Ž .Physica C 309 1998 251–256

Phase transition from a d to d q id superconductor2 2 2 2x yy x yy x y

Angsula Ghosh, Sadhan K. Adhikari )

Instituto de Fısica Teorica, UniÕersidade Estadual Paulista, Rua Pamplona 145, 01.405-900 Sao Paulo, Sao Paulo, Brazil´ ´ ˜ ˜

Received 19 August 1998; accepted 28 September 1998

Abstract

The temperature dependencies of specific heat and spin susceptibility of a coupled d 2 2 q id superconductor in thex yy x yŽ . Ž .presence of a weak d component are investigated in the tight-binding model 1 on square lattice and 2 on a lattice withx y

orthorhombic distortion. As the temperature is lowered past the critical temperature T , first a less ordered d 2 2c x yyŽ .2 2superconductor is created, which changes to a more ordered d q id superconductor at T -T . This manifests inx yy x y c1 c

two second order phase transitions identified by two jumps in specific heat at T and T . The temperature dependencies ofc c1

the superconducting observables exhibit a change from power-law to exponential behavior as temperature is lowered belowT and confirm the new phase transition. q 1998 Published by Elsevier Science B.V. All rights reserved.c1

PACS: 74.20.Fg; 74.62.-c; 74.25.Bt

Keywords: d 2 2 q id -wave superconductor; Specific heat; Susceptibilityx yy x y

w xThe unconventional high-T superconductors 1c

with a high critical temperature T have a compli-c

cated lattice structure with extended andror mixedw xsymmetry for the order parameter 2,3 . For many of

these high-T materials, the order parameter exhibitsc

anisotropic behavior. The detailed nature of anisotro-py was thought to be typical to that of an extended

Ž Ž . .s-wave, a pure d-wave, or a mixed sqexp iu d -wave type. Some high-T materials have singlet d-c

wave Cooper pairs and the order parameter hasw x2 2d symmetry in two dimensions 2,3 , which hasx yy

been supported by recent studies of temperature de-wpendence of some superconducting observables 4–

x13 . In some cases there is the signature of anextended s- or d-wave symmetry. The possibility of

) Corresponding author. Fax: q55-11-3177-9080; E-mail:adhikari@ift.unesp.br

a mixed angular-momentum-state symmetry wasw xsuggested sometime ago by Ruckenstein et al. 14

w xand Kotliar 15 . There are experimental evidencesbased on Josephson supercurrent for tunneling be-

Ž . w xtween Pb and YBa Cu O YBCO 16 , and on2 3 7w xphotoemission studies on Bi Sr CaCu O 172 2 2 8qx

among others which are difficult to reconcile em-ploying a pure s- or d-wave order parameter. These

w Ž . xobservations suggest that a mixed sqexp iu dw xsymmetry is applicable in these cases 18,19 .

w xMore recently Krishana et al. 20 reported aphase transition in the high-T superconductorc

Bi Sr CaCu O induced by a magnetic field from a2 2 2 8

study of the thermal conductivity as a function ofw xtemperature and magnetic field. Laughlin 21 have

suggested that the new superconducting phase is thetime-reversal symmetry breaking d 2 2 q id state.x yy x y

w xSimilar conclusion has been reached by Ghosh 22in a recent work from a study of the temperature

0921-4534r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0921-4534 98 00585-1

( )A. Ghosh, S.K. AdhikarirPhysica C 309 1998 251–256252

dependence of thermal conductivity. This has led tothe possibility of the transition to a d 2 2 q idx yy x y

phase from a pure d 2 2 phase of Bi Sr CaCu O .x yy 2 2 2 8

From a study of vortex in a d-wave superconductorusing a self-consistent Bogoliubov–de Gennes for-

w xmalism, Franz and Tesanovic 23 also predicted the´ ´possibility of the creation of a d 2 2 q id super-x yy x y

conducting state. Although, the creation of the mixedsuperconducting state d 2 2 q id is speculative,x yy x y

Franz and Tesanovic conclude that a dramatic change´ ´Žshould be observed in the observables of the super-

.conductor as the superconductor undergoes a phasetransition from a d 2 2 state to a d 2 2 q idx yy x yy x y

state. In this work we study the effect of this phasetransition on superconducting specific heat and spinsusceptibility in the absence of magnetic field. Thegeneral trend of observables under the d 2 2 tox yy

d 2 2 q id -wave phase transition of the supercon-x yy x y

ductor, as studied here, is expected to be independentof the external magnetic field. We recall that therehave been several studies on the formation of a

Ž .mixed sq id -wave superconducting state from aw xpure d-wave state 24–29 .

First we study the temperature dependence of theorder parameter of the mixed d 2 2 q id statex yy x y

Ž .within the Bardeen–Cooper–Schrieffer BCS modelw x 2 230,31 . The BCS model for a mixed d q idx yy x y

state becomes a coupled set of equations. The ratioof the strengths of the d 2 2- and d -wave interac-x yy x y

tions should lie in a narrow region in order to have acoexisting d 2 2- and d -wave phases in the casex yy x y

of d 2 2 q id symmetry. As the d 2 2-wavex yy x y x yyŽ .d -wave interaction becomes stronger, the d -x y x y

Ž .2 2wave d -wave component of the order parame-x yy

ter quickly reduces and disappears and a pureŽ .2 2d -wave d -wave state emerges.x yy x y

The order parameter of each of d 2 2 and dx yy x y

states has nodes on the Fermi surface and maychange sign along the Fermi surface. The s-waveorder parameter does not have this property. Becauseof this, many d-wave superconducting observableshave power-law dependence on temperature, whereasthe s-wave observables exhibit exponential depen-dence. We find that in the present coupled d 2 2 qx yy

id state the order parameter does not exhibit nodesx y

and change of sign along the Fermi surface andexhibits a typical s-wave like behavior. Conse-quently, the observables in the coupled d 2 2 q idx yy x y

state does not exhibit typical d-wave power-lawdependence on temperature, but rather a typical s-wave exponential dependence.

For a weaker d -wave admixture, in the presentx y

study we establish in the two-dimensional tight-bind-Ž . Ž .ing model 1 on square lattice and 2 on a lattice

with orthorhombic distortion another second-orderphase transition at TsT -T , where the supercon-c1 c

ducting phase changes from a pure d 2 2-wave statex yy

for T)T to a mixed d 2 2 q id -wave state forc1 x yy x y

T-T . The specific heat exhibits two jumps at thec1

transition points TsT and TsT . The tempera-c1 c

ture dependencies of the superconducting specificheat and spin susceptibility change drastically atTsT from power-law behavior for T)T toc1 c1

exponential behavior for T-T . We find that thec1

observables for the normal state are closer to thosefor a pure superconducting d state than to thosex y

for a pure superconducting d 2 2 state. Conse-x yy

quently, superconductivity in d 2 2-wave is morex yy

pronounced than in pure d -wave. Hence as temper-x y

ature decreases the system passes from the normalstate to a ‘less’ superconducting d 2 2-wave state atx yy

TsT and then to a ‘more’ superconducting statec

d 2 2 q id with dominating s-wave behavior atx yy x y

TsT signaling a second phase transition.c1

The profound change in the nature of the super-conducting state at TsT becomes apparent from ac1

study of the entropy. At a particular temperature theentropy for the normal state is larger than that for allsuperconducting states signaling an increase in orderin the superconducting state. In the case of thepresent d 2 2 q id state we find that as the tem-x yy x y

perature is lowered past T , the entropy of thec1

superconducting d 2 2 q id state decreases veryx yy x yŽ .rapidly not shown explicitly in this work indicating

the appearance of a more ordered superconductingphase and a second phase transition.

We base the present study on the two-dimensionaltight binding model which we describe below. Thismodel is sufficiently general for considering mixedangular momentum states, with or without or-thorhombic distortion, employing nearest and sec-ond-nearest-neighbor hopping integrals. The effec-tive interaction in this case can be written as

V syV cos k ybcos k cos q ybcos qŽ . Ž .k q 1 x y x y

yV sin k sin k sin q sin q . 1Ž .Ž . Ž .2 x y x y

( )A. Ghosh, S.K. AdhikarirPhysica C 309 1998 251–256 253

Here V and V are the couplings of effective d 2 2-1 2 x yy

and d -wave interactions, respectively. As we shallx y

consider Cooper pairing and subsequent BCS con-densation in both these waves the constants V and1

V will be taken to be positive corresponding to2

attractive interactions. In this case the quasiparticledispersion relation is given by

e sy2 t cos k qbcos k yg cos k cos k , 2Ž .k x y x y

where t and b t are the nearest-neighbor hoppingintegrals along the in-plane a and b axes, respec-tively, and g tr2 is the second-nearest-neighbor hop-ping integral.

We consider the weak-coupling BCS model intwo dimensions with d 2 2 q id symmetry. At ax yy x y

finite T , one has the following BCS equation

D Eq qD sy V tanh 3Ž .Ýk k q 2 E 2k Tq Bq

wŽ .2 2 x1r2with E s ´ yE qD , where E is theq q F q F

Fermi energy and k the Boltzmann constant. TheB

order parameter D has the following anisotropicqŽ .form: D 'D cos q ybcos q q iD sin q sinq 1 x y 2 x

Ž .q . Using the above form of D and potential 1 ,y qŽ .Eq. 3 becomes the following coupled set of BCS

equations2

D cos q ybcos q EŽ .1 x y qD sV tanh , 4Ž .Ý1 1 2 E 2k Tq Bq

2D sin q sinq EŽ .2 x y q

D sV tanh 5Ž .Ý2 2 2 E 2k Tq Bq

where the coupling is introduced through E . In Eqs.qŽ . Ž .4 and 5 both the interactions V and V are1 2

<assumed to be energy-independent constants for ´q< <yE -k T and zero for ´ yE )k T , whereF B D q F B D

k T is the usual Debye cutoff.B Dw xThe specific heat is given by 31

< < 22 E f 1 d Dq q2C T sy E y T 6Ž . Ž .Ý qž /T E E 2 dTqq

Ž Ž ..where f s1r 1qexp E rk T . The spin suscep-q q Bw xtibility x is defined by 12,13

2m2N

x T s f 1y f 7Ž . Ž .Ž .Ý q qT q

where m is the nuclear magneton.N

Ž . Ž .We solved the coupled set of Eqs. 4 and 5numerically by the method of iteration and calcu-lated the gaps D and D at various temperatures for1 2

Ž .T-T . We have performed calculations 1 on acŽ .perfect square lattice and 2 in the presence of an

orthorhombic distortion with Debye cut off k T sB DŽ .0.02586 eV T s300 K in both cases. The parame-D

Ž .ters for these two cases are the following: 1 SquareŽ .lattice— a ts0.2586 eV, bs1, gs0, V s0.731

Ž .t, and V s6.8 t, T s71 K, T s28 K; b ts2 c c1

0.2586 eV, bs1, gs0, V s0.73 t, and V s7.91 2Ž .t, T s71 K, T s47 K; 2 Orthorhombic distortionc c1

Ž .— a ts0.2586 eV, bs0.95, and gs0, V s0.971Ž .t, and V s6.5 t, T s70 K, T s25 K; b ts2 c c1

0.2586 eV, bs0.95, and gs0, V s0.97 t, and1

V s8.0 t, T s70 K, T s52 K. For a very weak2 c c1Ž .2 2d -wave d -wave coupling the only possiblex yy x y

Ž .solution corresponds to D s0 D s0 .2 1

In Figs. 1 and 2 we plot the temperature depen-dencies of different D’s for the following two sets ofd 2 2 q id -wave corresponding to models 1 and 2x yy x y

Ž Ž . Ž .above full line—models 1 a and 2 a ; dashed lineŽ . Ž ..—models 1 b and 2 b , respectively. In both cases

the temperature dependence of the D’s are verysimilar. In the coupled d 2 2 q id -wave as tem-x yy x y

perature is lowered past T , the parameter D in-c 1

creases up to TsT . With further reduction ofc1

temperature, the parameter D becomes nonzero and2

Fig. 1. The order parameters D , D in Kelvin at different1 2Ž . Ž .2 2temperatures for d q id -wave models 1 a full line andx yy x y

Ž . Ž .1 b dashed line for square lattice.

( )A. Ghosh, S.K. AdhikarirPhysica C 309 1998 251–256254

Fig. 2. The order parameters D , D in Kelvin at different1 2Ž . Ž .2 2temperatures for d q id -wave models 2 a full line andx yy x y

Ž . Ž . Ž2 b dashed line in presence of orthorhombic distortion b s.0.95 .

begins to increase and eventually both D and D1 2

first increases and then saturates as temperature tendsto zero. Recently, the temperature dependencies ofthe order parameter of the d 2 2 q is-wave super-x yy

conducting state has been studied, where at TsT ,c1

the transition from d 2 2 to d 2 2 q is state takesx yy x yy

place. In that case, below TsT the d 2 2-wavec1 x yy

component of the order parameter is suppressed, asthe s-wave component becomes nonzero. No suchsuppression of the d 2 2-wave takes place in thisx yy

case as the d component appears.x y

Now we study the temperature dependence ofspecific heat in some detail. The different supercon-ducting and normal specific heats are plotted in Figs.

w Ž . Ž .x3 and 4 for square lattice models 1 a and 1 b andw Ž . Ž .xorthorhombic distortion models 2 a and 2 b , re-

spectively. In both cases the specific heat exhibitstwo jumps—one at T and another at T . From Eq.c c1Ž .6 and Figs. 1 and 2 we see that the temperature

< < 2derivative of D has discontinuities at T and Tq c c1

due to the vanishing of D and D , respectively,1 2

responsible for the two jumps in specific heat. For apure d 2 2-wave we find that the specific heatx yy

exhibits a power-law dependence on temperature.However, the exponent of this dependence varieswith temperature. For small T the exponent is ap-

Ž .proximately 2.5, and for large T T™T it is nearlyc

2. In the d 2 2 q id model, for T )T)T thex yy x y c c1

Ž . Ž . Ž .Fig. 3. Specific heat ratio C T rC T vs. TrT for models 1 an c cŽ . Ž . Ž . Ž . Ž .and 1 b for square lattice: 1 a full line , 1 b dashed line , dx y

Ž . Ž .dotted line , normal dashed–dotted line . In all cases T f70 K.c

specific heat exhibits d 2 2-wave power-law behav-x yy

ior; for T-T the specific heat exhibits an s-wavec1

like exponential behavior. For the d-wave modelŽ . Ž .2 2d , C T rC T is a function of T and b. Inx yy s c n c c

Figs. 3 and 4 this ratio, for T s70 K, is approxi-cŽ . Ž .mately 3 2.5 for bs1 0.95 . In a continuum

d-wave calculation this ratio was 2 in the absence ofw xa van Hove singularity 12,13 . We also calculated

the specific heat for the pure d case. For squarex y

Ž . Ž . Ž .Fig. 4. Specific heat ratio C T rC T vs. TrT for models 2 an c cŽ . Ž . Ž . Ž . Žand 2 b for orthorhombic distortion: 2 a full line , 2 b dashed

. Ž . Ž .line , d dotted line , normal dashed–dotted line . In all casesx y

T f70 K.c

( )A. Ghosh, S.K. AdhikarirPhysica C 309 1998 251–256 255

lattice with V s9.0 we obtain T s67 K and1 cŽ . Ž . Ž . Ž . Ž .1.4C T rC T s1.82 and C T rC T ; TrTs c n c s n c c

for the whole temperature range. For orthorhombicdistortion bs0.95, with V s9.0 we obtain T s691 c

Ž . Ž . Ž . Ž .K and C T rC T s1.94 and C T rC T ;s c n c s n cŽ .1.5TrT for the whole temperature range. Thisc

power-law behavior with temperature in both thed-waves is destroyed in the coupled d 2 2 q id -x yy x y

wave and for T-T , we find an s-wave-like expo-c1

nential behavior in both cases. In both the uncoupledd-waves the order parameter D has nodes on theFermi surface and changes sign and this property isdestroyed in the coupled d 2 2 q id -wave, wherex yy x y

the order parameter has a typical s-wave behavior.In Fig. 5 we study the jump DC in the specific

heat at T for pure s- and d-waves superconductorsc

as a function of T , where we plot the ratiocŽ .DCrC T vs. T . For a BCS superconductor in then c c

Ž . Ž . Žcontinuum DCrC T s1.43 1.0 for s-wave d-n c. w xwave independent of T 12,13,31 . Because of thec

presence of the van Hove singularity in the presentmodel this ratio increases with T as can be seen inc

Ž .Fig. 5. For a fixed T , the ratio DCrC T is largerc n cŽ .for square lattice bs1 than that for a lattice with

Ž .orthorhombic distortion bs0.95 for both s- andd 2 2-waves. However, for a d -wave supercon-x yy x y

Fig. 5. Specific heat jump for different T for pure s- andcŽ . Žd-waves: s-wave solid line, square lattice , s-wave dashed line,

. Ž2 2orthorhombic distortion , d -wave dashed–dotted line, squarex yy. Ž2 2lattice , d -wave dashed–double-dotted line, orthorhombicx yy

. Ž .distortion , d -wave dotted line, square lattice , d -wavex y x yŽ .dashed–triple-dotted line, orthorhombic distortion .

Ž . Ž .Fig. 6. Susceptibility ratio x T r x T for square lattice vs.cŽ . Ž2 2TrT : pure d -wave solid line , pure d -wave dashedc x yy x y

. Ž . Ž . Ž .line , pure s-wave dashed–dotted line , models 1 a dotted lineŽ . Ž .and 1 b dashed–double-dotted line . In all cases T f70 K.c

Ž . Žductor DCrC T is smaller for square lattice bsn c.1 than that for a lattice with orthorhombic distortionŽ .bs0.95 . The jump in d -wave is smaller thanx y

that for s- and d 2 2-waves. At T s100 K, in thex yy cŽ .2 2s-wave d -wave square lattice case this ratiox yy

Ž .could be as high as 3.63 2.92 , whereas for d -wavex y

Ž . Ž .Fig. 7. Susceptibility ratio x T r x T in presence of orthorhom-cŽ .2 2bic distortion vs. TrT : pure d -wave solid line , purec x yy

Ž . Ž .d -wave dashed line , pure s-wave dashed–dotted line , modelsx yŽ . Ž . Ž . Ž .2 a dotted line and 2 b dashed–double-dotted line . In all

cases T f70 K.c

( )A. Ghosh, S.K. AdhikarirPhysica C 309 1998 251–256256

Ž .this ratio at 100 K is 1.15 1.25 for square latticeŽ .orthorhombic distortion .

Next we study the temperature dependencies ofspin susceptibility for square lattice and in the pres-ence of orthorhombic distortion which we exhibit inFigs. 6 and 7, respectively. There we plot the resultsfor pure d 2 2-, d -, and s-waves for comparison,x yy x y

Ž . Ž . Ž .in addition to those for models 1 a , 1 b , 2 a andŽ .2 b . In all cases reported in these figures T f70 K.c

For pure d-wave cases we obtain power-law depen-dencies on temperature. The exponent for thispower-law scaling is independent of critical tempera-ture T but vary from a square lattice to that with anc

orthorhombic distortion. For d 2 2-wave, the expo-x yyŽnent for square lattice orthorhombic distortion, bs

. Ž .0.95 is 2.6 2.4 . For d -wave, the exponent forx yŽ .square lattice orthorhombic distortion, bs0.95 is

Ž . 2 21.1 1.6 . For the mixed d q id -wave,x yy x y

d 2 2-wave power-law behavior is obtained for Tx yy c

)T)T . For T-T , one has a typical s-wavec1 c1

behavior.Ž 2 2In conclusion, we have studied the d qx yy

.id -wave superconductivity employing a two-di-x y

mensional tight binding BCS model on square latticeand also for orthorhombic distortion. We have keptthe potential couplings in such a domain that a

Ž .2 2coupled d q id -wave solution is allowed.x yy x y

For a weaker d admixture, as temperature is low-x y

ered past the first critical temperature T , a weakercŽ .less ordered superconducting phase is created in

Ž2 2d -wave, which changes to a stronger morex yy. Ž .2 2ordered superconducting phase in d q id -x yy x y

wave at T . The d 2 2 q id -wave state is similarc1 x yy x y

to an s-wave-type state with no node in the orderparameter. The phase transition at T from ac1

d 2 2-wave to a d 2 2 q id -wave is marked byx yy x yy x yŽ .power-law exponential temperature dependencies

of specific heat and spin susceptibility for T)Tc1Ž .-T . Similar behavior has been observed forc1

w x2 2d q is-wave state 24 . As the mixed state isx yy

s-wave like in both cases, from the present study itwould not be possible to identify the proper symme-try of the order parameter—d 2 2 q id opposedx yy x y

to d 2 2 q is—and further phase sensitive tests ofx yy

pairing symmetry in cuprate superconductors isneeded.

Acknowledgements

We thank Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico and Fundacao de Amparo a´ ´ ˜ `Pesquisa do Estado de Sao Paulo for financial sup-˜port.

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