Microwave conductivity of superconductors with a strong electron-phonon interaction

Solid State Communications, Vol. 89, No. 10, pp. 827831, 1994 Elsevier Science Ltd printed in Great Britain. All rights resewed 0038- 1098/94 $6.00+.00 003%1098(94)EOOSS-G

Pergamon

MICROWAVE

CONDUCTIVITY OF SUPERCONDUCTORS ELECTRON-PHONON INTERACTION.

WITH

A STRONG

0. V. Dolgovt, E. G. Maksimovt, A. E. Karakozovt, A. A. Mikhailovskyt. $P.N.Lebedev Physical Institute Russian AS, Moscow, 117924, Russian. tL.F.Vereshchagin Institute for High Pressure Physics Russian AS, Troitsk, Moscow region, 142092, Russian. (Received by V.M.Agranovich

, December 27

The conductivity and the surface resistance are calculated for systems with a strong electron-phonon interaction. It is shown that the real part of the conductivity has a maximum at intermediate temperatures in the clean limit in accordance with the experimental observation in high-T, superconductors. This maximum connects with the temperature dependence of the electron relaxation rate and is not the coherence peak. But this phenomenon can not be described properly in terms of a simple two-fluid model.

real part of the conductivity that occurs at intermediate temperatures. Strictly speaking it is a peak or even plateau in the surface resistance R, that are observed. This results in a peak of the conductivity only after the treating using the formula connecting R. with conductivity u. 2. The appearance of this peak have been interpreted usually in terms of a generalized two-fluid model. The conductivity in this model can be written as

1. Microwave properties of high-T, superconductors continue to be of interest both theoretically and experimentally[l-71. Values measured are the surface resistance R, and to lesser extent the penetration depth XL. Experimental results, in particular, at T < T, are still controversial. But they are comparatively reproducible at T = T, and at intermediate temperatures. There are a lot of experimental evidences that the microwave behavior of high-T= superconductors differs significantly from the predictions of the BCS theory. Firstly, the real part of the microwave conductivity cri does not show the coherence peak as the temperature is lowered below T,. The simplest explanation of this fact in the framework of the BCS can be done considering these superconductors in the clean limit. But this explanation seems to be doubtful in spite of that the main part of the observation had to do idled with samples in the clean limit. BCS theory predicts that the coherence peak should occur also in the NMR relaxation rate and this value does not depend on impurities in comparison with the conductivity. But no coherence peaks was observed also in the NMR relaxation rate for high-T, superconductors. This fact leads to the conclusion that the absence of the coherence peaks in both the NMR relaxation rate and the conductivity should have the common reason. The simplest way to find this reason is to introduce the smearing of the density-of-states singularity. Such smearing can occur, for example, due to an anysotropy of the gap-function. The another origin of this smearing can be an inelastic scattering of electrons due to the strong electron-phonon interaction which is important mainly near T,[8]. Just that very reason was used for the explanation of the absence of the coherence peaks for high-T, superconductivity in both the NMR relaxation rate and the conductivity[9,10]. But this explanation could be questioned in the light of some other results obtained in the experimental investigation mentioned above[l-51. It is the existence of the peak in the

e(w, T) = f(T)e,(w,

T) + (I-

f(T))e&,

T).

(I)

Here f(T) is the normalized density of the superconducting electrons which can be determined, for example, in terms of London penetration depth as

The value (1 - f(T)) is the density of the normal electrons. The functions Q,(u, T) and a,,(~, T) are the conductivities of the superconducting and normal electrons correspondingly. The function u,(w, T) is determined by the behavior of superconductivity condensate and can be presented as the temperature independent value

T) = u,(w)

&b(w) +ineZ m wm’

(3)

The phenomenological two-fluid representation for the conductivity(I) was introduced early[ll,l2] to describe the experimental data in the infrared region for T < T,. The conductivity of normal electrons o,(w,T) was also chosen as the temperature independent value equal u,(w, T = T,). After that the fitting of the experimental data has led to the function f(T) coinciding with that of given by the old Kasimir-Gorter two-fluid theory

(4) 827

Vol. 89, No. 10

MICROWAVE CONDUCTIVITY OF SUPERCONDUCTORS

828

We have proved this result[l3] using the microscopical calculation of the conductivity for the system with a strong electron-phonon interaction.It was shown recently[6] that the representation (1) for the conductivity can be proved also in the framework of the BCS theory with the value b,(w, T) given as U,(W,T) =

(l - f(T))neagscs(w,7,T), m

{

&(W-

W’)

+

&(W’)

2i7imp

+

W’)

-

E’(W’)

+

2i7imp

_

1

1 - n(w + w’)n(w’) - a(w + w’)a(w’) _ E(W + W’) + C(W’) +

(7)

2i7imp

1 + n(w + w’)nl(w’) + a(w + W’)a*(W’) &(W

Then the appearance of the peak in the conductivity is the result of the competition between the temperature dependence of the two functions that are the density of the normal electrons (1 - f(T)) and the relaxation rate 7(T). It is clear from the expression (7) that such type peak can appear only for very small w because the value of the relaxation rate 7(T) at T < T, is very small. The people uses usually this explanation for the conductivity peak in high-T, superconductors in the frame of some exotic mechanisms of the superconductivity[3,5] or for an exotic origin of the relaxation rate[l). It is not clear why such conductivity peaks can not exist in the usual s-wave superconductors with a strong electron-phonon interaction. The preceding microscopical calculations of the conductivity in the framework of Eliashberg equations has shown ether the presence of the coherence peak in the dirty limit of weak-coupling superconductors or the absence any peak in a strong-coupling case. But as we shall show just below these calculations was not made in the suitable limit. It is the conductivity in the clean limit and at very low frequency satisfying the condition (7), which should be calculated. 3. We shell use the well known microscopical expression for the conductivity of superconductors with a strong electron-phonon interaction[l6). This expression was used in our preceding works[9,13] as well as in a lot of others investigations[l4,15]. It can be written in the form w 4lr -U(W) = 5 dw’thg x w 0 WZ

J

x

-

&.

= w.

-

(5)

It has a more complicate form for T < T, including the coherence peak at small w, but the general behavior of this function is not so different in the BCS model from the Drude form. To avoid the coherence effects it is the Drude form which is used usually for the interpretation of the conductivity peak at intermediate temperatures in high-T, superconductors. In this framework origin of this peak is very clear and it is connected with a temperature dependence of the relaxation rate 7(T). Indeed, the function (6) at any given w has the maximum as a function of a temperature with T,,,,. determined by the condition 7(Tm.z)

E(W

1 - n(w - w’)n*(w’) + a(w - w’)o’(w’)

where the function gBcs(w, 7, T) depends on a frequency, a temperature and a relaxation rate 7. This function has the usual Drude form for T > T,

gecs(w,r,T) =

1+ n(w - w’)n(w’) - a(w - W’)U(W’)_

x

+

W’)

-

&*(W’)

+

+

2i7i,,

. {J”

w + w’ dw’thy x

+zRe

0

x

1 - n(w + w’)n(w’) - a(w + w’)o(w’) E(W

+

W’)

+

E(W’)

+

.

(8)

2i7hp

Here n(w) and o(w) are the functions equal

n(w) = wz(w) -

E(W)’

a(w)

=

fJ$,

(9)

thereby the real part of the function n(w) is the density of the electron states and E(W) is the excitation energy

E(W)=

Jw’z’(w) - d”(w).

(10)

The value 7imp represents the electron relaxation rate due to the scattering on impurities. z(w) and d(w) are, correspondingly, the renormalization function and the superconducting self-energy determined by the Eliashberg equations. In the weak-coupling case we can take

z(w) = 1,

b(w) = A

(11)

to get the BCS’ model. All differences in the superconducting properties of systems with a strong electronphonon interaction and the BCS model connect with a distinction the function z(w) and d(w) from that of given by (11). Moreover it connects mainly with the existence of imaginary parts of this function that is Im z(w) and Im 4(w). We have used the simple model for one spectral function of the electron-phonon interaction a*(w)F(w) determining the solution of the Bliashberg equations. This model is a lorenzian centered at ws and tails Do2 and a(w - w,,,)’ for low and high energies respectively. The constant of the electron-phonon coupling A = 2Jo m %a’(w)F(w) was chose equal 1.5l. Using the obtained solution for the functions z(w) and d(w) we have calculated the conductivity u(w) in accordance with the formula (8) for w = 5 x lo-‘, 2 x 10Ts, 3 x ’ We have used the energy units normalized characteristic phonon energy ws.

Vol. 89, No. 10 10e2(w/Ao

MICROWAVE

= 1.2 x 10-s,

4.6 x lo-‘,

CONDUCTIVITY

7 x 10S2).

As is the gap at T = 0. We have calculated surface

resistance

London

limit as

R,

which

can be wrote

Here

also the

in the local

= JZ&,

(12)

= 1+

p(w) results

early

of the systems

teraction

of these

F,(w).

contribution

and al(T)

are shown on Fig. 1 and 2. The value 7imp was chose

difference

in the be-

electron-phonon

in-

model arises from the imaginary

z(w) and 4(w).

Moreover

the largest

in this difference is given by the imaginary

part of the renormalization

function

.z(w). This function

as

W)

(13)

for R.(T)

calculations

the main

with a strong

and the BCS

can be presented

where

The

As we mentioned havior

part of the function R,-iX

829

OF SUPERCONDUCTORS

Z(W) = 1 + X + i-, where X is the constant

W

of electron-phonon

coupling

and

l?(w) is

equal zero. We would like now to emphasize ture dependence large resemblance

with that of observed

tally at least at the intermediate to T,. Indeed the surface crease, 0.8

that the temperaby experimen-

temperatures

resistance

exhibits

by near four orders of magnitude,

< g

tures diminishing with an increase

maximum

to high tempera-

w. The precise

its amplitude

and the value of the electron-phonon of this problem

posi-

and width depends

from the form of the function

tailed investigation

near

a2(w)F(w)

coupling.

The de-

will be published

Let us consider

details.

the appearance

- w’)[iV(w’)

Here N(w)

and f(w)

are Bose and Fermi functions.

the normal state the function of the electron-self-energy rate.

It follows from the numerical

can be proved

analytically

that

to ~1 at very low frequencies in the expression

can be written

and

c(w) = (1 + X)

(w + iF(w))z

where I’(w) and A(w)

ne2

+ n(w)n*(w

P

&*(W

+

W’)

-

Jo xx

A(w)

+ w’)]x

(14)

+

c(w) which

as

- AZ(w),

(17)

functions

s

(18)

= $$.

We can see from this expressions ues of frequency

&(W)

relaxation

and

dw’

1

XIm

energy

are renormalized

F(w)=

ch2 =

+ w’) + a(w)a*(w

In part

w in the superconducting

the excitation

the main contribution

2mT

I’(w) is the imaginary

using our approximation

exists due to only one term

= --

+ f(w - w’)]}.

that is the electron

for all frequencies

Let us consider

(1’3)

I’(w) has not such simple inter-

(8), that is q(w)

xRe[l

calculations

But the function

pretation case.

in

of (~1 in some

w’)[N(:q) +f(W + w’)]+

- w’)n(w

the near future. 4.

{n(w +

dw’a2F(w’)x

+sign(w

a rapid de-

and then disappears

of the frequency

tion of this maximum, considerably

moves

at the amplitude

x

and close

in the interval

< 1 and then shows a broad

T/T= N 0.5. This maximum

P(w) = ?r /=

R,(T) and 61 (T) has a

of the functions

that for the large val-

w >> A the behavior

of real and imag-

2i7i,,’

“3

0.2

0.4

06

1.Surface resistance

log R,(T)

0.0

0.8

1.0 Fig.

Fig.

+ cast.

t = T/T,.

2. Conductivity

10b4, 2~ 10-s,

Re a(T) for rimp = 0, w = 5 x

3 x 10b2(from

top to bottom).

t = T/T,.

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MICROWAVE CONDUCTIVITY OF SUPERCONDUCTORS

830

inary parts of e(w) is very similar with that of existing in normal metals

E(W)N (1+ X)w + Z(w).

(20)

But for a smaller frequencies the imaginary part of the function E(W) does not coincide with l?(w). For example, for w N A it is equal Im E(W) = (1 + X)&Z,

(21)

which is more lager value then I? at low and intermediate temperatures T < T,. We consider now the coherence factor L,,,, L w,wr= 1 + n*(w’)n(w t w’) t a*(w’)a(w t w’).

(22)

It can be written at very low w N 0 as

w” +

LO+’ = 1 t

f’” + A*

(wl* _ A* _

j+)*

+

4w~2F2

(23)

We can see that this factor in BCS model@ = 0) is very strong peaked function at w 11 A for all temperatures. This fact leads in this model to the appearance of the peak in ui(w N 0) at the T N T, in the dirty limit. But the coherence factor is smeared near T, in systems with a strong electron-phonon interaction where F > A leading to the disappearance of the coherence peak both in dirty and clean limits. The situation can be changed for such systems in the clean limit but only at the intermediate temperatures where F < A. The coherent factor Lo,,~ at these temperatures becomes again well peaked function at w’ N A with the amplitude N * and the width N ?. In this region of the frequencies w’ the value Im(l/c(w + w’) - I’ + %7;,,) for w 21 0 can be presented in the clean limit as 1

I/2 Irnb(W t w’) - E’(W’) = (I + x,a

<

(1 t X)a.

(25)

&s it was shown by Eliashberg et al.[17] the function r(T) can be presented as (27) where n N 5. It is easy to see that the function u(w 21 0) determined by (26) has a maximum at T = T,,,,, and

--.

maz-

n-l-2

(29)

1 -f(T)

N

J

se-+,

we can rewrite the expression for ci(w N 0) as

m (1 t X)dF

(31)

in accordance with the discussed above two-fluid model. The maximum of the value ui(w N 0) exists in this approach due to the temperature dependence of the

(26)

T

N 4 than we

in accordance with our numerical calculations. Taking into account that the number of normal electron l-f(T) can be determined as

(24)

After that we can estimate the value ui(w N 0) as

2A

Since our calculations give the ratio !j! have Tmao -N 0.5T,,

ul(wEo)2:* 1 -f(T)

and the condition determining the clean limit is 7imp

Fig. 3. Conductivity Re u(T) for w = 2 x 10s3, 7imp = 0, 0.05, O.l(from top to bottom). t = T/T,.

value (1 + X)fi which plays the role of the electron relaxation rate. But this results does not prove the simple two-fluid model for systems with a strong electron-phonon interaction. Firstly, the role of relaxation rate determining the clean limit is played by the value &? in accordance with our expression (25). As the results of our numerical calculation for the different values 7imp are shown on Fig. 3. Secondly our numerical calculations show that the peak Q(W) disappears at the very small frequencies w N f’(Tmoz) but not for w N (l-l- X) fl or (1 +A)&! as it could be assumed in terms of the two-fluid model. This result can be obtained also using the analysis of the expression (14) for m(w).

Acknowledgement-we are very grateful to A. Golubov, D. van Marel, I. Trofimov and S.V. Shulga for fruitful discussions.

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MICROWAVE CONDUCTIVITY OF SUPERCONDUCTORS

831

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