Phase fluctuations of the superconducting order parameter in high-Tc systems

Pergamon

Solid State Communications, Vol. 98, No. 4, pp. 313-316, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/96 $12.00 + .00 , S0038-1098(96)00047-6

P H A S E F L U C T U A T I O N S OF T H E S U P E R C O N D U C T I N G

ORDER PARAMETER

IN HIGH-TC SYSTEMS J.J. Vicente Alvarez and C.A. Balseiro Centro Atomico Bariloche e Instituto Balseiro 8400 S.C. de Bariloche, Argentina

(Received 7 July 1995; accepted in revised form 6 January 1996 by C.E.T. Gonfalves da Silva) Stoning with a microscopic Hamiltonian describing a superconducting system, we use the functional integral formalism to map the problem to the XY-model. The effective model describes vortex-line fluctuations with a coupling constant J that is given in terms of the microscopic parameters. The results obtained with this procedure, which is valid at all temperatures, are compared with those obtained in previous works. Keywords: A. superconductors D. electron-electron interactions D. phase transitions

During the last few years a big effort has been made to obtain a phenomenological description of high-To materials in the presence of external magnetic fields and external currents. From the experimental point of view, it is clear that, because of the relatively high values of Tc in the cuprates, the fluctuation effects on the vortex system are very much enhanced, leading to novel effects[l]. The magnetic field-temperature ( H - T ) phase diagram is much richer and more complicated than the predictions of the mean-field theory. Several phases are known, such as the vortex-glass, a 3D vortex liquid or a 2D vortex phase. The transition to the normal state as temperature is increased is likely to be a crossover rather than a well-defined transition. It has also been shown that anisotropy plays an important role in determining the H - T phase diagram. The different behavior of BiSrCaCuO and YBaCuO is attributed to the difference in the anisotropy ratio of these materials. From the theoretical point of view, different approaches have been used to describe the thermodynamics of the high-Tc materials[l]: effective elastic theories[2], a mapping of the vortex lines onto a two-dimensional interacting boson problem[3], networks of Josephson junctions[4] and the uniformly frustrated XY model[5]. The last two models allow a more complete description of the problem, in particular in the H = 0 case, and are to some extent equivalent. Josephson networks, however, allow a detailed description of the dynamics of vortices while the XY model can be used only to describe the thermodynamic properties of the system.

The XY model is the best studied and can be taken as a phenomenological model that describes the phase excitations of the order parameter in the superconducting state. Recently it has been rederived starting from the LandauG-insburg theory[6]. This procedure, which allows us to write the parameters of the effective XY model in terms of the Landau-Ginsburg parameters, is fully justified only close to Tc where the linearized Landau-Ginsburg theory is valid. In this work, we present an alternative way of deriving the effective Hamiltonian which is believed to contain the essential physics of the phase fluctuations. Our starting point is a microscopic I-Iamiltonian describing the superconducting material. We use the functional integral formalism and perform the saddle point approximation. We calculate fluctuations in the phase of the order parameter around the saddle point to end up with an effective action corresponding to that of the XY-model. The coupling constant J of the effective model is given in terms of the microscopic parameters. The results obtained with this procedure, which is valid at all temperatures, are compared with those obtained in previous works. For the sake of simplicity, we consider the simple electronic model that in the usual notation reads:

ko

q

where c ~ creates an electron with momentum k and spin or, ~q = Ek C,,+q/2rc_k+q/2t t t is a creation operator for 313

SUPERCONDUCTING ORDER PARAMETER IN HIGH-TC SYSTEMS

314

a pair with total momentum q , ek = e~, - # where e~, = -2(cos k~+cos k~) is the dispersion relation of the fermions, # is the chemical potential and U is an attractive interaction. In the last term on the right-hand side of equation (1) an implicit cutoffwD can be taken. Using the Hubbard-Stratonovich transformation, the interaction term in (I) is reduced to a quadratic form in fermion operators and the partition function of the system is given by

Vol. 98, No. 4

A(q, "r) = 6qOAO+ Ao (iOq - ~O(q2)) • i (~) At(q, 7") ~qoAo + Ae (z0_q - ~0_q)

where 0(q~) = ~k 0k0_k+q. Now the partitionfunction(2) reads: Z = f D0exp {-,Sell [0]}

(4)

where Sell [0] is an effective action of the phases• Expanding the effective action up to second order we obtain: Sell [0l = Sap+ ~ Jel/(q)OqO_q

(5)

q

z = f ~-Zl[At,A]e-

fo%"b ~ql a(q,r}f2

(2)

where

here S,~ = - In ZI[A~] is the contribution of the fermions at the saddle point, and \ ° ] J~/l(q) = A o 2 [ u - - ( k ~ q ~ ( 0 + ) ) : v + \/~t-q ~q \[ o + ~''avJ

'} and Zl

= where the subscript 4- indicates k 4- q/2, u~: = -~ (1 + Ei) - y~ q ( A ( q , ' r ) ~ (r) + A'(q,r)ff2q ('r))] }

here At(q, T) and A(q,'r) are the Hubbard-Stratonovich fields and/~ is the inverse of the temperature. The saddle-point approximation consists of replacing the Hubbard-Stratonovich fields by c-numbers independent of space and time A ~ ( q , T) = 6qoA?oand A ~ ( q , 7-) = 6qoAo. These c-numbers are taken to minimize the free energy F~ = - ~ I n Z I [ A ~ , A ~ ] . There is an arbitrary global phase in the solution of these equations, and the real solution is given by: = ~ Ao = V (~q-----O)~q~ AoU ~ k

tarda(Ek/2fl) 2Ek

(3)

which is the usual BCS equation for the gap or order parameter Ao. Here Ek ---- V/~k + A~ are the energies of the new fermion quasiparticles. The chemical potential is determined by fixing the number of particles with the equation n = -

(OFsp/OI~)T.

To go beyond this approximation, we propose a variation of the Hubbard- Stratonovich fields around their saddlepoint values. In doing this, we look for the low energy excitations. Choosing the order parameter to be real, we break its continuous phase symmetry, provided Ao # 0. According to Goldstone's theorem, this symmetrybreaking implies that a soft energy mode exists as q ~ 0. These modes are associated with the phase fluctuations of the order parameter. We propose a time-independent variation of the phases of the order parameter.

and v~ = -~ (I - ~-£~] and f+ = f(E±) is the usual fermi function. This last ~ m of the effective action implies that the partition function in (4) is a product of two contributions; one is the partition function obtained at the saddlepoint level and the other accounts for the phase fluctuations:

fo2xDO exp {--flEqJell(q)OqO_q} (6)

Z = Z liar]

Now DO = Hqd0q/27r, this second factor corresponding to the classical partition function of a system with a Hamiltonian H = EqJell(q)OqO_q. It is clear that Jell is nonzero only below the mean-field (BCS) critical temperature where Ao ~ 0 and that Jell(q --.*0) ~ O. To provide a link with the XY model, we expand Jell up to second order in q to obtain a low energy behavior H = JEqq20q0-q = J f d r [V0(r)] 2. We discretize this effective model by introducing a lattice parameter a, to obtain

H = Ja a-2 ~

(O, - Oj 2 ,

where d is the dimension of the system. This is the low energy expansion of an XY model: =

Ja'-'

E [i-cos(0,- 09]



(7)

As we show below, the interaction parameter or is always positive and the model hamiltonian (7) corresponds to a frustrated XY model. The following points deserve a comment. In our approximation, the amplitude of the order parameter is constant. In a vortex- like excitation, the order parameter goes to zero at the vortex core, but the XY model describes the phase excitations outside the core so the discretizafion should be made in a mesh with lattice parameter a of the

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SUPERCONDUCTING ORDER PARAMETER IN HIGH-TC SYSTEMS

order or greater than the coherence length ~o. The phase difference appearing in equation (7) should be a gauge invariant quantity and in the presence of a magnetic field it has to be replaced by 0i - 05 - 2~r/0o f~ A.dl, where 0o is the flux quantum and A is the vector potential. In Fig 1, the amplitude of the order parameter Ao and the coupling constant J versus temperature are shown for different values of the attractive interaction U. Both quantities go to zero at the mean- field critical temperature To. For small U, the gap A0 is much smaller than J at all temperatures and below T~ cs the thermodynamics is dominated by the quasiparticle excitations. In the clean limit we are describing this weak coupling regime corresponds to a type I superconductor. As U increases A0 increases and J decreases, as seen later in Fig2, and in the strong coupling regime the low energy excitations are phase excitations. In the absence of an external magnetic field, these low energy excitations are vortex loops or vortex-antivortex pairs in the two dimensional case. In this strong coupling regime, the system behaves as a type II superconductor. The coupling constant J is only weakly temperaturcdependent up to a temperature of the order of the critical temperature where it shows a rapid decrease to become zero at T~ cs. At T = O, the coupling constant J is given by:

2

Ek

where e~, = {Oek/Ok= and e~ = lmek/O2k=. The low temperature value of J as a function of 1/U is shown in Fig. 2. For large U, J goes as 1/U in agreement with the result obtained by a canonical transformation on the Hubbard hamiltonian[8]. Assuming a rectangular density of states of width 2W and an electron density n, we obtain # = - ( 1 - n)U/2, Ao = ~ n)V/2, and j = 3 n(2 - n) 2 U

0.04

ooi! •

,

.

,

•

,

315

.

,

....-

......'"• .. ... .."'"""

0.03

0.01

O.

o11

o:2

'

o13

d4'o:5

o6oo

uu

Figure 2: The low temperature coupling constant J vs the inverse of U, calculated for a square lattice with electron density n = .2.

For small U, the coupling J saturates and becomes proportional to the square of the Fermi velocity of the system [7] [9]. In this limit, for a two-dimensional system 1

2

J = .~pyv!

In 2D, the XY model is known to have a transition to a state of quasi- long- range order and finite superfluid density below a critical Kosterlitz- Thouless temperaturc[10]

TcKT "" O,89J. Above this temperature the phases arc disordered and the system has normal properties. As U increases from zero, there is a crossover between two regimes[ll]. In the weak coupling limit at low temperatures, J >> Ao and the pairbreaking, BCS- type excitations dominate the low temperaturc thermodynamic properties. In this regime, the phases exhibit a quasi- long- range order up to temperatures of

0.05

0.03 n=.2

0.04 0.02

0.03 <~ 0.02 0.01

[

0.00 0.00

tu:l , t 0.01

0.02

1

% o.ol I,.-

0.00

0.03

T

i

i

i

x

i

2

4

6

8

10

Ult

Figure 1: The order parameter A0 (dashed line)and the coupling constant J (full line) are calculated for differcnts values of U, in a square lattice with electron density n =

Figure 3: The critical temperature of BCS (dashed line) and the Kosterlitz- Thouless critical temperature (full line) vs U, calculated for a square lattice with electron density

.2.

'/~=

.2.

316

SUPERCONDUCTING ORDER PARAMETER IN HIGH-TC SYSTEMS

the order of the BCS critical temperature T~cs. It is only very close to this temperature, when J rapidly decreases to zero, that the phases become really disordered. In this regime, the Kosteditz- Thouless critical temperature T f T lies near T~cs, and the transition to the superfluid phase occurs with the appearance of the Cooper pairs. In the strong coupling limit, the low- temperature value of the coupling constant J is such that J << Ao and the thermodynamic properties at low temperatures are dominated by the phase fluctuations with Tfi T << T~cs. In this regime, when the temperature is decreased, first Cooper pairs without off-diagonal long-range order form and then, below the Kosteditz- Thouless transition temperature, the system becomes superfluid. In Fig. 3, T~ T is shown as a function of U, and for comparison Tf cs is also shown in the same graph. In summary, starting from a microscopic BCS hamiltonian, we have formulated a functional integral theory and evaluated the fluctuations in the phase of the order parameter around the mean-field saddle point. The problem can then be mapped onto an XY model describing the phase excitations. As a result, we obtained the coupling constant

Vol. 98, No. 4

J of the effective a,odd in temls of the microscopic parameters. We showed that J i s weakly temperature- dependent up to a temperature of the order of T~acs. The crossover between the strong and weak coupling regimes has been discussed for the two dimensional case. The KosteditzThouless transition occurs close to the mean- field temperature in the weak coupling regime, and as the attractive interaction U increases, it goes through a maximum and then decreases as 1/U. The amplitude fluctuations of the order parameter have been neglected. As a consequence, the physics at the mean- field critical temperature is not well described. In the strong coupling regime, T~cs should be interpreted as a crossover temperature at which a macroscopic number of Cooper pairs become stable rather than a phase transition temperature. To overcome this difficulty, one should include amplitude and phase fluctuations at the same footing.

Acknowledgments: One of us (C.A.B.) would like to thank B. Chacraverty for stimulating discussions. J.J.V.A. is supported by CONICET and C.A.B. is partially supported by CONICET.

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