Layered high-Tc superconductors in an AC field

PhysicaC 211 (1993) 455-466 North-Holland

Layered high-T, superconductors in an AC field A.Yu. Simonov’ Ames Laboratory - DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA

Received13April1993

Ananalysisof electrodynamicproperties of high-temperature superconductors with the AC magnetic field parallel to the layers is presented. The system is modelled as a superlattice of normal (N) and superconducting (S) layers, and the bulk superconductivity is maintained by the proximity effect. The expression for the system conductivity is evaluated. The AC losses in the region of frequencies s1 smaller than 24 (d is the order parameter value at zero temperature) are non-zero even at T=O; several peaks on the frequency dependence of the conductivity are predicted. Comparison of this result with experimental data is discussed.

1. Introduction

Measurements of the electrodynamic response in a finite-frequency field can give important information on the superconducting properties of materials and may play a certain role in establishing the validity of any theoretical model. These are also important for various applications. The electrodynamics of conventional, low temperature isotropic superconductors was successfully described by Mattis and Bardeen [ 11. The generalization of this approach for the superconductor with anisotropic energy gap was obtained by Clem [2]. High-temperature superconductors have extremely unusual properties, such as a very short coherence length <, a long mean-free path 13 c, and a low carrier concentration (and, consequently, a large penetration depth 2). In addition, these materials have a highly anisotropic layered structure. The ratio t/EF, where t is the hopping integral between layers and EF is the Fermi energy inside the layers is a good characteristic for the degree of anisotropy. In high-T, superconductors this ratio ranges from 10W2in La- and Y-systems down to 10d4 in Bi- and Tl-compounds. This crucial factor should be taken into account in any discussion of properties of superconducting cuprates. The applicability of the classical Mattis-Bardeen treatment is limited to a small range of temperatures near T,, where the layered structure becomes irrelevant. The electrodynamic properties have been investigated theoretically since the discovery of HTSC [3-l 21. Still, the anisotropy effects are considered only in a few papers [ 5,7,8,12]. The two-dimensional superconductor with an anisotropic energy gap was considered in refs. [ 5 ] and [ 12 1, but the effects of coupling between layers were neglected. The layered structure effect was considered in ref. [ 71 for the field perpendicular to the conducting planes. The optical conductivity for the model with five different layers in a unit cell was calculated in ref. [ 8 1; unfortunately, the model has too many parameters, and the conductivity evaluation is possible only numerically. Also, as is shown below, one of the terms in the current perpendicular to conducting planes is omitted in ref. [ 8 1. In this paper, the microscopic theory is used to evaluate the response of the layered superconductor to the AC magnetic field parallel to the layers. This is an interesting case, because the nature of the weak interlayer interaction is pronounced. There is a lot of experimental evidence which indicates that the Cu02 layers are responsible for the super’ New address: bit. Decisions and Systems, Inc., 8500 Normandale Lake Blvd, Suite 1840, Bloomington, MN 55437, USA. 0921-4534/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

456

A. Yu. Simonov /Layered HTSC in an ACfield

conductivity in cuprates, while both the chains CuO in YBazCu30, and BiO or TlO layers in Bi- and Tl-compounds respectively, behave like a normal metal (see refs. [ 13- 161). For that reason, we study a layered array of two alternating two-dimensional layers: a superconducting one with BCS pairing inside the layer and a normal metal layer with no BCS interaction. The carrier motion is described by the in-plane momentum. Only the case of identical two-dimensional energy bands c(p) = ~+(p-p~) in both layers is considered. The tight-binding approximation is used for the description of electron motion between the layers. The coupling between the layers is described by the hopping integral t e I&. Some aspects of this model as well as those of similar ones have been discussed in refs. [ 17-23 1. In particular, in refs. [ 17-201 the density of electronic states of such a system has been considered. The superconductivity turns out to be gapless [ 17-201 and the energy corresponding to the peak in the density of states can be much higher than the BCS value, 1.76T, [ 18,191. In the framework of the S/N model, the Raman spectra were studied in ref. [23]. The structure of the paper is the following. In section 2 we introduce the model and evaluate the energy spectra in the superconducting state. The expression for the electromagnetic kernel for the current flowing along the z-direction is obtained in section 3. The electromagnetic properties in the microwave region are discussed in section 4. The behavior of conductivity in the optical region is considered in section 5. Comparison of our results with experimental data is given in section 6.

2. Hamiltonian and the energy spectrum of an S/N superlattice We consider the system of infinite layers parallel to the (x, y) plane with spacing d and mark the two different types of layers in the unit cell with i= 1,2 and assume the presence of identical two-dimensional energy bands c(p) = ~&p-p,) in S- and N-layers. In the mean-field approximation the hamiltonian of the system under consideration is (fi=k,= 1)

where u?,,,(p) is the creation operator of an electron with momentump and spin cr in the ith layer of the nth elementary cell, A is the BCS coupling constant on the S-layer, N( 0) is the density of states at the Fermi surface inside the layer, N( 0) = (m,,/2rr), m,, is the effective electron mass insider the layer, and i labels the layer inside the unit cell (i= 1 for an S-layer and i= 2 for an N-layer). Note that the hamiltonian is the generalization to be used in the case of multilayers of the tunneling hamiltonian which was introduced by Cohen et al. (see ref. 1241). The normal G,( r, p, IV) and anomalous Flj( r, p, N) Green’s functions can be written as (see ref. [ 251) Gij(r,P,N--Ni)s

-i(T~u~,,,(p)I,U,,,j,,(p)lo)

F$(~,P,


N-N,)=

IrUL,,j,-o(-P)

3 IO> .

(2)

Making a Fourier transformation from the discrete variables N to the quasi-momentum q, 0 I q %2x and form 7 to the Matsubara frequencies w=nT( 2n+ 1 ), n=O, + 1, -+_ 2, .... we obtain the Gor’kov equations in matrix form:

A. Yu. Simonov / L.ayered

o-T*(q)

-T(q) w-

A

0

Gj(o,

P, q)

F*j(W

0

0

G,(W P. q) Ffj(o,P, q)

fij(a, -Gjl(-W,

0, T*(q)

0

A 0

0

T(q) x m+ ii

457

HTSC in an ACjieid

P,q)

61j

P, q) --PI -q)

s2j

=

f'Ij(~,P,q) -Gj2(-0, -P, -4) I(

0 0

0

6, '

0

62j

(3)

I

wherej= 1,2 marks the layers in the elementary cell, o+ = io ? t(p) and T(q) = t( 1 + exp (iq) ). It is convenient to go from momentump to the band energy Qp) =v&-p,). The order parameter A is given by the usual selfconsistency equation A=ATN(O)

; j TFT,(w,

6

q) .

(4)

Hereafter we will not indicate the dependence of A on the temperature T. Using eq. (3) one can determine the quasi-momentum spectrum in the superconducting state. It has two branches and

D2(<, q, A)=4c2F2+

$

+F2A2,

(5)

where F= ( T(q) I=2t cos(q/2). As for r=O and q=O the value of E2 is zero, superconductivity in S/N superlattice is gapless. The absence of the gap in the quasi-particles spectra leads to the non-zero density of states inside the “gap” (see refs. [ 17- 19 ] ). As we know, in classical superconductors the absence of a gap in the spectrum also causes losses in a weak AC field for any, even small frequency B +KT, even at zero temperature [ 261. For the case of weak (Josephson) interlayer coupling (t << T,) if the temperature is not very close to T,(A- T,), eq. (5) can be written as E, =Jm, E2 =,/m-.

(6)

In the absence of a tunneling hamiltonian, we have the usual spectra on superconducting layers E, = (t’-iAZ) l/2 and on the normal layers E,=<. We also mention that in the presence of the tunneling term, the association of the branches of the spectrum with the layer is no longer correct. Equation (3 ) is easy to solve. Hereafter we will need only the expressions for the diagonal Green’s function:

Gll(wpr,q)=

w_ (0:

- T’2)

(W2+Ef)(W2+E;)

’

w_(w: G22(07 5v q)=

--2)-o+A2 (&+Ef)(&+E;)

y

Ft,(wCq)=

-Ao_o+ (02+E:)(w2+E;)

’

Ft,(w,t,c7)=

AT2 (02+E:)(w2+E2’)

’

(7)

and Fia = F,. It is obvious that the value J2(q) = T2/A plays the role of the superconducting order parameter for the N-layer which is induced through the proximity effect from neighboring S-layers. The case of the superconductor-normal metal sandwich was discussed by McMillan [ 271. For the S/N and S/S’ multilayers,

A. Yu. Simonov /Layered HTSC in an AC field

458

the thermodynamic properties of the S/N superlattice are considered in refs. [ 18) and [ 191. If the temperature approaches T,, the coherence length in the direction perpendicular to the layers exceeds the interlayer distance. For such temperatures, the system loses its 2D character, and the approach employed in this paper is no longer valid. For simplicity we limited our consideration to the temperature range in which the temperature dependence of the order parameter can be neglected.

3. Response kernel for the current flowing along the z-axis To obtain the electrodynamic properties of a system under consideration we investigate the so-called response kernel Q( 9 k) which describes the current density induced by the external electromagnetic field A (r, t). In a Lorentz gauge, divA =O, and the relation between the vector-potential and current reads i(S, k) = -QW,

kM(Q,

k) .

(8)

We are interested in the frequency dependence of the kernel. In order to study this dependence, we will transform from real time to imaginary time T,where - 1/ TI ZI 1/T, and write the vector-potential as A& 7) =A,(k, I;i) exp(ik*r-i6r), where C=2njT, j=O, r!~1, f2, ... are the Matsubara frequencies for a Bose-system [ 25 1. If the magnetic field is applied parallel to the layers, the expression for the hamiltonian Xr in eq. ( 1) will be changed:

(9) Assuming that A, does not change sufftciently at interlayer distances of the order of dz 10 A, one can write the factor exp( i (e/c)JA=dz) as exp (i (ed/c)A,), and, after expansion in the vector-potential

&

d2p( 1+ $A=)

exp[ik-p] .

We will consider here only the London (local) case, which is quite natural for the high-temperature superconductors, and assume that k= 0. It means that we limited our description by the frequencies &?=aL?t_=kt+- t+/ 1. For the high-temperature superconductors with extremely large penetration depth (especially along the zdirection) and quite small L+=[ 281 this lower limit for the validity of our description is ai_< T, (for YBCO the values %-x 10’ m/s and the penetration depth when the screening current flows along the c-axis 1,~ 3 urn yields L$_N 1O’Os- *x 0.1 Kz 10e3 T, -CCT,). As a result of such a simplification, the expression for g in terms of quasi-particle energies on the S- and N-layers can be written as

We are interested in the frequency dependence of the kernel. Using the Green’s functions (7) and considering the tunnel hamiltonian .#$-eq. (9) as a perturbation, we can obtain the expression for the current [ 251 f

j(t)=e

s

dTeiYr(1i-%-(0, N(t) I, %F(T.)I> ,

-00

where v+ +0 and

(11)

459

A. Yu. Simonov / Luyered HTSC in an ACfield

is the operator of the number of particles in the ith layer of the nth elementary cell. In the lowest order in the hopping integral t, the expression for the kernel reads [25]

Q(i6)l(e2N/ml

I= (Ql(iCj) +Qz(i6) 1 ,

Q,(iC,)=TIdC[---z

x

&

F]i(iw)Mio+i&),

0

Qz(iC,)=7’j

z

d51;

dq

z Gii(ioVL(io+iC)

(12)

,

0

where N/m, = 2t*dN( 0) and the Green’s functions are given by the expression (7) (we did not indicate their dependence on 5:and q). The first term in eq. ( 12) describes the current of Cooper’s pairs, and the second one the current of a single quasi-particle. These expressions can be represented as diagrams as in fig. 1. One can see that these are the usual diagrams for Josephson tunneling [ 25 1, but the Green’s functions are different. In the static case and in the limit of weak coupling between the layers, our expression coincides with the expression obtained in ref. [ 171. We note that expression ( 11) was obtained under the assumption that the hopping integral is small, t << A. Nevertheless, we think that our model gives a qualitatively correct prediction at least when the ratio t/A is smaller than [ ( 1+$)/32] l/* [ 181. In view of the crude character of our model, this should not be a serious approximation. In order to obtain the frequency dependence of the kernel ( 12), we have to do the analytic continuation from the discrete values of & for the half- space Re C> 0 of the complex variable C (upper half-space of LLiC). We employ the method of contour integration [25]. As a result, the expression for the kernel can be written as Q,(n)=-&jdrjfjdotanh(g) 0

x

[ mvw)

-m(o)

)Fh(Q+Q)

+

(F?*i(w)

-%22(m)

Ff?(w+Q)

19

(13)

and there is a similar expression for Q2 (differing by replacing anomalous Green’s functio@s, #, by normal ones, G). In eq. ( 13) the index R(A) indicates retarded (advanced) Green’s functions, w&& are given by the expression:

c.J%*(~)=- i,

ucmuam(~_~m ,+im ,3, fiO

s T

T

Fl+l

-

52 Gil

G22

T

@ (a)

(b)

Fig. 1. The diagrams for the response kernel.

460

G%‘(~)=

A. Yu. Simonov /Layered HTSC in an ACfield

+ m.,dmkio w

-

m

2 u,,

o+E,,,+iO

(E;-~2-{2-&2A2)

1’ k ;

(E;+~‘Z_<*-&~A~)

1 .

(14)

Here the upper sign in front of i0 is for the retarded Green’s functions and the lower sign is for the advanced ones, respectively. The expressions for E L,2,D are given by eq. (5) and (Y, /I= 1, 2. The coefficients u and v are the generalized Bogolubov’s coefftcients. Contrary to ref. [ 23 1, they are different from the usual Bogolubov’s coefficients for the 3D isotropic superconductors. As a result, the difference of the corresponding advanced and retarded functions are GE - G$ = 2iIm Gz and FLt-F&$=ix

GCX-G&.=-in:

i

u,,v,,(G(o-E,)-6(o+E,)),

i (u&G(o-E,)+v&,,d(w+E,)), m=,

and, as one can see, using delta-functions, kernel can be written as Q(SL)/(e’N/m.)=-

(15)

one integration

can be done in a trivial way. The expression

for the

Jd~j~~2~~,tanh~)“‘~v2~~~~~~~~v2~)

-(tanh&)

-tanh(

-(tanh(g)

+tanh($))

(~:,~:2+~:,v:2+~1,~22v*,~22+~2,~2,~12v,2)

El -E2-B-i0 (~:*v:2+~:,~:2+~11~22v11~22+~21v21~12~12

El +E2-Q-i0

(16)

Using the identity J dx/ (x & i0) = P [J d-x/x] T iS( x), where P indicates the main value of the integral, one can re-write the expression for the real and imaginary part of kernel Q (52). In this paper, we will evaluate only the expression for the real part of conductivity along the z-axis, uZ= - Im [ Q(S2) ] /52 [ 29 1. For this reason, we keep in eq. ( 16) only the terms which contribute to the imaginary part of kernel. The expression ( 16) is the main result of this paper. It is long and complicated, but it is very easy to understand if we remember that our system has two energy bands El,z, which are given by the expression (5). One can seen that the term with (2E, - s2) describes the process of the breaking of the Cooper pair, where each electron has the energy E,, by the electromagnetic field quantum with energy S=2E,. Hereafter, we will describe all the quantities which are connected with these process by indices “A” and “B”, respectively. The term (El - E2 -G?) describes the process of the excitation of the electron with energy E2 to the band with energy E,. Since at zero temperature both bands are filled, the transition is not possible at T= 0 (index “C” indicates all the quantities connected with this process). The breaking of the electron pair, one with energy El and the other with energy E2, is described by the term (El + E2-8) (we denote this process by the letter “D” ). It is also easy to understand that for different frequencies, different terms are important. For some limiting cases the expression ( 16) will be discussed in the next two sections. Probably, it should be mentioned here that the expression for the conductivity, which was evaluated in ref. [ 81 for the very complicated model with live layers per unit cell contains only the Q2 term. Generally speaking, we can neglect the Cooper pair term Q, only if the temperature is close to T,, but in that case it is necessary to take into account the temperature dependence of the order parameter A. Another important thing is that near the critical temperature, the coherence length diverges and the layered system becomes effectively threedimensional. In this case, the Mattis-Bardeen approach is valid.

A. Yu. Simonov /Layered HTSC in an ACfierd

461

Our final remark in this section concerns the magnitude of the field in which all these expressions are correct. As was shown in ref. [ 171, the magnetic field changes the proximity effects and the Green’s functions (7) only in very strong fields, H2 IP zz &,/d& (0). But this is not the only limiting condition. We assume, that there are no vortices in the sample. The presence of vortices changes the behaviour of AC losses in type-II superconductors dramatically (see, for example, ref. [ 301). Nevertheless, due to the presence of a large surface barrier, some time is required for the thermally activated penetration of the vortices into the sample, and if the frequency of the field is high enough, the vortices cannot penetrate at all. The estimation for the field of the surface (Bean-Livingston) barrier disappearance gives in the anisotropic London model He, x H,,/fi [ 3 1 ] and for the Josephson model He, w ( tl ld) H,, [ 32 ] (H,, is the thermodynamic critical field). This value is much smaller that P. We think, that our model is applicable if the field is smaller than the field of the surface barrier disappearance.

4. Frequency and temperature dependence of conductivity: the microwave region In this section we will evaluate the expression for the kernel for the frequencies Q CKA for different temperatures. The induced superconducting order parameter on the N-layer d”= ~2/A~ A provides us with a new temperature scale. We will not consider the case when the temperature is close to T,, and limit our consideration to the cases when we can consider A as temperature-independent. In agreement with our previous calculations, we will consider a temperature below approximately 0.5 T, [ 19 1. Using our expression (5) for the energy spectrum, one can see that the minimal value of E, is A, and for E2 the minimal value is zero. The value of El is always larger than E2. That is why in expression ( 16 ) terms with +iO do not contribute to the imaginary part of the kernel for any frequency 9. If we consider the small frequency case, there are only two possible processes, which contribute to the conductivity. The first is the breaking of the pair of electrons with energy E2 by the quantum with energy Q (process “B”), and the second one describes the excitation of the electron from the low energy band E2 to the upper band E, (process “C”). Using the identity ~Scf(x))d~=C~S(x-xi)/ If’ (xi) I, where Xi are the roots of the equation f(xi) ~0, the expression for the conductivity at microwave region can be rewritten as

r2_ 2-

(A2-Q2)(A2-Q2+4~2) 4(S2-4P)

.

(17)

Here ooo=e2N/ml =2e2t2dN(0); 5, is the root of the equation 2E2=4, r2 yields the solution of the equation and, as usual, e(x) is the step-function, which is equal to 1 for positive x and zero elsewhere. It is evident that for small frequencies Q-X A, the value of t2 and, correspondingly, energies El and E2 are very large and the second term can be omitted for practically all temperatures. In the limit of weak (Josephson) coupling between the layers ( t K A), the expression ( 17 ) can be re-written as

E, - E,=Q,

A. Yu. Simonov /Layered HTSC in an ACfield

462 x

G(Q)

s====P%lq

= &

o Jl

-

(2P/LtO)2

If the frequency is small ($2~ ( t2/A)) temperature: u,(a)

= &G

(18)

.

we have a square-root dependence of the conductivity on $2 for low

z tanh(Q/4T)

e’tdN(0)

.

J

If TB 8, we have the s-power law in the frequency dependence of the conductivity and it decreases with temperature as T--I. If Q+ (8t*/A), then one can observe the logarithmic divergence of the conductivity, tanh(Q/4T).

(i= sh(l-(8t’/QA))

(19)

One can see that the behavior of the conductance in this frequency region reflects the behavior of the density of states [ 18,191. The result of numerical calculations of the frequency dependence of the conductivity for the different temperatures is presented in fig. 2. The weak (logarithmic) peak for s2x 8t2/A and the crossover from a t-power law for low temperatures to the t-power law for T2 t2/A are clearly seen. The reason of such an unusual decrease of the conductivity in the region of small frequencies is quite simple. As we mentioned previously, the process which leads to the appearance of AC losses at the low frequency region is the pair breaking of electrons with low energy E2. As T rises, the number of electrons with such small energy decrease, what leads to the decreasing of the conductivity as the temperature increases. Note, that our approach is valid if the frequency is not too small, SzB z.+/Izc. Nevertheless, in high-temperature superconductors there is a wide range of frequencies where the results obtained in this section are still valid.

5. Frequency and temperature dependence of the optical conductivity If the frequency of the external field is compared with the superconducting order parameter, A, the conductivity changes dramatically. As follows from our analysis, there are several regions of Q with quite different properties.

,

1o-’

,

,,,

,

,,,(

lo-’Q/A lo-*

_,

,4

10-l

Fig. 2. The frequency dependence of the conductivity for different temperatures for the value of the hopping integral t/d = 0.1 in the microwave region.

A. Yu. Simonov /Layered HTSC in an ACfieid

463

First we will consider the case of L&A. In these regions, the expression for the conductivity is the same as in the microwave region, but the first term, which describes the transitions between bands EC and ED is small (this term for the frequencies is proportional to t8) and it gives only a very small contribution to conductivity at zero temperature. If the temperature is not zero, the second term is in order. For Josephson coupling this term can be written as

up to some point very close to A, where One can see, that if (A-Q) >> t, this term increases as (A-Q)-“2 there is a maximum of this dependence, and after that it is practically constant. This maximum (but not the divergence) will be increasingly pronounced if the temperature rises. Such a behavior can be seen in fig. 3. If the frequency exceeds A (but 812A), the transition between bands E2 and E, (process “C”) is no longer possible (because of (E- E2) I A), but the new channels occur, a quantum of the electromagnetic field breaks the pair of electrons, one of them with energy E,, and another one with energy E2 (process “D”). The breaking of pairs of electrons when both are in the E2 bands is also possible, but as in the microwave region G?B t2/A, they give only a very small contribution to 0. The expression for the conductivity in this region can be written as

ldd[

00 * 0

u,(Q)=u~+u,=

+

(d,dz+d2&

Sztanh(Q/4T)

(~12~12~22~22+~:2~:2)

r(l_2T2,D)

=c,m:)

+~,*~,,~22~22+~,2~,2~2,~2,)

and the expressions for <,.2 are given in eq. (17). In the limit of Josephson coupling between the layers, the second term can be written as x

00 A

dgT2

%(Q) = 4Q(Q-A)

0

e( 1-2T2/ [A(Q-4 ,/l-2T’/[A(Q-A)]

I)

[tanh($)+tanh(g)],

(22)

0.10

I I

0.06

T=O.ZT,' I I

go.06

t=O.lA

I

a b

/

0.04

I

I 0.02

0.00

/ 4,

T=&; I,, 1 I,

n/A

’

Fig. 3. The frequency dependence of the conductivity for different temperatures for the value of hopping integral t/A= 0.1 at the optical region.

464

A. Yu. Simonov /Layered HTSC in an ACjield

and, as one can see, that integral goes to zero as (52-A) ‘I2 if &+A+. If the frequency exceeds A and goes to the value A+ &*/A, there is a logarithmic divergence on the dependence a(Q) which looks like u,(Q) x - QT$‘tA)

[tanh(&,)

+tanh($)]

ln( 1- A(t:A)).

(23)

This peak is quite close to the maximum at the frequencies slightly below A and it is possible that on the experimental curves it looks as if only one peak exists. If the frequency exceeds 24, there is the possibility of the electron pairs breaking from the E, band (“A” process). In this case it necessary to add an additional term to the expression for the conductivity:

~;=Q2/4+~2-A2/2-,/A4/4+~2(Q2-A2),

(24)

and & is the root of the equation 2E, =a. One can see that this term is similar to the analogous term with E2. Let us mention, that for such large frequencies, the temperature dependence of this term is very weak (B z+ T). One can see that this term goes to zero if 52+2A+, and if frequency exceeds 24, there is also a logarithmic divergence for the frequency 52=2A+ l&‘/A: (Tot* ozz - A5/2(Q_2A)

In ‘-

4t2522 A3(Q-24)

>.

For the high-frequency region (Q > 2A), the conductivity decreases as a-QV3. So, in the frequency dependence of the conductivity for an S/N superlattice there are four peaks: (a-c) three logarithmic peaks at 52~ (8t*/A), (A+8t*/A), and (2A2+ 16t*/A), and (d) a peak at frequencies slightly below A. The temperature dependence of the intensity of these peaks is quite unusual too. If the intensity of the logarithmic peaks is weak, the intensity of the peak (d) decreases with temperature and it disappears completely if T=O.

6. Discussion We have investigated the frequency dependence of the conductivity for a model system composed of an array of superconducting and normal-metal layers if the AC magnetic field is directed along the conducting plane. We assumed that the hopping integral t is small, t *: A. This seems to be the most important assumption in our consideration, because when we obtained the expression for the current, eq. ( 1 1 ), we considered the tunnel hamiltonian Xr as the perturbation. Considering high-T, superconductors, we would like to note that the interlayer distance in these materials is d2 cl (<, is the coherence length that is perpendicular to the layers) and t I T,. Generally speaking, for the quantitative description of the situation in real materials, it is necessary to calculate the next terms in the expansion over t. Nevertheless, our model can give the qualitative prediction of the frequency and temperature dependence of the conductivity. It is not clear enough if the BiO- and TlO-layers in bismuth- and thallium-based high-T, compounds are normal metal or if they are superconductors with a low critical temperature. In this case, we can use the S/S’ model. The gap in the density of states occurs in this case, but, due to the big difference between critical temperatures of different layers, this gap will be small [ 19 1. Artificially created superlattices is another object to which our results can be applied. Recent progress [ 33 1

A. Yu. Simonov /Layered HTSC in an ACjield

465

in atomic level molecular beam epitaxy provides the opportunity of the creation of superconductor-normal metal systems with monoatomic layers. In this paper only the case of an AC magnetic field parallel to the layers was considered. Nevertheless, it is possible to make some conclusions about the frequency dependence of the conductivity in the case of an AC magnetic field parallel to the z-axis (currents flows in the (&)-plane). Because of the gapless character of superconductivity in an S/N layered system, the AC losses appear at any finite frequency even at zero temperature. The electron band structure remains the same, as in eq. (5) the peculiarities in the frequency dependence of conductivity appear at the same places. The exception is, probably, only the peak at Q- t21A, which can be affected by non-locality effects (k# 0). This effect is much more important for the case of the AC magnetic field parallel to the c-axis. Generally speaking, effects of non-locality, which are not considered in this paper, can make the weak logarithmic singularities at 52%t2/A and B-A smoother, and the peak at 52~4 will be much more pronounced that the other two peaks. Now we have a wide range of experimental data supporting our model. In Raman scattering, measurement of the different materials (YBa2Cuj0, and T12Ca2Ba2Cu30io) gives evidence in favor of the absence of the gap in these compounds [ 34-361. These measurements are supported by the results of tunneling experiments [ 37-401. Unfortunately, there are only a few c-axis polarized measurements [ 4 1,42 1. All these experiments show that there are a number of peaks on the frequency dependence of the conductivity. The interpretation of this experimental data is difficult because of the large phonon contribution. One of the features of the present model is that it predicts the peak of the frequency dependence of conductivity near f&A. The identification of this peak with the peak observed in (&)-plane polarized experiments at frequency 9~ 150 cm-’ = 2.4T, in YBa2CuJ07 by Thomas et al. [ 431, Orenstein et al. [ 441 and Kamaras et al. [ 45 ] provides us with the value of 2A/ T, x 4.8 which is in agreement with the results of tunneling measurements [ 37-401. In conclusion, the layered structure of high-temperature superconductors is the important factor which strongly modifies the behavior of the frequency dependence of conductivity. The AC losses can be observed for any finite frequency. The onset of losses, which was predicted for isotropic BCS superconductors at 52= 24 is absent for a layered S/N superlattice. Fine structure inside the “gap” region is predicted. It is important to mention, that the frequencies, which correspond to the singularities a(Q) no longer coincide with the points A or 26 exactly. This picture is similar to the behavior of the density of electronic states, where the place of the peak is shifted to the region of energies E 2 A [ 18 1. As one can see, both tunneling experiments, which provide at low temperatures the density of states, and measurements of finite-frequency properties, give no opportunities of direct measurement of the order parameter A, which for the system under consideration has nothing to do with the gap in spectrum. This factor has to be taken into consideration in the interpretation of experimental data.

Acknowledgements I would like to express my gratitude to J.R. Clem for valuable discussions. Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. The work was supported by the DOE through the Midwest Superconductivity Consortium, Grant No. DE-FG02-90ER45427.

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