Optimum multilayer coating of superconducting particle accelerator cavities and effects of thickness dependent material properties of thin films

We revisit the field limit of a superconductor–insulator–superconductor multilayer structure for particle accelerator cavities (BML), taking into account thickness (d)-dependent critical temperature, normal resistivity, and normal density of states seen in many thin films. Resultant d-dependent thermodynamic critical field and penetration depth lead to the appearance of a peak in BML(d) which has been missed in the previous studies. The procedure shown in this note would be useful to evaluate BML based on properties of one’s own films.


© 2019 The Japan Society of Applied Physics
The maximum amplitude of the surface RF magnetic field B 0 is one of the key parameters of superconducting RF (SRF) cavities for particle accelerators. 1,2) B 0 is proportional to the electric field on the cavity axis and its improvement leads to a reduction of the accelerator length necessary for getting a target energy. In the last several decades, 2) SRF researchers have continuously pushed up the record value of B 0 . The state-of-the-art Nb cavities can reach B 0 ∼ 200 mT [3][4][5] which corresponds to the accelerating electric field ∼50 MV m −1 for Tesla shape cavities. 6) However, further improvements are thought difficult as long as the present bulk Nb technology is used. This is because the corresponding screening current density at the surface J 0 ∝ B 0 is already close to the depairing current density J d ∝ B sh ∝ B c at which the Meissner state becomes absolutely unstable and vortex dissipation necessarily leads to strong Q degradations or quenches. Here B sh and B c are the superheating field [7][8][9][10] and the thermodynamic critical field, respectively.
Thus, to develop technologies beyond the bulk Nb cavity is of importance for even more improvement of the SRF cavity performance. Using an s-wave superconductor with a higher B c pushes up the theoretical field limit (e.g., B sh ∼ 400 mT for Nb Sn 3 ), [8][9][10] but cavities made from such alternative materials have not yet broken the record field of Nb cavity. 11) This is probably because such materials are prone to have small lower critical fields B c1 , resulting in the dissipative penetration of vortices at B c1 < B 0 < B sh where the Meissner state is metastable and instead the vortex state is the stable state.
The multilayer structure 12) shown in Fig. 1 has attracted much attention as it may solve this problem. The idea is to coat superconducting substrate (Σ) with a thin superconductor layer (S) separated by an insulator layer (I) to avoid catastrophic vortex dissipation and to achieve fields as high as B sh (>B c1 ). Then it has been recognized that there are appropriate material combinations and layers thicknesses: 13) the penetration depth of the S layer (λ) must be larger than the sum of the insulator thickness (d I ) and the penetration depth of the substrate (λ Σ ). Then the screening current density in the S layer is suppressed by the counterflow generated by the substrate. This current suppression effect is pronounced as d decreases. On the other hand, to protect the substrate, a thickness d ∼ λ is necessary. This observation 13) results in existence of the optimum thickness d m . Theoretical calculations have shown that the field limit of multilayer structure B ML can exceed the intrinsic B sh when d ; d m . [13][14][15][16] Today, various experiments for demonstrating the multilayer superiority at d ∼ d m are ongoing (e.g., the third harmonic measurement of first flux penetration field, [17][18][19] RF measurement by using the quadrupole resonator, 20,21) mushroom-shaped cavities, [21][22][23] etc). However, the theoretical calculations carried out so far had aimed for understanding of the general properties of multilayer structure and not for providing with predictions for a specific experiment. To extract predictions from the theory and compare them with experiments, the material parameters of one's own films should be used. This is because superconducting properties of thin films are sensitive to the growth conditions, resulting in different J d , λ, and then B ML and d m .
In addition, it is known that superconducting properties of a thin film generally depends on its thicknesses below d ∼ 100 nm. 24) However, any theoretical calculation of B ML has not yet taken into account the d dependences of superconducting properties of thin films. This effect also shifts B ML and the optimum parameters, and should be taken into account for a comparison between the theory and experiments.  Table I. In this brief note, (1) we show an example of the way to obtain theoretical predictions by using real material parameters and (2) examine how d dependences of superconducting properties affect the field limit and optimum parameters.
Let us consider a model shown in Fig. 1. We assume T T c  , because the operating temperature of SRF cavities T  4 K is well below T c of S layer material candidates (e.g., Nb Sn 3 , MgB 2 , NbN, NbTiN, etc). Since the nonlinear Meissner effect which makes λ dependent on the screening current density is weak for s-wave superconductors at T T c  , we use the London equation to calculate the field and current distributions. Then the screening current density at the surface of S layer J 0 and the magnetic field at the S-Σ interface B i are given by 13,14) g m l = J B , 1 The Meissner state of the S layer becomes absolutely unstable when J 0 reaches the depairing current density J d . Then we find B 0 = μ 0 λ J d / γ 1 is the maximum field that the S layer can withstand. On the other hand, breakdowns of the substrate are triggered when B i reaches a threshold value B Σ , resulting in another limitation B 0 = B Σ /γ 2 . Then the field limit of the multilayer structure is given by Fortunately, we know μ 0 λ J d at T T c  : it corresponds with the superheating field for a semi-infinite superconductor at T T c  14) and is given by 0.84B c for a clean limit superconductor with the coherence length l  . [7][8][9][10] The effects of impurity concentration is small. 10) So we may write 14) m lJ B 0.84 . Here ρ is a normal resistivity, N is a normal density of states, and α = Δ/k B T c . The parameters with the subscript ref represent some reference values. For a concrete discussion on the effects of d-dependent material properties and for demonstrating how to use experimental data of real samples to extract theoretical predictions, we consider NbN-I-Nb multilayer structures in the following. The substrate is assumed a bulk cavity-grade Nb with λ Σ = 40 nm and B Σ = 180 mT. For the NbN layer, we use the data reported by Semenov et al. 25) as an example. Here we do not consider the origin of d-dependences of material properties and simply assume the superconducting properties of NbN films are well described by the BCS theory. 26,27) 16) in which λ and B c were assumed to be independent of d.
The solid curve shown in Fig. 3(b) is the cross Sect. of the contour map at d I = 280 nm, different from the dashed curve calculated under the assumption of the thickness-independent l l = ¥ ( ) and = ¥ B B c c ( ). The peak at d ; 10 nm comes from the following mechanism: (1) At d  20 nm, we have λ ; 200 nm, which satisfies λ < λ Σ + d I , at which the screening current density on the S layer increases as d decreases. Thus, the field limit decreases with d. (2) However, at d  20 nm, λ becomes to satisfy λ > λ Σ + d I at which the counterflow-induced current suppression effect is pronounced as d decreases, enhancing the field limit. (3) For even smaller d regions, the suppression of B c overwhelms the counterflow-induced current suppression effect and the field limit decreases with d.
The solid curve shown in Fig. 3(c) is the cross section at = d 30 nm I , slightly shifted from the dashed curve calculated under the assumption of the thickness-independent λ and B c . At such thin d I regions in which λ > λ Σ + d I , the counterflow-induced current suppression effect is always pronounced as d decreases in contrast to thick d I cases seen in the above, and the field limit of the S layer is enhanced as d decreases. For d  λ, however, the S layer is too thin to protect the substrate, resulting in breakdowns at the substrate. The optimum thickness is given by d = d m at which the S-layer-limited B ML = μ 0 λ J d /γ 1 equals the substrate-limited B ML = B Σ /γ 2 . If the condition λ > λ Σ + d I is satisfied and λ(d) and B c (d) rapidly converge to constants at l ¥ d ( )  , the following analytical formula to find d m is still useful (see Ref. 14 This yields the optimum d of the dashed curve in Fig. 3(c), providing with a good approximation of the true d m .
For a more realistic evaluation of B ML and the optimum thicknesses, we need to take into account the existence of defects on the surface; e.g., non-stoichiometric composition or impurities can locally suppress the depairing current density, 32) topographic defects locally enhance the screening current density, 33,34) etc. These effects can be expressed by replacing 14,16,33) h Here η( h <  0 1) is a phenomenological suppression factor. Shown in Fig. 3(d) is the effects of defects on the surface of NbN layer. The value of B ML continuously changes from η = 1 (ideal surface) to η = 0.9. As η decreases, the optimum shifts to a smaller d and the maximum B ML decreases.
Application of the above procedure to other materials data is straightforward. For NbTiN-I-Nb multilayer, 35)    ) (here T c * and d* are fitting parameters), 28) and the assumption of d-independent =´--N 5 10 J m 47 1 3 39) for simplicity. In both the cases, we have the peak like Fig. 3(b) originating from the thickness dependent λ and B c .
We used the real experimental data that exhibits the thickness dependent T c , ρ, and N to evaluate λ and B c (Fig. 2), and calculated B ML and the optimum thicknesses (Fig. 3). We found the effects of the thickness dependences are pronounced at thicker d I regions. When d I is large enough (λ < λ Σ + d I ), we have a peak in B ML (d) which has been missed in the previous studies, shown in Fig. 3(b). Assuming constant λ and B c is still valid at thin d I regions if λ > λ Σ + d I is satisfied and λ(d) and B c (d) rapidly converge at l ¥ d ( )  . Yet, since material properties of a thin film are generally sensitive to growth conditions, those of one's own films should be used to extract theoretical predictions. The procedure shown in this note might be useful to obtain λ and B c , and then B ML particularly when λ and B c of one's own films are not available.