23 Electronic Theory for Superconductivity 1481
whichturns outto be the low-temperature supercon-
ducting gap
0
, is reduced. For temperatures T < T
∗
c
the superfluid density is non-zero at ! =0,indi-
cating the presence of the Cooper-pair phase fluc-
tuations in the system. On the contrary, for T = T
∗
c
n
s
(! =0)/m
∗
= 0 and thus no dynamical Cooper-
pair phase fluctuations are present. Note that for
! > 0 the superfluid density is finite even for
T > T
∗
c
. At at a first glance this is surprising, since
the phase is not well-defined for = 0. However, the
corresponding action of the Cooper-pair phases also
contains a contribution from the time derivative of
the phase besides the stiffness term. While the total
action vanishes, each term on its own does not. An-
other interesting observation is that all curves merge
together at energies of about 2
0
.Thisisexpected,
since the superfluid density is a current–current cor-
relation function, which has a characteristic energy
scale at about 2
0
in the superconducting state.
Even slightly below T
∗
c
, n
s
(! =0)/m
∗
obtains a
significant finite value, leading to the Meissner ef-
fect, and there is a considerable redistribution of
weight from energies roughly above twice the low-
temperature gap, 2
0
, to energies below 2
0
.Thisre-
distribution increases with decreasing temperature.
Also, a peak develops slightly below
0
followed by
adiparound2
0
,thisstructurebeingmostpro-
nounced in the underdoped case. Since
0
is smaller
in the overdoped regime, n
s
(!)/m
∗
changes more
rapidly for small ! in this case. Of course it is not
surprising that 2
0
is the characteristic frequency
of changes in n
s
(!)/m
∗
related to the formation of
Cooper-pairs.
To summarize, we have analyzed the behavior of
the superfluid density n
s
(!, T, x), which controls
the doping dependence of the phase coherence, the
thermodynamic behavior, the penetration depth, the
Nernst effect, Cooper-pair phase fluctuations, etc.
Note that we find that the superfluid density plays
the most significant role in the underdoped cuprates
and determines the superconductivity. We find our
results to be consistent with the BKT theory.Also the
recently noted universal scaling relation by Homes
n
s
∝
DC
T
c
(23.140)
in hole-doped and electron-doped superconductors,
in which the dc conductivity
DC
is measured ap-
proximately at T = T
c
[97], is contained in the gen-
eral expression for n
s
. We immediately obtain in the
underdoped high-T
c
cuprates
n
s
(T =0)∝
DC
(T = T
c
) T
c
, (23.141)
asobservedin experiment.Notethat(23.139) is taken
in the limit T → 0 assuming a thermodynami-
cal phase transition and Fermi liquid-like behavior.
However, in the case that quantum fluctuations are
important close to T = 0,the simple formof (23.139)
is no longer valid because of logarithmic correction
terms [148–150].
Another interesting question in cuprates is
whether there exists a quantum critical point (QCP)
at around x =0.19 doping concentration. The QCP
should be reflected characteristically by a universal
behavior of some physical properties such as the op-
tical conductivity (independent of time and length
scales). A possible interplay of QCP and thermody-
namical points needs further study.
23.4.8 SIS and NIS Tunneling Spectra
Tunneling spectra are a good measure of the
superconducting state and the elementary exci-
tations in general. They shed light on the va-
lidity of our electronic theory. Here, we calcu-
late the tunneling spectra for SIS (superconductor/
insulator/superconductor) and NIS (normal-state
metal/insulator/superconductor).Letus see to which
extent the spin fluctuation model for superconduc-
tivity can explain the fine structures seen by tunnel-
ing spectroscopy below T
c
in the cuprates (remem-
ber the significant success of the BCS and Eliash-
berg theory of superconductivity due to electron–
phonon coupling resulting from the explanation of
the fine structures seen by tunneling spectroscopy
in conventional superconductors,which exhibits the
single-particle excitations spectrum [50]). It is im-
portant to relate these features to the spin fluctu-
ation frequency !
sf
and resonance frequency !
res
.
One expects that the remarkable dip structure seen
in the single-particle excitation spectrum of vari-
ous cuprates might be an intrinsic fingerprint of a