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Europhysi s Letters PREPRINT 5 0 Ma ros opi Aharonov(cid:21)Bohm E(cid:27)e t in Type-I Super ondu tors 0 2 Catrin Ellenberger, Florian Gebhard, Wolfgang Bestgen n a Fa hberei h Physik, Philipps(cid:21)Universität Marburg, D(cid:21)35032 Marburg, Germany J 6 2 PACS. 73.23.-b (cid:21) Ele troni transport in mesos opi systems. PACS. 74.25.Sv (cid:21) Criti al urrents. ] n PACS. 74.25.Op (cid:21) Mixed states, riti al (cid:28)elds, and surfa e sheaths. o PACS. 74.40.+k (cid:21) Flu tuations (noise, haos, nonequilibrium super ondu tivity, lo alization, c et .). - r p u s . Abstra t. (cid:21) In type-I super ondu ting ylinders bulk super ondu tivity is destroyed above t a the (cid:28)rst riti al urrent. Below the se ond riti al urrent the `type-I mixed state' displays m (cid:29)u tuation super ondu tivity whi h ontributes to the total urrent. A magneti (cid:29)ux on the - axis of the ylinder an hange the se ond riti al urrent by as mu h as 50 per ent so that d half a (cid:29)ux quantum an swit h the ylinder from normal ondu tion to super ondu tivity: n the Aharonov(cid:21)Bohm e(cid:27)e t manifests itself in ma ros opi ally large resistan e hanges of the o ylinder. c [ 1 v 4 Introdu tion. (cid:21) The super ondu ting ondensate an be used as a `quantum ampli(cid:28)er'. 2 6 Forexample,mi ros opi e(cid:27)e tssu hasquantuminterferen eofele tronwavesbe omema ro- 1 s opi ally measurable as (cid:29)ux quantization. In this work the ma ros opi ondensate is used 0 to make the Aharonov(cid:21)Bohme(cid:27)e t prominent on ma ros opi s ales. R J 5 Considerawireofradius witha urrent paralleltothewireaxis. Thesuper ondu tiv- 0 J / ity of a type-I super ondu torbreaksdown when the urrentex eeds the Silsbee urrent 1. t The Silsbee urrent is determined from the ondition that the magneti (cid:28)eld at the surfa e a H J = c0RH /2 c0 m of the wire is equal to the upper riti al (cid:28)eld , 1 , where is the speed of J >J - light. For 1, the so- alledintermediate state forms in the wire with alternatindg normal d and super ondu ting domains [1℄. The diameter of the super ondu ting domains shrinks d n and (cid:28)nally the intermediate state ollapses when be omes of the order of the oheren e o ξ length . However,super ondu tivitystill remainsin theformofsmalltemporal(cid:29)u tuations. c : In this `mixed state' for type-I super ondu tors, super ondu ting (cid:29)u tuations and ele tri al v and magneti (cid:28)elds o-exist. The mixed state was (cid:28)rst predi ted by L.D. Landau in 1938 [2℄ i X as asurfa e statefor hollowsuper ondu ting ylinders. Its existen ewaslater veri(cid:28)edexper- r imentally by I.L. Landau and Yu.V. Sharvin [3℄. As later shown by Andreev and Bestgen [4℄ a the (cid:29)u tuation indu ed super ondu tivity is destroyed when the thi kness of the (cid:29)u tuation ξ layer be omes smaller than the oheren e length , and super ondu tivity is ultimately de- J stroyed at the se ond riti al urrent 2 whi h was (cid:28)rst al ulated by Andreev [5℄. An H externally applied magneti (cid:28)eld a parallel to the wire axis ontributes to the suppression J J of the super ondu ting phase and redu es 1 and 2 a ordingly [6,7℄. (cid:13) EDPS ien es 2 EUROPHYSICSLETTERS In this work we investigate the stability of the mixed state in type-I super ondu tors for R ≪R H a hollow ylinder of inner radius i in a magneti (cid:28)eld a with an additional magneti Φ (cid:29)ux line of strength a on the ylinder axis. We showJth(Φat)the Aharonov(cid:21)BohmΦ e/(cid:27)Φe0 t leads to strong os illations of the se ond riti al urrent 2 a as a fun tion of a where Φ0 =hc0/(2e) isthe(cid:29)uxquantum. Thee(cid:27)e tisparti ularlymarkedforthe`one-dimensional' R ≪ ξ J (Φ ) mixed state for whi h i . In this ase, 2 a anJvar<y Jby(aΦlm0/o2s)t <50Jpe<r Jent(.0)If the applied urrent through the wire is (cid:28)xed at some value 1 2 2 , the Φ modulation of between integer (cid:29)ux quanta will lead to a periodi onset and breakdown of J Φ Ω V the super ondu ting urrent s as a fun tion of . The resistan e at applied voltage be omes V V Jz Ω= = ≈Ω 1− . s J J +Jz n J (1) n s (cid:18) (cid:19) Therefore,theAharonov(cid:21)Bohme(cid:27)e tleadstoperiodi (cid:29)u tuationsin theresistan eoftype-I super ondu tors in the mixed state. J ≃ J Model and super ondu ting urrent. (cid:21) In the mixed state with 2, the dynami s of the super ondu ting (cid:29)u tuations at the inner ylinder surfa e is des ribed by the time- dependent, linear Ginzburg(cid:21)Landau equations with Langevin for es. In dimensionless ylin- ρ,φ,z dri al oordinates they read [7(cid:21)9℄ ∂Ψ −Ψ+LΨ=f(ρ,ϕ,z,t) ∂t , (2) with the di(cid:27)erential operator 2 2 1 ∂ ∂ 1 ∂ iγ iδ ∂ iβ 2 L=− ρ − − ρ− − + ρ +iǫt ρ∂ρ ∂ρ ρ∂ρ 2 ρ ∂z 2 (3) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) and the Gaussian (cid:29)u tuations 1 hf(ρ,ϕ,z,t)f∗(ρ′,ϕ′,z′,t′)i= δ(ρ−ρ′)δ(ϕ−ϕ′)δ(z−z′)δ(t−t′) ρ (4) ǫ β γ δ The dimensionless parameters , , and are given by 2eξ 2eH ξ2 H ǫ= E γ = a = a ¯hν , ¯hc0 H , 4eξ3J eΦ Φ 2 β = ¯h 2R2 , δ = ¯hc0aπ = Φa0 . (5) 0 ξ = ξ(T) ν = 8k (T −T)/(π¯h) e is the oheren e length, B is the relaxation frequen y, and m E and aretheele tron hargeandmass,respe tively. istheele tri al(cid:28)eldwhi hdrivesthe J = J +Jz z ǫto≪tal1 urrent n s in -dire tion. For type-I super ondu torsJwza≪s sJhown in [4℄ that . Using the linearized Ginzburg(cid:21)Landau equations implies that s . Thegeneralexpressionforthetotallongitudinal(cid:29)u tuation urrentis al ulatedalongthe lines of [4,9℄ and gives the result ∞ 2eTk ∂λ nmk Jzs = πǫ¯hB dk ∂k Fnm(k), (6) nm Z X−∞ Catrin Ellenberger et al.: Ma ros opi Aharonov(cid:21)Bohm Effe t 3 with ∞ k 2 Fnm(k)= dτ exp τ − λnmk′ dk′ ǫ   . (7) Z0 k−Zτ    λ L k = k +ǫt k nmk z z is the lowest eigenvalue of as a fun tion of where is the wavenumber z λ nmk ve tor in -dire tion. The spe trum Rn=0 ,mank b∼eρo|bmt|aei−nαedρ2with the help of a variψational method, whi h starts from the ansatz for the radial part of . We n=0 α may restri t ourselves to the dominant ontribution, [10℄; is a variational parameter whi hdeterminestheradialextensionofthesuper ondu tingregion[4,9℄. Fromthevariation prin iple we then (cid:28)nd for λm,km =λo0p,mt,k β2 λ =−(m−δ)γ+4α(|m−δ|+1)− (|m−δ|+1), mkm 16α2 (8) with γ2+4k β β2(|m−δ|+2) m 2− − =0 8α2 8α3 , (9) −β(|m−δ|+1) k = m 4α . (10) The riti al urrent an be evaluated analyti ally in the vi inity of the riti al urve [8℄ Jz but for our further analysis expli it expressions for s are not required. We merely state that the experimental parameters an be tuned in su h a way that the (cid:29)u tuation urrent is severalper entofthe total urrent,i.e., the resistan e(cid:29)u tuationsa ordingto (1) shouldbe measurable. J (H ,Φ ) Se ond riti al urrent. (cid:21) Inourtheory, the riti al urrent 2 a a abovewhi hthe super ondu ting (cid:29)u tuations vanish is determined by the ondition [λ (β,γ,δ)]=1. Mmin m,km (11) λ If the minimum over all m,km was noti eably smaller than unity the orresponding super- urrentwouldbe omesolargethat theuseofalinearizedGinzburg(cid:21)Landauequationswould λ not have been justi(cid:28)ed. If the minimum over all m,km was substantially larger than unity Jz s wouldbe tiny. Our theory appliesto the transition regionwhere the super urrentis small but measurable. J Solving (11) for 2 leads to the se ond riti al urrent as a fun tion of the external magneti (cid:28)eld and applied (cid:29)ux in the interior of the ylinder. In SI units we (cid:28)nd: H Φ c2R2¯h 0 J 2 Ha Φ0 = β 2 4eξ3 (cid:18) 2 (cid:19) 3 c2R2¯h H 2 1 0 3 = eξ3 "A 1+s1−(cid:18)H a2(cid:19) 48A2 (12) 2 2  1/2 H A H 1 −(cid:18)H a2(cid:19) 161+s1−(cid:18)H a2(cid:19) 48A2# ,   where 1+(m−δ)γ A= 6(|m−δ|+1) . (13) 4 EUROPHYSICSLETTERS δ m J For a given (cid:29)ux , in (13) is the integer whi h gives the minimal 2 [8℄. Figure 1 gives a J γ δ three-dimensional plot of 2 as a fun tion of (external (cid:28)eld) and ( entral (cid:29)ux). β (γ,δ) R ≪ξ Figure 1 (cid:21) The normalizedse ond riti al urrent 2 for i . The se ond riti al urrent an be almost halved by the appli ation of a suitable ve - tor potential even if no magneti (cid:28)eld is present. This is a ma ros opi realization of the Aharonov-Bohm e(cid:27)e t. Con lusions. (cid:21) In this work we have studied the super urrent in a hollow ylinder of type-I super ondu ting material in the presen e of an external magneti (cid:28)eld and a (cid:29)ux line onthe ylinderaxis. We haveshownthat the se ond riti al urrent, abovewhi h(cid:29)u tuation super ondu tivity breaks down, periodi ally hanges as a fun tion of the applied (cid:29)ux, by as mu h as 50 per ent in the absen e of an external (cid:28)eld. These Aharonov(cid:21)Bohm os illations lead to a periodi hange of the resistan e of the ylinder, see (1). The latter are measurable be ause the urrents due to the super ondu ting (cid:29)u tuations are sizable, of the order of a few per ent of the total urrent. Therefore, a suitably tuned experiment with type-I super ondu ting ylindersshouldbeabletoverifytheAharonov(cid:21)Bohme(cid:27)e tbyma ros opi resistan e measurement. ∗∗∗ We thank F. Rothen and M. Pae h for useful dis ussions. Referen es [1℄ London F., Super(cid:29)uids I.(Wiley, New York)1950. [2℄ LandauL.D.,private ommuni ationwithD.Shoenberg;see: D.ShoenbergSuper ondu tivity. Cambridge: Cambridge UniversityPress, 1938; page 59. Catrin Ellenberger et al.: Ma ros opi Aharonov(cid:21)Bohm Effe t 5 [3℄ Landau I.L.andSharvin Yu.V.,Pis'maZh.Eksp.Teor.Fiz.,10(1969)192andJETPLett., 10 (1969) 121. [4℄ Andreev A.F. and Bestgen W., Zh. Eksp. Teor. Fiz., 10 (1973) 453 and Sov. Phys. JETP, 37 (1973) 942. [5℄ Andreev A.F., Pis'ma Zh. Eksp. Teor. Fiz., 10 (1969) 453 and JETP Lett., 10 (1969) 291. [6℄ Bestgen W., Zh. Eksp. Teor. Fiz.,76 (1979) 566 and Sov. Phys. JETP, 49 (1979) 285. [7℄ Bestgen W., Sol. State Commun.,36 (1980) 441. [8℄ Ellenberger C., Ph.D. Thesis, Marburg 2001. [9℄ Bestgen W., Z. Phys., 269 (1974) 73. [10℄ Bestgen W. and Rothen F., Low. T. Phys., 77 (1989) 257.