Condensed Matter Physics 2016 Lectures 29/11, 2/1: Superconductivity. References: Ashcroft & Mermin, 34 Taylor & Heinonen, 6.5, , 7.

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1 Condened Matter Phyi 6 Leture 9/ /: Suerondutivity. Attrative eletron-eletron interation. 3 year of uerondutivity 3. BCS theory 4. Ginzburg-Landau theory 5. Meooi uerondutivity 6. Joehon effet Referene: Ahroft & Mermin 34 Taylor & Heinonen blaboard

2 9: Diovery of uerondutivity Merury ha aed into a new tate whih on aount of it extraordinary eletrial roertie may be alled the ueronduting tate Reitivity R = in Hg below T = 4. K Suerondutivity an alo be detroyed by a magneti field larger than the ritial field H (T )=H ()( (T/T ) ) (Kamerlingh Onne 94) H. Kamerlingh Onne (Nobel rize 93)

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4 933: Meiner-Ohenfeld effet A uerondutor i a erfet diamagnet: Exel magneti field from the interior. An alied magneti field indue a ueronduting urrent on the urfae of the uerondutor. Thi reate an indued magneti field whih omenate the alied field. W. Meiner

5 933: Meiner-Ohenfeld effet London enetration deth B(x) =B() ex( x/ L ) A uerondutor i a erfet diamagnet: Exel magneti field from the interior. An alied magneti field indue a ueronduting urrent on the urfae of the uerondutor. Thi reate an indued magneti field whih omenate the alied field. W. Meiner

6 933: Meiner-Ohenfeld effet London enetration deth B(x) =B() ex( x/ L ) tye I A uerondutor i a erfet diamagnet: Exel magneti field from the interior. An alied magneti field indue a ueronduting urrent on the urfae of the uerondutor. Thi reate an indued magneti field whih omenate the alied field. W. Meiner

7 Suerondutor = erfet ondutor and erfet diamagnet

8 935: Diovery of tye II uerondutor J. Rjabinin & L. Shubniov B B mixed (vortex) hae * * * The magneti field enetrate a uantized * flux line (eah arrying flux = h/e ). * Suerurrent around the flux line rodue vortie. Flux inning (or uantum loing) mae oible table magneti levitation. B normal hae Meiner hae T T L. Shubniov

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11 Early theorie of uerondutivity London & London (935): Model of the Meiner effet uing laial eletromagnetim Ginzburg & Landau (95): Marooi order arameter theory ( effetive field theory )

12 Exerimental hint: the iotoe effet (95) E. Maxwell 95 T / M B / M

13 BCS: Mirooi theory of uerondutivity (957) For their jointly develoed theory of uerondutivity alled BCS theory

14 Bai ( laial ) idea: An eletron moving through the lattie interat with the ion and reate a region of relative oitive harge. Thi attrat another eletron. Retarded effet The firt eletron an travel a ditane of ~ 4 Å before the eond eletron get attrated by the net loal oitive harge from the erturbed ion. Thu: negligible effet from the e-e reulion between the eletron (auming a reened Coulomb otential).

15 Bai ( laial ) idea: An eletron moving through the lattie interat with the ion and reate a region of relative oitive harge. Thi attrat another eletron. Retarded effet Quantization: Attrative eletron-eletron interation mediated by honon The firt eletron an travel a ditane of ~ 4 Å before the eond eletron get attrated by the net loal oitive harge from the erturbed ion. Thu: negligible effet from the e-e- reulion between the eletron (auming a reened Coulomb otential). ba to the blaboard

16 Summary from lat leture:

17 Fröhlih Hamiltonian H = H H e 3 Shrieffer-Wolff tranformation ee only uarti interation term 4 H = H e ; M ~ ( ) (~)

18 Fröhlih Hamiltonian H = H H e 3 Shrieffer-Wolff tranformation ee only uarti interation term 4 H = H e M ; ~ ( ) (~) H H e e

19 ; He M ( ~ ) (~ ) = h/e H = H H e B(x) = B() ex( x/ L) B(x) = B() ex( x/ L ) () T B M/ M T / M/ M B / e Fröhlih Hamiltonian He H = H He H = = He H H B / M 3 = h/e ~ ~ = h/e H = H M ~ H = H M M (~ ( ) ) H = H ( ) (~ ) ) ; ; ( (~ ) ; H H Hee e<e ~ H H He e Shrieffer-Wolff tranformation / /interation M Bterm B // MM ee only TTuarti M = T / M H = H He 4 = H = H He H = H He ~ ~ H e M M ( (~ ) ) ) ( (~ ) ; ; H Hee ee He Attrative eletron-eletron interation for H e < ~ Cooer intability (L. Cooer 956) < ~ Formation of (wealy bound H ==HH HH Cooer e air < ~ H e tate of of two eletron maing u a boon) CM momentum when Lowet energy = : = = interation To exeriene the wea attrative be ; = = the eletron an t too far from eah other: H e (antiymmetri Hein wave funtion)

20 e H H e e H H e e = h/e H = He H = e = = = = ~ = HHe M ( e ; He B / M ) == == = () H= H He = < ~ < ~ (~ ) = = = = < ~ < ~ < ~ H = HT He M H = H H/ e () () < ~ H e ; ; ; () ; ; ; < ~ () () where " # et. = = where where et. " " # # et. where " # et. " # " # ; ; () where where where " et. et. where " ## ; et.et. ()

21 e H H e e H H e e = h/e < ~ < ~ H = HT He M H = H H/ e H = He H HHe M ( e ; e ) == == = = () H= H He (~ ) = = < ~ < ~ = = = = ~ = = = = = B / M He = < ~ () () < ~ H e ; ; ; ; () ; ; ; < ~ () () where " # et. = = where where et. " " # # et. where " # et. " # " # " # ; ; () where where where HBCS = ( ) et. et. where " where " ## ; et. et.et. ()