REVISTA MEXICANA DE FÍSICA 4S SUI'LEMENT() 1, 158-163 JUNIO 1999

Superconducting transition-temperature enhancement due toelectronic-band-structure density-of-states

V. Fuentes1, M. de Llanol, M. Grelher', and M.A. Solís3

1 Instituto de Investigaciones en Materiales2 Facultad de Ciencias3 Instituto de Fúica

Universidad Nacional Autónoma de México, 04510 México, D.F., Mexico

Recibido el5 de abril de 1998; aceptado e13] de agosto de 1998

We briefly review a simple statistical model of a boson-fcrmion mixture of unpaircd fermions plus linear-dispersion ..relation Cooper pairsthat leads to Bose.Einstein condensation (BEC) for all dimensions grcater than unity. (The "dispersion rdation" of a particle is its energyvs. momcntum rclation.) This contrasts sharply with "ordinary" BEC for a many-boson asscmbly of non-interacting hosons each movingin vacuum with n quadratic dispcrsion relation, which is well-known lO occur only for dimensions greater than two. The BEC criticallempcmtures Te are substantially higher than those of the BeS theory of supcrconductivity, for the same BCS model interaction betwecnthe fermions that gives rise to the Coopcr pairs, at both weak and strong couplings. Howcver, these results hold with an ideal~fermi-gas(lFG) density.of.states (DOS) for the underlying electron (or hole) carriers. We then show tha! even higher Te values are obtained in 20if a non-IFG DOS is employed which reflect~ the electronic band structurc of the quasi.2D copper.oxygen planes characteristic of cupratesuperconductors. The non-IFG DOS used are both a so-called Van Hove scenario (VHS) with a logarithmic singularity in the DOS, and aDOS with a power-Iaw-singularity associated with an extended.saddle-point (ESP) in the energy-momentum curve.

Keywords: Supcrconductivity; Bose-Einstein condensation; electronic densily of states

Recordamos brevemente un modelo estadístico simple de una mel.c1a de fermiones desapareados más pares de Cooper que satisfacen unarelación de dispersión lineal que les permite condensarse a la Base-Einstein (BEC) en dimensiones mayores que la unidad. (La relaciónde dispersión de un partícula es su energía en función de su momento.) Lo anterior contrasta notoriamente con la BEC "ordinaria" de unconjunto de muchos bosones que no interactúan entre sí, y moviéndose en el vacío con una relación de dispersión cuadrática, la cual es biensabido se da sólo para dimensiones mayores que dos. Las temperaturas críticas Te de la BEC son sustancialmente más elevadas que las dela teoría de la supcrconductividad BCS, para el mismo modelo de interacción BCS entre los fermiones que forman los pares de Cooper,en ambos límites de acoplamiento débil y fuerte. Sin embargo, estos resultados son para electrones (u hoyos) con una densidad de estados(DOS) igual a la de un gas ideal de fermiones (lFG). Mostramos que Te más altas aún son obtenidas en 2D si empleamos una DOS diferentede la IFG, que refleje la estructura de bandas electrónicas de los planos de cobre-oxígeno característicos de los cupralos superconductores.Las DOS diferentes de la IFG usadas aquí son la de el llamado escenario de Van Hove que contiene una singularidad logarítmica en la DOSy, una DOS con una singularidad potencial asociada a con un punto silla extendido (ESP) en la curva de energía vs. momento.

Descriptores: Supcrconductividad; condensación Bose-Einstein; densidad de estados electrónicos

PAes: 05.30.J; 71.20; 74.70; 74.72

1. Introduction

Consider [1) firsl un ideal quanlum gas in d dimensions 01'pcrmancnt U.e., number-conserving) bosons with a generaldispersion relation

For ordinary bosoos of mass m in vacuum s = 2 andes = n2/2m. while for a Cooper pair in the Fermi seas = 1 as discussed below. The boson numbcr dcnsity Hllin a "box" 01' lcngth L in d dimensions is defined as llIJ =N"I L", where lhe lolal number01' hosons NB = N ",o(T) +" [e13{ó,-"") _1]-1 wilh l' < Otheehemieal polentiaiL...ktO . , R - 'f3 == 11kllT, kll Ihe BolIzmann conslanl und T is lhe ahsolutelemperature. The summation implies an elcmentary integraleasily evaluated in terms oflhe usual Bose inlegrals [2) (wilh

(2)

1 100 ,,"-1Ya(z) == -r() d"-_-I--

(Jo zex-100 I

= "¡z -----t (,,).o ff z-+11=1

The last identificalion holds when" 2: 1, where ((,,) is theRiemann Zela-function 01'order ". The funcIion ((,,) < 00for a > 1, whilc the series g(1 (1) divcrges for (f S l. As T islowered down lOT" helow which NB.O(T) jusI eeases to benegligible compared wilh NB and simultaneously JlB == 0-.The condcnsate fraction for O < T < Te in el dimensionsis the fraclional number NB,o(T)/NB.o(O) 01'bosons in Ihek = O stale. Note that ¡iR == 0- over the entirc tempera-ture range 0< T < T, since Nu.o(T) = (e-13"" - 1)-1

z = e13J1ll the fugacity)

(1)wilh s> O.

SUPERCONDUCfING TRANSITION-TEMPERATURE ENHANCEMENT DUE TO. 159

where 8D == Ttwv/ko is the Debye lemperature, the dimen-sionless eoopling eonslanl,\ == g(Ep)V, Here g(Ep) is lhedensily of one-spin, eleelronie slales (DOS) at lhe Fermi sur-raee wilh energy E p, and V is lhe aHraetive slrength of lheBCS model interaction,

implies lhal eI'"B = NB,o(T)/[NB,o(T) + 11 < 1, andapproaches 1- aver this entire tcmpcrature range becauseN B,o(T) on eooling grows to a sizeable fraelion of N B whiehis maeroseopie. Sinee NB = NB,O(T) + NB,k>O(T) whileN B = N B,O(O), fm d > O one can wrile Ihe fraelional num-bcr as

NB,o(T) = 1 _ [2d-1 Ifd/2r(~2)nB]-1N B,O(O)

(3)

J-VVk,k' = lo

h'k2if Ep - rlWD < --,- 2m

h2k,2-- < Ep + rlWD2m -

otherwise

(8)

whcre the sum-to-integral conversion [2]

was employed, Using (2) lo evaluale lhe inlegral in (3) gives

where Vk,k' is lhe double Fourier lransfonn of lhe spaee-dependent pair ¡nteraetion bctween ferrnions, and k, k' arerelative wavcvcctors. The positive eoupling constant V rep-rescnts the net attractive effect of the eleetronMphonon inter-aetion which overwhelms the repulsive Coulomb interaction.

NJj,o(T)N13,o(O)

2.2. BEC lheory Te ror Iinear.dispersion.relation Cooperpairs

On lhe other hand, the number of bosons N B,O(O) aeluallyfmmed al T = O through Ihe Bes model inleraetion in thefermion gas is precisely NB,O(O) ~ g(Ep )hwD, where [4]

since g(e) is the numbcr of up-spin fermion states per unitenergy, lf all fermions were imagined paired, then nB /n =1/2, /lowever, since nEl = NB,O(O)/Ld = g(Ep)rlWD/Ld,(9) and (10) show lhat in fael

In ddimensions the numberoffermions is N = 2¿k O(kp-k), where O(x) is lhe Heaviside slep funclion, so Ihal us-ing the sUIll-toMintcgraleonversion again thejermion numberdensity n becomes

r(d/s) gd/,(I)x S((3C.)d/. (4)

Sinee lhis is negligible al T = Te, selling Ihe rhs of (4) lozero givcs a simple algcbraic equation whosc solution yieldsthe general Te formuLa

T _ C, [ S (21f)d r(d/2) ] ,Ide - ku 2"d/2 r(d/s) gd/,(I) nB (5)

Thc most familiar spccial case is [oc d = 3 and s = 2 whichgivcs the well-known formula [oc the Base-Einstein condcn-salion (BEe) transition temperature

2 h2 2/3 3 31~2 2/3T. _ 1r na ",' l no (6)e - mkn[((3/2)J2/3 - mko

where in lhe last slep we used ((3/2) '" 2,612,

2. Ideal fermion gas density of sta tes

N kdn= _= F- Ld 2d-2"d/2d r(d/2)'

E = (!:..) d ddk = (~) d/' LdEd/2-1g( ) - 2" dé 21fr,2 r(d/2)

(9)

(10)

for apure ullbreakable-pairon gas. In particular, [or d = 2and 3, with a(2) = 2/" and a(3) = 1/2 [5], and na/n =1'/2 and 31'/4, respeelively, (12) gives

a fraelion much Iess lhan 1/2 sinee typieally rlWDv « 1. This allows writing (5) with s = 1 as

[

,+> ] lidTe a(d) r¿up ,,--.-- nBTp = kR r(~)gd(l)

[V ] lid

= 2a(d) 2r(d)((d) ,

(1 1)

(12)

« Ep m

dTtwD _ vd4Ep = 4'

nan

(7)

Consider now a many-fermion system with attractive ¡n-teraetioos capable of pairing al least same of the fcernioos intoCoopcr paies (m "pairons"); these are cansidcred as "basan s"even through they do nat obey Base cornrnutation relationssinee lhey do ohe)' a Bose-Einstein dislribulion [SI if lhesepairs are of definitc ccntcr-of-mass (but llOt dcfinite relative)momenta.

2,1. BCS Theory Te

The BCS Te formula is the solulion of Ll(Te) = O, whereLl(T) is the BCS energy gap, whieh in lurn is Ihe solution ofthe Bes gap equation [3]. The well-known resullS is

Rey, Mex, Fis. 4S SI (1999) 158-163

160 V FUENTES. M.llE LLANO. M. GRETHER. AND MA SOLis

g(E)(1£

J(E - EF)' + u'(T)

2

ESPCJ '" 1/2

4

6

IFG

o0~5 1

X = e/E,

10g(

SUPERCONDUCTING TRANSITION-TEMPERATURE ENHANCEMENT DUE TO ... 161

3.2. BEC Te in Ihe VHS 4. Extended-saddle-point singularity DOS

Introducing the new integration variable y = x - 1, we have

When T --+ O, ~ --+ 00 and /1 = EF, or l' = 1, so lhat (20)bccomes

where Ihe VHS DOS has heen subS!iluled in lhe laS! equa-lion. LeIting x '" e/ EF, ji '" /1/ EF, ~ '" (3EF and sincev'" hwn/ EF, (18) reduces lo

RecaJling (13) for 2D, as weJl as (9) and n BNB,o(O)/L" = g(EF)rlJJJn/L", if N2,o(O) = 2NB,o(O),where N2,o(O) is Ihe number of paired fermions al T = O,lhe numherofpairable fermions N2(T)YIlS becomes

(27)

(30)

17 > O, (28)

Ng(e) = 2E

FJ(x)

f -1 (xO)"-1 + xo{1- 17) + (17 - 2)o = Xo - (2 _ 317 + a2)xo '

xJ(x) = ( ) 8(1 - Xo - x)1 - x u

+ J08(x + Xo - 1)8(1 - x),

Wilh lhe dimensionless funclion J(x) defined hy

where 8(y) = O ify < O, = 1 ify > O and = 1/2 ify = O.The curve J(x) rises linearly from zero al x = O and woulddiverge al x = 1 with a power-u-singularity; this rise is theninterrupled al x = I - Xo heyond which il flanens oul inlo a"plaleau" ofheighl Jo and widlh xo. The value of Jo is deler-mined by

We now consider an ESP energy dispersion curve suggest-ing a power-Iaw (as opposed to a logarilhmie) singularilyDOS [12]. These dispersion curves have aCluaJly heen ob-servcd in angle-rcsolved photoemission experiments [13] insuch superconductors like YBa2Cu306.9 and YBa2Cu,Os,

Since x'" e/ EF, we write lhis DOS as

which is a (normalizalion) condilion ensuring lhal the 10lalnumber offermions al T = O is N = Jo""de g(E)8(EF - E).Note thal from (29) Jo is expressihle solely in terms of Xoand 17, specifically

11 11-.0 X /1J(x) dx = (1 _ ,)" dx + Jo dx = 1, (29)o o :1 1-xo

( 18)

(19)

(20)

(21 )

(22)

N2(T)YIlS = - 2g(EF )EF

l.MV-1 In Iyldy _ .{J-1I-1 e)3(y+l-¡j) + 1

!.~+hWD In I £~ÉF IN2(T)YIlS = 2g(EF) de ~(,_") .¡.l-1íWD e + 1

1"+v Inlx-llN2(T)YIlS = -2g(EF )EF dx _ _ .{J-v eO(x-IJ) + 1

N2(T)YIlS --+ O7'-+0

N2,0(0)YIlS = -2g(EF )EF IVv dy8(ji - y) In Iyl

= -2g(EF )EF 10v dy In Iyl

= 2g(EF)EF(v -vlnv) = 2NB,0(0).

where k} = 2mEF/ñ', which subslituted inlo (24) leaves

Consequently

= 'V (O)/L2 = g(EF )EF(V - v In v)nB - 1 B,O L2 1

Further. as a --+ 00 and IO --+ 1, Jo --+ 1 and one recoverslhe IFG case. However, as a --+ O we find lhal lhere is a mio-irnum valuc ami n ~ 0.3819 beyond which Xo ceases to bereal. The expooenl a = 1/2 is suggesled in [12] on physicalgrouods,

4.1, BCS Te in ESP singularily

Consider lhe gap equation for lhe finile-lemperalure energygap t.(T) (14) huI with ao ESP DOS ofthe form (27), wherelhe dimensionless funclion J(x) sali5fies (28) lhrough (3D),except thal (28) is modified 10 he left-righl symmetrie aboutx = 1 as appropriale if T 2: O. See Fig. 2 for case17 = 1/2. Thus lhe BCS gap equalion becomes, where againx'" e/EF,

(23)

(25)

(24)

v - vln v2

nBn

n

so thal recaJling (9) and (lO) lhis gives

mL2 Edv - v In v)2rr2rrh2k}L'

FinaJly, using (13) we have

TYIlS ~-- 4vÍ3 TIFG_c_ = vI -In v --"IV '" 2_c_, (26)TF K2 TF

i.e., an enhancerncnt of roughly two for v = 0.05 which islypical for cuprates.

2 1.'I+V J(x)V =N 1-" dx j[Ep(x _ 1)]' + t.2(T)

1 V[Ep(x - 1)]' + t.'(T)x tanl ---------, (31)2kBTc

Rev. Mex. Fís. 45 SI (1999) 158-163

162 V. FUENTES. M. DE LLANO. M GRETllER. AND MA soLis

whcre we have used the syrnmetry of the integrand aoout Ep.To detennine T, one most solve L\(T,) = O; this reduces (31)lo

where /J '" ¡1EF, /' '" /'/ EF, amI as he rore x'" é/EF whilev'" '''''o/ EF.IfT = O. ji = 1 and (38) hecomes

= N Jo /.' dx = N Juv, (39),-v

/.

'+vN2(O)¡csp = N Jo 8(1 - x)dx,-v

Sinee 1111= NII(O)/ L2• while" = N/ L2• the lirst memberon the rhs 01' (13) ror 2D linally gives

so Lhatthe boson number al T := O is just

(33)

(32)

~ = (1J/2Tc dz tanh zNF Jo Jo z'

since the inlegrand is even. Integrating by parts gives

2 N /.t+v J(x) X - 1- = - dx--tanhx -.-,V EF I-v X - 1 2T,

with T, '" T,/TF and (as be rore) v '" r,wo/ EF being di.mcnsionlcss. The DOS r¡ses [ram zcro energy. and occorncsa plateau at x equals 1 - Xo; this plateau ror a = 1/2 isexhibited in Fig. 2 in the vicinity just bclow EF.

For:ro > v. J(x) = Jo, the constant given hy (30). In.troducing the new variahle z = (x - 1)/2T, we are then leftwith

case.

01.2. BEC Te in ES\' singularily

where g(o) is defined by (27) and (28). Since ror Xo > " theDOS g(r) is a constant aver thc entire integratian range, anchas

(44)

(41 )

(43)

(42)

(xo < v).

(xo > v).

(~) (nu) TJ~'G1/ H 1F=

1111 Nu(O)----n N

1 /.'-xo x= - (.)" dx + Jo:ro/2.2 l-v 1 - .1

/.

I-XO /.'N2(0) = N (x dx + N Jo dx.¡-ti 1-.r)O ¡-.ro

Finally. suhstituting (43) into (13) gives

Thus, (40) and (42) leave

with enh3ncement f3ctor )(2/1')("11/11). where "u/n mustbe determined by (43).

In Table 1 arc Iisted several values 01' Xo and Jo for dif-fcrcnt a values. In the last column wc givc TcESP Jrj FG, thercsulting cnhancclllcnt factor for thc ESP DOS. This is plot-tcd in Fig. 3 for sevcral values 01' Ihc exponent a, with theexponent a = 1/2 proposed in (12) being markel! in the ligoure.

TESP .~F = .,(Jo(4J:i/7f')VV

TIFG= .,(Jo -'-TF

or an enhancement by a factor vTo or T, with the ESP DOSover that with the IFG DOS. On the other hand, ir :ro < v.then g(é) consists 01' two parts, since (27) 3nd (28) in (37) rorT = O gives

(37)

(3K)

(35)

(34)

1,1+v 1:= N Jo - _ d;r 1I'-V exp ¡1(x - /,) + 1

j\' t f"+"~o 1iV:,!(T)mw := __ o ------dEEF ,,-hwo exp ¡1(é - 1') + 1

Wc now determine the critical tempcrature Te associatcd wilhBEC assuming a DOS resulting from un cxtcndcd-saddlc-point (ESP). For this we must recalculate Tlu/" as in (11)and then insert this into the 2D version or (13). The numberof pairable fermions is

. f"+"~o g(é)i\'2(T)ESP = 2 --~~--d'£.

,,-h~o exp¡1(é - 1') + 1

EF (v.) (4e')--- = In - T, + In - .j\'VJo 2 7f

Since g••dEF) = L2m/27fh2, ,\ '" Vg ••G(EF), EFr,2k}/2m and TI = N/ L2 = k}/27f, one finally ohtains

E V lvl2T

SUPERCONDUCTING TRANSITION.TEMPERATURE ENIIANCEMENT DUE TO ... 163

TABLE 1.The corresponding ESP DOS g(e) parar,.t:lcrs in (27) and(2R), TO. /0, ror .sOIne choiees oC G.

a Xo Jo r}!'SP /T~FG

0.3819 0.0 00 2.23245

0.4 0.009754 6.31065 2.14177

0.5 0.1038 2.7816 1.66784

1.5 0.486122 1.51615 1.23132

10 0.82775 1.14071 1.06804

,,=1/2

1.S

1.0

0.1 10 100cr

1000

S. Conclusions

Enhanccment faclOrS foc the supcrconducting transition tCffi-pcraturc Te in a Bose-Einstein condcnsation (BEC) picturccan he as high as aboul 32 for a strongly-coupled (A = 1/2)Bardecn-Cooper-Schrieffer (BCS) model inleraelion, overlhal 01' lhe familiar BCS lheory. Indeed, for weak coupling(.-\ ---t O) this eflhancement diverges, meaning (hai BECcan arise evcn roc arbitrari)y-weakly-couplcd Coopcr paies,which are basons.

1. v.e. Aguilera.Navarro, M. de Llano. and M.A. Salís. l:'llr. 1.PhY.f. (in press).

2. R.K. Pathria, Slarislical Mechanics, 2nd Edition. (Pcrgamon,Oxford, 1996).

3. D. Tilley and J. Tillcy, Superflllidity and Superconducliviry(Adam lIilger, New York, 1990).

4. R.M. Ziff et al., Phys. Rel'. 32 (1977) 169.5. S. Fujita and S. Godoy, Qualltum Stalislical Theory of SUI't'r.conductivity, (Plcnum Press, New York. 1996).

6. M. Casas el al., Physica e 295 (1998) 93.i. Y.J. Uemura el al .• Phys. Rev. Ull. 66 (1991) 2665; Nalun' 352

(1991) 605; Physica B 186-188 (1993) 223.8. v.e. Aguilera-Navarro el al .• in C01UJ.Malla Theories. Vol.

12 edited by R.V. Panat, (Nova Science Publishcrs, New York.1997).

FIGURE 3. Enhancemcnt of (he BEC Te using che ESP DOS. oyerthat with (he IFG DOS. fOf several values of the exponent a in (27)and (28), the value (j = 1/2 bcing that suggested in [12].

Much more modcrate cnhanccments obtain in cilherlhe BCS or lhe BEC pielures when non-ideal-Fermi-gas(IFO) eleelronie-densily-of-slales (DOS) eharaelerislie ofthe clcctronic.band-structure of quasi-2D systems like thecuprate superconductors are cmployed. Two such POS'sare lhe Van Hove seenario (VHS) logarilhmieally-divergentDOS, as well as lhe extended-saddle-poinl (ESP) power-Iaw-divergcncc DOS.

9. M. Casas l'l al., in Cond. Malla Theories, Vol. 13, edircd by J.da Providencia, (Nova SciCflce Publishers, New York, 1998).

lO. c.e. Tsuci el al .• Phys. Rev. Ull. 65 (1990) 2724.

11. J.M. Celillo, M. de Llano, and H. Rubio, Phys. Rev. B 48 (1993)597; for a reccnt review see R.S. Markiewiez, 1. Phys. Chem.Salid.>58 (1997) 1179.

12. AA Abrikosov el a[., Physica e 214 (1993) 73; AAAbrikosov el al .• Physica C 222 (1994) 191.

13. R. Manzke el al., Surfaa Sei. 269 (1992) 1066; lG. Tobin elal., Phy", Rev. B 45 (1992) 5563.

14. A.L. Fetter and lD. Waleeka, Quanlum Theory of Many-Particfe Syslems, (McGraw.Hill, New York. 1971), p. 581.

Rn'. M ••x. fú. 45 SI (1999) 158-163