Van Hove excitons and high-Tc superconductivity VII

PflYSICA

Physica C 183 (1991) 303-318 North-Holland

Van Hove excitons and high-Tc superconductivity VII Gap equation with pair breaking R.S. Markiewicz

PhysicsDepartmentand BarnettInstitute, NortheasternUniversity,Boston,MA 02115, USA Received 17 May 1991 Revised manuscript received 9 August 1991

Detailed calculations are presented for the superconducting gap and critical temperature of a van Hove superconductor. The calculations are modified to account for a number of complicating factors, including interlayer coupling, strong coupling, and •depairing effects due to real phonon scattering and spin scattering. The calculations are compared to experimental data on the isotope effect, spin-lattice relaxation time Tt, and magnetic penetration depth, and satisfactoryagreement is found. Power law corrections, particularlyin T~, are suggestiveof a gap zero.

1. Introduction T¢ = 2~a e x p ( - N / + There have been a number of suggestions [ 1-4 ] that high-T~ superconductivity is associated with the van Hove singularity (vHs) of the two-dimensional (2D) CuO2 planes. In an ideal 2D band, the densityof-states (DOS) has a logarithmic singularity when the Fermi level coincides with the vHs. This singularity modifies the form of the BCS equation for Tc [ 1,2,4,5 ], greatly enhancing Tc above the typical BCS values for constant-DOS materials, without requiring a large electron-phonon coupling parameter 2. In the ideal 2D case, T¢ is found from the BCS equation htoc

vl f-N ( E )

t a n h ( 2 ~ - ~ ) d-if, E

(1)

o

where V is the electron-phonon interaction potential, o9~ is a phonon frequency cutoff, and N(E) is the DOS, with E = 0 at the vHs. If the Fermi level is exactly at the vHs, then

N(E) =Noln(B/E) ,

(2)

where B is the effective bandwidth, and No is a normalization constant. In this case, an explicit formula for Tc can be found [ 1,4 ] if the hyperbolic tangent is appr9ximated by two straight lines, tanh(x) = m i n ( x , 1 ):

+ In2 ( ~ - ~ ¢ ) - 1) .

(3)

When 2o=No V is small, this has the approximate form Tc~B e x p ( - ~ ) l / 2 ) . This should be contrasted with the typical BCS result T¢~hwc e x p ( - 1 / 2 ) . The large prefactor (B~;>hogc) and weakened dependence on 2 combine to produce a significant Tc enhancement, even though the average 2 is small. Tsuei et al. [4] give an example, to represent YBa2Cu307_~ (YBCO): assuming 2o=0.12, B=5800 K, hoJ~=754 K, eq. (3) yields Tc=92 K. This should be compared to the BCS value for To, using the same parameters, but with a value of 2 averaged over the phonon energy range: htoe

2av=2O ~ l n ( B ) ~ d E =2oln h ~

=0.36 ,

(4)

o

for the present parameters. Plugging this value into the standard BCS equation gives To---53 K, due to the large value assumed for htoc. Thus, the vHs provides an additional enhancement of T~ by ~ a factor of two. The same model can also explain the anomalously small isotope effect and its strong doping dependence [4]. It is rather remarkable that such a simple treatment seems to describe the superconducting prop-

0921-4534/91/$03.50 © 1991 ElsevierScience Publishers B.V. All rights reserved.

304

R.S. Markiewicz / Van Hove excitons and high-To VII

erties of the high-To oxides, since their normal state properties appear to be a great deal more complicated. There should be very strong electron-electron scattering near the vHs, as well as a strong competition between superconductivity and charge or spin density wave ( C / S D W ) formation [3,6]. There is evidence that the carriers can be divided into two groups. Those carriers near the Brillouin zone boundary constitute the DOS peak, and are quasi-localized, while carriers in other parts of the Brillouin zone have a low DOS and are relatively weakly scattered (mostly by the quasi-localized carriers), and are the carriers observed in transport, optical conductivity, and Hall effect measurements. The quasilocalization of the vHs holes can give rise to the marginal Fermi liquid scenario [7,8], in which the renormalization factor Z--, 0. Hence, it might be supposed that the superconducting transition is associated with the condensation of the low-density holes, perhaps paired by slow fluctuations of the vHs holephonon system (CDW fluctuations) [ 6-9 ]. Such features cannot be properly incorporated into a simple BCS-like treatment of superconductivity, so in this paper, a Green's function approach [ 10] is introduced, which can treat such complications. However, it was found that even in the simplified picture of a single, logarithmic DOS, the predictions of the present model are quantitatively (but not qualitatively) very different from those of the previous calculations [ 1,2,4,5 ], based on the weak-coupling BCS formalism. As shown below, this is because the large values of Tc require a strong coupling formalism. This paper introduces a simple Ansatz for calculations in the strong coupling regime. Because of the many novel features of a van Hove superconductor, this paper is restricted to the simplified model, with the single-layer DOS given by eq. (2). Complications due to the quasilocalization will be ignored, and the renormalization factor will be set to 1. Within this restricted model, this paper discusses how the vHs modifies the superconducting state: how T~ is enhanced; how A (T) is modified; can the vHs model describe such features as the T-dependence of the London penetration depth, the isotope effect, and the nuclear relaxation rate. In analyzing these properties, it is necessary to see how they are modified by factors which can smear out the vHs, such as real scattering and interlayer coupling.

The organization of this paper is as follows: the Green's function formalism is described in section 2, and the modifications due to pairbreaking by real scattering in section 3. Section 3 also presents a calculation of the gap, A(T), while in section 4, several other properties are calculated, in particular the nuclear relaxation rate, T i- ~, the penetration depth, and the isotope effect. In section 5, the role of electronelectron scattering and interlayer coupling are discussed; section 6 compares the predictions of the model to experiment; and conclusions are discussed in section 7.

2. Strong coupling correction In this section, a Green's function calculation [ 10 ] of Tc will be introduced, to allow the incorporation of pairbreaking effects. This formalism reveals a shortcoming of previous T~ calculations, which persists even in the absence of pairbreaking. The use of the weak-coupling BCS-like equations ( 1 ) - (3) has been justified by noting that the phonon energy is small compared to the electronic bandwidth, ho9~<< B. However, in a Green's function calculation [ 10], it is clear that weak coupling holds when 2xkBT~<< ho9~, which is not satisfied in the present problem. Thus, even though 2 < 1, the system is in the strong-coupling limit. Here, an Ansatz will be introduced to modify the Green's function calculation so that it agrees with the exact strong-coupling result, in the absence of pairbreaking. For constant DOS, in the absence of pairbreaking, T~ is found as the solution of

1

~ 2nT~ = ~ = o ~o~ '

(5)

where h t o , = 2 n ( n + ll2)kBTc, tO,o=tOc. If Tc<
R.S. Markiewicz / Van Hove excitons and high-To VII

1 1 + 1.52 ;tort -2

40( (6)

1 1.5 + - 2 1 + r/2 '

1

--

~, 2xf. Tc

/, n=0

(K) 7'

zoc

//

I

4/_

1-

~¢ L\°/I 0

°<

150 30C

1

1,

4

8

T¢ (K)

X 12

Fig. 1. Effect of pairbreaking on T¢ in a conventional (constant density-of-states) strong-coupling superconductor. Solid lines = eqs. (8), ( 13 ); dashed line-Kresin's result [ 1i ]. Inset: ratio of T¢ for a van Hove superconductor to T~, the transition temperature of a conventional superconductor with the same value of Aav.Calculations assume eoc=754 K.

C-On

with f~ = 1 / ( 1 + (o9,/O9c) 2 ). In this case, the BCS result would still follow for Tc << ~o¢, ;%fr=2. However, for large Tc, only the first term in the sum would contribute, leading to Tc ~ FOgcx/~, F=

,/'2"A

O0

(S)

I ,'//

z'"

(7)

where q is an adjustible constant. The second correction involves the sharp cutoff at n = no in eq. ( 5 ) . The Eliashberg equations suggest a softer, q u a d r a t i c cutoff in frequency, o f the form:

~'eff

I

Tc

However, the second term leads to a m a x i m u m value for To whereas it is known that Tc increases without limit as 2 increases. Hence, the second t e r m must ultimately be cut off at sufficient large 2. The following cutoff will be used here:

1 2efr

305

V[2 ~x/(1 + 1.5/q)"

400 Tc(K)

(9)

E

~//l

///~'OOK

2oc

,.o[

-

,

,

// T~ o.sk

(10)

The correct strong coupling result [ 11 ] is given by eq. ( 9 ), with F = 0.18. By adjusting r/_~0.2, eqs. ( 7 ) ( 9 ) yield F = 0.15, a n d lead to a reasonable agreem e n t with Kresin's a p p r o x i m a t e analytic result [ 11 ] over the full range o f 2, as shown in fig. 1. The soft cutoff a n d the effective 2 will be a s s u m e d in the following. This m a y lead to some error when the D O S is not constant. However, the correction t e r m in eq. ( 7 ) depends only weakly on 4, so the uncertainty int r o d u c e d will be small c o m p a r e d to uncertainties in other parameters. F o r the van H o v e superconductor, the right-hand side o f eq. ( 8 ) m u s t be m u l t i p l i e d by ln(B/hog,). This equation can then be solved numerically, yielding the curves o f fig. 2. It can be seen that strong coupling corrections greatly reduce Tc - by approximately a factor o f 3 (inset). However, the high values o f Tc observed experimentally can still be o b t a i n e d with m o d e r a t e values o f 2 - the exact value o f 2 depends on the m a g n i t u d e o f the pairbreaking, but for a 90 K superconductor, a value 2av--- 1 is typical. It

,

I

_

4

Tc° ~'~ ~

-~m

0.06

°o

'

'Xo

Fig. 2. Effect of pairbreaking on T¢ in a van Hove superconductor. Solid lines = eq. ( 14); dashed line = weak coupling result, eq. (3). Inset shows ratio of Tes from eq. (14) (To) and eq. (3) (T¢o), for no pairbreaking, c~= 0. Parameters are assumed to be toc= 754 K, B= 5800 K. is o f some interest to note that the strong-coupling result corresponding to eq. ( 9 ) is

To

( l n ( B / ~ T c ) )l/2

~ro~v~

(ll)

3. Palrbreaking by real scattering 3.1. Comparison with previous work At the high temperatures at which these materials go superconducting, there is considerable scattering

306

R.S. Markiewicz / Van Hove excitons and high-T~ VII

by real phonons, as well as carrier-carrier scattering. These processes act to reduce T~. This reduction is independent of the mechanism of generating the large T~ values, and it has recently been estimated both for ordinary BCS superconductors [ 12,13 ] and for superconductivity associated with the "phenomenological polarizability" (PP) [ 9]. In this section, the analogous calculations will be carried out for a vHs superconductor. It is most interesting to compare the PP and the vHs superconductors. As discussed above, the quasilocalized carriers have been postulated to be the source of the phenomenological polarizability [ 7 ]. In this sense, the present calculation and the PP calculation are complementary: the present model treats all carriers on the same level, whereas the PP calculation [ 9 ] represents the opposed view, assuming that the preexisting PP acts to pair the low-DOS carriers. The correct answer should lie somewhere between. Two routes have been followed to calculate the effect of real phonon scattering on T~. One approach [ 12-14 ] uses the full Eliashberg equations [ 15 ], treating the scattering as giving rise to an imaginary part of the gap function A. The other approach [ 16 ] is also based on the Eliashberg equation, but it is noted that the scattering introduces a pairbreaking contribution into the equations for A, so that a formalism similar to that used for magnetic impurities [ 10 ] may be employed. This latter approach has regularly been used in the study of 2D localization effects [ 17 ], and in recent calculations on high-T~ materials [9 ], and will be followed here:

3.2. Pairbreaking For a constant DOS, a simple result is found in the weak-coupling limit [ 10]. In the presence of any kind of pairbreaking, T~ is reduced from its value T¢o in the absence of pairbreaking by a factor: In(T~o/Tc) =q/( ½+ y ) - g / ( ½) ,

(12)

where ~ is the digamma function, and y the pairbreaking parameter. This formula may be generalized to the strong-coupling regime by the Ansatz discussed above. Now T¢ is the solution to

1 2elf

-

Z

L

(13)

,=on+½+Y"

For electron-phonon (or electron-electron) scattering [16], y=ot/2nkBT~, with a=h/2z( 1 + 2 ) , and z the scattering rate. For YBCO, r can be estimated from the resistivity. The intrinsic resistivity is [ 18 ] pa=50 gf~-cm at T = 100 K, with pa=m/ne2z. Taking n ~ 6 × 102 ~c m - 3, rn _ 2too, with mo the free electron mass, then r(100 K)---2.4X10 -~4 s, so a ( 1 + 2 ) -~ 160 K. Since 2,v will be found below to be ~0.9, and r-~ocT, then a"~kaT, and this approximate value will be used in explicit calculations below. In fig. 1 are plotted curves of T~ versus ;t (eq. (13) ) for a variety of values of y. For the logarithmically singular DOS, the Tc reduction must be recalculated. Equation (13) is replaced by

1

~ ln(~B ~ , , + ½f~+ y '

-

n=o

(14)

\COn/

while the weak coupling result, eq. (12 ), becomes In(Too/To) =ln(B/ZnTc) [g/(½ + y ) -gt(½) 1 - [ 0 ( ½ + Y ) - 0 ( ½ )

],

(15)

where 0(x) =

I n ( n + ½) /-, n=O

n+X

The resulting curves of To(2) (eq. (14) ) are plotted in fig. 2. It is seen that the finite pairbreaking, y~0.56, leads to a further reduction of T¢. In the inset to fig. 1, it can be seen that, when T~ ~ 100 K, the vHs enhances Tc by a factor of ~ 2-3, over the situation with a constant DOS, but the same value of 2av. For smaller Tc the enhancement is larger. The full T-dependence of the gap function d can be calculated, following ref. [ 10 ]. Incorporating the logarithmic DOS and the strong coupling corrections, d is found from 1

2nT ~ 1 (_B'~. -

f~±

3

(16) 2

where u is the solution to A

(17)

R.S. Markiewicz / Van Hove excitons and high-Tc VII

and ( = ot/A. Figure 3 shows plots of A(T) for 20=0.3 and a variety of constant values of ot (solid lines). In reality, ot is strongly T-dependent, which should further enhance A at low temperatures. This is illustrated in fig. 3 by the dashed lines, which assume that the normal-state T dependence, ot = kBT, persists into the superconducting phase. However, real phonon scattering is known to be reduced in a superconductor (e.g., enhanced phonon thermal conductivity below To), so it is not clear that the linear decrease in t~ will persist into the superconducting state. Indeed, it was recently suggested that c~ocT 3 below Tc [ 19 ]. The dotdashed line in fig. 3 shows how A(T) is modified by this assumed T-dependence. The filled circles in fig. 3 show an approximate expression for A(0) (derived in the appendix): A=Be -~ ,

307

2O

i

a

2A

I

i

:200K

T '/'

T~

..5/// IC

i

I

2OO

i

Tc (K)

4OO

Fig. 4. 2d (0) ~kaTe as a function of To. Solid and dot-dashed lines: J ( 0 ) calculated from eq. ( 16); dashed lines: from eq. (18). For the dot-dashed lines, ct = kaT is assumed; for the other curves t~ is assumed constant. The dotted line is the weak-coupling BCS result.

(18) 4. Other properties

b=ln 2 - -~-,

4. I. Nuclear spin relaxation

c : ~-~ + ln2(~-~)- 2-125- ~ - ~ - ~-) •

The main motivation for introducing pairbreaking is the experimental observation [20] that the coherence peak is absent from the nuclear spin relaxation rate, T i-~. The explanation of the peak in T i- ~ was one of the early triumphs of BCS theory [21 ]. Now it is recognized that, because of the high values of To, there is strong pairbreaking due to electron scattering by real phonons, which suppresses the coherence peak in the oxide superconductors [ 9,12,13 ]. This same result follows for a van Hove superconductor. Figure 5 shows the frequency-dependent density-of-states, N(to), showing the opening of the superconducting gap. Here, N(og) is given by

Figure 4 compares this approximate form to the exact result, plotting values of 2A(O)/kBTc. As anticipated from the derivation, the approximate expression is valid when d << mc and ( < 1. Figure 4 also shows that the ratio 2d(0)/kaT~ is substantially enhanced over the BCS value of 3.53, particularly when the pairbreaking is of the form ot ~ kaT.

o. 4

[ B "~Imu

k ....

~ ~'~X.."'\ ',~" .a=OK

oo X

U(og)=Noln~;)

~ ,

(19)

where u is given by analytical continuation of eq. (17): 01

0

I

I

I

~ I'l

I

5O T (K) I00

Fig. 3. Gap function vs. temperature, for 2o= 0.3, and several fixed values of ot (solid lines), or for various 2 0, with a = kaT (dotted line) or t~ = 2kaT (dashed lines). Filled circles = eq. ( 18 ).

A =u(1-

1~_~)

•

(20)

From this density-of-states, the nuclear spin relaxation rate can be calculated:

R.S. Markiewicz I Van Hove excitons and high-T< VII

308 r

T

for the experimental observations. As illustrated in the insert to fig. 6, 7"1 has an activated temperature dependence, ,,, exp(z~/kaT), whereas experimentally, T i-' has a nearly power-law T-dependence [20,22]. Larger values of 2o provide some rounding of In( Tt ), but still disagree with experiment at low T; this data will be discussed further in section 6.

1

,°f

N(~)

4.2. Penetration depth

co (K) Fig. 5. Density-of-states N(o~) as a function of frequency, for a van Hove superconductor, assuming 20 = 0.4, a = 2kaT. Temperatures illustrated are 2, 4, 10,20, 40, 60, 80, 86, 88, 89, and 89.44 K. (T¢is 89.45 K).

Using this same formalism, a number of other properties of the superconductor may readily be calculated. The simplest electrodynamic property to calculate is the London penetration depth, 2L: c

2,- wpx/~, IO5

i

I

1.0 -

.
,llt,,~ ]

- -

.T i (T c) /

T,-'(Tc)

(22)

.#--.~/.

,o',t.~, "~'

-

4 n n e 2 / m , and

toE =

Q= ~, ln(iB~ f"

*~

I

where

I ,'fi

n=O

(23)

\ C O n / I + U 2"

r rc/r

0.5

0O0

'/J

/#

// I/" ,' I x o--o.?,,'J,/o.4 40 80 T (K)I2Q

Fig. 6. Spin-lattice relaxation rate, T F~(T), for a van Hove superconductor, assuming ot=kBT and 20=0.3 (dashed line) or 2 = 0.4 and a = 2kBT (solid line ) or a = 2kBT(T/T,) 2 (dot-dashed line). Inset: same data replotted as ln(T~ ) vs. TdT. Filled circles=data ofref. [21 ].

Figure 7 shows the calculated London penetration depth, 2L VS. T, plotted as a dashed line. The Maki formalism applies to dirty superconductors, so a correction is required to recover the clean-limit value, 2[. For this purpose, the simple result [23] is used: ~,2_~2 ,,, - , , ,

1+

h

(24) )

with h / z ~- 2kBT. This is plotted as a solid line in fig. 7. Because of the assumed T-dependence of or, the

oo

,do

10 5

2,,,

TF'oc J - ~ l n t ~ ) cosh-2 0

x[(im, l~u2)2+(im

1

2 ;

the results are shown in fig. 6. This figure should be compared with fig. 2 of ref. [ 13 ]: it can be seen that the suppression is similar to that of a BCS superconductor, but that strong suppression arises at considerably smaller values of 2,v for a vHs superconductor. Despite the absence of a coherence peak, the present model does not provide a complete explanation

,o

103 /

iOz ~" 0.0

I

0.5

T/T

c

I.O

Fig. 7. London penetration depth for a van Hove superconductor, assuming ot=2kaT and ,lo=0.4. Dashed line=dirty limit; solid line = clean limit.

R.S. Markiewicz/ VanHoveexcitonsand high-Tc VII dirty limit penetration depth vanishes as T ~ 0 , whereas the clean limit 2 L remains finite.

4.3. Isotope effect Experimentally, the isotope effect has been measured as a function of doping. In the simplest picture, this doping produces a shift of the Fermi level away from the vHs. To allow for this possibility, eq. (14) must be generalized to incorporate a shift 8E between the Fermi level and the vHs. If the DOS is N ( E ) =NOn(B~ (E+ BE) ), the frequency-dependent DOS becomes [ 24 ]: N(ogn) =

J to~+

n

-oo

1 - ~.,In(S

(')

.

(25)

n=o

)

~

f~

n+½+y'

dln(Tc) din(M) =

;t'= ( A - / z * ) / ( 1 + 2/~*+2/z*t(2) ) ,

dln(Tc) , ,~dln(T~) ½dln(toC) -rP d l - i ~ -

½F, -- flyF3 F,+F2 '

(27a)

where M is the isotope mass and

:

/"l-- n=O ~ In ~ + ~ ]

.=o

( toE '~ \~]n+½+y'

nq21~yy , f"

In the ordinary BCS formalism, the electronphonon interaction parameter ;t must be reduced by a renormalizedelectron-electron interaction parameter, #*. Kresin [ 11 ] showed that inclusion of/t* amounts to replacing 2 by 2', where

(26,

and the isotope effect slope a0 is found from

r2=

5. E x t e n s i o n s o f the m o d e l

5.1. Electron-electron scattering

8E 2

Thus, eq. (14) becomes:

ao--

tional to f l = - d l n ( a ) / d l n ( M ) allows for the possibility of such an isotope effect contribution. Note that since F,, F2, and/'3 are intrinsically positive, ao-< 0.5. Figure 8 shows that the isotope effect is qualitatively similar to that calculated in ref. [4]: ao has a minimum at BE=0, and increases relatively strongly for non-zero 5E. However, the rise in Oto is limited, and Oto cannot be larger than 0.5; moreover, there is not a sharp peak in ao at 8E-~tOc, and any peak in ao is relatively weak.

o~__~_~~ N(E)dE

-- Noln

2~r

309

and t(2), which is presented graphically by Kresin, is approximately t(2) ___1.6 e x p ( - 2 / 3 ) . Figure 9 is a plot of 2' versus 2 for a variety of values of p*. If /t* can be estimated for the vHs model (see below), then its effect can be approximately incorporated via this figure: by comparison to experiment, the calculations of sections 2 and 3 would yield a value for 2'; then fig. 9 allows extraction of the corresponding

(27b)

80'

(27c)

Tc (K)

I ~

x

[

x

x

~

0.4 t2o

and F3=½ ~__oln

~/d~+SE2

(n+l+y)2.

Usually, isotope effect calculations assume that the isotopic mass enters into the expression for Tc only through the Debye frequency, too, However, if the pairbreaking is due in part to scattering by real phonons, then there should also be an isotope effect associated with z, or oz. In eq. (27), the term propor-

0.2

\x "-,

[

I000

0.5

I "--2000 500~ 3.0 BE(K)

Fig. 8. Isotope effect coefficient ao, and critical temperature Tc vs. distance from Fermi level to vHs, BE, for several values of the pairbreaking slope p.

310

R.S. Markiewicz / Van Hove excitons and high-T¢ VII

)ke

.

in error. In reality, approximating the DOS by a single logarithm ignores the light-hole fraction of the carriers which appears to dominate the transport properties, and may be the carriers which directly condense into the superconducting state, coupled in part by slow fluctuations of the nearly localized van Hove holes. This could correspond to a negative value of/t*. Nevertheless, within the present simplified vHs picture, the present treatment of electron-electron scattering should be adequate.

.

0.2

0 Fig. 9. Reduced 2' vs. 3. parameter, for g*=0, 0.05, 0.10, 0.15, 0.20. (After ref. [11].)

electron-phonon parameter, 2. For example, if /t*-~0.15 is assumed, then i is approximately 80% larger than 2' in the range of interest. The logarithmic singularity of the vHs also serves to modify the effective value of/t*. Electron-electron interaction can be incorporated into eq. ( 1 ) by the substitution [25] V--, V - U~, where

v~ U~ = +./V'---~e 1 '

(28)

where V¢ is the unrenormalized electron-electron Coulomb repulsion, tom

E' --toe

tom is a high-frequency cutoff, and, for a constant D O S , / t - N ( 0 ) V¢>/t*-=N(0) U¢. If tOm is identified with B, then

X = ½Noln2(B/tO¢) •

(29)

This should be compared to the equivalent BCS result X = N , vln(B/tO¢), where, following eq. (4), N,v=Noln( eB/tO¢). Thus,

~m

X

Jirncs -- 2 ( X + 1 ) ' with X = In (B/tOc)= 2, for the parameters assumed above. Thus,/t* may be expected to be somewhat larger in a vHs superconductor. It must be stressed that it is here, in evaluating the electron-electron interaction, that the present, simplified version of the vHs model is most likely to be

5.2. Interlayer coupling The above discussion has been for an ideal two-dimensional system. However, in a real material, interlayer coupling acts to smear out the DOS divergence, and reduce To. The role of interlayer coupling has been discussed recently [ 26 ], and it was shown that there is no Tc reduction if T~> tz, the interlayer hopping energy. Moreover, the effective interlayer hopping in real materials is likely to be small enough to approximately satisfy this inequality. However, as T is reduced below T¢, a point will be reached at which T < tz. Below this temperature, the electrons will not sample the full DOS associated with the logarithmic singularity, and A(T) will saturate to a lower T = 0 value than predicted above, thereby reducing the predicted value of 2A (0)/T¢. The appropriate cut-offDOS may be calculated as in eq. (25), using N(E) from ref. [26]. If N ( E ) = ~N~, (Noln (B/E),

if E-< 2tz if E > 2tz,

(30)

with N~ = Noln ( eB / 2tz ) , then

N(to,)=No (ln( B )+ 2 [tan-'fz -~tan-led~]),~

(31)

0

with tz=2G/to,. Figure l0 shows the numerically calculated N(to,), along with some analytical approximations in the form

N(to,)m=Noln[B/(to~ + (2tJe)') I/m],

(32)

for m = 1, 1.5, 2. The approximation N(to,)L5 was used to generate the curves of 3( T, t~) shown in fig.

R.S. Markiewicz / Van Hove excitons and high-To VII

NoI X~. I-

m

0

I00 ~(K) 200

Fig. 10. Effective DOS, for finite t~=50 K. Solid line=N(co); dot-dashed l i n e = N ( E ) . Dashed lines are fits to eq. (32), for m=1,1.5,2.

40C A (K) 20C

'0

50 T (K) I00

Fig. 11. Gap ~J vs. T for finite interlayer coupling. All curves assume a=2kaTc; solid lines=2o=0.4, for various tz; dashed lines= tz= 800 K, for various 20.

11, illustrating how interlayer coupling suppresses superconductivity. Note that, once tz >> T¢, the shape of the z/ ( T) -curve is approximately independent of tz.

5.3. Additional corrections The vHs model is sufficiently complicated that even the above calculations omit a number of factors. This section will briefly enumerate some of the most important of these factors. In comparing the model to experiment, it should be kept in mind how these factors can modify the theoretical curves.

5.3.1. Anisotropy In the high-To cuprates, there is evidence for a smaller, BCS-like gap along the c-axis, and there is

311

also a suggestion of considerable in-plane gap anisotropy, at least in Bi-2212 [27]. Within the vHs model, such anisotropy would be expected [28]; however, it is not yet possible to calculate this anisotropy from first principles, or to rule out the possibility of a gap zero. As is well known, such a zero would have profound effects on many properties, changing temperature dependences from exponential to power law. Hence, it will be important to ascertain whether such power law corrections are required to explain the experimental data.

5.3.2. Renormalization The logarithmic singularity in the DOS signals that the theory has an infrared divergence, and that higher order corrections to the self energy actually make a comparable contribution to the terms retained. The role of such higher-order corrections has been studied in the related problem of one-dimensional metals [29], and Dzyaloshinskii [2] has applied parquet diagram and renormalization group calculations to the present problem. However, he uses an oversimplified band structure, which will lead to incorrect results. By including only nearest neighbor hopping, the vHs is constrained to fall at half filling of the CuO2 band. This leads to a square Fermi surface, with perfect nesting and hence a very strong density-wave instability. This density-wave instability would completely wipe out the Fermi surface, and eliminate the possibility of superconductivity. In reality, the vHs lies away from half filling, and nesting is reduced to such an extent that only short-range density-wave fluctuations are present, greatly enhancing the probability of a superconducting transition [6,30]. Once the possibility of a density-wave transition is reduced, the remaining role of renormalization is expected to be fairly minor. Typically, in the one-dimensional problem, the situation is as follows. There can be no phase transition at T > 0 K for a strictly one-dimensional metal; inclusion of fluctuation effects shifts Tc from the mean-field value to To=0, with significant short-range order below the meanfield To. Finally, inclusion of interchain coupling drives a three-dimensional transition at some finite temperature below the mean-field To. The present case is two-dimensional, and it is known that there can be a true (Kosterlitz-Thouless) superconducting transition in a two-dimensional system. Thus, the

312

R.S. Markiewicz/ Van Hoveexcitonsand high-To VII

transition temperature need not be greatly reduced from the mean-field value, even in the absence of interlayer coupling, and a weak interlayer coupling can further raise the transition temperature. Experiments on superconductor-insulator superlattices can shed some light on this problem. In sandwiches of Bi-2212: Bi-2201, a single cell of Bi-2212, separated by up to 5 layers of the low-T¢ Bi-2201 phase, has essentially the bulk T~ [ 31 ]. In YBCO-PrBCO superlattices, the single YBCO cell shows strong fluctuating superconductivity below 70 K, and zero resistance below 20 K [ 32 ]. In the latter case, Tc may be partly depressed due to pairbreaking from the PrBCO layers. In conclusion, the mean field calculation should be qualitatively correct, but there may be secondary modifications due to fluctuations, both superconducting and density-wave. One consequence of renormalization is that the logarithmic divergence will be cut off below some temperature T*. The analysis of section 6.4. (below) suggests that T*<< T~.

6. Comparison to experiment 6. I. Choice of parameters and energy gap

The adjustible parameters of the theory include the phonon frequency, oJc, electron-phonon coupling, 20, electron-electron coupling,/1", the bandwidth, B, the pairbreaking, a (T), and the interlayer coupling, tz. As discussed above, I will neglect/z*, assuming that it can be combined with 20 to produce an effective 2'. The above figures assumed a large value, tOc= 754 K, whereas theoretical estimates are generally smaller. Thus, Klein et al. [ 33 ] estimate a phonon frequency (tO2)1/2~430 K for YBCO, or 365 K for LSCO (calculated from tabulated values of ref. [33]). However, reducing o9c can be compensated for by increasing 2o, with only minor changes of the A(T) curve; smaller values of tOc will be included in the fits, below. Table 1 lists a representative set of parameters used, with resulting values for Tc, 2A(0)/ kaTc, and 2L(0). Above, I assumed the T-dependent pairbreaking parameter ct ~_kaT, based on normal-state transport properties. There is a caveat to using this data: the normal state transport appears to be associated with

a low hole density, not the large density expected for the full Fermi surface. While this can be understood in terms of the full vHs model, it is not clear that the scattering time extracted from the transport should be representative of the carders near the vHs (and hence yield the correct a ) . Nevertheless, the best fits to the data suggest that a is not much larger than this value. As mentioned above, a may have a stronger T-dependence within the superconducting state. However, the limited tests of a T3-dependence carried out (set b of table 1 ) suggest that this stronger T-dependence worsens agreement with the data, so most of the fits assume a linear T-dependence. Unfortunately, in attempting to compare the present model to experiment, it must be noted that the fundamental parameters of the high-T¢ materials are still not well understood. Thus, measured values of the gap range from 2A/kBTc~-2-10. The present model favors gap ratios considerably larger than the BCS ratio, 3.53, and hence I will use a value comparable to photoemission results for Bi-2212, of A_~25-30 meV, or R.j-2A/kBTc~-7-8 [34]. This value is in good agreement with recent tunneling data [ 35 ], A(0)-~ 23-32 meV. However, the recently reported optical gap [36] is only half as big, 2A=25 meV. This small optical gap could result from gap anisotropy [27,28]. The fitting strategy adopted here is to see what range of parameters can reproduce Ra, using 2 as a correction factor to give Tc = 90 K. From table 1, it can be seen that Ra is generally too large for a=2kBT~, but falls in the correct range for a=kBT¢. This is discussed further in section 7. A number of studies, including the recent optical results [ 36 ], find an anomalous T-dependence of A, with A remaining nearly constant until T--- T¢. As seen in fig. 3, the vHs model has the opposite effect, causing d to vary significantly at lower temperatures than expected from the BCS model. This experimental A (T) variation may not be representative- recall that d (0) disagrees with photoemission results. If real, it may be an effect of fluctuations (section 5.3) causing a depression of T~ below the mean-field value, but causing relatively small corrections at lower temperatures.

R.S. Markiewicz I Van Hove excitons and high- Tc VII

313

Table 1 Parameter sets Set #

toc

alT

tz

2o

Tc

2JIkaT¢

2L(0)

a b c d e f g h i i' j k 1 m n o

754 K 754 K 754 K 754 K 754 K 754 K 754 K 500 K 500 K 500 K 500 K 500 K 500 K 400 K 400 K 400 K

2 2(T]Tc) 2 2 2

0 0 800 K 800 K 0 400 K 800 K 0 0 200 K 400 K 800 K 1600 K 0 400K 800 K

0.4 0.4 0.65 0.8 0.3 0.4 0.5 0.5 0.4 0.45 0.5 0.65 1.3 0.45 0.6 0.9

89.45 K 89.45 K 86.2 K 103.2 K 88.95 K 94.7 K 90.5 K 84.0 K 89.25 K 90.15 K 87.4 K 84.2 K 93.05 K 83.57 K 84.17 K 88.65 K

10.3 10.3 8.75 8.8 7.7 6.6 6.35 10.5 8.2 7.3 6.85 6.6 6.6 8.5 7.15 7.0

970 A 970 A 1210 A 1250 A 910J~ 1040 A 1140/~ 1040 A 975 A 1020 A 1090 A 1200 A 1440 A 1000 A 1130,/~ 1275 ,~,

1 1 1

2 (T/Tc) 2

T f I =T,-,2 + b T 3 ,

6.2. Nuclear relaxation rate, T~ As discussed in section 4, the present model cannot reproduce the curvature in the In ( T~ ) versus 1/ T curves; inclusion of finite tz does not greatly improve the fit, fig. 12. M o n i e n a n d Pines [37,38] showed that T~ could be fit to a d-wave superconductivity. In agreement with this result, I find that a good fit to T~ ( T ) can be obtained by including an anomalous c o n t r i b u t i o n proportional to T 3. Figure 12 shows that the rate is well represented by the form

(33)

where the first term is given by eq. (21), above. The addivity of the two contributions to T i- i is plausible, since the vHs and the zero of A would fall at wellseparated parts of the Fermi surface [28 ]. The data in fig. 12 are plotted as In ( 1/ TI T) versus T / T c . In agreement with Barrett et al. [39], -this causes the experimental data to fall on a straight line. The theory does not produce such a line, but is approximately linear over the range of the data. 6.3. Isotope effect

io °,

//

TcTL(Tc) T T,(T)

/'

l//t" i

iJ ." /

I

tO-2

0.0

t

.'k/

0.5

T/T

c

1.0

Fig. 12. Nuclear relaxation time T~, for various data sets from table 1. Solid line = set a; dot-dashed line=b; dotted line = set d; long dashed line=set e; short dashed line=set h. Filled circles= data of ref. 121 ]; lines with cross-ties include a T3 contribution, from eq. (33), with b=8.8× 10-~ m s - 1 K - 3 .

In the high-To materials, the isotope effect is found to be small at the compositions of optimal Tc a n d to increase to values > 0.5 as the composition is varied to reduce Tc [40,41 ]. This variation has been interpreted as evidence for the vHs [4 ]. However, analysis of the LSCO a n d LBCO data [40] is complicated, because the largest values of ao may be associated with the incipient structural transition to a low-temperature tetragonal phase. The present calculations are in qualitative agreement with experiments on YBCO [41 ], in which T~ is reduced by substituting Pr for Y. The agreement is not quantitative, however. To match the experimental Oto-0.04 at optimal Tc requires fl~ 0.5, but then the increase in ao with de-

314

R.S. Markiewicz / Van Hove excitons and high-Tc VII

creasing Tc is too small. As Carbotte et al. [42 ] have shown, this could be due to magnetic pairbreaking, induced by the Pr. Indeed, it has been suggested that the Fermi surface is pinned at the vHs, so that ordinary hole doping (e.g., by oxygen depletion in YBCO) is brought about by microscopic phase separation [ 3, especially II, III ], with the second phase associated with the antiferromagnetic insulator. In this case, the T~ reduction is also due to magnetic pairbreaking, now associated with the second phase. Thus, both O-depletion and Pr-substitution would have nearly the same effect. The magnetic pairbreaker can be described by a modified form ofeq. (27): ½F, -

By,F3 (34)

a O - - El + F 2 _ y 2 F 3 ,

with y=yl q-y2 (additive pairbreaking), y~=a/ 2nkaTc is the real-scattering-induced pairbreaking discussed above, and ))2= oq/2nkaT~ represents the magnetic pairhreaking. It is assumed that ot~ increases with the decrease in hole content below the vHs, but is independent of M. To simulate Fermi surface pinning, it is assumed that ~ E = 0. Figure 13 shows that with these assumptions, the observed doping dependence of the isotope effect can be explained. The model leads to a number of predictions: ( l ) if magnetic pairbreaking is involved, the variation of other properties, such as d (T) and HEc, can be calculated as a function of doping with no additional parameters; (2) the variation of Oto with hole depletion should 1.0 s0

0.5

0.0~)

J

40

To(K)

80

Fig. 13. Isotopeeffectin Pr-dopedYBCO. Solid line derived from eq. (34), data from ref. [41 ].

be nearly the same for Pr substitution as for O depletion; but (3) there may be a totally different variation if Tc is reduced by hole overdoping, since in this case, the second phase appears to be an ordinary metal, and, as such, would not be expected to introduce extra magnetic pairbreaking.

6.4. Penetration depth In most penetration depth measurements, what is directly measured is the normalized temperature dependence, ~2=--2L( T)/~,L o - l, where 2L0 is '~.L evaluated at some reference temperature. The absolute values Of 2L(T) are then found by fitting the data to an assumed temperature dependence; these values are very sensitive to the assumed form of temperature dependence - see the discussion in Anlage et al. [43]. This can lead to substantial errors in ,~L(0), particularly since 2 L does not appear to follow a pure BCS T-dependence. In fitting the data, I have found that it is easy to obtain a good fit for ~2, but that the resulting absolute values, typified by ,~,L(0), are smaller than those reported in the literature. To obtain values Of AL(0) comparable to those reported, I have found it necessary to assume large values of tz. This makes sense: as tz increases, the DOS peak gets smeared out, leading to a more BCS-like temperature dependence (constant DOS); as the theoretical temperature dependence becomes closer to that assumed in the data analysis, so do the absolute values of 2 L. However, if a vHs-like temperature dependence had been assumed in analyzing the data, then the experimental value of 2 L(0) would be quite different. Moreover, a large value for & requires quite large values of the electron-phonon coupling to produce large Tc values (see table 1 ). Since there is no other reason for assuming such large values of tz, I prefer to fit only the T-dependent ~2, treating the experimental absolute magnitude as a free parameter. Nevertheless, it is clear that a parameter-free measurement Of AL(0) would provide an important constraint on the theory. In addition, the anisotropy of ~L must be accounted for. The materials show strong anisotropy between in-plane and c-axis properties - e.g., 2 c - 3 52ab in YBCO. The theory should agree best with 2ab , where 2ab(O ) is found [ 4 3 - 4 6 ] to b e _~1400 A;

R.S. Markiewicz / Van Hove excitons and high-To VII

However, in some instances the m e a s u r e d value should also be corrected for anisotropy [47]. The measurements o f ref. [44] are m a d e on rectangular plate-shaped samples, so the field penetrates a distance ;tab into the long faces, b u t a distance 2c into the short faces; hence the m e a s u r e d ~-L is a convolution o f 3.ab a n d 2c. Correcting for this a m o u n t s to adjusting ~,ab(O)--¢1180 A. This correction is incorp o r a t e d into the experimental curves, plotted in fig. 14. Figure 14 c o m p a r e s the m e a s u r e d [ 44 ] t e m p e r a ture d e p e n d e n c e o f the L o n d o n penetration d e p t h in the a,b-plane o f Y B C O with the predictions o f various d a t a sets o f table I. The quantity plotted is ~,~.-----,~L(T)--)~L(50 K ) , while the corresponding value of~lL(0 K) is listed in table 1. It can be seen that the vHs m o d e l generally provides a good app r o x i m a t i o n to the T-dependence o f ).L, b u t with smaller values o f )],L(0) than are found experimentally, unless some interlayer coupling is included in the fits. Taking aoc T 3 gives a worse fit, but this again depends on the absolute value o f ,~-LThe non-BCS features o f the present m o d e l should show up m o r e clearly at lower temperatures, since both the energy d e p e n d e n c e o f the D O S a n d the Td e p e n d e n c e o f a should lead to a stronger T-dependence of,~ L- F o r instance, the d a t a o f fig. 15 show a strong l o w - T t e m p e r a t u r e d e p e n d e n c e o f hE, which had been interpreted [43] as evidence o f either a

1°4

-

10 3

--

i

/. ," o

lo2~ ~

-"7" I

~ 0 ~ ~ 2 - -~o~"~o -

-'

,

, 0.7

,

,

i

i

i

0.8

i

i

i 0.9

i

i

i 1

T/Tc

Fig. 14. London penetration depth vs. T, for various data sets of table 1, with 52=3.L(T)--~.L(50 K). Diamonds=data of ref. [44 ]; dot-dashed line = set b; solid line = a (nearly the same for sets c, e, i,j, k); dashed line= 1.

10 z

315

I

I

I

J

I 5

L T c/T

'o10 0 J

10-z

5

Fig. 15. Low temperature London penetration depth ~L~-~L(T). Curves are all normalized to have the same value at Tc/T=2. Circles = data of ref. [ 43 ]; solid lines = set i of table 1 (nearly the same for sets a, e, h, l ); short dashed line = b; long dashed line = i'; dot-dashed line = set i, with power law correction ( eq. ( 35 ), with b ' = 0.05 J~). For all curves, 3-L0=2 L( T= 2 K), except the lower solid line, for which ~ . L 0 = ~ . L ( T= 18.6 K). small gap (Ra~-2.5) or o f zeroes in the gap. It can be seen that the theoretical curves (which are the same as in fig. 14) fit the d a t a at higher temperatures, but deviate systematically at lower temperatures. The nature o f the d e v i a t i o n is that the theoretical curves have a stronger T-dependence than the experimental curves. This is the opposite o f what would be expected for a BCS superconductor, where '~'L(T)--,~.L(0 ) falls exponentially with temperature, and arises both from the logarithmic energy dependence o f the D O S a n d from a ~ T (which are probably correlated). Hence, cutting off b o t h o f these Tdependences would i m p r o v e agreement with experiment. Figure 15 (lower solid line) shows that good agreement can be achieved simply by cutting off the T-dependence at T * = 1 8 . 6 K (i.e., letting ,~.L0----~,L(T*)). This a m o u n t s to increasing 2 L ( 0 ) by only 4 0 / k . Such a cutoff could be caused by higher o r d e r loga r i t h m i c corrections, as discussed in section 5.3.2, but within the present model, this cutoff can be generated by simultaneously eliminating the pairbreaking (letting a ~ T 3) and b r o a d e n i n g the D O S peak (t= # 0). However, fairly large t= values are required to p r o d u c e changes in the superconducting properties. This is because c-axis b r o a d e n i n g smears out the D O S over an interval 4t=, whereas the F e r m i function averages the D O S over an interval 4kBT, a n d superconductivity over ~ 2A ( T ) . Hence interlayer

316

R.S. Markiewicz / Van Hove excitons and high- T~ VII

coupling reduces superconductivity only when t~> Tc at high temperatures [26 ], or 2t=>A(0), at low temperatures. Hence, to produce sufficient DOS smearing requires a minimum t2=200 K (long dashed line in fig. 15 ). This can be compared with the calculated [48] value for YBCO, 4 t ~ 150 meV, or t=-~430 K. The agreement is reasonable, particularly when it is recalled that strong correlation effects can easily reduce LDA bandwidth calculations by a factor of ~ 2. (Note that t: is expected to be considerably larger in YBCO than in other superconductors, because of the c-axis path along the apical oxygens and the chains• ) In light of the fact that a T 3 correction was required to fit T, (T), it might be asked whether the present data are compatible with such a power-law correction. The dotdashed curve in fig. 15 incorporates such a term: 2L = 2 c ~ + b ' T 3 •

(35)

It is seen that this term can describe the low-T parts of the curve (eq. (35) should not be trusted too close to T¢). Hence, the present data cannot be used to rule out the possibility of a gap zero, although they do not seem to require it. Not~.. In a preliminary account [ 49 ] of these calculations, I showed that a good fit could be obtained to the earlier data of Anlage et al. [46], using the s a m e parameters as in fig. 14, without any low-Tcorrections. However, Anlage et al. [43] have discovered that their measured '],L values depend on the substrate on which their films are deposited. In particular, the data taken on films deposited on yttria stabilized zirconia (YSZ) (e.g., the data reported in ref. [46 ] ) are distorted because of the T-dependent dielectric constant of YSZ. Hence, the data analyzed here [43 ] should be more representative of the true 2 L of YBCO.

7. Results and discussion

Figure 16 is a scatter plot of the predictions of the various parameter sets of table 1 for R4 and 2L(0). The box shows the approximate experimental limits on these parameters. As discussed above, I do not put much faith in the absolute values of 2 L ( 0 ) . From

1500 I I

1400

I 1300v

12o0

! o k I

-

•

c

~

'.

g~n

ilOO

-

I000

-

900

', \ J\ \ f

.

4

".".

~

.

.

.

.

6

" \ ",

~-~

. ~rn

~

8 2A/kBTc

. . . .

h

"'"'"a I

10

,

,

,

L

12

Fig. 16. Scatter plot of fits to 2A(O)IkBTc and ~L(0). Letters refer to the data sets o f table l; box delineates approximate experimental range for the parameters. Lines indicate variation of parameters with tz for fixed values o f ~oc, a: solid line: ~oc= 754 K, a=kBT~; dot-dashed line: oJ~=500 K, a=kBTc; dashed oJ~=400 K, a = kBTc; dotted line: ~ c = 754 K, a=2kBTc.

line:

table 1 and the fits in figs. 12-15, the following conclusions can be tentatively made. ( l ) The nuclear relaxation rate and (possibly) the penetration depth seem to require anomalous T 3 corrections, suggestive of gap zeroes, which could follow from a gap anisotropy similar to, but larger than, that found in ref. [28]. (2) The pairbreaking parameter, a ~ k a T in the normal state, continues to fall of as a power law ~ k a T " ( n ~ 1-3) in the superconducting state. It also contributes significantly to the isotope effect, suggesting that it is at least partly electron-ph0non in origin. The power-law temperature-dependence is surprising, since one might have expected that a would be exponentially cut off below the gap [ 50 ]. It could be that this constitutes additional evidence for a gap zero. (3) The magnetic penetration depth suggests a value of tz ~ 200 K for YBCO, which is broadly consistent with theory [48]. This is small enough that only very low-Tproperties are affected ( T < ~ 20 K), and is likely to be smaller in other high-Tc materials with larger anisotropy. Hence, in most theoretical estimates, it should be adequate to assume t=~0. (4) The best theoretical estimates assume mc---400-500 K [ 33 ]. The present calculations cannot fix a value of o~c, since changes in ~Occan be com-

R.S. Markiewicz / Van Hove excitons and high-T~ VII

pensated by changes in ;to. Nevertheless, assuming toe=500 K leads to quite reasonable values for ;t,v= 1.4. This is to be c o m p a r e d to the L D A calculated value [ 51 ] ; t - 1 in YBCO. It is interesting to note that the calculated value o f ;t was greatly enhanced by the p r o x i m i t y o f the v H s to the F e r m i level. Thus the m a i n conclusion is that the superconducting properties o f the cuprates can be described within the v H s m o d e l using quite reasonable p a r a m eter values, o9~~ 500 K, ;tav ~ 1.4, a n d small t:. The near absence o f the coherence peak in T, can be u n d e r s t o o d as due to real pairbreaking. The detailed t e m p e r a t u r e - d e p e n d e n c e o f T~ is suggestive o f a gap zero. Large gap a n i s o t r o p y m a y contribute to a n u m ber o f other a n o m a l o u s features, including the small value observed for the gap both optically, a n d in Ram a n and tunneling measurements. This possibility will be explored in a future publication.

± ).eft

317

=ln~A) AI-A2,

(A2)

with A1-~ In ( 2 u c ) - ( ~ , A2 =A21 +A22 ,

A22 "" --<~

-- ~),

and

Uc

1

A21--

+

i

lnudu

I would like to t h a n k S. Sridhar, J.P. Franck, M.R. Beasley a n d C.P. Slichter for stimulating discussions o f their data. This work was s u p p o r t e d by the Dep a r t m e n t o f Energy under subcontract from Intermagnetics General Corporation, Publication 483 from the Barnett Institute.

Appendix: gap at T = 0 At T = 0, the gap equation, eq. ( 1 6 ) , b e c o m e s

Uc ;tell --

Ul

duln

(

"

)

Au(I-(/~)

×

~

1

(1+u2)3/2

,

(A1)

where ue= o)c/A,

{0,/ ul =

~2_ 1 ,

i._ l if~> 1 ,

and a sharp cutoff at u~ is assumed. This can be solved approximately, in the limit A << o)~ a n d ~ << 1, as

l+u2

1 -0.9552-0.1075+

Acknowledgements

[~/ 1 l]+;lnudu u

1 ~ln2uc.

In the above expressions, terms o f higher o r d e r than (2 were omitted. The numerical factors in the last line are the result o f numerical integration. Solving eq. ( A 2 ) f o r A yields eq. (18).

References [ 1] J. Labbe and J. Bok, Europhys. Lett. 3 (1987) 1225. [2] I.E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 93 (1987) 1487 [Sov. Phys.-JETP 66 (1987) 848]. [3] R.S. Markiewicz, Physica C 153-55 (1988) 1181; ibid., J. Phys. Condens. Matt. 1 (1989) 8911 (I); ibid., 8931 (II); ibid., 2 (1990) 665 (III). [4] C.C. Tsuei, D.M. Newns, C.C. Chi and P.C. Pattnaik, Phys. Rev. Lett. 65 (1990) 2724; C.C. Tsuei, Physica A 168 (1990) 238. [ 5 ] J.E. Hirsch and D.J. Scalapino, Phys. Rev. Len. 56 ( 1986 ) 2732. [6] R.S. Markiewicz, Physica C 169 (1990) 63 (V); ibid., Int. J. Mod. Phys. B 5 (1991) 2037. [7] R.S. Markiewicz, Physica C 168 (1990) i95 (IV). [8] C.M. Varma, P.B. Littlewood, S. Schmin-Rink, E. Abrahams and A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989) 1996. [9] Y. Kuroda and C.M. Varma, Phys. Rev. B 42 (1990) 8619. [10] K. Maki, in: Superconductivity, vol. 2, ed. R.D. Parks (Dekker, NY, 1969) p. 1035. [ 11 ] V.Z. Kresin, Phys. Lett. A 122 (1987) 434. [ 12] B.W. Statt, Phys. Rev. B 42 (1990) 6805. [ 13] P.B. Allen and D. Rainer, Nature 349 ( 1991 ) 396. [ 14] M. Fibich, Phys. Rev. Lett. 14 (1965) 561,621.

318

R.S. Markiewicz / Van Hove excitons and high-To VII

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