Maximum screening fields of superconducting multilayer structures

It is shown that a multilayer comprised of alternating thin superconducting and insulating layers on a thick substrate can fully screen the applied magnetic field exceeding the superheating fields $H_s$ of both the superconducting layers and the substrate, the maximum Meissner field is achieved at an optimum multilayer thickness. For instance, a dirty layer of thickness $\sim 0.1\; \mu$m at the Nb surface could increase $H_s\simeq 240$ mT of a clean Nb up to $H_s\simeq 290$ mT. Optimized multilayers of Nb$_3$Sn, NbN, some of the iron pnictides, or alloyed Nb deposited onto the surface of the Nb resonator cavities could potentially double the rf breakdown field, pushing the peak accelerating electric fields above 100 MV/m while protecting the cavity from dendritic thermomagnetic avalanches caused by local penetration of vortices.

It is shown that a multilayer comprised of alternating thin superconducting and insulating layers on a thick substrate can fully screen the applied magnetic field exceeding the superheating fields Hs of both the superconducting layers and the substrate, the maximum Meissner field is achieved at an optimum multilayer thickness. For instance, a dirty layer of thickness ∼ 0.1 µm at the Nb surface could increase Hs ≃ 240 mT of a clean Nb up to Hs ≃ 290 mT. Optimized multilayers of Nb3Sn, NbN, some of the iron pnictides, or alloyed Nb deposited onto the surface of the Nb resonator cavities could potentially double the rf breakdown field, pushing the peak accelerating electric fields above 100 MV/m while protecting the cavity from dendritic thermomagnetic avalanches caused by local penetration of vortices.

I. INTRODUCTION
The maximum magnetic field H which can be screened by a superconductor in the vortex-free Meissner state has attracted much attention, both from the fundamental and applied perspectives [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] . This problem is particularly important for the Nb resonator cavities 2 which have extremely high quality factors Q(2K) ∼ 10 10 −10 11 up to the breakdown fields H b ≃ 200 mT at which the screening current density J approaches the depairing limit 1 , J d ≃ H c /λ 0 , where H c ≃ 200 mT is the thermodynamic critical field of Nb and λ 0 ≃ 40 nm is the London penetration depth. The lack of radiation losses and vortex dissipation in the Nb cavities (clean Nb has the highest lower critical field H c1 ≃ 180 mT among type-II superconductors) enables one to probe the high-field nonlinear quasiparticle conductivity 17 which can be tuned by alloying the surface with impurities [18][19][20] .
The screening field limit of Nb could be exceeded by using s-wave superconductors with higher H c and the critical temperature T c , but such materials are prone to the dissipative penetration of vortices at H ≃ H c1 < H N b c1 . To address this problem it was proposed to coat the Nb cavities with multilayers of thin superconductors (S) with high H c > H N b c separated by dielectric (I) layers 11 (see Fig. 1). This approach is based on the lack of thermodynamically stable parallel vortices in decoupled S screens of thickness d s < λ, which also manifests itself in a strongly enhanced H c1 in thin films predicted by Abrikosov 5,6 and observed in Refs. 7-10.
The maximum field H m screened by N superconducting layers of thickness d = N d s ≫ λ is limited by the superheating field H s of S-layers 11 , for example, H s ≃ 0.84H c = 454 mT for Nb 3 Sn at T ≪ T c . Here the Meissner screening currents at H = H s become unstable with respect to infinitesimal perturbations of electromagnetic field and the order parameter, while the magnetic barrier for penetration of vortices vanishes [12][13][14] . This paper addresses the limits to which the maximum screening field can be increased by S layers with given d s , λ and H s deposited onto a thick Nb substrate with given λ 0 and H s0 . It is shown that: 1. The maximum H m can be reached at an optimum multilayer thickness d m which depends on the materials parameters of S layers and the substrate; 2. The optimized S-I-S multilayer can screen the field exceeding both superheating fields, H s and H s0 ; 3. S-I-S multilayers arrest thermomagnetic avalanches caused by local penetration of vortices at defects and do not let them develop into global flux jumps, which otherwise quench the cavity at fields well below H m .

II. SUPERHEATING FIELD, DEPAIRING CURRENT DENSITY AND INSTABILITY OF MEISSNER STATE
Distribution of a low-frequency ( ω ≪ k B T c ) rf magnetic field h(x) in a multilayer can be described by the London equations λ 2 h ′′ 1 = h 1 at 0 < x < d and λ 2 0 h ′′ 2 = h 2 at x > d, where the prime denotes differentiation with respect to x. Given that h(x) in a multilayer calculated numerically from the Eilenberger equations is close to the result of the London theory 13 , I first disregard the nonlinear Meissner effect 21 which makes λ dependent on J at J ∼ J d . Solutions of the London equations are (see also Refs. 15,16): Here h(x) and the rf electric field E = −µ 0 ωλ 2 h ′ are continuous at x = d, k = (λ − λ 0 )/(λ + λ 0 ), and g = exp(d/λ), both for a single S film and for a stack of S-I layers with d i ≪ d s . For a S-I bilayer, the low-field surface resistanceR s is determined by the total Joule rf power, H 2R where σ and σ 0 are the quasiparticle conductivities in S layers and the substrate, respectively, and q i accounts for dielectric losses. Using Eqs. (1)-(3), and R s = µ 2 0 ω 2 σλ 3 /2 in the local dirty limit 3 yields 22 where R s and R s0 include the residual resistances 3 , and H/λ at the surface is lowered by the counterflow induced by the substrate 16 . The Meissner state in S layers and the substrate is stable with respect to infinitesimal perturbations [12][13][14] if the current densities are smaller than the respective depairing limits, These conditions define the field region of the vortex-free state: Shown in Fig. 2 where µ = H s λ/(λ + λ 0 )H s0 . Substituting d m back to one of Eqs. (5) yields the maximum screening field  7) give H s = 0.84H s ≃ 840 mT and H m ≃ 872 mT at d m = 1.78λ = 356 nm. The case of the two-band superconductor MgB 2 with H c ≃ 230 mT is more complicated as the rf dissipation can occur at a smaller field H d ≃ H c ξ σ /ξ π at which the screening current decouples two weakly coupled σ and π bands 27 . For the typical ratio of coherence lengths, ξ σ /ξ π ≃ 0.2 − 0.3, the band decoupling field H d ∼ 50 mT is consistent with the rf breakdown field of 42 mT at 4 K observed on MgB 2 /Al 2 O 3 bilayers on the Nb substrate 8 .

III. PENETRATION OF VORTICES AT DEFECTS AND THERMOMAGNETIC STABILITY OF MULTILAYERS.
The maximum screening field H m at which the Messier state becomes absolutely unstable with respect to infinitesimal perturbations can hardly be reached under realistic operating conditions which require that the accelerating resonator cavities remain stable with respect to penetration of vortices, strong transient electromagnetic perturbations of charged beams, and local field enhancement at surface defects. In the multilayer approach 11 I layers are instrumental to assure the necessary stability margin with respect to local penetration of vortices at surface defects, which can otherwise trigger thermomagnetic flux jumps 29,30 particularly at low temperatures T ≪ T c and extremely high screening currents J ∼ J d at which the cavities operate. Misinterpretation of this issue has lead to speculations that neither I layers nor the enhancement of B c1 is necessary, so a few µm thick Nb 3 Sn film coating of the Nb cavities could just be protected against penetration of vortices by the Bean-Livingstron surface barrier 15 . This assumption contradicts a vast body of experimental data on magnetization of high-κ type-II superconductors for which inevitable materials or topographic defects at the surface cause premature local penetration of vortices at H c1 < H < H s , or even H < H c1 due to grain boundary weak links 31 .
The maximum screening field H m at which the Meissner state is stable with respect to penetration of vortices can be evaluated from Eqs. (6)-(7) with effective H s and H s0 depending on the operating conditions. For instance, in a multilayer with h(d) < H c10 ≃ 180 mT at H = H m , a vortex entering through a defect in the S layer cannot penetrate further into the bulk Nb. Let a surface defect cause local penetration of vortices as the current density, J(0) = H ′ (0) = βH s /λ reaches a fraction β 1 of J d in S layer with d s < λ and ξ ≪ d s . If J(0) is not too close to J d , the London theory shows 6,11 that the energy barrier U = ǫ 0 ln(1/β) per unit length of a vortex in the S layer at J = βJ d coincides with the bulk Bean-Livingston surface barrier 5,6 at H = βH sh ≫ H c1 and (ξ/d s ) 2 ≪ 1, where ǫ 0 = φ 2 0 /4πµ 0 λ 2 . The criterion J(0) < J d /2 assures a reasonable protection against penetration of vortices caused by low-angle grain boundaries in polycrystalline Nb 3 Sn or pnictides 31 , or local field enhancement at typical topographic defects in the Nb cavities 2 .
At H s → H s /2 and H s0 → 170 mT, Eqs. To inhibit dissipative penetration of vortices, S film can be subdivided by I layers into N thinner S layers with d s = d m /N . At h(d) < H c10 , even if a vortex penetrates at a defect in the first S layer, it could not propagate into the next S layer and further in the bulk Nb where it can cause a thermomagnetic avalanche 30 . As H(t) reaches the critical value βH s at a week spot, a vortex line cannot penetrate parallel to the surface but first originates at a defect as a small semi-loop as depicted in Fig. 3. The vortex semi-loop expands under the action of the perpendicular Meissner current until it hits the I layer where most part of the dissipative vortex core disappears in a loss-free flux channel connecting two short vortices of opposite polarity. Because of the magnetic flux compression in the I layer and the substrate, the energy of a perpendicular vortex ≃ d sǫ0 ln(L/ξ) diverges with the lateral film size L, while the energy of the vortex-antivortex pair ≃ d sǫ0 ln(u/ξ) grows with the distance u, whereǫ 0 ∼ ǫ 0 (Ref. 33). This vortex-antivortex pair expands during the positive rf cycle and contracts and annihilates as H(t) changes sign. The upper limit of the pair size u m can be estimated neglecting the long-range vortex-antivortex attraction described by the last term in the dynamic equation, ηu/2 = φ 0 J(t) −ǫ 0 /u, where η = φ 2 0 /2πξ 2 ρ n is the viscous drag coefficient, and ρ n is the normal state resistivity. Hence, u m ∼ 2φ 0 H/ληω ∼ βf ρ n / √ 2πκµ 0 λν at H = βH s and ω = 2πν, giving u m ≃ 4 µm and the rf power 32 q ∼ dJ 2 φ 2 0 /η ∼ β 2 f 2 dφ 0 B c ρ n /κµ 2 0 λ 2 ∼ 2 µW for Nb 3 Sn at ρ n = 0.2 µΩm, λ = 100 nm, β = 1/2, f = 0.84, κ = 20, ν = 2 GHz and d s /λ = 0.2. Taking into account attraction of antiparallel vortices in S layer reduces u m and q. Penetration of vortex semi-loops at a defect appears more realistic than the model of a long vortex parallel to the surface for which the ill-defined notion of H c1 = 0 associated with the magnetic flux trapped in I layer 15 is irrelevant to the rf dissipation 29 .

IV. DISCUSSION
Localization of the rf power in a thin S layer inhibits expansion of multiple vortex loops in the bulk and stops dendritic thermomagnetic avalanches 30 that are particularly pronounced at the extremely high screening cur-rent densities J ∼ J d and low temperatures T ≪ T c in the materials like Nb 3 Sn, NbN or pnictides with low ρ −1 n and thermal conductivity k. The multilayer thus significantly reduces vortex dissipation as compared to the bulk Nb 3 Sn, yet a thin Nb 3 Sn layer with d ∼ 100 nm may only slightly increase the thermal impedance of the cavity wall, The optimum number of S layers for particular materials is determined by a balance between reduced vortex dissipation and suppression of superconductivity at the S-I interfaces. Here H s of ideal S layers with d s > (ξλ) 1/2 remains the same as in the bulk 12 , contrary to the assertion 15 that H s is reduced at small d s . This claim was based on the artifacts of the London model discussed above and on taking into account only one right vortex image in Fig. 4 of Ref. 15 instead of summing up an infinite chain of vortex-antivortex image dipoles which ensure that vortex currents do not cross the film surface. If this effect is properly taken into account 6,11 H s in a thin film (d < λ) is the same as in a thick film (d > λ). Moreover, the GL simulations of Ref. 15 for a Nb 3 Sn film on Nb show that H m (d) reaches the maximum H m (d m ) ≃ 1.08H s at d m ≃ 1.15λ but remains larger than H s in the whole region λ < d < 2λ for which the numerical results were presented.
High-field rf performance of the Nb cavities can be boosted by depositing not only materials with higher H s but also alloyed Nb-I-Nb multilayers which can increase H m and benefit from a significant raise of Q(H) with H in a wide field region [17][18][19][20] . A polycrystalline Nb multilayer may be tuned by alloying and heat treatment to reduce the residual resistance 3,17,18 , and is also less prone to the current-blocking grain boundaries than the A-15 or pnictide compounds 31 . Enhancement of the vortex penetration field by a dirty Nb/Al 2 O 3 bilayer deposited onto the Nb cavity was observed in Ref. 36.
In conclusion, optimized multilayers can significantly increase the Meissner screening field while inhibiting dissipative penetration of vortices. Implementation of such multilayer coatings could potentially double the accelerating field gradients of superconducting resonators as compared to the existent high-performance Nb cavities. This work was supported by DOE HEP under Grant No. DE-SC0010081.