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We present results for the resonant frequency shift and quality factor of disordered Nb SRF cavities driven out of equilibrium by the resonant microwave field. The theory is based on the nonequilibrium theory of superconductivity for the current response to the electromagnetic field at the vacuum-metal interface. We are able to accurately predict the observed frequency shifts with a precision of order several Hz over the full temperature range $0 < T \le T_c$, including the negative frequency shift anomalies that are observed very near $T_c$. The origin of these anomalies is shown to be the competition between the normal metal skin depth and the London penetration depth which diverges as $T\rightarrow T_c^-$. An analytic approximation to the full current response, valid for $|T-T_c|\ll T_c$, accounts for the negative frequency shift near $T_c$. The non-monotonic dependence of the quality factor on the quasiparticle scattering rate is related to the pair-breaking effect of disorder on the superfluid fraction, and thus the London penetration depth., Comment: 7 pages, 3 figures

Recently reported anomalies in frequency shift of order $\delta f\sim 0.1 -10\,\mbox{kHz}$ for Niobium SRF cavities in a narrow temperature region near $T_c\simeq 9\,\mbox{K}$ are sensitive to the surface preparation and Nitrogen doping. We developed methods for calculating the surface current response of Nb SRF cavities, as well as the resonant frequency shift and quality factor, as functions of temperature, frequency and disorder based on the Keldysh formulation of the theory for nonequilibrium superconductivity coupled with Maxwell's theory and boundary conditions for the response functions and electromagnetic field. We show that the anomaly is sensitive to impurity disorder. Our results for the anomaly in the frequency shift are in good agreement with experimental results for Nb with an anisotropic gap on the Fermi surface and inhomogeneous non-magnetic disorder. We also show that the quality factor as a function of the impurity scattering rate is maximum for in SRF cavities with intermediate disorder, $\hbar/\tau\Delta\sim{\cal O}(1)$ with the maximum $Q$ decreasing with increasing frequency., Comment: 6 pages, 3 figures

Niobium is one of the most studied superconductors, both theoretically and experimentally. It is tremendously important for applications, and it has the highest superconducting transition temperature, $T_{c}=9.33$ K, of all pure metals. In addition to power applications in alloys, pure niobium is used for sensitive magneto-sensing, radio-frequency cavities, and, more recently, as circuit metallization layers in superconducting qubits. A detailed understanding of its electronic and superconducting structure, especially its normal and superconducting state anisotropies, is crucial for mitigating the loss of quantum coherence in such devices. Recently, a microscopic theory of the anisotropic properties of niobium with the disorder was put forward. To verify theoretical predictions, we studied the effect of disorder produced by 3.5 MeV proton irradiation of thin Nb films grown by the same team and using the same protocols as those used in transmon qubits. By measuring the superconducting transition temperature and upper critical fields, we show a clear suppression of $T_{c}$ by potential (non-magnetic) scattering, which is directly related to the anisotropic order parameter. We obtain a very close quantitative agreement between the theory and the experiment.

Niobium superconducting radio-frequency (SRF) cavities for high energy accelerator applications have been greatly improved in terms of the quality factor $Q$ by techniques such as Nitrogen doping. However, the mechanisms leading improvement in $Q$ are still not fully understood. Quite recently the SRF group at Fermilab measured anomalies in the frequency shift of N-doped SRF Niobium cavities near the transition temperature. Here we report our theoretical analysis of these results based on the microscopic theory of superconductivity that incorporates anisotropy of the superconducting gap and inhomogeneous disorder in the screening region of the SRF cavities. We are able to account for frequency shift anomalies very close to $T_c$ of the order of fractions of a kHz. Our results for the frequency shift and Q are in good agreement with the experimental data reported for all four N-doped Nb SRF cavities by Bafia et al. We also compare our theory with an earlier report of on a Nb sample measured at 60 GHz. We also show that the quality factor calculated theoretically has a peak of upper convexity with the largest $Q$ at intermediate levels of disorder. For strong disorder, i.e. the dirty limit, pair breaking in the presence of disorder and screening currents limits the $Q$., Comment: 11 pages, 6 figures

Niobium is one of the most researched superconductors, both theoretically and experimentally. It is enormously significant in all branches of superconducting applications, from powerful magnets to quantum computing. It is, therefore, imperative to understand its fundamental properties in great detail. Here we use the results of recent microscopic calculations of anisotropic electronic, phonon, and superconducting properties, and apply thermodynamic criterion for the type of superconductivity, more accurate and straightforward than a conventional Ginzburg-Landau parameter $\kappa$ - based delineation, to show that pure niobium is a type-I superconductor in the clean limit. However, disorder (impurities, defects, strain, stress) pushes it to become a type-II superconductor.

Niobium is commonly used for superconducting quantum systems as readout resonators, capacitors, and interconnects. The coherence time of the superconducting qubits is mainly limited by microwave dissipation attributed to two-level system defects at interfaces, such as the Nb/Si and Nb/air interface. One way to improve the Nb/air interface quality is by thermal annealing, as shown by extensive studies in 3D superconducting radio frequency (SRF) cavities. However, it is unclear how the microstructure and chemistry of the interface structures change during heat treatment. To address this knowledge gap, we comprehensively characterized Nb films deposited on Si wafers by physical vapor deposition, including (1) an Nb film from a transmon and (2) an Nb film without any patterning step, using an aberration-corrected transmission electron microscope. Both Nb films exhibit columnar growth with strong [110] textures. There is a double layer between the Nb film and Si substrate, which are amorphous niobium silicides with different Nb and Si concentrations. After in-situ heating of the heterostructure at 360{\deg}C inside the microscope, the composition of the double layers at the Nb-Si interface remains almost the same despite different thickness changes. The initial amorphous niobium oxide layer on Nb surface decomposes into face-centered cubic Nb nanograins in the amorphous Nb-O matrix upon heating.

We report results for the superconducting transition temperature and anisotropic energy gap for pure Niobium based on Eliashberg's equations and electron and phonon band structures computed from density functional theory. The electronic band structure is used to construct the Fermi surface and calculate the Fermi velocity at each point on the Fermi surface.The phonon bands are in excellent agreement with inelastic neutron scattering data. The corresponding phonon density of states and electron-phonon coupling define the electron-phonon spectral function, $\alpha^2F({\bf p},{\bf p}';\omega)$, and the corresponding electron-phonon pairing interaction, which is the basis for computing the superconducting properties. The electron-phonon spectral function is good agreement with existing tunneling spectroscopy data except for the spectral weight of the longitudinal phonon peak at $\hbar\omega_{\text{LO}}=23\,\mbox{meV}$. We obtain an electron-phonon coupling constant of $\lambda=1.057$, renormalized Coulomb interaction, $\mu^{\star}=0.218$ and transition temperature $T_c=9.33\,\mbox{K}$. The corresponding strong-coupling gap at $T=0$ is modestly enhanced, $\Delta_0=1.55\,\mbox{meV}$, compared to the weak-coupling BCS value $\Delta_0^{\text{wc}}=1.78\,k_{\mbox{b}}\,T_c= 1.43\,\mbox{meV}$. The superconducting gap function exhibits substantial anisotropy on the Fermi surface. We analyze the distribution of gap anisotropy and compute the suppression of the superconducting transition temperature using a self-consistent T-matrix theory for quasiparticle-impurity scattering to describe Niobium doped with non-magnetic impurities. We compare these results with experimental results on Niobium SRF cavities doped with Nitrogen impurities., Comment: 24 pages, 11 figures with Tables