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Estimation of the influence of ionic dielectricity on the dynamic superconducting order parameter.

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Ann. Physik 7 (1998) 174-200
Estimation of the influence of ionic dielectricity
on the dynamic superconducting order parameter
M. Peter', M. Weger2, and L.P. Pitaevskii3
' DPMC, Geneva University, Geneva, Switzerland
The Racah Institute of Physics, The Hebrew University of Jerusalem, Israel
Department of Physics, The Technion, Haifa, Israel
Received 29 April 1998, accepted in final form 25 June 1998 by B. Muhlschlegel
Abstract. We consider the influence of an w-dependent ionic dielectric constant ~ ( w on
) the properties of a superconductor. Assuming that the pairing interaction is proportional to c2 we have
solved the Eliashberg equations for this case, both for imaginary and real frequencies. The interaction potential depends on a coupling constant J. and on a longitudinal phonon frequency Q. The dielectric constant is assumed to be independent of wavevector q, and to depend on frequency
where wlong,wvans are the frequencies
through the expression: ~ ( w =
) (0'- wtng)/(w2- w:,,,,),
of optical phonons of the dielectric. We find that along the imaginary frequency axis (but not for
real frequencies) the weighted phonon propagator can be modeled by an appropriate choice of a cutoff frequency and an effective coupling constant. The influence of ~ ( w on
) ,'"I the gap d(w), and the
renormalization function Z(w) are studied and it is found that these quantities increase significantly
with the dielectric constant.
Keywords: High temperature superconductivity; Eliashberg equations; Dielectricity
1 Introduction
The problem of the origin of high-temperature superconductivity is still one of the
most topical problems of condensed matter physics. The quest continues to find an
interaction strong enough to produce a superconducting phase transition at temperatures much higher than what is observed in "normal" metals. There are serious reasons to believe that antiferromagnetic exchange can indeed serve as a stronger pairing mechanism than the conventional phonon-mediated interaction [ 11. Moreover, if
an interaction is found which is sufficiently strongly coupled to the electrons, then
can strong coupling theory describe the superconducting state, or do fluctuations
make this approach ineffective, so that some new method of calculation has to be
used [2]? Can this method be handled with presently available computer power?
This problem was addressed by several groups for the case of a Hubbard Hamiltonian, where the pairing is due to a magnetic interaction. Specifically, a mean-field
approach with equations analogous to the Eliashberg equations, was considered in detail by Dahm and Tewordt [3] and by Schmalian, Grabowski et al. [4].The equations, analogous to the Eliashberg equations [5], are coupled non-linear integrd
equations, involving the superconducting gap function, and two renormalization func-
M. peter et al., Estimation
of the influence of ionic dielectricity
175
/
tions, one describing the frequency dependence of the self-energy, the second describing its momentum dependence [6]. In spite of the complexity of the problem, solution of these equations along the ,imaginary frequency axis is relatively
suaightfomard. However, many physical properties depend on the real frequency behavior of the gap and renormalization functions; solutions along the real-co axis are
much more difficult to obtain, because the kernel of the integral equations possesses
poles near the real-co axis. Nevertheless, Dahm and Tewordt [3] and Schmalian et al.
[4]were able to obtain detailed solutions, using ingenious algorithms. In the present
work, we consider a phonon-mediated interaction, rather than the magneticallymediated one. In the past, it was believed that such an interaction cannot give rise to
superconductivity above 20-30 K. Cuprates are more complex than “conventional”
superconductors in many ways, their near-antiferromagnetism being just one feature.
Most existing theories can be classified into 4 rather general categories:
(i) Theories which assume that the mean-field approximation is broken down completely. The degrees of freedom for charge and spin are usually separated (bosons
and spinons) [2].
(ii) Mean-Field theories based on the Nambu-Gorkov-Eliashberg formalism [5], but
with the interaction between the electrons being mediated by some magnetic interaction, such as paramagnons [3, 4, 71, or some Coulomb interaction, such as plasmons, excitons, etc.
(iii)Mean-Field theories based on the phonon-mediated interaction, just as the original BCS Nambu-Gorkov-Eliashberg formalism [5, 81.
(iv) Theories assuming a Bose-Einstein condensation, according to Schafroth. The interaction causes the formation of bosons at high temperature, and these bosons
undergo a Bose-Einstein condensation at the superconducting transition temperature. These bosons may be lattice or spin bipolarons [9].
Because of the complexity of observable phenomena, and the diversity of properties
of different substances, it is very difficult to build a scheme which takes into account
all aspects of the phenomenon, and, as a preliminary step, one can try to account for
different factors separately, having in view to estimate their relative importance. In
the present paper, we take into account one such effect. We present a model based
on a “nearly ferroelectric” scenario, which we believe fits the perovskites, and use
the Lyddane-Sachs-Teller picture for the dielectric constant ~ ( c o ) .This model yields
an interesting behavior for the functions Z(co), d(co), etc. which characterize the
superconducting state. The effect of a very large and highly-dispersive ionic dielectric constant was (to our knowledge) not considered before. This dielectric constant
modifies the Eliashberg equations in an essential way. We solve these modified equations along the imaginary-co axis, and extend the solutions to the real-co axis by a
method developed by Carbotte et al. [lo]. Because of the pole of the dielectric function near the real axis, we believe that this method is appropriate here. We investigate the influence of dispersion of the dielectric constant of the ionic background due
to the longitudinal and transverse optical phonons in the framework of the general
Eliashberg scheme.
While we attribute the pairing to phonons, we do not claim that the electron-electron interaction can be ignored. The electronic relaxation rate, obtained from a Drude
fit, is found to be: l/z=aw, where a is close to 1 (about 0.8 or so). This fit holds
Over a frequency range from about 30 meV to about 1 eV [ll]. Since phonon frequencies extend to about 90 meV only, this behavior can not be attributed to pho-
176
Ann. Physik 7 (1998)
1o 3
10-2
Wt
Fig. 1 The calculated electron-electron scattering rate as function of energy (from Ref. 13). Below
wmns x 20 meV, FLT applies. Above it, l l t , increases very rapidly and FLT breaks down
Inset: Experimental data of Timusk et al. [ 111 on YBa2Cu408 , slightly underdoped (T‘=84 K). At 85
K the curve is qualitatively similar to the calculated one, and starts to rise above about 200 cm-’. In the
superconducting state (T= 10 K), the cutoff increases by 2 A. At ambient, the cutoff at co,t
is smeared
out
nons. Thus the scattering process is electronic. Even at low frequencies, measurements of Bonn et al. [12] of the surface impedance in the superconducting state, are
not consistent with a relaxation process due to phonons.
Since the ionic dielectric constant eion((o) is very large at low frequencies,
O<O-,M
15 meV, and is small at higher frequencies, the Coulomb interaction is
screened-out for (o <coWms.
We show the electron-electron scattering rate T ~ ~ ( o ) - ’calculated by the Golden rule [13] in Fig. l. We see that there are two energy regions one below (ohns and one above, with a sharp, but continuous, transition in between.
The sharp drop in the relaxation rate as the frequency drops is indeed .observed experimentally - in Fig. 1 we show in the inset the results of the work of Timusk et al.
on YBa2Cu408 [ll].
2 Implication of a large ionic dielectric constant
We have recently explored the influence of an ionic dielectric constant due to ions
whose electrons do not directly participate in the superconducting state (e.g. the apex
oxygen, the alkaline earths, the rare earths, ions like Bi, Hg, etc.) on the strength of
superconducting pairing [6]. The mechanism considered here is the diminution of the
electronic shielding (by the conduction electrons) of the electron-phonon interaction
caused by the ionic dielectric constant, which “preempts” the electronic screening. It
is clear that the influence of dielectricity on superconductivity will be felt not only
through the weakening of the electronic shielding, but also through other channels,
M.peter et al., Estimation of the influence of ionic dielectricity
177
/
Fig. 2 a) The unit that may be responsible for the enormous ionic dielectric constant of the cuprates.
The apex oxygen is close to the chain copper, which is monovalent, i.e. with a 3d” configuration. The
low 3d” -+ 3d9 4s excitation energy causes a dz2-2pzhybridization of the oxygen orbitals. A slight motion of the Ba or Cu(1) ions may change this to the more common sp3 hybridization. Thus the lowfrequency phonons involving motion of the heavy Ba and Cu(1) atoms cause a very large apex-oxygen
polarization, giving rise to the enormous observed ionic dielectric constant. The same argument applies
when T1, Hg ions replace the Cu(1) , and Sr, Ca replace the Ba. b) The interaction between two charges
a distance r apart is very different in the region of the planar oxygen, and in the region of the planar
copper-apex oxygen, because of the extreme inhomogeneity of the material
but we have shown [14] that the diminution of the electronic shielding may be the
dominant effect, not automatically compensated through other contributions.
The role of the dielectric constant is far from obvious. Maksimov [I51 discusses
the issue of double-counting of the dielectric constant, which may give rise to entirely erroneous results. Allen and Cohen [I61 point out the need to consider the
screening of the “bare” electron-ion potential by the background dielectric constant
on the same footing as the screening of the electron-electron interaction. Here we
consider the possibility that the interaction between planar oxygen (and copper) ions
and the conduction electrons is shielded by the Ba (or Sr, Ca) atoms, in conjunction
with the neighboring (apex) oxygen which possesses a very high polarizability, as
well as rare-earth atoms such as La and their neighboring oxygens.
The ionic dielectric constant of the cuprates is very large, E~ z 30-50 with a dispersion at a rather low frequency (owans
% 15-25 meV) [ 171. This is the frequency of
low-lying transverse-optical (TO) modes, involving predominantly the motion of the
barium (or other alkaline earth) atoms. This very large polarizability is probably due
178
Ann. Physik 7 (1998)
to the apex oxygen O(4) in conjunction with the barium (or other alkaline earth ion),
and the chain copper (Cu(1)) or some other ion such as T1 or Hg, with a d” configuration (Fig. 2); the chain co per is probably mainly monovalent, with a 3d” configuration, as are Hg+2 and Tl+’; moreover, the d+s excitation energy of these ions is
very low. Therefore, the d” configuration may hybridize with the apex oxygen to
form dz2-2pzhybrids; motion of the Ba ion, or the d” ion, may change this to the
more “normal” sp3 hybrids, and this may give rise to the abnormally large polarizability. The “parent” material of the perovskites is SrTiO3, which is “almost77ferroelectric. The original work of the IBM group concentrated on this aspect [18]. We
are not concerned here with the microscopic mechanism responsible for the large polarizability [17a], but accept it as an experimental fact. We denote the center responsible for the large polarizability by: “Ba”. Cuprates without an apex oxygen such as
Nd-Ce-Cu-0 possess considerably lower values of T,. Our work should be applicable
to them as well, although we do not explicitly specify the unit responsible for the polarizability in such materials. Thus, we denote the dielectric constant that describes
the shielding of the interaction between the planar oxygen ions and the conduction
electrons by the “Ba” unit, ~ ~ - ~ q),
~ -whereas
~ ~ ( the
o ,interaction between two conduction electrons is shielded by the ionic dielectric constant Eel-Ba-el(W, q). Here, “Ba”
intervenes as an element otherwise exterior to the superconducting system. It provides for an additional interaction, not accounted-for in the analysis of a “conventional” system.
~ - ~ ~~ l - B arises
~ . ~ l from the different location
The large difference between E O - ~ and
in space of the interacting charges, in the extremely inhomogeneous material. The
first is in the vicinity of the planar oxygen, and the second is in the vicinity of the
planar copper-apex oxygen. We illustrate this in Fig. 2 b.
In such a model, we obtain a modified electric interaction of the form:
Whether the potential V ( o ,q)o-e will be diminished or strengthened by the dielectric
contribution from the “Ba” depends on the relative strength of the two dielectric
constants. When they are equal, the potential becomes weaker except at q=O, where
it retains the value ZV(E,) [19], E dropping out. When the two dielectric functions
are no longer the same, as a result of the inhomogeneity of the system, this is no
longer the case. In Ref. 14 we show that E O - ~in~ materials
- ~ ~
like the cuprates is considerably smaller than E ~ ~ - ~ ~ - ~ ~ .
The treatment of the inhomogeneous medium is in the spirit of the central cell correction of the Kohn-Luttinger theory of shallow donors in semiconductors [20]. In
that theory, space is divided into a region close to the donor (“central cell”) where
~ = l and
,
a more distant region, where the background dielectric constant E is the
average, measured dielectric constant of the insulating semiconductor. Our ~ ~ - is ~
analogous to the E in the central cell of the Kohn-Luttinger theory, and our ~ ~ ~ is analogous to the E of the distant region of that theory.
~ ~ ( o . the ionically-screened ThomasThe dielectric function E ~ ~ - ~ q~) - determines
Fermi cutoff:
~
~
~
~
-
~
d. peter et al., Estimation of the influence of ionic dielectricity
179
/
gTF becoming smaller means that V O - ~ ~ q( O) becomes
,
larger, particularly at small values
of q. The small value of E O - B ~prevents
-~~
the compensation of this increase. In this work,
we take this E to be 1. We discuss this estimate in detail in Ref. 14. In that paper, the
influence of the dielectric constant has been considered in the static limit, as a function
of k and k‘ (or of k-k‘ =q).The purpose of the present work is the discussion of the effect
of a dielectric constant where the q-dependence is neglected, but with the inclusion of
dynamic efsects. For small q-values, we approximate E ~ ~ - ~ ~ q)
- ~ by
~ ( its
o , q = 0 value
~(0).
Thus, we have simplified our model by assuming that the relevant interaction is
multiplied by a dielectric constant E(O) to be taken in the form suggested by Teller,
Sachs and Lyddane [21]:
ulongis the frequency of the longitudinal optic phonons (LO), which is about
80 meV in the cuprates.
We consider coupling with phonons of frequency 52. For the cuprates, 52 z
40meV [22]. The mode 52 involves vibrations of the Cu-0 bonds in the (Cu02),
planes.
For q = O , the potential Vo.el is thus multiplied by &(a).
Since the coupling constant
is proportional to V&el, we take a weighted phonon propagator (WPP) given by:
D,(O) = E 2 ( 0 )
0 2
O /2 I
- 52
(4)
We use the subscript “w” (weighted) to distinguish this WPP from the “bare” phonon
propagator DO(m)=Q/(w2-Q2). A is a dimensionless coupling constant, which is calculated, for example, in band-structure calculations [23]. We denote the static limit of
E, i.e. E(O), by e0.
3 Gap functions of imaginary frequency
In Fig. 3 a we show the WPP, as function of the imaginary frequency, plotted in units
of im/mIong for two values of c0, namely c0= 1 and E O = ~ . Also we chose ~ l , , , , ~ =
E ~ 1,
= and A = 1. The WPP for E O = ~ is much higher as 040 than for EO= 1 (KEO),
but narrower, as seen by comparison with the scaled e0= 1 curve.
We solved the Eliashberg equations [5] for this WPP. We show the solutions, i.e.
the gap d and renormalization function 2 as function of the imaginary frequency in
Fig. 3(b). All calculations are carried out for low temperatures (TKT,) (except for
the determination of T,, of course). The results are summarized in Table I (Sect. 5).
The ratio 2d(O)/T, is found to be close to that found by Carbotte (Ref. 10, Table l),
which is interesting in view of the different form of the potential. It reflects the fact
that potentials which can differ strongly along the real axis, look remarkably similar
for the Matsubara frequencies. From this imaginary axis calculation, which is simple
and well behaved, one sees that the &-functioncreates an anomalously large 2 at low
frequencies, while the critical temperature T, and gap d are not increased to the same
degree. This result is expected, since T,K&
(for very strong coupling) while
9
180
4
Ann. Physik 7 (1998)
3
2
1
1
2
3
4
1
2
3
4
lo--
4
3
2
1
b)
Z(0) M l+&; thus the increase in effective coupling brought about by E increases
Z(0) more than T, and d(0). We notice therefore that the enhanced Kernel influences
d(w) and Z(o) in different ways. Note that when the cutoff is in the k-channel (rather
than the o-channel), Z is decreased and d increased significantly [ 6 ] .Observation of
the ratios of d(o) to Z ( o ) can therefore be of interest in estimating the nature of the
interaction, and particularly the role of ~ ( o ) .
4 Gap functions of real frequency
We have calculated the functions d(w) and Z(o) and the tunneling density of states,
for real frequencies, by analytic continuation of the above functions for imaginW
frequencies. To do this, we have used the method due to Marsiglio, Schossmann, and
Carbotte [lo] (Fig. 4). In the case discussed by Carbotte, one series of poles has to
be subtracted at S Z + ~ - O(with
~ ~relative
~ ~ strength
~ ~ ~ 1). In our case, these poles take
M.peter et al., Estimation of the influence of ionic dielectricity
181
/
h e strength ~ ( 0 )In~addition,
.
we now have a series of quadratic poles at
utrans+i.u~atsu~ara.
The strength of these quadratic poles comes out to be:
m e extension to poles higher than quadratic would be straightforward which gives
considerable generality to our approach since we can now use this extended method
for any Kernel which can be satisfactorily described by a Padde approximation.
In addition, we have extended the method to functions which depend not only on
0, but also on the variables k and q, making in this way contact with our earlier
work where we discussed in some detail the dependence of the gap function on these
variables.
Our first calculation (Fig. 4a) is done with essentially the same parameters as used
previously by Marsiglio, Schossmann and Carbotte (Ref. 10, Fig. 1 therein). We obtain essentially the same results as these authors, which can be taken to be a calibration of our version of their method.
The upper curves show the real and imaginary part of the gap function, A =
A’+iA”.
Next follows the approximate single particle tunneling density of states N(E), given by
N ( E ) = N ( 0 ) . Re{E/(E2 - A 2 ( E ) ) i } .
(6)
w/Q
A’lQ
-6
6
I
-6
-4
-2
2
4
6
0
I
A(kr.w)
w/Q
182
Ann. Physik 7 (1998)
NO/N@F)
-a
-12
o
-4
a
4
12
Z’
~
-12
-a
10WIWt
0
-12
4
8
1
!&,A,
a
-a
2
-4
O
i
l2
WIOt
b)
4
8 -4--
Z’
-6
-4
-6
-4
)--y
-2
w/R
0
2
4
6
0
2
4
6
C)
Fig. 4 The functions A’ =Re[d(w)], A”=Im [d(w)],the quasiparticle density-of-states N(E)!
Z=Re[Z(w)l, Z”=Im[Z(w)], and the spectral density (A(kFw) along the real-w axis. a) c0= 1, 1=1$
T=0.01 R. b) &0=4,1= 1, wlong=R. T=O.Ol R
M. Peter et al., Estimation of the influence of ionic dielectricity
men is shown the renormalization function Z = Z'
Finally follows the spectral function given by
~ ( ko)~= ,- I m { o / z ( o ) [ 0 2
183
+ iZ" . We note that Z(0) = 1+1.
- ~~(o)]}/n
(7)
study the influence of dynamical dielectric functions on the superconducting gap
and on the related functions, we then proceeded to solve Eliashberg equations for different values of weans,wlong(which determine E ) , Q, ;Iand the integration step.
Fig. 4b uses the same parameters as Fig. 3a, full line and Fig. 3b, right. We can
see here the influence of a relatively high dielectric constant. The WPP is much higher than for E= 1, but, for our (physically reasonable) values of otrms,
along it is also
considerably narrower. Since the strength of the gap depends on the WPP over the
entire frequency range, we expect for the Lap function only a modest increase. We
have indeed found that the increase of E by a factor of 4 increases A(0) by the same
amount as an increase of ;1 by a factor of 2.6. If two sets of values lead to the same
&O), we call them equivalent values. The functions d(io) and Z ( i o ) for ~ = are
l
plotted on the left side of Fig. 3 b. Since, as noted previously, A and Z are affected in
different ways by E, it is of interest to compare the case of high E (Fig. 4b), with the
case of increased 2. Fig. 4 c shows therefore A and Z as function of real frequency
for ;1=2.6.
Comparison of Fig. 4 b and c is revealing. Both figures show a relatively smooth
underlying structure (positive in the center, with sign change at a distance of several
phonon energies), and a set of sharp peaks coming from the poles in the complex frequency plane. These two structures correspond to the main steps of Marsiglio's numerical method: First, find A and 2 along the imaginary axis. Next, extend this result
to the real axis by admitting poles situated symmetrically around the real w-axis. At
this stage, d and Z resemble our smooth underlying structure. The final step is the removal of the advanced poles, this brings on the sharp peaks. From this, it becomes
clear why the underlying structures in Fig. 4 b and c resemble each other, except for
the spacing of the phonon peaks which, in the case of high E , is given by the (rather
small) value of otrans.
Another interesting feature is the behavior of Z(o) in the two cases under discussion. In spite of their markedly different magnitude for the Matsubara frequencies,
they show again a very similar underlying structure. In the real frequency domain,
the large value of Z is limited to the frequencies below wnmS.
As to the physical meaning of this calculation, it might consist of a reminder that
the superconducting gap equations depend on integrals over all phase space. Therefore, even prominent features like a double pole can not strongly alter the underlying
structure of the gap functions. Sharp features of the interaction potential and of the
propagators will add sharp features to the gap functions that could be observed with
spectroscopic methods, for example. But the thermodynamical properties, with above
all the critical temperature, will depend mostly on the overall strength of the interaction, rather than on their specific form, since they are obtained from the gap functions along the imaginary Matsubara frequencies.
TO
184
Ann. Physik 7 (1998)
5 Similar WPP’s for different E(W)’S
If two WPP’s have the same value at o = O , we call them similar. In Fig. 4 we present the result of a calculation for two similar WPP’s, with different values of E and
different widths in co space. Their relative shapes are as in Fig. 3 a but their heights
are scaled to be one. The case of I = 1 and E = 1 leads to Fig. 4a; in Fig. 5 we show
the case of I=0.07 and ~ = 4 .Comparing the two figures shows that the shapes of
the two curves are similar but scale with mmans.From Table 1, which summarizes our
numerical results, we see that the critical temperatures are respectively, T,=53.4 and
T,=20.6. So we see that in this pair of similar interactions, the one with I = 1 has a
higher T, than the one with I=0.07, ~ = 4 If. we compare, however, a pair with
1=0.07 and E = 1 versus I=O.O7 and ~ = we
4 find that E is very efficient in raising T,
by a factor of nearly lo5.
Z’
w/Q
0
1
2
3
4
5
6
&Q
0
1
2
3
4
5
6
Fig. 5 Red, I d , N(E), ReZ, I d , A
along the real-o axis. 1=0.07,&=4
Fig. 6 The WPP (full
line) and its Lorentzian fit
(dots) for a case of very
strong coupling; for the
WPP, 1= 1, &()= 5.7,
olong=f2.
For the Lorentzian fit, l e f f = 3 2 ,
lo
ia’A%z
Ao,12=0.25Q
M. Peter et al., Estimation
185
of the influence of ionic dielectricity
/
E/ut
N Q N E r)
Fig. 7 Red, Imd, N(E),
ReZ, ImZ and A for the WPP
of Fig. 6 a) and its Lorentzian fit b), along the real-w
axis
-15
-12
-9
-6
-3
0
3
6
9
12
15
-15
-12
-9
-6
-3
0
3
6
9
12
15
186
Ann. Physik 7 (1998)
w/n
2
3
4
5
2
3
4
5
2t1
N@)/N(EF)
Lj.
-5
-4
-3
-2
-1
0
&.
1
E/n
2
3
4
Z'
-5
Z"
5
w/R
-4
-3
-2
-1
0
1
2
6
2
0
I
1
3
4
5
w/n
2
3
4
5
If two sets of values are both similar and equivalent (Sect. 4), then we call them
matching sets.
Figs. 6 , 7 show two such matching sets. Not surprisingly they have very similar
WPP's. The values of A as function of w/w,,,, are also similar, as are values of T,.
The peaks due to the poles are situated differently, of course. And, there is again the
difference in the behavior of Z(0). As before, Z(o) shows a high peak around o = O .
Finally, in Fig. 8 we show a case where the three frequencies in our model are all
different:
There are now more poles in the calculation, and one is moving away from the
original standard case while still conserving a relatively simple underlying structure.
Note the large value of Z (Z'(0)=3) in spite of the very small value of A(A=0.2). A
further important step is the replacement of the Einstein spectrum by a more realistic
frequency distribution. We have done some calculation for this situation also and verified the expected smoothing.
Table 1 gives a summary of our calculations. The values of 2d(O)/T, are reasonable throughout. They increase with increasing effective coupling. In the last column
we give (for the cases with E = 1) the values of T, from the Kresin-Gutfreund-Little
formula [24] :
T, = 0.18 Q J A / ( 1
+2 . 6 ~ ~ )
187
M.peter et al., Estimation of the influence of ionic dielectricity
N
t
*+
+
I
o.
0.4
0
I
0
20
40
60
80
Fig. 9 The phonon density-of-states F(mph) (left) possessing large peaks at mvans( z 15 meV), Q
( w 3 5 4 0 mev), and miong( X 7 0 mev), and the Eliashberg function a* (Wph) F ( m p h ) (right), with a
large peak at Q (from Ref. 22)
Table 1
Fig.
SZ (meV) 1
3a
40
40
40
40
40
40
40
10
40
10
40
40
3b (left)
3b (right)
4a
4b
4c
5
6 (dots)
6 (line)
7a
7b
8
1
2.6
1
1
1
2.6
0.07
16
0.5
16
0.5
0.2
&
T,(K)
A (mev)
53.4
93
118.1
20.6
9.55
29
27.2
37.8
83.5
71.9
53.8
25
21.7
14
Z(0)
2A/Tc
T'-KGL(K)
1.93
8.75
3.03
2.04
4.15
7.43
5.34
4.25
83.5
134.5
-
7.8
18.0
3.66
6.94
7.01
6.03
83.5
-
1
1
4
1
4
I
4
1
5.6
1
5.6
4
6 Discussion
6.1 The role of the ionic dielectric constant
A high
E,
generally present in perovskites, can contribute to a high T,, by shielding
p*. At low values of A, the large E changes the scenario from weak coupling to effec-
tive strong coupling, and in this way the increase in T, is very dramatic. This is in
line with the suggestion of Marvin Cohen in 1964 regarding doped SrTiO3 [25], and
the work of Baratoff, Binnig, Bednorz and Miiller around 1980 [18]. When A is large
to start with, the effect of a large E is not so large, but there is still an enhancement
of T, by a factor of 2 to 3. This enhancement does not take place in a homogeneous
medium, but requires a very large inhomogeneity with several highly different dielectric constants characterizing the ionic screening of electron-ion and electron-electron
interactions.
Considering the rather 'wild' shape of our effective interaction, with a double pole
at otrans,
it is remarkable that we obtain reasonable results for the gap functions.
188
Ann. Physik 7 (1998)
A double pole is not common in physics; however, there is no basic physical principle that prohibits its occurrence. We are aware of one case in which a double pole
has been considered in plasma theory by V. Silin [26].
Our model calculation allows to make small changes of the interaction along the
Matsubara frequencies and to appreciate the important consequences for such properties as tunneling and spectroscopy, particularly at low temperatures.
6.1.1 The double pole
In the present theory, two phonon branches play a major role, namely 52 and otrans.
If we had to deal with simple poles only, we could describe the scenario by the “ordinary” Eliashberg equations, with effective coupling constants for phonon modes at
frequencies muan, and 52. For the sake of an argument, if we would use a phonon
propagator given by: e(w).D0(o),then, for the special case along = 52, the effective
is given by: l , ~ =52/cotrm,,
l.
and the effective
coupling constant for the mode cot,,,
coupling constant for the mode 52 is zero. (If along > 52, then the effective coupling
constant for the mode 52 turns out to be negative, while the one for the mode awns
is larger). In any case, we would still deal with the standard well-known solutions of
the Eliashberg equations. Since we deal here with second-order poles at o=cotra,,,
this is no longer the case, and our solutions are essentially diferent from the standard, well known ones.
There is experimental evidence, from infra-red measurements and some microwave
measurements, that the dielectric constant of the cuprates (and organic metals) is insindeed anomalously large (30-60)(171. Optical investigation of Y 1-xPrxBa2C~307
gle domain crystals shows optical reflection data putting into evidence phonon
modes. But it turns out that these data are obtained from ordinary (Al-oxide) grown
crucibles, and that the much cleaner zirconate crucibles of A. Erb [27] produce crystals which show a more metallic approach. These results show that the problem is
complex, that the model with a STL dielectric constant makes some experimental
sense, but also that the manifestation of such a dielectric constant is strongly impurity-dependent. This poses the problem of why very different forms of optical conductivity curves appear in samples of similar structure and similar T,. [Normally,
Y 1-xPrxBa2C~307
is nearly insulating. Exceptionally-well prepared crystals of this
material are metallic and superconducting. Probably the insulating nature of imperfect
crystals is due to trivalent Pr replacing divalent Ba and, vice versa, the charged disorder destroying the metallic state, since its distance from the (Cu02), plane is less
than the Debye screening length pTF-’ z 4 A. We believe that this supports our contention that the superconductivity is associated with an inherently metallic state; and
that cuprates differ from “normal” metals mainly by their abnormally large screening
length.]
Low-energy spectroscopies are particularly relevant to the issues treated in this
work. They involve neutron diffraction, tunneling, ARPES measurements, Raman
spectroscopy, etc., and transport properties. Some such measurements on organic
superconductors, suggesting a particularly important role of a 2 meV phonon [28],
are especially intriguing. For these substances also, calculations must be carried out
for real frequencies, since for the Matsubara frequencies (and hence for the thermodynamic functions), interactions of different shape give rise to remarkably similar results.
189
M.Peter et al., Estimation of the influence of ionic dielectricity
6.2 The large value of Z ( 0 )
we find that Z(o) is extremely large for very low frequencies, o<ouans.
This large renormalization means that the band is extremely flat at the Fermi level. The group velocity
at the Fermi level is given by: V = v/Z(O),where v is the unrenormalized group velocity, given by band-structure calculations. The range in k over which the renormalized
bands are flat, is given by ouans/V. Thus Ak=o,,,Z(O)/v, and A M B z x o,,Z(O)/AE
where A E = v . k B Z is the half-bandwidth. Z ( 0 ) could be 20 (Fig. 7b), and
waans z 15 meV, thus Z(0)muansis about 0.3 eV. Since the unrenormalized half-width
of the band, in the “comer” region of the Fermi surface, is about 0.5 eV, this means that
the band is flat over about half the Brillouin zone. This is indeed observed experimentally, and denoted as the “Extended Van Hove Singularity”. The “conventional” interpretation of the EVHS is that it is due to renormalization arising from the electron-elecwon (rather than electron-phonon) interactions [29]. At the present moment, we do not
know how to distinguish between these two causes of the extreme band flattening.
Since Z(o) falls sharply at o=ouans,
the slope of the E vs. k curve there becomes
very steep, and the group velocity there is very large. This is in agreement with the
curves found by ARPES.
6.3 The question of high T,
In the present picture, the maximum electron-phonon matrix element I,,, diver es as
the dressed Thomas-Fermi screening parameter i j becomes
~ ~
small: Z,,,=27Ze
/@TF.
The average (Z2) entering McMillan’s expression for the coupling constant:
A= ( I 2 ) N ( E ~ ) / M Qis2given approximately by I&,, multiplied by the area on the FS
where I ( q ) is large; in 3-D, this area is given by n&, thus ( I 2 )does not diverge. In
2-D, the area is given by: 4 ~ ~ ( 2 7 c / cthus
) , ( I 2 )diverges like 1 / 4 for
~ ~small ~ T F
This gives rise to an anomalously large T, [14].
A dilemma with this mechanism is, that the direct electron-ion potential is also
screened by the background dielectric constant. In a homogeneous system, this screening removes the above-mentioned divergence altogether. Therefore, T, of a homogeneous 2-D electron-gas, immersed in a medium with a large dielectric constant, is
small [30]. However, as discussed in Sect. 2, in an inhomogeneous system, the screening of the (bare) electron-ion potential (by the dielectric constant E ~ ~ ~ - Bis~difeerent
- ~ I )
from the screening of the electron-electron interaction (by the dielectric constant cel~ ~ - ~ and:
l ) , Zma= ( 2 ~ Z e 2 / E i o n - ~ , - e 1 ) / ( qIf~EeI-Ba-el
~ / ~ ~ is
) . very much larger than
~ i ~ , , - ~ , - then
~l, I
, is enhanced. We made some preliminary estimates that suggest that
the inhomogeneity in the cuprates is indeed large enough to bring-about the required
[ 141.
increase in I,,
Actually, when we consider the role of E O - B , - ~ ~ ,which is somewhat larger than
one, and do the proper averaging over q, we find [14] that the increase is about a factor of two smaller. This applies to a 2D FS, where the small-q electron-phonon scatterings fill a large part of phase space. For a 3D FS, the small-q region of phasespace is so small, that there is hardly any enhancement of T, from this channel [14]
and the enhancement observed in superconductors like Bal+K,Bi03 is probably due
to the reduction of the p* term, to be discussed below.
The actual value of T, that we obtain for 3, x 1, E = 4 is about T, x 100 K (and T, x 50 K
with the corrections of Refs. 6, 14). For the measured value of E ( x 20-40) T, is
’ con-
9
.
190
Ann. Physik 7 (1998)
siderably higher. This estimate is very crude, but we believe that it can be trusted to within
about a factor of 2.
The actual increase over values, predicted by “conventional” Eliashberg theory, is
much larger. The reasons for this are as follows:
(i) Some band-structure calculations yield the value of J. M 1.7 in YBCO [23]. However, a large part of this J. comes from the chains, where the density-of-states is
large. For the (CuOz), planes, Jarlborg finds J. M 1 [31]. For this value of A, with
c0= 1, we cannot use the KGL formula (8), which applies only when A>2, but must
use the McMillan formula [32] which yields a value of T, about a factor of 2 smaller. But, for the large eo, Leff is very large, rendering the more-favorable KGL formula (8) valid. The data of Fig. 5 show how good the strong-coupling formula is for
e0=4, even for the small value of 1=0.067.
(ii) The bare Coulomb interaction U in the cuprates is very strong, U M 6 eV. Therefore, the bare Coulomb parameter p is very large, about p M 3. As a result, the effective Coulomb parameter p* given by: p*=,d[l+pln(E&2)] is large, p* M 0.35. This
value of p* reduces T, by about a factor of 3. This factor of 3 can be obtained from
the McMillan expression for T, [32]. The KGL formula does not apply for such a
large value of ,u*, but numerical solution of the Eliashberg equation [6] substantiated
this estimate.
The dielectric constant
reduces the Coulomb interaction to [ 131:D(w)=p/e(w)
M 0.06-0.1. Since this P ( o ) is cutoff at wtritns,
we have: ,ii*
M ji, which is still small and
has a negligible effect on T,.
Support for this estimate comes from tunneling measurements on Bal,KxBi03
[33], where the McMillan-Rowel1 structure indicates a rather “normal” electron-phonon coupling, but p* is experimentally observed to be zero, although from theoretical
considerations applicable to the “normal” scenario, it should have been very strong.
[Formally, we can calculate the effect of the Coulomb interaction on T, by subtracting from D,(w) the Coulomb term, and solving the Eliashberg equations along
the imaginary axis with this combined propagator. The effect of the Coulomb term
on T, is small. DJO) increases like EO to eg (depending on the behavior of wlong)
while b(0)decreases like e0, thus we have an eg to ei ratio, which is enormous. This
illustrates why “local field” effects are so large; even in Pb alloys, J. can be an order
larger than the bare p; here the difference is much larger.]
Thus, the order-of-magnitude increase in T, in the cuprates, over “conventional”
metals, results from a combination of factors - an inherent increase coming from the
large EO, the applicability of the very-strong-coupling expression for T,, although the
“bare” coupling constant A is not very large; and elimination of the very large reduction of T, caused (in “conventional” theory) by the Coulomb interaction.
6.4 Tunneling density of states
The gap d o observed in tunneling is the solution of the equation d(do)=do[lo]
rather than d(0). Therefore it depends critically upon the real-o axis solution. There
is no a-priori reason why do must be close to d(0). Our calculation must be extended over a wider range of parameters to find out the systematic behaviour of d o
Also, because of the very large value of AeE, the gap d may have a back-effect on
%ans,
and this also can cause drastic changes in do [34].
M.Peter et al., Estimation of the influence of ionic dielectricity
191
We find, for the parameters of Fig. 8, that do is very large (2d,,/T,w 10.55). A
very large value of do is in accord with experiments, particularly on organic superconductors [35].
6.5 CD Waves and relationship with polaronic models
In this work we use a “metallic” picture, although the cuprates are “doped insulators”, as demonstrated by their very large ionic dielectric constant EO. Doped insulators are susceptible to the formation of a polaron gas, as studied in detail by Mott
and Alexandrov [9]. We follow their picture in attributing to the apex oxygen, with
its high polarizability, a major role. However, since the binding energy of polarons in
underdoped samples is about 0.12-0.3 eV, and the Fenni energy in optimally-doped
samples is about 0.75-1 eV, we believe that the metallic picture is somewhat more
appropriate for optimally-doped samples. The formation of a CDW with q=0.24 A-’
in overdoped BSCCO [36] can be accounted-for by the metallic picture [14]. The
feature of the bands at the FS sometimes denoted “extended Van Hove singularity”
[37] follows from the metallic picture, in which Z is extremely large (Figs. 7, 8), just
as it does from the polaronic picture.
Here, since we use a BCS picture, there is a factor of ,,/in the expression
for T, [14], and as a result the maximum T, is about 100-200 K. Moreover, since we
need small q values, the q+O approximation employed in the present work applies
only to a 2D FS. For a 3D FS, the actual values of T, will be considerably smaller
[14]. In the polaronic picture, the factor ,,/is absent, therefore, in principle,
much larger values of T, are possible, and a 2D FS is not essential. We believe that
the closeness of the observed properties of the superconductivity of the cuprates to
BCS superconductivity, favors the metallic picture.
In many cuprates a charge-density wave with a wavelength lStrii,=25 A is observed. It leads to the formation of stripes with an alternating crystal structure.
Namely, there are stripes of width W x 8-10 A with a low-temperature tetragonal
(LlT) structure, and stripes of width Lw 17-15 A with a low-temperature orthorhombic (LTO) structure. The sum of the widths W+L= Astripe is always 25 f 1 A. The LTO
stripes are apparently metallic and superconducting, and the L?T stripes are Mott insulators. This structure has been observed in BSCCO, LSCO, and TBCCO; however,
it is usually observed in overdoped and underdoped materials, and not in optimally
doped BSCCO and YBCO. The structure is observed by neutron and X-ray diffraction, EXAFS, electron diffraction, and STM, by several groups [36].
We might attribute this structure to a very strong charge-density-wave, with wavevector 2n/&~p,=0.24A-’. Since the cuprates are on the borderline of the Mott metal-to-insulator transition, the regions of space where the electronic CDW density is
low, are a Mott insulator, while the regions where the electronic density is high, remain metallic.
Because of the long, mesoscopic wavelength &tripe, these maxima and minima of
the CDW can be considered as “bulk” metal and insulator, in contrast with conventional, “microscopic” CDW’s.
The wavevector 2n/AS~,,,=0.24 A-’ is equal to the dressed Thomas Fermi screening parameter GTF. Since the electron-phonon matrix element I(q) has an enormous,
sharp peak at q = QTF, it is not surprising that a CDW forms with this wavevector.
192
Ann. Physik 7 (1998)
As the temperature changes, the widths L and W change, but their sum
Lstripe=L+W always remains constant. Also, it is the same in several different cuprates (BSCCO, LSCO, TBCCO). Thus, the wavevector 2 7 ~ / / 2 ~ ~ r i ~ is
~ =the
& -invar~
iant quantity. 4 T F is a single parameter that accounts for the various anomalous
properties of the cuprates in the phonon picture.
Thus, the observation of a CDW singling-out ~ T can
F
be regarded as a “smoking
gun” that provides extra weight to the specific phonon mechanism proposed in the
present work.
Positron annihilation experiments [38] have shown that YBaCuO, for instance,
provides only one positron lifetime, which seems to suggest absence of CDW. On
the other hand, recent N M R work has been interpreted as a strong indication for the
presence of CDW’s.
6.6 Shijit of the phonon frequencies in the superconducting state
Pintschovius et al. [28] observed a huge (8%) shift in the frequency of a 2 meV phonon in the organic superconductor (BEDT-TTF)2Cu(NCS)2in the superconducting
state. Such shifts are predicted by the Balseiro-Falicov theory [39] as arising from
the mutual repulsion of the superconducting gap and phonon states. Thus, the upward push indicates a gap state below 2 meV. The superconducting gap is 10 meV
[35]. This would be expected to push the phonon frequency down. However, the
states-inside-the-gap that we find in the present work (Figs. 4b, 7a, 8), at energies
much below 2d0, fall in the range 1-2 meV, and thus can account for the upward
shift of the phonon frequency. The very-strong coupling that we consider here accounts very well for the large magnitude of this effect. The states-in-the-gap are thus
one possible mechanism for the zero-bias-anomaly, observed in tunneling experiments in these materials [35]. The zero-bias-anomaly indeed commences at about
2 meV. The neutron-diffraction experiments of Toyota et al. [40] show clearly the
two structures, which are superpositions of phonon and superconducting gap states.
6.7 Normal state properties
The normal-state properties of the cuprates are highly anomalous; this applies to
transport as well as magnetic properties. The normal state resistivity of the cuprates
is abnormally low [41] and, in many cases, linear in T. In some cases, such as
NCCO, and in organic metals, it is proportional to ?. The Hall constant is temperature dependent and, in some cases, proportional to 1/T [42]. This behavior is cited as
evidence for a non-Fermi liquid behavior (category (i), Sect. 1). The NMR relaxation
rate 1/T, and Knight shift are anomalously strong. This is frequently regarded as indicating magnetic interactions (category (ii), Sect. 1) [4, 71.
For a phonon-mediated interaction the behavior of both transport and magnetic
properties is expected to be entirely different. In particular, the normal-state and
superconducting properties are related by the Hopfield relation C431:
~ 1Peter
.
et al., Estimation of the influence of ionic dielectricity
193
/
where the resistivity is given by: p = (4n/G2)1/z, where Gp is the (unshielded) plasma frequency a2=4nne2/m, and 3, is the &Millan coupling constant, related to T,
by: T, % (GV1.49) exp[-(1+3,)/;1], where SZ is the phonon frequency. Thus, a high
mansition temperature T, requires a large value of 1, causing in turn a large normalstate resistivity p. This correlation between p and T, was established well before the
BCS theory. However, in the High-T, superconductors this relation between resistance and T, does not hold. The breakdown of the Hopfield relation is usually attributed to a non-phonon mechanism of superconductivity [4]. We attribute it here to the
difference between the values of the coupling constant for real and virtual processes,
as we discuss later in this section. Also, anisotropic scattering, and hence indirectly
dielectricity, may also be involved. Amongst other normal state properties, optical
conductivity is of particular interest since it should reflect the nature of the dynamical dielectric constant.
6.7.1 Knight shift and the relaxation rate
The concept of ionic dielectricity implies that at energies close to EF (within less
away), the electron-electron interaction is screened-out, and at larger enerthan owans
gies they are not. This behavior is illustrated in Fig. 1 [13]. Thus, the electron-electron interactions at energies above about 15 meV are extremely strong. This behavior
is in accord with the accepted theory of Pines et al. [44], Chubukov [45], etc., where
the cutoff frequency is denoted osf(sf standing for spin fluctuations). Empirically osf
is about 14 meV. In our theory, the cutoff frequency otrans
has a different origin
(namely, the low-lying transverse optical phonons, which determine the dispersion of
the ionic dielectric constant), but the value of owans
M 15 meV is the same. Therefore, we predict that the behavior of the magnetic properties, such as the NMR
Knight shift, relaxation rate l/Tl, etc. which are dominated (at temperatures of about
100 K and higher) by the energy-range above the cutoff, are as given by the “conventional” theory. We differ from the conventional theory in the low-energy range,
below the cutoff, where we predict that the electron-phonon coupling is extremely
strong, and serves as the primary pairing mechanism.
Since the pairing is mediated by phonons of frequency SZ (E35-40 meV), and the
coupling strength is large only for frequencies below wmnS(E 15-20 meV), this large
coupling constant affects only virtual processes, where the energy-transfer
AE 5 Otrans and thus AE << SZ. For real processes of phonon emission or absorption,
AE = SZ and therefore the coupling constant is small. In conventional BCS-Eliashberg theory, the coupling constant is the same for real and virtual processes. The
Hopfield relation (9) follows from this. In the present work, this is no longer the
case, and our work is thus an extension of the BCS-Eliashberg theory into a new regime.
6.8 Coulomb interaction and its consequences
In “conventional” metals, interactions are screened at a distance of the screening
length qTF-’<0.5A. This distance is smaller than the Bohr radius ao, the inverse Fermi wavevector kF-l, the lattice constant d, etc.; therefore interactions are considered
local, and the local density approximation (LDA) is so successful. In “exotic” metals,
the ionic dielectric constant
is very large, and consequently the dressed Tho-
194
Ann. Physik 7 (1998)
mas-Fermi screening parameter C ~ T F = J 4 7 c e 2 N ( E ~ ) / ~isg ve
small, thus the
screening length Cj& x 4 A is large - much larger than ao, kF-? d etc. Therefore,
atomic changes at a distance of x 4 A from the (CuOz), plane can have a drastic effect on T,. We suggest here (Fig. 2a) a specific cause why the replacement of chain
Cu, Hg, or T1 by Bi, reduces T, by an order-of-magnitude. The question of action at
a distance has gained new interest with the work of Blackstead and Dow [46]. These
authors have commented a number of observations on PrBaCuO which show the
planes unperturbed, the chains perturbed by ions situated far away. They conclude
that superconductivity in these substances occurs in the chains - but it could also be
that the range of the ionic interactions is abnormally large because of the present
mechanism. Action at a distance is related to low q scattering. Such scattering, be it
from our mechanism, from electron-electron interaction [47] or from nesting features
[48]has been invoked to explain high T,, and also to explain d-state pairing [49-511.
The effect of the ionic dielectric constant on the electron-electron interaction can
be viewed as a modification of a vertex correction. Vertex corrections were investigated in some detail by Pietronero et al. [52].
6.9 The symmetry of the order parameter
The prediction, and observation, of the d-wave symmetry of the order parameter, is
one of the strongest arguments in favor of a paramagnon-mediated interaction, rather
than a phonon-mediated one [l, 71. However, a phonon-mediated interaction can give
rise to d-wave pairing, when it is anisotropic, and accompanied by a repulsive Coulomb interaction [53]. The electron-phonon interaction that we consider here is extremely anisotropic, and the scattering of the electron when it emits or absorbs a phonon (real or virtual) is redominantly forward. The scattering angle is given by:
dB = &/kF = qTF/ ~ ( 0 ) k F
x 0.3 radians. For absolute forward scattering
(dB = 0), the states with s-wave and d-wave scattering are degenerate, and an infinitesimal repulsive Coulomb interaction favors the d-wave state over the s-wave one.
For our case of dB x 0.3 radians, a rather small Coulomb interaction causes the dwave state to be more stable than the s-wave state [51]. Thus, the d-wave state
“wins” in the phonon scenario, because of the large value of E.
If this interpretation is correct, then if the scattering angle dB is increased, for example by impurities or defects in the sample, the d-wave symmetry may turn into swave symmetry without a significant reduction in T,. This may account for the data
of Klein et al. [54] and several other groups who claim to see a gap with s-wave
symmetry. In the more-widely accepted Pines picture [7], destruction of the d-wave
symmetry should destroy superconductivity altogether; in the Bulut Scalapino [531
picture, suppression of the s-wave state (by the Coulomb interaction) involves a drastic reduction in c. In the model described in the present work, suppression of the
state with s-wave symmetry and the appearance of a state with d-wave symmetry,
does not involve a significant reduction in T, [51].
s“
6.I0 Relationship with the bipolaron picture
High-T, superconductivity due to a very strong electron-phonon coupling is postulated in the bipolaron model [9], substantiated by the Uemura plot [55] which sug-
M. Peter et al., Estimation of the influence of ionic dielectricity
195
gests that T, cc n,, consistent with a Bose-Einstein condensation of these bipolarons.
m e existence of the pseudogap (in underdoped samples) up to z 300 K is suggested
as further support of bipolarons with a binding energy in this energy range.
The question whether the bipolaron or the metallic picture are more appropriate,
depends on whether the binding energy of polarons is larger or smaller than the Fermi energy of the metallic state. In the former case, the metal will decompose into a
polaron gas, and in the later case - the polarons will dissolve to form a metal. A
large enhancement of the effective mass takes place in both pictures and, in the metallic picture, it is represented by a large value of Z(0) (see Fig. 7, for example).
In the cuprates, the polaron binding energy and the Fenni energy are comparable
(of order 0.5 eV), therefore we are in the transition region. Therefore it is important
to understand both limiting situations; the bipolaron scenario is described in the excellent book by Alexandrov and Mott [9]. In the present work, we attempt to describe the metallic scenario.
In the metallic scenario, the pseudogap is attributed to the dielectric gap, which is
(Fig. 1 and Ref. 13). In underdoped materials, the ionic dielectric
of order ovans
constant E is larger than in the optimally-doped ones (where the conduction electrons
screen E to a certain degree), therefore the increase in the pseudogap in underdoped
materials is to be expected.
Also, the ratio 2d0/Tc is seen to increase with E (Table 1); thus in underdoped
samples it should be larger than in optimally-doped ones; this is also in accord with
experiment.
Thus, both the bipolaron and metallic scenarios seem to contain elements characteristic of strong electron-phonon coupling; it is necessary to consider the fine details
of both scenarios to determine which one is better. The present work attempts to look
into the details of the metallic scenario.
’ 6.11 Numerical analysis
The Eliashberg equations can be solved along the real-o axis, either directly as was
done by McMillan [32], Tewordt [3], Schmalian et al. [4], etc., or by solving them
first along the imaginary-o axis as was done by Bergmann and Rainer [56], and then
extending the solutions analytically to the real-o axis as was done by Carbotte et al.
[lo]. In the present work, we use the second approach. We chose this alternative because the solutions along the imaginary-o axis are extremely stable, and even the enormous renormalization of L ( o ) (from 1 to about 20) leaves a nearly-lorentzianweighted phonon propagator (Fig. 6). Thus the calculation of T, is straightforward
and simple. The analytic continuation sometimes causes numerical instabilities, but
because of the stability of the solutions along the imaginary-co axis, we know that
these instabilities are a numerical artifact, and not a “physical” phenomenon. The
amount of work involved, both in writing the algorithms, and in running them on the
computer, is appreciable, and requires a computer with a very large CPU. It may be
instructive to try also the first computational method, of direct solution along the
real-w axis, and compare the results, as well as the extent of numerical work required.
The algorithm of Marsiglio, Schossmann and Carbotte [lo] has the merit to take
the very stable solution along the imaginary axis as a starting point, to construct a so-
196
Ann. Physik 7 (1998)
lution for the real frequencies which is equally stable. A second integral equation has
to be solved to add causality - this equation is reasonably stable and quickly solved.
Mean-field equations like the ones employed in this study do have well-known
shortcomings - fluctuations are largely neglected. Other, more sophisticated models
(like the t-J model) treat fluctuations quite rigorously. However, the size of such calculations makes the modeling of bandstructures, and of dynamics, impractical. For
our discussion of the influence of dielectricity on superconductivity we have chosen
to use Eliashberg-like equations because they make possible experimentation with the
dynamic properties of superconductors. Extension to realistic band structures and
symmetries of the superconducting gap is not included here, but has proven [6, 511
to be quite feasible.
7 The isotope effect
The isotope effect was the “smoking gun” for the phonon mechanism in “conventional” superconductors. In “exotic” superconductors, the situation is more complicated. In optimally-doped cuprates, there is almost no isotope effect [57]. In underdoped cuprates, there is a positive isotope effect for the oxygens, T, K M a , with a
being sometimes even larger than the “classical” value of 0.5. For the copper, there
is a negative isotope effect with a = -0.3.
It is possible to account for this behavior by postulating a non-phonon pairing
mechanism, thus accounting for the absence of the isotope-effect for optimally doped
samples, and postulating an isotope-effect of the doping in underdoped materials
[581.
Alternatively, with a phonon pairing mechanism, the Coulomb term
p* = p / ( 1 p . In Ep/w0,h) has a negative isotope effect which is known to compensate (partially or fully) the normal positive isotope effect in “conventional” superconductors [32].
In our model, an approximate expression for the maximum value of T, is given by
[14]:
+
(
~
c
=) o .~l ( e~ 2 /~2 n , ) ~ ~ ( l / J d e ) w t r a n , / ~ .
(10)
Here a, is the Bohr radius in the c-direction, m the band mass of the electrons (assumed constant), A 0 the angle of scattering of the electrons by the phonons, and M
is the ion mass. We see that in the factor
SZ the isotope-effect of the mode 0
is compensated, leaving us with an isotope-effect for the mode wtrans,and possibly
an isotope effect of d0.
This expression does not take into account the strong Coulomb interactions that
dominate at large vdues of o.These interactions are characterized by a cutoff frequency wsf (sf standing for spin fluctuations), which is roughly estimated to be about
14 meV [45].
Now, when wsf is smaller than w-, (that we estimate as 15-20 meV), then, in
the region os.<
co c wtrans,strong spin-fluctuations destroy the pairing, and consequently the transition temperature is reduced to:
m.
-M.peter et al., Estimation of the influence of ionic dielectricity
197
The reduction is not large, since wsf is just about 25% smaller than cotrans-cosfdoes
not have an isotope effect. Therefore, T, will have an isotope effect only if A 0 has
one.
The Fermi surface in the cuprates is rather anisotropic, with a Van-Hove singularity close to the FS in the [loo] direction [59]. If the scattering angle is limited by this
VHS, it has no isotope effect, and consequently T, should have no isotope effect.
This is in line with the suggestion of Abrikosov [49]. We suggest that this may be
the situation in optimally-doped materials.
In underdoped materials, E is larger, and consequently the scattering angle may be
limited by E :
2
2
Now, E = ~ , c o l ~ ~possesses
~ / w ~an~isotope
~ ~ effect. Consequently, A 8 causes a positive isotope effect for along,
and a negative isotope effect for cowms. Since along
is
dominated by the light oxygen atoms, and w,
by the heavy atoms, like copper, the
experimental behavior is accounted for, at least qualitatively. (As the present work
shows, the above formula for (T,),,, is simplistic.)
Obviously, this is only one possibility out of several ones.
7.1 Pressure dependence
The above formulas for T, can provide an indication of the expected pressure dependence of T,. Pressure increases cotran,,and thus decreases E , and these two effects
compensate each other to a large degree. Therefore we do not expect in general a
very large pressure dependence of T,. When there is a Van-Hove singularity near the
FS, it manifests itself in an increase of the band mass m and thus of T,. If pressure
brings the VHS closer to the FS, it should increase T,. This seems to be the case in
HgBa2Ca2Cu308where pressure increases T, from 135 to 164 K [60]. This is in line
with the explanation of Freeman’s group [61].
8 Conclusions
Experimental evidence exists for the existence of dielectrics of essentially the SLT
type, as postulated for our present work. However this form for ~ ( w can
) be changed
with impurity content [27], and the influence of such change on superconductivity
should be further investigated. Other relevant experiments are positron lifetimes (for
CDW’s) [38] and different low energy spectroscopies (neutrons, tunneling, Raman
spectroscopy, ARPES [29, 371 etc.) and transport properties. For a choice between
different models, experiments involving quantities depending on real frequency are
crucial since for the Matsubara frequencies (and hence for the thermodynamic functions), interactions of different shape give remarkably similar results.
The high ionic dielectric constant E that is generally present in the perovskites
(and in organic metals as well), can contribute to a high superconducting transition
temperature T,, by shielding the Coulomb interaction ,u (=N(EF)U),and by increasing
the effective electron-phonon interaction 1.This effect is large when 1is small (weak
coupling), but appreciable also when 1 is large (strong coupling). As already dis-
198
Ann. Physik 7 (1998)
cussed in Ref. 14, the effect of the background dielectric constant is compensated in
a homogeneous system, but in an inhomogeneous one, there are several different dielectric “constants”, and the effect is not compensated.
Numerical calculations:
We have extended the conventional calculations with a phonon propagator and a constant coupling by the use of weighted phonon propagators, implying in particular the
multiplication by the square of a model (STL,) [21] dielectric constant. Besides carrying out calculations aimed at investigation of the specific idea of the present paper,
i. e. the importance of dielectricity in high T, superconductivity, we have also generalised the elegant method of Schossmann, Marsiglio, Carbotte [ 101 to interaction kernels which can be represented as a sum of poles of different order in the frequency
plane; i.e. we can treat a rather general class of interactions. Along the imaginary
axis, the resulting gap functions are remarkably similar to each other. Even along the
real axis, there appears a rather universal smooth w-dependence of the gap functions,
to which there are added sharp spikes corresponding to the poles which are added to
establish causality.
The algorithm of Marsiglio, Schossmann and Carbotte [lo] has the merit to take
the very stable solution along the imaginary axis as a starting point, and to construct
a solution for the real frequencies which is equally stable. A second integral equation
has to be solved to add causality - this equation is reasonably stable and quickly
solved.
Physical consequences of strong dielectricity
We find therefore that strongly varying interactions will bring no great change to the
thermodynamical quantities, including T,, except by alteration of the familiar effective parameters from the BCS model values. A noteworthy exception is the strong increase of Z(O)/d(O). This increase could lead to some flattening of energy bands
(EVHS). Through the diminution of the Thomas-Fermi vector, we obtain a strong
peak of Z(w,q) for low values of q. Hence, strong dielectricity will favor low angle
scattering, with its known consequences like, for instance, d-wave superconductivity.
The dynamical response will show characteristic changes, affecting different experimental properties and allowing in principle to measure the importance of the contribution of dielectricity (McMillan-Rowel1 peaks in tunneling, for instance). The values of T, increase both through increased (electron-phonon) coupling and through
shielding of p.
Action at a distance
The large value of the electronic screening length qFk e 4 A implies that atomic
changes at a distance x 4 A from the (CuOz), planes can have a drastic effect on T,,
i.e. there is an action at a distance. This feature has been pointed out by P.W. Anderson [2] serving as a basis for his mechanism. Our interpretation is somewhat more
pedestrian. The recent discovery that YPrBaCuO is superconducting if the atoms in
the chain region are well-ordered [27, 461 also supports the “action at a distance”
scenario.
Forward scattering
The small value of ~ T causes
F
the electron-ion potential to be highly peaked at small
q values, i.e. the electron is scattered predominantly in the forward direction upon
M. Peter et al., Estimation of the influence of ionic dielectricity
199
emission or absorption of a phonon. This causes the electrical resistivity due to electron-phonon scattering to be small. Also, it makes possible a gap parameter with dwave symmetry even if the pairing is due to exchange of phonons, as is discussed
elsewhere [49-511.
We benefited from Discussions with D.J. Scalapino, W. Kohn, I. Bozovic, V.Z. Kresin, J. Kuebler,
and A.J. Freeman. M.W. thanks the U.S.-Israel BSF for supporting this research.
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