quasi-two-dimensional bose condensation and high tc superconductivity

Volume 135, number2 PHYSICSLETTERSA 13 February1989

QUASI-TWO-DIMENSIONAL BOSECONDENSATIONAND HIGH T. SUPERCONDUCTIVITY

Wen-Zu LICCAST(WorldLaboratory),andPhysicsDepartment,ZhejiangUniversity, Hangzhou,PRChina

Jian-BinWU, FengCHEN,Jin-ChangTANGPhysicsDepartment,ZhejiangUniversity,Hangzhou,PRChina

and

Kai-JiaCHENGNorth WestNuci. Tech.Instituteof China,Xian, PRChina

Received16 August 1988; revisedmanuscriptreceived22 November1988;acceptedfor publication5 December1988Communicatedby D.Bloch

Startingfromthequasi-two-dimensionalBosecondensation,westudytheinfluenceof layerthicknessandoxygen-vacancyonthecritical temperatureT~in realhigh T~superconductors.Theresultsarequalitatively consistentwith knownexperiments.

Since the discoveryof high T~superconductors, solvethisproblem,Andersonet a!. [5] did notmakegreatprogresshasbeenmadein the theory of su- theassumptionof a well-developedinterlayerband,perconductivity.Theresonatingvalencebond(RYB) butconsideredJosephson-likecouplingbetweenCu—[1—3] is the most favourableone. There are three 0 planes; this naturally drived a 2e condensate.kinds of excitations in the RVB vacuum:an un- Kivelson et al. [6] andWen [7], however,specu-bondedspin (spinon),an empty site (holon), and lated that the RVB statemay contain some addi-a chargedelectron.Theyareconstrainedin the Cu— tional structureswhich allow the h/2e flux to exist0 plane.It is the crucial propertyof the RVB vac- evenin thechargee condensate.Therearestill otheruumthat spinonsare neutralfermions, andholons approaches,oneof them [8,91 conjecturedthat theare chargedbosons.In real RVB superconductors holonsare anyonsobeying half-integerstatistics,athereis a net numberof holons,andthetotal num- pairofholonsform aboson,thusthesystemcanhaveberis conserved.Thus,if theholonsarebosons,Bose Bosecondensation.Nowadays,the problemof “2econdensationis possible.It is naturalto regardthis flux quantum” receivesextensiveattention;wewillas the mechanismfor superconductivity.Because discussit elsewhere.In thispaper,however,only Bosetwo-dimensionalBose condensationcannot occur, condensationis considered.the three-dimensionalquantum fluctuation effect The structure of YBa2Cu307, Bi2 Sr2CaCu308,

shouldbe taken into account.Wen et al. [4] con- etc. tellsusthateachlayermight includeseveralCu—sideredfree bosonswith weak dispersionalong the 0 planes, the coupling among them should bec-axis, which wasshownto lead to Bosecondensa- strongerthanthatbetweentwo nearestneighborlay-tionof a“(2 +e)-dimensionalBosegas”.Thisseems ers. On the otherhand,oxygen-vacancyshouldin-to contradictwith the magneticflux measurements fluencethe condensationgreatly. Thetreatmentsinof the new oxide superconductorswhich show that refs. [4,5,8] did not takethesefacts into account.the flux quantumis h/2e,andnot h/e. In order to Wethusstudyherethe Bosecondensationof a gen-

0375-9601/89/s03.50© ElsevierSciencePublishersB.V. 137(North-HollandPhysicsPublishingDivision)


erallayeredmodelin which eachlayermightinclude m ____________

severalCu—Uplanes(in somecasesincludingM—0 ~ ~ exp(/3~~~)—

planes,M ~ Cu). Thebosonscanmove freely in the mitlayer,while the hoppingof them betweenlayerscan 2ML3[exp(fl~f~)— 1]beignored.Basedonthismodel,‘the oxygen-vacancydistribution is introducedin a layer to studyits in- take m/M—.0.2, L3—~l0—~°m3, fl~’—~0.0l eV,fluenceon T~. y—~lO’°cm3.

As stated above, the Hamiltonian for a boson (ii) If /3~‘—.

moving freely in a layercanbe written:m _______

H=P~/2m+P~/2m+P~/2M+V(z), ~L~exp(fl~f?)—l

V(z)=0, ze[0,L] , mf3~’ 12ith2L ~e12_l Y~

=~, $[0,L], (1)

whereL is the thicknessofalayer;m andMare the takem~’me,L—~102 A, y—~l0’~cm3.effectivemassesof a bosonmovingparallelandper- (iii) For /~ >> 1?, that is, L-+co, dfl~/dL—0.pendicularto thelayerrespectively.The eigenvalues Experimentally[10], .A~ 1 021 cm ~, thusthe con-are dition (4) canbe easily satisfiedin real supercon-

ductorsfor Le(0, oo). This meansthat T~increases~(p)= ~?+P~/2m+P~/2m, with L. WhenL—+cc,

~ it2h212/2ML2 (1=1,2...) . T~=h2(.A~/2.6l)213/2itkB(m2M)~3.

Thedensityof bosonsat temperatureTmay be ex- In the confirmed high T~oxide materialspressedas La

2_~Ba~Cu04,YBa2Cu3O7,Bi2Sr2CaCu3O8and

Ti2Ca2BaCuO8 [11], thethicknessofthe layersvar-

~=~Jdp exp{fl[~(p)—~u]}—l. ies from 6 to 30 A, the Ta’s vary from 40 to 120 K,which coincideswith our prediction.

The condensationoccursat ~=0, thus If L—~0,the Bose condensationtemperaturevan-ishes (from eq. (2)). But in that casethereexistsa

2ith2.A~L 7 exp(P~~?) ), (2) transitiontosuperfluid(Kosterlitz—Thoulessphase)

Pc = ~ in(m \exp(fl~~?)—l [9]. It is quite different from Bose condensation.

= 1/kBT~T~is the critical temperaturefor Recently,Fisheret al. [12] studiedthe weakly in-condensation. teractingBose gas underthe dilute limit, obtained

the correspondingtransition temperatureTKT

d$~ / 2 ___________ + 27L46h2) 4ith2n0/2mln(ln y’) wherey=n01~,l0is the scat-

dL = — — L exp(/J~~) — 1 m tering length betweenbosons.—1 Whenthereexist oxygenvacancies,the motionof

+ ~ ~‘? ) (3) holonsis complicated.We know thatan oxygenva-\, m / exp(Pcbi) — 1 cancy of an oxygen-octahedroncan split the Cu

obviously, d/3~/dL<0when 3d~2_~2and Cu 3d~2levels further [13,14], thussuppressingthe holon-siteenergyon a defectedox-

i m ygenoctahedron.Wedenotethe suppressedvalueofthe boson-siteenergy as4~.We assumethat the ox-exp(/J~~?)—ih

2ygen-vacancydistributionalongthec-axisin a layer

(i) Whenfl~~?>>l, isf(z), ze[0, L]. The interactionof a boson-oxy-

gen-vacancyin a layeris

138


L n(z) andthe equationfor the Bosecondensationtemper-

Hin=4ojdz ~ ‘b~(z)b~(z), atureT~is0 2it~h

2Lflc/mflc(it2h2/2ML2_y)

whereb1 (z) is thebosonannihilationoperatorat site

(~,z),n(z)=n’f(z), n’ is the total numberof ox- —lnIexp[/3~(it2h2/2ML2—y)]—1I. (8)

ygenvacanciesina layer.~‘ correspondsto thesum- Concerningwith the realhigh Tc oxide supercon-mation over defected unit cells. Under random ductors,we notice that thereexistboson—bosonandapproximation, boson—spinoncouplings.In thecaseof a diluteBose

gas,thecouplingsaresmall.To zerothorder,we con-H

1,, = —40ô ~ ~f(z)b7(z)b~(z), (5) sider theBosecondensation,the couplingsare onlytakeninto accountin theeffectivemassesof bosons.

ô=n’/N, andN is the total numberof unit cells in In YBa2Cu3O7.....~Cavaet al. [18] observedthata layer. Taking a Fourier transformation, T~hadabruptdecreasesat the points ö—~0.20 and

4 L 0.40,anda plateaunearô—~0.35. ThepolycrystallineH1~=—~— ~f(m)b,,~bn[ôm+,,÷,,’ 2/+l resistivity increaseswith ö, exceptfor öe(0.2,0.4),

4it in which regiontheresistivityhada suddendecrease

(fig. 1 a). The JR spectrum[16] anddiffuse streaks2t+1 ôm_n+n’, 21+1 [15] of the materialindicatedthat the oxygenva-

+~,n—n_n~ 21+I]/(21+ 1) . (6) canciesin the materialweremainlyconcentratedonthe centralCu—O chainof a layer,and theywere or-

Experimentaldata [5—7] indicate that the oxygen dered along the chainwhen ö—~0.35. According tovacanciesaremainly concentratedat thecenterof a the statedexperimentaldata,it is reasonableto re-layerin Cu-oxide high T~superconductors,so it is gard that the oxygen vacanciesare randomly dis-reasonableto assume tributed on the Cu—U chain for ö�(0, 0.2). When

ö> 0.2, the randomnessis basedon the ordereddis-aexp(—aIL/2—zI) tribution at c5~0.35.2[l—exp(—aL/2)]

T~(°K)

If L is not very large, andonly the low energybe- 100 a p

havior is takeninto account,onehas(a)

H1~~_~±~[—~,f(l)+~f(3)]bt(k)b1(k) (7) 80

4it k

where60

a(2a+2ite~’72/L) (c) 10

L(l_e~U2)(a2+it2/L2)’

a(—2a+6ite~2) 40 (b) 8

ft3 L(l—e~2)(a2+9it2/L2)’ 6 3

thusthe modified first band-energyis 20

~‘1(p)=p~1/2m+h

2it2/2ML2—y(ó), _______________________________

40c5a ( 5(2a+2ite~’~

2/L) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (ô)= 4it (1 — e~~zI~~/2)!I\ — 3 (a2 + it2/L2) OxygeflvaCancy content 8

Fig. 1. Curves(a), (b) aretheexperimentaldatacorresponding— 2a+6ite~’-”2/L\ to 1’, androom-temperatureresistivity versusoxygen-vacancy

+ 3 (a2+9it 2/L2) ) ‘ contentrespectively.Curve(c)correspondstothecalculatedre-

sistivity alongthea-axisat roomtemperature.

139


Fioryet al. [19] pointedoutthat theelectricalre- e~(~)0.1019)

sistivity at room temperaturewas dominantly de-terminedby the carrier—oxygen-vacancyscattering, tothe other contributionscould be ignored. We thussupposethat the electricalresistivity at room tem- 8

tothedisorderdistributionoftheox-

Using the formula of conductivity [20], 6

aaa=’o(S)ta(S)e2/m’=p;c,~

(a=a, b indicatingthedirectionsperpendicularandparallel to the central Cu—O chain respectively)wheren ~(5) is thedensityofholons,andm’, M’ are 2

the effectiveholon massescorrespondingto the mo- 1

tionsparallelandperpendiculartothe layer,m= 2m’,0 0.1 0.2 0.3 0.4 0.5 0.6 5

MM . Ta(S) is themeanscatteringtimedueto ho-lon—oxygen-vacancyinteraction.We here approxi- Fig. 2. Thecalculatedcurveof thedensity of holonsversusJ.mately take 9_—..Jm’/2me.

ta( ~ ‘ic T m —‘ , crease,which might be regardedas the result of co-‘,.j B room herent scatteringof holons due to the ordered

= ô, 5,,= 5, 0<5<0.2, distribution ofoxygenvacancies[21]. Thisquestionis underour investigation.

= 15—0.351, 5>0.2. (9) If the bosonsin our model are holons, n0=n’0,

V0, Troom arethe unit cell volumeandroom temper- m=m’, M=M’. Substitutingeq. (10) into (8), weaturerespectively.Forpolycrystalline,p= (p~+Pbb) / have2, B/Tc=A/TclnIeA/Tc_lI,

PaaP, 0<S<0.2, it

2h~/2M’L2—y(S)

— 2p 5>02 A= kB

— 1+1(5—0 35)/S ~B=27th2Lp~(S)(kaTroom/m’)”2

fromthe knowndata (fig. lb) ofp versus5, onehas 1/3 2

the S dependenceof p,~,(fig. lc). Hence X (J/vo) /kBe (11)/k T /m’ for YBa

2Cu3O74,v0=abc, a=3.8699A, b=3.8667B room . (10) A, c~L=1l.688A. Troom~300 K. As oxygen va-

- canciesare stronglylocalized to the central Cu—OMaking useof fig. lc, andtaking Toom=300 K, we chain, we choose a~18/L, 12/L; 40i~t0.10 eV,calculatethe curve ni~ S (fig. 2). m’ 8.5 me, M’ 37 me. Solving eqs.(11) numer-

Formula (10) indicatesthat when S—eO, n~-+0. ically, onehasthecurvesshownin fig. 3.This radical result was due to the rough approxi- Now, if we supposethat the bosonin our modelmationof the holon—oxygen-vacancycrosssection. is formed by a pair of holons, one should takeIn fact, thecrosssectionshouldbe independentof S n0 = n ~,/2,m= 2m’, M= 2M’. In order to keep thefor small dopingof oxygen-vacancy,n~(S)should calculatedcurves in fig. 3 invariant, one shouldincreasewhenS—eO. Hencein fig. 2, n’0 decreases choose4~—~0.10eV, m’ —~2.2m~,M—~20m~.when5~(0, 0.25)or 5>0.45.It might be explained Taking into account the small S behavior ofas the effect of randomdistribution of oxygen—va- n ~(5), as discussedbefore,we know that (from fig.cancy.For Se(0.25, 0.45), n~(S)hasa suddenin- 3) to curve (a), T~decreasesto 60 K whenSvaries

140

Volume135, number2 PHYSICSLETTERSA 13 February1989

T~)°K) ygen-vacancyon the critical temperatureof super-conductivity, but did not find the plateauwhen

180 Se(0.3,0.35). We believethat this is dueto the ox-

ygen-vacancydistribution alongthec-axis. To curve5, (a) in fig. 3, a~18/L, to curve (/1), however,

vacanciesdistributedoff the centreof a layer for

4 curve (fi). In fact, thesamplesusedin ref. [16] were1 0 obtainedfrom slowly cooling in oxygen gas, while

thoseuse.din ref. [18] wereobtainedfrom quench-120 ing. It is well known that the quenchingprocessis

favourablein preventingtheoxygenvacanciesofthecentralCu—O planefrom diffusing.

160 a~12/L. That meansthat thereexist moreoxygen100 Discussion.(i) In our calculations,thechosenval-

uesof holonmassarem’ 8.Sme,2.2mecorrespond-ing to the mechanismsof holon- and biholon-con-

80 $ densation respectively. These values are in the

f ertimatedregionsgiven in refs. [4,9]. To decidewhichonecorrespondstotherealcase,afurtherstudy

60 . is necessary.As we know, m’ 1/I, M’ -~1/I’; t andt’ arethe hoppingenergiescorrespondingto themo-tion in the Cu—Oplaneandbetweenthe nearestCu—

40 0 planeswithina layer:t’ /t—~0.1. Thisis muchlargerthanin Huang’streatment[22], which only consid-ers interlayer hopping, the effect having been ig-

20 noredinour paper.Mattheissetal. [22] pointedoutthatthematerialYBa

2Cu3O7wasself-doped,andthe______________________________________ central Cu—Ochainwasthesourceof holons.Thus

0 0.1 0.2 0.3 0.4 0.5 5 it is reasonableto assumethat thehoppingeffectbe-

Fig. 3. Curves(a), (fi) are thecalculatedresultsof 7’. versus tweenCu—Oplaneswithin a layeris moreimportantoxygen-vacancycontentcorrespondingto a= 18/L and I 2/L than thatbetweenlayers.respectively. (ii) Up to now,no experimentdirectlymeasured

Ao. Kasowskieta!. [14] howevercalculatedtheelec-from 0 to 0.3, andhasa plateaufor Se(0.3,0.35); tronic propertiesof oxygenvacanciesin La2CuO385.to curve (fi), however,doesnotexist sucha plateau Their results suggestedthat oxygen-vacancybrokefor S in this region. Both curvesindicatethat T~has up the2 eV widepartially filled conductionbandintoabrupt increaseswhenS~0.4, which correspondto narrowerbands,at the point f, the nearestenergythe pointssatisfyingS’s’ — y<0. They haveno phys- level to the Fermi surface is raised about 0.1 eVical meaning,becauseif ‘?— y<0, the motion of higherthanthat in La2CuO4.Thisvalue is closetocarrierswill be localizedin the centralCu—O plane the chosenvalueof ~ in our paper.of alayer, theeffectivethicknessof thelayeris L/ (iii) It wasconjecturedthat spinonsandholons6, not L, thusT~will be suppressedto 40 K, like in are topological solitons [3,23]. If this is so, thethe well-known superconductorLa2....~Ba~CuO4. structureof the Kosterlitz—Thoulessphasefor LTakingthis “effective-thickness”effectintoaccount, tendingtozeroshouldbemorecomplexthanthatofcomparingwith curves (a) in fig. 1, we know that the ordinaryBose gassystem.On the otherhand,ifcurve (a) in fig. 3 qualitatively coincideswith the the 3-dimensionalquantumfluctuationscould notexperimentaldata. beavoided,whatwouldthe KT phasebefor this sit-

Kuzmanyet a!. [161 observedthe influenceofox- uation?What is the role of spinonsin the conden-

141


sation?Work is in progresson theseequations.In [61S. Kivelson,D.S. RoksharandJ.P.Sethna,preprint.

summary,westudiedthequasi-two-dimensionalBose [7] Xiao-GangWen, Chargeebosons,neutralfermionsandh/2eflux-field theory,preprint.

gas condensationand the influenceof layer-thick- [8] V. KalmeyerandR.B.Laughlin,Phys.Rev.Lett. 59 (1987)nessonthecriticaltemperature.In thecaseofanox- 2095.

ygen-vacancydistribution we studied the depen- [9] J.M.KosterlitzandD.J.Thouless,J.Phys.C 6 (1973) 1181.

dence of T~on oxygen content.The results are [101 Z.Z. Wang, J. Clayhold and N.P. Ong, Phys.Rev.B 36qualitativelyconsistentwiththeexperimentsrelated (1987)7222.

[11] J.G.BednorzandK.A. Muller, Z. Phys.B 64 (1986)193;to YBa2Cu3O7. M.K. Wuetal., Phys.Rev.Lett. 58 (1987)908;

R.M. Hazenetal., Phys.Rev.Lett. 60 (1988) 1659;

Wewould like to acknowledgefruitful discussions C.W. Chuetal., Phys.Rev.Lett. 60 (1988)941.

with ProfessorsG. BaskaranandS. Kivelson, and [12] D. Fisherand P.C. Hohenberg,Phys. Rev. B 37 (1988)4936.

stimulating conversationswith Dr. Ying He-pin, [13] C. Michel andB. Reveau,Rev. Chim. Miner. 21 (1984)

Zhang Jian-BouandProfessorZhu Xue-Tiang.We 407.are also grateful for the constructivecommentsto [14]V. Kasowski, Y. Hsu and F. Herman,Phys. Rev. B 36

the first versionof thispapergiven by the referee. (1987)7248.[15] D.J.Werdaretal.,Ploys.Rev. B 37 (1988)2317.[16] H. Kuzmanyetal., SolidStateCommun.65 (1988) 1345.[17] J.D.Fitzgeraldetal.,Phys.Rev. 60 (1988)2797.

References [18] R.J. Cava, B. Batlogg, C.H. Chen, E.A. Rietman, S.M.ZahurakandD.Werder,Phys.Rev.B 36 (1988)5719.

[19] A.T. Fiory,M. Gurvitch,R.J.CavaandG.P.Espinosa,Phys.[11 P.W. Anderson,Science235 (1987) 1196. Rev.B 36 (1987)7262.[21G. Baskaran,Z. Zou and P.W. Anderson, Solid State [20] R. Kubo, Statistical mechanics (North-Holland,

Commun.63 (1987)973; Amsterdam,1965).P.W. Anderson,G. BaskaranandT. Hus,Phys.Rev.Lett. [21] P.A. Lee,Rev.Mod. Phys.57 (1985)287.58 (1987)2790. [22] L.F. MattheissandD.R.Hamann,SolidStateCommun.63

[3] S. Kivelson,D.S.RoksharandJ.P.Sethna,Phys.Rev.B 35 (1987)395.(1987) 8865. [23]P. W. Andersonet al., Fermionsandtopologyin thetwo-

[4] Xiao-GangWen andRuiKan,Phys.Rev.B 37 (1987)595. dimensionalquantumantiferromagnet,preprint.[5] J.M. Wheatley,T.C. Hsu,andP.W. Anderson,Phys.Rev. [24] K. HuangandE. Manousalcis,Phys.Rev.B 36 (1987)8302.

B 37 (1988) 5897. [25] 5. Kivelsonetal.,Phys.Rev.B 36 (1987)7237.

142

quasi-two-dimensional bose condensation and high tc superconductivity

Documents