Tuning microwave losses in superconducting resonators

In the BCS model, the quasiparticle DOSs $N(\epsilon)$ vanishes at energies $|\epsilon|\lt\Delta$, even if weak scattering on nonmagnetic impurities present [18, 83, 90]. It is the feature of $N(\epsilon)$ which ensures the exponentially small $R_\mathrm{BCS}(T)$ and $R_i = 0$ in equation (2). Yet, many tunneling measurements of $N(\epsilon)$ [58, 59] have shown that $N(\epsilon)$ differs from the idealized DOS which diverges at $\epsilon = \Delta$ and vanishes at $\epsilon\lt\Delta$, as shown in figure 1. In the observed $N(\epsilon)$ the gap singularities at $\epsilon = \Delta$ are smeared out and subgap states with a finite $N(\epsilon)$ appear at $\epsilon\lt\Delta$. Such DOS has often been described by the phenomenological Dynes model [58] in which

$$\begin{align} N(\epsilon) = \mbox{Re}\frac{N_n(\epsilon-i\Gamma )}{\sqrt{(\epsilon-i\Gamma )^2 - \Delta^2}}. \end{align} \tag{ 10 }$$

Here the damping parameter Γ quantifies a finite lifetime of quasiparticles ${\sim} \hbar/\Gamma$, resulting in a finite DOS $N(0)\simeq \Gamma N_n/\Delta$ at the Fermi level. Tunneling conductance measurements on Nb [91, 92] and Nb₃Sn [93] have indeed revealed a finite $N(\epsilon)$ at $\epsilon \lt\Delta$ (a review of tunneling measurements of $N(\epsilon)$ and applications of equation (10) are given in [59]). The physics of subgap states is not fully understood (see e.g. reviews [59, 94] and the references therein). Many mechanisms of subgap states have been suggested in the literature, including inelastic scattering of quasiparticles on phonons [95], Coulomb correlations [96], anisotropy of the Fermi surface [97], inhomogeneities of the BCS pairing constant [98], magnetic impurities [90], spatial correlations in impurity scattering [90, 99], diffusive surface scattering [100] or quasiparticles trapped by inhomogeneities of $\Delta(\textbf{r})$ [101]. The phenomenological equation (10) captures the observed broadening of the DOS peaks at $\epsilon\approx \Delta$ but it can hardly describe the low-energy tails in $N(\epsilon)$ due to, for example, energy-dependent electron–phonon relaxation times [102]. Details of exponential or power-law energy tails in $N(\epsilon)$ at $|\epsilon|\ll \Delta$ can be essential for the behavior of $R_i(T)$ at ultra low temperatures [101]. Yet, the widely used equation (10) in which all microscopic mechanisms are included in a single parameter Γ is rather useful to address qualitative effects of the DOS broadening on R_s .

Zoom In Zoom Out Reset image size

The broadening of the DOS gap peaks reduces T_c and Δ in a way similar to the pairbreaking effect of magnetic impurities [90]. For instance, T_c decreases with Γ and vanishes at $\Gamma_c = \Delta_0/2$, where Δ₀ is the gap at T = 0 and $\Gamma = 0$ [45, 103]. For weak DOS broadening $\Gamma\ll k_\mathrm{B}T_{c0}$, we have [45, 103]

$$\begin{align} T_{c}=T_{c0}-\frac{\pi\Gamma}{4}, \kern64.3pt \end{align} \tag{ 11 }$$

$$\begin{align} \Delta\simeq\Delta_{0}-\Gamma-\frac{\pi^{2}\Gamma T^2}{6\Delta_0^{2}}, \quad T\ll T_c. \end{align} \tag{ 12 }$$

A finite DOS at  = 0 in the Dynes model yields a quadratic temperature dependence of $\Delta(T)$ instead of the BCS behavior of $\Delta(T)\simeq \Delta_0-\sqrt{2\pi k_\mathrm{B}T\Delta_0}\exp(-\Delta_0/k_\mathrm{B}T)$ at $T\ll T_{c}$ [83]. The DOS broadening increases the magnetic penetration depth λ at $T\ll T_c$ in the dirty limit as following [45, 103]:

$$\begin{align} \frac{1}{\lambda^{2}}=\frac{2\mu_{0}\Delta}{\hbar\rho_{n}}\tan^{-1}\frac{\Delta}{\Gamma}, \end{align} \tag{ 13 }$$

This reproduces the BCS result $\lambda^2 = \hbar\rho_n/\pi\mu_0\Delta$ at $T = \Gamma = 0$ [83]. The dependence of λ on the mean free path l_i at $T = \Gamma = 0$ is given by [11]:

$$\begin{align} \frac{1}{\lambda^2}=\frac{1}{a\lambda_0^2}\left[\frac{\pi}{2}-\frac{\cos^{-1}(a)}{\sqrt{1-a^2}} \right],\quad a \lt 1 \end{align} \tag{ 14 }$$

$$\begin{align} \frac{1}{\lambda^2}=\frac{1}{a\lambda_0^2}\left[\frac{\pi}{2}-\frac{\cosh^{-1}(a)}{\sqrt{a^2-1}} \right], \quad a \gt 1 \end{align} \tag{ 15 }$$

where $\lambda_0 = (m/\mu_0n_0e^2)^{1/2}$ and the parameter $a = \pi\xi_0/2l_i$ quantifies scattering on impurities, so that $a\ll 1$ and $a\gg 1$ correspond to the clean and the dirty limits, respectively. As l_i decreases, $\lambda(l_i)$ increases from λ₀ in the clean limit $l_i \gg \xi_0$ to $\lambda \simeq \lambda_0 (\xi_0/l_i)^{1/2}$ in the dirty limit, $l_i \lt \xi$. In turn, the coherence length $\xi(l_i)$ decreases from ξ₀ at $l_i \gg \xi_0$ to $\xi\simeq \sqrt{l_i\xi_0}$ if $l_i \ll\xi_0$ [18]. In the BCS model, weak scattering of electrons on nonmagnetic impurities does not affect $\Delta(T)$, T_c and $B_c(T)$ of a superconductor with a spherical Fermi surface [18, 90]. This is no longer the case for strong impurity scattering and anisotropic Fermi surface [90]. Strong electron–phonon coupling gives rise to a power-law temperature dependence of $\lambda(T)$ at $T\ll T_c$ due to the contribution of low-energy phonons [104].

The observed temperature dependence of $R_s(T)$ on s-wave superconductors follows equation (2) with $\Delta = c_1 k_\mathrm{B}T_c$, where c₁ is slightly higher than the BCS prediction $c_1 = 1.76$ due to the effects of strong electron–phonon coupling [84]. Many extrinsic mechanisms of the residual surface resistance have been pointed out in the literature, including lossy oxides or metallic hydrides in Nb [7, 8], trapped vortices [62–70], grain boundaries [105–107], or TLSs [5, 108, 109]. The effect of metallic hydrides has been well documented for Nb cavities [110, 111] and films [112–114]. Formation of metallic hydride precipitates from over-saturated solid solutions of H interstitials is characteristic of Nb [115]. In films, R_i can be increased by the surface roughness and absorption of noble gases [112–114]. These intrinsic factors can be mitigated by high temperature (600 ^∘C–800 ^∘C) annealing [8] of by pushing trapped vortices out by temperature gradients [64, 80, 116, 117].

A residual resistance produced by the subgap states can be obtained by incorporating the Dynes DOS into the BCS expression for $R_s(T)$ [44, 45]:

$$\begin{align} R_s=R_1\sinh\left(\frac{\hbar\omega}{2k_\mathrm{B}T}\right)\int_{-\infty}^\infty \frac{[n(\hbar\omega+\epsilon)n(\epsilon)+m(\hbar\omega+\epsilon)m(\epsilon)]\mathrm{d}\epsilon}{\cosh(\epsilon/2k_\mathrm{B}T)\cosh[(\epsilon+\hbar\omega)/2k_\mathrm{B}T]},\nonumber\\ \end{align} \tag{ 16 }$$

$$\begin{align} n(\epsilon)=\mbox{Re}\frac{(\epsilon-i\Gamma )}{\sqrt{(\epsilon-i\Gamma )^2 - \Delta^2}},\qquad m(\epsilon)=\mbox{Re}\frac{\Delta}{\sqrt{(\epsilon-i\Gamma )^2 - \Delta^2}}, \end{align} \tag{ 17 }$$

where $R_1 = \mu_0^2\omega\lambda^3/4\hbar\rho_s$ and $n(\epsilon)$ and $m(\epsilon)$ are real parts of retarded normal and anomalous quasiclassical Green's functions [83]. Equation (16) with $\Gamma = 0$ reproduces $R_\mathrm{BCS}(T)$ in equation (2). Complex conductivity in the Dynes model and arbitrary mean free path was calculated in [103]. In addition to the BCS part $R_\mathrm{BCS} \propto \exp(-\Delta/k_\mathrm{B}T)$, equation (16) accounts for a residual $R_i(T)$ that is not exponentially small at $T\ll T_c$ [45]:

$$\begin{align} R_i(T)=\frac{\mu_0^2\omega^2\lambda^3 \Gamma^2}{2\rho_n(\Delta^2+\Gamma^2)}\biggl[1+\frac{4\pi^2k_\mathrm{B}^2T^2\Delta^2}{3(\Delta^2+\Gamma^2)^2}\biggr],\quad k_\mathrm{B}T\lt \Gamma \end{align} \tag{ 18 }$$

where $\Delta(\Gamma)$ is given by equation (12). According to equation (18), getting $R_i\simeq 4$ nΩ at 1.5 GHz for Nb with λ = 40 nm and $\rho_s = 1$ n$\Omega\cdot$m requires $\Gamma \simeq 0.03\Delta$. The finite R_i at T = 0 results from the Dynes model assumption that Γ is independent of and T. In the case of a power law $N_s(\epsilon)$ at $\epsilon\ll \Delta$ (see, e.g. [94, 101]), one could expect a power law $R_s\propto T^n$ at ultralow temperatures. Shown in figure 2 is the Arrhenius plot of $R_s(T)$ calculated from equations (16) and (17) for different ratios of $\Gamma/\Delta_0$. At higher temperatures $\ln R_s(T)$ follows the BCS linear dependence on $\Delta/k_\mathrm{B}T$ and levels off as T decreases. The latter is indicative of a residual resistance resulting from the broadening of the DOS gap peaks. One can see that at low temperatures $R_s(T,\Gamma)$ increases as Γ increases but at higher T this trend reverses and $R_s(T,\Gamma)$ decreases as Γ increases.

Zoom In Zoom Out Reset image size

The non-monotonic dependence of R_s on Γ, shown in figure 2, can be understood by noticing that R_i in equation (18) increases with Γ. In turn, the reduction R_s with Γ at higher T comes from the reduction of the logarithmic factor in equation (2). In the BCS model with $\Gamma = 0$ the factor $\ln(C_1k_\mathrm{B}T/\hbar\omega)$ in $R_\mathrm{BCS}$ occurs because the square root singularities in the DOS at $\epsilon = \Delta$ and $\epsilon = \Delta\pm\hbar\omega$ merge at ω → 0 and produce a pole in the integrand of equation (16). However, if $\Gamma\gt0$, the gap singularities in $n(\epsilon)$ and $m(\epsilon)$ broaden into peaks of width ${\sim}\Gamma$ cutting off the divergence in equation (16) at ω = 0. At $\Gamma \gt\hbar\omega$ but $\Gamma\ll k_\mathrm{B}T$ integration of these peaks in equation (16) at $\epsilon\simeq\Delta$ yields a logarithmic term similar to that in equation (2) but with the energy cutoff Γ instead of $\hbar\omega$. The smearing of the DOS gap peaks can be qualitatively taken into account by the following replacement in equation (2):

$$\begin{align} \ln\frac{k_\mathrm{B}T}{\hbar\omega} \to \ln\frac{k_\mathrm{B}T}{\Gamma}. \end{align} \tag{ 19 }$$

Hence, broadening the peaks in the DOS reduces $R_s(T)$ at temperatures at which R_i is negligible. At 2 K and 1 GHz we have $k_\mathrm{B}T/\hbar\omega \sim 10^2$, so even weak broadening with $\Gamma = 0.02\Delta$ causes a two-fold reduction of R_s . Broadening the peaks in the DOS can be used to reduce the rf losses by pairbreaking mechanisms, as discussed below.

It is unclear how the Dynes parameter Γ could be tuned, but R_s can be reduced by magnetic impurities, the concentration of which can be varied by material treatments [45, 118, 119]. The spin-flip scattering on magnetic impurities reduces T_c , smears the gap singularities in the DOS and decreases the quasiparticle gap [21]:

$$\begin{align} \epsilon_\mathrm{g}=\left(\tilde{\Delta}^{2/3}-\Gamma_p^{2/3}\right)^{3/2}. \end{align} \tag{ 20 }$$

Here $\Gamma_p = \hbar/2\tau_s$, where the spin-flip scattering time τ_s is inversely proportional to the volume density of magnetic impurities [21, 90]. If the Dynes broadening of the DOS is disregarded and only the effect of magnetic impurities is taken into account, the quasiparticle gap $\epsilon_\mathrm{g}$ in equation (20) is smaller than the order parameter $\tilde{\Delta}$ [21]:

$$\begin{align} \tilde{\Delta} = \Delta_0-\frac{\pi}{4}\Gamma_p, \qquad \Gamma_p\ll\Delta. \end{align} \tag{ 21 }$$

Here Δ₀ is the order parameter in the absence of magnetic impurities. The broadening of the DOS peaks increases with $\Gamma_p$ as shown in the inset of figure 3.

Zoom In Zoom Out Reset image size

The microwave conductivity and the factors $n(\epsilon) = \mbox{Re}\cosh\theta$ and $m(\epsilon) = \mbox{Re}\sinh\theta$ in equation (16) were calculated in [45, 118, 119] by solving the Usadel equation which takes into account the magnetic pairbreaking [83, 90]:

$$\begin{align} \epsilon\sinh\theta+i\Gamma_p\cosh\theta\sinh\theta=\tilde{\Delta}\cosh\theta.\end{align} \tag{ 22 }$$

The effect of magnetic impurities on R_s was calculated in [45]. The results are shown in figure 3, where $R_s(\Gamma_p)$ is plotted as a function of $\Gamma_p$ at different temperatures. There is a minimum in $R_s(\Gamma_p)$ resulting from interplay of the DOS broadening which reduces R_s as $\Gamma_p$ increases, and the reduction of the quasiparticle gap $\epsilon_\mathrm{g}$ which increases R_s with $\Gamma_p$. The position of the minimum in $R_s(\Gamma_p)$ shifts to larger $\Gamma_p$ as T increases. Thus, incorporation of a small density of magnetic impurities in a superconductor can noticeably (by ∼ 30%–40%) decrease the surface resistance at low temperatures. The conditions for the minimum in $R_s(\Gamma_p)$ can be evaluated using the Abrikosov–Gor'kov theory of weak magnetic scattering in which T_c vanishes at $\Gamma_p = \hbar/2\tau_s = \Delta_0/2$ [90]. The latter implies that magnetic impurities suppress superconductivity if the spin flip mean free path $l_s = v_\mathrm{F}\tau_s$ becomes of the order of the size of the Cooper pair, $\xi_0 = \hbar v_\mathrm{F}/\pi\Delta_0$. The values of $\Gamma_p \simeq (0.01$–$0.02)\Delta$ in figure 3 correspond to $l_s\sim 10^2\xi_0$ if no bound quasiparticle states on magnetic impurities occur [90]. Magnetic impurities associated with oxygen vacancies in the native surface oxide of Nb have been revealed by tunneling measurements [91].

Another tunable pairbreaking mechanism of reducing R_s involves a thin metallic (N) layer coupled to the bulk superconductor (S) by the proximity effect as shown in figure 4. Such an N layer models a generic surface oxide structure of superconducting materials, particularly a 1–2 nm thick metallic NbO suboxide sandwiched between the dielectric NbO₂—Nb₂O₅ oxides at the surface and the bulk Nb. This model was investigated in [45, 46] in which the Usadel equations were solved to calculate a position-dependent quasiparticle density of states $N_n(\epsilon, x)$ across a thin N layer coupled to the bulk superconductor, and their effect on the surface resistance. The DOS profile in the N layer can be inferred from scannong tunnely spectroscopy [120, 121].

Zoom In Zoom Out Reset image size

The DOS profile and R_s are controlled by the parameters α and β which quantify the thickness of N layer and a resistive N–S interface barrier, respectively:

$$\begin{align} \alpha=\frac{\mathrm{d}N_n}{\xi_sN_s},\qquad \beta =\frac{4e^2}{\hbar}R_BN_n\Delta d. \end{align} \tag{ 23 }$$

Here d is the thickness of N layer, $\xi_s = (D_s/2\Delta)^{1/2}$ is the coherence length in the bulk superconductor with nonmagnetic impurities, D_s is the electron diffusivity proportional to the conductivity $\sigma_{n,s} = 2e^2N_{n,s}D_{n,s}$ in the normal state, N_s and N_n are the respective DOS at the Fermi surface in the normal state, the subscripts n and s label the parameters of the N layer and the S substrate, respectively, and R_B is a contact resistance between N and S. The properties of the structure shown in figure 4 can be tuned by material treatments which change the thickness and conductivity of the N layer and the interface resistance R_B . For instance, $R_B(T)$ can either increase or decrease with T depending on the heat treatment which can change R_B by several orders of magnitude, as was shown for the YBCO-Ag interface [122, 123]. Complexities of the Schottky barrier between different materials are not fully understood [124], but the dependence of $R_s(T)$ on the interface resistance can be used to optimize R_s .

The thickness of the N layer and the interface resistance which control the strength of proximity coupling of the N layer with the S substrate, strongly affect the DOS profile at the surface. Shown in figure 5 are the DOS in the N layer much thinner than the proximity length $\xi_n = (\hbar D_n/2\Delta)^{1/2}$ and the DOS at the S side of the N–S interface calculated in [45] for different values of β at $\Gamma = 0$. For strong coupling $\beta\ll 1$, the DOS in the N layer has a sharp peak at $\epsilon\simeq \Delta$ and drops to zero below the minigap energy $\epsilon_0\lt\Delta$ characteristic of N–S proximity-coupled structures [125, 126]. If $\beta\ll 1$ the proximity effect makes the N layer superconducting with $\epsilon_0\approx \Delta$. As R_B increases, the minigap in the N layer decreases and the DOS approaches N_n for a decoupled N layer at $\beta\to\infty$. Here $\epsilon_0(\beta)$ at $\Gamma = 0$ is determined by the equation [45]:

$$\begin{align} \beta=\frac{\Delta}{\epsilon_0}\left(\frac{\Delta-\epsilon_0}{\Delta+\epsilon_0}\right)^{1/2}. \end{align} \tag{ 24 }$$

Hence, ₀ decreases as β increases: $\epsilon_0\simeq (1-2\beta^2)\Delta$ at $\beta \ll 1$ and $\epsilon_0 = \Delta/\beta$, at $\beta\gg 1$. In a weakly-coupled N layer $(\beta\gg 1)$ the minigap $\epsilon_0 = \Delta/\beta$ expressed in terms of R_B using equation (23) is independent of superconducting parameters:

$$\begin{align} \epsilon_0=\frac{\hbar}{4e^2N_n\mathrm{d}R_B},\qquad \beta\gg 1. \end{align} \tag{ 25 }$$

The account of the Dynes parameter Γ smoothens the peaks in $N_n(\epsilon)$ and $N_s(\epsilon)$ shown in figure 5 and produces low-energy tails in $N_n(\epsilon)$ extending to the region $0\lt\epsilon\lt\epsilon_0$ of 'mini subgap' states in the N layer [45]. Here, a thin metallic layer can significantly affect $N_n(\epsilon)$ and $N_s(\epsilon)$, resulting in a variety of temperature dependencies of $R_s(T)$.

Zoom In Zoom Out Reset image size

Shown in figure 6 are the Arrhenius plots of $\ln R_s(T)$ as functions of $\Delta/k_\mathrm{B}T$ calculated in [45] for a thin N layer with α = 0.05, $\hbar\omega\lt\Gamma$ and different values of β varying from β = 0.1 (weak resistive barrier) to β = 30 (strong resistive barrier). In the limits of $\beta\ll 1$ and $\beta\gg 1$ the qualitative behaviors of $\ln R_s(T)$ are similar to those shown in figure 2: as T decreases, the slope of $\ln R_s(T)$ changes from the bulk Δ at high T to zero at low T, the residual resistance at $\beta\gg 1$ being much larger than at $\beta\ll 1$. The latter reflects higher rf losses in the normal layer as the proximity-induced superconductivity in the N layer weakens with the increase in R_B . As a result, R_i at $\beta\gg 1$ is dominated by the N layer fully decoupled from the S substrate, whereas at $\beta\ll 1$, the N layer is strongly coupled with the S substrate and the structure shown in figure 4 behaves as a single superconductor with an effective Γ. At intermediate values of β = 2–4, a change in the slope of $\ln R_s(T)$ occurs around $\Delta/k_\mathrm{B}\simeq 8$–10 due to switching from a thermally activated resistance controlled by the bulk gap Δ at high T to $R_s(T)$ controlled by the smaller minigap ₀ in the thin N layer. As the temperature decreases further, $R_s(T)$ approaches a temperature-independent residual resistance.

Zoom In Zoom Out Reset image size

The surface resistance can be reduced by tuning the parameters of the N layer. For example, figure 7 shows $R_s(T,\Gamma,\beta)$ for a thin dirty N layer at two temperatures and different values of Γ. The minimum in $R_s(\beta)$ results from the interplay of two effects. The first one causing the increase of R_s with β results from decreasing the proximity-induced minigap ₀ in the N layer as the interface resistance R_B increases. The second effect causing the initial decrease of R_s with β results from the change in the DOS around the N layer, as shown in figure 5. Here, a moderate broadening of the DOS peaks due to the combined effect of finite quasiparticle lifetime $\hbar/\Gamma$ and the metallic layer reduces R_s in a way similar to equation (19). Moreover, in the case of $\Delta/k_\mathrm{B}T = 4$ shown in figure 7(a), the minimum in R_s at $\Gamma = 0.01\Delta$ with the N layer is below $R_{s0}$, suggesting that one can engineer an optimal DOS to reduce R_s below its value for an ideal surface.

Zoom In Zoom Out Reset image size

The pairbreaking effect of current on the DOS was addressed theoretically in the sixties [19–21, 127] and then observed by tunneling measurements [128]. Shown in figure 8(a) is the DOS in the clean limit $l_i\gg\xi_0$ and $\Gamma = 0$ calculated in [27]. Here, the current turns the DOS singularity at $\epsilon = \Delta$ into a finite peak and reduces the quasiparticle gap $\epsilon_\mathrm{g}$ at which $N(\epsilon_\mathrm{g}) = 0$. The gap $\epsilon_\mathrm{g}(J)$ is smaller than Δ and decreases with the current density J, whereas Δ² proportional to the superfluid density is independent of J at $J\lt J_\mathrm{g}$ and T = 0, where $J_\mathrm{g} = en_0\Delta_0/p_\mathrm{F}$ is the critical current density at which $\epsilon_\mathrm{g}$ vanishes [19–21]. In a clean superconductor, the gap $\epsilon_\mathrm{g}$ is anisotropic and depends on the angle between J and the momentum of a quasiparticle.

Zoom In Zoom Out Reset image size

In the current-carrying state impurities round the cusps in $N(\epsilon)$ as shown in figure 8(b). Here, the quasiparticle gap decreases as J increases but remains finite as J reaches the depairing current density J_c . Thus, impurities preserve the gapped state all the way to the pairbreaking limit, unlike the case shown in figure 8(a). There is a critical concentration of impurities above which the gap $\epsilon_\mathrm{g}$ at $J = J_c$ opens, so that a superconductor at $J = J_c$ is in the gapless state if $a = \pi\xi_0/l\lt a_c$ and in a gapped state at $a = \pi\xi_0/l\gt a_c$. The critical value of $a_c = 0.36$ corresponds to the mean free path smaller than $l_c = \pi \xi_0/a_c \approx 8.72\xi_0$ [27]. The dependence of $\epsilon_\mathrm{g}$ on the mean free path at a > 0.36 is shown in figure 8(c). The critical gap $\epsilon_\mathrm{g}(a)$ at $J = J_c$ increases monotonically with a and approaches $\epsilon_\mathrm{g}(\infty) = 0.323\Delta_0$ at $l_i\ll \xi_0$. Scattering of quasiparticles on impurities makes $\epsilon_\mathrm{g}$ isotropic and independent of the direction of J.

The dependencies of $\epsilon_\mathrm{g}$ and Δ on J in the dirty limit are given by [21]

$$\begin{align} \epsilon_\mathrm{g}^{2/3}=\Delta^{2/3}-\left(\frac{J}{J_0}\right)^{4/3}\Delta_0^{2/3} \end{align} \tag{ 26 }$$

$$\begin{align} \Delta(J)=\left(1-\frac{\pi J^2}{4J_0^2}\right)\Delta_0,\qquad J\ll J_0, \end{align} \tag{ 27 }$$

where $J_0 = \phi_0/\sqrt{2}\pi\mu_0\lambda^2\xi\sim J_c$. The current-induced DOS broadening makes the surface resistance dependent on J in a way similar to the dependence of R_s on other pairbreaking parameters considered above. At $T\ll T_c$ and $\hbar\omega\ll\Delta$, the exponentially small density of quasiparticles affects neither T_c nor the dynamics of the superconducting condensate, but the rf superflow causes temporal oscillations of the DOS and $\epsilon_\mathrm{g}(t)$, resulting in a field dependence of $R_s(B_a)$. Usually, R_s is expected to increase with the rf field due to current pairbreaking, electron overheating, penetration of vortices, etc [14, 129]. Yet, a significant reduction of R_s by the rf field has been observed in alloyed Nb cavities [33–41] in which the decrease of R_s by ${\simeq} 20\%$–$60\%$ at 2 K extends to the fields $B_a\simeq 90$–100 mT at which the density of screening currents at the surface reaches ${\simeq} 50\%$ of the pairbreaking limit, $J_c\simeq B_c/\mu_0\lambda$. The reduction of R_s by a microwave field is not unique to Nb: similar effects have been observed in thin films [130–132]. Particularly, a reduction of $R_s(B_0)$ with the dc field B₀ superimposed onto a low-amplitude rf field in Pb, Sn, Tl, and Al has been known since the fifties [133–139].

The microwave suppression of R_s can be understood as follows [44]. The rf field $B(t) = B_a\cos\omega t$ causes temporal oscillations of the DOS between the singular $N(\epsilon)$ at $B(t) = 0$ to a broadened $N(\epsilon, B_a)$ at the peak field $B(t) = B_a$, as shown in figure 8. Thus, the peak in the DOS $\langle N(\epsilon)\rangle$ averaged over the rf period is smeared out within the energy range $\epsilon_\mathrm{g} \lt \epsilon \lt \Delta_0$. In the dirty limit equation (26) yields the width of the averaged gap peak $\delta\epsilon = \Delta_0-\epsilon_\mathrm{g} \sim (B_a/B_c)^{4/3}\Delta_0$ at $B_a\ll B_c$. As a result, the current-induced broadening of the DOS can be roughly accounted for by replacing the materials-related broadening parameters Γ or $\Gamma_p$ with the current-induced broadening δ in equation (19). This yields a logarithmic decrease of R_s with B_a :

$$\begin{align} R_s(B_a) \sim \frac{\mu_0^2\omega^2\lambda^3\Delta}{\rho_nk_\mathrm{B}T}\ln\left[\frac{CT B_a^{4/3}}{T_cB_c^{4/3}}\right]e^{-\Delta/k_\mathrm{B}T}, \end{align} \tag{ 28 }$$

where $C\sim 1$. This qualitative picture gives an insight into a mechanism of microwave reduction of $R_s(B_a)$ in the region of the parameters $(\Gamma/\Delta)^{3/4}B_c\ll B_a\ll B_c$ and $\mbox{max}(\Gamma,\hbar\omega)\ll k_\mathrm{B}T$ relevant to the experiments [33–40]. The behavior of $R_s(B_a)$ is affected by material treatments, yet the qualitative equation (28) describes well $Q(B_a)$ observed on Ti-doped Nb cavities [35]. The effect of current-induced DOS broadening on R_s was pointed out by Garfunkel [140] who calculated $R_s(H)$ was biased by a strong parallel dc field H in the BCS clean limit.

A theory of nonlinear surface resistance $R_s(B_a)$ in strong microwave fields must take into account both the temporal DOS oscillations and nonequilibrium effects [83, 141–144]. Many previous works have focused on nonequilibrium states caused by absorption of high-frequency photons at weak fields $B_a\ll (\omega/\Delta)^{3/4}B_c$ and $\hbar\omega \gt k_\mathrm{B}T$ for which the effect of rf current on $N(\epsilon)$ is weak [145–147]. In this case $\sigma_1(B_a)$ can decrease with B_a as the quasiparticle population spreads to higher energies $\epsilon \gt k_\mathrm{B}T$ [132]. This mechanism is similar to that of stimulated superconductivity [148, 149] and can explain the reduction of σ₁ with B_a observed in Al films at 5.3 GHz at 350 mK [132]. However, this approach is not applicable to low-frequency and high-amplitude rf fields with $\hbar\omega\ll k_\mathrm{B}T$ and $B_a \gt (\omega/\Delta)^{3/4}B_c$ for which $R_s(B_a)$ is determined by the time-dependent DOS and a nonequilibrium distribution function $f(\epsilon,t)$ driven by an oscillating superflow. A nonlinear surface resistance $R_s(B_a)$ in the dirty limit and $\lambda\gg\xi$ and $\hbar\omega \ll k_\mathrm{B}T$ was calculated in [150] by solving the time-dependent Usadel equations, taking into account temporal oscillations of $N(\epsilon,t)$, $\epsilon_\mathrm{g}(t)$ and $f(\epsilon,t)$ under strong rf field. Here R_s is obtained from the relation $H_a^2R_s/2 = \int_0^\infty\langle J(A(x,t)E(x,t)\rangle \mathrm{d}x$, where $J[A(x,t)]$ is the current density calculated for exact solutions of the Usadel equations, $\langle \ldots \rangle$ denotes averaging over rf period and $E(x,t) = -\dot{A} = B_a\omega\lambda e^{-x/\lambda}\sin\omega t$. This theory, in which $R_s(B_a)$ can decrease with B_a even for the equilibrium Fermi distribution of quasiparticles, captures the field dependence of $R_s(B_a)$ observed on Nb cavities [33–41].

Strong rf fields can drive quasiparticles out of thermodynamic equilibrium making $f(\epsilon,t)$ different from the Fermi–Dirac distribution $f_0(\epsilon) = (e^{\epsilon/k_\mathrm{B}T}+1)^{-1}$. Generally, $f(\epsilon,t)$ is determined by kinetic equations taking into account the current pairbreaking and scattering of quasiparticles on phonons and impurities [83]. The deviation from equilibrium depends upon the rate $1/\tau_\epsilon$ with which the rf power absorbed by quasiparticles is transferred to the crystal lattice. The time τ is determined by inelastic scattering of quasiparticles on phonons and recombination of quasiparticles into Cooper pairs [102]. If $\omega\tau_\epsilon\ll 1$, quasiparticles adiabatically follow the temporal DOS oscillations so $f(\epsilon,t)\to f_0(\epsilon)$, but the density of quasiparticles $n(t) = 2\int_0^\infty f_0(\epsilon) N[\epsilon,J(t)]d\epsilon$ varies in response to the instant changes of $N[\epsilon,J(t)]$ shown in figure 8. If $\omega\tau_\epsilon\gg 1$, quasiparticles do not have enough time to equilibrate so their density does not change much during rapid oscillations of $N(\epsilon,t)$. Due to slow power transfer between electrons and phonons at $\omega\tau_\epsilon\gg 1$, quasiparticles become hotter than the lattice, at it has been established in thin film electronic applications, for example, superconducting bolometers [151].

The rf periods ${\sim} 0.1$–1 ns are typically much longer than the electron–electron scattering time τ_ee and the condensate relaxation time $\tau_\Delta\sim\hbar/\Delta$. In this case, the quasiparticle energy relaxation time τ is limited by inelastic electron–phonon collisions for which $\tau_\epsilon(T)$ at $T\approx T_c$ is given by [83]:

$$\begin{align} \tau_\epsilon=\frac{8\hbar}{7\pi\zeta(3)\gamma k_\mathrm{B} T_\mathrm{F}}\left(\frac{c_s}{v_\mathrm{F}}\right)^2\left(\frac{T_\mathrm{F}}{T}\right)^3. \end{align} \tag{ 29 }$$

Here γ is the electron–phonon coupling constant, c_s is the speed of longitudinal sound, $T_\mathrm{F} = \epsilon_\mathrm{F}/k_\mathrm{B}$ is the Fermi temperature and $\zeta(3)\approx 1.2$. The time $\tau_\epsilon(T)$ increases rapidly as T decreases. For Nb₃Sn with $c_s/v_\mathrm{F}\simeq 10^{-3}$, $T_\mathrm{F}\sim 10^5$ K, $T_c = 17$ K and $\gamma\simeq 1.5$ [152], equation (29) yields $\tau_\epsilon\sim 10$ ps at T_c and $\tau_\epsilon\sim 6$ ns at 2 K. Hence Nb₃Sn at 1 GHz is in a quasi-equilibrium state near T_c but can be in a non-equilibrium state at 2 K. For Al with $c_s\simeq 5.1$ km s⁻¹, $v_\mathrm{F}\simeq 2030$ km s⁻¹, $T_\mathrm{F} = 1.36\times 10^5$ K, $T_c = 1.2$ K and γ = 0.43, [84, 153], one obtains $\tau_\epsilon\sim 0.4\, \mu$s at T_c . Generally, τ depends on energy , which becomes essential at low temperatures [102]. The electron–phonon relaxation time τ can be reduced by nonmagnetic and magnetic impurities [154–156] or by a thin proximity coupled metallic suboxide layer which reduces the minigap at the surface and allows more effective energy transfer from the quasiparticles to phonons. These effects can expand the temperature range of the quasi-equilibrium state.

The nonlinear conductivity controlled by the nonequilibrium kinetics of quasiparticles in strong electromagnetic fields is beyond the scope of this review. Here, we focus on the field dependence of R_s due to the temporal current broadening of the DOS affected by the Dynes parameter Γ, magnetic impurities or a proximity coupled N layer at quasi-equilibrium, $\omega\tau_\epsilon\lt 1$. Interplay of the current and material broadening of the DOS can produce a multitude of field dependencies of $R_s(B_a)$ [45, 46, 157–160]. Unlike the Dynes parameter in the bulk, tuning the layer thickness d and conductivity σ_n of the metallic suboxide, the contact resistance R_B and the bulk conductivity σ_s by material treatments can be used to optimize $R_s(B_a)$.

Shown in figure 9 are examples of $R_s(B_a)$ curves calculated in [46] for different thicknesses of the N layer and two interface barrier parameters β = 0.1 and β = 1 being around the minimum of R_s in figure 7. The dashed line shows the microwave suppression of $R_s(B_a)$ caused by the current broadening of the DOS for an ideal surface with d = 0 [44]. For β = 0.1, the dip in $R_s(B_a)$ gets less pronounced as the N layer thickness increases, $R_s(B_a)$ increasing with field at α > 0.1. This is consistent with weakening the induced superconductivity and reduction of the minigap in the N layer as it becomes thicker. Yet the crossover of $R_s(B_a,\alpha)$ curves at low fields in figure 9(a) implies that removing the N layer increases $R_s(B_a)$, in agreement with figure 7. This crossover disappears at a larger contact resistance as shown in figure 9(b).

Zoom In Zoom Out Reset image size

Figure 10 shows how the field dependence of $R_s(B_a)$ changes as the conductivity ratio $\sigma_n/\sigma_s$ is varied at a fixed thickness of the N layer for three values of the interface barrier parameter β [45]. Given that R_B and $\sigma_n/\sigma_s$ can be changed significantly by heat treatments [122–124] and alloying with nonmagnetic impurities, the results shown in figures 9 and 10 may account for the variability of the Q-factors of Nb cavities.

Zoom In Zoom Out Reset image size

Reduction of microwave losses by optimizing the DOS using pairbreaking effects may shed light on the mechanisms behind the improvement of the rf performance of Nb cavities after a low-temperature baking (by 100 ^∘C–200 ^∘C for 2 h) [28–32], medium temperature baking at 290 ^∘C–390 ^∘C for 3 h [32], high-temperature (800 ^∘C) annealing [7] and infusion of Nb with impurities. The latter has caused much interest since the discovery of microwave reduction of $R_s(B_a)$ after alloying the Nb cavities with nitrogen, titanium, oxygen and other impurities [33–41]. Alloying with nonmagnetic impurities could reduce the high-field rf losses since the quasiparticle gap $\epsilon_\mathrm{g}$ which ensures an exponentially small R_s does not close up to $B_a = B_s$. This may also pertain to the baking effect, which reduces the high-field Q-slope by diffusive redistribution of impurities, particularly interstitial oxygen or hydrogen in a thin ${\sim} 20$ nm layer at the surface [28–32]. The length $L = (Dt)^{1/2}$ over which impurities diffuse from the oxide surface layer to the bulk during the time t gives $L\simeq 17$ nm for the interstitial oxygen at 120 ^∘C and t = 48 h taking the diffusivity D from [161]. Uncovering the mechanisms by which material treatments affect superconducting properties requires compositional analysis of the Nb surface using multiple tools such as transmission electron microscopy (TEM), Auger electron spectroscopy (AES), x-ray photoelectron spectroscopy (XPS), electron energy loss spectroscopy (EELS) and atom probe microscopy [8, 162–169] combined with STS and $Q(B_a)$ measurements to reveal the effects of different treatments on the DOS and R_s .

There are several scenarios by which the infusion of impurities over a few µm from the surface could contribute to the field-induced reduction of $R_s(B_a)$. 1. Impurities mostly reduce the DOS broadening in the entire layer of rf field penetration ${\simeq} 2\lambda\sim 100$ nm, which reveals the microwave reduction of $R_s(B_a)$ of the BCS model [44, 150]. 2. The impurity infusion primarily modifies the surface oxides, for instance, by shrinking the metallic suboxide layer [44]. 3. The appearance of magnetic impurities and TLSs in the oxide layer and N–S interface [5, 108, 109, 170]. To determine which of these scenarios is more relevant, the R_s measurements are to be combined with scanning tunneling spectroscopy (STS) to measure the DOS at the surface and link it with the behavior of $R_s(B_a)$. This has been implemented by several groups, starting with the pioneering work [33] which showed that Ti infusion significantly reduces the lateral distribution of local values of Δ. The effect of N-doping on the DOS at the Nb surface was addressed in [120, 121]. In particular, the analysis of STS spectra in [121] using the model of [45] gave an insight into the effect of N infusion on the properties of the metallic suboxide.

The results of [121] summarized in figure 11 indicate that the effect of the nitrogen infusion gives rise to the following: 1. Slightly reduces the average superconducting gap $\bar{\Delta}$ while significantly reducing the spatial inhomogeneities of Δ and the Dynes parameter Γ. 2. Reduces the thickness of metallic suboxide from ${\approx} 2$ nm down to ${\approx} 1.2$ nm. 3. Reduces spatial inhomogeneities of the Nb suboxide thickness and the interface contact resistance parameter β close to the optimal values 0.3–0.4 corresponding to a minimum R_s shown in figure 7. It seems that these effects of the nitrogen doping bring the DOS toward its optimum which minimizes R_s , while reducing the effect of such extrinsic factors as the lateral inhomogeneity of superconducting parameters characteristic of the surface of Nb resonators.

Zoom In Zoom Out Reset image size

Material mechanisms which cause the modifications of the oxide structure in Nb require further investigations. It was found that heat treatment in a temperature range sufficient to dissociate the natural surface oxide not only causes a significant reduction of the residual resistance down to $R_i\simeq 1$ nΩ but also reduces the thermally-activated BCS part of R_s [32]. This observation seems consistent with the theoretical results of [45] which show that both R_i and $R_\mathrm{BCS}$ can be reduced if d_n and R_B are reduced from their respective values on the right side of the minimum in figure 9. Furthermore, recent TEM investigations indicate that nitrogen doping passivates the Nb surface by introducing a compressive strain close to the Nb/air interface, which impedes the diffusion of oxygen and hydrogen atoms and reduces the surface oxide thickness [169]. This conclusion seems consistent with the analysis of STM data shown in figure 11.

Microwave losses in amorphous dielectrics at low temperatures can be dominated by the presence of TLS in the material [60, 61]. These defects exist in a glassy state in which light atoms or trapped electrons or dangling atomic bonds can tunnel between two neighboring positions in a disordered atomic structure. Such TLS can occur in amorphous oxide layers on the surface of Al and Nb as well as at interfaces between a superconducting film and a dielectric substrate. TLS have attracted much attention as a source of noise and decoherence in superconducting quantum devices at low temperatures. Here, TLS can not only contribute to the residual surface resistance but also result in decreasing $R_i(E)$ with the rf electric field E [108, 109, 170, 171]:

$$\begin{align} R_i^\mathrm{TLS}(E)=R_{i0}^\mathrm{TLS}\frac{\tanh(\hbar\omega/2k_\mathrm{B}T)}{[1+(E/E_c)^2]^q}, \end{align} \tag{ 30 }$$

where q is close to $1/2$ in the standard tunneling model [60, 61], although the q values well below $1/2$ have also been observed [170]. The factor $R_{i0}^\mathrm{TLS}$ is proportional to the TLS density of states and depends on the resonator geometry, $E/E_c = \omega_R\sqrt{\tau_1\tau_2}$, where $\omega_R = 2d_0E/\hbar$ is the Rabi frequency, $d_0\simeq ea_0$ is a dipole moment proportional to the tunneling distance a₀, and $\tau_1(T)$ and $\tau_2(T)$ are the TLS energy relaxation and dephasing times, respectively. One can see that $R_i^\mathrm{TLS}(E)$ decreases with E at $E\gt E_c = \hbar\sqrt{\tau_1\tau_2}/2d_0$ as the TLS becomes saturated by the microwave field.

TLS losses in superconducting films can depend strongly on the dielectric substrate, for instance, the losses in Nb on SiO₂ are significantly higher than in Nb on Al₂O₃ [61]. Of particular interest are intrinsic TLS losses coming from the native oxides in Nb or Al separated from the losses in external dielectric components [172–174]. Fitting $R_i(B_a)$ measured on Nb cavities at 1.5–2 K and 1.3 GHz with equation (30) gave $E_c\approx 0.2$ MV m⁻¹ [109], much higher than the parallel electric field at the equatorial cavity surface $E\simeq \omega B_a\lambda\simeq 3\times(10^{-2}$–1) V m⁻¹ at $B_a = 0.1$–10 mT. The decrease of $R_i(B_a)$ with B_a observed in [109] was attributed to TLS on the inner cavity surfaces near the orifices, where the perpendicular rf electric field can reach a few MV m⁻¹. Yet, a much lower $E_c\simeq 56$ V m⁻¹ was observed on a 150 nm thick Nb stripline resonator grown on a Si substrate and coated with aluminum oxide [108].

TLS in AlO_x or Nb₂O₅ oxides have been commonly associated with dangling atomic bonds and oxygen vacancies [173, 174], although the true atomic origin of TLS has not been fully understood [61]. TLS may also result from common environmental impurities, such as nitrogen, carbon or hydrogen, which get dissolved in the material during the film growth and deposition [8]. For instance, the formation of metallic hydride precipitates in Nb and their contribution to rf losses is well-known [8, 110–114]. If the hydrogen bound to the oxygen vacancy in Nb₂O₅ does behave as TLS [175], one may expect R_i to increase after neutron or proton irradiation [176] which produces hydrogen irradiation defects and lattice disorder. In any case, the manifestations of TLS and quasiparticle contributions to the surface resistance are quite different. At GHz frequencies and T = 1–2 K the TLS residual resistance $R_i^\mathrm{TLS}\propto \tanh(\hbar\omega/2k_\mathrm{B}T)$ increases with ω and becomes independent of T at mK temperatures. This distinguishes the TLS microwave reduction of $R_i^\mathrm{TLS}(B_a)$ from that of the quasiparticle surface resistance $R_s\propto \exp(-\Delta/k_\mathrm{B}T)$ which decreases exponentially as T decreases.

In addition to the reduction of dissipative conductivity σ₁, the proximity effect can be used to tune the inductive conductivity σ₂ and either increase or decrease the kinetic inductance L_k which defines the energy $L_kI^2/2$ of flowing supercurrent I. Here, large L_k is desirable in transmons [1] and kinetic inductance photon detectors [2–5], whereas small L_k can be useful to reduce the readout or reset times $\tau_r = L_k/R_0$ in quantum memories, qubits and photon detectors, where R₀ is the resistance of the readout circuit. The influence of the proximity effect on L_k is most transparent for thin-film S-N bilayers or nanowires, which have been used in single photon detectors [177–179].

Consider an N-S bilayer shown in figure 4 with the thicknesses d_n and d_s smaller than the respective coherence lengths ξ_n and ξ_s . This corresponds to the Cooper limit [180, 181] in which the superconducting order parameters $\psi_{N,S}$ are uniform through the N and S layers, although ψ_N can be different from ψ_S due to the decoupling effect of R_B . Such N-S bilayers have been studied in the literature (see e.g., a review [182] and the references therein). In the case of $R_B = 0$ and $\Gamma\ll k_\mathrm{B}T_c$ the critical temperature of the bilayer decreases exponentially with the N layer thickness [180]:

$$\begin{align} T_{c}=T_{c0}\exp\left(-d_nN_n/\gamma_s d_sN_s\right), \end{align} \tag{ 31 }$$

where $T_{c0} = 1.13\Theta\exp(-1/\gamma_s)$ is the critical temperature of the S layer, γ_s is the BCS pairing constant and Θ is the Debye temperature. At $R_B = 0$ a dirty N-S bilayer has a uniform superconducting order parameter determined by the Usadel equation (22) with $\Gamma_p = 0$, $\epsilon\to \epsilon+i\tilde{\Gamma}$ and the composite parameters [183]:

$$\begin{align} \tilde{\Delta}=\frac{d_sN_s\Delta_s+d_nN_n\Delta_n}{d_sN_s+d_nN_n},\quad \tilde{\Gamma}=\frac{d_sN_s\Gamma_s+d_nN_n\Gamma_n}{d_sN_s+d_nN_n}. \end{align} \tag{ 32 }$$

The pairing potential $\Delta_s$ is nonzero in the S layer and vanishes $(\Delta_n = 0)$ in the N layer [181, 182]. The composite $\tilde{\Delta}$ is determined by the BCS gap equation with the effective coupling constant

$$\begin{align} \tilde{\gamma}=\frac{d_sN_s\gamma_s+d_nN_n\gamma_n}{d_sN_s+d_nN_n}. \end{align} \tag{ 33 }$$

Equations (32) and (33) describe both the N-S bilayer with $\gamma_n = \Delta_n = 0$ and a bilayer of two different superconductors S and S^ʹ for which the index n refers to the S^ʹ layer with nonzero γ_n and $\Delta_n$. For the N-S bilayer with $\tilde{\Gamma}\ll k_\mathrm{B}T_c$, equation (33) yields equation (31) because $T_c = 1.13\Theta\exp(-1/\tilde{\gamma})$ and $\gamma_n = 0$. Unlike the dissipative conductivity σ₁, the effect of weak Dynes broadening of the DOS on σ₂ and L_k is negligible [183].

The current I flowing along a strongly coupled bilayer of width $w\ll \lambda_s^2/d_s$ in response to the vector potential A is a sum of phase-locked currents in both films: $I = -\mu_0w\left(d_n/\lambda_n^2+d_s/\lambda_s^2\right)A$, where $\lambda_{s,n}^{-2} = (\pi\mu_0\sigma_{s,n}\tilde{\Delta}/\hbar)\tanh(\tilde{\Delta}/2k_\mathrm{B}T)$ in the dirty limit [183]. The kinetic inductance per unit length $L_k = -A/I$ is then:

$$\begin{align} L_k=\frac{\hbar\coth(\tilde{\Delta}/2k_\mathrm{B}T) }{\pi w\tilde{\Delta}(d_s\sigma_s+d_n\sigma_n)} \end{align} \tag{ 34 }$$

For a single S film, equation (34) yields $L_{k0} = \mu_0\lambda_s^2/wd_s$ [5].

Deposition of the N layer can either decrease or increase L_k . Shown in figure 12 is $L_k(d_n)$ calculated from equation (34) for different diffusivity ratios $D_n/D_s = \sigma_nN_s/\sigma_sN_n$ at $\gamma_s = 0.5$, $T\ll T_c$ and $\tilde{\Delta}(d_n) = 1.74k_\mathrm{B}T_c(d_n)$, where $T_c(d_n)$ is given by equation (31). If $D_n\lt D_s/\gamma_s$ the kinetic inductance monotonically increases with d_n due to suppression of T_c by the proximity effect. For a more conductive N layer with $D_n\gt D_s/\gamma_s$ the dependence of L_k on d_n becomes nonmonotonic due to the interplay of two effects: 1. The decrease of L_k with d_n due to the additional inertia of Cooper pairs induced in the N layer (the term $d_n\sigma_n$ in the denominator of equation (34)). 2. The increase of L_k with d_n due to the proximity effect reduction of $T_c(d_n)$. If $D_n\gg D_s$ the optimum thickness $\tilde{d}_n$ and the minimum inductance $\tilde{L}_k$ at $d_n = \tilde{d}_n$ are given by:

$$\begin{align} \tilde{d}_n\simeq d_s\frac{\gamma_sN_s}{N_n},\qquad \tilde{L}_k\simeq \frac{3L_{k0}D_s}{\gamma_sD_n}\ll L_{k0}. \end{align} \tag{ 35 }$$

Deposition of highly conductive Ag or Cu films onto a superconducting film can significantly decrease L_k . For instance, a thin Cu film of thickness $d_n = \tilde{d}_n$ on top of a NbN film could decrease L_k at $J\ll J_d$ by some two orders of magnitude because $\sigma_\mathrm{NbN}/\sigma_{Ag}\sim 10^{-2}-10^{-3}$. The effect of the mean free path and subgap states on L_k was investigated in [160]. Increasing the contact resistance weakens the S-N proximity coupling and reduces the contribution of the N layer to the kinetic inductance, so that $L_k\to L_{k0}$ at $\beta\gg 1$. Both $\sigma_2(J)$ and $L_k(J)$ increase with J due to current pairbreaking [5, 157, 160, 184, 185], which manifests itself in a nonlinear Meissner effect and intermodulation [129, 186–188]. Nonlinear screening of a dc parallel field and the breakdown of superconductivity in N-S bilayers has been investigated in [189–192].

Zoom In Zoom Out Reset image size

The kinetic inductance can increase greatly in polycrystalline films with weak-linked grain boundaries, particularly granular Al films which have been used in transmons and photon detectors [193–195]. Weakly-coupled grain boundaries characteristic of polycrystalline Nb₃Sn [54–56] can increase L_k and amplify the nonlinear Meissner effect in Nb₃Sn coplanar thin film resonators [196]. Here, a polycrystalline film can be regarded as a network of weakly-coupled planar Josephson junctions, each of which adds a nonlinear kinetic inductance $L_J = \phi_0/2\pi I_c\cos\chi$ inversely proportional to the tunneling Josephson critical current I_c and dependent on the superconducting phase difference χ on the junction [197].

At weak rf fields, the vortex velocities are small $v\ll v_0$, so $\eta = \eta_0$ is independent of v and the effect of vortex mass at GHz frequencies is negligible. As shown in figure 18, the dimensionless surface resistance r_i calculated in [270] for pinning centers is distributed randomly over the film thickness (see figure 17(a)). The global surface resistance $\bar{r}$ obtained by averaging $r_i(\beta,k)$ for different random pin configurations with the same volume pin density is shown in figure 18(b). The vortex moving in a particular kth pinning landscape produces a unique $r_i(\beta,k)$ which can vary rather non-systematically with the rf field, reflecting many metastable positions of the curvilinear vortex in a random pinning potential. Figure 18(b) shows the result of averaging over ten different random pin distributions with the same $n_i = 1.67\lambda^{-3}$. The low-field $r_i(\beta,k)$ fluctuate strongly but converge to the same value at high fields. Here $r_i(\beta,k)$ at low fields is strongly affected by pinning, whereas $r_i(\beta,k)$ at higher fields is mostly limited by the vortex drag and the effect of mesoscopic pinning fluctuations weakens. The averaged $\bar{r}_i(\beta)$ shown in figure 18(b) first increases with β and levels off at β > 0.2 as the low-field $\bar{r}_i(\beta)$, which mostly results from pinning hysteretic losses, crosses over to a drag-dominated $\bar{r}_i(\beta)$. A similar low-field dependence of $R_i(H)$ has been observed in Nb cavities [268, 269].

Zoom In Zoom Out Reset image size

The LO decrease of $\eta(v)$ with v can radically change the nonlinear dynamics of a trapped vortex and $R_i(B_a)$ at high fields and frequencies [270, 271]. Taking the LO effect into account raises the following questions: 1. What happens if the tip of the vortex moves faster than v₀ while the rest of the vortex does not? 2. How is the LO instability affected by pinning? 3. How does the decrease of $\eta(v)$ with v manifest itself in the dependencies of R_i on B_a , ω and the pinning strength? For a vortex segment pinned by a single strong pin, these issues were addressed in [271], and the effect of artificial pinning centers in films under a dc magnetic field and transport current was investigated in [292, 293]. The effect of LO instability is quantified by the control parameter α (not to be confused with α given by equation (23)) [270, 271]:

$$\begin{align} \alpha=(\lambda\omega/2\pi v_0)^2. \end{align} \tag{ 47 }$$

For $v_0 = 0.1$–1 km s⁻¹, equation (47) yields $\alpha = 4\times(1$–$10^{-2})$ at λ = 100 nm and 2 GHz. The increase of α with ω indicates that manifestations of the LO mechanism become more pronounced at higher frequencies.

Shown in figure 19 is $r_i(\beta)$ calculated for random bulk pinning in a film of thickness $l = 10\lambda$ at $\zeta_n = 0.8$, $n_i = 0.25 \lambda^{-3}$, and different values of α [270]. At the lowest frequency γ = 0.01 and α = 1, the elastic skin depth $L_\omega = \lambda/\sqrt{2\pi\gamma}\approx 4\lambda$ is about half of the vortex length. In this case, the surface resistance decreases with the rf field due to the LO decrease of the vortex viscosity with v. A similar descending field dependence of $R_i(B_a)$ caused by the LO effect was obtained for a vortex pinned by a single defect [271], the results being independent of the elementary pinning force if the pin is spaced more than L_ω from the surface.

Zoom In Zoom Out Reset image size

At higher frequencies, the behavior of $r_i(\beta)$ changes, as shown in figure 19(c). Here $r_i(\beta)$ becomes nonmonotonic, the peaks in $r_i(\beta)$ shifting to lower fields as the LO parameter α increases. At the peak of $r_i(\beta)$ at $\beta = \beta_m$, the maximum velocity of the vortex tip v_m becomes of the order of v₀, but no LO vortex jumps occur because of the restoring effect of the vortex line tension. The increase of $r_i(\beta)$ with β at $\beta\lt\beta_m$ is mostly due to the increase of $L_\omega\sim[\epsilon/\eta(v)\omega]^{1/2}$ caused by the decreasing $\eta(v)$. At $\beta\gt\beta_m$ the elastic skin depth L_ω becomes of the order of the vortex length and $r_i(\beta)$ decreases with β due to the decrease of $\eta(v)$ with v [270, 271]. The peak velocity v_m of the vortex tip increases with B_a and can significantly exceed v₀ at high fields.

As β exceeds β_m , the dynamics of the vortex tip change from nearly harmonic oscillations at $\beta\lt \beta_m$ to highly anharmonic relaxation-type oscillations at $\beta\gt \beta_m$, the amplitude of oscillations increases greatly at $\beta\gt \beta_m$ [270]. Bending distortions along the vortex are mostly confined within the elastic skin depth L_ω which increases with v due to LO reduction of $\eta(v)$ and eventually becomes larger than l at $v_m\gt v_0$. The nonlinear dynamics of the vortex at $\beta\gt\beta_m$ becomes dependent on the vortex mass, the peaks in $r_i(\beta)$ shifting to higher β as the ω increases [270, 271].

The LO velocity dependence of $\eta(v)$ can produce the residual surface resistance which decreases with the rf field amplitude. Such field-induced reduction of $R_i(B_a)$ results from the interplay of vortex elasticity and the LO decrease of the viscous drag with the vortex velocity. The decrease of $R_i(B_a)$ with B_a can contribute to the negative $Q(B_a)$ slopes observed on alloyed Nb cavities [33–41]. Unlike the decrease of $R_i^\mathrm{TLS}(B_a)$ or the quasiparticle surface resistance with the rf field, the vortex contribution to $R_i(B_a)$ scales with the density of trapped magnetic flux. The field-induced reduction of $R_i(B_a)$ produced by trapped vortices becomes more pronounced at higher frequencies, which appears to be consistent with the experiment [39] on N-doped Nb cavities. As was shown in [271], the LO mechanism can account for $R_s(B_a)\propto B_a^{-2}$ observed on Nb cavities [28] which cannot be explained by the weaker field dependence of $R_i^\mathrm{TLS}(B_a)$.

The significance of vortex losses in high-Q resonators brings to focus approaches for decreasing R_i by material treatments. The vortex losses at low fields can be reduced by increasing the volume density and strength of pinning centers. Consider first a reduction of R_i by alloying a superconductor with nonmagnetic impurities without changing the pinning defect structure. This was addressed in [270–272] by incorporating the dependencies of $\lambda = g\lambda_0$, $\xi = \xi_0/g$, and the pinning parameter $\zeta_n = 2\kappa^2 u_p/\pi\varepsilon\xi = \zeta_{n0}g^5$ on the mean free path l_i into equations (40) and (41). Here, the conventional factor $g = (1+\xi_0/l_i)^{1/2}$ interpolates the dependencies of superconducting properties on the mean free path [18], ζ_n is evaluated for small dielectric inclusions for which $u_p\sim B_c^2r_0^3/\mu_0$ is independent of l_i , and the index 0 labels the parameters of a clean material. The vortex viscosity η and the LO parameter α at $l_i\lt\xi_0$ can be evaluated as $\eta\simeq\eta_0g^2l_i/\xi_0$ and $\alpha\simeq\alpha_0g^2\xi_0/l_i$ using equation (43) with $D = l_iv_\mathrm{F}/3$ and τ being independent of l_i (see also [294]). Hence, alloying the material with $l_i\lt\xi_0$ results in the following:

• Enhances the pinning parameter $\zeta_n\sim\zeta_{n0}(\xi_0/l_i)^{5/2}$ as the vortex core diameter $\xi\sim (l_i\xi_0)^{1/2}$ and the line tension $\varepsilon\sim\varepsilon_0l_i/\xi_0$ decrease. This increases the elementary pinning forces and allows a softer vortex to better accommodate the pins.
• Weakly affects the Bardeen–Stephen vortex drag coefficient η₀ which becomes independent of l_i at $l_i\ll \xi_0$.
• Facilitates manifestations of the velocity dependence of $\eta(v)$ in the residual resistance $R_i(B_a)$ as the LO parameter $\alpha\sim\alpha_0(\xi_0/l_i)^2$ increases.

The field dependence of $r_i(B_a)$ calculated in [272] for bulk pinning in a film with different mean free paths and the Bardeen–Stephen η₀ is shown in figure 20. The overall behavior of $r_i(B_a)$ is similar to that shown in figure 18(b): the dip in $r_i(B_a)$ at low field occurs if the sheet Meissner current B_a is smaller than the rf depinning field $B_p\sim \mu_0J_pL_\omega$. At $B_a\lt B_p$ the vortex undergoes small-amplitude oscillations impeded by pinning which reduces rf losses. At $B_a\gt B_p$ the surface resistance increases significantly as the net Lorentz force exceeds the pinning force, and the amplitude of vortex oscillations is primarily determined by the balance of Lorentz and viscous drag forces. The transition from the pinning-dominated to viscous drag dominated regimes shifts to higher fields as pinning becomes more effective upon alloying the material. Yet alloying increases $r_i(B_a)$ at higher fields, although r_i drops a bit as l_i is decreased from $0.1\xi_0$ to $0.05\xi_0$. The latter reflects a nonmonotonic dependence of R_i of a freely moving vortex segment on l_i : $R_i\propto l_i^{-1/2}$ at $L_\omega \gt\lambda$ and $R_i\propto l_i^{1/2}$ at $L_\omega \lt\lambda$ [80].

Zoom In Zoom Out Reset image size

Figure 21 shows the effect of the mean free path on the microwave reduction of $R_i(B_a)$ caused by the LO velocity dependence of $\eta(v)$ for a vortex pinned by a single defect spaced by $l = 3\lambda_0$ from the surface [271]. One can see that the decrease of $R_i(B_a)$ with B_a becomes stronger and shifts to lower fields upon alloying the material. This reflects the increase of the LO control parameter $\alpha\propto (\omega\xi_0/l_i)^2$ as ω increases and l_i decreases. A maximum in $R_i(B_a)$ in figure 21 appears at high frequencies at which L_ω becomes smaller than l, similar to the evolution of $R_i(B_a,\omega)$ shown in figure 19. The descending dependence of $R_i(B_a)$ at high fields can be masked by overheating [270].

Zoom In Zoom Out Reset image size

Low-field rf losses of trapped vortices can be reduced by making pinning more effective using designer pinning structures which have been very effective in increasing J_p in superconductors for dc magnets [295, 296]. However, the artificial pinning centers in high-Q resonators can only be used in the form of dielectric precipitates or nano pores because metallic pins (for example, α-Ti ribbons in NbTi [295]) can produce high rf losses. Consider an upper limit of J_p for strong pinning by small dielectric precipitates or pores of radius $r_0\simeq \xi$ which chop vortex lines into short segments of length $\ell\lt\lambda$. If the ends of each vortex segment are fixed by the strong pins, J_p is determined by the pin breaking mechanism [297, 298] in which vortices bow out under the action of the Lorentz force and escape as the tips of two antiparallel vortex segments at the pin reconnect at the critical current density $J_p\sim \varepsilon/\phi_0\ell$ which increases with the pin density $n_p = \ell^{-3}$ [74]. However, too many dielectric pins would block the current-carrying cross section so $J_p(\ell)$ is determined by the interplay of pinning and current blocking [298]:

$$\begin{align} J_p\simeq \frac{\phi_0}{2\pi\mu_0\lambda^2\ell}\ln\frac{\ell}{2\xi}\left(1-\frac{4\pi r_0^3}{3p_c\ell^3}\right). \end{align} \tag{ 48 }$$

Here the factor in the parenthesis accounts for the reduction of the current-carrying cross section by dielectric pins and p_c is the percolation threshold which varies from $p_c = 1/2$ in two dimensions to $p_c\simeq 2/3$ in the 3D isotropic limit [300]. The interplay of pinning and current blocking yields a maximum of $J_p(\ell)$ at $\ell_m\simeq (16\pi/3p_c)^{1/3}r_0\simeq 3r_0$, an optimal volume fraction of pins $p_m\simeq 9\%$–$12\%$ and the maximum $J_{pm}\simeq (0.2$–$0.3)J_d$, where the spread of numerical values comes from the shapes of pins and the effect of crystalline anisotropy [298]. The maximum in J_p at $p\sim 10\%$ was also revealed by TDGL numerical simulations of vortices interacting with metallic pinning centers [299]. The optimized pinning with $J_p\simeq 0.2J_d$ allows a superconductor with sparse trapped vortices to withstand without significant losses the applied rf field $B_m\simeq \mu_0J_{pm}\lambda\simeq 0.2B_c$. Yet, even such idealized pinning structures can only provide rf breakdown fields much smaller than $B_b\simeq B_c$ observed on Nb cavities [16, 17]. Thus, pinning can only reduce the residual surface resistance at $B_a\ll B_c$.

The reduction of R_i by increasing the volume fraction of dielectric pins can come at the expense of higher dielectric and quasiparticle rf losses because dielectric precipitates increase the composite magnetic penetration depth $\bar{\lambda}$. This follows from the effective medium theory [301] which gives the conductivity $\bar{\sigma} = (1-p/p_c)\sigma$ [300, 301] of a composite comprised of dielectric spherical precipitates of radius r₀ and the volume fraction $p = 4\pi r_0^3/3\ell^3$ embedded in a matrix with conductivity σ. At $T\ll T_c$, we have $\sigma_1\ll\sigma_2$ so $\sigma\to i\sigma_2 = i/\mu_0\omega\lambda^2$ and $\bar{\sigma} = (1-p/p_c)\sigma = i/\mu_0\omega\bar{\lambda}^2$. Hence, $\bar{\lambda}(p)$ increases with p and diverges at the percolation threshold $p = p_c$:

$$\begin{align} \bar{\lambda}=\frac{\lambda}{\sqrt{1-p/p_c}}. \end{align} \tag{ 49 }$$

According to equations (2) and (49), the surface resistance $\bar{R}_\mathrm{BCS}\propto \bar{\lambda}^3$ increases with p. For a small fraction of dielectric pins $p\ll p_c$, we have $\bar{\lambda} = \lambda (1+3p/4)$ and

$$\begin{align} \bar{R}_\mathrm{BCS}=\left(1+3\pi r_0^3/\ell^3\right)R_\mathrm{BCS}. \end{align} \tag{ 50 }$$

At the optimum pin spacing $\ell\simeq 3r_0$ the composite surface resistance $\bar{R}_\mathrm{BCS}$ is increased by about $30\%$ relative to $R_\mathrm{BCS}$. Vortex trapping depends strongly on the sample geometry and is most pronounced in thin films in which the perpendicular $B_{c1}$ is greatly reduced. Vortex losses in thin film coplanar resonators were eliminated by producing an array of microscopic pinholes of radius $r_0\gt\xi$ which fully absorb the dissipative vortex and block vortex motion [71–73]. Such columnar pins result in a very high depinning current density $J_p\sim J_c$ [302, 303]. At the same time, the pinholes can increase the TLS losses at the edges [73].