ecture Phy 798S Spring 6 Steven Anlage The heart and oul of uperconductivity i the Meiner Effect. Thi feature uniquely ditinguihe uperconductivity fro any other tate of atter. Here we dicu oe iple phenoenological approache to decribing the Meiner effect quantitatively. The ondon equation The brother F. and H. ondon wrote down two iple equation which conveniently incorporate the electrodynaic repone of a uperconductor. Thee equation decribe the icrocopic electric (E) and agnetic (h) field inide a uperconductor. Here h i the icrocopic flux denity, and B will be the acrocopic averaged flux denity. See Appendix (p. 435 of Tinkha). To derive the firt ondon equation, think of the net force acting on the charge carrier in a noral etal (Drude odel): ( v) d dt v ee τ (Note that thi i a OCA equation. It aue that only the local electric field influence the drift velocity. A uch, it require the ean free path be le than the agnetic penetration depth, l MFP < ). Here v i the average or "drift" velocity of the charge carrier of charge e, i it a, E i the local electric field, and τ i a phenoenological cattering tie for the carrier which decribe how long it take the cattering to bring the velocity of the carrier to zero. In a noral etal in teady tate, the drift velocity achieve a contant value, eaning that the electric force and cattering force balance, leading to: v e Eτ / If there are n carrier per unit volue, the current denity can be written a J ne v, o J ne τ E
which i Oh' aw (J σe) with the conductivity σ ne τ/. Oh law ay that an electric current in a noral etal i a conequence of an applied electric field. To odel a uperconductor, we hall uppoe that there i a denity of uperconducting electron, n, and they do not have their velocitie reduced to zero by ean of cattering. (See Tinkha p. 5 for why τ doe not give perfect conductivity) Fro the above equation, thi ean that the electron will accelerate in an applied electric field! v / ee, giving rie to the firt ondon equation: ne E or ( Λ ) E Strictly peaking, thi equation only hold for ac current and electric field, ince it predict very large current for large tie at dc. The firt ondon equation ay that in order to create an alternating current (i.e. a non-zero / ) it i neceary to etablih an electric field in the uperconductor. Thi ha iplication for the finite-frequency loe in uperconductor. If any un-paired electron (quaiparticle) are around, they will be accelerated by the electric field and caue Ohic diipation. Hence a uperconductor ha a all but finite diipation when illuinated with a finite frequency electroagnetic wave at teperature above zero Kelvin. Superconductor are only diipation-le at zero frequency, or at finite frequency at zero teperature (for a fully-gapped uperconductor). We define a new quantity, Λ a, E E µ where Λ µ /(n e ). We have alo introduced an iportant new length cale, the (ondon) agnetic penetration depth,. It i defined a, n e To get a deeper inight into the firt ondon Equation and thi new length cale, tart with the Maxwell equation for the icrocopic field (Apere aw),
B µ J + µ and ignoring the diplaceent current (thi i uually appropriate in uperconductor becaue we often conider only frequencie ω < ~ THz roughly), take the tie derivative of both ide and ue the firt D ondon equation, to obtain, J µ E Now take the curl of both ide, E The electric field curl around the tie-varying agnetic field (Faraday law) E to get t Integrating both ide with repect to tie yield B + B. Now ue the vector identity B ( B) B And the fact that B to arrive at B B Thi equation adit olution of the general for B( x) ± x / B e, o the ondon penetration depth repreent the exponential creening length of the agnetic field in the uperconductor. Thi length cale i alo coonly referred to a the agnetic penetration depth for obviou reaon. Thi equation 3
how that the agnetic field i excluded fro the bulk of a uperconductor, and decribe the reult of the Meiner effect. A iilar reult can be derived for the electric field: E E, howing that it i creened on the ae length cale. The ondon penetration depth can be etiated for Al, which ha a total carrier denity of n 8. x free electron/c 3, where we find.5 n. Thi i an iportant icrocopic length cale in uperconductor. The Second ondon equation J Start with ondon firt equation, ne ne J + E B and ue Faraday law ( ne E E and take the curl of both ide: ) to write. Thi i a cobination of Faraday law and enz law. It ay that a conductor will develop a current to oppoe a change in flux in the aterial. Thi equation doe not predict the Meiner effect of coure, only diaagnetic repone to tie-varying field. The ondon thought that uperconductor could be decribed by iply aking the quare-bracketed ter equal to zero: n e J B B (the ondon hypothei) Or, Λ (the econd ondon equation). Thi i not a rigorou derivation of the econd ondon equation. degenne, for exaple, ue an energy iniization arguent to arrive at the ae reult. Thi equation tate that a agnetic field applied to a uperconductor create a creening current, uch that the curl of that current i oppoitely directed to the field. It ay that dc current are controlled by agnetic field, a oppoed to noral etal where they are controlled by electric field, J n σe. 4
Thi equation only applie under liited condition: ) B ut be all and treated a a perturbation, naely B << B c. ) The uperfluid denity hould be unifor in pace. 3) ocal electrodynaic, naely >>. / ξ + / One can derive both ondon equation by tarting with the following iple expreion relating the fp vector potential to the uper-current repone, A. Taking the tie-derivative of both ide Λ yield ondon firt equation, while taking the curl of both ide yield the econd ondon equation. However one ut chooe an appropriate gauge for the vector potential, and the tandard convention i the ondon gauge: a) A J A b) Λ i obeyed on all urface and in the bulk of the uperconductor,, c) A deep (any penetration depth) inide the uperconductor. Thee condition can break down in a nuber of ituation, including near a uperconductor/noral boundary in a current-carrying wire where the divergence condition on J i not atified, or in a ultiply-connected uperconductor where the uperconducting order paraeter can no longer be taken to be purely real. 5