Formation of Cooper Pairs as a Consequence of Exchange Interaction

Transcription

1 Stanislav Dolgopolov omation of Coope Pais as a Consequene of xhange Inteation Abstat: Analyzing the exhange enegy of two ondution eletons in a ystal at many-body appoah we find that the exhange enegy may be negative and, thus, the singlet state may be favoable. A full ovelap in the eal spae of the wave funtions of two ondution eletons leads to a deepe exhange enegy. hus the exhange inteation auses a bond between two ondution eletons in the eal spae. he singlet bond is possible beause the singlet eletons ae in aveage lose to positive ions than single eletons. If ondution eletons, befoe the paiing, ae put on the emi sufae in the momentum spae, then evey pai may exist pemanently in time. he motion of ondution eletons in the ystal may pevent the fomation of Coope pais, beause the kineti enegy of the motion is usually lage than the binding enegy in the pai. Condution eletons as standing waves have zeo momenta, hene thei momenta ae synhonous; theefoe the fomation of Coope pais is moe pobable than in ase of non-zeo momenta. he appoah of standing waves explains the invese isotope effet and many othe fats about supeondutos. Consideing the eletoni pais as bosons we find that a futhe neessay ondition fo supeondutivity is a non-zeo tempeatue of the ose-instein-condensation. Keywods: Coope pai; exhange inteation; Pauli xlusion Piniple; singlet state; supeondutivity; standing wave; ose-instein-condensation.. Intodution and motivation. he knowledge of oot auses of supeondutivity SC would explain many mysteious fats about all lasses of supeondutos. Howeve, a unified solution emains still an open question, uent theoies ae not univesal and explain many effets ambiguously []. he mainsteam theoies assume that SC is a esult of the eleton paiing at a mean-field appoximation, the spin odeing plays a pat fo the pai fomation [], [3], [4]. vey spin odeing is elated with the exhange inteation, whih influenes the total enegy of the eletons inteating with evey patile of the ystal. Moeove the exhange inteation may in itself ause binding states in quantum systems at a many-body appoah [5], [6]. heefoe the many-body appoah is moe appopiate to define the eleton states than the meanfield appoximation, and the ole of the exhange inteation seems to be uially impotant fo the pai fomation. In the wok is shown that the Pauli xlusion Piniple and its assoiated exhange inteation may in piniple lead to binding states of ondution eletons, whih unde etain onditions beome supeonduting.. omation of supeonduting pais. Nomally the spins of ondution eletons in a ystal ae unodeed beause the themal flutuations and own motion of eletons destoy the spin odeing. hus the spin of evey ondution eleton e is andom to spins of all othe eletons. his state of eleton e is designated as unpaied o single. If the spins of eletons e, e fom a singlet in thei ovelap aea in the eal spae, then the state of the eletons is designated as paied. vey unpaied ondution eleton has its auate spatial wave funtion desibing the position of the eleton in the ystal. Knowing the auate wave funtions of unpaied eletons e, e we an ompute thei exhange enegy. If two eletons e,e fom a singlet, then thei oveall position-spae wave funtion, is symmeti:,. whee, ae auate wave funtions of unpaied e,e ;, ae adius-vetos of e,e. he sum of the diet and exhange enegies D+J we find substituting, fom q.. into the integal with an oveall enegy opeato Oˆ, : D J, Oˆ,, Oˆ, Oˆ, If the exhange inteation of eletons e, e is vey weak, then thei spins emain unpaied and fully andom. In this ase the exhange enegy is negligible and the oveall enegy of the unpaied eletons is equal to the diet tem D as it should be fo the unpaied eletons with auate wave funtions, :., D Oˆ,.3

2 he exhange enegy J fom q.. is J Oˆ,.4 If the exhange inteation of eletons e, e is not negligible, then thei exhange tem J is not zeo. A singlet state of e, e is favoable, if thei exhange enegy J is negative [7]. J takes into aount the modifiation of the initially unpaied wave funtions esulting fom the paiing. his wave funtion modifiation influenes all inteations of e, e in the ystal; hene we must ompute the exhange enegy J fo Oˆ, as a sum of all inteations of e, e inluding thei kineti enegy, epulsion of e, e fom evey ondution eleton and attation of e, e to evey ion. If two eletons fom a tiplet, then thei oveall position-spae wave funtion is antisymmeti. he tiplet state of e, e is favoable, if thei exhange enegy J is positive. We assume that, have an ovelap in the eal spae and ontain simila atom obitals fo sample s- obitals; momenta of e, e along the ystal ae equal o zeo. In this ase, ae not othogonal as obitals of the gound state in H -moleule o in Helium atom; hene the ovelap integal appeaing in q..4 is not zeo:.5 he wave funtions of ondution eletons fade out slowly with the distane and an ove many points of lattie, thus the wave funtions of many eletons may ovelap in a shaed eal spae, so we onside at fist the limiting ase that, almost oinide in the eal spae, i.e.:.6 elow we will see that this assumption is tue beause a maximal ovelap in the eal spae of two paied wave funtions is enegetially favoable in ompaison to a patial ovelap. Using q..6 and non-othogonality of, in q..5 we an use fo q..4:, Substituting q..7 into.4 we obtain:.7 J Oˆ,.8 We see that q..8 is equal to q..3, i.e. in the ase of the full ovelap of non-othogonal wave funtions, the exhange enegy of two singlet eletons is equal to the oveall enegy of two initially unpaied eletons: J,.9 We may define that the eleton enegy outside of the ystal is zeo. hen the eleton enegy, inside the ystal should be negative, othewise the states of single eletons in the ystal ae instable. hus the exhange enegy J of e, e is also negative and the paied state is favoable in ompaison to the unpaied state. his onlusion has a lea physial meaning. he exhange tem takes into aount that the aveage distane between two singlet eletons deeases [8], [9], what ineases the epulsion between eletons. Conside a small aea aound one of ions in the ovelap aea of e, e in the eal spae; due to the xlusion Piniple two singlet eletons ae loated in this small aea with a pobability highe than two eletons with paallel spins, beause the eletons with paallel spins avoid eah othe and annot be put into a small aea i.e. the pobability that is little. If two eletons ae unpaied, then thei spins ae equipobably paallel o antipaallel, hene the eletons avoid eah othe, but do it weake than the eletons with paallel spins. hus the pobability to obseve in this small aea two unpaied eletons is lage than this pobability fo two eletons with paallel spins, and smalle than this pobability fo a singlet. heefoe the singlet eletons ae in aveage lose to the ion than two unpaied eletons. In simple wods two singlet eletons ae

3 lose to eah othe, the ions ae loated somewhee between eletons, theefoe the singlet eletons ae unavoidably lose to ions. he exhange tem takes into aount this deease in distane between eletons and ions. Conside that the ovelap aea of e, e in the eal spae is negligible i.e. integal is small, then the exhange enegy in q..4 is negligible. In this ase thee is no advantage of the singlet state, sine the eletons ae sepaated in the eal spae. hus the lage the ovelap, the geate the enegy advantage of the paiing. Consequently two paied wave funtions tend to a full oinidene in the eal spae and emain togethe in equilibium. hus the assumption in q..6 is justified. inally two eleton waves stay togethe beause the singlet state with a full ovelap in the eal spae edues thei total enegy. It is possible to show that the singlet paiing of some ondution eletons is favoable fo the whole ystal. We define all pats of the oveall enegy of two unpaied ondution eletons, :. he kineti enegies of eletons e and e, K e, K e ;. he potential enegy of epulsion of eleton e fom all ondution eletons in the ystal, P e, e ; 3. he potential enegy of epulsion of eleton e fom all ondution eletons in the ystal, P e, e ; 4. We must oet double ounting the epulsion between e, e, so we subtat the potential enegy of epulsion between eletons e, e, P e, e ; 5. he potential enegy of attation of eleton e to all ions in the ystal, P e, I ; 6. he potential enegy of attation of eleton e to all ions in the ystal, P e, I. he exhange enegy of e, e in q..9 is a sum of the points -6: J, K e K e P e, e P e, e P e, e P e, I P e, I. he points -6 ae a list of the ystal enegy tems, whih ontain the paied eletons e, e. If the ystal has many singlet pais, then the enegy of eah pai e, e ontains the points -6 howeve, we must again oet double i i ounting the epulsion between eletons of diffeent pais. he total enegy of the many-body ystal ontains additional enegy tems: 7. he kineti enegies of single ondution eletons; 8. he potential enegy of epulsion between single ondution eletons; 9. he potential enegy of attation of single ondution eletons to ions;. he potential enegy of epulsion between ions. he points - ae a full list of all ystal enegy tems. he single, emaining unpaied, eletons don t hange thei states; hene the ystal enegy tems in the points 7- emain unhanged. In the points -6 the oveall enegy of evey singlet pai e, e is lowe than the enegy of two unpaied eletons e, e i i i i due to the negative exhange enegy J. hus the singlet paiing of some ondution eletons inevitably leads to the enegy loweing of the whole ystal, the maosopi state an exist. One an obtain the same esult by exploing the many-body ystal Hamiltonian H... and the total ystal wave n funtion as a podut of nomalized auate wave funtions of evey single [] and paied eleton: ystal... i i i i... m m m m... H... n... i i i i... m m m m... Whee: i i, i i nomalized auate wave funtions of paied eletons; m m, m m auate wave funtions of unpaied eletons;... n adius-vetos of all eletons and ions. he Hamiltonian H... is a sum of opeatos fo enegies: n. he kineti enegies of paied and single ondution eletons;. he potential enegy of epulsion between all ondution eletons paied and single; 3. he potential enegy of attation to ions of all ondution eletons paied and single; 4. he potential enegy of epulsion between ions. All onlusions fom qs.-. ae valid if the eletons e, e ae two equal unning loh waves []:, t, t u exp i t i k.. nomalized 3

4 Howeve, it is a ae event that the momenta k of two unning waves ae equal pemanently in time. he momenta of eletons may be pemanently equal if befoe paiing eah eleton is a standing wave, whih is a sum of two equipobable loh waves popagating in opposite dietions:, t, t u exp i t i k u exp i t i k.3 he total momentum of eah eleton as standing wave is zeo, hene the momenta of eletons ae synhonous, so the paiing is possible despite the fat that the kineti enegy of eletons may be lage than thei paiing enegy. he oveall enegy of two unpaied eletons, is usually not abitaily small; onsequently the exhange enegy J in q..9 is also not abitaily small. he sign of the exhange tem J is elated with the sign of the enegy inement esulting fom the paiing. his enegy inement is elated with wave funtion modifiations and is not neessaily negligible if J is not negligible; theefoe the binding enegy in the singlet pai is also not negligible: Oˆ, Oˆ, A J.4 Whee, ae nomalized auate wave funtions of e, e afte paiing; A is a positive mateial onstant. Sine the binding paiing enegy is not abitaily weak, the unpaied nomal state of e, e is instable. Howeve, the paied state of e, e is pemanent in time only if extenal enegies tempeatue, adiation, magneti field ae weake than. he binding enegy is oughly elated to a paiing tempeatue *: k *. If the wave funtions of two ondution eletons in the ystal fo sample two s-eletons pemanently oinide in the eal spae and fom a pemanent singlet, then the eletons ae simila to the eletons in the gound state of Helium. he diffeene is that in the ystal the wave funtions ove many ions and the pai an move in an extenal potential, sine all ystal aeas ae equipotential fo the pai. In the gound state of Helium the singlet state is favoable despite the fat that the epulsion of eletons is maximal; the inease in attation of the singlet s-eletons to the Helium nuleus exeeds the inease in epulsion and in kineti enegy. he eletoni pai is stable like a valene bond in multi-atom systems, so the pai doesn t fom/lose any bonds in the ystal and doesn t absob/adiate any hemial enegy. vey standing wave is limited in the eal spae, theefoe a stable singlet of two standing waves an be onsideed as a zeo-spin boson. osons an fom the ose-instein-condensate C below a etain tempeatue C. If all eletoni pais ae in the C gound state, then the exitation enegy of evey pai is oughly elated to k C, whih is not zeo if the bosoni density is not zeo. If all extenal influenes ae weake than k C, then the pais emain in the C gound state and annot absob/adiate any enegy in the ystal; as a esult the total enegy and momentum of all pais don t dissipate, the pais flutuate without esistane despite the fat that the single eletons wee standing waves befoe paiing. hus the paiing of ondution eletons and C of the pais lead to the zeo esistivity likewise woks the supefluidity in Helium-4; the paiing enegy of eletons k * in Helium-4 is huge, wheeas k C is small. In an extenal magneti field H the ystal obtains an additional enegy density w,5 H ; the enegy of the singlet eletons splits. If the magneti enegy split H is weake than the binding enegy k * and exitation enegy k C, then the pai flutuates in the field H as a fee patile with a hage -e and zeo spin. Consequently thee ae no obstales to edistibute the non-dissipative flutuations of the pais into non-dissipative uents ompensating the additional magneti enegy w Meissne effet. If the binding enegy pe one eleton in q..4 is lage than the insulating band gap g of the ystal, then eletons an leave the valene band at a tempeatue *, hene the eletons may pai up despite the band gap. A doping in the ystal may edue the band gap and, thus, give ise to SC. his doping effet is obsevable in upates [], in ion-based supeondutos [3], in semiondutos [4]. A neessay ondition fo SC in metals is that the eletons, befoe paiing, ae lose to the emi sufae in the momentum spae. We show this assuming that the paiing ous when the enegy of the single eleton has a etain value λ. If the themal enegy doesn t exeed the binding enegy in q..4, then the onentation of the pais is not zeo and in the enegy spetum of single eletons ous a gap aound the value λ. he gap is thin, sine is usually small and the density of eletons with lose enegies is limited by the xlusion Piniple. If λ is notably less than the emi level, then thee ae single eletons with the enegy lage than λ. hese single eletons may dop to the level λ due to enegy flutuations and may, thus, fom new pais. he onentation of the paied eletons is limited by the thin gap aound λ ; theefoe the new pais eplae the aleady existing pais, whih lose the paied state. hus eah eleton is not pemanently paied, but it beomes peiodially unpaied. Duing evey swithing of states the eleton absobs/loses enegy, in the unpaied state the eleton is esistive, theefoe the momentum of the eleton and of the pai dissipates. hus the state with λ < annot keep a supeuent, despite the fat that the paiing is possible. If λ = then evey pai may exist pemanently in time, beause below a tempeatue the 4

5 single eletons annot oveome the enegy gap and annot eah the paiing level λ = ; as a esult the new pais don t aise and don t eplae the existing pais. Hene the swithing of states doesn t ou and the total momentum of the pais doesn t dissipate. hus the supeonduting paiing ous only fo single eletons in an enegy gap with as the uppe limit. Only suh pemanent pais ae supeonduting. At a tempeatue above the themal enegy is suffiient to satte single eletons though the enegy gap to the paiing level λ, theefoe new pais may aise and eplae the existing ones, the state beomes dissipative. 3. Paiing of standing waves. We found that the binding enegy in the singlet pai e, e is maximal if the ovelap integal is maximal, i.e. the wave funtions oinide in the eal spae. he enegy gap of supeondutos has ode of magnitude -3 ev, the emi level has ode of magnitude a few ev. Consequently the eleton motion an split the pai in the eal spae. he enegy of vey slow eletons is usually muh lowe than the emi level; hene the slow eletons annot fom supeonduting pais. wo eletons an fom a pai if thei momenta ae synhonous, but it is a ae event fo unning waves. he eletons as standing waves have zeo-momenta, hene thei momenta may be synhonous and the paiing is possible despite a lage kineti enegy. A standing wave ous as a esult of efletions of a unning wave fom a peiodi potential. he ondition of the standing wave in a ystal is the agg ondition [5]: n 3. Whee: n intege; length of the loh wave in q..; one of lattie paametes. Unde agg ondition the eleton beomes a set of standing waves with a zeo total momentum [6]. At n= in q. 3. the length of the standing wave is maximal:. A ystal has some values,, et. depending on the ystal axis and, thus, some values. ah value is linked to the enegy : h / m h 8 m 3. Whee m is the inetial mass of fee eleton. Not all mateials have ondution eletons with shot values λ= and with its assoiated. If the emi level of a ystal is low, then λ values ae lage than shot values and eleton enegies ae lowe than oesponding ; so the states with shot λ= ae empty and shot standing waves don t ou. In some metals is lose to it is equivalent that λ =. Pobably in some ystals the fomation of pais is possible at n lage than in q. 3.. o sample at n= the length of standing waves is λ =. vey at of the paiing is enegetially favoable; hene the enegy is emitted. hus the momentum of eah pai ineases fom zeo to a value p m.5. his momentum p should be added to the momenta of paied eletons in the momentum spae. heefoe the kineti enegies of eletons fom diffeent pais ae not equal, but distibuted into a spetum with the width. o instane, if the kineti enegy of nomal eletons was befoe paiing, then afte the paiing the kineti enegy beomes +. So the spetum of paied eletons is above the spetum of single eletons in the momentum spae. Note: the same onlusion follows fom the exhange enegy: the exhange tem fo kineti enegy is positive, i.e. the kineti enegy gows by paiing, wheeas the total enegy falls the same elation is valid fo all singlet bonds in hemisty. hus the paied eletons an ovelap with nomal eletons in the eal spae. he density of paied eletons is S, whee S is the density of states of ondution eletons. he spetum of nomal eletons obtains a oesponding gap - aound the value λ=, ae limits of the gap. he gap is not negligible if the themal enegy is insuffiient to destoy the pais. As shown above a neessay ondition fo SC is that is the uppe limit of the gap: =. he enegy gap is -, whee the gap bottom should be below λ= othewise new pais aise and eplae the existing ones, enegy dissipates. he density of the singlet eletons N s is limited by the enegy gap: N s S d 3.3 hus the enegies and states of single eletons below the gap - stay unhanged as assumed fo q... he eletons befoe supeonduting paiing must be lose to the emi sufae, i.e. the value must be lose to i.e.. eally, the enegy gap is muh less than ; theefoe if is signifiantly less than, then the uppe gap limit is also less than ; as shown above this ase is not supeonduting beause the pais ae not pemanent in time. o this eason Au, Ag, Cu whee signifiantly [7] ae not supeondutos. If 5

6 is signifiantly lage than, then thee ae no eletons with and the gap doesn t ou. o this eason in some stutues with a low a doping may aise the aie density and its assoiated up to the level whih is onstant, if doesn't hange. hus the doping may lead to SC, ineases. If the ystal is ovedoped, then is too lage;, vanishes. his doping effet explains the dome fom of phase diagams of supeondutos [8]. A double dome fom is possible due to the fat that the ystal has some lattie paametes depending on the ystal stutue. hus a lage value suppesses. If, then oesponds to the paiing enegy in q..4; howeve, the doping influenes on both maximum is not always pinned exatly to. and, so the - We an speify the enegy C k C is a mateial speifi onstant as a minimum themal enegy, whih is neessay to satte single eletons fom the bottom of the supeonduting gap to the paiing level, whee new pais aise and eplae the old ones. So we know about the tuning: if 3.4 C k C k if if and and hus Ck is an enegy aea between the SC - gap bottom and λ=. A gowing loses ontinuously the aea Ck by putting thee single nomal eletons. At = the aea Ck is oveed by single eletons. Knowing the density of single states S λ aound the SC-gap, we an alulate and C - paamete in Ck. he paiing enegy in q..4 is to find by investigating the wave funtion modifiations esulting fom the singlet paiing and leading onsistently to expeimental -values. he SC - gap bottom is to find fom qs 3.5, 3.6: C k if if C k he single-eleton-distibution on the SC-gap bottom is the emi-dia funtion f λ,. he single-eletononentation N at just below is: N S f, d S d 3.9 exp k At = the single-eleton-onentation N is: N S d 3. New pais don t aise at, so N is independent of. Hene N=N and q. 3.9 is equal to q. 3.: S d S d 3. exp k is alulable fom qs 3.7 o 3.8, λ= is known fom ystal stutue; hene, knowing S λ, we an alulate fom q. 3.. C paamete in Ck we find fom and using Ck = λ= -. Calulations with q. 3. show:,5 A. o emi liquids S λ is popotional to, then C depends slightly on the level and. Substituting the anges =.5-3 ev and =. - 5 K into q. 3. we find λ= and the ange C units;. C depends on the S λ -slope aound the level : the lage ds/d λ, the smalle C. On the zone edge λ= ds/d λ,5 may be lage than the -slope, theefoe C may be smalle than by a few units, i.e. C 3-7. hese C - values ae onsistent with expeiments. he isotope substitution is a way to tune by tuning to λ= based on the fat that depends on the effetive mass of eleton m* and eleton density N [9], wheeas λ= in q. 3. depends only on the lattie paamete. h 8m * 3N /3 3. 6

7 he isotope effet is a onsequene that the enegy of phonons is popotional to M -,5 M - mass of ion. he deease in M aises the enegy of phonons; theefoe the eleton-ion inteation and its assoiated efletion of eletons fom ions may intensify. his intensifiation is equivalent to the inease in the effetive mass m* and, thus, to the deease in, wheeas is almost unhanged. If the initial value is lage than λ= it is usual fo metals, then the deease in M pulls down lose to λ= ; hene gows the isotope oeffiient α>. If the initial value is less than λ=, then the deease in M pulls down away fom λ= ; hene may vanish α<. One an onlude that in ase the isotope effet may be weak α <,5. hus the diffeent values and sign of α [] ae a esult of the diffeent initial positions to λ=. Othe ways to tune by tuning to λ= ae: eleti field [] sine depends on the eletoni density; film thikness [], [3], [4], [5] sine the mutual of layeed stutues depends on laye thiknesses; the high pessue [6], [7], [8] sine depends on the distane between atoms. A futhe sample of the tuning is the alkali metals Li, Na, K, b, Cs. Only Lithium is supeonduto at ambient pessue [9] and only Lithium has =3,9 ev alulated by q. 3. in b-stutue, =3,49 Å elatively lose to 3, ev [3] at ambient tempeatue. he next andidate in supeondutos afte Lithium is Cesium: =,33 ev alulated by q. 3. in b-stutue, 6,4 3 5,3,,54 ev alulated by q. 3.; Cesium is eally supeonduto unde high pessue [3]. he high pessue ineases the density of ions, so m* ises and dops to ; theefoe gows both in Li and in Cs. he othe alkali metals ae not supeondutos and thei values ae lage than moe signifiantly than in Li and in Cs table. We note that and ineases both ae equally popotional to and., hene without the modifiation of m* an isotopi -edution able. Compaison of enegies and λ= fo alkali metals. λ= ae alulated by q. 3. fo lattie paametes and in b ystals. Lage -values ae not onsideed, sine they oespond to smalle λ= values. o Li is used the expeimental value at ambient tempeatue; fo othe alkali metals ae used values alulated by q. 3. oesponding oughly to the expeimental values. - fo fo fo fo ev ev ev ev ev Li Na K b Cs he desibed appoah explains the ombined isotope and high pessue effet in lithium [3]. In lithium-6 the high pessue and light isotope pull below the level λ=, so stats to diminish at a etain pessue p. In heavy lithium-7 emains above λ= at p, hene the ineasing pessue ontinues to pull down towad λ= ; ontinues to gow. As a esult the sign of d /dp above the pessue p is diffeent fo 6 Li and 7 Li. A pefet onduto annot fom the Coope pais, beause the eletons pass though the lattie without efletion, the standing waves don t aise, the eletoni wave pakets ae unlimited in the eal spae, hene a oelation of wave funtions in aodane with q.. is impossible beause of a finite speed of eleton-eleton inteation; so the exhange enegy and tends to zeo. hus the exhange enegy and the assoiated paiing enegy should be elated with the stength of the eleton-ion efletion via the potential enegy of eletons in q... A deepe potential enegy of eletons leads to a deepe J in q.. and, thus, to a stonge paiing enegy in q..4. On the othe hand, a deepe potential enegy means a deepe potential on eah ion, whih inteats/eflets ondution eletons moe stongly. So the singlet bond is stonge if the efletion of the unpaied eletons is stonge; hene may also be lage, but unde the ondition that is kept. he desibed appoah is onsistent with the fat that the high tempeatue supeondutos ae layeed stutues and poo ondutos in the nomal state. In some layeed stutues is possible to ombine two pooly ompatible things: a lage effetive mass m* elated to the stong eleton-ion inteation/efletion and a lage up to the value λ=. his is beause in thin films is lage than in bulk [33], wheeas the eleton efletion and m* in-plane may emain almost unhanged. In a 3-dimensional stutue is diffiult to ombine a lage m* > 5 m and a few ev. hus in quasi -dimensional systems an be highe. he paiing enegy in the poposed model is elated athe to the lattie potential than to the aie density. his enables SC at elatively low aie densities. A sample is supeonduting bismuth at ambient pessue, a semimetal with a low

8 aie density, N 3 7 /m 3 [34]. of bismuth is 5 mev, hene oesponding λ = h m m. hus in bismuth ae woking the long values of =.5 λ.39-8 m i.e. 8, 9 and simila. hese long standing waves exist only due to the high ystal puity, whee the eletoni mean fee path is muh lage than λ. hus SCpais emege fom the long standing waves on the emi level. Sine the eleton's waves ae long ange, the mean distane between eletons in one pai may be lage than the eleton's wave length λ -8 m, so the paiing is possible at a aie density less than λ -3 8 /m 3. o emi metals we an estimate the elation between C-tempeatue and paiing tempeatue, C /*. / 3 Assuming N S and 3.5 k *, using the well-known equations S m 3/ / fo s /3 emi liquid and N 3.35 N / m k fo bosoni gas we obtain: * C /3 C s 3.3 s o emi metals usually >>, hene C >* and SC depends athe on * than on C, i.e. =*< C. o stongly oelated systems may be ode of magnitude, so C may define SC, i.e. = C <*. If the paiing tempeatue * is lage than C i.e. *> C, then at above C the eletoni pais may be pemanent but non-supeonduting. he non-supeonduting pais ae obseved in [35]. Conside is lose to the SC-gap bottom and notable below λ= that is possible, fo example, in undedoped upates. hen the gap - in q. 3.3 is small and the singlet density N s is also small. Hene the oesponding C may be lowe than the paiing tempeatue *, beause * depends athe on the lattie potential than on N s. If the C is a neessay ondition fo SC and C <*, then = C ; so is also lowe than *. hus at between and * thee ae pemanent pais without SC. his may be elated with the pseudogap in some supeondutos. 4. Conlusion and disussion. he above agumentation shows that the exhange inteation may in itself ause the eletoni paiing in a ystal. hus the non-zeo paiing binding enegy is a esult of the Pauli xlusion Piniple. uthe neessay onditions fo SC: the pemaneny in time of evey eletoni pai povided with paiing of standing waves on the emi sufae; the non-zeo tempeatue of the C gound state of the eletoni pais. he appoah of the exhange enegy is lealy appliable when the waves, ontain s-obitals, beause the s-obitals envelop eah ion and the singlet paiing leads to a onvegene of eletons to ions. In ase of p-, d-, f- obitals the desibed appoah woks if the obitals envelop neaest neighbo ions. In this ase the singlet paiing depends on the obital oientation and on fatos influening the distane between ions pessue, doping et. he appoah of standing waves is elated with the agg-efletion, whih may fom diffation pattens in the ystal. his explains why the hage density ode pe-exists the supeondutivity in upates [36], [37], [38]. ollowing the poposed appoah we an define main ways to a highe in new supeondutos: A. he value should be tunable lose to λ= i.e. λ is tunable to shot. he tuning is possible by doping, pessue, film thikness, eleti field et.. he mateial should have a lage value m*, sine m* is elated to deepe lattie potentials in q.. ausing a deepe exhange enegy. Howeve, the ondition = λ= should be kept. A possible way to ombine lage values and m* is the low dimensionality; C. he mateial should be homogeneous miosopially, beause impuities/defets suppess by satteing the egula standing waves and values and. he appoah of standing waves is not appliable to systems with heavy femions, whee is muh smalle than λ=. ut in this ase the kineti enegy of eletons on the emi sufae may be smalle than the binding enegy in the pai; hene the pai may aise and exist pemanently in time ausing SC. One an wite oughly fo naow band systems: C k 4. heefoe a tuning of down may ause a highe in systems with heavy femions obseved in [39]. 5. efeenes. [] J.. Hish, M.. Maple,. Masiglio, Supeonduting mateials lasses: Intodution and oveview, Physia C, Vol [] adeen, John; Coope, Leon; Shieffe, J.. heoy of Supeondutivity. Physial eview 85:

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