A new integrating understanding of superconductivity and superfluidity
ABSTRACT An integrating theoretical scenario of superconductivity and superfluidity has been built. It reduces to the special BCS superconductivity mechanism for conventional superconductor and to a new theory for high transition temperature superconductors, which can explain the recent angleresolved photoemission experiments and the earlier nuclear spinlattice relaxation rate experiments. Both experiments suggest the existence of pairing carriers and the normal state energy gap well above the transition temperature of superconductivity. A powerful and workable experiment designed to validate this scenario is also put forward for the experimenter.
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Article: The theory of superconductivity
Advances in Physics. 01/1959; 8(29):144.  SourceAvailable from: D. H. Lu[Show abstract] [Hide abstract]
ABSTRACT: We review our angleresolved photoemission spectroscopy (ARPES) results on different layered oxide superconductors, and on their insulating parent compounds. The low energy excitations are discussed with emphasis on some of the most recent issues, such as the Fermi surface and remnant Fermi surface, the pseudogap and dwavelike dispersion, and lastly the signatures in the ARPES spectra of multiple electronic components, many body effects, and the superfluid density. We will focus on systematic changes in the electronic structure which may be relevant to the development of a comprehensive picture for the evolution from Mott insulators to overdoped superconductors.Journal of Electron Spectroscopy and Related Phenomena. 08/2001;
Page 1
A new integrating understanding of superconductivity and superfluidity
Jie Han (jiehan@yahoo.com)
An integrating theoretical scenario of superconductivity and superfluidity has been built. It reduces to the special BCS
superconductivity mechanism for conventional superconductor and to a new theory for high transition temperature
superconductors, which can explain the recent angleresolved photoemission experiments and the earlier nuclear spin
lattice relaxation rate experiments. Both experiments suggest the existence of pairing carriers and the normal state energy
gap well above the transition temperature of superconductivity. A powerful and workable experiment designed to validate
this scenario is also put forward for the experimenter.
_______________________________
BCS theory has been the dominant theory in the research
realm of superconductivity mechanism since 1957. The
situation did not have any substantial variation until the
discovery of high transition temperature cuprate
superconductors in 1986. Since then, because of the great
difficulties that BCS theory has met in explaining some new
experimental results of cuprate superconductors, various
theories based on different interaction pairing mechanisms
have been put forward in order to obtain such a high
transition temperature. However, all these theories met
difficulties in explaining some anomalous phenomena, such
as null isotope effects, anomalous nuclear spin lattice
relaxation rate, anomalous angleresolved photoemission
spectroscopy (ARPES) etc. In the author's point of view,
although these theories were based on interactions other than
BCS interaction, they, in fact, were still theories that were
developed in the framework of BCS theory. i.e., some
interactions lead to the formation of pairing carriers and
energy gaps in superconductivity states.
There are enough facts that make the author of this article
believe that the BCS pairing mechanism is not always a
necessary and sufficient condition for the occurrence of
superconductivity. Of course, in any case, pairing
mechanism should be a necessary condition. In fact, as early
as 25 years ago, the discovery of the reentrant
superconductors (1, 2, 3) appeared to be an early
experimental confirmation that BCS pairing mechanism was
not a sufficient condition for superconductivity. In those
years, a pair breaking theory (induced by magnet ordering)
was put forward to explain the reentrant phenomenon. The
great success of BCS theory excludes any possibility of
calling into question BCS superconductivity theory. Even
though, at the beginning of BCS theory, M. R. Schafroth (4,
5) and C. G. Kuper (6) pointed out that BCS theory could not
establish the Meissner effect and the vanishing of the ideal
resistance, (when T≠0). The recent discoveries of high
transition temperature superconductors (HTS) force
scientists to question whether BCS theory needs correction,
particularly in the realm of HTS. In this article, a new
superconductivity theoretical framework is suggested, which
reduces to BCS theory for conventional superconductors and
to a wholly new mechanism for HTS that explains
significant anomalous phenomena mentioned above. It
should be pointed out that the present work concentrates on
building a general framework of superconductivity, so that
low and high temperature superconductors can be treated in
the same theoretical framework without generating so many
anomalies. At the same time, pairing mechanism remains an
open question, leaving just enough space for further
development of superconductivity theory. However, from
this simple theoretical framework, one may expect many
specific conclusions to explain those anomalous effects and
clarification for further development of superconductivity
theory.
A longstanding prejudice is that particle pairing transition
temperature Tp has always been regarded as the
superconductivity transition temperature Tc. But, with the
application of this new theoretical framework, any previous
or later calculation of transition temperature Tp based on
interaction pairing mechanism should first be compared with
a new characteristic temperature Tb (transition temperature
for superfluidity) in order to get the real superconductivity
transition temperature Tc, which then can be compared with
the experimental results of Tc. The following will be, first,
the suggested theoretical framework that stipulates the
relation among Tb, Tp, Tc. Second, an explanation and
application of this relation. Third, experimental evidence of
it, fourth, a direct experimental design to demonstrate this
theoretical framework for the experimenter, and finally a
conclusive review and prospect.
Unified criterion for superconductivity and superfluidity.
It was generally accepted that superconductivity comes
from the superfluidity of the charged particles, which leads
scientists to regard superfluidity transition temperature Tb
(identical with BoseEinstein condensation temperature) as
the critical transition temperature for superconductivity in
terms of the work of F. London. But, for element super
conductors, C. Kittel (7) pointed out that the Bose
condensation temperature calculated for metallic carrier
concentrations is of the order of the Fermi temperature
(104∼105 K). This rules out any possibility of regarding Tb as
Tc. Later in this paper it is pointed out that Boson
concentrations of element superconductors are not of the
order of general metallic carrier concentrations. In fact, the
concentrations of Bosons should be ~(105~104) of metallic
Page 2
carrier concentrations. (8, 9) Kittel's calculation of Tb needs
substantial correction.
A critical analysis shows that there exist three significant
transition temperatures. Their practical denotations are as:
Tb, superfluidity transition temperature for Boson or
Bosonlike systems, which is identical with BoseEinstein
condensation temperature of the system. Tp, the critical
transition temperature of forming particle pairs for Fermion
systems, which means that, when T>Tp, all pairing Fermions
will be broken from pairing state. Of course, pairing particle
interaction should not be considered as the electron phonon
interaction only. Other attractive interactions can also serve
as pairing mechanisms; the real Tp is obtained after
considering all possible pairing interactions. Tc is the critical
transition temperature of
superfluidity. For a Boson system, one has, Tc=Tb, which
means that the superfluidity transition temperature is
identical with the BoseEinstein condensation temperature.
For a Bosonlike system formed by pairing Fermions, a
formula that stipulates the relations among Tb, Tp, Tc. is
provided here: (attention: Tb should be obtained after
formation of carrier pairs).
Tc = Minimum [Tb, Tp] (1)
An appropriate application of the formula above can easily
explain many significant experimental results concerning
BCS element superconductors and HTS superconductors.
Explanation and application of Formula (1).
According to the BCS framework, not all free electrons will
become Cooper pairs and therefore become superconducting
electrons. The interaction that leads to the formation of
Cooper pairs can only act in the energy range of magnitude
order Eg (BCS energy gap) near EF (Fermi energy), which
means that the maximum electron number density that can
form electron pairs is about (Eg/EF)n, where n is the total free
electron number density; at Tp, only a small part of the total
free electron number density n can form the socalled
Copper pairs. In my view, electron pair number density is a
constant for T<Tp, and at T=Tp, only those free electrons with
energy above the Fermi energy EF(Tp) can form Cooper pairs
(8). The number of these electrons is:
?
E
gfN
F
=
)(
superconductivity and
(2)
EE
d
F
E∫
∞
)(
This is a general formula. f(E) is the Fermi distribution
function, and g(E) is the state density. It is natural to
consider a case with
kT
p
helpful in obtain NF from equation (2):
1)(
<<
TE
pF
which is very
( )
0
( )
0
( )
0
( )
0
04. 1
2
2ln3
24
2ln
2
3
2
FF
FF
F
E
N kT
E
N kT
E
kT
E
N kT
N
pp
pp
≈≈
+≈
π
(3)
From equation (3), the number density of Cooper pairs is:
kT
p
b
04
Where, n is the number density of free electrons and nb is the
most important quantity for calculation of Tb and Tc. It also
should be mentioned that these approximations should be
reasonable because of the smaller value of kTp /EF(0) and the
larger value of the denominator of f(E). It is believed, for
Boson and Bosonlike systems, the transition temperature of
superfluidity is identical
condensation temperature of the system. Because of the
difficulties in obtaining
temperature for interactive Bosonlike systems, We use the
BoseEinstein condensation temperature of ideal Boson
systems to replace that of interactive Bosonlike systems and
hope to get a reasonable result as in the case of 4He. Then,
( )n
E
F
2 ln3
=
n
(4)
with the BoseEinstein
BoseEinstein condensation
one gets the transition temperature formula for this system:
n
T
32
612. 2
?
()
b
b
mk
b
322
2 h
π
=
(5)
Where, h: Plank constant, mb: Boson mass, nb: Boson
number density, k: Boltzman constant. We know that:
( )
(
3
2
mm
Take (4) and (6) into (5), and one finally has the formula .
(
(
3 612. 2
b
m
k
Here the constant H ≈ 4.42×1016 J1/3K1/3S2/3 (SI units).
The following are a few calculated consequences for some
typical superconductors.
Sample one: BCS element superconductor: Aluminum. We
have: n=18.06×1028 m3, mb=2me=18.22×1031 kg, Tc= 1.14
K. Assume that Tp= Tc, then, we get Tb (Al)=12.53 K. (If
mass correction induced by strong coupling has been
considered, a lower Tb can be obtained.) It’s reasonable that
one uses Tc=1.14 K to replace Tp for Al. because Tp<Tb, so
)()
32
2
2
32
2
2
30
nnE
b
F
ππ
hh
==
(6)
)
) ()
32
p
31
b
92
32
p
31
92
3 / 1
94
2
32
32
32
2ln75
π
. 02
b
T
m
n
HT
n
T
==
πh
(7)
2
Page 3
we have Tc=Minimum [Tp,Tb] =Tp. for other element
superconductors, one has the same conclusion, i.e.,
Tc=Minimum[Tp,Tb] =Tp. One also should pay attention to
the fact that Tb of element superconductors is far lower than
104∼105K of Kittel's results (7). More data of Tb and Tp for
BCS element superconductors are given in table one.
Table one: Tp (=Tc), Tb, n, nb for element superconductors (SI Units)
element
n×1028
nb×1023
Tp
Tb
element
n×1028
nb×1023
Tp
Tb
Sample two: high Tc cuprate superconductors. For La2
xSrxCuO4, n≈6×1027 m3, (10), mb=2meff≈8me=72.88×1031 kg.
(11) Tc=38 K, then easily one has, Tb≈3.395Tp
Tp=Tc, one has Tb=38.37 K>Tp, and nb=3.39×1025 m3, It can
be seen that the assumption Tp=Tc is reasonable. But it also
should be pointed out that for La2xSrxCuO4, generally one
has Tp∼Tb∼Tc for different x (i.e., different n and mb). One
can get two conditions for it. The first is Tp∼Tb, but Tp<Tb.
The second is Tp∼Tb, but Tp>Tb. For the two conditions
above, one can obtain basically the same results for
superconductivity transition temperature Tc. However, for
the isotope effect, one can obtain significant discrepancy
because the isotope effect is induced by pairing interaction,
thus, for the two conditions above, one has Tp∝Mα (if
applicable); here α is the isotope effect exponent. For the
first condition, because Tp<Tb, one has Tc=Tp∝Mα. For the
second condition, one has Tp>Tb, therefore, Tc=Tb∝M2α/3,
which gives an isotope exponent 2α/3. Finally we see that
even for the same structure of La2xSrxCuO4, the isotope
effect exponent is not a single value. Instead, it has two
values, 2α/3, and α, which is in agreement with the
experiments of M.L.Cohen et al. (12)
Now, let's see what can be got for Y1Ba2Cu3O7 (Tc=90K).
One has n≈9×1027 m3. (10,13). But for mb, there is a large
selection scope, from mb≈6me to mb≈20me. It is considered
that mb≈6me is a reasonable one (10, 14). Then one gets
Tb=4.089Tp
Now, it is known that Tp≥Tc=90 K, so Tb<Tp. Thus, for
Y1Ba2Cu3O7, in any case one always has Tc=Tb<Tp. This is a
significant discrepancy from
superconductors. Here the isotope effect exponent is 2α/3
(if Tp∝Mα). Finally, for Y1Ba2Cu3O7, Tc=Tb=90K,
Tp=103.27K, nb=7.92×1025 m3. What follows are the
significant facts that support this larger Tp=103.27 K, and
especially explain the anomaly of nuclear spinlattice
relaxation rate for superconductor Y1Ba2Cu3O6.67 with
Tc=60 K. For this sample, n≈8×1027 m3, but for this
superconducting phase, it is considered that a larger effective
W
37.08
0.09
0.01
0.6
Tl
10.50
13.85
2.39
18.20
Hf
18.08
0.83
0.12
2.79
In
11.49
20.31
3.40
23.48
Ir
28.24
1.13
0.14
3.42
Hg
8.52
22.43
4.15
25.10
Ti
22.64
2.92
0.39
6.45
Sn(w)
14.48
24.00
3.72
26.25
Cd
9.28
3.11
0.56
6.73
La
8.10
31.89
6.00
31.73
Zr
17.16
3.75
0.55
7.62
Ta
27.75
35.90
4.48
34.33
Ru
58.88
5.25
0.51
9.53
Pb
13.20
44.97
7.19
39.90
Zn
13.10
5.49
0.88
9.82
V
36.10
47.06
5.38
41.12
Ga
15.30
7.16
1.09
11.72
Tc
49.28
75.40
7.77
56.31
Mo
38.52
8.22
0.92
12.85
Nb
27.80
76.17
9.50
56.69
2/3, again let
2/3, one easily sees that, for Tp>68.35 K, Tb<Tp.
conventional element
mass (15) mb≈18me should be used. Then Tb=2.762Tp
again, one finds Tb<Tp, if Tp>22.07 K. Finally we have
Tb=Tc=60 K, Tp=101.26 K, nb=2.24×1026 m3. Now, let's see
what one can say for nuclear spinlattice relaxation
experiments of Y1Ba2Cu3O7δ with superconducting transition
temperature of 60 K and 90 K respectively. For experimental
results, see W. W. Warren et al. (16) and M. Takigawa et al.
(17). It is known that when Tp is arrived, because of the
formation of carrier pairs, the state of conducting carrier will
produce a deep change, which definitely, will cause the
corresponding change of nuclear spinlattice relaxation rate
(T1T)1. For La2xSrxCuO4 and Y1Ba2Cu3O7, although Tp may
not be Tc, for the sake of Tp∼Tb∼Tc, the anomaly of curve
(T1T)1 vs. T is not so obvious. Nevertheless, for 60 K
Y1Ba2Cu3O6.67 superconductor, because Tc=60 K, Tp=101.26
K. Curve (T1T)1 vs. T will produce a distinct anomaly at
Tp=101.26 K, instead of at Tc=60 K. Here, the nuclear spin
lattice relaxation rate experiment is easily explained in this
new superconductivity theoretical framework without any
anomaly. This provides substantial support for this
theoretical framework.
The recent ARPES experiments (18, 19) reveal evidence of
an energy gap in the normal state excitation spectrum of the
underdoped cuprate superconductor Bi2Sr2CaCu2O8+δ, and a
sharp peak feature of superconducting state for underdoped
or overdoped samples. These contradict previous theories.
But, with the application of the new scenario mentioned in
this paper, this puzzle is explained. In fact, underdoped
samples generally correspond to the case: Tc=Tb<Tp, which
means that carrier pairs and energy gaps preform before the
occurrence of superconductivity. That is, carrier pairs form
without longrange coherence
superconducting realm. In the mean time, overdoped
samples generally correspond to the case: Tc=Tp<Tb, (i.e.,
standard BCS case) which means that when energy gap and
carrier pairs are formed, the system will go into the
superconducting state automatically. By the way, the
optimally doped Tc is obtained when Tb~Tp. As for the sharp
peak feature of superconducting state, this paper suggests
2/3, and
well above the
3
Page 4
that it might be the natural consequence of BoseEinstein
condensation. It even is hoped that further study of the peak
intensity evolution with temperature will reveal quantitative
evidence of BoseEinstein condensation. Qualitatively,
when temperature increases from 0 K to Tc, the sharp peak
will go down from the maximum intensity to that of normal
state.
The difference between superconducting electron pairs
and general carrier pairs.
At temperature Tp, although carrier pairs are formed with the
concentration nb, it doesn’t mean that these carrier pairs
become superconducting electron pairs. In fact, at
temperature T (T<Tp, Tb), the number density of
superconducting electron pairs nS, is given by
nS/nb=1(T/Tb)3/2. For the case: T<Tp<Tb, when T increases
from zero to Tp, nS tends to zero because of broken pairs and
superconductivity disappears. For the case T<Tb<Tp, when T
increases from zero to Tb, nS tends to zero and
superconductivity disappears even though carrier pairs still
exist.
Experiment designed to verify this integrating scenario.
The thought to consider superconductivity as a result of
superfluidity of charged carrier pairs has been suggested in
the early years of superconductivity theory. But according
to the scenario outlined above, it can be seen that Tc and Tb
are not always the same temperature, sometimes, Tc = Tp, as
in the case of BCS conventional superconductors, and
sometimes, Tc = Tb, as in the underdoped HTS cuprate
superconductors. These
temperatures make understanding the mechanism of
superconductivity even more complex. However, the great
success of BCS theory cannot deny that the superfluidity of
the charged carrier pairs is a control factor for the occurrence
of superconductivity. Therefore, a workable and decisive
experimental design is suggested for the experimenters to
validate the viewpoint presented in this paper. The
experimental principle is simple. Because the most direct
proof of the pairing effect of charged carriers is the Giaever
effect, this paper suggests a new Giaever sandwich like
Al/Al2O3/HTS. Where HTS may be the underdoped
Y1Ba2Cu3O6.67, or Bi2Sr2CaCu2O8+δ (both should have Tc~60
K) it is suggested that the similar Giaever effect can be seen
in the temperature range T~(70~80 K). Because the
temperature range chosen is just in the range Tb=Tc<T<Tp.
this experiment should be an executable one under current
experimental conditions, though it can be a difficult one due
to the very reactive surface of underdoped samples. A
further consideration may be the new Josephson sandwich
formed by the underdoped cuprate superconductors
mentioned above. Because of the conventional Josephson
effect observed under the condition T < Tc = Tp <Tb, it is
reasonable to expect some new consequences from this new
Josephson sandwich in the temperature range Tc = Tb< T <
Tp. Of course, it will deepen the current understanding of the
two competing transition
pair tunneling effect and the coherence effect among carrier
pairs. They are combined in the case of conventional
superconductors and are difficult to distinguish from one
another. The third consideration is a low temperature
specific heat experiment at appropriate temperature.
Conclusion, discussion and prospect of this new scenario.
Based on the essential understanding that there should exist a
unified theory for the conventional BCS superconductors
and the high transition temperature cuprate superconductors,
a new theory framework is put forward, which reduces to
BCS pairing theory for convention superconductors.
However, for cuprate superconductors, it develops into a
very new theory that suggests the origin of the anomalies in
the normal state of cuprate superconductors is the same as
that which induces Cooper's
superconducting states. This viewpoint is in agreement with
many other theories, such as RVB theory, Luttinger liquid
theory, marginal Fermi liquid theory, etc. From this new
theory framework, many significant conclusions can be
drawn.
(a) Superconductivity transition temperature depends on the
delicate balance among some competing factors from
which pairing effect and BoseEinstein condensation
effect are two important factors. Which factor becomes
the control or dominant one depends on the samples
(doping) and it is this sample dependence that makes the
research on the mechanism of superconductivity more
complex.
(b) Because this theoretical framework stands above pairing
theory and BoseEinstein theory, the properties of the
superconducting state, e.g., zero resistance and Meissner
effect, will result from this theory unambiguously.
(c) From formulae (1) and (4), one sees that a larger Tc
requires a larger Tp and Tb, the cuprate superconductors
are samples with larger Tp, and as for a larger Tb, one
needs a larger nb, which requires a smaller EF(0). A
smaller EF(0) means a larger effective mass, which
means heavy Fermions. This theory appears to reveal
its potential in application to heavy Fermion
superconductors.
(d) Many people know that element superconductors with
lower carrier concentrations n have higher transition
temperatures Tc, which appears difficult to understand.
In fact, these superconductors generally have Tp and
EF(0) such that the ratio Tp/EF(0) has a larger value.
Therefore, a larger effective carrier pair concentration nb
can be obtained even though it has a smaller n.
(e) Unusual energy gap. We know that for Y1Ba2Cu3O7
(Tc∼90 K), the experimental value of the energy gap is
2∆ab(0)/(KTc)=6∼8, where, 2∆ab(0) is energy gap at
temperature T=0 K. According to the scenario in this
paper and BCS theory, it is found that the significant
quantity is 2∆ab(0)/kTp, rather than 2∆ab(0)/kTc, because
Tc<Tp, for Y1Ba2Cu3O7, one can get a smaller value of
pairing effect of
4
Page 5
the quantity 2∆ab(0)/kTp. This finally clarifies the so
called unusual energy gap phenomenon. i.e., Eg induced
by pairing effect scales with Tp instead of Tc, which
solves this question. In fact, ARPES experiments
suggest the lack of scaling of the energy gap with Tc.
(f) From formulae (4) and (7), one understands that Tp and
Tb both depend on nb and n sensitively, which strongly
implies the possibility of the application of high
pressure effect on the enhancement of Tc. This is
especially true for the cases for which Tc=Tb<Tp, such
as HTS superconductors. On the other hand, the high
pressure effect will deepen the current understanding of
the superconductivity mechanism.
(g) Because the low temperature specific heat of the ideal
Boson is ~T3/2 (when T < Tb), which is different from
that of the free electron (~T), then if carrier pairs are
considered as Bosonlike particles, further measurement
of specific heat will give meaningful results.
The author believes that this theoretical framework clarifies
many key problems in the realm of high temperature
superconductors and this new scenario will stimulate further
research in this field, especially in the pursuit of new Tp that
originates from interactions other than electronphonon
interaction. Otherwise, in the case of underdoped
superconductors, Tp may not be Tc, even when one has a
correct Tp.
Acknowledgements: The author hopes to thank Dr. Claude
Jacques, Dr. George Chapman of NRC for help in preparing
this paper, Professor A. M. Tremblay of The University of
Sherbrooke for instructive and stimulating discussion.
(h) It is believed that the curious properties of the
superconducting state come from the corresponding
properties of carrier pairs, i.e., carrier pairs present
themselves as a BoseFermi statistical duality, which
will merit further study.
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5