aa r X i v : . [ phy s i c s . acc - ph ] O c t Superconducting nano-layer coating without insulator ∗ Takayuki Kubo † KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801 Japan
Abstract
The superconducting nano-layer coating without insula-tor layer is studied. The magnetic-field distribution and theforces acting on a vortex are derived. Using the derivedforces, the vortex-penetration field and the lower criticalmagnetic field can be discussed. The vortex-penetrationfield is identical with the multilayer coating, but the lowercritical magnetic field is not. Forces acting on a vortexfrom the boundary of two superconductors play an impor-tant role in evaluations of the free energy.
INTRODUCTION
The multilayer coating is an idea for pushing up the fieldlimit of superconducting (SC) accelerating cavity [1]. Ac-cording to the recent theoretical studies [2, 3, 4, 5], differ-ences among the vortex-penetration fields of semi-infiniteSC, SC thin film and multilayer SC are due to those of cur-rent densities or slopes of magnetic-field attenuation whichare proportional to the force pushing a vortex into SC. Forthe case that a penetration depth of the SC layer is largerthan that of the SC substrate, the current density or theslope of magnetic-field attenuation in the SC layer is sup-pressed, and the vortex-penetration field of the SC layer ispushed up.A combination of SC materials with different penetrationdepths is essential, but the insulator layer seems to be un-necessary. In this paper, the SC nano-layer coating withoutinsulator layer is studied. First the magnetic-field distribu-tion is computed. Then a set of forces acting on a vortex isderived, which is different from that of the multilayer coat-ing model, because of the existence of the boundary of twoSCs as shown in Fig. 1. By using the forces, the vortex-penetration field and the lower critical magnetic field arediscussed.
MAGNETIC-FIELD DISTRIBUTION
Let us consider a model with an SC layer with a pene-tration depth λ and a coherence length ξ formed on anSC substrate with a penetration depth λ and a coherencelength ξ . All layers are parallel to the y - z plane and thenperpendicular to the x -axis. The magnetic field is appliedparallel to the layers. We assume the SC layer thickness d is larger than their coherence lengths ( d ≫ ξ , ξ ). Theproximity effect between the two SCs. is neglected. ∗ The work is supported by JSPS Grant-in-Aid for Young Scientists(B), Number 26800157. † [email protected] Figure 1: A contour plot of the magnetic field around avortex core near a boundary of two semi-infinite SCs. Theboundary is indicated by a vertical white-line on x = 0 .Penetration depths of left and right SCs are
200 nm and
100 nm , respectively.The variational approach is an easy way to understandhow the magnetic-field and the current density distributein the system. Defining magnetic-fields in the SC layerand SC substrate as B ( x ) and B ( x ) , respectively, cur-rent densities in the SC layer and SC substrate are givenby J ( x ) = − µ − dB /dx and J ( x ) = − µ − dB /dx ,respectively. Then the total energy of the system as a func-tional of B k ( k = 1 , ) is given by E [ B k ] = Z dydz X k =1 Z L k L k − dx (cid:16) B k µ + µ λ k J k (cid:17) , (1)where L ≡ , L ≡ d , L ≡ ∞ . Variation of the abovefunctional vanishes when conditions d B k dx = B k λ k , (2) B ( d ) = B ( d ) , (3) λ dB dx (cid:12)(cid:12)(cid:12)(cid:12) d = λ dB dx (cid:12)(cid:12)(cid:12)(cid:12) d , (4)are satisfied. Eq. (2) is the London equation in each SClayer. Eq. (3) and (4) are the continuity condition of themagnetic field and the vector potential, respectively. Theabove formalism can be easily generalized to an n SC-layersystem ( n ≥ ).igure 2: The magnetic-field distribution in the SC nano-layer coating without insulator layer. A solid curve rep-resent the magnetic-field distribution given by Eq. (5) and(6). A dotted curve represents a naive exponential decayfor comparison with the correct curve.Solving the above equations, we obtain B ( x ) = B cosh d − xλ + λ λ sinh d − xλ cosh dλ + λ λ sinh dλ , (5) B ( x ) = B ( d ) e − x − dλ , (6)where B ≡ B (0) . The result is identical with themagnetic-field distribution of the multilayer SC with d I → . Fig. 2 shows how a magnetic field attenuates in the sys-tem. The slope of attenuation is suppressed in commonwith the multilayer SC with insulator layer. FORCES ACTING ON A VORTEX
Suppose there exist a vortex with the flux quantum φ =2 . × − Wb parallel to ˆz at x = x , where materialsof the SC layer and the SC substrate are assumed to be ex-treme type II SCs, namely, λ k ≫ ξ k ( k = 1 , ). This vor-tex feels three distinct forces f S ( x ) , f B ( x ) and f M ( x ) ,where f S is a force from the surface, f B is that from theboundary of two SCs, and f M is that from a Meissner cur-rent due to an applied magnetic-field. Vortex in the SC Layer ( ξ ≤ x ≤ d − ξ ) Let us consider the case that a vortex is located at theregion ξ ≤ x ≤ d − ξ . The forces, f S and f B , canbe evaluated by introducing image vortices to satisfy theboundary conditions (BCs): the zero current normal to thesurface and the continuity condition of the vector potentialat the boundary of two SCs. For simplicity we consideronly two dominant images, an image antivortex with flux φ at x = − x and an image vortex with flux φ ≡ λ − λ λ + λ φ (7)at x = 2 d − x . The former and the latter correspond toBCs at the surface and the boundary, respectively. Then we find f S ( x ) = − φ πµ λ x ˆx , (8) f B ( x ) = − φ φ πµ λ ( d − x ) ˆx , (9)where ˆx = (1 , , . The surface attracts the vortex andprevent the vortex penetration, and the boundary pushesthe vortex to the direction of the SC material with a largerpenetration depth, and thus prevent the vortex penetrationif λ > λ . The force, f M , is obtained by evaluating theMeissner-current density at the vortex position x = x (seeRef. [4, 5]) and is given by f M ( x ) = B φ µ λ sinh d − x λ + λ λ cosh d − x λ cosh dλ + λ λ sinh dλ ˆx , (10)which always pushes the vortex to the inside. Vortex in the SC Substrate ( d + ξ < x < ∞ ) For the case that a vortex is at d + ξ < x < ∞ , inorder to satisfy BCs at the surface and the boundary, weintroduce following two images: an image antivortex withflux φ ≡ λ λ + λ λ λ + λ φ (11)at x = − x and an image antivortex with flux φ at x =2 d − x . Then we find f S ( x ) = − φ φ πµ λ x ˆx , (12) f B ( x ) = − φ φ πµ λ ( x − d ) ˆx . (13)The surface and the boundary, if λ > λ , prevent the vor-tex penetration as the above. f M can be obtained in muchthe same way as before: f M ( x ) = B ( d ) φ µ λ e − x − dλ ˆx , (14)which pushes the vortex to the inside. VORTEX-PENETRATION FIELD
The vortex-penetration field, at which the Bean-Livingston barrier disappears, can be obtained by balanc-ing the forces acting on a vortex at the surface ( x = ξ ).Substituting x = ξ into Eq. (8), (9) and (10), and im-posing the condition f S ( ξ ) + f B ( ξ ) + f M ( ξ ) = , weobtain B v = φ πλ ξ cosh dλ + λ λ sinh dλ sinh dλ + λ λ cosh dλ , (15)where the contribution from | f B ( ξ ) | ( ≪ | f S ( ξ ) | ) is ne-glected. For a thin SC layer, the vortex-penetration field isenhanced up to B v | d ≪ λ ≃ ( φ / πλ ξ )( λ /λ ) . As ex-pected, the SC nano-layer coating model without insulatorlayer also enhance the vortex-penetration field as well asthe top SC layer of the multilayer SC (see Fig. 3).igure 3: The vortex-penetration field of the SC nano-layer coating without insulator layer as a function of SClayer thickness. LOWER CRITICAL MAGNETIC FIELD
The lower critical magnetic field B c is the appliedmagnetic-field at which a stable position of the vortexchanges from the surface to the inside of the SC. In otherwords, the applied magnetic-field at which the free-energydifference between the system with a vortex at the surfaceand that with a vortex at the stable position inside the SCvanishes. The free-energy difference between different vor-tex positions can be evaluated by integrating forces actingon the vortex. We show only a part of the results. Fig. 4 shows thefree energy difference inside the SC layer for the case thatzero or small magnetic field is applied. Since both the sur-face and the boundary pushes a vortex to the surface, thefree energy increases as a vortex approaches the boundary.Thus the boundary is not necessarily a stable position of thevortex in contrast to the insulator-side surface of the mul- For example, let us derive B c of the SC thin film with a penetrationdepth λ , a coherence length ξ and a thickness d ( ξ ≪ d ≪ λ ) . Theforces acting on a vortex at x from both the surfaces is approximatelygiven by contributions from two images, f S ≃ − ( φ / πµ λ )[1 /x − / ( d − x )] . Note that the second term, which can be neglected for acalculation of the vortex-penetration field [4, 5], is necessary to evaluate B c . The force due to the Meissner current [4, 5] is given by f M =( B φ /µ λ )[sinh( d/ λ − x /λ ) / cosh( d/ λ )] . The stable position ofthe vortex is x = d/ . Then the energy difference between the systemwith a vortex at the surface x = ξ and that with a vortex at x = d/ isgiven by ∆ G = − Z d ξ dx ( f S + f M ) ≃ φ πµ λ ln dξ − B φ µ d λ . The condition ∆ G = 0 yields B c ≃ φ πd ln dξ , which corresponds to the result given in Ref. [7]. It should be noted that the boundary can be a metastable positiondue to the energy gap at the boundary. For simplicity, let us considerthe case that the SC layer is replaced with a semi-infinite SC, namely,a system that consists of two semi-infinite SCs [8] with a boundary at x = d . In this case, the free energy at x = d − ξ and x = d + ξ are given by G = ǫ + ( φ φ / πµ λ ) ln κ and G = ǫ − ( φ φ / πµ λ ) ln κ , respectively, where ǫ k = ( φ / πµ λ k ) ln κ k Figure 4: The free energy difference (in an unit of g ≡ φ / πµ λ ) between a system with a vortex at the surfaceand that with a vortex inside the SC layer with thickness
30 nm .tilayer SC. To know the stable position and evaluate thelower critical magnetic field, the free energy calculationsof entire region including an energy gap at the boundaryare necessary [10].
SUMMARY
We studied the SC nano-layer coating without insulator. • The magnetic-field distribution is identical with thatof the multilayer SC with an insulator layer thickness d I → . • The boundary of two SCs introduces a force thatpushes a vortex to the direction of material with largerpenetration depth. • The vortex-penetration field is identical with that ofthe multilayer SC with d I → , because the differ-ence between the systems with and without insulatoris negligible at the surface. • The boundary is not necessarily a stable position ofthe vortex, in contrast to the insulator-side surface ofthe multilayer SC. To know the stable position andevaluate B c , the free energy calculations of entire re-gion including an energy gap at the boundary are nec-essary.The SC nano-layer coating without insulator might be apossible idea to enhance the vortex-penetration field with-out sacrificing B c . Results and detailed discussions on B c will be presented [10]. ( k = 1 , ) is the free energy of an isolated vortex in each SC. Thus theenergy gap at the boundary is given by ∆ G gap = G − G = φ πµ ( λ + λ ) ln κ κ , which can make a valley of the free energy. The insulator-side surface of the multilayer SC attracts a vortex, andthe free energy decreases as a vortex approaches the insulator-side sur-face. As a result the insulator-side surface becomes a stable position ofthe vortex even if the applied magnetic-field is very small [9].
EFERENCES [1] A. Gurevich, Appl. Phys. Lett. , 012511 (2006).[2] T. Kubo, Y. Iwashita, and T. Saeki, Appl. Phys. Lett. ,032603 (2014).[3] T. Kubo, Y. Iwashita, and T. Saeki, in Proceedings ofIPAC’13, Shanghai, China (2013), p. 2343, WEPWO014.[4] T. Kubo, Y. Iwashita, and T. Saeki, in
Proceedings ofSRF2013, Paris, France (2013), p. 427, TUP007.[5] T. Kubo, Y. Iwashita, and T. Saeki, in