4. The critical magnetic field

Transcription

1 4. The critical magetic field

2 The critical curret deity ad the critical magetic field The upercurret deity ha a limit: J C Whe the upercoductor i applied with a magetic field, a upercurret i geerated o a to maitai the perfect diamagetim. If the curret deity eeded to cree the field exceed J C, the upercoductor will loe it upercoductivity. Thi limit of the field tregth i called the critical magetic field H C. Note the differece betwee the flux deity, B ad field tregth, H

3 Free eergy The magetiatio deped o the applied field ad temperature but ot the hitory. The traitio i a reverible proce the Gibb free eergy g ( T, H ) Why Gibb free eergy? a i a fuctio of temperature ad field. Becaue the ytem i i cotact with a thermal reervoir, (T=cotat) a preure reervoir (P=cotat), ad a magetic-field reervoir (H=cotat) I thi cae, the ytem will miimize the Gibb free eergy

4 Below T C The free eergy of the upercoductig tate < The free eergy of the ormal tate g T g T (,0) < (,0) The free eergy icreae whe applied a magetic field: H a g ( H ) a = µ 0 0 MdH a Area of the M-H curve Here we eglect the effect of demagetizig factor

5 (, ) (,0) g T H = g T µ MdH (,0) Free eergy a 0 a 0 = g T + µ HdH 0 H H H = g ( T,0) + µ a a 2 a a g g ( T,0) ( T,0) g ( T, H ) The ormal tate i o-magetic ad the applied field doe ot chage the free eergy much. (, ) > (, ) g T H g T H a H C g ( T, H ) Whe the upercoductivity i detryed. 2 HC µ 0 = g T,0 g T,0 2 ( ) ( ) H a

6 The temperature depedece Phae diagram H Normal tate H 0 SC H C T C Experimetal reult: 2 T HC = H0 1 = H0 1 t T C T 2 ( )

7 The cryotro Curret I Gate(Ta) Whe i C i applied, the field may detroy the upercoductivity of the gate ad reduce the curret I A electroic witch i C Cotrol(Nb)

8 The M-H curve Deep iide the upercoductor M B H C H H C H area: the free eergy cotributed by M For a perfect pecime, the curve are reverible

9 The hyterei: o-ideal pecime Reidual magetizatio: trapped flux Ill defied H C Hyterei Trapped flux

10 Meauremet of B C: pick-up coil S: oleoid to apply the field LR circuit Whe the witch i cloed, the ballitic galvaometer i deflected. The deflectio of G i proportioal to B.

11 Ta at 3.7K

12 Meauremet of magetizatio VSM The pecime i moved to ad fro betwee pick-up coil A ad B. The galvaometer wig with a amout proportioal to the magetizatio M.

13 Itegratig method E= E E A B d d µ 0( H + M) µ 0H dt dt E dm dt Itegrator: V V dt M out i cf: AC uceptometer

14 Homework 1. Ue the defiitio of Gibb free eergy G = U TS + PV µ 0HaM to how the Gibb free eergy chage due to a magetic field H a i give by. ( a) 0 H a g H = µ MdH 0 a

15 5. Thermodyamic

16 Etropy From We have 2 HC µ 0 = g T,0 g T,0 2 ( ) ( ) µ 0,0, 2 ( 2 2 H ) C Ha = g( T ) g( T Ha) G = U TS + PV µ 0HaM ( ) ( ) µ ( ) dg = du d TS + d PV d H M = SdT + VdP µ MdH 0 a 0 a

17 etropy thu g = T =µ p, Ha dh C 0HC dt H C i temperature depedet ice > dh C dt < The upercoductig tate i more order tha the ormal tate 0 H H 0 SC Normal tate H C T C T

18 At T C, H C =0 = Accordig to the third law of thermodyamic, = = at T=0 Sice H 0 i fiite, thi require that dh C 0 dt = at T=0 Etropy v temperature

19 2 d order phae traitio At T C = g g = T T Both g ad it derivative g T are cotiuou A 2d order traitio Feature: o latet heat ( ) L = vt a jump i pecific heat C = vt T

20 Specific heat ( ) 2 2 d HC dhc µ = = 0 C + C C vt vt H T dt dt At T C, H C =0 Jump at T C ( ) C C vt dh µ C = T C 0 C dt T Rutger formula 2 C

21 Latet heat at o-zero field ( ) 0 = = dh C L vt vtµ HC dt H H 0 Normal tate 1 t order traitio > SC 2d order traitio T C T

22 Adiabatic magetizatio Adiabatic proce, =cotat dq = TdS = 0 Adiabatically detructig the upercoductivity by applyig a large field ca lower the temperature Adiabatic demagetizatio I ued for ordiary magetic material i which the etropy decreae with applicatio of field.

23 Specific heat The pecific heat of a metal i cotributed from the lattice ad coductio electro C = C + C The propertie of the lattice do ot chage at the traitio latt el ( ) ( ) C C = C C el el For a ormal metal T C = Clatt + ( Cel) = A γt + θ 3 θ: Debye temperature γ: Sommerfeld cotat

24 At T ~T C C C > C becaue > ds dt > < ds dt At low temperature C

25 C T A T θ 2 = + 3 At T<T C, oe ca meaure C for ormal tate (H a >H C ) γ C/T Slope=A/θ The electroic part for upercoductig tate ca be determied / ( C ) = ae el b kt The form of the thermal activatio : hit of a eergy gap b/e γ T C A T approache T C, b rapidly decreae to zero T 2

26 Thermal coductio Irreverible procee The thermal coductio i a metal i motly cotributed by the coductio electro Superelectro have egligible iteractio with the lattice The thermal coductivity i upercoductig tate i much maller tha that i ormal tate. Thermoelectric effect Due to zero reitace of the upercoductor, the thermal e.m.f. hould be zero. Therefore, Peltier ad Thomo coefficiet are zero. I fact, thermoelectric effect may appear i type-ii upercoductor

27 Seebeck(thermoelectric) effect e.m.f. due to temperature gradiet uder coditio of zero electric curret Peltier effect A T 1 B V T 2 Heat flow followig a electric curret acro a iothermal juctio Thomo effect Electric curret travere a temperature gradiet T 1 A B curret Heat flow T 2 curret

28 Homework 1. Apply Sommerfeld theory to how that the electroic pecific heat of the metal i proportioal to the temperature, i. e. Cel = γt i which the cotat γ i proportioal to the deity of tate at Fermi urface. 2. Do the ame calculatio by aumig a eergy gap formig at the Fermi urface ad how that Cel = e / kt ae at low temperature, kt<<