Journal of Modern Physics, 2013, 4, 6-12
http://dx.doi.org/10.4236/jmp.2013.44A002 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Some Implications of an Alternate Equation for the BCS
Energy Gap
Gulshan Prakash Malik1*, Manuel de Llano2
1Theory Group, School of Environmental Sciences, Jawaharlal Nehru University, New Delhi, India
2Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México Apdo, México City, México
Received January 12, 2013; revised February 15, 2013; accepted March 1, 2013
Copyright © 2013 Gulshan Prakash Malik, Manuel de Llano. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
A set of generalized-BCS equations (GBCSEs) was recently derived from a temperature-dependent Bethe-Salpeter
equation and shown to deal satisfactorily with the experimental data comprising the Tcs and the multiple gaps of a vari-
ety of high-temperature superconductors (SCs). These equations are formulated in terms of the binding energies
of Cooper pairs (CPs) bound via one- and more than one-phonon exchange mechanisms; they con-
tain no direct reference to the gap/s of an SC. Applications of these equations so far were based on the observation that
for elemental SCs
 
,,W T
12
WT
01 0
W 0 at T = 0 in the limit of the dimensionless BCS interaction parameter

. Here ∆0 is
the zero-temperature gap whence it follows that the binding energy of a CP bound via one-phonon exchanges at T = 0 is
01
2W. In this note we carry out a detailed comparison between the GBCSE-based


1
WT and the BCS-based energy
gap for all and realistic, non-vanishingly-small values of λ. Our study is based on the experimental
values of Tc, Debye temperature Θ, and ∆0 of several selected elements including the “bad actors” such as Pb and Hg. It
is thus established that the equation for provides a viable alternative to the BCS equation for . This sug-
gests the use of, when required, the equation for

T0TT

WT

T
c
1


2
WT

T
which refers to CPs bound via two-phonon exchanges, for the
larger of the two T-dependent gaps of a non-elemental SC. These considerations naturally lead one to the concept of
T-dependent interaction parameters in the theory of superconductivity. It is pointed out that such a concept is needed
both in the well-known approach of Suhl et al. to multi-gap superconductivity and the approach provided by the
GBCSEs. Attention is drawn to diverse fields where T-dependent Hamiltonians have been fruitfully employed in the
past.
Keywords: BCS and Generalized-BCS Equations; T-Dependent Equations for BCS Gap and for Binding Energy of
Cooper Pairs; Elemental and Composite Superconductors
1. Introduction


0NV   is the dimensionless BCS interact-
The BCS equations (BCSEs) [1] for the temperature-
dependent energy gap and the critical tempera-
ture Tc of a superconductor (SC) are, respectively:


12
22
12
22







0
1
tanh 2
1d
B
k
B
kT
 

 (1)
2
0
where
tion parameter corresponding to a net attractive interac-
tion V that brings about the formation of Cooper pairs
(CPs), N(0) being the density of states at the Fermi sur-
face; Θ the Debye temperature of the SC, and kB the
Boltzmann constant.
These equations have recently been generalized [2] to
cater to composite superconductors (CSs) by following
an approach based on the Bethe-Salpeter equation (BSE).
While the BCSEs are based on the one-phonon exchange
mechanism for the formation of Cooper pairs, general-
ized-BCS equations (GBCSEs) invoke the mechanism of
tanh
1d
c
T
x
xx 

 (2)
*Present address: B-208 Sushant Lok I, GurgaonHaryana, India.
C
opyright © 2013 SciRes. JMP

G. P. MALIK, M. DE LLANO 7
multi-phonon exchanges, besides the one-phonon ex-
change mechanism, for the formation of the pairs. Addi-
tionally, while BCSEs characterize an elemental SC by a
single Debye temperature, GBCSEs characterize a CS by
multiple Debye temperatures which take into account its
anisotropy.
For a binary CS characterized by two gaps such as
MgB2, GBCSEs are [2]:
 
11
1
2
1
2
ta
1
c
B
kW
c
W
nh 2
d
B
x
kT
x
Tx


 (3)
 
 
12
2
22
2
2
1
2
2
2
2
tan
1d
ta
d
c
B
c
B
kW
c
W
kW
c
W
h 2
nh 2
B
B
x
kT
x
Tx
x
kT
x
Tx








(4)
 
 
1
2
1
0
2
0
tanh
1d
tan
c
B
c
B
k
c
c
k
c
c
Tx 2
h 2
d,
Bc
Bc
xkT
x
x
kT
x
Tx








(5)
where (3) is obtained from Equation (34) in Ref. [2] by a
redefinition of the variable of integration and refers to the
situation where CPs are bound via one-phonon ex-
changes; (4) is a generalization of (3) to the situation
where CPs are formed via two-phonon exchanges; (5) is
obtained from (4) by demanding that 2 at T = Tc;
i is the interaction parameter due to the ith species
of ions in the binary, to be distinguished from λi, which
denotes the interaction parameter of the same species in
its free state, a similar distinction applies to i and Θi.
Note that we have written 1,2
0W

cT

c

c

as


1,2 for the rea-
son that will become apparent via (12) below.
cT

In the above equations, in our earlier work [2-4], 1
has been referred to as the binding energy of CPs bound
via one-phonon exchanges. Indeed, that is how it was
first introduced in the appropriate BSE in [2]. However,
it turned out that
W
1obtained via (3) very nearly equals
∆ obtained via (1). Since we know that the energy re-
quired to break up a CP is 2∆ because of “Pauli block-
ing” of states between F and F we should
refer to
W
,E E
1
2W as the binding energy of the pairs bound
via the one-phonon exchange mechanism. Likewise, the
binding energy of pairs bound via two-phonon exchanges
is 2 2W. Even though obvious, we note here that since
1,2 also vanish when
2W1,2 W0, the equations used
in our earlier work (obtained by putting 1,2 0W

) are
correct.
The motivation for this note is primarily to establish
that insofar as elemental SCs are concerned, use of (3) is
equivalent to the use of (1) not only at T = 0 as follows
from our earlier work, but for all T ≤ Tc. Such a demon-
stration is deemed desirable prior to suggesting the use of
(4) whenever a detailed variation of the larger of two
gaps of a CS with T is required. This is so because the
equivalent of (4) via the variational approach of BCS is
not yet available. In Section 2, this exercise is interest-
ingly found to shed light on an alleged universal relation
of BCS theory, viz., 0c
2 3.53.kT

 Additionally, it
brings into focus the rather unfamiliar feature of
T-dependence of λ in the theory of superconductivity.
This feature is briefly discussed further in Section 3 in
the context of multi-gap superconductivity. We recall
that multi-gap superconductivity is usually qualitatively
addressed in the BCS theory via the seminal approach of
Suhl, Matthias and Walker [5]. This approach is juxta-
posed with the approach provided by the framework of
GBCSEs. Finally, Section 4 sums up our study.
2. A Comparative Study of BCSE (1) and
GBCSE (3)
1) We note that BCSE (2) for Tc follows not only from
(1) by demanding that ∆ = 0 at T = Tc, but also from (3)
by demanding that 10W

. This already suggests a con-
nection between 1
2) It is readily seen that when T = 0 (tanh = 1), (1) and
(3) yield, respectively:
W and ∆.

0sinh 1
B
k
 (6)

and

01
2.
exp 11
B
k
W


 (7)
Hence

001
2exp1, when0.
B
Wk   (8)
Based on this observation, (5) and the equations cor-
responding to (3) and (4) at T = 0 were recently used [3,4]
to deal with the experimental data related with the Tcs
and the multiple gaps of a variety of high-temperature
SCs (HTSCs). Specifically, it has been shown in these
papers that if one identifies 1 in (3) with the smaller
gap of an HTSC, and
W
2 in (4) with the larger gap,
then given any two parameters from the set
W


,,T12 1c


0TT
one can calculate the remaining pa-
rameter without any arbitrariness. This is similar to what
the normal BCS equations achieve for a simple SC.
3) We now propose to go beyond the approximations
which led to (8). Specifically, we carry out below a de-
tailed comparison of the implications of (1) and (3) for
all c

 and realistic values of λ (i.e., not → 0).
Our study is based on the experimental values of Tc, Θ,
and ∆0 of several selected elements including the “bad
actors” such as Pb and Hg.

G. P. MALIK, M. DE LLANO
Copyright ciRes JMP
8

T 

c
T 
4) With the input of Tc and Θ into (2), we first deter-
mine c for the elements listed in Table 1. With
known, we calculate ∆0 for these elements via (6),
and 01 via (7). The results are given in Table 1,
along with the experimental values for ∆0 (as well as the
gap-to-Tc ratio
eliminate λ from these equations, which leads to R = 3.53.
We now invert (6) and (7) to obtain
© 2013 S.
W
0
2Bc kT R). It is thus seen that: a)
generally, 010 ;W b) Both 01 W

expt
and ∆0 fall short of
0, the mismatch between them being greater for
the “bad actors”; c) The values of 01 2
B
c
WkT
calcu-
lated via (3) are closer to the values of


0Bc
than the values of
expt
2kT


0cal
2Bc kT obtained via (1).
5) Using the values of , ∆0 and

c
T 
01
W

, we now
calculate the dimensionless ratios 0 and T

0101 for all the elements under consideration,
the former via (1) and the latter via (3), for all
WTW
c. These lead to two clusters of curves,
plotted in Figure 1, which are found to be more or less
similarly compact. The thick dashed-curve between the
two clusters in the figure corresponds to a relation ob-
tained via the Gorter-Casimir two-fluid theory of super-
conductivity, see e.g., Ref. [6]:
01tTT 
 


4
01
s
N t
ss
nt Nt, (9)
where Ns is the density of superconducting electrons.
Inclusion of this curve is suggested by the observation
that:
 
01
00
01
s
Wn 
001
TT
TWT

 ,
and
  
0110. s
Wn 

001
11tt
tWt


6) Our considerations so far have been based on the
λ-values that were determined with the input of Tc and Θ
of each of the elements. An assertion of the BCS theory
is that it has no disposable parameters, which means that
the same λ must occur in both (1) and (2). Hence one can

0
0
1
arcsinh B k
   (10)

01
01
1.
ln 12 B
WkW   (11)
The values of λs obtained with the input of




expt W
001
and Θ into (10) and (11), respectively,
are also given in Table 1. It is thus seen that for each of
the listed elements,

 
W  
001
, though the for-
mer is always marginally greater than the latter. A sig-
nificant feature of these λ-values is that for all the listed
elements





001
,.
c
WT   
(12)
Using these values of


 0 and

W 
01 in (2) to
calculate the values of Tc for the elements listed in Table
1, the results are, in degrees Kelvin: Cd (0.47, 0.47, 0.42),
Pb (8.7, 8.1, 7.2), Hg (5.41, 5.14, 4.15), Sn (3.9, 3.83,
3.72), In (3.56, 3.46, 3.41), Tl (2.6, 2.52, 2.38), Nb
(10.19, 9.88, 9.25), where the first entry in the parenthe-
ses for any element denotes Tc corresponding to



0

,
the second entry to


W 
01 , and the third entry denotes
the experimental value. It is thus seen that: a) there is a
mismatch between the calculated and the experimental
values of Tc; b) the mismatch is more pronounced for the
“bad actors”, a circumstance which is responsible for
violation of the universality of R; and c) The Tcs calcu-
lated via


W 

 
01 are invariably closer to the experi-
mental values than those calculated via .
0
7) We believe the inequality in (12) to be important,
and while we will discuss it further in the next section in
the context of a CS (YBCO) which is characterized by
two ∆0s but one Tc, let us first delve on its origin. We

Table 1. Experimental values of Tc, Θ and W
00
 are taken from Poole [6]. The

c
T 

is calculated with the input of Tc
and Θ into (2), while

0
  and




W0  are calculated with the input of Θ and W
00
expt into (10) and (11), re-
spectively. Also, ∆0 and


c
T  and Θ into (7) and (8), respectively. W0 are calculated with the input of

SC

c
TK


K


00 0
p exp
meV, 2Bc WkT 
ex


c
T 


0



0
W 



00
meV, 2Bc kT
00
meV, 2
B
c
WWkT
Cd 0.42 210 0.072, 3.98 0.15770.16080.1607 0.064, 3.52 0.064, 3.53
Pb 7.2 96 1.33, 4.29 0.36820.39570.3849 1.099, 3.54 1.17, 3.78
Hg 4.15 88 0.824, 4.61 0.31450.34300.3372 0.632, 3.53 0.658, 3.68
Sn 3.72 195 0.593, 3.7 0.24480.24770.2466 0.566, 3.53 0.575, 3.59
In 3.41 108 0.541, 3.68 0.27920.28260.2804 0.519, 3.53 0.533, 3.63
Tl 2.38 79 0.395, 3.85 0.27560.28240.2802 0.362, 3.53 0.372, 3.62
Nb 9.25 276 1.55, 3.89 0.28400.29200.2894 1.407, 3.53 1.449, 3.64

G. P. MALIK, M. DE LLANO 9
δPb
δCd
δHg
δSn
δTl
δIn
δNb
wPb
wCd
wHg
wSn
wTl
wNb
wIn
r
0
0
tr
Reduced temperature
1
Reduced Dels and│W│s
1





Figure 1. Plots of T0  obtained via (2), and of obtained via (4). λ in each case is taken to be WT W
0c
T  as
given in Table 1. The upper cluster of curves corresponds to


T0 


, and the lower to

WT W
0. The thick dashed
curve between the two clusters corresponds to

t
4
1

s
n, see (9).
recall that it was pointed out by Pines [7] that λ can be
decoupled into a product of two rather disparate quanti-
ties N(0) and V. The first of these is the density of states
at the Fermi surface
22
3
0,
2πB
kv

 N
where γ is the Sommerfeld parameter, v the gram-atomic
volume of the element, and V is the algebraic sum of the
attractive interaction between electrons owing to the lat-
tice and the repulsive Coulomb interaction. We now ask
as to how V might change for an SC if, starting from T >
Tc, its temperature is progressively lowered to T = 0.
We recall that Cooper showed a long time ago
that—and this is the basis of virtually all theories that
attempt to explain superconductivity—pair-formation will
take place however small the net attraction between elec-
trons may be, causing a lowering of the energy of the
system and hence superconductivity to arise. This sug-
gests that for all T > Tc, V is positive, changes sign at T =
Tc and continues to (negatively) increase so as to attain
its maximum value at T = 0. Such T-dependence of λ is
required in addition to the T-variation because of the tanh
term in (1) in order to cause closure of the two gaps of a
binary at the higher Tc, as will be seen in the next section.
There is, however, another interesting possibility: V is
attractive even for T > Tc (i.e., there is no change of sign
at T = Tc), but (negatively) keeps increasing down to T =
0. If this be so, then there must be CPs without there be-
ing superconductivity. This suggests the existence of
“preformed CPs” before superconductivity sets in [8,9];
whether or not this feature has a bearing on the so-called
pseudogaps in HTSCs seems still to be an open question.
8) In connection with the T-dependence of λ, we note
that in a recent study [10] it has been shown that one can
account for the empirical law that relates the critical
magnetic field


c

H
T of an elemental SC with

0Hc
by assuming that V varies linearly with temperature.

G. P. MALIK, M. DE LLANO 10
9) Assuming that λ in (1) varies linearly with t, we have

 

0
2,
c
Eq
tTt    

0cc
TT 



. (13)
from Figure 2, the cluster of curves corresponding to
the latter is more compact than the cluster for the for-
mer. This is one of our main results because of its uni-
versality unlike, strictly speaking, the alleged universal-
ity of the constancy of the gap-to-Tc ratio.
Using the values of λs in Table 1, α is found to have
the values 7.381 × 10−3, 3.819 × 10−3, 6.867 × 10−3, 7.796
× 10−4, 9.971 × 10−4, 2.857 × 10−3, and 8.649 × 10−4, for
Cd, Pb, Hg, Sn, In, Tl, and Nb, respectively.


We note in passing that the cluster for 01 01 WTW

2.6
1
s
nt t
is reproduced rather well by .
Assuming a similar variation for the λ in (3)
 



01
4
10
Eq
tW 
 

1
1
,
c
cc
Tt
WTT





01t

, (14) 3. Dealing with Binary SCs via BCSEs
Compared with GBCSEs
An important feature of our considerations concerned
with elemental SCs above is that we have been willy-
nilly led to the idea of T-dependent coupling constants.
We now draw attention to the need for such an idea even
for composite (i.e., non-elemental) SCs that are charac-
terized by two ∆0s but one Tc. For the sake of concrete-
ness, let us consider the case of YBCO for which the
we find that α1, for the same elements in the same order,
has the following values: 7.143 × 10−3, 2.319 × 10−3, 5.47
× 10−3, 4.839 × 10−4, 3.519 × 10−4, 1.933 × 10−3, and
5.388 × 10−3.
For , we now solve (1) for 0using
(13), and (3) for
T

01
WT 01 W using (14). As seen
δPb
δCd
δHg
δSn
δTl
δIn
δNb
wPb
wCd
wHg
wSn
wTl
wNb
wIn
r
0
0
1

Reduced Dels and│W│s
tr 1
Reduced temperature


Figure 2. Plots of T0 obtained via (2) with variable λ given in (11), and of WT W
0 obtained via (4) with λ given in
(12). The upper cluster of curves corresponds to




T0, and the lower to

WT W
0
.

s
nt
26
1.
. The thick dashed curve corre-
sponds to

G. P. MALIK, M. DE LLANO 11
characteristic set of parameters is given
by [3].

01 02
,,,
c
T 

92 K,410 K

5.5 meV,20.0 meV, (15)
It was pointed out by Suhl et al. [5] that itinerancy of
electrons between the s- and the d-bands and the overlap
between them can cause an SC to have two gaps via (1)
and, in general, two critical temperatures via (3) within
the framework of BCS theory. This happens because λ is
now given by a quadratic equation. Indeed, this is the
approach that has frequently been invoked, albeit qualita-
tively, to deal with two-gapped SCs.
Let us therefore first apply this approach to YBCO.
The λs corresponding to the two gap values in (15) cal-
culated via (10) are found to be 0.391 and 0.75, leading
to Tc = 36.2 and 123 K, respectively. On the other hand, λ
calculated with the input of Tc from the set (15) into (2)
is found to be 0.615, which leads to the value of 14.4
meV for the larger gap. Note that we are working in the
“BCS-Suhl” scenario and not in the “Eliashberg-Suhl”
scenario. Therefore we posit that a value of λ exceeding
the minimum upper bound of 0.5 via the Bogoliubov
criterion [11] is unacceptable. Furthermore, even if one
ignores the difference between 0.75 and 0.615, we need
to close both the gaps at the higher Tc i.e., 123/92 K. We
assert that this cannot be brought about without invoking
T-dependence of the λs. This will be explicitly shown
elsewhere.
We note that, as has been shown [3,4], none of the λs
invoked in the framework of GBCSEs to deal with the
gap-values and the Tc of any HTSC violates the Bo-
goliubov constraint mentioned above. In the context of
YBCO, since it has already been shown as to how
GBCSEs account for its superconducting properties, we
need to make just one additional remark which is: for
closure of the two gaps of YBCO one requires, again, the
λs in the theory to be T-dependent.
Having willy-nilly been led to the not-so-familiar do-
main of T-dependent approach here, it seems pertinent to
note that such an approach has factually been employed
in diverse fields in the past by: 1) Bogoliubov, Zubarev
and Tserkovnikov, as discussed in [11, p. 250]; Sheahen
[12], and in Ref. [10] in the context of superconductivity;
2) by Weinberg [13], Linde [14], and Dolan and Jackiw
[15] in the context of finite-temperature behavior of a
class of relativistic field theories (RFTs) to address the
question of restoration of a symmetry which at zero
temperature is broken either dynamically or spontane-
ously; 3) in Ref. [16] for a model RFT; 4) for an expla-
nation of the legions of unidentified solar emission lines
in Ref. [17], and 5) in QCD to explain the masses of dif-
ferent quarkonium families and their de-confinement
temperatures in Refs. [18,19].
4. Conclusions
1) We believe to have presented in detail a convincing
case for regarding (3) as a viable alternative to (1).
2) The above finding suggests the use of both (3) and
(4) whenever a detailed variation of the two gaps of a CS
with T is required.
3) While our main motivation for this note was to
draw attention to the above feature, we have collaterally
been led to the somewhat not-so-familiar domain of
T-dependent Hamiltonians. This led us to draw attention
to diverse fields where T-dependent Hamiltonians have
fruitfully been employed in the past.
4) The well-known approach of Suhl et al. [5] to
multi-gap superconductivity was recalled. It was pointed
out that if experiment dictates that the two gaps of an SC
close at the same (higher) Tc, then both Suhl et al.’s and
the approach provided by GBCSEs need to invoke
T-dependence of the interaction parameters.
5) We conclude by noting that considerations of this
note find an immediate and important application in the
study of the thermal conductivity (κ) of high-Tc SCs via
the Geilikman [20] and Geilikman-Kresin [21] or, equi-
valently, the Bardeen, Rickayzen and Tewordt theories
[22]. Specifically, we are now enabled to investigate the
variation of κ with T both in the scenario where the two
gaps of say, MgB2, close at the same Tc and in the sce-
nario in which they do not. We recall that thermal con-
ductivity is a non-equilibrium phenomenon and that it is
measured under conditions of no electric current. Since a
thermal current tends to drag a small electric current with
it, this current must be balanced by an equal and opposite
supercurrent. For these reasons measurement of κ re-
quires a rather elaborate experimental set up. A pertinent
question therefore is: Could the cumulative effect of the
stresses caused by such a set up lift the degeneracy of the
two gaps closing at the same Tc? A preliminary study
shows that this is indeed so i.e., the experimental data on
κ(T) of MgB2 is better explained in the scenario in which
its two gaps close at different temperatures. These find-
ings will be reported elsewhere.
5. Acknowledgements
GPM thanks Prof. D. C. Mattis for correspondence hav-
ing a bearing on this note. He thanks Ms. Amudhakumari
for an interesting discussion—during EUROQUAM2010
—about the contents herein. MdeLl thanks UNAM-
DGAPA-PAPIIT (México) for partial support from grant
IN106908.
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