Altering magnetic response of superconductors by rotation
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Altering magnetic response of superconductors by rotation
Jun-Ping Wang
Department of Physics, Yantai University, Yantai, P.R. China (Dated: August 24, 2020)It is generally believed that, at a certain temperature below the critical one, magnetic responseof a superconductor (SC) is determined solely by its intrinsic properties. Here we show that themechanical rotation of a SC can easily change the values of the critical fields at which the super-conductivity is destroyed (type-1 SC) or the vortices penetrate into (exit from) the material (type-2SC). This is due to a superposition of the Meissner current induced by the external field, and thespontaneous current on the surface of the SC induced by the mechanical rotation. As a result, thecritical fields of SCs can be increased or decreased, depending on the relative orientation of rotationand the external field.
Introduction . Superconductivity is characterized notonly by perfect conductivity, but also by unusual mag-netic response [1, 2]. In the presence of an external mag-netic field, a supercurrnet is generated near the surface ofthe superconductor (SC), which prevents the field frompenetrating the interior of it. The external field can de-stroy the superconductivity as it reaches the thermody-namic critical value H c for a type-1 SC. To a type-2 SC,magnetic field penetrate into the material in the form ofquantum vortices as the field equals to the lower criticalvalue H c , and vortices exit from the SC accompanied bythe destruction of superconductivity as the external fieldincreases to the upper critical value H c .On the other hand, a spontaneous magnetic field canbe produced by a rotating SC [3]. The physical pictureis that, as the most of superconducting electrons followthe movement of the rotating body exactly, there aresuperconducting electrons lag behind near the surface.Those lagged electrons generates a weak current. Thiscurrent produces a magnetic field, which is homogeneousin the interior of the SC, B L = − mce Ω = 2 cγq Ω . (1) B L is the London field, m and e are the mass and chargeof the electron, c is the speed of light, Ω is the angularvelocity of rotation. Here the absolute value of electriccharge q and inverse mass γ of two electrons are intro-duced. The induced current provides a contribution tothe effective magnetic moment of the SC - the ”LondonMoment”. The magnetic field induced by the rotation ofthe SC has been verified in experiments, both in conven-tional and high temperature SCs [4–8].An interesting issue is to investigate the magnetic re-sponse of a rotating SC. As we shown in this work,magnetic response of a rotating SC is quite differentfrom that of a static one. Effect of superposition ofrotation-induced supercurrent (London current) and thecurrent induced by the external field (Meissner current)can change the magnetic response of a SC dramatically.Depending on the relative orientation of rotation and theexternal field, the critical fields of a SC can be increased or decreased. The physical effects predicted here are ex-perimentally observable, especially near the transitiontemperature T c , at which the critical fields are compa-rable with the London field (1). A simple example . Let us consider a superconductingsphere which rotates with constant angular velocity Ω inan uniform external magnetic field H ext , and the direc-tion of rotation is parallel to the external field, H ext || Ω .To determine the distributions of the supercurrent andthe magnetic field, we use the London theory and startwith the following equations: ∇ × v = γqc h , (2) ∇ × h = − πnqc ( v − Ω × r ) . (3)Here v is the velocity of superconducting electrons, h isthe magnetic field, n is the density of the electron pairs.Outside the sphere, ∇ · h = ∇ × h = 0 and h ( r → ∞ ) → H ext . In the case considered here, it is reasonable toassume that the magnetic field h outside the sphere canbe written as h r = ( H ext + 2 Mr ) cos θ,h θ = ( − H ext + Mr ) sin θ,h φ = 0 . (4)Here M is a constant, and can be regarded as the inducedmagnetic moment of the sphere. Inside the sphere, theequation for velocity of the superconducting electrons v can be deduced from (2) and (3): ∇ × ∇ × ( v − Ω × r ) = − β ( v − Ω × r ) . (5)Here β = 4 πnγq /c , and β − is the London penetra-tion depth. We have used the relations ∇ × ( Ω × r ) =2 Ω , ∇ × ∇ × ( Ω × r ) = 0. Assuming that the velocityof the superconducting electrons has only the azimuthalcomponent v = v e φ , the equation (5) can be solved: v = (cid:20) Ω r + Ar (sinh βr − βr cosh βr ) (cid:21) sin θ e φ . (6) A is a constant to be determined. Substituting equation(6) into equation (2), then the magnetic field inside thesphere is obtained, h r = cγq (cid:20)
2Ω + 2 Ar (sinh βr − βr cosh βr ) (cid:21) cos θ,h θ = cγq (cid:20) −
2Ω + Ar ((1 + β r ) sinh βr − βr cosh βr ) (cid:21) sin θ,h φ = 0 . (7)With the continuity of magnetic field at the boundary ofthe sphere, the constants M and A can be determined, A = −
32 ( H ext − B L ) γqRcβ sinh βR , (8) M = − R H ext − B L ) (cid:20) β R (1 − βR coth βR ) (cid:21) . (9) B L = 2 c Ω /γq , R is the radius of the sphere. Thus thedistributions of the magnetic field h and the velocity field v of the superconducting electrons in the whole space aregiven.The supercurrent J can be extracted from (6) and (8)by taking the rotation of the sphere into consideration, J = − nq ( v − Ω × r ). The current is confined near thesurface of the SC with a depth β − , and equals to thesum of the Meissner current J M induced by the externalfield H ext , and the London current J L induced by themechanical rotation of the sphere with angular velocity Ω : J = J M + J L , (10) J M = − nq A r (sinh βr − βr cosh βr ) sin θ e φ , (11) J L = − nq A r (sinh βr − βr cosh βr ) sin θ e φ . (12)Here A = − H ext γqRcβ sinh βR , (13) A = 32 B L γqRcβ sinh βR . (14)Note that the directions of these two current are opposite.Except for a layer of the depth β − near the surface of thesphere, the magnetic field inside the sphere (7) equals tothe field induced by the rotation of the superconductingsphere with angular velocity Ω . The magnetic field out-side the sphere (4) is the superposition of external field H ext and the field generated by the induced magneticmoment M of the sphere. It is clear from (9) that thisinduced magnetic moment of the sphere is the result of (d) (e)(a) (b) H ext = 2 B L H ext || - (c)(f) H ext = 0.5 B L H ext = B L H ext || FIG. 1: A rotating superconducting sphere in the externalfield. Except for a layer of the London depth near the surface,the magnetic field inside the sphere is uniform B L = 2 c Ω /γq (London field). The magnetic field outside the sphere is thesuperposition of the external field and the field generated bythe induce magnetic moment M of the sphere. The inducemagnetic moment M is the result of the combined effect ofexternal field and the rotation. From left to right: the ex-ternal field increases from H ext = 0 . B L , B L to 2 B L . Thetop row shows the cases that the field is parallel to the an-gular velocity. (a) Magnetic field lines converge toward andpenetrate through the sphere. (b) The Meissner current iseliminated by the London current and the total supercurrentvanishes. The magnetic field is uniform in the whole space.(c) Magnetic field lines tend to spread out around the sphere.The bottom row shows the cases that the field is antiparallelto the angular velocity. In all three cases, (d)-(f), magneticfield lines are repelled by the sphere strongly comparing to thestatic cases. Note that the direction of the field reverses at theequator of the sphere. Parameters used: London penetrationdepth β − = 10 − cm , radius of the sphere R = 1 cm , angularvelocity Ω = 10 sec − , London field B L = 1 . × − Gauss . the combined effect of external field and the rotation: M = M M + M L , (15) M M = − R H ext (cid:20) β R (1 − βR coth βR ) (cid:21) , (16) M L = R B L (cid:20) β R (1 − βR coth βR ) (cid:21) . (17) M is the total moment, M M , M L are the magnetic mo-ments generated by the Meissner current J M and theLondon current J L respectively. In figure (1) we illus-trate the distributions of the magnetic field for a rotatingsuperconducting sphere in the external field.The effect of mechanical rotation of the sphere on itsmagnetic response originates from the London current J L . It can be verified from (12) and (14) that the Londoncurrent J L equals to the current generated by a fictitiousexternal field H fic = − B L = − c Ω /γq . Except for alayer of the penetration depth β − near the surface, themagnetic field generated by the London current inside thesphere, B L , is uniform. Outside the sphere, the Londoncurrent generates a magnetic field equals to that of asphere with London moment M L .Due to the occurrence of the London current J L , onemay speculate that a rotating superconducting spherewith angular velocity Ω in the external field H ext isequivalent to a static sample in the external field H ext + H fic . Here H fic = − B L is the fictitious field to producethe London current J L . We will show in the next sectionthat this is indeed the case if the demagnetization factor N of the sample is zero. To a type-1 superconductingsphere with N = 1 /
3, it is equivalent to a static samplein the external field H ′ and Z d r ( g s ( H ext , Ω ) − ( g n ( H ext )) = Z d r ( g s ( H ′ ) − g n ( H ′ )) . (18) g s and g n are Gibbs energy densities of the supercon-ducting state and normal state respectively. Be aware ofthe perfect diamagnetism in a static sample, it can beproved easily that H ′ = q H ext + 2 B L − H ext B L . (19)In most cases H ′ < H ext , since B L is small and H ext > B L /
3. In the above example, the critical field for a type-1 superconducting sphere changes as H ′ = H c . As theexternal field equals to H ext = 2( H c + B L / /
3, it canbe verified from (4) and (7) that the field at the equatorof the sphere equals to the critical field H c . This meansthat, the small part of the sphere of a type-1 SC near theequator tends to be in the normal state. As a comparison,the field near the equator of the sphere reaches the criticalvalue H c when the external field increases to 2 H c / General consideration . To study general aspects ofthe magnetic response of a rotating SC, we consider theminimal model of the superconductivity in the rotatingframe f s = γn ~ (cid:20) ∇ θ + q ~ c ( A − cγq Ω × r ) (cid:21) − H c π + 18 π ( ∇× A ) . (20) f s is the free energy density, θ is the phase of the orderparameter, A is the vector potential. Here we take thedensity n as a constant and omit the free energy densityof the normal state in the absence of the magnetic field.The first term in equation (20) is the kinetic energy, the second term is the condensation energy, and the fieldenergy is presented in the last term. The free energydensity (20) can be rewritten as follows: f s = γn ~ ∇ θ + q ~ c A eff ) − H c π + 18 π [( − B L ) −∇× A eff ] . (21)Here B L = c ∇ × ( Ω × r ) /γq = 2 c Ω /γq is the Londonfield, A eff = A − cγq Ω × r (22)is the effective vector potential. The energy density (21)is equal to the Gibbs energy density difference for a SCbetween the superconducting state and the normal state,both under the external field H fic = − B L . Based onthis equivalence, the rotating SC (20) and (21) can beregarded as a static SC under the fictitious field H fic ,and some insightful results have been obtained in [9].Taking into account the external field H ext , the Gibbsenergy density of a rotating SC is g s = f s − π H ext · [( ∇ × A eff ) + B L ] . (23)The demagnetization factor here is set to zero, N = 0.It can be realized in, for example, a long cylinder in anaxially applied magnetic field. Then H is uniform inwhole space. The Gibbs energy density of the normalstate under the external field H ext is g n = H ext π − π H ext · H ext = − H ext π . (24)The Gibbs energy density difference of the rotating SCbetween the superconducting state and the normal state,both under the external field H ext , can be obtained from(21), (23) and (24): g s − g n = γn ~ ∇ θ + q ~ c A eff ) − H c π + 18 π [( H ext + ( − B L )) − ( ∇ × A eff )] . (25)The equations which determine the motion of supercur-rent and the distribution of the magnetic field can nowbe obtained by minimizing the energy difference, δ Z d r ( g s − g n ) = 0 . (26)It is clear from (25) and (26) that a rotating super-conducting SC with angular velocity Ω in the exter-nal field H ext , and a static SC in the external field H ext + H fic = H ext − B L , are physically equivalent.Equations (25) and (26) describe the underly physics ofa rotating SC in the external field. The effect of rotationof the SC on its magnetic response is reflected in thefictitious field H fic = − B L . The critical fields, at which TABLE I: The comparison of supercurrents, distributions of the field and critical fields in three casesSC in an external field Rotating SC Rotating SC in an external fieldVariables H ext Ω H ext , Ω Supercurrents J M J L J M + J L Magnetic field inside the SC B = 0 B L = 2 c Ω /γq B L = 2 c Ω /γq Magnetic field outside the SC H ext + field generated by J M field generated by J L H ext + field generated by J M , J L Critical field for type-1 SC H c = H c ( T ) H c = H c (Ω , T ) H c = H c ( T ) ± c Ω /γq Critical fields for type-2 SC H ci ( i = 1 ,
2) = H ci ( T ) H ci ( i = 1 ,
2) = H ci (Ω , T ) H ci ( i = 1 ,
2) = H ci ( T ) ± c Ω /γq superconducting-to-normal phase occurs in a type-1 SC,or the vortex phase transitions occur in a type-2 SC,can be increased or decreased, depending on the relativeorientation of the external field and the angular velocityof the rotation of the SC: H c → H c − H fic = H c + B L , (27) H ci → H ci − H fic = H ci + B L , ( i = 1 , . (28)Vortex physics in a rotating type-2 SC under externalfield is different from that in a static one. If the externalfield is parallel to the angular velocity of the rotation,both of the lower critical field and the upper critical fieldare increased the amount by B L . As the external fieldequals to H c + B L , vortices start to penetrate into theSC. Since there exists the London field B L inside theSC, the flux through a vortex is the sum of the quantumflux Φ = hc/q and the flux from the London field, Φ =Φ + B L ∆ S , where h is the Planck constant, ∆ S is theeffective area of a vortex. With increasing field, vortexlattice form and distance between vortices decreases. Thevortices exit from the SC as the external field increasesto H c + B L .If the external field is antiparallel to the angular veloc-ity of the rotation, both the lower and the upper criticalfield decreased the amount by B L . As the external fieldequals to H c − B L , vortices penetrate into the SC. Thequantum flux carried by vortices here is antiparallel tothe London field inside the SC. As a consequence, thetotal flux inside the SC is decreased the amount by theinteger multiples of quantum flux Φ . With increasingfield, invasion of magnetic field in the form of vortices willeliminate the London field and there is no flux inside theSC. As the external field increases further, magnetic fluxinside the SC reverse and parallel to the external field.Vortices exit from the SC as the external field equals to H c − B L .Equations (20) and (25) are derived in the rotatingframe. And it can be verified that the start equations(3) and (5), which are used to study the rotating super-conducting sphere in previous section, can be deducedfrom these equations. Choice of the reference frame doesnot change the conclusions about the magnetic responseof the SC. Of course, all physical observable quantitiesshould be translated into the experimental static frame.In the present case, the physical picture in the experi-mental static frame and that in the rotating frame fixed on the SC is different in the supercurrent velocity andpossible global rotation of the vortex lattice in a type-2SC. Discussion and Conclusion . London field generatedby the mechanical rotation of the SC is generally weak.For the angular velocity Ω ≈ sec − , the London fieldis of the order of 10 − Gauss . To improve the observabil-ity of the effects predicted in the present work, the experi-ments can be performed near the transition temperature T c . Since the critical fields of the SCs decreases withthe increasing temperature, e.g., H c ( T ) ∝ − ( T /T c ) for type-1 SCs, it can be expected that the Londonfield becomes comparable with the critical fields H c , or H ci ( i = 1 ,
2) near the transition temperature T c . Thenthe effects predicted in this work could be significant andobservable.Here we concentrate on the combined effects of ex-ternal magnetic field and mechanical rotation on theSCs. If we set angular velocity of the rotation to zero,Ω →
0, classical magnetic response of SCs is recovered.On the other hand, in the absence of the external field, H ext = 0, rotational response of the SCs is also nontriv-ial [9]. It was shown that a type-1 SC experiences thesuperconducting-to-normal phase transition at a criticalrotation frequency. To a type-2 SC, there exist two criti-cal rotational frequencies at which the vortex phase tran-sitions occur. In table (I) the comparison of these threecases is presented.Finally, validity of the free energy expression in therotating frame (20) should be addressed. In (20), onlythe relative velocity difference between superconductingelectrons and the crystal lattice of positively charged ionsin the rotating frame is taken into account. In fact, in-fluence of fictitious forces in the rotating frame on thesystem should also be considered. Here we discuss thepossible influence of fictitious forces in an axisymmetricSC, in which the velocity of the superconducting elec-trons has only the azimuthal component. There are threekinds of fictitious forces. Two of them, the Coriolis force ∝ v ′ × Ω and the centrifugal force ∝ − Ω × Ω × r ,contribute nothing to the energy of the superconductingelectrons. v ′ is the velocity of superconducting electronsin the rotating frame. The third force, the Euler force ∝ r × d Ω /dt can accelerate or decelerate the supercon-ducting electrons, depending on the angular accelerationof the rotation, and does contribute to the energy (20).To guarantee the validity of the free energy (20), one mayadjust the angular velocity of the rotating SC slowly toensure d Ω /dt →
0. A more reliable method to avoid theeffect of the Euler force is to set the sample into rotationbefore cooling it into superconductive state.In conclusion, we investigate the magnetic response ofthe SCs under rotation. Depending on the relative orien-tation of rotation and the external field, critical fields ofthe SCs, at which the superconducting-to-normal phasetransition (type-1 SC) or the vortex phase transitions(type-2 SC) occur, can be increased or decreased. Theresults in the present work show that mechanical rotationhas influence on the quantum phase transitions drived bymagnetic field in the SCs.
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