Unveiling London Moment in Superconductors
Explore the intricacies of the London moment in rotating superconductors, discussing its formation, observations, and implications. The dynamics of superfluid and normal components, along with the Generalized London equation, shed light on this phenomenon. Discover the Meissner effect and kinetic energy considerations in this fascinating study.
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THE LONDON MOMENT OF A ROTATING SUPERCONDUCTOR: SOME UNSUSPECTED SUBTLETIES Tony Leggett Department of Physics University of Illinois a tUrbana-Champaign Many-body/ASC-CENS Workshop in honor of Willi Zwerger Technische Universit t M nchen 6 Sept. 2019 Acknowledgement for extended discussions: J. G. Hirsch (very much work in progress !)
ZF 2 What is the London moment? ? ? or R If cylinder is superconducting and is rotated, then in the bulk of the sample a spontaneous magnetic field ?0will be generated: ?0= 2? ? (so since ? < 0,?0 ?) ? Why? Electrons (both normal and superfluid, i.e. Cooper pairs) would like to rotate with lattice. For normal component, ??= ? ? without problem. For superfluid, ??= ?? ? +? /2?? ? So need? ? = ? ? ? ? ? = ? ? ? = 2? ? ?0 ? ? However, ? ? must 0 as ? ? falls off as ? ? , ? ? = 1 ? ? ?/??0, ?2? 1/2 ? London penetration depth = ?0?? London moment Meissner effect as viewed from rotating frame. Apparently most recent experiment: Hendricks et al., JLTP 4, 209 (1971) on Sn (type-I) Field is very small (3000 Hz 10??)
When is London moment (not) observed? ZF 3 Solid hollow spin-up yes yes Meissner yes no* *that is, magnetometer in hole observes ? 0, consistent with ? = ?0 in bulk. kinetic energy Statics of London moment well understood: need to minimize KE as observed from rotating frame, leading to generalized London equation ? ? = ?? ? ?? ? cylindrical polars consistently with Maxwell s equation ?2? ? =1 ? ?? ??? ?? 1? ? = ?0 ? and subject to ? ? = ? ? ? ?=?= ?. This gives 1 ? ? ?? ??? ?? = ? 2? ? = ? 2? ??? with the approximate solution ? ? ? ? =1 2?0? + ? ? ,? < ? =1 2?0 ?? ?2? +0 ? > ? where ? ? = 2??exp ? ? /? 2??exp ?/? R-r
ZF 4 ? ? ? ?0? ? ? ? ? = 0 ? = ? (surface) 1 2?0 ?2? ?? Note: total energy in rotating frame = magnetic energy + ?.?. ?? =1 2? ?? ? ?2??, which relative to its ? ?,? = 1 ? = 0 value 1 1?0 2? 2? 4?0 2?0?0 is note this is an incredibly tiny fraction (~10-12) of the characteristic thermal energy Now, question (Hirsch): what is the kinetics of the formation of the London moment (in particular under Meissner conditions)? can discuss (a) initial nucleation (b) development with change of temperature 1/2 ??? ?2 ?0 ? = ? ? ?
Take ? ?? to be correction to ? 0 value 1 and ?? in rotating frame, then in equilibrium ??? = 0, ??, ? = ?? ? /? ??? = ???2? ? /? 2??0, measure ?? ZF 5 Quite generally*, at London level, Maxwell: ?2? ?? ??2 = ?0? ?? Generalized London: ? ?? = ???2? ?? /? + ???? dynamics of superfluid controlled entirely by that of normal compontent. Consider step in temperature ?old ?new which conserves total local current: ??? = 0 + = ??? = 0 + ??, ??new ???:? = 0 + = ?????,? = 0 ??old = 2?? ? ? exp 2/?old thereafter ?? ??new= ind.of ?.:???new ?0??new?2/? 1/2. Total work Wtot done on normal component by Faraday electric field ?? arising from ??/??: Wtot= ?? ?? ?? ???? ?+ 0 where from Maxwell and gen*. London ?2 ?? ??2 *neglecting displacement-current terms and terms 0 ?/? ??? ?? ?? total rate of change of ???? 2 ?new ?? = ?0
ZF 6 After several integrations by parts: Wtot= ? ? = ? ? = 0 + + other ? ? = ? ? = 0 + independently of details of normal-component dissipation mechanisms. ( everything consistent). Prima facie simplest ansatz: local relaxation, ?? ??? ?? relax= ???? /?0 then ???? = 2??oldexp 2/?old exp ?/?old+2??newexp ?/?new exp ?/?new What are ?old, ?new? ?2 ??2 ??? ?? ?? ?new ?? 2 ?? = total however relax+???2 ??? ?? ??? ?? ?? ?? ?? total= ? so ?2 ??2 ??? ?? ?? ?0 ?? 2 ?tot?2?0/? 1/2 2 ?? = ?0 relax hence for any ?? ~? ?/? ? 2 ?new ? 2 ?0 2 ? ? = ?0 2 OK for old term: what about new one?
ZF 7 winding no. Back to question (a) (nucleation): Recall: ???? = ?? ?? /?+? /2?? So far, have set ? = 0. But is this right? What determines ? ? Plausible conjecture: under Meissner conditions, ?is determined by fitting to rotation of cylinder at point where superconductivity first nucleates. i.e. if nucleation occurs at radius ?, then ? = nearest integer to 2???2/ . So, choice ? = 0 corresponds to nucleation at origin. But coldest point is ? = ? (surface)! What if nucleation were to occur on surface + propagate inwards? Then since winding no. conserved ??? = ?? ? /?+ ? /2?? for all ? singularity at origin. For hollow cylinder, can accommodate at cost of extra KE metastable state But what about solid case? Conjecture: normal core forms near ? = 0. ( pseudo-London state) Is there an analogy for Meissner effect?
ZF 8 A further question (Hirsch): is ? ? transition in rotating cylinder 1st or 2nd order? Within London approxn, if nucleation occurs at origin, should be 2nd order. However, superconductor used in experiments ?? is type-I! need to go beyond London (at least to GL level) Question: independently of rotation, in presence of thermal gradient, is ? ? transition 1st or 2nd order? ? ? = 0 Definitely work in progress
