PHYSICAL REVIEW B 1 JANUARY 1997-IIVOLUME 55, NUMBER 2 Scaling in the BCS to Bose crossover problem in different partial waves Sadhan K. Adhikari and Angsula Ghosh Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, 01.405-900 Sa˜o Paulo, São Paulo, Brazil ~Received 15 August 1996! The BCS superconductivity to Bose condensation crossover problem is studied in two dimensions inS, P, andD waves, for a simple anisotropic pairing, with a finite-range separable potential at zero temperature. The gap parameter and the chemical potential as a function of Cooper-pair bindingBc exhibit universal scaling. In the BCS limit the results for coherence lengthj and the critical temperatureTc are appropriate for high- Tc cuprate superconductors and also exhibit universal scaling as a function ofBc . @S0163-1829~97!02501-0# ig er e re n un as n o id he e tio tu h e ll t b ov c h ar ee ie n s g d te rsal ble a wo- ient - o- een of d al i- in er- g for , in ob- , d- tial ge the of av- ge the r nd de- A collection of electrons~Fermions!, interacting via a weak residual interaction at very low temperature and h density, exhibits pairing instability. Overlapping Coop pairs1 are formed spontaneously according to the Barde Cooper-Schreiffer~BCS! theory of superconductivity.2 For usual superconductors, the BCS theory yieldsjkF;1000 and 2D/(kBTc)53.52 in agreement with experiments, whe kF is the Fermi momentum,kB the Boltzmann constant,D the gap parameter,Tc the critical temperature, andj the coherence length. In the opposite limit of a strong residual interaction, no overlapping diatomic bosonic molecules emerge as bo states of fermion pairs.3,4 At sufficiently low density and temperature, these composite objects act as an ideal g bosons withjkF;0, which may undergo a phase transitio In three dimensions this phase transition is the usual B condensation.3,5 In two dimensions one can have a superflu transition under appropriate conditions.6,7 Hence the two ex- treme limits of the same system exhibit two interesting p nomena. Leggett3 emphasized the importance of the abov mentioned BCS superconductivity to the Bose condensa crossover problem. This problem has gained new impe after the discovery of high-Tc cuprate superconductors wit certain general properties.5,7–10 These superconductors hav a very small coherence lengthjkF;10,8,9 and exhibit a lin- ear scaling betweenTc and TF known as the Uemura scaling,11 where TF is the Fermi temperature. The sma value ofj suggests that its superconducting phase migh understood as one in the above-mentioned cross regime.5,7,8,10 Many high-Tc superconductors have a conducting stru ture similar to a two-dimensional layer of carriers,8,10,11 which suggests the use of two-dimensional models. T mathematical complications of the crossover problem also simpler in two dimensions than in one or thr dimensions.5,7–9Hence, in order to understand the subtlet of this problem, the present study is limited to two dime sions. In the S-wave crossover problem in three dimension Leggett consideredN electrons, each of massm and of spac- ing l , interacting via a weak short-range potential of ran r 0(! l ). The scattering lengtha satisfiesuau@r 0. By varying the ratiol /a, the BCS and the Bose limits could be attaine When suitably scaled, most of the properties of the sys 550163-1829/97/55~2!/1110~4!/$10.00 h n- - d of . se - - n s e er - e e s - , e . m are insensitive to the details of the potential and are unive functions of the dimensionless variablel /a. In three dimen- sionsl;kF 21 and hence one could use the equivalent varia 1/(kFa). For theS-wave problem in two dimensions with very-short-range attractive potential, the presence of a t body bound state in vacuum is the necessary and suffic condition for pairing instability.4,8 Hence the convenient di mensionless variable could beB2 /EF with B2 the two-body binding in vacuum andEF[kBTF[\2kF 2/(2m) the Fermi energy. The usual treatment of BCS employs a potential in m mentum space with a constant value for momentum betw 2m(EF2ED)/\ 2 and 2m(EF1ED)/\ 2 and zero elsewhere with ED the Debye energy. This implies a moderate range the interaction. This potential is a physically motivate phonon-induced electron-electron one.2 The objective of the present work is to find out to what extent the univers nature4,7,8 of the solution of the crossover problem is mod fied after introducing a smooth potential of medium range place of the above potential. For some of the high-Tc mate- rials, experiments suggest non-S-wave Cooper pairing.8,9 This is why we have also extended the discussion of univ sality to the crossover problem inP and D waves, for a simple anisotropic pairing. The energyB2 is not really the ideal variable for studyin universality in the crossover problem in two dimensions non-S partial waves, because inP andD waves one could have Cooper pairing, and hence BCS superconductivity the absence of a two-body bound state in vacuum.8 The ap- propriate reference variable for studying the crossover pr lem in all cases, including the three-dimensional problem12 is the Cooper-pair bindingBc . For the zero-rangeS wave model of Ref. 8,B25Bc . Previously, there have been stu ies of the crossover problem in terms of the poten strength or scattering length. We studied the crossover problem with a finite-ran separable potential and, especially, the dependence of universal behavior of the crossover problem on the range the interaction potential. We found that the universal beh ior of the crossover problem does not significantly chan with the change of the range of potentials. We calculated zero-temperature chemical potentialm and gap paramete D in the entire crossover region, andj andTc in the BCS region for different values of the range parameter. We fou robust universal scaling in each case, valid over several 1110 © 1997 The American Physical Society ng f o e al o o al th . nt f al o th el s by c- d s th ed ition . pic o- y . at eter e ned ir e - tion e 55 1111SCALING IN THE BCS TO BOSE CROSSOVER . . . cades of Cooper-pair binding. We did not find similar scali when m, D, Tc , and j were considered as a function o potential strength or of bindingB2. 12 The present model als produced an appropriateTc /TF ratio and a smallj in the BCS region in accordance with experiments on high-Tc cu- prate superconductors. The two-body problem and the Cooper and BCS mod all exhibit ultraviolet divergences for zero-range potenti and require renormalization to produce finite results.8 For non-S partial waves, the nature of these divergences is m complicated and there is no general prescription for ren malization. The present study with finite-range potenti leads to a well-defined mathematical problem without necessity for renormalization. We consider a two-body system, each of massm, in the center-of-mass frame.8 The single-~two-! particle energy is given byeq5\2q2/2m (2eq), whereq is the wave number We consider the attractive separable short-range pote Vpp852l f pf p8. The angular momentum (L) dependence o all the variables will not be explicitly shown. This potenti leads to pairing instability for anyl, L, and f p . 12 In even ~odd! partial waves pairing occurs in singlet~triplet! state. The correspondingt matrix is Tpp8~2E!5 f pf p8F2l212( q f q 2~2E22eq! 21G21 , where 2E is the parametric relative energy. The condition f a bound state at energy 2E52B2 is l21[( q f q 2~B212eq! 21. ~1! The Cooper-pair problem for two electrons above filled Fermi sea for this potential is given by1,2 l215(q.kF f q 2(2eq22Ê)21, with Cooper binding Bc[2EF22Ê. Using Eq.~1!, the Cooper problem is written as ( q f q 2~B212eq! 212 ( q.kF f q 2~2eq22Ê!2150. ~2! Leggett3 provided a generalization of the BCS mod valid for a crossover from large to small coherence length zero temperature. The finite-temperature (T) version of this problem is given by the BCS gap and number equations4 Dp52( q Vpq Dq 2Eq tanh Eq 2kBT , ~3! N5( q F12 eq2m Eq tanh Eq 2kBT G , ~4! with Eq5@(eq2m)21uDqu2#1/2 andm (ÞEF). At finite tem- peratures the coupled system of equations~3! and~4! is only valid in the weak-coupling BCS region characterized positivem/EF . In the strong-coupling Bose region, chara terized by negativem/EF , due to the existence of preforme composite bosons at finite temperatures aboveTc the number equation~4! breaks down.5,7 Actually, the physical proces changes as one moves from the BCS to the Bose limit. In ls s re r- s e ial r e at e BCS limit the formation of Cooper pairs atT5Tc signals superconductivity, whereas in the Bose limit the preform composite bosons may undergo a superfluid phase trans at T5Tc under appropriate conditions.6,7 At zero tempera- ture Eqs.~3! and~4! are valid in the whole crossover region8 As in Ref. 8, we consider the simplest case of anisotro pairing in two dimensions whereDq has the angular dependence;exp~iLu!. With the present potential,Dq of Eq. ~3! then behaves asDq[D f qexp (iLu). Equation~3! then becomes l215( q f q 2 1 2Eq tanh Eq 2kBT , ~5! whereEq5@(eq2m)21D2f q 2#1/2. Using Eqs.~1! and~2!, Eq. ~5! can be rewritten as ( q.kF f q 2 eq2Ê 2( q f q 2 Eq tanh Eq 2kBT 50. ~6! Equation~6! is valid independent of the existence of a tw body bound state in vacuum. Equations~4! and ~6! can be explicitly written as E EF ` deq f q 2 eq2Ê 2E 0 ` deq f q 2 Eq tanh Eq 2kBT 50, ~7! E 0 ` deqF12 eq2m Eq tanh Eq 2kBT G52EF . ~8! Equations~7! and ~8! permit the following analytic solu- tions for S-wave (L50) zero-range potential given b f q51. At T50, D5(2BcEF) 1/2 and m5EF2Bc/2. 8 At T5Tc (D50) in the BCS limit (m'EF) we find the ana- lytic solutions m12kBTcln@2cosh(m)/(2kBTc)#52EF and Tc /TF5A2exp(g)(Bc /EF)1/2/p where g50.577 22. In this case 2D/(kBTc)52p/exp(g)'3.528. Next Eqs. ~7! and ~8! are solved numerically inS (L50), P (L51), andD (L52) waves with form factors f q5qL@a/(q21a)# (L11)/2 wherea is the range parameter We studied the crossover problem in the entire domain T50 and calculated the dimensionless order param D[D/EF (12L/2) and the chemical potentialm/EF as functions of Bc /EF for different L and a. The order parameters ar shown in Figs. 1~a!, 1~b!, and 1~c! for S, P, andD waves, respectively. In Fig. 2 we exhibitm/EF in these cases. The coherence length, or the pair size in the BCS region, defi by j25^cqur 2ucq&/^cqucq&, with the zero-temperature pa wave functioncq5Dq /(2Eq), was numerically calculated using r 2[2¹q 2 . The calculated (jkF) 2 are shown in Fig. 3 as a function ofBc /EF . The S-wave Pippard coherenc length@5\kF /(pmD)# is also shown in Fig. 3 for compari son. We also calculatedTc in the BCS domain by setting Dq50 in Eqs. ~7! and ~8!. The calculatedTc /TF is also plotted in Fig. 1 for different partial waves. Before presenting a discussion of the results we men two limitations of the present model. First, for non-S waves the zero-range limit (a→`) cannot be taken because of th appearance of strong ultraviolet divergences. ForS waves, this limiting solution is analytically known.8 Second, the size ture 1112 55SADHAN K. ADHIKARI AND ANGSULA GHOSH FIG. 1. log10~2D! and log10(Tc /TF) vs log10(Bc /EF) plots for ~a! S (L50), ~b! P (L51), and~c! D (L52) waves, denoted by solid and dash-dotted lines, respectively, whereD5D/EF (12L/2) . Tc is calculated in the weak-coupling or the BCS regime as the finite-tempera version of the Leggett equations are valid only in this regime. ia u o a lid a n th e b n ant ity to l- of the two-body bound state in vacuum with this potent behaves asB2 21/2 (a21) whenB2/2 ,(.)a2. Hence the size of this bound state as well as the Cooper pair becomes realistic for B2/2.a2. Consequently, for very large Bc /EF , the ideal Bose limit of nonoverlapping bosons is n realized for smalla. Hence the present study is limited to potential of intermediate rangesa/EF;1–10. From Fig. 1 we find that bothD andTc /TF exhibit uni- versal behavior as functions ofBc /EF in different partial waves. In the BCS region they exhibit linear scaling va over about four to five decades. The scaling exponents roughly constant for all L and a: D;(Bc /EF) 1/2, Tc /TF;(Bc /EF) 1/2. For an ideal Bose gas there is no co densation in two dimensions and hence one might think Tc should reduce to zero asBc /EF increases in the Bos region. However, because of a weak residual interaction tween bosons, this system may undergo a superfluid tra tion with quasi-long-range order below a fixed smallTc /TF independent ofBc /EF . 7 The Tc for this transition can only l n- t re - at e- si- be found in numerical model studies. Hence allTc /TF of Fig. 1 should reduce quickly and attain a small const value asBc increases in the Bose region. 7 We find that both Tc /TF andD increase with decreasinga and L. The ratio 2D/(Tc /TF)[2DEF L/2/(kBTc) increases asa decreases and/or L increases. For example, fora/EF51 ~2, 5! this ratio is 4.93 ~4.31, 3.87! for L50, 7.07 ~5.29, 4.21! for L51, and 10.0~6.37, 4.39! for L52. The corresponding universal gap-to-Tc ratio 2uDkF u /(kBTc) for a/EF51 ~2, 5! is 3.50~3.52, 3.53! for L50, 3.54~3.53, 3.51! for L51, and 3.53 ~3.47, 3.35! for L52. From Fig. 2 we find that the zero-temperaturem has a linear dependence onBc for all a and L almost over the entire crossover region. The minor deviation from linear occurs for smallm. We present this dependence up Bc /EF540. For largerBc /EF , m is essentially given by the zero-range analyticS-wave solution:m5EF2B2/2. From Fig. 3, for alla andL, we have the universal sca ing (jkF) 2;(Bc /EF) 21 valid over three decades ofBc in - ge k e to tia u po n en e a the oo ific end not eral pa- - p- nto e- eled 55 1113SCALING IN THE BCS TO BOSE CROSSOVER . . . the BCS domain. The correspondingS-wave Pippard coher ence length satisfies the same scaling. The parameter (jkF) decreases asL and/ora decreases. The analytic zero-ran S-wave solution has this scaling.8,9 For a fixedBc /EF , Fig. 1 leads to a scaling ofTc with TF for all a and L. This scaling was observed by Uemuraet al.11 for high-Tc super- conductors. From Fig. 1 we find forS, P, andD waves that Uemura’s experimental valueTc /TF.0.05 leads to a Bc /EF in the domain 0.01–0.001, which implies the wea coupling BCS limit. From Fig. 1 we find that, for a/EF55, Tc /TF50.05 leads to Bc /EF50.0032~0.0079, 0.0135! for S (P, D) waves. From Fig. 3 the above Cooper-pair bindings givejkF.9 in all partial waves. This implies a universal correlation betwe Tc /TF andjkF in all partial waves. In conclusion, we studied the BCS superconductivity Bose condensation crossover problem in different par waves for a finite-range separable potential. We found rob scaling relations involving the order parameter, chemical tential, coherence length and critical temperature as a fu tion of Cooper-pair binding. In the BCS domain the pres FIG. 2. m/EF vs Bc /EF plots for different partial waves and a. The curves are labeled by partial wave~s! anda. - n l st - c- t results may simulate typical high-Tc values for the coherenc length j andTc . They also exhibit theTc versusTF linear correlation ~at a fixed Bc /EF) as observed by Uemur et al.11 The consequence of these findings in describing high-Tc superconductors in two dimensions is not all t obvious. Though we have exhibited the results for a spec separable potential model, we verified that the general tr is maintained as form factors are changed. Hence we do believe our findings to be so peculiar as to have no gen validity. A preliminary study of the S-wave three- dimensional crossover problem employing the same se rable potential as a function ofBc /EF also leads to similar universal scaling.12 A detailed account of that will be re ported elsewhere. We thank Dr. M. de Llano and Dr. T. Frederico for hel ful discussions and Conselho Nacional de Desenvolvime Cientı́fico e Tecnolo´gico, Fundac¸ão de Amparo a` Pesquisa do Estado de Sa˜o Paulo, and John Simon Guggenheim M morial Foundation for financial support. FIG. 3. (jkF) 2 vs Bc /EF plots for different partial waves and a: dashed lines,S- (L50) wave results; dotted lines,P- (L51) wave results; dash-dotted lines,D- (L52) wave results; and solid lines, theS-wave Pippard coherence lengths. The curves are lab by partial wave~s! anda. ys. 1L. N. Cooper, Phys. Rev.104, 1189~1956!. 2J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.108, 1175~1957!; J. R. 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