o ( ACTA PHYSICA POLONICA Α No. ό E OM O EEGY GA A CIICA EMEAUE O SUECOUCO IMIE Y UCUAIOS O ESIY O SAES M GŁAYSIEWIC GONcZAREK A M MULAK Isiue o ysics Wocaw Uiesiy o ecoogy Wyeże Wysiaiiskiego 7 5-37 Wocaw oa (Received Ocfober 21' 1999; revised version February 8' 2000) e s-aie CS suecouco wi e eeco esiy o saes eeig o eegy i e iciiy o emi ee is cosiee e eis- ece o suc ucuaios o eeco esiy o saes may e eee o a oe siguaiies wiey iscusse i e ieaue I e ae ee yes o sma ucuaios o e eeco esiy o saes iouce aou is aeage ackgous aue ae aayse oeia iage a oga- imic oes I oe o cacuae umeicay e eegy ga a ciica emeaue e omaism o aameic CS ga equaios is aie e osiie ucuaios (eaks coeso o e icease i occuie saes ume iucig a ige ciica emeaue O e oe a egaie ucuaios wic ecease e ume o occuie saes ea o a owe ciica emeaue Suc ecease i ciica emeaue ca ea a a seciic coice o aamees o e ecease i suecouciiy e es- ece o ucuaios is eecio i e sae o eegy ga as a ucio o emeaue I e iciiy o Τ = a au ecease o icease i e eegy ga o e oigia CS aue accoig o e sig o e uc- uaio is osee I u e ucuaios o o cage e eaiou o e eegy ga ea 1 ACS umes 7— . Intrdtn Siguaiies o e eeco esiy o saes (EDOS) a ei iuece o suecoucig oeies ae ee ecey wiey iscusse i e ieaue wii e a oe sceaio eg [1 3 ] e easo o e eisece a mecaisms o ig-emeaue suecouciiy is o cea u oay a i (139 1040 M Glαdz Gnzαr M ΜέΙα seems to be possible that fluctuations of EOS may turn to be crucial in its explanation. Within the Van Hove scenario we try to answer a profound question, how the physics of interacting electron liquid is modified by presence of Van Hove singularities near the Ferini level, which were originally classified in two and three diinensions by Leon Van Hove [5]. From the point of view of cuaes physics the two-diinensional singular- ity, i.e. a saddle point, which is associated with a crossover from electron-like to hole-like conduction, might be extremely important. In the paper [3] the density of states with a superposition of one-dimensional eemum and two-dimensional saddle point (connected with a logarithmic divergence) has been studied and an analytical approximation for the CS gap equation within the weak coupling the- ory was proposed. As a result the values of between 40 Κ and 140 Κ as well as the isotope parameter between 0.18 and 0.28 were established. In turn, in the pa- per [4] the logarithmic EOS with Coulomb repulsion was investigated when the electronic interaction was eomaise to obtain the effective repulsive seuoo- eia In this case a high critical temperature was also achieved. In the review [1] many theoretical and experimental approaches associated with Van Hove scenario are discussed. According to them a reasonable comprehension of the high teinper- ature superconductivity phenomenon could be related to some unusual properties of cuaes with singularities in EOS Before the discovery of cuaes an einpirical rule was stated that the best way to find out a high temperature superconductor was to look for a material with a peak in EOS Recently, Mioic and Caoe have proposed some in- teresting models of EOS with peaks close to E for Α15 compounds in order to explain the anomalous behaviour of their normal and superconducting states, respectively [6, 7]. They have considered fluctuations of EOS in the oeia and triangle shapes analysing the problem within the isotropic approxiination. Band structure calculations have confirmed the possibility of such sharp peaks in EOS for materials with the Α15 crystal structure [8], as well as some recent angular resolved ooemissio spectroscopy (AES experiments have shown that many high temperature superconductors possess a sharp peak in EOS near Fermi energy [1]. In this paper we model such simple fluctuations and the numerically calcu- lated critical temperature together with the energy gap dependence vs. tempera- ture. The starting point of further calculations is the CS gap equation. e moe Let us consider an s-paired CS superconductor with a sinall fluctuation added to constant background's EOS Ν ( The modified EOS has the fr Below, we focus our interests on three particular types of fluctuations: • oeia type: e om o Eegy Ga a Critical Temperature ... 1041 e aoe equaio eeses a sueosiio o Lorentzian eaks o EDOS, oigiay iscusse i [ 7] e us emasise a e ieesig emoy- amic oeies o e sysem wi suc Lorentzian EDOS wee cacuae i e ae [9] • iage ye ee EDOS as iage eaks ea e emi ee • ogaimic ye wee e ucuaio is eesee y e ogaiiic iegeces ocaise o cosa EDOS. is ye o ucuaio may e oe ou i ig eieaue suecoucos [1 ] I e osuae oms o EDOS is e eegy eee o e emi ee xi e eig o e ucuaio Eqs. ( a (3 o e gow ae o e si- guaiy (Eq ( Β — is a-wi R2 — e eegy isace (si o e ucuaio om e emi ee a is e ume o eaks o siguaiies un- : der consideration. Since only small fluctuations are taken into account the average ume o quasiparticles Ν is esee a cosa wee a eoe -imesioa ecos o Β , a χ , eseciey : We begin with more general form of the BCS eegy ga equaio akig EDOS as o-symmeic wi esec o e emi ee wee λ = Ν(ρ is e aiig aamee Ioucig e symos (c [1-13] a iegaig Eq ( y as wi wo iee iega ucios e ga equaio ca e wie i e oowig om a cosequey 1 M Gaysiewic Gocαek M Muak I a ou cacuaios e oowig coiios ae assume i u es o e imi o e iegaio iea (│→wicuω/Τ is oe eace y iiiy e aoe oceue is uy cosise wi e assumio gie y Eq (5 oe a i Eqs ( a (9 a ew aamee Τ/Δ(Ο as ee iouce ow e imis Τ = Ο (τ = οο a Τ = Τ (τ = ae iesigae a ae some ageaic asomaios oe may oai eseciey a I is coeie o iouce e oowig susiuios wic ae useu o umeica cacuaios e us emasise a e aamee τ akes aues om eo o iiiy . rntzn EOS eow Eq (11 wi (1 is soe umeicay y e euaioa meo o e ucuaio gie y ( eaig wi Eqs ( a (1 we emoy e eicie umeica agoim wee Equaios (13 (1 u o e ey coeie o cacuae e eegy ga e- eece s emeaue I u o eie e ew ciica emeaue ase o Eqs (9 a ( we ay e agoim as oows e om o Εεgy Ga a Ciica Τemρeaυe 1043 wee isy e us cosie e case o a sige oeia eak e eegy ga em- eaue eeece is oe i igs 1 a 3 ea Τ → oe may osee is ai cage ese esus emai i ageeme wi e uamea CS eaio Δ(/Τ ig 1 e eemay oms o e eegy ga cacuae o e oeia EOS s euce emeaue o a ew cose aues o b. aamees x a r ae ie a χ = 1 r = aamee b akes e aues 5 3 1 e oigia CS cue is eesee y e oe ie (b 0). ig e eemay oms o e eegy ga cacuae o e oeia EOS s euce emeaue o a ew cose aues o x. aamees b a r ae ie a b = 5 = aamee x akes e aues 5 3 1 -3 -5 e oe ie eeses e case o x = Ο (e CS cue 1 M Głαysiewic Gonczarek' M Mulak ig 3 e eemay oms o e eegy ga cacuae o e oeia EOS s euce emeaue o a ew cose aues o aamees a x ae ie a b = 5 o e ue ee cues = 1 aamee akes e oowig aues 3 e e ie coesos o e oigia CS cue = iay e owe ee cues coeso o = —1 aamee is equa o 3 eseciey ig euce ciica emeaue cacuae o e oeia EOS s aa- mee aamee = Ο a x akes e aues 3 1 -1 - -3 ig 5 euce ciica emeaue cacuae o e oeia EOS s aa- mee aamee s 1 a akes e aues 5 1 3 e om o Eegy Gα a CiicαΙ Τemρeαυτe 1045 e us emasise a e aeaace o e aeau o e eegy ga i e ey sma egio cose o Τ wic oe ca osee i e ises o igs 1 3 is ue o e eaio ewee e eegy ga a ea caaciy As i is sow i [1] e ea caaciy C Δ/9Τ so i e oaie esus C — Ο a Τ —+ a e i aw o emoyamics is aso uie e cues coesoig o iee b aamees a ie x a ae eice i ig 1 I u e cues o iee x aamees u ow a ie b a = ae gie i ig wee ey ae isiue symmeicay wi esec o e cassica CS cue I e aamee is o equa o Ο e e eegy ga s emeaue as a eemum a maimum o osiie x a a miimum o egaie x, eseciey (see ig 3 I is ceay see a e asic aamees o ucuaio n a ae a sog a saigowa iuece o e aue o ciica emeaue wic is sow i a iec mae i ig e ciica emeaue gows u a osiie x a age b, a goes ow a egaie x, we ee sma Τ cou e oaie we b is eoug age I e eow Susecio 31 some iiguig aayica esus coesoig us o is case ae iey esee O e oe a e iuece o e aamee o Τ is summaise i ig 5 3.1.