k (but not necessarily much larger).

Transcription

1 Dolgopolov Sanislav Russian Fdraion Sank-rsburg Inracion bwn lcrons as wav packs and suprconduciviy In h work h suprconduciviy is plaind using h rprsnaion of valnc lcrons as packs of h wav funcions in crysals. Hr is provd ha h oal nrgy of wo inracing wav packs dpnds on h as shif bwn hm. A minimum of h oal nrgy is achivd whn wo lcron s wavs ar in opposi ass. Th minimum of h oal nrgy lads o a quanum bond bwn valnc lcrons in crysals wha inducs h suprconduciviy. Main ffcs rlad o h suprconduciviy and h high mpraur suprconduciviy ar brifly plaind on h bas of h quanum bond bwn valnc lcrons.. Inracion bwn lcrons as wav packs Hr w invsiga lcrons as packs of sanding wavs and prov ha rpulsion bwn wo valnc lcrons in a crysal dpnds on h as shif bwn hir wav funcions. A minimum of h rpulsion is nrgy-wis mor probabl han a maimum. I will b plaind why h minimum of h rpulsion is no obsrvabl undr usual condiions. W assum ha a h momn = lcron is a wav pack Ψ () and lcron is a wav pack Ψ () of sanding wavs propagaing along ais X. Th sanding wavs ar plan in h spac clos o h ais X conncing h cnrs of h wav packs Ψ () and Ψ (). Th cnr of h pack Ψ () is in = and of h pack Ψ () is in =. Boh packs ar idnical only hir compl ass ar diffrn. vry sanding wav pack can b rprsnd as a sum of wo idnical progrssing wavs propagaing in opposi dircions wihin h ara whr h lcron is locad. This rprsnaion is on of h soluions of h Schrödingr quaion for lcrons in a priodic ponial lik h ponial in crysals ( ): ( ) u( ) p( i ik) u( ) p( i ik) u( ) cos( k)p( i ) (. ) up i ( ) ik( ) up i ( ) ik( ) u cosk p i (. ) Whr : ω cyclic frquncy k wav numbr im shif of h compl ass of Ψ () and Ψ () u priodic funcion of wih a priod qual o h laic consan R ( ). Th funcion u is ral and rprsns approimaly a soluion of h Schrödingr quaion for lcron in a D bo ( 3 ). Th wav packs hav a fini lngh along ais X. This mus b plaind mor daild: in a prfc conducor a valnc lcron dos no inrac wih h laic h wav pack is fr and no localizd somwhr in h crysal bu blongs o h whol crysal hus h lngh of h pack is comparabl wih crysal dimnsions. In an insulaor h valnc lcron inracs wih h laic h wav pack is localizd somwhr in h crysal and hav a lngh ordr of magniud of h laic consan (on h lngh is locad h grar par of h lcron s dnsiy). This mans ha in mos ral crysals h wav pack of h valnc lcron is largr han h laic consan bu smallr han h macroscopic crysal dimnsions wha is namd as fini lngh. Sinc h wav packs hav a fini lngh w can dfin an ara wih boundaris ( ) as h ovrlapping ara of Ψ () and Ψ (). Th ovrlapping ara ( ) is a 3-dimnsional spac locad mainly bwn h cnrs of h wav packs. W assum ha insid h ovrlapping ara h ampliuds of boh wav packs ar qual ousid h ovrlapping ara h ampliud of on of h wav packs dominas absoluly in is ara. In addiion w assum ha h lngh of ( ) is largr han h wav lngh k (bu no ncssarily much largr). All mad assumpions ar acly fulfilld for wo idnical squar packs conaining a fini numbr of h wav

2 lnghs hrfor w will s ha all assumpions do no disurb h main principls o b dscribd in h work. Th wav dnsiy and h nrgy pcaion valu of ach wav pack Ψ () and Ψ () in h ovrlapping ara can b found from q. (.) and (.): 4u cos k (. 3) 4u cos k (. 4 ) * i d 4u cos k d (.5 ) * i d 4u cos k d (.6 ) W mark ha sinc h packs ar plan wavs h ovrlapping ara wih boundaris and is 3-dimnsional. can b considrd as a quanum Th common wav funcion in h ovrlapping ara sa. consiss of wo funcions Ψ () and Ψ (). If h mos par of h charg dnsiy of ach lcron is ousid h ovrlapping ara hn wo wav packs ar pracically indpndn of ach ohr and can inrfr lik as a sum of Ψ () and Ψ () and compu from q. (.) fr lcrons consqunly w can approima and (.): i i u cosk W can prov ha cosk (.7 ) is rally roughly a sum of Ψ () and Ψ () in cas whn h mos par of h charg dnsiy is ousid h ovrlapping ara. Th Hamilonian H for wo inracing lcrons can b wrin: Whr: p p H m m p and p opraors of momnum of lcrons and m mass of lcron p Th kinic nrgy rms ponial nrgy of inracion bwn wo lcrons. m m p in h Hamilonian ar idnical sinc h lcrons ar idnical wavs. If h mos par of h charg dnsiy of ach lcron is ousid h ovrlapping ara hn w can wri: (.7.) p p h d h d h d h d ( ) m m 8 m d 8 m d 8 m d 8 m d (.7.) Th q. (.7.) is h opraor samn ha h oal kinic nrgy of wo wakly ovrlappd lcrons is roughly a sum of h individual kinic nrgis. Th ponial nrgy dpnds on h disanc bwn lcrons. If h main par of h lcron s dnsiy is ousid h ovrlapping ara hn h lcrons inrac as wo chargd srs and h ponial nrgy is roughly qual: 3 4πε 4πε 4πε 4πε (.7. ) Th q. (.7.3) is h opraor samn ha h oal ponial nrgy of wo wakly ovrlappd lcrons is roughly a

3 sum of h individual ponial nrgis. Subsiuing q. (.7.) and (.7.3) ino (.7.) can b wrin: H h 8 m d ( ) 4πε d (.7.4) Th q. (.7.4) shows ha h common wav funcion can b approimad in ha cas as h sum. Th ysical sns of h q. (.7.4) is ha h wav funcions of wakly inracing lcrons rmain approimaly indpndn of ach ohr. For ampl if on of wo lcrons absorbs an rnal nrgy and changs is wav funcion h wav funcion of h scond lcron rmains roughly sady. Hr mus b nod ha if h ovrlapping ara covrs a larg par of h lcron s dnsiy hn h ovrlappd ampliuds of h funcions Ψ () and Ψ () ar larg h oal ponial nrgy and h oal kinic nrgy canno b rprsnd as a sum of h individual ponial and kinic nrgis. Consqunly canno b handld as a sum wha has h ysical sns sinc wo closly ovrlappd wav funcions ar dpndn of ach ohr. Th smallr is h ovrlapping ara of wo lcrons h mor acly is h q. (.7.4) and (.7). Th wav funcion rprsns an lcron s wav funcion wih a wav dnsiy pcaion. W can find a minimum of h nrgy and nrgy and a minimum of h dnsiy funcion of h as shif bwn Ψ () and Ψ (). Sinc h nrgy of boh wav packs ousid h ovrlapping ara is indpndn of h as shif a minimum of h nrgy as a corrsponds o h minimum of h oal nrgy of wo lcrons. Blow i will b provd ha h kinic nrgy and ponial nrgy minimal undr h sam condiions for h as shif as condiions of h minimal full nrgy h minimum of can b considrd as h ground sa nrgy of h sysm of wo lcrons. ar. For ha rason Th pcaion valu of h nrgy of h common wav funcion Ψ in h ovrlapping ara can b found from q. (.7): 4u cos k * i d cos k cosk 4 u cosk coskcos coskcos d d (.8 ) Th wav dnsiy of h common wav funcion Ψ in h ovrlapping ara can b found: 4u cos k cos k cosk coskcos 4u cosk coskcos (.9) Th valu in q. (.8) has a minimum as funcion of h as shif k and 4 u cosk coskcos d in q. (.8) is minimal. Th funcionu of h as shif hrfor i can b rplacd in h ingral by a consan valu whn h rm is posiiv and indpndn U u cons which can b 3

4 compud in vry concr cas. Thn h rm can b compud: coskcos d 4U cos cosk sink k (.) 4 u cosk Th valu ( - ) is largr han h wav lngh k and w assum bforhand ha h absolu valu cos 4k k. Thn h rm sink k 4k in q. (.) is nglibibly small and w can wri : coskcos d 4U cos cosk (.) 4 u cosk Th q. (.) and hav a minimum whn ru is on of wo condiions: n and cosk k n (. ). cos OR n and cosk k n (. ) 3 4. cos 3 Whr n n n 3 n 4 ar arbirary ingr numbrs or zro. Now w s ha h prior assumpion abou cosk is jusifid. Th condiions (.) and (.3) corrspond o oscillaions of Ψ () and Ψ () in opposi ass. asy o s ha h valu is maimal if Ψ () and Ψ () ar in qual ass i.. : ( n and k n ) OR ( n and k n ) (.4) Thus h oscillaions of Ψ () and Ψ () in opposi ass ar nrgy-wis mor probabl. Comparing q. (.8) and (.9) w s ha a minimal nrgy h ovrlapping ara a maimal corrsponds o a maimal corrsponds o a minimal wav dnsiy in. Obviously an ara of h maimal lcron s dnsiy rpls ach lcron mor innsivly han an ara of h minimal dnsiy. Sinc h ovrlapping ara is locad mainly bwn wo lcrons a wakning of h rpulsion bwn wo lcrons occurs if Ψ () and Ψ () ar in opposi ass. Thus h minimal nrgy of h oal wav funcion of wo lcrons corrsponds o a wakr rpulsion bwn hm wha is quivaln o an ffc of som addiional aracion. Th ysical sns of h wakr rpulsion by h minimal nrgy if Ψ () and Ψ () ar in opposi ass hn h valu conncing wo lcrons bu h valu ddydz y z can b plaind h following way: is minimal in h spac clos o h lin is consan ovr all spac around wo lcrons: ddydz y ddydz y ddydz y cons B (.5) ousid ( ) insid ( ) W assumd ha ousid h ovrlapping ara h ampliud of on of h individual wav packs dominas absoluly in is ara and is indpndn of h scond wav pack hn w can wri: y ddydz y ddydz y B (.6) ddydz ousid( ) ousid( ) ousid ( ) 4

5 B is consan bcaus ousid h ovrlapping ara boh individual wav packs ar indpndn of ach ohr and of h as shif bwn hm. As a rsul h ingral of h common wav dnsiy insid h ovrlapping ara is also consan: y ddydz y ddydz y B B B (.7) ddydz 3 insid ( ) ousid ( ) y This mans ha h common wav dnsiy dos no lav h ovrlapping ara and is rdisribud from h lin conncing cnrs of h wav packs o disan aras wihin h ovrlapping ara. Th wav dnsiy y z (and h ngaiv charg) is rdisribud ino a broadr spac around h lin conncing wo lcrons wha lads o a wakr ponial nrgy of rpulsion of h lcrons. Th rdisribuion of h common wav dnsiy is possibl du o h suprposiion of Ψ () and Ψ () in h disan aras for h rason ha h frons of boh lcron s packs ar acly paralll only locally in h spac clos o h lin conncing h lcrons. If Ψ () and Ψ () ar in qual ass hn h common wav dnsiy is concnrad mainly in h spac bwn wo lcrons and h ponial nrgy of h rpulsion is maimal. Now w prov ha h ponial nrgy of h common wav funcion rally is minimal by condiions (.) and (.3). onial nrgy of h lcron s funcion in h ovrlapping ara can b found ( 4 ): d d 4u cos k cos k cosk d 4u cosk coskcos coskcos (.8) Whr: is h funcion of h lcron s ponial nrgy in h fild of wo valnc lcrons. h lcrons rpl h common lcron s dnsiy. is compud in q. (.9). is posiiv bcaus Th q. (.8) for and (.8) for h full nrgy ar mahmaically idnical as funcions of k and hrfor condiions (.) and (.3) of h minimum ar valid for h minimum. Th funcion d d lcrons is rlad o h valu ( 5 ) also dpnds on h as shif bwn Ψ () and Ψ (). Sinc h kinic nrgy of d d h kinic nrgy in h ovrlapping ara also dpnds on h as shif. Obviously paks and dprssions of Ψ () and Ψ () in qual ass ar muually amplifid in opposi ass h paks and dprssions of Ψ () and Ψ () ar muually waknd hrfor in qual ass h d valu d and kinic nrgy ar largr han in opposi ass. Now w prov ha h kinic nrgy of h common wav funcion rally is minimal by condiions (.) and (.3). Th kinic nrgy of h lcron s dnsiy in h ovrlapping ara can b found ( 6 ): 5

6 h 8 m 4k u sin k du 4 cos k d du 4k u sin k d d d d Whr: m is h mass of lcron. is from q. (.7). h 8 m sin k sin k cos k cosk sin kcos cosk sin k cosk cos cosk sin k sin k d coskcos cosk (.9) By subsiuing condiions (.) or (.3) ino q. (.9) w find ha undr boh condiions. Sinc h kinic nrgy is no ngaiv w can affirm ha h condiions for h minimum (.) or (.3) ar valid for h minimum. Thus w provd ha h nrgis as funcions of h as shif bwn Ψ () and Ψ () ar minimal undr idnical condiions ha is whn h wav funcions Ψ () and Ψ () ar in opposi ass (in ohr words Ψ () and Ψ () ar asymmric). Thrfor on can say ha h asymmric sa of Ψ () and Ψ () corrsponds o h ground sa nrgy of h sysm of wo lcrons if boh wav funcions ar sanding wav packs. I is possibl o show ha for wo idnical progrssing wavs running acly in on dircion is valid h sam principl as for wo sanding wav packs: a minimal common nrgy and a minimal common wav dnsiy ar whn Ψ () and Ψ () in opposi ass. Sinc h common wav dnsiy is locad mainly in h spac bwn wo lcrons a minimal rpulsion is mor probabl han a maimal rpulsion. Th lcrons ar usually no sanding and no moving acly paralll hrfor h as shif bwn Ψ () and Ψ () flucuas vry quickly bwn zro and influncd by rnal filds and nrgis. Th as shif k is influncd by lcron s moions; h as shif is influncd by absorpions of rnal nrgis which chang h frquncy. If all valus k or bwn zro and ar quiprobabl hn h avrag nrgy pcaion wav funcions of h common wav funcion Ψ is qual o h sum of h individual nrgis of h individual ; h avrag common wav dnsiy wav dnsiis is qual o h sum of h individual. I is asy o prov by compuing h avrag valu h avrag common dnsiy from q. (.9): from q. (.8) and d 4u cosk coskcos d (.) coskcos d 4u cosk (.) Whr is k or. Usually all valus ar quiprobabl bcaus of many rnal influncs on h lcrons. In his cas h avrag 6

7 rpulsion bwn wo lcrons corrsponds o h inracion of wo chargs - hrfor usually w do no obsrv any dparurs from h Coulomb s law. Nighboring aoms in crysals build a bond which is nrgy-wis mor probabl for h aoms as whol. Usually h crysal bond rndrs impossibl an asymmric sa of wo nighboring lcrons. For ampl in a covaln bond wo valnc lcrons ar in qual ass ( 7 ) hrfor h asymmric sa of wo nighboring lcrons is impossibl. If h valu n hn from q. (.8) follows ha from q. (.9) follows ha. As a rsul by n wo lcrons inrac as wo chargs - (lik in cas whn all valus of h as shif ar quiprobabl).. Th inracion nrgy of wo lcrons Hr w will find h nrgy rlad o h as shifing bwn wo lcrons. In chapr w found ha wo lcrons as sanding wav packs Ψ () and Ψ () hav a minimum of h common nrgy whn hir as shif is qual o n or o n [condiions (.) and (.3)]. By n h lcrons inrac as chargs - and h valu is no minimal. Thrfor hr is som addiional aracion nrgy (as-dpndn nrgy) rlad o h muual posiion of h ass Ψ () and Ψ (). Th as-dpndn nrgy rducs h rpulsion bwn lcrons hrfor i is ngaiv. Usually w canno obsrv any dparurs from h Coulomb s law bcaus h as shif of Ψ () and Ψ () flucuas vry quickly bwn zro and influncd by rnal filds and nrgis. W can prss h as-dpndn nrgy as a diffrnc: min by n (.) Whr min is h minimal valu which can b found from condiions (.) or (.3). Thn q. (.) is: n k n OR n k n by n (.) Basd on q. (.) w can dfin h absolu valu as a minimal nrgy which is ncssary o chang h as shif bwn Ψ () and Ψ () by /. is qual o an nrgy which changs h as of on of wo lcrons by /. W can compu h minimal nrgy which shifs h as of on lcron by /. W assum ha a h momn = wo lcrons ar sanding wav packs Ψ () and Ψ () synchronizd in h sa of h minimal oal nrgy. lcron absorbs an rnal nrgy 7 and says afr ha (as bfor ha) a sanding wav pack hrfor h coordina X of h lcron is consan and h as shif of h wav funcion occurs only du o h chang of h cyclic frquncy ω: (. 3) W dfin h communicaion priod as duraion of h chang of h signals Ψ () and Ψ () bwn wo lcrons: (. 4 ) V Whr: V sparaion bwn lcrons (bwn cnrs of h wav packs) propagaion vlociy of h lcron s wavs forming h sanding wav packs Ψ () and Ψ ().

8 In crysals h maimal vlociy of h lcron s wavs V is qual o h Frmi vlociy V f. Th maning of h communicaion priod can b plaind h following way: A h momn = lcron and lcron ar synchronizd in opposi ass. Th lcron snds is signal (conaining informaion abou h as of h lcron ) o h lcron. lcron rcivs h signal and insanly snds is own signal (conaining informaion abou h as of h lcron ) back o lcron. If during h ravl of h signal hr and of h signal back h as of h lcron dos no chang hn wo lcrons say synchronizd in opposi ass. If during h ravl hr and back h as of h lcron shifs by / or mor hn h asymmric synchronizaion of wo lcrons is los. Th as shif of lcron during h communicaion priod can b found from q. (.3) and (.4) : V (.5 ) Sinc h minimal as shif is qual o / h minimal dsynchronizing nrgy is from q. (.5) : In chapr 3 w will s ha h nrgy V (. 6 ) is oo wak o b obsrvd undr usual condiions. 3. Th inracion nrgy of wo aoms in a crysal Hr w invsiga inracion of wo nural aoms in a crysal ach of hm conains on valnc lcron. W will s ha a sa of lcron s wav funcions Ψ () and Ψ () in opposi ass can b nrgy-wis mor probabl han h sa in qual ass wha lads o a bond bwn nural aoms in a crysal a sparaions much largr han - m. I will b plaind why h bond bwn nural aoms is usually no obsrvabl. Th as synchronizaion of wo aoms occurs rahr du o h inracion of hir valnc lcrons bcaus h wav funcions of h inrnal lcrons ar limid in h spac by h siz of h aom. W assum ha boh aoms ar moionlss. Thir valnc lcrons ar localizd in h crysal and can b rprsnd as sanding wav packs in form of q. (.) and (.). If h lcron s wav funcions Ψ () and Ψ () ar in qual ass hn h lcron s dnsiy concnras in h spac bwn aoms a posiiv charg prvails in h spac clos o h nuclus hrfor wo aoms arac as a whol o h ara of h high lcron s dnsiy ( 8 ). Thir valnc lcrons rpl from h ara of h high lcron s dnsiy bu h aracion of aoms as a whol prvails. Th covaln bond iss du o h principl. Th ypical radius of h covaln bond is ordr of magniud - m. Th shorr disancs ar impossibl bcaus of h ion rpulsion. If Ψ () and Ψ () ar in opposi ass hn h lcron s dnsiy is rdisribud from h lin conncing h aoms ino a broadr spac hrfor again h posiiv charg prvails in h spac clos o h nuclus bu now h lcron s dnsiy is minimal in h spac bwn aoms. As a rsul h aoms as a whol rpl ( 9 ) bu h valnc lcron of on aom aracs o h prvailing posiiv charg of h scond aom. Th rpulsion of aoms as a whol prvails. Th nrgy of h rpulsion r of wo aoms as a whol (rlad o h spaial disribuion of h lcron s dnsiy) can b gnrally approimad by a rial funcion: Whr: sparaion bwn aoms B( ) r ( 3. ) ( ) B () and ( ) ar posiiv fini funcions (spcific for vry marial). I is possibl o show ha by sparaions much largr han - m is largr han. Rally: for rpulsion of wo 8

9 poin-lik chargs bu in our cas h disribuion of h lcron s dnsiy is spaial. By incrasing dimnsions of h ovrlapping ara and ampliuds of Ψ () and Ψ () hr dcras hrfor wo aoms bgin o inrac as nural paricls consqunly h rpulsion nrgy nds o zro fasr han in cas of wo poin-lik chargs. Indpndnly of h as shif bwn Ψ () and Ψ () by sparaions m h inracion nrgy of wo nural aoms (rlad o h spaial disribuion of h lcron s dnsiy) is ngligibly small in comparison o h inracion nrgy of wo poin-lik chargs. For ha rason: ( If Ψ () and Ψ () ar in opposi ass hn h full nrgy h ngaiv as-dpndn nrgy of h valnc lcrons from q. (.6): m) ( 3. ) a of h asymmric sa of wo aoms is rducd by B( ) V a r ( 3. 3) ( ) Sinc( ) h full nrgy a is ngaiv and minimal somwhr by m (s figur ) hrfor a wak bond bwn wo aoms is possibl whn h wav funcions of hir valnc lcrons ar in opposi ass. Th bond w dfin as h asymmric bond and is nrgy as h nrgy of h asymmric bond. On ohr way o show h isnc of h asymmric bond is an analysis of h as-dpndn nrgy. Th nrgy is rlad o h as shifing and rducs h common nrgy of wo lcrons by h as-dpndn ponial nrgy (rlad o h spaial rdisribuion of h lcron s dnsiy) and also by h as-dpndn d kinic nrgy rlad o h funcion d. Th q. (3.3) can b wrin: a a r r ( 3.4) Whr h nrgis and can b prssd lik in q. (.). and ar ngaiv. Th rm r is a dirc consqunc of h spaial disribuion of h lcron s dnsiy; consqunly r is rlad o h ponial nrgy. If by larg sparaions nds o zro slowr han hn nds o zro slowr han r consqunly in q. (3.4) h ngaiv rm nds o zro slowr han h posiiv rm r and h nrgy a has a ngaiv minimum. Thus w mus show ha nds o zro slowr han by larg. Th raio of o can b simad h 9

10 following way: w subsiu q. (.8) (.9) ino q. (.) applid for and. Afr all mahmaical ransformaions h rsul is: ( 3.5 ) Whr : kinic nrgis of h wav funcions of lcron and of lcron in h ovrlapping ara ( ) ponial nrgy of h wav funcion of lcron in h ovrlapping ara ( ) in h fild of lcron ponial nrgy of h wav funcion of lcron in h ovrlapping ara ( ) in h fild of lcron. Sinc h ovrlapping ara ( ) is locad in h middl bwn wo lcrons h raio of h kinic nrgy o h ponial nrgy of ach lcron can b roughly simad as Thus h raio (3.5) is ordr of magniud: m V. 4πε ~ m V 4πε In crysals as V can b usd h Frmi vlociyv f ~ 6 m/ s. Thn h raio (3.6) for crysals is: ( 3.6 ) ~ 9 ( 3.7 ) As pcd whn m ; consqunly h kinic nrgy nds o zro slowr han h ponial nrgy; h ysical planaion for his is vry simpl: h kinic nrgy is lss dpndn of h sparaion. Hr should b nod ha from q. (3.7) follows ha by ~ 5 m (ordr of magniud of h aom siz) ~ and ~. This is an addiional argumn ha h raio (3.7) is corrc bcaus i is in lin wih h rprsnaion of h aom in h crysal as harmonic oscillaor (sinc a propry of a harmonic oscillaor is ). By sparaions ordr of magniud - m h posiiv rm r in q. (3.4) (rlad o h ponial nrgy) prvails h aoms rpl a bond bwn aoms in asymmric sa Ψ () and Ψ () is impossibl. By largr sparaions h rm (rlad by larg mainly o h kinic nrgy ) prvails. Thus w s again as in q. (3.3) ha h nrgy of h asymmric bond a is ngaiv and minimal somwhr by m as a rsul a bond bwn nural aoms is possibl whn h wav funcions of hir valnc lcrons ar in opposi ass. Usually w canno obsrv h asymmric bond bwn nural aoms bcaus h as shif flucuas vry quickly bwn zro and undr influnc of moions and rnal nrgis (mpraur magnic and lcric filds oons) which disurb h as synchronizaion of Ψ () and Ψ () bfor i bcoms sabl. For ampl a sparaion bwn wo aoms radiaion T : 9 m and V ~ 6 m/s in q. (.6) corrsponds o a wav lngh of h hrmal h c h c T 9 m 4 6 m ( 3. 8) V Thrfor h hrmal radiaion can disurb h asymmric synchronizaion of valnc lcrons a sparaions 9 m. A shorr disancs ohr yps of bonds ar nrgy-wis mor probabl. Thus ffcs of h

11 asymmric synchronizaion ar obsrvabl only for som marials and undr spcial condiions. 4. Condiions for h isnc of h asymmric bond in crysals Hr w prov ha h asymmric bond in a crysal is impossibl if h lcrons ar progrssing wavs (and no sanding wavs). W will also dfin ncssary condiions for h isnc of h asymmric bond in a crysal. A valnc lcron in a crysal can b rprsnd as a progrssing or a sanding plan wav ( ) of h wav funcion Ψ(y). Thrfor poins of laic wih valnc lcrons can build h dscribd asymmric bond. If h moion of h paird lcrons is causd by h asymmric bond hn h lcrons say cohrn and h bond is sabl. If h lcrons ar progrssing wavs and h wav moion is causd by ohr influncs hn h as synchronizaion is los a disancs ordr of magniud - m bcaus of h moion (cp h cas whn wo lcrons run acly paralll). To prov i w compu h as shif bwn Ψ () and Ψ () causd by h lcron s wav moion during a communicaion priod : Whr is a shif of on of h lcrons during a communicaion priod. k ( 4. ) is found in q. (.4) : can b found as : V V ( 4. ) h ( 4. 3 ) k m V If during on communicaion priod h as shif hn h synchronizaion of wo lcrons is los. Th sparaion of a sabl asymmric bond can b found by subsiuing q. (.4) (4.) (4.3) and ino q. (4.): h ( 4.4 ) 8m V As vlociy of lcron s wavs V in a crysal w can ak h Frmi vlociy m/s. Finally for a sabl asymmric bond is ordr of magniud: V f which is ordr of magniud 6 ~ m ( 4. 5 ) A longr disancs h asymmric bond is dsroyd by h chaoic lcron s moions a disancs ordr of magniud - m h asymmric bond canno is for h rason ha ohr yps of bond ar nrgy-wis mor probabl. Thus h synchronizaion of lcron s wavs in opposi ass canno sar if h lcrons in a crysal ar progrssing wavs bcaus h as shif of wo progrssing wavs is no sabl and dpnds on h vlociy vcor V which is influncd by nrgy flucuaions in h crysal. Consqunly h asymmric bond can occur in wo cass:. Valnc lcrons ar localizd by ions of laic and canno mov in h crysal.. Valnc lcrons ar fr bu so-calld Brgg-rflcion iss in h crysal and progrssing wavs bcom sanding du o h ordrd rflcion on h priodic ponial of laic. Sricly spaking h q. (.) and (.) ar soluions of h Schrödingr quaion for lcrons undr on of hs wo condiions. Gnrally a valnc lcron mus b rprsnd as a sum of sanding and progrssing wavs.

12 Th condiion of h Brgg-rflcion ( ) is: Whr : n ingr lngh of h lcron s wav Ψ(y) n R ( 4. 6 ) R laic consan By falling mpraur vibraions of h poins of laic bcom wakr consqunly R bcoms mor sabl and disurbancs of h sanding wavs wakn. A h sam im h hrmal nrgy disurbing h as synchronizaion dcrass hrfor h valnc lcrons can b synchronizd a longr sparaions. By a criical low mpraur T c h possibl sparaions bwn valnc lcrons corrspond o h sparaion of h minimal asymmric nrgy a (figur ); as a rsul h valnc lcrons in h crysal bcom paird. Th asymmric bond dos no influnc main propris of h crysal bcaus h nars aoms rmain conncd by h usual bond of h crysal. nrgy flucuaions in h crysal and rnal nrgis disurb h asymmric synchronizaion of h lcron s wavs. 5. Suprconduciviy Hr w show ha h abov dscribd synchronizaion of valnc lcrons in opposi ass can plain h isnc of h suprconduciviy. Main ffcs rlad o h suprconduciviy will b brifly plaind on h bas of h asymmric bond. As shown abov wo valnc lcrons in a crysal can build h asymmric bond. In an rnal lcric fild h pair of synchronizd lcrons ravls in h crysal from poin o poin of laic. Th asymmric synchronizaion lass if h nrgy of h asymmric bond in q. (3.3) is lowr han h aracion nrgy of ach valnc lcron wih is a ion of laic (s figur h nrgy lvl = is h boom of h conducion band). A pair of h synchronizd g lcrons and h laic canno chang h nrgy porion lss han a valu : a g ( 5. ) Whr: g is absolu valu of h inracion nrgy of a normal lcron and is ion of laic. For mals g is clos o zro for smiconducors and insulaors valnc band. nrgy gap of suprconducor. g is approimaly qual o h nrgy gap bwn conducon band and

13 a can b simad in q. (3.3). W can assum ha V V f and a larg sparaions r : Vf a r ( 5. ) Whr is h avrag sparaion bwn synchronizd lcrons (corrlaion lngh of suprconducor). Th condiion (5.) can b plaind h following way: h nrgy of a suprconduciv lcron is lowr han h minimal nrgy of a normal valnc lcron hrfor h suprconduciv lcron canno los any nrgy by inracion wih poins of laic. Now w rmmbr abou h minimum nrgy of h common wav funcion. Th minimum nrgy mans ha h sysm of wo lcrons mus radia by h ransiion ino h asymmric sa of Ψ () and Ψ () from a sa wih anohr posiion of h as shif. Obviously h radiaion is h nrgy which can b absorbd back and disurb h asymmric synchronizaion. A pair of asymmric lcrons can mov in h crysal wihou rsisanc (suprconduciv) unil on of wo lcrons absorbs a porion of rnal nrgy largr han. Afr h nrgy absorpion h asymmric bond is dsroyd boh lcrons bcom normal and inrac wih h laic suprconduciviy disappars. As h disurbing rnal nrgy can b h hrmal nrgy oons moving paricls magnic and lcric filds inducing criical currns c. If disurbing influncs work a h sam im hir impacs on h as shif bwn Ψ () and Ψ () ar summd up. Two lcrons in h asymmric bond can b considrd as a quanum sysm. Two idnical lcrons in on quanum sysm occupy fr quanum sas; hrfor wo asymmric lcrons ak opposi spins. In addiion in a quanum sysm h oal momnum of h lcrons mus b zro (ohrwis h sysm radias). Th quanum sysm of h paird lcrons has 6 dgrs of frdom: 3 ranslaional + roaional + oscillaory (oscillaions along h lin conncing lcrons). In h hrmodynamics is provd ha vry ranslaional or roaional dgr of frdom prcivs k T of h hrmal nrgy an oscillaory dgr of frdom prcivs on kt of h hrmal nrgy. Thn h oal hrmal nrgy for on quanum sysm lcron-lcron is: kt 3 kt kt 35 k T Thus a pair can prciv3 5kT. Th asymmric synchronizaion and suprconduciviy disappar as soon as h lcron s pair absorbs a porion of h hrmal nrgy largr han hrfor w can find h criical mpraur of h suprconduciviy by us of q. (5.) and (5.): (5.3) 35 ktс a g r g (5.4) Normal fr lcrons usually do no hav h addiional roaional and oscillaory dgrs of frdom. Obviously i plains why h hrmal capaciy of lcrons incrass by ransiion ino h suprconduciv sa. Low mpraur suprconducors ar usually mals hrfor hir valnc lcrons ar fr in h crysal and bcom sanding packs du o h Brgg-rflcion. Th lcrons as sanding packs coninu o communica bcaus vry sanding wav pack conains progrssing wavs propagaing in opposi dircions. Th condiion (5.) is ncssary for suprconduciviy bu no sufficin. g of prfc mals is clos o zro hrfor inracion of normal lcrons wih poins of laic is vry wak consqunly h rflcion of lcron s wavs on h poins of laic is wak and vry low mpraur can disurb h sanding wavs. Thus conducion lcrons say progrssing wavs a mpraur clos o zro whr h nrgy 3 5 k T is comparabl wih h nrgy of zrooscillaions of h crysal. Th zro-oscillaions disurb h asymmric bond and h suprconduciviy is impossibl. Obviously for h suprconduciviy is ncssary ha h inracion nrgy of valnc lcrons wih poins of laic g would b largr han h nrgy of zro-oscillaions in h crysal. Ohrwis ach lcron is localizd in h с 3

14 whol crysal and canno b considrd as a wav pack wih a fini lngh. On h ohr hand if h nrgy g is largr han h nrgy of h asymmric bond a hn h nrgy gap is ngaiv and h lcrons canno bcom suprconduciv. Th nrgy on h valu a canno b arbirary larg bcaus i dpnds in q. (5.) and a small ohr yps of bond bcom mor probabl. Firs of all h covaln bond (i.. h symmric sa of h wav funcions) can b mor probabl a shorr disancs bwn aoms. Thrfor no vry marial is suprconducor. Th wav funcion of vry valnc lcron propagas in all dircions (which can b quiprobabl) hrfor h asymmric bond of on lcron can swich from on o h ohr lcron on a sr of h radius. Thus h asymmric bond builds a 3-diminsional communiy of muually conncd lcrons in h crysal. Th communiy can covr h whol macroscopic crysal. Thus h quanum ffc obains obsrvabl macroscopic propris. Th Missnr ffc can b plaind by h macroscopic bhavior of h corrlad lcrons. All lcrons hav small vibraions in h crysal. In an rnal magnic fild all chaoic vibraions of h lcrons ar influncd by h Lornz forc which ris o cra currns compnsaing h rnal magnic fild (h Lnz rul). Sinc mos valnc lcrons ar bound in h 3-diminsional communiy hy canno mov indpndnly of ach ohr lik singl lcrons wih a zro rsisanc. Thrfor h 3-diminsional communiy cras a common currn which compnsas h rnal magnic fild. This procss can b calld h invrs ffc of h London momn. Th crysal laic says moionlss bcaus h bound lcrons mov wihou rsisanc around a common macroscopic ais. Th principal conras o h singl lcrons wih zro rsisanc is ha ach singl lcron movs indpndnly of ohr lcrons nighboring microscopic currns of h singl lcrons compnsa ach ohr and only surfac currns ar no compnsad whil h bound (corrlad) lcrons mov around on common ais as a whol wha cras a common currn in h bulk. To dmonsra h approach in h abl is h nrgy gap = 35kT c and a compud by q. (5.) for a low mpraur suprconducor aluminium. Aluminium is akn bcaus i is a good conducor ( g ) and a wak paramagnic (µ ) hrfor dviaions from h hory ar minimal. For suprconducors wih a srongr lcric rsisanc w mus ak ino accoun h inracion nrgy of valnc lcrons wih poins of laic g which rducs h nrgy gap in q. (5.). In addiion h inracion of valnc lcrons and poins of laic influncs h Frmi nrgy of lcrons wha affcs h nrgy of h asymmric bond a low mpraurs can influnc ssnially h inrnal nrgis of crysals. a via h vlociy V f. Magnic propris Tabl aramr Valus for aluminium Criical mpraur of suprconducor T c 75 Corrlaion lngh ( ) m 5-6 Frmi vlociy V f m/s ~ 6 nrgy gap of suprconducor =35kT c mv 35 a compud by quaion (5.) mv ~ High mpraur suprconduciviy Hr w show ha h abov dscribd asymmric bond can plain h high mpraur suprconduciviy. In HT suprconducors works h sam principl as in LT suprconducors: aniasd synchronizaion of lcron s wav funcions cras h asymmric bond bwn valnc lcrons h paird lcrons do no inrac wih poins 4

15 of laic by nrgy porions lss han h diffrnc. a Th Brgg-rflcion of lcron s wavs aks plac in HT suprconducors as wll bu dos no play a ky rol. I mus b plaind mor daild: vry valnc lcron in a crysal can b considrd as a sum of sanding and progrssing wavs (in ha cas in h q. (.) and (.) on of wo ponnial rms obains a consan muliplir D ). In a good conducor in h normal sa h ampliud of h progrssing wav prvails hrfor h lcron is fr in h crysal. In chapr 4 is provd ha h asymmric synchronizaion of wo progrssing wavs is impossibl. In a crysal wih a larg lcric rsisanc h lcron is parly localizd nar is ion of laic hrfor h ampliud of h sanding wav prvails. Th fac is ha HT suprconducors ar usually bad conducors in h normal sa; hir valnc lcrons ar lss mobil in h crysal and br coupld wih ions of laic. Thus h lcrons ar alrady o a dgr sanding wavs. This maks a synchronizaion of lcrons much asir bcaus wo sanding wavs do no scar bfor hir synchronizaion bcoms sabl. A coupld lcron lavs h valnc band and bcoms a progrssing wav a a highr mpraur han an lcron in a good mal; consqunly h suprconduciviy of a bad conducor is possibl a a highr mpraur. In addiion h br coupling of valnc lcrons wih poins of laic lads o a mor innsiv Brgg-rflcion of h wav funcion on ach poin of laic consqunly ach lcron s wav pack bcoms mor spac-saving and can communica wih ohr packs a shorr disancs. This incrass h asymmric nrgy In spi of h fid locaion of valnc lcrons h spd of h signal chang Ψ () and Ψ () in q. (5.) is ordr of magniud V f bcaus small ampliuds of h wav funcions propaga by h vlociy V f far away from h main locaion of ach lcron wha maks possibl h synchonizaion of h wav packs Ψ () and Ψ () as a whol. Th condiion of q. (5.) is valid for HT suprconducors: bu now g is larg hrfor h corrlaion lngh mus b ssnially shorr han in LT suprconducors. Using q. (.6) (3.3) (5.) w can prss : V f 4 r g g a g a. ( 6. ) Th q. (6.) shows ha for h suprconduciviy a larg g rquirs a small. On h ohr hand a shor disancs h ohr yps of bond comp wih h asymmric bond. Firs of all h covaln bond (i.. symmric sa of wav funcions) can b mor probabl a shor disancs bwn aoms. Thrfor no vry marial is suprconducor. I is asy o prss g from q. (6.): V f g r ( 6. ) 4 Whr r is a vanishingly small a larg sparaions. For ampl by = 5-9 m and V f = 6 m/s h valu g may rach ordr of magniud V. Tha plains why som bad conducors and smiconducors ar suprconducors. Du o h asymmric bond h paird lcrons ovrsp h ponial g g and ransi from h valnc band dircly o h suprconduciv sa hus h nrgy no longr affcs h lcron s moion. On could call i a unnling ffc. Obviously a similar procss occurs in a Josson juncion. Th hrmal nrgy ohr nrgy yps and magnic fild cra flucuaions of h as shif bwn Ψ () and Ψ () in HT suprconducors and rduc in h sam way as in LT suprconducors. 5

16 Rfrncs. Robr L. Sproull Modrn hysics Moscow Nauka 974. Ginzburg V. L. Andryushin. A. Suprconduciviy Alfa-M 6 [ 8.5 q. 8.3] [ 8.5 p.66] 3 [ 8.5 p.66 and 5.4 p.5] 4 [ p.58 q. (D6)] in q. (D6) is h im-indpndn soluion () o prov ha h rplacmn is admissibl. 5 [ p.58 q. (D6)] 6 [ p.58 q. (D6)] 7 [ 7.5 p.37] 8 [ 7.5 p.37] 9 [ 7.5 p.37] [ 8.5] [ 8.5 q. (8.7)] [ Chapr 3] in h work is usd h im-dpndn soluion. I is asy 6