ZHEN YE: Fluctuation Effect on the Excess Conductivity of Superconductors 193

phys. stat. sol. (b) 171, 193 (1992)

Subject classification: 74.40

Department of Physics, University of Ottawa’)

Fluctuation Effect on the Excess Conductivity of Layered Superconductors

BY ZHEN YE

By using thermo field dynamics (TFD) and Ginzburg-Landau theory, the fluctuation effects on conductivity are studied for layered superconductors above critical temperature. It is argued that the broadness of the resistivity is due to the decoupling of electron tunneling among layers. The results are compared with experiments.

Mit der Thermofelddynamik (TFD) und der Ginzburg-Landau-Theorie werden die Fluktuationsein- flusse auf die Leitfahigkeit fur Schichtsupraleiter oberhalb der kritischen Temperatur untersucht. Es wird angenommen, daD die Breite des Widerstands durch die Entkopplung der Elektronen, die zwischen den Schichten tunneln, hervorgerufen wird.

1. Introduction

The CuO, planes in the layered superconductors are considered to be important for high T, superconductivity. This leads to strong anisotropy of their superconductive properties. In some cases their behavior can be described as anisotropic three-dimensional, whereas in other cases as quasi-two-dimensional. The cross-over between quasi-2D and 3D behaviors takes place when some characteristic length, i.e. the coherence length, along the c-axis is of the order of the interlayer space a [l]. While the mechanism of high temperature superconductivity continues to be a mystery, many people are looking at the properties in the normal state. The consensus is that, if the correlation length is of the order of the lattice spacing and the system is nearly two-dimensional, the fluctuation effect must play a very important role in defining the properties of the superconducting state in these materials. The significant rounding of the resistance versus temperature curve above the critical temperature in high T, superconductors is one of the examples [3]. The excess electrical conductivity due to the thermal fluctuation has been studied intensively for a bulk superconducting system [5 ] . In the present paper, we look at this property in the layered superconductors with interlayer electron tunneling.

Considerable effort has been devoted to elucidate the properties of quasi-2D super- conducting systems. In our previous work [2], a realistic tunneling model has been developed to explain the anomalous behavior of the temperature dependence of magnetic penetration. We argued that the Meissner effect may disappear in the vicinity of the critical temperature. Our results agree with experiments qualitatively. For a more general model [6, 71 including interlayer and intralayer pairings, a possible first-order phase transition in layered superconducting state was predicted [8]. In [6], we also studied the fluctuation effect on the intralayer order parameter at the temperature where interlayer pairing disappears. All our

I) Ottawa, Ontario, Canada K I N 6N5.

13 physica (b) 171/1

194 ZHEN YE

works are in the frame of U(l) symmetry breaking. It is therefore an aim of this short communication to study the fluctuation effect on the conductivity above the critical temperature in our tunneling model. Because the interlayer order parameter is more important in many respects than intralayer pairing, in the present paper we consider the interlayer Cooper pairs only. The results obtained here can explain a quite common feature in HTSC materials: broadness of resistivity near T,. Our approach goes as following: we first use the TFD method to find London penetration depths and coherence lengths, out of which we define the Ginzburg-Landau free energy; then we follow a standard fluctuation treatment method to calculate the resistivity near the critical temperature.

2. Model Hamiltonian

The interaction Hamiltonian we are considering is

Hint = (-4 c t [w: (4 4 Y i (x, i + 1) W?(X> i + 1) w L(X, 2) 1

+ v:(x> V q X , i - 1) Y r ( X , i - 1) wl(x, 41, (1) where x is the two-dimensional coordinate and yo is the Heisenberg electron field or “bare” field. So different layers are communicated with each other by the electron tunneling. The order parameter is defined as

(2) A ( T ) = i(yI(x, i) w+(x, i & 1)). Simple calculation leads to

which is same as that in bulk BCS superconducting systems. In order to find out the London penetration depths and coherence lengths, we here make use of the TFD method [2,9]. Readers may also refer to an easy essay by Weinberg [lo]. Here we outline the TFD treatment. As U(l) symmetry is broken, the Ward-Takahashi relation and Goldstone theorem lead us to the following quasi-particle fields: the quasi-electrons cp,, the plasmons P,, and the gapless Goldstone bosons x. The energy spectrum of the quasi-electrons is

E k = v.$ + A 2 cos’ k,a, (4) where &k is two-dimensional electron kinetic energy. Here we see that when T approaches T,, E goes from 3D to 2D. For the case of temperature greater than T,, in order to keep our argument self-consistent, we need introduce a third direction energy term like t,k:, where t o N T - T,.

By considering the rearrangement of the phase symmetry, we have the following dynamical mappings for y, and current J,:

in which we included the electromagnetic field A. Here 2 is the residue of the Goldstone boson Green’s function [2], and u = 2A/A, c( - ia) is the so-called c-function [4], b(x) is the ghost field used to fix the gauge of the electromagnetic field A,. We have to restrict our

Fluctuation Effect on the Excess Conductivity of Layered Superconductors 195

Fock space by requiring that for any physical state Jp) the following condition is satisfied:

(x - b)(-’ Jp) = 0 . (7) The symbol (-1 means the annihilation part of operators x and b. So we see from (6) that the effect of Goldstone bosons is eliminated from observables by including the electromag- netic interaction. This is known as Anderson-Higgs-Kibble mechanism. However, we can see from (5) that the field x, as a phase factor, can appear independently of the electromagnetic field. So the Goldstone boson will modify the phase ordering which is not observable. In our layered model, the energy spectrum of the Goldstone boson field x is

o ;=l v,(k; 2 + k;) + f (ad)2 k: . (8) Here we see that the Goldstone mode becomes two-dimensional as temperature approaches T,, which introduces extra fluctuations. From (6) we see that in order to find the London penetration depths and coherence lengths, i.e. the range of the c-function, we simply calculate the coefficients between current J, and Goldstone field x. Since the electromagnetic field does not affect the dynamical mappings [ll], we could simply ignore the field A,. After tedious calculation, we find [2] for T close to T,,

The coherence lengths are approximately calculated as [ 121

3. G-L Theory and Results

The anisotropic behaviors of magnetic penetration and coherence lengths have been studied in [2,12]. Now we consider the properties of layered superconductors in the framework of the G-L function F with an anisotropic “effective mass” [13],

where B = curl A , q ( r ) is the order parameter, mi = (ma, mb, m,) is the “effective mass” tensor and ma = m6 = ma-b, mZ = m,. From the above equation (11) we have

Comparing these equations with (3), (9), and (lo), we find, as T is close to T,,

and - 1

m,-, (T) = mO,a-b = const, m C ( T ) = mo,, I( 1 - :)I .

Relation (14) can be also obtained by calculating the linear response function as we did in [2] where we have near T,,

j 3 = ( A 2 ) j 1 , 2 . (15)

196 ZHEN YE

Now we consider the fluctuation in the superconducting state due to the spatial and temporal variation of cp. In the simplest approximation, the time dependence of y(r, t ) will be described by the Langevin equation [14]

where zo = (nh)/(k(T - T,)) is the relaxation time and F , the random force. Following the method in [15], we easily have

a+L' 4mi

where fpk is the Fourier transformation of cp, zk = zo/(l + C k?t?) . In deriving (17), we assumed Fr to make (q$(t)) = const and the Wiener-Khintchine theorem has been used. According to the response formula we have the conductivity tensor oij,

m , I -

The current Ji can be calculated from

eh i2mi

J , = ~ (q* aicp - cp aicp*).

For a space-uniform case, we have

Making use of (17) and (18), we then find

For the stationary case, i.e. o + 0, we have

Uij(0) = sij

Fluctuation Effect on the Excess Conductivity of Layered Superconductors 197

After taking (14) into consideration, we finally get

Therefore, we get anisotropic behavior of the conductivity for the layered superconductors. We see in layered superconductors, the thermal fluctuation will broad on the disappearance of a - b direction resistivity, i.e. e(T),- , - (TIT, - 1). Its temperature dependence behaves like that in two-dimensional superconductors. This result can well explain the observations on several high T, layered superconductors, e.g. Bi-Sr-Ca-Cu-0 compounds [3], while the resistivity along the c-axis will have a sharp drop when temperature decreases to T,.

In summary, we discussed a tunneling model for layered superconductors. For temperature close to T,, the Goldstone mode becomes almost two-dimensional, the superconducting layers become decoupled. This introduces excess fluctuation which can explain many peculiar features in high T, superconducting states [2,6,12]. The calculation of the thermal fluctuation effect on the excess electrical conductivity was given with the use of TFD and Ginzburg- Landau theories.

Acknowledgements

1 sincerely thank Prof. H. Umezawa’s advice on the subject. The work was supported by NSERC. Canada.

References

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[4] Z. YE and H. UMEZAWA, Phys. Letters A 162, 63 (1992). [5] L. G. ASLAMAZOV and A. I. LARKIN, Phys. Letters A 26, 238 (1968). [6] Z. YE, H. UMEZAWA, and R. TESHIMA, Phys. Rev. B 44, 351 (1991). [7] Z. YE, H. CHU, and H. UMEZAWA, Phys. Letters A 154, 421 (1991). [8] Z. SCHLESINGER, R. T. COLLINS, F. HOLTZBERG, C. FEILD, G. KOREN, and A. GUPTA, Phys. Rev.

[9] H. UMEZAWA, H. MATSUMOTO, and M. TACHIKI, Thermo Field Dynamics and Condensed States, B 41, 11 237 (1990).

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(Received December 11, 1991)