Effect of the energy-dependent density of states on a strong-coupling superconductor

PHYSICA ELSEVIER

Physica C 231 (1994) 341-351

Effect of the energy-dependent density of states on a strong-coupling superconductor Yasushi Yokoya Department ofApplied Physics, Facultyof Science, Science Universityof Tokyo, Shinjuku-ku, Tokyo 162, Japan Received 17 June 1994; revisedmanuscript received 19 July 1994

Abstract

The effect of the inclusion of the energy dependence in the electronic density of states (EDOS) on quantities in the superconducting state is studied for strong-coupling superconductors, within the framework of the Eliashberg theory. In order to see how the effect depends on the shape of the EDOS around the Fermi energy when the coupling strength is increased, the critical temperature and the energy gap are calculated with two typical shapes of the Lorentzian EDOS for various electron-phonon coupling strengths. It is shown that the more the coupling strength is increased, the more significantly the energy dependence of the EDOS affects these quantities. The temperature variation of the quasiparticle density of states is calculated for various parameters of the Lorentzian EDOS to examine the temperature dependence of the phonon structures in the tunneling characteristics. 1. Introduction

The electronic states of the strong coupling superconductors have been extensively studied for unusual superconductors (A 15 compounds, high-T¢ copper oxides, superconducting C6o etc. ). In particular, it has been known that the anomalous features of the A15 compounds in the superconducting state [ 1-3 ] result from the effect of the energy dependence in the electronic density of states (EDOS) near the Fermi energy [ 4-12 ]. The high-To copper oxides (cuprates) have a wider phonon band ( ~ 100 meV) than traditional superconductors have. In recent years, many band-structure calculations for the high-T¢ cuprates have been carried out [ 13-15]. These calculations suggest that, in cuprates, a considerably rapid change of the EDOS occurs on the scale of the phonon energies. Therefore, taking account of the energy dependence of the EDOS is expected to be necessary for the analysis of physical properties of high-T¢ cuprates. From this point of view, there have been several works on high-T¢ cuprates which, for example, discuss the anomalous isotope effect by taking account of the energy dependence of the EDOS of these materials in the Eliashberg theory [ 16 ]. A central purpose of this paper is to examine to what extent deviate the results of the constant EDOS approximation in the conventional Eliashberg theory from those of the Eliashberg theory taking the energy dependence of the EDOS into account, for the case of materials in which the energy variation of the EDOS occurs on the scale of phonon energies. High-T¢ cuprates are appropriate materials for this study for reasons stated above. For high-T~ cuprates, a lot of both theoretical and experimental works have been made [ 17-24 ]. In our study, on the basis of the isotropic Eliashberg theory including the energy dependence in the EDOS, we calculate the superconducting critical temperature Tc and the superconducting energy gap Ao, and examine the effects of this 0921-4534/94/$07.00 © 1994 ElsevierScienceB.V. All rights reserved SSDI0921-4534(94)00494-3

342

Y.

Yokoya / Physica C 231 (1994) 341-351

inclusion on them, by using a model spectral function which models ot2F( 09 ) of Bi2212. Numerical analyses are made for the cases that the EDOS has a peak structure or a hollow structure just at the Fermi energy, and employ the Lorentzian EDOS for simplicity. The last two decades, the Eliashberg theory including the energy-dependent EDOS N(~) has been examined extensively to explain anomalous features in A 15 compounds [ 1-3 ]. Especially Carbotte et al. have investigated the problem by solving the Eliashberg equations [ 8-12 ]. In Section 2 of this paper, starting from the imaginaryaxis Eliashberg equations with the energy-dependent EDOS which is due to Mitrovic and Carbotte [ 11 ], we extend the "MSC equation", which is an exact analytic-continuation method of the imaginary-axis quantities to the real-axis ones in the conventional constant EDOS theory [25 ], to the case involving an energy-dependent EDOS. In Section 3, based on these equations, numerical calculations are carried out on the critical temperature T~ and the energy gap Ao in order to discuss the effects of the inclusion of the energy dependence in the EDOS. In Section 4 the quasiparticle density of states at finite temperatures is calculated in order to examine how the feature of phonon structures in the tunneling density of states depends on the type of EDOS around the Fermi energy and we discuss their temperature dependence. The results are summarized in Section 5.

2. Formalism We start from the isotropic Eliashberg equations in the imaginary-axis approach involving the energy dependent EDOS N(e) [ 11 ], which are given by

1 ~ i z~,,=~m__~_oo{)t(n-m)-~*(°gc)O(°gc-I~,,,I)} (~)nmO)n "~-

1

~o

?m=~_oo2(n-m)

2n=__ ~1 ~ .

.

.

2(n--m)

.

i

N(e)

N(e)

'~m

de N(0----~eb2 + (e_ a#+)7 )2 + 2 ~ ,

(1)

(7_)m

(2)

de N(0--~ o32 + (e_ a/t+)~m)2 + ~2 '

deN(0)~Zm+(e__~#+)Tm)2+ff2m ,

(3)

with 2(n-m)=

d ~ a 2 F ( Q ) ~ 2 + (COn--0)m)2"

(4)

o

where fl= 1/kBTand 7C

w.=~(2n+l),

co.is the nth Matsubara frequency, n=0,+_l, +2,...

(5)

Here the pairing energy z~. and the renormalized frequency &. are related to the gap function A(iog.) and the renormalization function Z(iog.) by the relations z~. - A (ito.)Z(io~.) and &. - oJ.Z(ito.), and2. is the diagonal part of the self-energy which does not necessarily reduce to a trivial shift in the chemical potential for a general shape of the energy-dependent EDOS. The EDOS N(E), here, is a single-spin electronic density of states and the electronic energy e is measured with respect to the bare band-structure chemical potential and 8/z is the shift of the chemical potential resulting from the interactions. The function oFF(to) is the electron-phonon spectral function, and/~* is the Coulomb pseudo-potential with a cut-offfrequency o9c. The extension of the MSC equations to the case involving the energy-dependent EDOS can be made straightforwardly and the extended MSC equations are written as

343

Y. Yokoya/ PhysicaC 231 (1994)341-351

3(o))=i'N(E)idQol2F(a){Kz(~,a-o))+Kz(~,Q+o))} --oo

-

0

,Y~__o #*(o)~)O(w~- Io),. I ) deN--~

-

O~-N~o52+(~__Sfl+2m)2+~z m

d.~a~f(Q) os~(o)_~)_(e_SU+2(o)_~))~_2~(o)_~) {n(~)+f(~-o))} ~(o)+~)

]

+ o52(to+g2)- ( e - 8/t+2(o)+g2))2-f2(o)+Q) {n(g2) +f(~+o))} , •

oS(O)) =co+

m(e)

f

ae ~

j d~ol2F(Q){K~(e, ~ - o ) ) - K ~ ( e , ~ + o))}

• N(E) f (h(o)--g2) {n(g2)+f(£2--o))} ae N---~ j d,Qot2F(~2) o52(o)_~2) _ (E_ 8#+2(o)_~2) )2_~2 (o)_~2)

---oo

0

os(o)+~)

]

+ o52(O)+~) _ (~_ 8/./+2(O)+Q) )2 32(O)+Q) {n(Q) +f(g2+o))} ,

(7)

• N(e) [" ae-~--(6-~j d~a2F(~2){K~(e,~2-o))+Kz(e,~+o))}

2(o))=--at)

+

(6)

• N(e) f

a~N----~J

0

dQot2F(.Q) ~--731l+7"(o)--Q) {n(g2)+f(g2--o))} oS~(o)_Q)_ (~_ 8U+g(o)_a))~_ff~(o)_~)

~ - 8#+2(o)+~) ] + o52(o)+g2)- ( e - 8U+2(o)+g2) )2-,]'2(o)+g2) {n(£2) +f(g2+o))} ,

(8)

where the kernels Kz, K,~ and K~ are given by 2 ~

z~,.

£2

2 oo

os,n

o),,,

K,~(E,a) = ~ Eo osL + (~- 8~'+2m)~+3L a2+o)L ' 2 oo e-Sg+2m KI(e, 12) = ~ m~=oo52 + (¢_ 8/t+2,~)z+~'~ Q2+oSL '

(9) (10)

(11)

respectively. Heref(g2) and n (Q) are the Fermi and Bose thermal factors, respectively. From these MSC equations (6-8), the real-axis quantities ~(o)), o5(o)) and 2(o)) are obtained by using the imaginary-axis quantities which solve the imaginary-axis Eliashberg equations ( 1-3 ). We define the energy gap do by the equation Re,~(Ao) = Re o5(Ao) , and the expression for the quasiparticle density of states is given by the formula

(12)

IT.Yokoya/ PhysicaC231(1994)341-351

344

.,~(o9)=-

1

Im

[i

. N(e )

~b(w)

]

O~N----~o~(o9)_(~_SU+2(o9))~_,~(o9) J .

The effective electron-phonon coupling constant 2 related to

a

(13)

2F(o9) by the formula

oo

0

will be used as a parameter which measures the strength of the electron-phonon interaction. In discussing the effects of the energy variation in EDOS in the following sections, this parameter 2 is varied to see how the effects depend on the coupling strength. In the following numerical calculations, for simplicity, the Lorentzian EDOS centered at the Fermi energy

1 (

X(~)=m(0)]-~x

a2x~ ]

l+a2+e2

(15)

will be used. Here the parameter a is the width of the Lorentzian and x is a parameter which measures the prominence and the type of the structure of the EDOS at the Fermi energy: If the value of x is positive, the EDOS has a peak at e = 0 ("convex type"). On the contrary, if it is negative, the EDOS has a hollow structure at e = 0 ("concave type"). The conventional constant EDOS approximation is recovered by x = 0 or a-+ oo. This Lorentzian EDOS has been already used extensively in works on the A 15 compounds by Carbotte et al. [ 9-12 ]. One of the merits of using the symmetric Lorentzian EDOS is that the energy integration for e can be carried out analytically [ 11 ]. Another merit is that, owing to the symmetric form of the EDOS, the chemical potential does not shift, 8/z = 0, and the diagonal self-energy)~ can be set equal to zero. If we use the EDOS given in Eq. ( 15 ), the imaginary-axis Eliashberg equations ( 1 ) and (2) are rewritten simply as [ 11 ]

. L= l~_X?~m=,.:oo{2(n_m)_u.(o9c)O(o9c_logml) 1 n ~

CO.=O9.+ l~_x~m=}.~oo2(n--m)

( l +a +x )~a + ~

(l+x)a+~

~

' 3~

(16)

~bm

a + ~

~ "

(17)

In our paper, these equations are used to determine the critical temperature To. Correspondingly, our MSC equations are written as follows: A(o9) = i

d'Oa2F(Y2) {K~(O-o9)+g~i(O+o9)}

0

1 ~ ~ ( l + x ) a + ~ l + x Pm=O +

I

l~in

f

df2ot2F(g2)

z~m

/t*(o9c)0(o9¢--Io9ml)

[(l+x)a-ix/Co2(o9-a)-,~2(og-a)A(o9-a){n(a)+f(a-o9)} a_ix/~z(o9_O)_z~=(og_i2) x/&2(og-g2)-A2(o9-I2)

0

+ ( 1 +x)a-ix/~2(o9+ta) -~'2(o9+O) Z(og+O){n(t2) +f(O+~o)}],

a-ix/go2(og+Ea)-z~2(og+t2)

x/goz(o9+t2)-Az(o9+t'2) I

~(o9) =o9+ f dY2oz2F(y2){K'~(Y2-og)-K~- (0+o9)} 0

(18)

Y. Yokoya/ Physica C 231 (1994)341-351

+~

345

oo

1

]= j" d.Qa2F(.Q) ( 1+x)a-ix/¢32(co-~2) -zT2(co-.Q) ¢3(co-.Q)(n(.Q) +f(.Q-co)} . [ ~ ~ - ~ ~ %/~2(~°-Q) -~2( ¢°-~2 ) 0

+ (1 + x ) a - i x / 0 3 2 ( o j + f 2 ) - • 2 ( 0 9 + t 2 ) a - ix/t~2 (to + ~2) - zT2(to + O)

cb(o~+t2){n(Q) +f(12+to)}] ~ + - ~ - ~ i'

(19)

where the kernels K~ and K~ are given by

K~(E2)-

1 2x ~, ( l + x ) a + ~ I + ~ fl m=O a + ~

,~m

.(2 g22+o9 2 '

~

1 2~ ~ ( l + x ) a + ~

~m

(20)

O,)m

(21)

3. Numerical analysis of the critical temperature and the energy gap In the numerical calculations we use a model of the electron-phonon spectral function ot2F((.o) constructed from the inelastic neutron scattering (INS) data for Bi2212 [27 ]. In this construction of a2F(og) the sum of the phonon modes relevant to the atoms in the CuO2 plane is extracted. We take this function to model the form of a2F(to) for Bi based cuprates qualitatively well, since tunneling studies for Bi2212 are analyzed successfully by use of this model a2F(og) function [ 28 ] and the qualitative feature of the infrared optical responses of highTc cuprates seems to be explained on utilizing it [29 ]. In Fig. 1 we show the model functions a2F(og) for various values of the coupling strength. The model functions A-D are such that, if the constant EDOS approximation is employed, they give four different values of the critical temperature To: 60.0 K (broken line), 80.2 K (dashed line), 100 K (dash-dotted line) and 120 K (solid line), respectively. These values of T¢ are the familiar ones; they can be observed in some high-T¢ cuprates [ 30 ]. In Table 1 are shown the values of 2, T¢ and Ao, which are calculated from these four model functions A-D with the use of a constant EDOS. In Fig. 2, to compare the energy scale of the variations, we show the EDOS N(og) i

i

i

i

i

i

.............

model

i A

. . . . .

model

B

. . . . . .

model

C

-

model

D

-

"~'3 ~2

,

J

.. .

I

I

2o

.

.

.

I

.

.

.

4'0

>.x.

(meV)

I

//¢

6'0

I

•

8O

Fig. 1: Elec~ron-phononspectral functionsol2F(to) which give four different values of the critical temperature. The broken line is for To=60.0 K (model A), the dashed line for 80.2 K (model B), the dash-dotted line for 100 K (model C) and the solid line for 120 K (model D).

Y. Yokoya / Physica C 231 (1994) 341-351

346

Table 1 The values of 2, the critical temperature T¢ and the energy gap do, which are calculated from four models of ot2F(co) with the use of the constant electronic density of states Model

2

(Tc)N(0) (K) (Ao)N(o) ( m e V )

A

B

C

D

2.60 60.0 13.4

3.63 80.2 19.6

4.84 100 26.4

6.23 120 33.7

1.4

o

O

1.2

i

i

............ model model ..... model - model

,,~ ":~x-'~"~

A B C D

<

...i

.. . . . . . . . . . .

v

1.0 ,.o

"-'

0.8

>

1

2

3 / ~ max

4

0.6~

~

1'0

'

15'

a / l~ max

Fig. 2. The normalized Lorentzian electronic density of states N(to)/N(O ) is shown together with an electron-phonon spectral function a2F(to) (model B) as a function of the normalized frequency to/I2=~. The upper three curves are for the parameter x = - ~ and the other three curves are for x = 3. The solid curves, the dashed curves and the broken curves are for the half widths a = ~m=x, 2~m,=, 512m~. Fig. 3. The normalized critical temperature (Tc)N(o/(Tc)~(o) as a function of the normalized half width a of the Lorentzian electronic density of states ( a / 1 2 ~ at the temperature T = 0 . t Tc for four models o f a 2 F ( a 0 .

together with o u r a2F(co) as a function of co normalized to the maximum phonon frequency Oma~ (78 meV). The parameter x is taken as x = 3 and - 3 for the convex and the concave type of EDOS, respectively. The parameter a is varied as ~2m~,, 2~2m~ and 5~2m~, for each type of the EDOS. We take the value of the Coulomb pseudo-potential p* and its cut-off frequency coc as 0.15 and 500 meV, respectively. Now we turn to a discussion of the effect of the energy-dependent EDOS N(~) on quantities such as the critical temperature and the energy gap. In Fig. 3 the critical temperatures (Tc)Nt() calculated with both types of EDOS, N(e), are shown for four different values of 2 as a function of the parameter a; the upper four curves are those of the concave type EDOS and the lower four curves are of the convex type EDOS. We can see that the deviation of the critical temperatures (T¢)N(,) from (T¢)N(o) becomes significant as the parameter a decreases, and the magnitude of the deviation increases as the effective coupling strength 2 increases. In several band-structure calculations [ 13,14 ] for the high- T¢ cuprates, the EDOS changes its value considerably around the Fermi energy. Moreover, their calculations suggest that the width a of Lorentzian EDOS is a few times the maximum phonon frequency £2maxif we want to imitate the structure of the EDOS around the Fermi energy by the Lorentzian. For such a value of a, the deviation of ( T¢)~vto from (Tc)~v(o) in our calculation is significant and amounts to nearly 20°/0 of (Tc)Nto) in the cases of model C and D. In Fig. 4 the normalized energy gap (Ao)N(()/(Ao)~vto) is shown

Y. Yokoya / Physica C231 (1994) 341-351

347

3

i

i

model

•

I

I

'

1 . 5 ~,

'

//

............ model A

-':,\\ ',\'\

. . . . .....

~'~"

~ ",~\'\, - \ -

- -

~"

i, " ' " -"'IS-7 ///"

1 .....

model B " model C model D ".

..................

hi \

, I

i

I

, ~,,,,.~

'\" ~

i model

, -

C

-

T/rc = 0.1 -.

I

'

I model' D

-"

T, To:O,

1

i

0"50

"-'~ 2 -

1

"

I",V

"

B

5

10 a / f2

max

5

0

100 co

(meV)

200

300

Fig. 4. The normalized energygap (,4o)Nt+>/(Ao)N
as a function of the parameter a. The x values are the same as those chosen in the (Tc)Nt+) curves in Fig. 3. The whole tendency of the deviation of (gO)N(+) from (A0)NtO) is very similar to that of the critical temperature in Fig. 3 except for the largest value of 2 (model D). As is clearly seen from the upper solid curve for the concave type of EDOS, the normalized energy gap (Ao)Nt+)/(Ao)N(O) for the model D has a smaller value than those of others in the small a region. The reason of this exceptional behavior for model D can be understood by looking into the behavior of the quasiparticle density of states in the normal state Nn(to) and in the superconducting state ~Ts(to). Here, the quasiparticle density of states in the normal state is obtained by solving the MSC equation while setting the value of the gap function equal to zero. The curves of the quasiparticle density of states of the normal state ~Tn(to) (dashed line) and that of the superconducting state ??s(to) (solid line) are drawn together for a comparative study in Fig. 5. Here, the dotted curves are the Lorentzian EDOS N(o9)IN(O) employed in the calculations. Since the mass enhancement of the electron becomes large as the electron-phonon coupling strength increases, the shoulder of the curve of the quasiparticle density of states in the normal state (dashed line) in the small to region moves toward the left as 2 increases. On the other hand, the value of the energy gap Ao increases contrarily as 2 increases in the case of the concave type EDOS N(e). Therefore once the dent of the curve of the quasiparticle density of states in the normal state goes inside the gap region, the value of Ao no longer increases as the half width a decreases. In Figs. 6 and 7, are shown curves of the critical temperature (T¢)N(+) and the energy gap (Ao)Nt+) for values of the parameter x = 1, 3, 100, - ½ , - ~ , - 2 -1oo ~ with the use of model B a s o t 2 F ( t o ) . From these figures we find expectedly that both the ratios (T¢) ~¢(,)/(To) U(0) and (Ao) N(+)/ (do)u(o) come to deviate strongly from 1 as the parameter x increases. From these analyses in (T¢)N(+) and (Zto)N(,), we can extract the conclusion that for a given value of T¢ and do, the electron-phonon coupling strength which realizes these T¢ and do values is smaller for the concave type EDOS and larger for the convex type EDOS than that expected for the usual constant EDOS theory. Massidda et al. [ 13 ] have calculated the band structure for the

Y. Yokoya / Physica C 231 (1994) 341-351

348

'

1,41

,

~....1

~ \ \, ~ 1

r

x

,

-100/201

i

,

model B

'

I

'

model B

x=-100/201

'\\~'~

~ =-3/7

~.1.2 r,,,\,,~

i

1.5 ~ /

T~ Tc = 0.1

~=_1,3

~"

",,,\,~

x = -1/3

~.g1"0 f ~

...........................

• ' " ' •x = 100

~

o.%

1'o

15

a~ ~max

0 " 5 0~

'

x = 100

5i

,

1 0I

'

15

a~ ~max

Fig. 6. T h e n o r m a l i z e d critical t e m p e r a t u r e (T¢)~v(o/(T¢) Nto) as a f u n c t i o n o f the n o r m a l i z e d h a l f w i d t h a o f t h e L o r e n t z i a n e l e c t r o n i c d e n s i t y o f states a/12~x f o r v a l u e s o f the p a r a m e t e r x = l, 3, 100, - -~, - {, - ~ w i t h the use o f m o d e l B as ot 2F(o)).

Fig. 7. The n o r m a l i z e d energy gap (~Jo)~<,)/(~J0)N
Y-Ba-Cu-O. Their calculation seems to suggest that the structure of the EDOS near the Fermi energy resembles a concave-type Lorentzian EDOS of our case. If we estimate the half width a of the hollow structure at the Fermi energy given their band-structure calculation, it amounts to a few times the maximum phonon frequency t2m~x. In such a case, from the result of our calculations, the value of 2 is expected to be smaller than that of 2 from the usual Eliashberg theory with a constant EDOS.

4. Temperature dependence of the quasiparticle density of states

Having described the behavior of the energy gap and the critical temperature as a function of the half width a, now let us see the temperature dependence of the quasiparticle density of states in the superconducting state. In the following calculations, model B is used as ot2F(to). In Fig. 8 are drawn curves of the normalized quasiparticle density of states in the superconducting state 375(to)/ Nn(to) calculated at four temperatures, T/Tc=O.1 (solid line), 0.7 (dashed line), 0.85 (broken line) and 0.95 (dotted line), in the energy region near the gap edge. In Fig. 9 we show the parts of these curves exhibiting phonon structures by magnifying them. In both figures, curves are drawn in the top frame for the constant EDOS, in the middle one for the convex-type EDOS ( a = 212m~xand x = 3 ), and in the bottom one for the concave-type EDOS (a = 2Qma~ and x = - ~ ). As pointed out in Fig. 4, for the convex-type EDOS (middle frame) the position of the gap edge for the curve at T = 0.1 Tc (solid line) shifts toward the left relative to that for the constant EDOS (top frame) although the electron-phonon spectral function a2F(og) is the same in both cases. The reason is that the number of electrons interacting with phonons in the case of the convex-type EDOS is smaller than that in the case of the constant EDOS. On the contrary, the position of the gap edge for the concavetype EDOS (bottom frame) shifts toward the right relative to that for the constant EDOS because much more electrons are available to interact with phonons. In Fig. 8 it can be seen that the temperature dependence of the peak position differs from type to type of EDOS. In particular, for the concave-type EDOS the peak position of the curve at T = 0.7 Tc (dashed line ) shifts unusually to a higher energy region than that of the curve at T = 0.1 T~ (solid line) does. This behavior of the

Y. Yokoya /Physica C 231 (1994) 341-351 3

I

I~

-

I

I

I

~

I

I

I

constant

I

1.6~

EDOS

-

,r .,;.')t

1 .........,"/

.

,

' constant EDO'S

... 1.0

~0

I -

[

"~ 2_I~"

I

I

convex

I type

"

"....... :

0.8 1.4

I EDOS

-

~

--~:_--_-

I

......

~ I convex type EDOS

a = 2~qmax

X=3

0 a-'c-I"

/

-

2

o

,

1.4L~ k

2

1

349

I

I

I

I

,, k I

~

concave

\

a = 2'~max

I type

::>"

I EDOS

-

1.4

L "~

a = 212 m a x

..-;+,"

..... .-+" j , // - _++. / - ' - [ -- "f" ~ ]

I

,o

I

,,o

I

+'o

8'0 lOO

50

100

150

200

to (meV) to (meV) Fig. 8. The normalized quasiparticle density of states ~7~(to)/Nn (to) in the low-frequencyregion with the use of model B a s ot2F(to). All three frames are for temperatures T~ Tc= 0.1 (solid line ), 0.7 (dashed line ), 0.8 5 (broken line) and 0.9 5 (dotted line ). The top frame is for the constant electronic density of states, the middle frame for the convex type and the bottom frame for the concave type, respectively. Fig. 9. The normalized quasiparticle density of states ~7+(to)/Nn (to) in the phonon region with the use of model B as o?F(to). All three frames are for temperatures T/Tc=O.1 (solid line), 0.7 (dashed line), 0.85 (broken line) and 0.95 (dotted line)• The top frame is for the constant electronic density of states, the middle frame for the convex type and the bottom frame for the concave type, respectively. peak position differs from the result o f the usual Eliashberg theory with a constant EDOS. Next, by the curves in Fig. 9, we discuss the intensity o f the phonon structures in the tunneling density of states. By looking into the curves at T = 0.1 Tc (solid line), we find that the structures for the convex-type o f EDOS are most pronounced among all types o f EDOS, while the energy gap is the smallest for it. On the other hand, for the concave-type EDOS (bottom frame) the phonon structures are less pronounced, while the energy gap is the largest o f all types. In the usual Eliashberg theory, both the energy gap Ao and the strength of the phonon structures become large as 2 increases. The present result, however, seems to differ from this usual correlation between the size o f the energy gap Ao and the strength o f the phonon structures when 2 is increased. The origin of this different situation is traced back to the difference o f the effective number o f electrons interacting with the first harmonic phonons. The shape of the energy-dependent EDOS seems to affect also the temperature dependence o f the phonon structures. In particular, the temperature dependence o f the phonon structures is unique for the convex type o f EDOS (middle frame). For this type o f EDOS, the phonon structures in the quasiparticle density o f states are still seen near Tc (at T = 0 . 9 5 T c ) . On the other hand, the phonon structures already disappear at T = 0 . 7 T c for the other two types o f EDOS (top and bottom frame ). The result o f recent tunneling studies for high-T¢ cuprates [ 31,32 ] seems to be explained by such a characteristic situation for the concave-type EDOS. In experiments on these the intensity o f the phonon structures in the differential current-voltage characteristic is considerably weak although the energy gap is large. It would be necessary for the quantitative comparison with tunneling spectroscopy to consider the energy dependence o f the normal-state quasiparticle density of states in the calculation o f the tunneling rate.

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5. Conclusions In the framework o f the isotropic Eliashberg theory with the energy-dependent EDOS, the critical temperature and the energy gap were calculated for various values o f the electron-phonon coupling strength 2. It is found that the inclusion o f the energy dependence in the EDOS leads to a significant deviation o f the critical temperature Tc and the energy gap A0 from those o f the conventional constant EDOS theory when the width of the Lorentzian EDOS is small and when the effective electron-phonon coupling strength 2 becomes large. The value o f 2 determined from the conventional Eliashberg theory with the constant EDOS is expected to be modified by including the energy-dependent EDOS as follows: The value of 2 for the concave-type EDOS becomes larger than that from the constant EDOS approximation. On the other hand, the value o f 2 for the convex-type EDOS becomes smaller than that from the constant EDOS approximation. From a study o f p h o n o n structures in the quasiparticle density o f states calculated with the energy-dependent EDOS, it is revealed that the behavior o f the p h o n o n structures strongly depends on the shape o f the EDOS. The size o f the energy gap and the intensity o f phonon structures does not necessarily correlate, although that is the case in the constant EDOS Eliashberg theory: For the convex-type EDOS, the intensity of p h o n o n structures increases while the energy gap A0 decreases compared with that for the constant EDOS. On the other hand, for the concave-type EDOS, the intensity of p h o n o n structures decreases while the energy gap Ao increases from that for the constant EDOS. The feature of the smearing o f phonon structures at finite temperature is quite different for the concave- and the convex-type Lorentzian EDOS. In particular, it is noticed that remnants o f phonon structures are still seen near T¢ for the convex-type EDOS although they are not for the concave-type EDOS. To summarize, the inclusion of the energy dependence of the EDOS in the isotropic Eliashberg theory is expected to give noticeable effects if we want to calculate various physical quantities of materials where the electron-phonon coupling is strong and the energy variation o f the electronic density o f states occurs on the scale o f phonon energies.

Acknowledgements The author would like to thank Y. Shiina for valuable discussions and Y. Oi N a k a m u r a for a careful reading o f the manuscript and helpful comments.

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