Binding energy of a Cooper pairs with non-zero center of mass momentum in d-wave superconductors

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Marylou Fitzgerald
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1 Bndng energ of a Cooper pars wth non-zero center of mass momentum n d-wave superconductors M.V. remn and I.. Lubn Kazan State Unverst Kremlevsaa 8 Kazan Russan Federaton -mal: gor606@rambler.ru PACS: z; g Kewords: Hgh-T c superconductors deparng current The bndng energ of Cooper pars has been calculated for the case of d-wave smmetr of the superconductng gap n laered cuprate superconductors. We assume that Cooper pars are formed b the short range potental and then derve the bndng energ n the form = ( cos a + ( cos a + ( sn a +( sn a where s a total momentum of the par. Numercal solutons of the self-consstent sstem of the ntegral euatons for uanttes ( ( and ( ( along dfferent lnes n plane have been obtaned. Ansotrop of the deparng total momentum (or deparng current has been calculated.

2 . Introducton The problem of the bndng energ of a Cooper par n the BCS model (phonon parng mechansm was dscussed b Cooper n hs poneerng paper [] and later on b Casas et al. [2]. It was founded that =0 c.e. Cooper pars are destroed ver fast when the start to move. Ths behavor of Cooper pars s n strong contrast to usual bosons. The crtcal value of momentum at whch Cooper pars are self destroed can be called as deparng momentum. In d-wave HTSC superconductors the energ gap s gven b = 0 =0 (cos a cos a / 2. As one can see the gap s zero along the dagonals n the Brlloun zone.e. when = ±. Therefore one can epect that deparng momentum (or deparng current n d-wave superconductors should be also strongl ansotropc. To our nowledge ths problem was not dscussed et. The goal of present letter s to cover ths feld. In secton 2 we shall derve the ntegral euaton for the order parameter (or for smple eplanaton the bndng energ usng Hubbard proecton technue. The results of numercal soluton derved euaton wll be gven n secton Model and Integral uatons We start from the so-called snglet correlated band model for laered hole doped cuprates [3]: σ σ H = t ( + + G ( ( ( σ 2 2 where the frst term s a netc energ the second s superechange nteracton and the thrd s denst-denst nteracton or pseudo-coulomb nteracton. We prefer to spea about the denst-denst nteracton because the ndrect nteracton uaspartcles va optcal phonons can be taen nto account va renormalzaton of parameter G. It s assumed that carrers move over ogen postons n CuO plane. Strong echange couplng between ogen and copper spns has been taen nto account durng the dervaton of Hamltonan ( [see Ref. 3]. Thus the smbol corresponds to cooper-ogen snglet state. The euatons of moton are gven b The coeffcent and = H D t = + = H = + D D are calculated usng followng eualt [4 5]: { l H } = l { l } { l H } = Dl { l } where the angular bracets denote the thermodnamc averagng. After calculaton of the antcommutators n both sde n partcular we have obtaned D δ G = t + l l l l l l l l l l P P P. (2. (3 Here s P = +. For d-wave superconductors the sum n (3 s zero so we can drop t. Performng the Fourer transform n. (3 n the usual wa R = e and ntroducng the total momentum of the par = + 2 we get: The order parameter s determned as follows: N = ε = ε ( ( ( ( G + NP (4 = +. (5.

3 It s nterestng to note the echange part of the ernel n. (5 namel ( contans the total momentum. Ths nterestng feature was not ponted out before because the most part of nvestgatons n ths feld were done tang nto account the denst-denst nteracton onl. Then we appled Green s functons method and completed the sstem of the euatons ( ε + + = P 2π. + ε = 0 ( + + Solvng ths sstem we get the energ of Bogolubov s uaspartcles spectrum: ε ε + ( = ± ε + + ε +. (6 2 4 Further usng the well nown formula we fnd the correlaton functon: ( d lm + = e β + + ε 0 + ε ε + P ( ( ( e e P f f + = = β β where f ( = s Ferm functon. Substtutng the correlaton functon n the euaton (5 and we get the e β + ntegral euaton whch must be solved self-consstentl = ( ( + ( G( ( f ( f ( 2. (7 2 The energ dsperson of uaspartcles n hole-doped HTSC s wrtten as ε = 2t( cos a+ cos a + 4t2cos acos a+... (8 where hoppng ntegrals t t 2 and chemcal potental μ were taen n accordance wth Norman s analss [6] who etracted all these parameters from the photoemsson and neutron scatterng data. The Fourer-transform of the echange potental s gven b: = cos a+ cos a. (9 ( ( Note ths form of Fourer-transform s vald for an short-range nteracton. As an eample one ma assume that ts role plas the superechange nteracton between of copper spn then 00meV. The soluton of euaton (7 for the case = 0 was descrbed earler [7]. Below we shall assume that the echange nteracton domnates. Therefore the Coulomb-le term n euaton (7 wll be dropped. 3. Numercal results and Conclusons uaton (7 belongs to the class of the separable ntegral euatons. It s eas to see that ts general soluton ma be wrtten n the form: = ( cos a + ( cos a + ( sn a +( sna. (0 After substtuton (0 n the euaton (7 we arrve to the sstem of euatons: ( = ( cos a+ cos( a ( f ( 2 f ( 2 ( = ( cos a+ cos( a ( f ( 2 f ( 2. ( ( = ( sn a+ sn ( a ( f ( 2 f ( 2 ( = ( sn a+ sn ( a ( f ( 2 f ( N The sstem was solved numercall. Resultng dependences of ( ( and ( ( on a where a s the lattce constant are shown n Fg. and Fg. 2 where for short we have dropped the smbol (. 2

4 Fg.. Dependences on a along a dagonal of Brlloun zone ( =. Fg.2. Dependences on a along as = 0. at ( a a Fg.3. Ansotrop of the deparng value a n and Fg.4. Temperature dependences at when plane. and have mamum value. In Fg. 3 (blue lne we show the ansotrop of the deparng total momentum.e. the values of when the bndng energ of a Cooper pars wth non-zero center of mass momentum becomes zero. The deparng current can b easl h estmated usng the smple relaton: = enυ = en. m e From Fg.3 we see that the deparng current n d-wave superconductors n the nodal drecton (usual superconductng gap s zero! appromatel twce smaller than n the ant-nodal drecton where the usual gap functon has a mamum. The uanttes ( ( ( and ( /see. (0/ are monotoncall decreasng functons of temperature and all of them vansh at the same crtcal temperature ( T c. In our calculatons T c = 40K. In Fg. 4. we show temperature dependences of ( ( ( and ( at where and have mamum value. As one can sees that temperature dependences are ute dfferent from those whch are well nown for Cooper pars wth zero total momentum. Let us shortl dscuss a varant when the ernel. (5 does not contan the tem le (. We note t can be specfc ust for decouplng procedure whch we have used above. In Fg. 3 (red lne we show ansotrop of the deparng total momentum when we have chosen the Fourer transform nteracton n the form V ( = V 0 cos( a+ cos( a wth V = 200meV. As one can see the red and blue curves are ver 0 smlar snce the deparng momentum s small. Thus we conclude that presented deparng momentum pcture on Fg. 3 s not senstve to the decouplng procedure. Fnall we would le to pont out that f one starts from t - model Hamltonan nstead of ( (whch to our opnon s more relevant to the electron-doped superconductors then one arrves to the same concluson as wrtten above because the netc energ tem does not enter the ntegral euaton for the order parameter wth d-wave smmetr.

5 Ths wor s partall supported b the Swss Natonal Scence Foundaton Grant # IB and the Russan Foundaton for Basc Research Grant # a. References. Cooper L.N. Phs. Rev. Lett ( Casas M. Futa S. de Llano M. Puente A. Rgo A. Sols M.A. Phsca C ( remn M.V. Solovanov S.G. Varlamov S.V. TP 85(5 963 ( Roth L.M. Phs. Rev ( Plada N.M. Han R. Rchard.-L. Phs. Rev. B ( Norman M.R. Phs. Rev. B ( remn M.V. Laronov I.A. TP Lett. 62(3 92 (995.

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as: 1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors