0 in the current response function. The result is given by (Rainer and Sauls, 1994) K ω ; τ , T = π σ D i c 2 τ ∫ − ∞ + ∞ d ϵ tanh ϵ − ω / 2 2 T 1 D + R + D − R + 1 / τ ϵ 2 + Δ 2 − ω 2 / 4 D + R D − R + 1 − tanh ϵ + ω / 2 2 T 1 D + A + D − A + 1 / τ ϵ 2 + Δ 2 − ω 2 / 4 D + A D − A + 1 + tanh ϵ + ω / 2 2 T − tanh ϵ − ω / 2 2 T 1 D + R + D − A + 1 / τ ϵ 2 + Δ 2 − ω 2 / 4 D + R D − A + 1 , (4) where σ D = 2 3 N f e 2 v f 2 τ = n e 2 τ / m * is the Drude result for the d.c. conductivity for charge e quasiparticles, with Fermi velocity, v _f , Fermi momentum, p _f , and density of states, N f = 3 2 n / p f v f , where n is the electron density and m* = p _f /v _f is the quasiparticle effective mass. The denominators in Eq. 4 are defined by D ± R / A ≡ Δ 2 − ϵ ± ω / 2 ± i δ 2 (5) and are retarded (advanced) functions defined by the analytic continuation to the real axis indicated by +i δ (−i δ) and δ → 0⁺. The response function K(ω) is directly related to the bulk microwave conductivity. σ = σ 1 + i σ 2 = i c 2 4 π ω K ω , (6) where σ ₁ = Re σ(ω) and σ ₂(ω) = Im σ(ω) are the real and imaginary parts of the a.c. conductivity, respectively. In the normal state, Δ → 0, the conductivity, σ(ω), defined by Eqs 4, 6 reduces to σ n ω = σ D 1 − i ω τ . (7)

Superconductivity leads to significant changes in the cavity resonance frequency, as well as the quality factor, that are sensitive to disorder and temperature, which we calculate to predict and analyze experimental data using Eqs 3–5 and boundary conditions for the EM field at the vacuum–metal interface.

SRF cavities for accelerator applications adopt the Tesla geometry, i.e., axially symmetric with an oval shape designed in part to eliminate sharp corners which are sources of field emission (Padamsee et al., 2008). Here, we consider cylindrical cavities of radius R and length L. This model is chosen in order to simply our theoretical analysis, in particular, in implementing the boundary conditions on the EM field for the lowest TM mode at the vacuum–metal interface. The theoretical results depend on the response function, the local boundary conditions on the field at the vacuum–metal interface, and an overall geometric factor, G. The latter geometric factor depends on the field distribution within the geometry of the cavity and is independent of the temperature and material properties of the cavity walls. Thus, we correct our result obtained using the TM₀₁₀ mode of the cylindrical cavity by replacing the value of G for the cylindrical cavity with the geometric factor computed for the lowest TM mode of the Tesla cavity.

For the cylindrical cavity, the lowest frequency TM resonance is the TM₀₁₀ mode with the vector potential along the cylinder axis A ( r , ω ) = A 0 J 0 ( ω ρ / c ) z ̂ in the gauge ∇  ⋅ A = 0. The Fourier component of the electric field is axial, E ( r , ω ) = ( i ω / c ) A 0 J 0 ( ω ρ / c ) z ̂ , and the magnetic field circulates azimuthally, B ( r , ω ) = A 0 ( ω / c ) J 1 ( ω ρ / c ) φ ̂ . J ₀(x) (J ₁(x)) is the cylindrical Bessel function of order n = 0 (n = 1). The perfect conductor boundary condition, n ̂ × E | S = 0 , where n ̂ is a local unit vector normal to any point on the surface, S, determines the field distribution in the interior to the cavity and the eigenfrequency of the cavity based on the cavity geometry. In particular, we have J ₀ (ωR/c) = 0 and, thus, a resonant frequency of ω = x ₀₁ c/R, where x ₀₁ ≃ 2.405 is the first zero of J ₀(x). The electric field is the maximum along the axis of the cavity and vanishes on the cylinder wall, while the magnetic field vanishes along the axis of the cavity and is maximum on the cylinder wall, B max = C A 0 R , where C ≡ x 01 J 1 ( x 01 ) ≃ 1.248 . The geometric factor for a cylindrical cavity of length L and radius R reduces to (Padamsee et al., 2008) G = Z 0 ω / c ∫ V d 3 r | B r , ω | 2 ∫ S d S | B r S , ω | 2 = Z 0 x 01 2 L L + R , (8) where Z ₀ = 4π/c ≈ 377 Ω is the vacuum impedance.

However, for a vacuum–superconductor interface, the vector potential, and, thus, a tangential electric field as well as a tangential magnetic field, penetrates into the superconductor, but is confined to the interface within a distance of order the London penetration depth, of order λ _L ≃ 50 nm for moderately disordered Nb. The penetration of the field over distance scales of order λ _L ≪ R leads to a small shift in the resonance frequency, δω, which depends on the material properties of the superconductor via the current response function K(ω). Penetration of the electric field into the superconductor also leads to the dissipation of the microwave field by quasiparticles scattering from the interface and from the random distribution of impurities, thus limiting the quality factor, Q.

Both δω and Q are determined by an eigenvalue equation obtained from the boundary conditions on the vector potential and its derivative at the vacuum–superconductor interface. In particular, for the TM₀₁₀ mode on the vacuum side, A ( ρ ) = A 0 J 0 ( ω c ρ ) . However, on the superconducting side, the confinement of the field to distances of order λ _L ≪ R allows us to neglect the curvature of the interface. Thus, for ρ ≥ R, we obtain A ( ρ ) = A 0 ′ exp { − Λ ( ω ) ( ρ − R ) } , where Λ ( ω ) ≡ K ( ω ) − ( ω / c ) 2 . The ratio of the two boundary conditions reduces to Λ ϖ = K ϖ − ϖ / c 2 = ϖ c J 1 ϖ R / c J 0 ϖ R / c , (9) where J ₁(x) = −dJ ₀(x)/dx. It should be noted that the response function is in general a complex function of ω, which is analytic in the upper half of the complex frequency plane; thus, ω → ϖ, with Im ϖ > 0. Equation 9 is the key equation for the complex eigenfrequency, ϖ, that determines the resonance frequency, the penetration depth, and the quality factor, all of which become functions of temperature, disorder, and frequency. The results reported here support and agree with our analysis of disordered SRF cavities based on Slater’s method (Ueki et al., 2022).

Before presenting the numerical results for the frequency shift and quality factor of disordered SRF cavities, we discuss the physics underlying the dependence of the cavity resonance frequency on disorder and temperature in both the normal and superconducting states.

In the limit ωτ ≪ 1, σ _n ≃ σ _D determines the dissipation of microwave power and the penetration depth of the EM field in the normal metal. In particular, on the metallic side of the vacuum–metal interface (x > 0), A n x , ω = A 0 Re e i q ω x , (10) where q ( ω ) = i 4 π σ D ω / c = ( 1 + i ) / δ ( ω ) with δ ω ≡ c / 2 π σ D ω = λ L 0 ω τ / 2 . (11)

It should be noted that λ L 0 = c / ω p is the clean limit result for the London penetration depth at zero temperature, defined here in terms of the plasma frequency, ω p ≡ 8 π 3 N f e 2 v f 2 . The penetration of the EM field in the normal metal is then given by λ n = 1 A 0 ∫ 0 ∞ d x A n x , ω = Re i q ω = δ ω 2 . (12)

In the superconducting state, A s ( x , ω ) = A 0 e − x / λ L , and thus, the penetration depth of the EM field is given by the London length λ s = 1 A 0 ∫ 0 ∞ d x A s x , ω = λ L . (13)

For pure Nb, we adopt the values of λ L 0 ≃ 33 nm , T c 0 = 9.33 K , and Δ₀ = 1.55 meV in Table 1 (Zarea et al., 2022). Here, we consider N-doped Nb SRF cavities with fundamental frequency f = ω/2π = 1.3 GHz and a transition temperature of T _c = 9.0 K (Bafia et al., 2021). The combination of quasiparticle scattering by non-magnetic impurities and gap anisotropy leads to pair breaking and accounts for the suppression of T _c with a quasiparticle scattering rate of τ ≈ 2 × 10⁻¹³ sec [c.f. Fig. 12 of the work of Zarea et al. (2022)]. Thus, ωτ ≃ 1.6 × 10⁻³ ≪ 1 as we assumed. The normal metal penetration depth with this level of disorder is then λ _n ≈ 910 nm. Disorder also suppresses the superfluid fraction, n _s /n ≈ 0.4, and, thus, increases the zero-temperature London penetration depth to λ _L ≈ 52 nm. For the f = 1.3 GHz TM₀₁₀ mode of a cylindrical cavity, the radius is R = 8.83 cm. Thus, the difference in the penetration depth of the EM field for the normal and superconducting states implies a maximum increase in the cavity resonance frequency of order δf ≈ f (λ _n − λ _s )/R ≈ 12.6 kHz, which is close to the maximum frequency shift for an N-doped Nb Tesla SRF cavity with f = 1.3 GHz, i.e., δf _expt ≃ 12.5 kHz (Bafia et al., 2021). The geometric factor for an f = 1.3 GHz cylindrical cavity with δf = 12.6 kHz and radius R = 8.83 cm corresponds to G _cyl = 268 Ω, and thus, L ≃ 1.44 R compared to G _Tesla = 270 Ω.

Although the maximum shift is positive, very near T _c , the cavity resonance frequency exhibits a negative frequency shift of order δf ≳ − 1 kHz over a narrow temperature range near T _c [c.f. Fig. 4 of the work of Ueki et al. (2022)]. This anomaly can be qualitatively understood based on the divergence of the London penetration depth as T → T _c , i.e., λ L ( T ) = λ L / 1 − T / T c . Thus, the temperature window over which the frequency shift is expected to be negative is | Δ T | / T c < ( λ L / λ n ) 2 ≈ 3.3 × 1 0 − 3 , which is in rough agreement with the experimentally observed width of the temperature anomaly of |ΔT|/T _c |_expt ≃ 3.8 × 10⁻³ at f = 1.3 GHz [c.f. Fig. 4 of the work of Ueki et al. (2022)]. More detailed and accurate analysis of the temperature dependence of the cavity resonance and quality factor, as well as an approximate analytical formula for the frequency anomaly very near T _c , is given in the following.

The EM response of cavity resonators, as well as 2D co-planar waveguide resonators, is often expressed in terms of the complex surface impedance, Z(ω). For SRF cavities, the surface impedance resulting from the response of screening currents in the superconductor is defined by the ratio of the tangential fields at the interface. Z / Z 0 ≡ E ‖ R / H ‖ R = J 0 ϖ R / c J 1 ϖ R / c = ϖ c Λ ϖ − 1 . (14)

Thus, the solution to the boundary value problem, Eq. 9, directly determines the complex impedance. Equations 6, 9, 15 give Z(ω) directly in terms of the complex conductivity Z = Z 0 ω 4 π i σ ω , (15) and thus, Z s / Z n = σ n ( ω ) / σ s ( ω ) . This result is valid in the limit c/ω ≫ λ _s,n , which is well satisfied for SRF cavities at GHz frequencies. The real and imaginary parts of Z _s/n = R _s/n − iX _s/n are the surface resistance and surface reactance of the metal in the superconducting (s) and normal (n) states, respectively. For SRF cavities in the normal state with a moderate level of disorder, we have ωτ ≪ 1, in which case σ _1n ≃ σ _D and σ _2n ≃ (ωτ)σ _D ≪ σ _1n . Thus, X n ≃ R n ≃ Z 0 ω / 8 π σ D . (16)

The change in the reactance upon cooling through superconducting transition generates a shift in the resonance frequency, δ f ≡ f s − f n = f 2 G X n − X s , (17) while the surface resistance below T _c determines the quality factor. Q s = G R s , (18) where G is the geometric factor of the cavity, which is given for the TM₀₁₀ mode of a cylindrical cavity by Eq. 8.

The effects of disorder on quality factor Q are important for the development of SRF cavities for both accelerator and quantum applications. Both the quality factor and resonance frequency of the cavity can be calculated directly from Eqs 4, 9. In particular, for a cylindrical cavity of radius R, Eq. 9 reduces to 1 Q = 2 R Im 1 K ω , (19) δ ω = − ω R Re 1 K ω (20) in the limit δω ≪ ω and ω/Q ≪ ω, where the complex eigenvalue defined by the solution of Eq. 9 is written as ϖ ≡ ω + δω − iω/2Q, with ω = x ₀₁ c/R being the TM₀₁₀ mode frequency of the ideal cavity.

Figure 1 shows the dependence of Q on the quasiparticle scattering time in units of τ _Δ ≡ ℏ/2πΔ(T) for an f = 2.6GHz Nb SRF cavity at T = 0.2T _c . Thus, in the local limit, i.e., sufficient disorder such that λ _L ≫ ξ, the maximum Q occurs for intermediate disorder with τ ∼ τ _Δ. Indeed the non-monotonic dependence of Q on τ is due to the pair-breaking suppression of the superfluid fraction, n _s ∝ 1/λ ², by quasiparticle-impurity scattering. For strong disorder, τ ≪ τ _Δ, the quality factor drops rapidly and approaches the Q of the normal state for τ → 0. In the clean limit τ → ∞, the local approximation for the current response function breaks down for Nb. This is not an issue for the current state-of-the-art SRF cavities for accelerator applications, but it may be relevant to consider ultra-purity cavity technology for quantum sensors and processors, the case in which non-local electrodynamics of SRF cavities will be relevant.

Figure 2 shows the negative frequency shift very close to T _c , which is expanded in Figure 3. The region of negative frequency shift is where the London penetration depth, λ L ( T ) ≈ λ L / 1 − T / T c , exceeds the normal-state penetration depth, λ _n . The experimental data are reasonably well described by theory, given the fact that the scattering time τ was fit to the full temperature-dependent shift.

Here, we provide an approximate analytic expression for the response function and the frequency shift anomaly near T _c . From Eq. 6, it should be noted that the real part of the current response function in Eq. 4 is proportional to the out-of-phase component of conductivity, and thus, the penetration depth in the superconducting state is Re K ω = 4 π ω c 2 σ 2 ω → T → T c − ℏ ω ≪ 2 Δ T 1 λ L T , τ 2 , (22) where λ L ( T , τ ) ≃ λ L / 1 − T / T c for T ≲ T _c . The imaginary part of K(ω) is proportional to the dissipative component of conductivity. Im K ω = − 4 π ω c 2 σ 1 ω → T → T c − ℏ ω ≪ 2 Δ T − 1 2 λ n ω , τ 2 , (23) with λ _n (ω, τ) given by Eqs 11, 12; i.e., we drop corrections of order ( Δ ( T ) / π T c ) 2 to the normal-state Drude conductivity.

Thus, for temperatures just below the superconducting transition, but still below the pair-breaking continuum, ℏω ≪ 2Δ(T), we obtain an approximate analytic form for the frequency shift and quality factor using Eq. 9 in the limit δω/ω ≪ 1 and 1/Q ≪ 1. In particular, the frequency shift of the superconductor relative to the ideal cavity frequency for |T − T _c |≪ T _c becomes δ f s = − f R eff Re 1 λ L T , τ 2 − i 2 λ n ω , τ 2 − 1 2 , (24) where R _eff = R G _Tesla/G _cyl and R is the radius of a cylindrical SRF cavity with a TM₀₁₀ mode frequency of f = 2.6 GHz. The corresponding shift for the normal metallic state just above T _c is δf _n = −(f/R _eff) λ _n . Thus, the observable frequency shift of the superconducting cavity relative to the normal state is δf = δf _s − δf _n . This approximate result reproduces the negative frequency shift near T _c , including a good estimate of the temperature and a minimum value of the negative frequency shift, but deviates from the full theory as the temperature drops further. Thus, the origin of the negative shift is the competition between the large normal-state penetration depth for disordered Nb and the London penetration depth which competes with λ _n only as T → T c − .