Influence of order parameter phase fluctuations on the physical properties of quasi-two dimensional superconductors

Physica C 295 Ž1998. 150–169

Influence of order parameter phase fluctuations on the physical properties of quasi-two dimensional superconductors E.P. Nakhmedov

a,b,)

, Yu.A. Firsov

c

a

c

Istanbul Technical UniÕersity, Faculty of Sciences and Letters, Department of Physics, Maslak, 80626, Istanbul, Turkey b Institute of Physics, Azerbaijan Academy of Sciences, 370143, H. CaÕid Street 33, Baku, Azerbaijan A.F. Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 Politechnicheskaya 26, St. Petersburg, Russia Received 29 April 1997; revised 24 September 1997; accepted 27 September 1997

Abstract The influence of the order parameter’s phase fluctuations on the Meissner effect is studied for strong anisotropic layered superconductors in the presence of a weak magnetic field, H, less than the lower critical field Hc1 Ž H - Hc1 .. The Josephson coupling between the layers can be realized in quasi-two dimensional Žquasi-2d. superconductors when the interlayer tunnelling integral J H satisfies the condition J H- k TcŽ2. - e F , where e F and TcŽ2. are the Fermi energy and the mean-field transition temperature for a single superconducting layer, respectively. The system of equations of motion is obtained for the order parameter phases w j Ž™ r . at the r-th point of the j-th layer. This system of coupled equations is investigated by applying the self consistent phonon approximation ŽSCPA. method. There exists the plasmon mode in the Josephson coupled layered superconductors in the frame of the SCPA approximation. The square of the transverse effective velocity, Õph,H , of collective excitations become proportional to the interlayer phase–phase correlator ²cos Ž w j y w jy1 .:. When this correlator approaches zero, correlations of the superconducting phases on the different layers disappear, i.e. the phase transition from quasi-2d superconducting state to a pure 2d state occurs at the critical temperature Tc1. The transverse rigidity of the system and the plasmon’s effective velocity Õph,H vanish at this temperature. Tc1 is less than TcŽ2. and ŽTcŽ2. y Tc1 . ; TcŽ2. Ž k TcŽ2.re F .. In the temperature interval of ŽTc1 y T . ; Tc1Ž k Tc1re F . ln Ž k Tc1rJ H . below Tc1 , the fluctuations of the order parameter’s phases become essential in the quasi-2d superconductor. In this interval the ratio of the longitudinal l 5 and transverse l H components of the London penetration depths, l 5rl H , should exhibit strong temperature dependence, unlike the prediction of the usual Ginzburg–Landau phenomenological theory. l 5 and l H diverge at different critical temperatures, namely at Tc2 ' TcŽ2. and Tc1, respectively. At Tc1 - T - Tc2 the phases w j on the different layers become non-correlative and the Kosterlitz–Thouless vortices appear in each superconducting layers. q 1998 Elsevier Science B.V. PACS: 74.40.q k; 74.80.Dm; 74.60.-w; 74.20.De; 74.60.Ec

)

Corresponding author. Tel.: q90 212 2853209; Fax: q90 212 2856386; E-mail: [email protected]

0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 7 6 1 - 9

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

151

1. Introduction Recently, the superconductivity of low dimensional systems has attracted a great deal of interest Žsee w1–8x and references therein.. The class of low dimensional superconductors includes A-15 compounds with chain like structures w9x, the artificial quasi-one-dimensional Žquasi-1d. materials, i.e. the second crystals w10x which consist of parallel placed superconducting chains in di-electric matrices, also large class of naturally occurring layered materials w1–4,9x and artificial superlattices w11x. The layered superconductors such as dichalcogenides of transition metals Že.g. NbS 2 , TaS 2 , TaSe 2 etc.., the layered compounds Že.g. MoS 2 , WSe 2 , TaS 2 etc.. intercalated by atoms of alkaline and alkali rare metals are investigated in detail. Another class of low-dimensional superconductors is formed by organic superconductors, the quasi-1d salts of tetramethyltetraselenafulvalene ŽTMTSF. 2 X ŽBechgaard salts. w5x and the quasi-two-dimensional salts of dis-ethylendithiotetratiafulvalene ŽET. w12x. The discovery of high temperature superconductors w13x considerably enlarges the class of layered superconductors. The anisotropy of Bi- and Tl-based superconductors such as Bi 2 Ba 2 CaCu 2 O 3 and Tl 2 Sr2 Ca 2 Cu 3 O 10qx increases with concentration of Ca. The critical temperature of these superconductors increases as Tc ; n = 40 K where n is the number of Ca atoms or CuO 2 layers per unit cell and n s 1,2, . . . . The common characteristic features of these crystals are strong anisotropy of both structural and physical properties. The conductivity of these crystals along and perpendicular to a preferred axis differs by a few orders of magnitude. The anisotropies of the critical current and magnetic field of such low dimensional crystals vary greatly, e.g. the ratio of the upper critical fields, Hc25 rHc2H , for an external field directed perpendicular and parallel to layers in intercalated compounds varies from 3 to 50, w3x. For the Tl-based high-Tc superconductors this ratio varies from 20 to 200, w4,14x. Experimental studies of the coherence length j in the Bi- and Tl-based ˚ superconductors reveal extremely small values for perpendicular to layers components of j , j 5 ; 0.3–1.0 A w4,14,15x. Therefore, the model consisting of periodically placed superconducting layers with Cooper pairs tunnelling between them is used to study the strongly anisotropic superconductors. Such a model has been studied by many authors w16–18x. However, the fluctuation of the phase of the order parameter at T - Tc has been ignored in these papers. The order parameter’s phase fluctuations were shown to have an essential influence on the physical properties of quasi-1d superconductors w6,19–21x. There is no superconducting phase transition in a two-dimensional Ž2d. system w22,23x. The existence of strong fluctuations of order parameter’s phase destroys the long range order in single superconducting layers. The destruction effects of the long wavelength fluctuations become weaker in the Josephson coupled quasi-2d superconductors due to the tunnelling of the Cooper pairs from one layer to another. Therefore, the long range order in the system sets in at a non-zero temperature. There exist, of course, the fluctuations of the order parameter’s modulus. However, it can easily be shown that such fluctuations are essential in the vicinity of the superconductivity transition temperature Tc , when t s < Tc y T
E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

152

renormalized by the phase correlator cos Ž w j Ž ™ r . y w jq1 Ž ™ r . . which can be calculated by applying the self-consistent phonon approximation Žsee Eq. Ž30... We also study the influence of quantum fluctuations of the order parameter’s phases on the Meissner effect at a weak external magnetic field H, H - Hc1 . For a magnetic field aligned with the superconducting planes, the transverse component of the penetration depth l H ŽT . diverges at a new critical temperature Tc) Žsee Eq. Ž36.. less than TcŽ2. ŽTcŽ2. is the mean field transition temperature for a single superconducting layer. due to the quantum phase fluctuations. The longitudinal component of the penetration depth l 5 ŽT . is not affected by the phase fluctuations in our approach Žsee last paragraph of Section 4.. The paper is organized as follows. In Sections 2 and 3 the problems are formulated by taking into account the dynamics of phases of superconducting order parameters. In Section 4 the quantization of phases is performed and phase–phase correlators are calculated in the framework of self-consistent phonon approximation. The new results, such as the renormalizations of the transverse component of the penetration depth l H ŽT . ŽEqs. Ž39. and Ž40... and of the Josephson coupling between superconducting layers ŽEqs. Ž30. – Ž36.. are presented in this section. A brief survey of non-linear phase excitations is given in Section 5, and Section 6 deals with the conclusions.

¦

;

2. The problems According to the Ginzburg–Landau theory the layered superconductors are described by the following free energy functional w16x: Fst  w 4 s NsŽ2. Ý

H

d 2r

j

½

"2 8 m5

Ew j

2

Ew j

ž / ž / Ex

q

Ey

2

q

Ý

E H Ž g . 1 y cos Ž w j Ž x , y . y w jqg Ž x , y . .

gs"1

5

,

Ž 1. where, w j Ž x, y . is the phase of the order parameter D j , i.e. D j s < D j
™

"2 Eel s

8

2

2 X

™

™X

H d r H d r Ý K Ž ryr . ij

w˙ iq1 Ž ™ r . y w˙ i Ž ™ r.

w˙ jq1 Ž ™ r X . y w˙ j Ž ™ rX . ,

Ž 2.

i, j

where a dot over w denotes the time derivative of the phase. The physical meaning of the Fourier transform of the parameter K i j Ž ™ r y™ r X . in Eq. Ž2. is the compressibility. The Eq. Ž2. can be qualitatively understood as follows. According to the first Josephson equation w˙ j y w˙ jX s Ž2 er" .Ž Vj y VjX ., i.e. the time derivative of phases

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

153

difference creates an electrostatic potential difference Ž Vj y VjX . between the layers j and jX . The additional electrostatic energy of charged layers can be expressed as, 1 Eel s Ý d ™ r d™ r X Ci j Ž ™ r y™ r X . Viq1 Ž ™ r . y Vi Ž ™ r . Vjq1 Ž ™ r X . y Vj Ž ™ rX . , Ž 2X . 2 i, j

H H

where Ci j Ž ™ r y™ r X . are the specific coefficients of electrostatic induction. For i s j they mean the specific capacitance coefficients. After expressing of the potential differences Viq1 Ž ™ r . y Vi Ž ™ r . in Eq. Ž2X . by phase derivation we get Eq. Ž2. for Eel . The coefficient of compressibility K Ž ™ q,q H . which is the Fourier transform of the coefficients K i j , has the 2 xy1 w dimension erg cm . The dispersion of K Ž ™ q,q H . in space is defined by the effects of screening Coulomb ™ interactions. K Ž q,q H . have been calculated in w20x for quasi-one dimensional systems. We shall not present the obvious expression of K Ž ™ q,q H . for the layered superconductor and we take K Ž ™ q,q H . s K s const. So, to study the dynamics of phases of superconducting order parameters we take Eqs. Ž1. and Ž2. as operators of kinetic and potential energy, correspondingly. Starting from the Langrangian of the model under investigation, Ls

K j 52 Ž 0 .

r . y w˙ j Ž ™ r.. Ý H d 2 r " Ž w˙ jq1Ž ™

8

2

y Fst w 4

Ž 3.

j ™

it is possible to write the Hamilton function H in the presence of the magnetic field H as:

°

X

H s Ý H d r~ Ý ¢

e i q H Ž jyj . P j Ž ™ r . P jX Ž ™ r.

2

X

j

q

j ,qH

Ý

1 y cos q H

ž

E H Ž g . 1 y cos w j y w jqg q

gs"1 ™

K j 52 Ž 0 . 2e "c

q NsŽ2.j 52

"2

Ž 0.

8 m 5 j 52 Ž 0 .

Azd z

ja H

=r w j y

/

2

2 ej 5 Ž 0.

™

A

™

"c

q j 52 Ž 0 . a H

/

ž H Ž r . y H / ¶•, ™ ™

Ž jqg . a H

H

ž

™

™

2

ext

8p

ß

Ž 4.

™

where A s  A 5 Ž™ r, j ., A H 4 is the vector potential; P j Ž™ r . is the dimensionless ‘momentum’, canonical conjugate to ‘coordinate’ w j and

Pj Ž™ r. s

1 dL s " d w˙ j

K j 52 " 4

™

™

™

Ž 2 w˙ j Ž r . y w˙ jq1Ž r . y w˙ jy1Ž r . . .

The new dimensionless coordinates, ™ r s  x s xrj 5 Ž 0 . , y s yrj 5 Ž 0 . 4 are introduced in Eqs. Ž3. and Ž4., where j 5 Ž0. s "Õ FrpD0 is the value of the coherence length inside a superconducting layer at zero temperature. The classical ™ equations of motion for the phases w j , obtained from Eq. Ž4. in the absence of the vector-potential AŽ™ r ., are defined by a set of coupled sine-Gordon type non-linear equations. These equations describe the order parameter phases oscillation in the infinite number set of Josephson coupled layers the characteristic dimensions of which are more than the Josephson length: 2 w¨ j Ž ™ r ,t . y w¨ jq1 Ž ™ r ,t . y w˙ jy1 Ž ™ r ,t . s

NsŽ2.

E 2w j

m 5 K j 52 Ž 0 .

Ex2

q

E 2w j E y2

y8

NsŽ2. "2 K

E H sin Ž w j y w jq1 . q sin Ž w j y w jy1 . .

Ž 5.

Let us introduce the characteristic scale of frequency v as

vs

ž

NsŽ2. m 5 K j 52

1r2

/

Ž 6.

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

154

and the dimensionless parameter of anisotropy dcl which is assumed to be less than unity:

dcl2 s

8 EH " 2rm 5 j 52

s

ž

2

4g J H

/

p 2k TcŽ2.

ln g s c s 0.577.

;

Ž 7.

We shall define v as v s 2pak TcŽ2.r", where

a s a 0t

1r2

a0 s

;

1 2g

žŽ

1r2

p 2 " 2rm 5 . K

/

ts

;

TcŽ2. y T TcŽ2.

.

Ž 8.

Here a is the dynamic parameter and a - 1. Then Eq. Ž5. can be rewritten as 2 w¨ j Ž ™ r ,t . y w¨ jq1 Ž ™ r ,t . y w˙ jy1 Ž ™ r ,t . s v 2 Dw j y dcl2 Ž sin Ž w j y w jq1 . q sin Ž w j y w jy1 . . .

Ž 9.

It is rather difficult to solve this system of non-linear equations. Therefore, we shall solve Eq. Ž9. in the harmonic approximation.

3. Meissner effect in the layered superconductors ™

To study the Meissner effect it is necessary to find the dependence of the current jŽ™ r . on the vector-potential AŽ r .. For high-Tc superconductors this dependence is believed to be local one due to a relatively small value of the coherence length. The expression for the current for bulk superconductors in the coordinate space is given as ™™

™

ja Ž r . s y

e ) 2 Ns ms c

™

Aa Ž r . q

e ) 2 Ns " 2 Ns m ) c m )k T

H

Ew Ž ™ r . Ew Ž ™ rX .

¦

Era

ErbX

;

Ab Ž ™ rX . d3 rX ,

Ž 10 .

where the first and second terms represent diamagnetic and paramagnetic contributions, correspondingly. The ™ ™ ™ existence of paramagnetic term ensures the current to be gauge invariant as AŽ™ r . ´ AŽ™ r . q =x Ž™ r .. For the layered superconductors, characterized by the model free energy functional, Eq. Ž1., the current components, parallel Ž5. and perpendicular ŽH. to the superconducting layers are defined by the following expressions: ™ ™ j5 r , j

Ž

¦ ¦

.s

e"

2 m5 j 5

jH Ž™ r, j. s y e y

e" 2 m H aH 2

e2

™

NsŽ2. =™r w j Ž ™ r. y

NsŽ2.

NsŽ2.

1

mH c 2

Ý

m5 c

1 2

Ý

™

;

NsŽ2.A 5 Ž ™ r, j. ,

Ž 11 .

g sin Ž w j Ž ™ r . y w jqg Ž ™ r..

gs"1

;

cos Ž w j Ž ™ r . y w jqg Ž ™ r . . A H Ž™ r, j. .

gs"1

Ž 12 . ™

The bracket ² . . . : in these equations denotes the Gibbs averaging with Hamiltonian, Eq. Ž4., where linear in A corrections are taken into consideration. Such corrections in the Hamiltonian are necessary for the gauge ™ invariance. The diamagnetic contribution to the current in the linear in AŽ™ r, j . approximation can be presented as ™ j 5dia ™ r, j

Ž

dia ™ jH r, j

Ž

. sy . sy

e 2 NsŽ2. mH c

™

A5 ,

e 2 NsŽ2. 1 mH c 2

Ž 13 . r . y w jqg Ž ™ r . .;A H Ž ™ r, j., Ý ¦cos Ž wj Ž ™

gs"1

Ž 14 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

155

™

where terms linear in A in the Hamiltonian are also taken into account. In Eq. Ž14. m H is the transverse component of the effective mass, m H s " 2r8 E H a2H , and a H is the interlayer distance. Taking the Fourier transforms of Eqs. Ž13. and Ž14. with ensuring the gauge invariance, we get:

~°A Ž q . y q

c

™ ™ j5 q

™ ™ 5

™ 5

Ž . s y 4pl2 ¢ 5

c

™

jH Ž q . s y

4pl2H

~°A

¢

™

H

Ž l 5rlH . q H A H¶ 2 2 ™2 ß•, q 5 q Ž l 5 rl H . q H

™ ™ q5 A5 q

Ž q . y qH

2

Ž l 5rl H . q H A H¶ 2 2 ™2 ß•. q 5 q Ž l 5rl H . q H

™ ™ q5 A5 q

Ž 13a.

2

Ž 14a.

Here l 5 and l H are the longitudinal and the transverse components of London penetration depth: 1

l25

s

4p e 2 NsŽ2.

1

l2H

s

c 2 m 5 aH

,

4p e 2 NsŽ2. c 2 m H aH

Ž 15 .

²cos Ž w 0 y w 1 . :s

4p e 2 NsŽ2. c 2 m H aH

eySa Ž1 ,T . .

Ž 16 .

Thus, the phase correlator cos Ž w j Ž ™ r . y w jq1 Ž ™ r . . must be calculated to study the temperature dependence of the penetration depths l 5 and l H . We shall study below this correlator in self-consistent phonon approximation ŽSCPA..

¦

;

4. Self-consistent phonon approximation The system of equations Ž9. becomes linear for small values of w j and it can be diagonalized in q-representation:

™

™ ™ ™ 2 w¨ Ž q H ,q 5 . s yv Ž q H ,q 5 . w Ž q H ,q 5 . .

Ž 17 .

™ . The proper frequency of oscillations v Ž ™ q H ,q 5 in the harmonic approximation is given by ™ v Ž q H ,q 5 . s vV q ,

Ž 18 .

™

where

Vq s ™

dcl2 q

q 52 2 Ž 1 y cos q H qa2H rl25 .

1r2

.

Ž 19 .

Eqs. Ž18. and Ž19. obtained for the collective excitations’ frequency for layered superconductors are low-lying plasmon-mode w34–37x. Following Ref. w35x we include into Eq. Ž19. the small parameter a 2H rl25 < 1, which characterizes the Meissner screening due to in-plane currents. The quantum character of phases w j Ž™ r . should be taken into account to study the quantum™fluctuations for quasi-2d superconductors in the harmonic approximation. In the absence of vector potential, AŽ™ r, j . s 0, expanding of cos Ž w j y w jqg . in Hamiltonian, Eq. Ž4., up to the second order and taking the Fourier transforms of w j and P j as

wj Ž ™ r. s

1

'N

™™

Ý

™

q5 , q H

wq e i q 5 rqi q H j ™

Ž 20 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

156

we get:

½

Hˆ s Ý ™

q

"2

P q Pyq ™

™

q NsŽ2.

K j 52 Ž 1 y cos q H qa2H rl25 .

Ž 1 y cos q H qa2H rl25 . V q2wq wyq ™

4 m5

5

™

.

Ž 21 .

Quantization of phases is performed by expressing w™ q and P ™ q as a linear superposition of the Bose operators of † ™: creation, b™ , and annihilation, b q q

wˆ q s ™

ž

byq q bq†

1r2

A 2 2 4V ™ q Ž 1 y cos q H qa H rl 5 .

V q Ž 1 y cos q H qa2H rl25 .

/

,

Ž 22a.

.

Ž 22b.

1r2

™

ž

byq y bq†

Pq s i ™

A

/

† † ™ and The dimensionless constant A is chosen to eliminate the off-diagonal terms as b™ byq™b™ q byq q in the Hamiltonian:

As

4

(K Ž " rm . 2

NsŽ2.j 52

s 8pa 0

k TcŽ2.

ž / eF

ty1r2 .

Ž 23 .

As a result the Hamiltonian gets the following form: 1 † Hˆ0 s Ý " vV ™ q Ž bq bq q 2 . .

Ž 24 .

™

q

For accurate study of the quantum character of the Bose operators it is necessary to take into account the ™ ™ † commutation relation ™ between b™ q and b™ q in the expansion of cos Ž w j Ž r . y w jqg Ž r . . . Therefore, the last term in Eq. Ž4. written for A s 0 can be expressed in the following form by using the Baker–Hausdorff identity expŽ Hˆ1 q Hˆ2 . s expŽ Hˆ1 .expŽ Hˆ2 .exp y 12 Hˆ1 , Hˆ2 :

½

yNsŽ2.j 52

Ý Hd

2

5

™

™

rE H Ž g . cos Ž w j Ž r . y w jqg Ž r . .

j, g

s yNsŽ2.j 52 Ý

Ž0.

2

H d rE

H

Ž g . eySa

Ž g ,0.

1

j, g

™

q

2

†™ q

Ł e A b Ł eyA ™

) ™ b™ q q

™

q

q

†™

) ™

q Ł eyA q b q Ł e A q b q , ™

™

™

™

q

Ž 25 .

q

where A™ q Ž j, g . s i

ž

1r2

A 4V ™ q Ž 1 y cos

q H qa2H rl25

™ ™

./

e i q H jqi q 5 P r Ž 1 y e i q H g . .

Ž 26 .

We also used here the fact that the commutator 12 w Hˆ1 , Hˆ2 x s SaŽ0. Ž g,0. is constant and is given as SaŽ0. Ž g ,0 . s

1

2

Ý 2

A™ q Ž j, g . s A

™

q

1

1 y cos Ž q H g .

Ý 2V N ™

q

Ž 1 y cos

™

q

q H qa2H rl25

1

.

2

.

Ž 27 .

Applying this procedure we get for the Hamiltonian, Eq. Ž4., in the harmonic approximation the same expression as Eq. Ž24. but with replacing the frequency V ™ q by

Vq s ™

Ž0. dcl2 eySa Ž g ,0. q

q 52 2 Ž 1 y cos q H qa2H rl25 .

1r2

,

Ž 28 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

157

i.e. in the classic Eq. Ž19. for V ™ q the small parameter of theory dcl is renormalized due to the quantum fluctuations as

dcl2 ™ dqu2 Ž 0 . s dcl2 exp  ySaŽ0. Ž g ,0 . 4 . The physical meaning of the additional term of expŽySaŽ0. Ž g,0.. is an average of cos Ž w j Ž ™ r . y w jqg Ž ™ r . . at T s 0 over the ground state, i.e. in the absence of bosons. It can be shown Žsee Appendix A. that in the self-consistent phonon approximation yS aŽ0.Ž g ,0.

e

ˆ

Tr eyH 0 r k T cos Ž w j Ž ™ r . y w jqg Ž ™ r..

½

s lim

yHˆ 0 r k T

Tr  e

T™0

4

5 '¦cos w Ž r . y w

Ž0 .

™

Ž

j

jqg

r . .;0 . Ž™

Ž 29 .

Thus, at T s 0

dqu2 ' d 2 s dcl2 cos Ž w j Ž ™ r . y w jqg Ž ™ r..

¦

Ž0 .

;

0

.

Ž 30 .

We show in the Appendix A that for T / 0 the replacement SŽ g,0. ´ SŽ g,T . should be performed, which is 1 .  wŽ . q ŽT . y 1x4 y1 is equivalent to replacement of the factor 12 in Eq. Ž27. by Ž N™ q q 2 , where N™ q s exp " vrk T V ™ Ž . the Planck distribution function for Bose particles with frequency vV ™ q . Besides, V ™ q T should be defined by the formula of Eq. Ž28. with replacement of SaŽ0. Ž g,0. ™ SaŽ0. Ž g,T .. The phase transition in this model corresponds to ‘melting’ in the w j Ž™ r . ‘phase field’. The critical temperature Tc is that when the transverse rigidity of the system vanishes, i.e. d ™ 0. As it is seen from Eq. Ž30., the character of ODLRO as T ™ Tc is defined by the correlator cos Ž w j Ž ™ r . y w jqg Ž ™ r . . . Although the expression for the correlator of

¦ ; ²cos Ž wj Ž r . y wjqg Ž r . : is given in the Appendix A we can get the expression for ¦cos Ž wj Ž r . y wjqg Ž rX . .; by ™

™

™

™

the same method:

™X

j

™

r X . .;s eyS a Ž g , r yr ;T . , Ž™

™

¦cos Ž w Ž r . y w

jqg

Ž 31 .

where SaŽ0.

™X

™

Ž g ,r y r ;T . s

1 y cos ™ q5 Ž ™ r X y™ r . q qH g

4pa 0 k TcŽ2. 1

t

1r2

Ý N

eF

™

q

V q Ž 1 y cos q H qa2H rl25 . ™

ž

N™ qq

1 2

/

.

Ž 32 .

Ž0. Ž Ž . Ž . The frequency V ™ g,0. ™ q T in the expression of N™ q is defined by Eq. 28 but with replacement of Sa ™ Ž0. Ž ™X Sa g,r y r;T .. To know the character of ODLRO in the quasi-2d superconductor, the correlator SaŽ0. Ž g,r™X y™ r;T . must be calculated. According to the Yang’s criterion w22,23x, the existence of ODLRO in the system at T - Tc is defined by the following behaviour of the correlator, Eq. Ž31., ™

lim X X <™ ryr™ <™` , < jyj <™`

¦cos Ž w Ž r . y w j

j

X

r X . .;s const / 0. Ž™

The value of this correlator does not exceed unity and should smoothly vanish as T ™ Tc . Transforming the sum in Eq. Ž32. into the integration we get: SaŽ0.

™X

2pa 0

™

Ž g ,r y r ;T . s

Ž

k TcŽ2.

1r2 1 y TrTcŽ2.

=

.

eF

1

d qx

1

d qy

p

Hy1 2p Hy1 2p Hyp

1 y cos q x Ž xX y x . q q y Ž yX y y . q q H g

V q Ž T . Ž 1 y cos ™

q H qa2H rl25

.

d qH 2p coth

ž

"v

/

Vq Ž T . . ™

2kT

Ž 33 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

158

For " vr2 k T ; pa 0 - 1 we can expand coth function in Eq. Ž33. and get an asymptotic expression for SaŽ0. Ž g,r™y™ r X ;T .. For < ™ r y™ r X
ž

/

Since the correlator SaŽ0. Ž1,T . does not depend on a in the first approximation, this result can be obtained in the classic approach by using the functional Eq. Ž1. in SCPA. Substituting Eq. Ž34. for the correlator SaŽ0. Ž1,T . into the expression for the renormalized Josephson coupling d 2 s dcl2 expwySaŽ0. Ž1,T .x we get 1y TrT cŽ2.

ds

1

'2p

Ž '2p dcl .

1

1y TrT c)

s

4'2 g J H

ž

'2p

p 3r2k TcŽ2.

1y TrT cŽ2.

/

1yTrT c)

.

Ž 35 .

The new critical temperature Tc) is introduced as 1 1 1 s q . ) Ž2. k Tc pe F k Tc

Ž 36 .

We can now express SaŽ0. Ž1,T . as, SaŽ0. Ž 1,T . s

1 kT

p

1

e F 1 y TrTc)

ln

1

ž / 2pdcl2

Ž 34X .

.

SaŽ0. Ž g,r™y™ r X ;T . can be easily calculated inside one layer, i.e. for j s jX we get:

°1 kT ~

SaŽ0. Ž j y jX s 0, x y xX ;T . s

1

ln Ž2.

< x y xX <

p e F 1 y TrTc 1 kT

¢p

1

e F 1 y TrTcŽ2.

ž

j5 x y xX

j5

j5

;

d

) < x y xX < ) j 5 ,

Ž 37a-b.

2

/

;

X

< xyx <-j 5 .

These expressions make it possible to find the correlator ²cos Ž w j Ž x . y w j Ž xX ..: as:

° ²cos Ž wj Ž x . y wj Ž xX

ž ..: ~

/

X

< xyx <

s

¢exp

½

kT

1

j5

p e F Ž1yTrT cŽ2..

d

kT

1 y

j5

;

p e F Ž 1 y TrTcŽ2. .

ž

x y xX

j5

) < x y xX < ) j 5 ,

Ž 38a-b.

2

/5

;

X

< xyx <-j 5 .

As it is seen from Eq. Ž38a-b., the value of the phase correlator for a single plane decreases as a power law at long distances. Such behaviour of correlator means the ‘quasi long range order’ sets up in a 2d system which takes place in the Kosterlitz–Thouless model w22x. However, the arbitrary non-zero d gives rise to setting a true ODLRO at < x y xX < ) j 5 dy1 . The decrease of the correlator ²cos Ž w j Ž x . y w jX Ž xX ..: 0 with distance is shown in Fig. 1. So, the quantum phase fluctuations are intensified with weakening the interlayer coupling, J H ™ 0, as a result of which ODLRO is destroyed. For small values of coupling, when J H - k TcŽ2., destruction of ODLRO occurs at the new critical temperature Tc) - TcŽ2.. To study the temperature dependencies of the penetration depths l 5 and l H , we introduce the coefficient of anisotropy h 2 by using Eqs. Ž15. and Ž16.:

h2s

l25 l2H

s

m5 mH

Ž0.

eySa

Ž1 ,T .

ž

f 2p

aH a5

2

/

EH E5

Ž0.

eySa

where E 5 s e F and a 5 is the in-plane lattice constant.

Ž1 ,T .

,

Ž 39 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

159

The substitution of Eq. Ž34. into Eq. Ž39. gives

ž

h 2 s 2p

aH a5

2

JH

2

/ž /ž eF

2

4'2 g J H

p 3r2k TcŽ2.

/

k Tr e F

p 1yTrT c)

.

Ž 40 .

So, the anisotropy of the penetration depths rapidly increases when T ™ Tc) . Furthermore, the destruction of the superconducting phase by a magnetic field occurs at different temperatures depending on the orientation of the external magnetic field. For a magnetic field perpendicular to layers, l 5 diverges at T s TcŽ2.. Nevertheless, for a magnetic field parallel to layers the quantum phase fluctuations strongly affect l H and l H ™ ` at T s Tc) - TcŽ2. Žsee Fig. 2.. This conclusion should be more accurately analyzed with taking into consideration the fluctuations as phases, as well as the modulus of the order parameter. Although the simplified study of the problem by Tsuzuki, w38x, displays a leading role of the order parameter phase fluctuations to set up of ODLRO, the problem needs further investigation. It should be noticed that even weak magnetic field directed along the c-axis Ž H 5 c . of a layered superconductor creates 2d pancake vortices, interactions of which, via the Josephson coupling, form vortex lines w1,2,30,39x. Such a vortex line, unlike the Abrikosov vortex, is not a rectilinear object, rather it has a slack

X X Fig. 1. The character of variation of the phase correlator K j j Ž x y x . s ²cosŽ w j Ž x . y w j Ž x ..: in distance. C0 is the saturate value of this X correlator at a large distance, when < x y x < ) j 5 dy1 and C0 ; Ž J H rk TcŽ2. .

2 k Tr e F) p 1y TrT c

.

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

160

Fig. 2. The temperature dependencies of the transverse l H and longitudinal l 5 components of the penetration depth. l 5 and l H diverge at 2

Ž2. y2 Ž2. p the different critical temperatures TcŽ2. and Tc) , correspondingly, as ly2 5 ; Ž 1 y TrTc . and l H ; Ž 1 y TrTc . dcl

k Tr e F 1y TrT c)

.

structure. Therefore, the transverse thermal fluctuations in the vortex position should essentially renormalize l 5 ŽT . giving rise to a non-trivial temperature dependence of l 5 ŽT . w40,41x.

5. Effects of non-linear phase excitations We calculated above the correlators K ji Ž ™™ r ,r X . s cos Ž w j Ž ™ r . y wi Ž ™ rX . .

¦

;

Ž 41 .

in the framework of SCPA Žsee e.g. w6,19,42,43x. which determines the properties of a system under consideration. Eq. Ž16. for l H and Eq. Ž36. for Tc) were also obtained in this approximation. These correlators decrease as a power law with < j y i < ™ 0 and < ™ r y™ r X < ™ ` or < j y i < ™ ` and < ™ r y™ r X < ™ 0 Žsee Eqs. Ž34., Ž37a-b. ™ ™ and Ž38a-b... The correlator ²cosŽ w j Ž r . y w jq1Ž r ..: saturates in the SCPA to the following value: 2kT

d

2 kT

1

pe F 1yTrT cŽ2.

´

JH

ž /

1

p e F 1yTrT c)

k TcŽ2.

and vanishes as T ™ Tc) . It is necessary to investigate the influence of non-linear excitations of phases on the results obtained Žsee w28–33x.. After replacement of the scales of time and length from Ž v .y1 and j 5 to the

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

161

Josephson’s scales Ž vdcl .y1 and j 5rdcl ' lJ , correspondingly, the system of equations Ž9. gets the following form w39x: 2 w¨ j y w¨ jq1 y w¨ jy1 s Dw j y sin Ž w j y w jq1 . y sin Ž w j y w jy1 . .

Ž 42 .

These equations contain non-linear dynamic excitations as well as static topological defects. However, studies even of static topological defects are encountered with many difficulties, so that the system of equations

Dw j s sin Ž w j y w jq1 . q sin Ž w j y w jy1 .

Ž 43 .

cannot be solved exactly. To study the influence of topological defects ŽTD. of phases on the behaviour of the correlators, such as Eq. Ž41., one needs either the exact solutions of Eq. Ž43. or such approximate solutions that contain the topological defect properties. Our approach above really means the existence of strong thermal displacements of centres of mass w jŽ0. and weak harmonic oscillations dw j around them. Therefore, sine terms in Eq. Ž42. can be expanded as sin Ž w j y w jq1 . ™ cos Ž w jŽ0. y w jŽ0. "1 .

yS aŽ0.Ž1,T .

;Ž dw y dw . s e

¦

j

j"1

Ž dwj y dwj " 1 . ,

where the average of cosine terms is calculated in SCPA approximation. The mean field type approximate method can be used to study the non-linear system of Eq. Ž43. Žsee e.g. w44x.. Applying this method, Eq. Ž43. can be reduced in the following form:

Dw j s 2²cos w :sin w j ,

Ž 44 .

where ²cos w : s ²cos w j : s ²cos w j " 1 :. This equation is known as Ž2 q 0. d type sine-Gordon equation. There exist several classes of exact solutions w44–47x for this equation. ŽThe second derivatives with respect to coordinates in Eq. Ž42. can be omitted for dot-like contacts, the cross-section of which is less than l2J . Such a system of dot-like Josephson contacts is solved exactly.. We shall give here the qualitative analysis of behaviour of correlator K j j Ž™ r y™ r X ., following the Doniach approach w44x, to the problem of melting by means of dislocations in the system of weak linked layers Že.g. in smectics.. Returning back to the old scale of length j 5 we introduce the following parameter: 2²cos w :dcl2 s 2 d˜ 2 . If we neglect the interlayer coupling, d˜ ™ 0, Eq. Ž44. is reduced to the form:

ž

E2 Ex2

E2 q

E y2

/

w s 0.

Ž 45 .

This equation contains besides the harmonic oscillations of phases also the Kosterlitz–Thouless’ static topological defects. These defects are called vortex and antivortex with topological charges "1 surrounding of which gives "2p correction to the phase w w24–27x. The solution of Eq. Ž45. corresponding to these topological defects is ™ ™

=w Ž ™ r y™ ri . s "

n = Ž™ r y™ ri . <™ r y™ ri < 2

,

where ™ n is the unit vector perpendicular to the layer, ™ ri are the vortex coordinates. The energy of interaction ™ V Ž ra y™ rb ., of two vortices placed at points ™ ra and ™ rb of a given layer, depends logarithmically on the distance <™ ra y™ rb <: ™

™

ž

V ra y rb ; e F 1 y

ž

/

T TcŽ2.

/

ln

<™ ra y™ rb <

j5

.

Ž 46 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

162

A regular lattice of vortex–antivortex pairs appears at T s 0, i.e. long range order sets up. However, thermal excitations of phases destroy the ODLRO with increasing T and quasi-ODLRO sets up at some temperature. The correlator K Ž™ r y™ r X . begins to decrease as power law up to the temperature T s TKT ŽT KT is the critical temperature of Kosterlitz–Thouless’s topological phase transition.: K Ž™ r y™ r X. ;

j5

ž

h ŽT .

/

. Ž 47 . <™ r y™ r X< The critical index h ŽT . contains both phonon and vortex contributions, i.e. h s hph ŽT . q hvortex ŽT . w26,27x. In the above case discussed by us Žsee Eq. Ž38a-b.. kT hph s g 0 , Ž 48 . e F Ž 1 y TrTcŽ2. . where g 0 is some numerical multiplier. The value of hvortex ŽT . is small at low temperatures w26,27x, however it increases with T and becomes compatible with hph in the vicinity of TKT . Therefore, the behaviour of K Ž™ r y™ r X . will be similar to that ™ ™X y1 obtained above in SCPA Žsee Eq. Ž34.. at < r y r < - j 5 d . Taking into account the transverse couplings between the phases w j and w j " 1 by mean field theory we reduce the system of equations for phases to the sine-Gordon equation Žsee Eq. Ž44.. which changes the character of interaction of vortex and antivortex at large distances Žinside of single plane.. As a result a confinement occurs:

˜ F 1y V ™ ra y™ rb ; de

ž

ž

/

T TcŽ2.

/

<™ ra y™ rb <

j5

,

if

<™ ra y™ rb <

j5

)

'3 d

,

Ž 49 .

i.e. the study of the non-linear effects by the mean field method gives the results which are similar to those obtained by the SCPA method. The correlator K Ž™ r y™ r X . in this case includes the parameter d˜ 2 s dcl2 ²cos w 0 : instead of d 2 s dcl2 ²cos Ž w 0 y w 1 .: in the SCPA method Žsee Eq. Ž30... The value of ²cos w 0 : should be calculated here by using the sine-Gordon equation, whereas ²cos Ž w 0 y w 1 .: was calculated in the SCPA approximation. Notice that the saturation value of the correlator K i j Ž™ r y™ r X . for a quasi-one dimensional superconductor has been calculated by two methods, namely by SCPA w6,19x and by using sine-Gordon type self-consistent equation w21x. Both methods give the same dependence on d but with different constant multipliers. Nevertheless, the collective excitations studied by these two methods differ strongly. The system of non-linear equations Ž42. has a solution which corresponds to another type of topological defect w1,2,28–35x. Here we shall particularly discuss the parallel interplanar vortex loops, proposed by Friedel in w28,7x. Following the dislocation theory the energy of such loop with diameter R is estimated as: UŽ R. ; p R

f 02 2

16p l 5 l H

ln

R

j5

,

Ž 50 .

where f 0 s 2p "r2 ec is the magnetic flux, l 5 and l H are the London penetration depths. In our case l 5 and l H are given by Eqs. Ž15. and Ž16. where the transverse component of the effective mass m H should be replaced by 2 E H a 2H

UŽ R. ; p R

²cos w : f 2

2 a 2H J H

²cos w : . "2 "2 e F Substituting Eqs. Ž15. and Ž16. into Eq. Ž50. with taking into consideration Eq. Ž51., we get m )y1 s H

" 2 NsŽ2. Ž T . aH

ž

1 m 5 m )H

1r2

/

ln

R

j5

;p

R aH

J H ²cos w :1r2 1 y

ž

T TcŽ2.

/

ln

R

j5

.

Ž 51 .

Ž 52 .

Here we take into account that NsŽ2. ŽT . s N Ž2. Ž0.t ; t s 1 y TrTcŽ2. ; N Ž2. Ž0. f ay2 and e F s p Ž " 2 N Ž2. Ž0.rm 5 .. 5

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

163

Notice that the terms ²cos w :1r2 and t , existing in Eq. Ž52., reflect the effects of phase fluctuations. For J H ²cos w :1r2 < e F the energy of the vortex loop is seen to be much less than the vortex–antivortex interaction energy given by Eq. Ž46. w28x. According to Friedel w28x, the vortex loops are created rather easily than the vortex–antivortex pairs which give rise to the following dependence of Tc on J H : Tc) ;

TcŽ2. 1 q ln Ž e FrJ H .

.

Ž 53 .

Let the defect have a pancake form, i.e. it describes the 2p phase variation around the circle with diameter R. The energy density d erd R along a circle at arbitrary cross-section of defect can be roughly estimated as energy of a one-dimensional kink. Choosing the cross-section along the x direction we get de dR

"2

`

; NsŽ2.

Hy`

dx

2

Ew

8 m5

ž / Ex

q E H ²cos w : Ž 1 y cos w . .

Ž 54 .

Crossing from x to a dimensionless variable xrlJ , where

ly1 J s

ž

1r2

1 "

8 m 5 E H ²cos w : 2

/

,

we obtain de dR

"2 ; 8 m5

NsŽ2. Ž T .

1

lJ

Zs

1 a5

J H ²cos w :1r2 1 y

ž

T TcŽ2.

/

Z.

Ž 55 .

The dimensionless integral Z in Eq. Ž55. is a finite constant. The same result can be obtained by integrating the Leibbrandt’s solution for infinite ‘fold’ Žsee fig. 7 in w45x. in the perpendicular to ‘fold’ direction. Closing this fold into circle with diameter of R and integrating along a circle we could get the expression as Eq. Ž50. except the logarithmic term in it. To restore the logarithmic term the dependence of the phase w on y in Eq. Ž54. should be taken into consideration. Korshunov w29x has shown, by using a more rigorous approach and by taking into account the interaction of different vortex rings, that the superconducting phase transition for J H / 0 occurs at the critical temperature, Tc , ln Ž Tc y TKT . ;

TKT JH

.

According to this expression Tc takes the finite limit for J H ™ 0, in contrast to the Friedel’s result Žsee Eq. Ž53.., where Tc) decreases with J H . The conclusion that Tc should be varied in the interval of TKT - Tc - TcŽ2. has been also confirmed by Horovitz w31,32x. Another dependence of Tc on J H has been given by Hikami and Tsuneto w48x, for a quasi-2d planar magnetic: Tc s TKT 1 q

ž

p ln Ž e FrJ H .

2

/

.

The origin of this dependence is an increase in energy with J H of the interplanar parts of vortex loops that bind together the Kosterlitz–Thouless vortices on different planes. This result is relevant, as it is shown in w44x, for a small value of J H until the creation of vortex rings lying between the neighbouring layers is less favourable. The topological defects, which are created due to "2p variations in the relative phase of neighbouring layers, can be generally taken out of the system of equations Ž9.. For the rather large number of layers Ž N 4 1. to solve this system of equations is very difficult. For N s 3 with cyclic boundary condition as w 3 s w 1 the

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

164

problem is reduced to the problem for SQUID. The sine-Gordon equation is obtained for the phase differences w 1 y w 2 . For N s 4 with cyclic boundary condition w4 s w 1 , a system of three equations for w 1 , w 2 and w 3 can be reduced into two equations by introducing the new parameters u s w 1 y w 2 and Õ s w 2 y w 3 :

D u y 2 d 2 sin u y d 2 sin Ž u q Õ . y sin Õ s 0, D Õ y 2 d 2 sin Õ y d 2 sin Ž u q Õ . y sin u s 0. The solution of these equations permits us to study the possibility of formation of ‘dislocation loop’ type defects between two neighbouring layers.

6. Conclusions for experimentalists The results obtained in this paper can be used to study the layered superconductors with open Fermi surfaces as a corrugated cylinder. The degree of corrugation Žor anisotropy. is defined by the small parameter J H re F ; J H rJ 5 < 1. Under this condition we can use the free energy functional, Eq. Ž1. or the Hamiltonian, Eq. Ž4., which describes a tunnelling of Cooper pairs between the layers. The case arising under the condition k Tc - J H < e F corresponds to a usual strong anisotropic layered superconductor. We studied here another case, J H - k Tc < e F . The small parameter is dcl s J H rk Tc - 1 and the coherence length in the direction perpendicular to superconductor layer j H F a H . The fluctuations of the order parameter phases performed in the frame of the Ginzburg–Landau formalism are shown to have a considerable influence on the physical properties of layered superconductor. To study the phase’s fluctuations the system of sine-Gordon type coupled equations Žsee Eq. Ž9.. should be considered. This system of equations describes the infinite set of Josephson contacts the area of which is much larger compared to l2J with lJ being the Josephson length

lJ s

ž

"2 8 EH m 5

1r2

/

, a5

eF JH

.

This system of non-linear equations also describes the different static non-homogeneouties such as topological defects in the system of phases w j Ž™ r . and different type of collective excitations, also particle like dynamic excitations such as coupled solutions. We studied the system of equations Ž9. for some simple approximations. According to the results obtained, there exits ODLRO, i.e. ²e iŽ w jy w j " 1 . : s ey

1 2

²Ž w jy w j " 1 . 2 :

/0

for rather low temperatures. However, mean square displacement ² Ž wj y wj " 1 . 2 : s

4 kT

p

1

e F 1 y TrTc)

ln

k Tc JH

increases with temperature and it diverges as T ™ Tc) . Thus, there exists a phase transition of quasi-2 d ™ 2 d. A ‘melting’ in the system of phases w j Ž™ r . is accompanied with vanishing of the transverse component of collective excitation velocities. In the frame of our approximation, Tc) is defined as Tc) s TcŽ2. Ž1 q k TcŽ2.rpe F .y1 , where TcŽ2. is the critical temperature of 2 d superconductors estimated by means of BCS theory. Interprating of Tc) as a crossover transition temperature from quasi-2d to 2d, we can estimate it by equating the correlation length perpendicular to layer, j H ŽT . s j H Ž0. Ž1 y TrTcŽ2. .y1 r2 , to an appropriate spacing a H , i.e. j H ŽTc) . , a H . Using the data for YBa 2 Cu 3 O 7y d cuprates w49x, such as j H Ž0. , 3 A0 and a H , 12 A0 , we find that Tc) rTcŽ2. ; 1

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

165

2

y Ž j H Ž 0 . ra H . , 0.93. This estimation is consistent with estimation carried out according to the expression for Tc) . We get from Eq. Ž36.: Tc) rTcŽ2. s Ž1 q k TcŽ2.rpe F .y1 , 1 y k TcŽ2.rpe F . Using the experimental data for TcŽ2. ; 1008 K and e F ; 15008 K w49x, we get Tc) rTcŽ2. ; 0.96. It should be noticed that the considered approximation is insufficient to find the dependence of Tc on J H Žsee Section 5.. However, our conclusion about the existence of two critical temperatures Tc1 and Tc2 Žwhich correspond to Tc) and TcŽ2., correspondingly. in quasi-2d superconductors seems to be reasonable and is confirmed by experimental measurements. At T s Tc1 s Tc) ODLRO vanishes and parallel to layers magnetic field penetrates into the superconductor, i.e. the London penetration length l H diverges with T ™ Tc1 as ln l H ;

kT

1

ln

e F 1 y TrTc1

k Tc 2

1 y

JH

2

ž

ln 1 y

T Tc 2

/

.

While the parallel component of penetration depth l 5 Žfor perpendicular to layers magnetic fields. diverges with T ™ Tc2 as l 5 ; Ž1 y TrTc2 .y1 r2 . The divergence of l H and l 5 at different critical temperatures was observed by the measurements on CdBa 2 Cu 3 O 7yy monocrystal w50x. According to Ref. w50x parallel and perpendicular magnetic fields penetrate into the superconductor at temperatures of 76 K and 79 K which correspond to Tc1 s Tc) and Tc2 s TcŽ2., correspondingly, so Tc)rTcŽ2. , 0.96. Let the logarithm of the penetration depths ratio, l H rl 5 , be written as ln

lH l5

s const q g 1

kT

1

e F 1 y TrTc1

, T - Tc1 .

Ž 56 .

In our approximation g 1 , lnŽ k Tc 2rJ H . Žnotice that the criterion of correctness of the theory is k Tc2 ) J H .. According to the Ginzburg–Landau theory the logarithm of l H rl 5 should be constant. Thereby the origin of the second term in Eq. Ž56. is fluctuations in the system of phases w j Ž™ r .. Thus the strong dependence Žnear Tc1 . of the penetration depth’s anisotropy on the temperature is predicted in the theory presented above. In our approximation Tc2 y Tc1 s TcŽ2. y Tc) s ŽTcŽ2. . 2re F which is about a few degrees centigrade for most of the high-Tc superconductors. In the interval of temperature as Tc1 - T - Tc2 the phase coherence on the nearest neighbouring layers is destroyed, i.e. ²e iŽ w jyw j " 1 . : s 0 and the specific topological defects, such as dislocation loops etc. defined by the system of non-linear equations of Eq. Ž9., disappear. However, the Kosterlitz–Thouless type defects of vortices and antivortices still exist, since for the following equation takes place for each single plane:

e F Ž 1 y TrTc 2 . Dw j s 0.

Ž 57 .

The prefactor under the Laplacian in Eq. Ž57. defines the Kosterlitz–Thouless critical temperature, T KT , for the topological phase transition. For Tc1 - T - Tc2 the critical temperature T KT varies from Tc1 to zero, i.e. in this interval of temperature the vortex–antivortex pairs are unbounded, so the quasi-ODLRO is destroyed due to the thermal excitations and possible motion of vortices. The existence of similar temperature dependence for the upper critical magnetic field anisotropy, Hc2HrH c25 , can be shown on qualitative reasoning. Really, according to Eq. Ž4. the following expressions take place for the magnetic field H parallel to the x-axis: EH

2e s

Ey

"c

EH

Ý gs"1

1 sy

Ez

NsŽ2. Ž T . E H

l25

¦ž

f 0 Ew j 2p E y

¦ž ;/

yAy

sin w jqg y w j y

.

2e "c

Hja jqg a Ž

.

H

H

Azd z

;/

,

Ž 58 .

Ž 59 .

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

166

™

Let us solve in common these equations and the system of equations Ž9. with including the vector potential A under the sine functions in Eq. Ž9.. Representing the sine term in Eq. Ž58. as

ž

sin w jy1 y w j y

2 ea H c"

/

A j ²cos Ž w j y w jy1 . :

where w is a smooth function and w j corresponds to the collective excitations, it can be shown that the renormalization of the Josephson coupling constant E H occurs as E H ™ E H ²cos Ž w j y w jy1 .:. This is equivalent to the renormalization of the transverse effective mass m H : my1 H ™

E H a2H "2

eySa Ž1 ,T , H . .

Ž 60 .

The similar renormalization with correlator ²cos Ž w j y w jy1 .: s expwySa Ž1,T .x has been encountered in Eq. Ž16. for l H . Here however, this correlator should be calculated in the presence of an external magnetic field Žsee e.g. w51–54x.. In this case an external magnetic field should affect the structure of the topological defects and the ‘phason’ velocities in the parallel and perpendicular directions. Following Ref. w17x

Ž Hc52 .

2

;

m5 mH

and Hc25 acquires an additional multiplier expwySa Ž1,T, H .x which depends on temperature. Therefore, the temperature dependencies of Hc25 and Hc2H should be different, as it takes place for l 5 and l H at H - Hc1. Hence the logarithm of the ratio Hc25 rHc2H depends on the temperature that is absent in the Ginzburg–Landau theory. The calculation of the correlator ²cos Ž w j y w jy1 .: s expwySa Ž1,T, H .x will be given in the next paper.

Appendix A Following Eq. Ž25. which corresponds to the tunnelling term in the Hamiltonian, Eq. Ž4., the factor expwySaŽ0. Ž g,0.x in Eq. Ž25. physically means an average value of cos Ž w j y w jqg . over the ground state T s 0. There are no phonons in this case, hence the contributions of the terms in the bracket in Eq. Ž25. are neglected. Let us study the phonon contribution to the correlator ²cos Ž w j y w jy1 .:. For this purpose we pick out the exact diagonal part of the expression 1 2

½Ł ™

q

†™

) ™

†™

) ™

e A q b q eyA q b q q Ł eyA q b q e A q b q ™

™

™

™

™

q

5

Ž A1.

as

Ł ™

q

½

< 2 q† b™ 1 y < A™ q b™ qq

< A™ <4 q

Ž 2! .

2

† q

2

žb / Žb . ™

™

q

2

y

< A™ <6 q

Ž 3! .

2

† q

3

žb / Žb . ™

™

q

3

5

q PPP .

Ž A2.

x To find the average value of this part we must use the Gibbs average. Denoting expwy" v ™ q rk T s a ™ q we get ey" v q r2 k T ™

† ™ 1. yŽ " v ™ q r k T .Ž b ™ q b qq 2

Tr  e

y" v ™ q r2 k T

4 se

ž

1 q a q q a q2 q PPP ™

™

/s

1 y a™ q

' Z0 .

Ž A3.

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

167

Then the trace of the arbitrary term of Eq. ŽA2. is equal to

½

Tr e

` y" v ™ q r2 k T

se

Ý Ns0

n

† q

žb / Žb .

† ™ 1. yŽ " v ™ q r k T .Ž b ™ q b qq 2

™

™

n!

n! n

† q

¦

™

N

5

;

q

n!

n

n

™

N N a™ q Ž y1 .

n!

n Ý Ž y1. a qN

™

Ž y1.

žb / Žb .

`

s ey" v q r2 k T

n

q

n

(N PPP Ž N y n q 2. Ž N y n q 1. (N Ž N y 1. PPP Ž N y n q 1.

™

n!

Nsn

a qN

`

™

n Ý Ž y1.

s ey" v q r2 k T ™

n!

Ž n! .

Nsn

2

N Ž N y 1 . PPP Ž N y n q 1 . .

Ž A4.

So, we can calculate the trace of whole Eq. ŽA2. as: ey" v q r2 k T

`

Z0

ns0

™

`

a qn < A q < 2 ™

n

Ý Ž y1. Ý

Nyn a™ q

ey" v q r2 k T

`

™

n Ý Ž y1.

Z0

< A™ <2n q

Ž n! .

Ns0

™

Ž n! .

Nsn

s

ž

2

n

/

2

N Ž N y 1 . PPP Ž N y n q 1 .

`

a qn

Ý

™

a qNyn N Ž N y 1 . PPP Ž N y n q 1 . .

Ž A5.

™

Nsn

The last sum in Eq. ŽA5. obviously is dn

`

Ý

d a qn Ns0

n!

a qN s

Ž1 y a q .

nq1

.

Ž A6.

Substituting Eq. ŽA6. in Eq. ŽA5. we get ey" v q r2 k T

`

™

Ý

Z0

Ž y1.

n

Ns0

< A™ <2n q

ey" v q r2 k T

a qn

™

™

n!

Ž1 y a q . ™

nq 1

s Z0 Ž 1 y a ™ q.

`

Ý Ns0

Ž y1.

n

< A™ <2n q n!

aq

n

™

ž

1 y a™ q

/

.

Ž A7 .

It is easy to see that in Eq. ŽA7. the last term can be written as n

aq

1

™

ž

1 y aq

™

/

s e

" v™ q rk T

y1

s N™ q

and ey" v q r2 k T Z0 Ž 1 y a ™ q.

s 1.

Therefore, the Gibbs average of the first term in the sum of Eq. ŽA5. is equal to `

Ý Ž y1. Ns0

n

< A™ <2n q n!

y< A q < N™ qse ™

2

N™ q

.

Ž A8.

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169

168

Hence, Tr

½Ł ™

q

†™

) †™ ™

5

2

e A q b q eyA q b q s Ł ey< A q < ™

™

N™ q

s eyÝ q < A q < ™

™

2

N™ q

.

Ž A9.

™

q

† Ž . Let us consider now the contribution from the linear and quadratic in b™ q and b™ q terms of Eq. A1 to the ™ Ž . correlator. We single out one arbitrary mode e.g. q1 from Eq. A1 : † q b™ q

ŁeA

) ™

eyA q b q ´

™

™

™

Ł eA

† q b™ q

™

™

™ q/q 1

q

† q b™ q

Ł eyA

) ™

e A q bq ´ y

™

™

™

) †™ ™

† yA q b q A™ , q 1 b™ q 1e

Ł eyA

† q b™ q

) †™ ™

† A q bq A™ . q 1 b™ q 1e

™

™

™ q/q 1

q

It is clear that the trace over all ™ q /™ q1 gives eyÝ q / q 1 < A q < ™

™

™

2

N™ q

žA

† † q 1 b™ q1 y A ™ q 1 b™ q1

™

/ s 0.

Since each term of the sum in the exponent of this expression is as small as 1rN, where N is the number of elementary cells per unit volume, we can write approximately eyÝ q / q 1 < A q < ™

™

™

2

N™ q

s eyÝ q < A q < ™

™

2

N™ q

.

Therefore, the contribution of the quadratic term is

Ž Aq .

2

™

† q1

žb /

2

™

1

Ž 2! .

Łe

2

™

† ) ™ A™ q b™ q yA ™ q bq

e

Ž Aq . ™

q

1

2

Ž 2! .

™

q/q 1

† q1

žb /

2

™

2

Ł eyA

™

† q b™ q

™

™

q/q 1

) ™

2

† yÝ q / q 1 < A q < e A q b q ´ 12 A2™ q 1 b™ q1 e ™

ž /

™

™

™

2

N™ q

,

Ž A11.

™ Ž . where N™ q is the Planck distribution function. So far, as q1 is an arbitrarily chosen mode, Eq. A11 should be ™ summed over all q1. As a result, taking into account the factor expwySa Ž g,0.x arising due to zero quantum oscillation, we get the factor

eyÝ q < A q < ™

™

2

1 Ž N™ qq 2 .

.

This term is caused by the thermal oscillations. The remaining non-diagonal square Žquadratic. terms as † † yS a Ž g ,T . A™ q1 A ™ q 2 b™ q 1 b™ q2e

Ý ™

™

™

qsq 1/q 2

are taken as perturbative terms. Thus the above obtained factor explains the origin of the appearance of the term dcl2 expwySa Ž g,T .x instead of dcl2 is the tunnelling term. Also Sa Ž g,T . is defined as Sa Ž g ,T . s

4pa 0 kTcŽ2. 1

t

1r2

eF

1 y cos Ž q H g .

ÝV N ™

q

Ž 1 y cos q H qa2H rl25 .

™

q

ž

N™ qq

1 2

/

Ž A12.

Ž . Ž . Ž . Ž . where, the frequency V ™ q is defined by Eq. 28 with replacing Sa g,0 ™ Sa g,T . Therefore, Eq. A12 is the Ž . self-consistent equation for Sa g,T .

References w1x w2x w3x w4x

G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 Ž1994. 1125. K.H. Fischer, Supercond. Rev. 1 Ž1995. 153. L.N. Bulaevskii, Adv. Phys. 37 Ž1988. 443. A.I. Socolov, Physica C 174 Ž1991. 208.

E.P. NakhmedoÕ, Yu.A. FirsoÕr Physica C 295 (1998) 150–169 w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x w37x w38x w39x w40x w41x w42x w43x w44x w45x w46x w47x w48x w49x w50x w51x w52x w53x w54x

169

D. Jerome, H.I. Schulz, Adv. Phys. 31 Ž1982. 299. ´ Yu.A. Firsov, Proceedings VII Winter School of A.F. Ioffe Physical-Technical Institute, USSR Academy of Sciences, 1975, p. 286. J. Friedel, J. Phys.-Condens. Matter 1 Ž1989. 7757. Yu.A. Firsov, V.N. Prigodin, Chr. Seidel, Phys. Rep. 126 Ž1985. 245. M. Weger, I.B. Goldberg, Solid State Phys. 28 Ž1973. 2. V.N. Bogomolov, Usp. Fiz. Nauk 124 Ž1978. 171. B.Y. Jin, J.B. Ketterson, Adv. Phys. 38 Ž1989. 189. S.S. Parkin, E.M. Engler, R.P. Schumaker, Phys. Rev. Lett. 50 Ž1983. 270. J.G. Berdnotz, K.A. Muller, Z. Phys. 64 Ž1986. 189. ¨ O. Laborde, P. Monceau, M. Potel, J. Padiou, P. Gougeon, J.C. Levet, H. Noel, Physica C 162–164 Ž1989. 1619. T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer, R.B. van Dover, J.V. Waszczak, Phys. Rev. B 38 Ž1988. 5102. E.W. Lawrence, S. Doniach, in: E. Kanda ŽEd.., Proceedings of the Twelfth International Conference on Low Temperature Physics, Kyoto, 1970, Keigaku, Tokyo, 1971, p. 361. R.A. Klemm, A. Luther, M.R. Beasley, Phys. Rev. B 12 Ž1975. 877. L.N. Bulaevskii, V.L. Ginzburg, A.A. Sobyanin, Physica C 152 Ž1988. 378. Yu.A. Firsov, G.Yu. Yashin, Zh. Eksp. Teor. Fiz. 72 Ž1977. 1465. K.B. Efetov, A.I. Larkin, Zh. Eksp. Teor. Fiz. 66 Ž1974. 2290. J.H. Schulz, C. Barbonnais, Phys. Rev. B 27 Ž1983. 5856. C.N. Yang, Rev. Mod. Phys. 34 Ž1962. 694. T.M. Rice, Phys. Rev. A 140 Ž1965. 1889. J.M. Kosterlitz, D.J. Thouless, in: D.F. Brewer ŽEd.., Prog. Low Temp. Phys. B7, North-Holland, Amsterdam, 1978, p. 373. P. Minnhagen, Rev. Mod. Phys. 59 Ž1987. 1001. J. Villain, J. Phys. 35 Ž1974. 27. J. Villain, J. Phys. 36 Ž1975. 580. J. Friedel, J. Phys. 49 Ž1988. 1561. S.E. Korshunov, Europhys. Lett. 11 Ž1990. 757. J.R. Clem, Phys. Rev. B 43 Ž1991. 7837. B. Horovitz, Phys. Rev. Lett. 67 Ž1991. 378. B. Horovitz, Phys. Rev. B 47 Ž1993. 5947. Yu.M. Ivanchenko, Phys. Lett. A 183 Ž1993. 102. S.N. Artemenko, A.N. Kruglov, Physica C 173 Ž1991. 125. L.N. Bulaevskii, M. Zamora, D. Baeriswyl, H. Beck, J.R. Clem, Phys. Rev. B 50 Ž1994. 12831. L.N. Bulaevskii, D. Dominguez, M.P. Maley, A.R. Bishop, B.I. Ivlev, Phys. Rev. B 53 Ž1996. 14601. H.A. Ferting, S. Das Sarma, Phys. Rev. B 44 Ž1991. 4480. T. Tsuzuki, J. Low Temp. Phys. 9 Ž1972. 525. L.N. Bulaevskii, M. Ledvij, V.G. Kogan, Phys. Rev. Lett. 68 Ž1992. 3773. V.G. Kogan, M. Ledvij, A.Yu. Simonov, J.H. Cho, D.C. Johnston, Phys. Rev. Lett. 70 Ž1993. 1870. Y.Q. Song, W.P. Halperin, L. Tonge, T.J. Marks, M. Ledvij, V.G. Kogan, L.N. Bulaevskii, Phys. Rev. Lett. 70 Ž1993. 3127. Y. Suzumura, Prog. Theor. Phys. 62 Ž1972. 342. S. Nakajima, Y. Okabe, J. Phys. Soc. Jpn. 42 Ž1977. 1115. S. Doniach, in: Ordering in Two Dimensions, 67 Ž1980.. C. Leibbrandt, Phys. Rev. B 15 Ž1977. 3353. A.B. Borisov, V.V. Kiseliev, Physica D 31 Ž1988. 49. A.B. Borisov, A.P. Tankeev, A.G. Shaganov, Z. Fiz. Tverd. Tela 31 Ž1989. 140. S. Hikami, T. Tsuneto, Prog. Theor. Phys. 63 Ž1980. 387. D. Wu, S. Sridhar, Phys. Rev. Lett. 65 Ž1990. 2074. F. Minghu, X. Jiansheng, X. Zhuan, Z. Zhanchun, S. Dunming, C. Zuyao, Q. Yitai, Z. Qirui, Solid State Commun. 68 Ž1988. 827. M.V. Feigelman, V.B. Geshkenbein, A.I. Larkin, Physica C 167 Ž1990. 177. K.B. Efetov, Zh. Eksp. Teor. Fiz. 76 Ž1979. 1781. D. Feinberg, C. Villard, Phys. Rev. Lett. 65 Ž1990. 919. A.D. Yoffe, in: Festkorper Problem XIII, Adv. Solid State Phys. 1 Ž1973.. ¨