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1
Resonance-enhanced high transition temperature superconductors within
the BCS scheme
Je Huan Koo*
Department of Electrical and Biological Physics, Kwangwoon University, Seoul 01897,
Republic of Korea
ABSTRACT
The high transition temperatures for cuprate and pnictide superconductors hinders
explanation in light of the Bardeen-Cooper-Schrieffer (BCS) models of superconductivity.
One major reason for this hindrance can be resolved via another distribution of electrons. We
have found an anyonic distribution when the site created by deleting a hole is occupied by an
electron termed the Kwangwoon distribution. The resonance between superconducting
electrons under conventional BCS scheme and independent charge density waves (CDWs)
may drive this high transition temperature superconductivity (HTSC).
PACS numbers:74.72.Mn; 75.30.Fv; 72.15.Nj
Keywords: Superconductors; Cuprates; Pnictides
* Corresponding author.
E-mail addresses: djhkoo@gmail.com, koo@kw.ac.kr
Tel:+82-(0)2-940-8257, Fax:+82-(0)2-6442-2269
2
1. Introduction
The behavior of the low-temperature superconductors reported by Onnes [1] can be
satisfactorily described by Bardeen-Cooper-Schrieffer (BCS) theory [2]. In 1986, Bednorz
and Müller [3] investigated the copper oxide compounds in quasi-two-dimensional (2D)
electronic structures and discovered high-temperature superconductors. However, the linear
dependence on the electrical resistivity to temperature, the very high superconducting
transition temperature, and the origin of the pseudo-gap requires a clear explanation to
achieve full agreement on the theoretical model of HTSC. Some models have been proposed
to explain the HTSC phenomena such as the s = 3/2-hole composite model [4,5], the
ferromagnetic cluster theory [6,7], the spin fluctuations [8-10], and a resonating valence bond
[11-13]. The theoretical approach to HTSC is under development both in the Heisenberg
antiferromagnetic model [14] using Green’s functional formalism and in the attractive
Hubbard model [15] using the mean-field theory.
Cuprate superconductors exhibit 2D superconductivity in the CuO2 plane, and Fe-As
compounds exhibit two-dimensional superconductivity in the Fe-As plane. Thus, the lower
dimensions of these systems are related in some way to superconductivity. The higher
transition temperatures of HTSCs are relatively higher than conventional BCS
superconductors, which suggests another mechanism or a much more complicated BCS type
mechanism. As a counterproposal, another distribution of electrons (different from the Fermi-
Dirac distribution) can be applied to heavy fermion superconductors [16] within pure BCS
schemes. In this paper, we describe another distribution of electrons applied to HTSC.
2. Theory
First, let us consider anyonic distributions in the case of electrons occupying the site
3
of deleted holes. A new anyonic distribution can be derived and described as the Kwangwoon
distribution:
tanh 2
1
1 exp( )
1
1 exp( )
FD FD
Kwangwoon hole electron anyon
B
FD
hole
B
FD
electron
B
f f f f
kT
f
kT
f
kT
−
= − =
=−
+−
=−
+
(1)
Here,
is the energy,
is the chemical potential,
B
k
is the Boltzmann constant,
T
is the
temperature, and FD means Fermi-Dirac distribution (Fig. 1).
We first consider HTSCs analogous to BCS superconductors. The BCS-type
Hamiltonian [2] for low transition temperature superconductors is given by:
, , , , ' , ' ', ' ,
, , ', , , '
1
( ) ( )
2
BCS k k k BCS c k q k q k k
k k k q
H a a U U a a a a
+ + +
+−
= + +
(2)
where the BCS-type electron-electron interaction is given by:
2
22
, , '
2
( ) ( )
q
BCS k q k q
g
U
+
=−−
(3)
In the above equations,
q
is the phonon energy,
g
is the coupling constant of
electron-phonon interactions,
designates the spin states,
,k
is the electron kinetic
energy,
,k
a
is an annihilation operator, and
c
U
is the Coulomb interaction.
The reduced BCS-type Hamiltonian thus becomes:
4
**
, , ,
, , ', , , '
' ' '
'
( ) ( )
,
BCS k k k k k k k
k k k k
k k k q
k kk k kk BCS c
k
kkk
H a a a a a a b
V b V U U
b a a
+ + +
− −
−
= − + −
= − = +
=
(4)
when used with the BCS approach, in which the brackets <> denote the average of the mean-
field and
k
is the superconducting gap.
Using the Bogoliubov transformation [17], the operators are given by:
*01
*01
k k k k
k
k k k k
k
a u v
a v u
+
++
−
=+
= − +
(5)
Here, the operator,
0k
, corresponds to a quasiparticle composed of an electron,
()k
, with
amplitude
k
u
and a hole
()k−
with amplitude
k
v
.
The HTSC gap equation is given by:
' ' ' '
'
' ' '
22
' ' '
[1 2 ( )] / 2
( ) 1/[1 exp( / )] tanh( /(2 ))
( ) ( )
k kk k k k
k
k k B k B
k k k
V f E E
f E E k T E k T
E
= − −
= +
= +
(6)
Using similar methods [17] and the above equations for high transition temperature
superconductors, the resulting superconducting gap is given by:
5
22
22
0
2
122
[2 tanh{ ( ) / (2 )} 1]
1()
(0)| |
1.14 7
1 2 ln sinh 2 (1.2) 0
( ) 8
B
BCS c
BB
d k T
NV
N U U NV
NV NV NV
k T k T
−
+ −
=+
+=
− + − − =
(7)
Here,
is the phonon energy, and
(0)N
is the density of the states at the Fermi level.
Let us now consider the spin relaxation rate in the superconductors.
From Hebel and Slichter [18], this is given as:
1
'2
2
0
1/
/ 2 [ ( , , )] (1 ) ( )(1 ( ))
/ , / , /
s n s
B B B
RT
R R x f x f x dx
x
x E k T E k T k T
=
= + −
= = =
(8)
These parameters are described in detail in [18], and s and n indicate superconducting and
normal, respectively. In the case of a Fermi-Dirac distribution, f(x), this has a peak below the
transition temperatures. However, this might have no peak in the case of an anyonic
distribution as
( ) tanh 2
x
fx=
, which is in line with the HTSC experiments [19].
Let us next consider the superconducting coherence length in HTSCs.
We can regard the coherence length as the diameter in an orbital so that from Bohr's
conjecture,
2 , 2 , h
r r n mv
= = =
and
n
is an integer,
h
is the Planck constant,
m
is the
mass of the electron, and
v
is the velocity of an electron. This becomes:
6
22
22
0
22
22
0
2
122
[2 tanh{ ( ) / (2 )} 1]
1()
[2 tanh{ ( ) / (2 )} 1]
()
(0)| |
1.14 7
1 2 ln sinh 2 (1.2) 0
( ) 8
2
B
B
F
BCS c
BB
F
d k T
NV
d k T
vv
N U U NV
NV NV NV
k T k T
hh
r NV
mv mv
−
+ −
=+
+ −
=+
+=
− + − − =
= = =
(9)
where
F
v
is the Fermi velocity.
We next consider spin gaps and pseudo gaps in HTSC.
These can be approximated by:
(10)
3
2
1
tanh( ) 0
3
1 2 1,
x
x x x for x
e for x
−
−
−
the spin-gap is given as
2
2
3
0
2
24 2
3
1
[ ( ) {1 2 }]
2 3 2
( ) 1 ( ) 2 ( 1)
4 96 ( )
B
B
kT
spin gap BB
kT
B
BB
ed
k T k T
k T e
k T k T
−
−
−
= − − −
= − − − −
(11)
where
is the Debye-cut off energy as shown in Figure 2; pseudo gaps are given
as
7
22
22
0
2
122
22
22
[2 tanh{ ( ) / (2 )} 1]
1()
1.14 7
1 2 ln sinh 2 (1.2) 0
( ) 8
2 tanh{ ( ) / (2 )} 1 0 , 0
2 tanh{ ( ) / (2 )} 1 0,
PG B
PG
PG pseudo gap
PG
B PG B
PG B BCS c c PG
B BCS c
d k T
NV
NV NV NV
k T k T
k T U U for T T T
k T U U
−
−
+ −
−= +
=
+ + − − =
+ − +
+ − +
00
:superconducting transition temperature
: on-set temperature of pseudo-gap
c
c
PG
for T T
T
T
. (12)
Let us then consider the linear resistivity given as
22
55
tanh 2
11
1/ 1/[ ] 2 tanh ( ) 1
2
~ ( ) ( ) 0
:
:
:
:
:
F
current density B
BF
B
HTSC
eEd
J J nev E
kT
JkT eEd
E ne vd kT
T for T
T electron T hole
resistivity
conductivity
V Ed voltage
E electric field
chemical potenti
−
= = =
= = = −
−
→
−
=
al
(13)
where
n
is a number density,
v
is velocity, and T5 resistivity is cancelled by electrons
and holes.
3. Discussion
When an external force
0
0it
Fe
applied to the bulk causes normal non-superconducting
materials to resonate, electrons inside the resonating material can become bosons with
angular frequencies in the region of
0
. If the external force is an electric or optical force in
the presence of isotropic 3-dimensional damped harmonic oscillations, then we obtain
8
2
2sin
d r dr
m b kr qE t
dt dt
+ + =
(14)
where
m
is the mass of an electron,
b
is a frictional coefficient,
k
is an elastic constant,
q
is a charge,
E
is an electric field, and
r
is a positional vector. This may be solved as
follows [20]:
2 2 2 2
0
0
cos( )
1
[( ) ( )
r A t
qE
Amb
m
k
m
=+
=
−+
=
(15)
At lower temperatures, the band gap [21] plays the role of a superconducting gap:
22
0
22
0
22
0
022
0
Re
coth( )
2
1()
(0)
||
()
coth( )
2
11
(0) 2 ()
R
B
R
BCS c
R
B
R
R sonance g
E
kT d
NV E
V U U
E
kT
NV E
E E E Superconducting gap
+
=−
+
=+
+
=+
= = =
, (16)
where
is a delta function and is Planck’s constant divided by
2
. The
superconducting gap and resonance energy are shown in Fig.3,4, where the BCS-type gap is
given as
2
52.5(1 ( ) ),
30
R BCS T
E = −
and the resonance gap is given by
02B
kT
=
and the
effective or pure BCS-type superconducting temperatures are as
100 , 30
BCS
cc
T K T K==
.
There is two track mechanism when this process occurs in an HTSC. This mechanism
includes the superconducting part with a Kwangwoon distribution and the CDW part.
Between these resonances inside materials, there is ~100 K of magnitude, and the effective
9
superconducting temperature will increase from ~10 K to ~100 K as the effective temperature
equals the superconducting temperature plus the resonance energy. Since CDW is pinned [22],
a pinned CDW confined in a quantum well transits from the 1st to the 2nd level via resonance
as shown in Fig. 5.
We estimate that the 1st level of the quantum well is
22
10
22
| ~ (10 0.1) , | ~ (0.1 0.001)
88
L nm L nm
hh
eV eV
mL mL
==
−−
(17)
where
22
2, 1,2,3,...
8n
nh En
mL ==
m
is the mass of an electron participating a CDW, and
L
is
the size of pinned CDW.
The clue of this resonance mechanism in HTSC is based on Inosov et al. [23] where they
assign the spin resonance to some collective mode.
We next consider the magnetic field dependence in normal metallic states [24].
From our previous work [25], the relation between magnetic field,
H
, and temperature is
given as
11
eff
B B B
k T k T H
(18)
where
is a constant,
B
is the Bohr magneton, and diagonal and off-diagonal
conductivity are given as
2
01
,
()
1[tanh ( ) 1]
[ * * * ]
: threshold constant value
xx xy
T
xx BB
B
T
J H J
VEH
E
deV eV
k T k T
J J H J k T H
eV
==
=
−
+
(19)
where
0
J
,
1
J
are constants.
10
4. Perturbations
We next consider gaps in view of the Josephson junction picture as a counterproposal to the
perturbations.
Herein, we address Josephson junctions on the axis of energy.
In the case of direct current, if
1
is the probability amplitude of electrons in the conduction
band on one side of a junction (the work function), and
2
is the amplitude on the field-
emitted electron band located outside the surface of the material, then the time-dependent
Schrödinger equation,
H
t
i=
can be applied:
12
21
;i T i T
tt
==
. (20)
Here,
T
represents the effects of transfer interactions across the work function along the
axis of energy. In this case,
T eV
=−
, where
V
is the external voltage,
is the work
function, and
t
is the specific time. We then have the following:
1 2 1 2
1 2 1 2
( ) ( )
( ) ( )
iT
t
iT
t
+ = − +
− = −
(21)
The amplitudes are
11
22
2
()
()
1 1 2 cos(2 )
iTt iTt
iTt iTt
n e e
n e e
Tt
−
−
=+
=−
= +
, (22)
where
i
n
is the number electron densities,
12
12
,nn
nn tt
= −
, and
is a normalized
factor.
From the relationship given as
11
** 1
1 2 1 2
22
1 2 2
Re{ }
[]
Re{ }
t
Jt
+
=
+
, (23)
the resulting pseudo-Josephson direct current is
0tan( )J J Tt=
, (24)
where * indicates the Hermitian conjugate, and Re denotes the real part.
The pseudo-Josephson effect on alternating current is given by
12
2 1 1 2
;i T eV i T eV
tt
= − = +
, (25)
where
eV
is the potential energy. In this case, the amplitudes are
()
1 1 1
()
2 2 1
2
11
()
()
1 1 2 cos(2 )
iTt i Tt eVt
iTt i Tt eVt
n e e
n e e
Tt eVt
−−
−−
=+
=−
= + −
, (26)
where
i
n
is the number electron densities,
12
12
,nn
nn tt
= −
, and
1
is a normalized
factor.
The resultant pseudo-Josephson for alternating current is
1
01
0
sin (1 )sin( )
cos cos( )
sin 2cos(2 )(1 )sin( ) .
cos 2cos(2 )cos( )
eV
Tt Tt eVt
T
JJ Tt Tt eVt
eV
Tt Tt eVt Tt eVt
T
JTt Tt eVt Tt eVt
+ − −
=+−
− − −
= − −
. (27)
We next consider perturbation theory using the Josephson formalism along the axis of energy
(Fig. 6).
For potentials as a junction,
0 1 0 1
0 1 0
: , ( )
::
Potentials V V V V
Josephson JunctionV V V
, the energies for V0 and V1
are
12
0
00
1
2
00
2
000
* ( / )
sin 2cos(2 )(1 )sin( )
1[]
cos 2cos(2 )cos( )
Josephson
t
For Perturbations
EV
EV
E E dt J J
eV
Tt Tt eVt Tt eVt
T
E E dt
t Tt Tt eVt Tt eVt
→
=
− − −
= − −
, (28)
Where
1
:
T eV V
V voltage
=−
Here,
0
E
is the unperturbed energy, and
E
is the energy change in the presence of
perturbation.
The superconducting gap via our new approach is given as
02
3
00
3
2
2
0
1[tan ] ,
1:Coulomb Repulsions
4
:dielectric constant
:,
:
t
F
BCS c
c
F
B
E Superconducting Gap dt T t
t
T U U
e
Uq
Fermienergy
i t Matsubara relation
kT
= = =
=+
=
(29)
where the Matsubara relation between time and temperature has been described [26].
We next consider gap symmetries in the presence of Kwangwoon distributions. There are
electron-hole configurations like charge density wave (CDW) systems [27], and thus the gap
symmetries of HTSC become
13
22
0
cos2 cos2
[cos cos ]
HTSC x y
d x y
A k x B k y
k x k y
= +
= +
(30)
where
x
k
,
y
k
are wavevectors along x or y-axis, and
0
,
d
are gaps independent of
wavevectors. Under consideration of electrons and holes, the resulting gap becomes
22
22
0
2 [cos cos ]
2 [cos cos ] .
2
HTSC
d x y
d x y
k x k y or
k x k y or
= +
−
(31)
5. Conclusion
In summary, the high-temperature superconductivity found in cuprates and pnictides might be
ascribed to another distribution of carriers, i.e., the Kwangwoon distribution under the
Bardeen-Cooper-Schrieffer (BCS) scheme. Spin gaps are different from pseudo gaps. Spin
gaps might originate from the difference between the gap in the lower temperature limit and
the other gap in the higher temperature limit. Pseudo gaps are a kind of superconducting gap
for electrons, but pure superconducting gaps can be considered as those gaps suitable for
holes. The resonance between superconducting electrons under a conventional BCS
(Bardeen-Cooper-Schrieffer) scheme and independent CDWs may drive the high transition
temperature superconductivity (HTSC). The experimental clue for this may correspond to the
so-called magnetic resonance modes in cuprates [28]. Nematic orders [29] occur for HTSC,
and these can be explained by electron-hole spin density waves (SDWs) where there are
electron-CDW and hole-CDW. The absence of a pseudo-gap in n-type materials [30] is
mainly attributed not to Kwangwoon distribution of electrons and holes but to a Fermi-Dirac
14
distribution of electrons in Eq. (12). Let us consider the pressure-induced room temperature
superconductor [31] via resonance using Eq. (16). It is given as
0
0
0
0
0
22
22
0
22
22
0
22
022
:Specific Pressure Energy
coth( )
12()
(0)
coth( )
12()
(0)
||
()
coth( )
12
(0) 2 ( )
~10
~ 300 :
P
R
BP
R
R
BP
R
BCS c
PR
B
PR
SC
res
E
E
kT E
NV E
E
kT Ed
NV E
V U U
EE
kT
NV EE
Though BCS T K
K resonance p
+
=−
+
+
=−
+
=+
+
=+
Re
~ (10 300)
eff
SC
R sonance g
eak
TK
E E E Superconducting Gap
+
= = =
(32)
In the presence or absence of magnetic fields, nematic phases are observed [32] in HTSC and
these can be explained as the resonance between ions and ionic part of CDW
because CDWs are electron-ion coupled modes.
Acknowledgments
This work was funded by a research grant from Kwangwoon University (2020).
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17
Fig. 1. Two cases are shown as two different operators including creation and annihilation; a
vacuum is denoted by |0>. Occupation is
|0 , |0a a a a
++
.
18
40 60 80 100 120 140 160
0
10
20
30
40
50
60
70
80
Theory
Spin Gap [K]
Temperature (K)
Fig. 2. A spin gap is depicted where the Debye cut-off is 150 K.
19
020 40 60 80 100
0
20
40
60
80
100
120
140
160
180
200
220
Superconducting Gap [K]
Temeperature [K]
at Tc=100 K
and TBCS
c=30 K
Fig.3 The superconducting gap is shown.
20
020 40 60 80 100
0
50
100
150
200
250
300
BCS gap [K]
Temperature [K]
BCS gap
Resonance gap
(a)
(b)
Fig.4 Superconducting gap structures are calculated (a) and are depicted (b).
21
Fig. 5. CDW is confined as (a) energy levels (b) wavefunctions.
22
Fig. 6. Junction along the axis of energy.
