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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 angle-resolved 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 angle-resolved 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 re-entrant 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 long-standing 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 Bose-Einstein 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 ~(10-5~10-4) of metallic

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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 Bose-Einstein

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 Bose-Einstein 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 so-called

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

NkT

E

N kT

E

kT

E

NkT

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

Bose-Einstein 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 Bose-Einstein

Bose-Einstein 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 .

(

(

3612 . 2

b

m

k

Here the constant H ≈ 4.42×10-16 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=18.22×10-31 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

2 ln75

π

. 02

b

T

m

n

HT

n

T

==

πh

(7)

2

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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×10-28

nb×10-23

Tp

Tb

element

n×10-28

nb×10-23

Tp

Tb

Sample two: high Tc cuprate superconductors. For La2-

xSrxCuO4, n≈6×1027 m3, (10), mb=2meff≈8me=72.88×10-31 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 La2-xSrxCuO4, 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∝M-α (if

applicable); here α is the isotope effect exponent. For the

first condition, because Tp<Tb, one has Tc=Tp∝M-α. For the

second condition, one has Tp>Tb, therefore, Tc=Tb∝M-2α/3,

which gives an isotope exponent 2α/3. Finally we see that

even for the same structure of La2-xSrxCuO4, 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∝M-α). 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 spin-lattice

relaxation rate for superconductor Y1Ba2Cu3O6.67 with

Tc=60 K. For this sample, n≈8×1027 m-3, 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 spin-lattice relaxation

experiments of Y1Ba2Cu3O7-δ 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 spin-lattice relaxation rate

(T1T)-1. For La2-xSrxCuO4 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 long-range 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

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that it might be the natural consequence of Bose-Einstein

condensation. It even is hoped that further study of the peak

intensity evolution with temperature will reveal quantitative

evidence of Bose-Einstein 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 Bose-Einstein 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 Bose-Einstein 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

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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 electron-phonon

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 Bose-Fermi statistical duality, which

will merit further study.

References

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2. M. Ishikawa, Φ. Fischer, Solid St. Comm. 23, 37 (1977)

3. W. A. Fertig, M. B. Maple, Solid St. Comm. 23, 105 (1977)

4. M. R. Schafroth, Phys. Rev. 111, 72 (1958)

5. M. R. Schafroth, Phys. Rev. 100, 463 (1955)

6. C. G. Kuper, Adv. in Phys. 8, 1 (1959)

7. C. Kittel, Intro. To Solid St. Phys. 6th Ed. P340, (John Wiley & Sons. NY. 1986)

8. L. N. Cooper, Phys. Rev. 104, 1189 (1956)

9. J. R. Schrieffer, Theory of superconductivity. (Revised printing 1983) and his Nobel lecture

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11. V. Z. Kresin, S. A. Wolf, ibid. 287

12. M. L. Cohen et al., ibid. 733

13. J. Bardeen, D. M. Ginsberg, M. B. Salamon. ibid. 333

14. Z. Schlesinger et al., Phys. Rev. B 41, 11237 (1990)

15. G. A. Thomas et al., Phys. Rev. Lett. 61, 1313 (1988)

16. W. W. Warren et al., Phys. Rev. Lett. 62, 193 (1989)

17. M. Takigawa, et al., Phys. Rev. B 43, 247 (1991)

18. A. G. Loeser et al., Science 273, 325 (1996) and references therein.

19. A. Damascelli et al., J. Electron Spectr. Relat. Phenom. 117-118, 165 (2001) and references therein.

5