A Relativistic BCS Theory of S-Wave Superconductivity

Transcription

1 A Relativisti BCS Theoy of S-Wave Supeondutivity Damien BERTRAND Jan GOVAERTS John MENDY UC-FYNU / Supeondutivity Goup ouvain-la-neuve - June 4

2 Outline Ginzbug-andau fundamentals Oiginal theoy Covaiant model Expeiment in N BCS fundamentals Oiginal theoy Relativisti BCS Detailed analysis Stategy Homogeneous situation Fist ode oetions Peliminay Conlusions & Pespetives DB June 4

3 Ginzbug-andau theoy e β F dv i A ψ h αψ ψ 4 ( B B ext Poweful fo desibing supeondutivity aound T DB June 4

4 A ovaiant model G fee enegy with magneti AND eleti field New phase diagam as a funtion of T AND E B J. Govaets D. Betand and G.Stenuit Supeond. Si. Tehnol. 14 ( DB June 4

5 Expeimental esults ( E B ƒ I Nothing! D. Betand FYNU Annual Repots 1 & (unpublished DB June 4

6 Hypothesis Seening by «nomal eletons» but what does that mean??? A loo at othe models that admit eletostati potential «The eletostati and the themodynami potential ompensate eah othe to a geat extent esulting into an effetive potential ating on the supeonduting ondensate.» P. ipavsy et al. Phys.Rev.B ( Need a miosopi undestanding DB June 4

7 DB June 4 BCS Theoy Oveview Attative inteation between eletons F ε ε η l l l V n H σ η DB June 4

8 BCS gap equation V l V ε < ε othewise E ( ε ε F ² Eg ² Above the gap exitation of Bogoliubov quasi-patiles γ u v DB June 4

9 Relativisti BCS Hamiltonian H i [ ] µ ψ γ ψ ψ γ ψ mψψ µψγ ψ eaµ ψ γ ψ g( ψγ 5ψ ψγ 5ψ Fee Dia Field Eletomagneti oupling Chemial Potential Contibution Sala (s-wave oupling DB June 4

10 Finite tempeatue field theoy Path integal fomalism: F( q' t'; q t q' e ih ( t' t q W q( t e is[ q] S dt [ pq& H ] Themodynamis patition funtion: Z( β T e βh β1/t dq W q e [ ψψ βh * ] e q S E [ β ] Eulidean ation in tems of an imaginay time τ it alulated between and β. DB June 4

11 DB June 4 Hubbad-Statonovith tansfom Quati tem in the hamiltonian is illed by the intodution of an auxiliay field : 5 1 ψ gψ γ g H H Then the integation of femioni field an be pefomed and the patition funtion beomes ( [ ] * 1 3 * ln det ( ] [ ] [ det exp ] [ 1 g eff g x d d S H H x d d Z τ β ψ ψ ψ ψ ψ ψ τ W How to alulate that???

12 Stategy fo a solution Dividing -Homogeneous situation : - Without e-m field : ( x µ ( x µ A ( x µ ln det ln T det ln [ (1 1 T ln (1 ] 1 Easy if the homogeneous hamiltonian an be diagonalized T [ ] DB June 4

13 DB June 4 The homogeneous ase without extenal field Deomposition in tems of the eation & annihilation opeatos of the Dia field & Appopiate Bogoliubov tansfom fo both inds of opeatos with a geneal fom ( s b ( s d ( os ( ~ sin ( s b s b s B θ θ New enegy levels fo quasi-patiles opeatos: ( ( ( ( µ ω µ ω E E D B The auxiliay field ats as an enegy gap!

14 Revisiting G potential Fom the analysis of the homogeneous ase one dedues: Z βh β S eff T e e ( Atually this is G potential fo the ode paamete! S DB June 4

15 G potential away fom T G potential in y^4 good aound T It is possible to find othe analyti funtions that fit well even below T Cuent job (> soy fo the bad pitue!!! S DB June 4

16 Fist ode oetions Reall that we must alulate 1 Geen s funtion of onstituted of the oelation funtions T ψ ( x ψ i j i 1 ψ ψ τ ψ ( x 1 j τ One it is done we obseve that the effetive ation eeps the same aspet but with hemial potential depending on the position! µ (x and ode paamete (x We need seond ode oetions to see the gadients DB June 4

17 Conlusions & pespetives A pomising model fo undestanding s-wave supeondutivity Aleady inteesting esults fo homogeneous situation: Fits well with y^4 theoy aound T Diffeent analyti funtions fom usual y^4 theoy BUT valid even fa below T Fist ode oetion is obtained and seond ode oetion is in pogess DB June 4