J. Mod. Phys., 2010, 1, 328-339
doi:10.4236/jmp.2010.15047 Published Online November 2010 (http://www.SciRP.org/journal/jmp)
Low Dimensionality as a Factor Stimulating Formation of
the Cooper-Like Pairs Characteristic for Superconductors
Stanisław Olszewski*, Tomasz Roliński
Institute of Physical Chemistry, Po lish Academy of Sciences, Kasprza ka, Warsaw, Poland
Received August 10, 2010; revised October 1, 2010; accepted September 16, 20 1 0
Abstract
The electron gas examined in a very thin potential tube exhibits some special kind of the excited pairs mak-
ing them similar to the Cooper pairs. The coupling energy of the pair can be calculated as an amount of en-
ergy required to transform the excitation energy of a coupled pair into the one-electron excitation energy. For
an extremely thin potential tube the coupling energy of the pair tends to infinity. The gas energy is unstable
with respect to the pair excitation which provides a kind of gap near the Fermi level. A decisive part of the
gap energy is due to the electron-electron interaction. The gap is attained on condition the length of a thin
potential box exceeds some critical value. In the next step, a coherence length in the gas is obtained. This
length, combined with a critical magnetic field representing a transition from a superconducting to a normal
state, allows us to calculate the penetration depth of the magnetic field for the singlet and triplet excitations.
The penetration depth together with the critical magnetic field and energy gap can provide us with a critical
current, as well as critical temperature for the superconducting state.
Keywords: One-Dimensional Superconductors, Electron-Electron Interaction, Cooper-Like Pairs
1. Introduction
Discovery of the superconducting behaviour in the nearly
one-dimensional compounds, as well as advances in
semiconductor technology of the wire-like structures,
provided a special impact of interest in the one-dimen-
sional many-electron systems. One of the main features
of the superconducting state described by the Bardeen-
Cooper-Schrieffer (BCS) theory is that two one-electron
states are occupied simultaneously by an electron pair
called the Cooper pair. The electrons in the pair have the
same absolute value of momen tum, but an opposite spin.
Moreover, if the electron momenta are considered in the
space of the wave vector k

, their directions in the pair
are exactly opposite. This property is characteristic also
for the excited states of the pair. A factor coupling the
electron pair together is usually attributed to the crystal
lattice: it is assumed that some phonons can provide an
attractive potential between electro ns pu tting th e pair in a
kind of the bound state. If some gap exists between the
ground state and an excited state of the Fermi sea, the
transport of the pair occupying the excited state can
exhibit the properties of a superconducting behaviour.
The superconductivity in one dimension has attracted
much theoretical attention [1-4] especially after the
experimental discovery of superconducting effects in the
one-dimensional organic conductors [5-9]. But an interest
came also because of the early claim that a high tempera-
ture superconductivity can be realized in one-dimen-
sional materials [10]. In this context an interesting
indication concerning an increase of superconducting
correlation due to dimensionality change in quasi-one-
dimensional conductors has been also done [11]. Simulta-
neously, the nature of supeconductivity in quasi one-
dimensional systems seems to remain controversial since
its discovery [12].
An important remark on superconductivity in one
dimension has been done by Friedel who suggested that
the coupling potential characteristic for the electron pair
is due to the electron-electron interaction and is not a
phonon-mediated effect [13,14]. The purpose of the
present paper is to point out that, in fact, a pair-like
excitement of the electron gas typical for the Cooper
pairs is not necessarily limited to a situation due to the
lattice potential and an interaction provided by a
phonon-like coupling. In particular, it is demonstrated
that the coupling of an electron pair, and an excitement
of the coupled pairs, can be an effect strictly connected

S. OLSZEWSKI ET AL. 329
with the low dimensionality of the electron gas. More
accurately, the pairing is due to the properties of the
Coulomb and exchange interaction between electrons
which become sound on condition the space occupied by
the electrons is narrowed to a very thin potential tube.
In the detailed calculations the Coulomb and exchange
interactions in one-dimensional systems are studied in
the framework of the Hartree-Fock approximation. It can
be shown that these interactions can give a strong
coupling energy of the electron pair having the same
absolute momenta, but opposite spin, on condition the
transversal cross-section of the gas potential tube is
tending to zero. The calculation of the excitation energies
of the one-dimensional electron gas is also based on the
Hartree-Foc k method.
Evidently, in a physical practice the electron gas can
never become strictly one-dimensional, nevertheless this
kind of situation is roughly approached when electrons
are enclosed in a very long and very thin potential tube.
Simultaneously, because the Pauli principle and the
Fermi statistics should be obeyed also in this limiting
case, the magnetic properties of the electron ensemble
become here of importance. These effects are studied on
the same footing as applied to the electron interactions in
a non-magnetic gas.
2. One-Electron and Two-Electron
Excitation Energy of an Almost
One-Dimensional Electron Gas
The electron gas energy in the approximation by Hartree
and Fock is a sum of the kinetic energy and the Coulomb
and exchange energy due to the interaction between the
electron particles. The interaction of electrons with a
positive core can be neglected because it can be con-
sidered as a constant term which remains unchanged in
course of the electron excitation process. The potential
tube is assumed to be so narrow that only the kinetic
energy excitations of the longitudinal motion of electrons
along that tube should be taken into account, with no
allowed transversal kind of the electron transitions.
In a ground state of the gas its one-electron levels are
assumed to be doubly occupied by electrons of an
opposite spin between the lowest level of =1n and
the highest occupied level F nn =. In consequence,
there exists no net magnetic moment of such a gas; see
Figure 1.
An electron excitation from some level g n located
within the interval 1
g
F
nn (1)
to a level e
n above F n, so
>,
eF
nn (1a)
(a) (b)
Figure 1. Pattern of the energy levels filling the Fermi sea of
a non-magnetic one-dimensional electron gas: (a) a scheme
before excitations, (b) a scheme after a pair excitement
from the level
F
n to the level 1
F
n. Full circles denote
the occupied electron states.
requires an excitation of the kinetic energy by the
amount (see e.g. [15])

222
2
=8
kine g
e
h
Enn
mL
 (2)
Here L is the length of the gas potential tube.
Simultaneously, the electron interaction energy due to an
excitation from g n to e n is changed by the amount
(see e.g. [16])


==/2
=1
=
=2 2
;
in N
F
eeninini ni
ee gg
i
in
g
nnn nnn
eg ggeg
EJKJK
JJK


 


 
 (3)
J and
K
are the Coulomb and exchange integrals,
respectively, the plus sign before
K
in the last step of
(3) refers to a singlet excited state of the gas, the minus
sign—to a triplet state. In the absence of the magnetic
field the quantum levels of a non-magnetic gas of N
electrons in their ground state are doubly occupied by
electrons having opposite spin, so /2= NnF.
More detailed formulae are:
 
2
22
12 12
12
=,
ijij e
Jdrdrr r
rr
 

 
 (4)
 
2
12 1122
12
=,
ijij ije
Kdrdrrrrr
rr
 

 

 (5)
where
1/2
2
()=sin()cos ,
lq llq
n
rNJurl
LLz






 (6)
with

S. OLSZEWSKI ET AL. 330
=(,,),nlq

 (6a)
is a real wave function of a free electron vanishing at the
boundary of the potential box. lq
N is the normalzation
coefficient of the ),(

r-dependent part of )(r


. l J
is the Bessel function of the first kind and index l
extended over the interval
,<<0 Rr (7)
where R is the radius of a circular cross-section of the
cylindrical potential tube, lq u is the argument giving
qth zero of the function l J at R
r
=. Regularly, the
radius R is assumed so small that excitations along the
potential tube are only considered. This means that only
quantum number n can be changed in course of an
excitation process and the number 0=l as well as
1
01 2.405= 
Ruulq remain constant for the whole of
calculations.
In a thin potential tube any Coulomb integral nm J
)=( mn  has a large component [17]
2
2ln,
eR
L
 (8)
and the same component enters exchange integral nm K
)=( mn . This implies that at 0R the expression in
(3) given by the terms enclosed within the square
brackets converges. A difficulty comes from the
remainder terms in (3 ) which are
.
nnn nnn
eg ggeg
JJK (3a)
Any integral nn
J has its large component [17]
2
3ln
eR
L
 (8a)
instead of (8). In effect, for the plus sign before g
n
e
n
K
the dominant terms in expression (3a) become at small
R equal to:


222
23ln2ln =ln.
eee
RRR
LLL
 (3b)
This is a large positive number. On the other hand, for
the minus sign before g
n
e
n
K, the expression (3a) gives
the dominant terms


222
23ln2ln =3ln,
eee
RRR
LLL
  (3c)
which is a large negative number at small R.
In effect, for a plus sign before g
n
e
n
K characteristic
for a singlet excited state, the expression in (3) diverges
at 0R giving
,
ee
E (9)
which provides us with an infinite (positive) excitation
energy at 0R.
But a different situation can exist, however, when not
a single electron but an electron pair occupying the same
one-electron level Fg nn

is excited to a level
Fe nn >, so after an excitation both electrons occupy e
n.
The change of the kinetic energy associated with such
transition is twice of that represented in (2):

222
2
=2 .
8
kine g
e
h
Enn
mL
 (10)
Simultaneously, the electron-electron interaction
energy of an electron pair located first on level g n, next
on level e
n, is changed by the amount [16]:
=.
pair
eennnn
ee gg
EJJ (11)
But each Coulomb integral having its both indices the
same, contains the same term (8a) as its component [17].
This property makes (11) a convergent result also at
0R.
A full change of the electron-electron interaction
energy of the gas due to the pair excitement from g
n to
e
n becomes:
 
==
=1 =1
==
=4 2
.
in in
FF
eein ininin
ege g
ii
in in
gg
nnnn
ee gg
EJJ KK
JJ




(12)
The number of nm
J )=(mn  entering (12) with a
plus sign is equal to the number of mn
J )=(mn



entering with a minus sign, and the same property
concerns the number of nm K and mn K having
opposite sign. Also be cause a s ingle e
n
e
n
J entering (12)
with a plus sign is combined with a single g
n
g
n
J having
a minus sign, the electron excitation energy (12) is a
fully convergent result also at 0R. In summary, for
a thin potential tube, an excitation of some special kind
of the electron pairs requires much lower energy than the
one-electron excitations. In a limiting process of the
cross-section of the potential tube tending to zero, any
one-electron excitation requires an infinite quantity of
energy, but this does not apply to a two-electron
excitation of the kind discussed above. This property is
used in the forthcoming sections of the paper.
3. Positive Change of the Kinetic Energy and
Negative Character of the
Electron-Electron Interaction Change
in Course of a Pair Excitation
The change of the kinetic energy of the electron pair in
course of transition from the level g
n to the level e n
is evidently a positive quantity, because >
eg
nn; see
(10). But the change of the electron-electron interaction

S. OLSZEWSKI ET AL. 331
of the same pair for an excitation from g n to e n is
coupled with a negative change of energy. For example,
at small R the Coulomb interaction energy of the pair
located on level 1 is [17]

2
11 0,2
2
=2
3ln3ln3ln2ln3ln2
e
JA
L
eRL
L



 
(13)
whereas the same interaction energy on level 2 is

2
22 0,4
2
=2
3ln3ln3ln2ln3ln2,
e
JA
L
eRL
L



 
(14)
where 0.681

is a term descending from the
normalization process and

is the Euler constant.
Since
0,2 2
=(2),
n
A
Si n


 (15)
where Si is the integral sinus, a difference of the
Coulomb energy in level 2 and level 1 becomes a
negative result:
2
22 11
22
2
211
ln 2242
1
=ln 2(0,59).
e
JJ L
ee
LL





 
 



 


(16)
Here the integral sinus entering (15) has been calcu-
lated with a satisfactory accuracy from the formula
cos
() 2
x
Si x
x

 (17)
because
x
equal to

multiplied by an even integer
number is a sufficiently large argument to apply the
approximate formula (17).
A still more approximate approach to the change of
the electron interaction part of the excitation energy of an
electron pair excited from some level F n, which can be
considered as a Fermi level, to a level Fe nn>, gives
also a negative value:
 
2ln 2ln 2.
eeeF
e
Enn
L
 

(18)
For 2=
e
n and 1=
F
n this gives
22
ln2=0.69
ee ee
ELL  (18a)
instead of (16).
A superconducting character of a system requires an
instability of the Fermi energy with respect to the pair
excitation. This is attained for a full excitation energy
=,
kin ee
EE E

 (19)
where kin E

of (10) is added to ee E of (18), on
condition ee E

predominates over kin E.
With the aid of the term
26
2
== 0.02610
8e
h
f
cm
me

 (20)
we can look for such cr LL = that

 
22
22
2
=
1
2lnln=0.
8eFeF
e
E
he
nnn n
mLf
L







(21)
This gives for cr
LL = the equation
 
22
=2ln ln
eF
cr eF
nn
Lf
nn

 (22)
where
=
eF
nn n (23)
and 0>n

. In order to give a negative excitation
energy of the electron pair the length L of the potential
tube should be larger than some critical length cr L
calculated in (22). In Figure 2(a) we plot cr L as a
function of F n for three integer values of n.
Here it should be noted that in calculating cr L in (22)
the electron interaction of the ‘bare’ pair submitted to an
excitation with the electron pairs occupying deeper levels
in the Fermi sea has been neglected. The correction due
to that interaction effect is calculated in Section 4.
4. Pair Interaction with the Fermi Sea
Section 3 examined only the Coulomb electrostatic
energy of a ‘bare’ electron pair; no exchange energy is
then involved because electrons have an opposite spin. In
reality, such a pair is interacting with the Fermi sea. Our
aim is to calculate a correcting term equal to the change
of the pair interaction due to the presence of other
electrons in the gas in course of an excitement of the
electron pair. The index of an excited level of the pair is
assumed to be 1

F
n, and the level on which the pair is
located before excitation let be F n; see Figure 1. The
interaction energy of the pair located in 1
F
n with the
Fermi sea is equal to

=1
,1 ,1
=1
22
in
F
inin
FF
i
JK



 (24)
(the level F n is emptied), whereas a similar interaction
energy of the pair located in the level F n is

S. OLSZEWSKI ET AL. 332
(a)
(b)
Figure 2. Critical length cr L of the electron gas potential
tube. A formation of the gap at the Fermi level requires the
tube length >cr
LL. (a) cr L for an excitation of a single
electron pair with neglected pair interaction with the Fermi
sea [see (22)]; (b) cr L calculated for an excited electron
pair interacting with the Fermi sea [see (32)]. The curves
are plotted for =1n (lowest cr L), 2, 3 (highest cr L).

=1
,,
=1
22 .
in
F
in in
FF
i
JK


 (25)
The factor of 2 before the integral
J
within the
brackets is due to the fact that the Coulomb energy
concerns an interaction which involves both kinds of
spin, whereas the exchange integral concerns an inter-
action only between electrons having spin of the same
kind [16].
In effect, the correcting term representing the change
of the electron interaction energy connected with the pair
transition from the level F n to the level 1

F
n
becomes
 
=1 =1
,1 ,,1,
=1 =1
=42 .
in in
FF
corr
inti nini nin
FFF F
ii
EJJKK




(26)
For large F n the first sum in (26), which can be
approximated by [17]

=1 2
,1 ,
=1
1,
2
in
F
in in
FF
iF
e
JJ nL


 (27)
becomes negligibly small in comparison with the second
sum in (26) which becomes approximately equal to [17]:
=1 2
,1 ,
=1 ()ln(2).
in
F
ininF
FF
i
e
KKn
L


 (28)
In total, the correction of energy due to the pair
interaction with the Fermi sea calculated from (27) and
(28) is a negative term:

2
2ln2.
corr
eeFe
En
L
 (29)
Evidently, for large F n, the excitation energy of a
‘bare’ electron pair calculated for 1= 
Fe nn [see (18)
and (18a) for a special case of 2=
e
n and 1=
F
n] is
negligibly small with respect to corr
ee
E, so

2
=2ln2.
corr
eeeeFe
EEn n
L

 (30)
This is a negative term because the difference of the
exchange integrals entering (26) has a minus sign. The
interval n

is that given in (23), where 1>n can be
also admitted.
Consequently, a critical length of the potential tube
descending from the requirement that
==0
kin ee
EE E

 (31)
[see (22)] becom es


 
22
22
=.
2ln2 ln2
eF F
cr FF
nn n
Lff
nn n

 (32)
In the formula for cr
L the number 1
F
n is
assumed. The length cr L is plotted in Figure 2(b) as a
function of F n for three values of n: 1, 2, and 3.
5. Energy Gap at the Fermi Level and the
Coherence Length
At length crext LLL 2== [see (32)] the difference
E

entering (31) plotted as a function of L attains its
minimum; see Figure 3.
For ext LL > there exists a slow increase of
E

to
the value 0E

 attained at very large L. The
absolute value of
E

at ext LL = can be considered as

S. OLSZEWSKI ET AL. 333
the energy gap for the pair excitation spectrum. This is:

2
4
=2
2ln 2
== =.
F
e
LL
gee ext F
nn
me
EE En
h



 (33)
A plot of g E versus F n is presented in Figure 4.
The energy g E in (33) can be referred to a distance
0

called the coherence length [18,19]:
02
=.
F
g
v
E

 (34)
The velocity F v at the Fermi level is estimated as
equal to
=,
2F
Fe
nh
vmL (35)
Figure 3. The excitation energy 
E
of the electron pair
[see (31)] in a one-dimensional non-magnetic electron gas
plotted versus the length L of the gas potential tube. The
minimum position ext L is attained at 2cr L, where cr L is
a critical length at which =0E; see (32).
Figure 4. Energy gap
g
E
(in eV) for the pair excitation of
a non-magnetic gas [see (33) and (30)]. The lowest
g
E
is
for =1n, the highest one—for =3n.
since F v satisfies the following equation for the Fermi
energy of the non-interacting gas:
222
2
==.
2 8
F
F
Fe
e
nh v
Em
mL (36)
The formula (34) combined with (35) for
==2
ext cr
LL L taken from (32) gives

2
022
1
ln 2 2F
F e
n h
nn me
  (37)
where
26
2= 80.20910cm.
e
hf
me

 (38)
A plot of the function )(
0F
n

for several Fe nnn

=
is done in Figure 5.
6. Relation between the Critical Length cr
L
and the Coherence Length 0

These parameters can be compared together giving an
especially simple relation, on condition an excitation of
the electron gas having large F n is considered. Si-
multaneously, the change n

of F n is assumed to be
a relatively small number. In this case, because of (32),
(37) and (38), we obtain the ra t i o:
02
21
=
cr
Ln

  (39)
which means a proportionality between cr
L and 0

,
with the factor of proportionality dependent on n

.
Figure 5. Coherence length 0

[see (37)] plotted for
several pair transitions =


eF
nn n as a function of F n.
The lowest 0

is for =

3n, the highest one - for =

1n;
see also (39) and Figure 2.

S. OLSZEWSKI ET AL. 334
7. Effect of the Magnetic Field on a
Superconducting Behaviour of a
One-Dimensional Many-Electron System
A well-known property of three-dimensional supercon-
ductors is a destruction of the superconducting state due
to an external magnetic field. A similar property can be
expected in a one-dimensional system. In this case the
magnetic field can do a reversal of the electron spin at
least at some levels near the Fermi level F
n. In effect,
the excitations of electron pairs near that level are
stopped and the low-energy one-electron transitions,
absent in a non-magnetic gas case, become then possible.
An example of the pattern of levels of this kind is
presented in Figure 6.
The transition energy of a single electron between
level F n and 1
F
n becomes a finite, i.e. convergent,
result at 0R in this case. Assuming the electron-
electron interaction as only important part in the ex-
citation energy, and the kinetic part of that energy
considering as negligible, the transition energy from the
level F n to the level Fe nn > becomes:
=1
,,
=1
=1 =2
,, ,,
=1 =1
=( )
()().
in
F
paramagnetic
een in i
ee
i
in in
FF
nininini
FF eF
ii
EEJ K
JK JJ


 



(40)
The first two sums on the right-hand side of (40) come
from the interaction between electrons having the same
kind of spin, the last sum is due to the interaction
between electrons having opposite spin. Approximately,
the expression (40) becomes [17]
(a) (b)
Figure 6. Pattern of the energy levels filling the Fermi sea of
a partly magnetized electron gas: (a) a scheme before
excitation; (b) a scheme after a single-electron excitation at
the Fermi level. Full circles denote the occupied electron
states.



  


2
22
2
1
21
2
ln[21]22
ln 2.
eF paramagnetic
ee F
eF
eF FF
F
nn e
EnL
nn
ee
nnnLn L
e
nn
L




 

(40a)
The end result in (40a) holds on cond ition
=,
eF F
nn nn

 (41)
and F
n is assumed to be a large number.
The density of energy due to the presence of the
magnetic field in a sample can be referred to the energy
difference between the paramagnetic and non-magnetic
state. This is expressed by the formula (see e.g. [20,21])
2
18 =
p
aramagneticnonmagnetic
c
paramagneticnon magnetic
ee ee
EE
H
EE







(42)
where

is the vo lume of th e metal sa mple:
2
=.RL   (43)
The energy difference in (42) can be limited to a
difference of the electron excitation energies near the
Fermi level. Because of (40a) and the relation

2
==2ln2
non magnetic
eeeeF e
EEnn
L


(44)
[see (30)], the formula (42) becom es



2
2
2
ln 22ln 2
18 =
ln 2
=.
FF
c
F
nn nn
e
HL
nn
e
L

 







(45)
The expression (45) is evidently a positive result for
any 0>n

.
The size of c H can be expressed in terms of
parameters R and L entering the right hand side of
(45). For physical reasons a constant
8
210cmR  (46)
can be put equal roughly to the length of the order of an
atomic radius estimated for the atoms entering the whole
of the atomic chain forming the core of a one-dimen-
sional system [22]. On the other hand, L should be not
smaller than cr L calculated in Section 4. A plot of c H
versus F n is done in Figure 7 (see the singlet (s) curve).
8. Penetration Depth in One-Dimensional
Superconductors and Its Kinds
With the action of the magnetic field on a supercon-

S. OLSZEWSKI ET AL. 335
Figure 7. Critical magnetic field c H (in oersteds) calcu-
lated from a difference of the energy density between the
paramagnetic and non-magnetic electron gas; see (45) for
the singlet states (curve s); for the triplet states see (62)
(curve t).
ductor is associated the penetration depth

. If the field
outside a superconductor is equal to 0 H, its decrease
measured inside a sample on a distance
z
from its
surface is represented by the formula [23,24]
 /
0
=)( z eHzH  (47)
where
1/2
2
2
=4e
tot
mc
ne




(47a)
and
2
2
=F
tot
n
nRL

(47b)
is a formula for the electron density within a quasi one-
dimensional sample, valid on condition F n is a large
number. This

is independent of the strength of the
magnetic field.
However, there exists also another penetration depth
II
 which can be associated with the coherence length.
This parameter is useful in classifying the superconductors,
namely those belonging to the second kind of super-
conductors. According to the Ginzburg-Landau theory,
the free energy of a superconductor of the second kind
can be defined with the aid of two parameters,

and

[23]. The first parameter is coupled with the
coherence length 0

with the aid of the equ ation
2
2
0
||= ,
2e m


 (48)
whereas the second parameter is defined by a critical
magnetic field c H and

:
2
2
4
=.
c
H


 (49)
If we put
||
=,
part



(50)
the penetration length II
 becomes
1/2
2
2
=.
4e
II
part
mc
e





(51)
In this formula part

replaces tot n present in (47a).
A comparison of II  and 0

, important in classifying
the properties of superconductors, is discussed below;
Section 9.
A plot of

calculated from (47a) is done in Figure
8, a similar plot of II  obtained from (51) is presented
in Figure 9, curve
s
. This dependence concerns a
singlet excited state of the electron pair.
9. Singlet-Singlet and Triplet-Triplet
Excitations of the One-Dimensional
Electron Gas
Transitions of the electron pairs in a non-magnetic gas
discussed in Section 3 and Section 4 concerned the
electrons of opposite spin forming the singlet states; see
e.g. [16]. But a situation may exist when an electron pair
of the same spin is located near the Fermi level, for
example on the levels F n and 1
F
n; see Figure 10.
This situation should b e classified as a triplet state. In the
next step, one of electrons of the pair can be promoted
from 1

F
n to a level qnF, where 1>q, leaving
the electron on F n unchanged:
).,(1),( qnnnn FFFF 

(52)
The electron-electron interaction energy of the pair in
its ground state 1),(

FF nn is equal to
,=1),( 1,1,  F
n
F
n
F
n
F
nFF
pair
ee KJnnE (53)
on condition the interaction with the remainder of
electrons in the gas is neglected. A convergent result is
attained for (53) at 0R, because the Rln
divergencies entering J and
K
cancel together. But
the same property concerns also the interaction energy of
an excited pair, say that obtained when one of electrons
is promoted from the level 1
F
n to some level
qnF

:
.=),( ,, F
nq
F
n
F
nq
F
nFF
pair
ee KJnqnE   (54)
Evidently, a difference of two convergent energies in
(54) and (53), which is the change of the electron-

S. OLSZEWSKI ET AL. 336
Figure 8. Penetration depth

of the magnetic field
calculated from (47a) is the same for the singlet and triplet
states; see (57).
Figure 9. Penetration depth II

characteristic for a
superconductor of the second kind calculated for the singlet
and triplet states [see (51)]. c
H and 0

are different for
singlets and triplets; see (61) and Figure 7.
(a) (b)
Figure 10. Pattern of the energy levels in a one-dimensional
electron gas having near the Fermi level an electron pair in
the triplet state. (a) the gas before an excitation of the pair ,
(b) the gas having an excited triplet pair.
electron interaction energy due to the transition between
1

F
n and qnF

, is also a convergent result at
0R.
A full change of the electron-electron interaction
energy in a triplet state associated with a one-electron
transition from the level 1
F
n to the level qnF

is
obtained when the interaction energy of the pair with
other electrons in the gas is taken into account. This
calculation replaces the difference between energies in
(54) and (53) by the change of the interaction energy
similar to that given in (40) and (40a):
 
=
,1,, 1,
=1
=2 .
in
F
triplet
eenqin inqin i
FFF F
i
EJJKK
 





(55)
The first sum in brackets represents a contribution of
the Coulomb integrals. This can be omitted at 0R
in comparison with the second sum in (55) which is due
to the exchange integrals. A full excitation energy is
approximately equal to:
 
2
1ln2
triplet
eeF e
Eqn
L
 (56)
The result in (56) is a negative number on condition
1>q.
The critical length cr L for the triplet states can be
obtained on the basis of (56), for example for 2=q. In
this case we have a promotion of an electron from
1

F
n to 2

F
n. Because of (56) the formula which
replaces (32) becomes
22
(2)(1)
=ln(2 )
2=,
ln(2)
triplet FF
cr F
Fcr
F
nn
Lf
n
nfL
n


(57)
so cr L in the triplet and singlet state remain approxima-
tely the same.
Another situation is for 0

. A comparison of triplet
ee
E
in (56) calculated for 2=q, so
1,=1=1)(= 



qnqnn FF (58)
with

2
==2ln2
singletnon magnetic
eeF e
EE n
L

 (59)
calculated in (30) for a singlet transitions having 1=n

,
shows that for the same n the lowering of the
excitation energy due to the electron-electron interaction
calculated in a triplet state is only a half of the result
obtained for a singlet state. Consequently to (33), a
reduction of the energy gap for the triplet-triplet tran-
sitions is obtained to the value

S. OLSZEWSKI ET AL. 337
11
==,
22
triplet singlet
g
gg
EEE (60)
where g E is that given (33). In effect, the coherence
length
,22= 000 singlettriplet   (61)
because the formula in (34) can be applied also for
triplet
0
 . The results obtained in (57) and (61) indicate that
the ratio cr L/
0

calculated in (39) becomes for the
triplet transitions twice as large as that obtained for the
singlet transitions.
An estimate of the critical magnetic field for the triplet
states takes into account the fact that the electron-
electron interaction energy near the Fermi level is of a
paramagnetic character, so it does not vary much upon
the action of an external magnetic field. In effect, the
expense of the field energy is expected to be done mainly
in order to change the kinetic energy:

22
2
22
21
1== .
88
FF
c
e
nn
Eh
HmL




(62)
Here cr LL = from (57) has to be substituted into
(62), as well as in  in (43). A plot of c H of this
kind done versus F n is given in Figure 7. This plot is
there compared with that done for the singlet states
obtained with the aid of the formula (45). In the next step,
triplet
0
 of (61) and c
H
of (62) can be applied to the
calculation of II
 for the triplet states following the
formulae (48)-(51); see Figure 9. In Figure 11 we
compare the plots of tot n entering (47a) and (47b) with
part

obtained for singlet and triplet transitions cal-
culated from (48)-(50).
10. Critical Current and Critical
Temperature for One-Dimensional
Systems
A critical current for destruction of the superconducting
state can be obtained from the ratio (see e.g. [19,25]):
c
c
H
J

 (63)
where c H is a critical magnetic field and

is a
penetration depth. We can calculate c
J separately for
the singlet and triplet states. The

of two kinds, that
of (47a) as well as II
 of (51) can be applied. The plots
of the dependencies of c J in (63) on F n are presented
in Figure 12.
The size of a critical temperature c T can be estimated
from g
E assuming the approximate formula:
Figure 11. Electron gas density tot n in the potential tube
[see (47b)] compared with the particle density
p
art

entering the calculation of II

for (a) the case of the
singlet states (s), (b) the triplet states (t); see (50). A
characteristic point is that tot n is not essentially different
from
p
art

.
Figure 12. Critical current intensity c
J
for the one-
dimensional superconductors calculated as a function of
F
n; see (63). 10.8 /OerstedA cm . The curves with the
indices s and II
s referring to the singlet states are
calculated for c H from (45) and

, II

taken from
(47a), (51), respectively; the curves with the indices t and
II
t referring to the triplet states are calculated for c H
from (62) and

, II

indicated above. For large
F
n the
curve s merges with II
s.
g
c
B
E
Tk  (64)
where B k is the Boltzmann constant. The plots of c T
are done on Figure 13.

S. OLSZEWSKI ET AL. 338
Figure 13. Critical temperature c T calculated from (64) as
a function of
F
n. Curve s: temperature c T for singlets;
curve t: temperature c T for triplets.
11. Summary
An attempt is done to approach the problem of super-
conductivity in quasi one-dimensional systems basing
mainly on an analysis of the electron-electron interaction
and kinetic energy effects; an important problem of the
many-electron wave fu nction is neglected here.
A well-known fact is that the geometry of the volume
in which the electron charges are enclosed can influence
considerably the interaction between them; see e.g. [26].
In particular, a one-dimensional volume provides us with
the logarithmic divergencies in the interaction energy
between the charges.
But in considering the superconductivity effects, the
excitation energy of an electron system, and not the
absolute value of the system energy, plays a dominant
role. Consequently—in the first step—the excitation
energy of a single electron pair, located in the vicinity of
the Fermi level is considered. In an extremely thin
potential tube the total energy of the pair tends to diverge.
Nevertheless, this divergence is precisely cancelled in
the case of a pair excitation. This holds on condition the
pair occupies a single orbital quantum state, equally after
and before the excitation process. This requirement
implies that the absolute values of the electron momenta
in the pair should be equal, but the spin states in that pair
are opposite. Evidently, the presence of pairs of this kind
leads to the singlet states.
An important property of the pair is that the Coulomb
part of its excitation energy from a lower orbital quantum
level to a higher level is a negative number. With a
positive value of the kinetic excitation energy between
the pair levels, we obtain a kind of competition between
the changes of the electron-electron interaction and
kinetic energies. Above some critical length of the
potential tube, the interaction part predominates over the
kinetic part, and the pair excitation is connected with a
lowering of the energy of the system.
This elementary result obtained for a single electron
pair can be extended to an excitation of the many-
electron system present in an almost one-dimensional gas
volume. In particular, the interaction energy of the pair
submitted to an excitation with the remainder of the gas
electrons is taken into account. A characteristic point is
that the size of the energy gap obtained when ex-
citements of an isolated electron pair are only considered
becomes much larger for the case of the pairs interacting
with the Fermi sea. The critical length cr
L for an
isolated pair at large F n is also very large, but for an
interacting gas cr L is reduced to about 12 1010  cm.
Beyond of singlets, also excitements of the pairs
representing the triplet states are taken into account. The
parameters of the electron gas examined in the paper
include the coherence length, critical magnetic field,
critical current intensity and critical temperature. The
plots of these parameters are done in dependence on the
electron number present in the linear gas sample, which
is approximately equal to F
n2, where F n is the index
representing the orbital quantum number at the Fermi
level. For a macroscopic 8
10
F
n we obtain: g
E
equal to about 5 10eV, 2
010


cm, 1
c
H Oe,
6
10


cm, 6 10
II
s


cm and 7 510
II
t



cm,
62
10 A/cm
c
J for singlets and 72 10 A/cmfor triplets,
2
10
c
T

K for singlets and a half of that value for
triplets.
12. References
[1] N. P. Butch, M. C. de Andrade and M. B. Maple, “Re-
source Letter Scy-3: Superconductivity,” American Jour-
nal of Physics, Vol. 76, No. 2, 2008, pp. 106-118.
[2] Y. P. Huang and D. W. Wang, “Quantum-Phase Dia-
grams of Fermionic Dipolar Gases in a Planar Array of
One-Dimensional Tables,” Physical Review A, Vol. 80,
No. 053610, 2009, p. 10.
[3] N. Belmechri, G. Abramovici and M. Herititier, “Phase
Diagram and Critical Fields of Organic Quasi-1d Super-
conductors in an Applied Magnetic Field,” Physica B,
Vol. 404, No. 19, 2009, pp. 3131-3133.
[4] K. Kajiwara, M. Tsuchiizu, Y. Suzumura and C. Bour-
bonnais, “Mechanism of the Singlet to Triplet Supercon-
ductivity Crossover in Quasi-One-Dimensional Organic
Conductors,” Journal of the Physical Society of Japan,
Vol. 78, No. 104702, 2009, p. 11.
[5] D. Jerome, A. Mazaud, M. Ribault and K. Bechgaard,
“Superconductivity in a Synthetic Organic Conductor
26
(TMTSF) PF,” Journal de Physique Lettres (Paris),

S. OLSZEWSKI ET AL. 339
Vol. 41, No. 4, 1980, pp. L95-L98.
[6] K. Bechgaard, C. J. Jacobsen, K. Mortensen, H. J. Peder-
sen and N. Thorup, “The Properies of Five Highly Con-
ducting Salts:2
(TMTSF) X ,6
X=PF,6 AsF

,6 SbF

,
4
BF and 3 NO, Derived from Tetramethyltetraselena-
fulvalene (TMTSF),” Solid State Communications, Vol.
33, No. 11, 1980, pp. 1119-1125.
[7] D. Jerome, “One Dimensional Organic Superconductors:
beyond the Fermii Liquid Description,” Journal de Phy-
sique IV (Paris), Vol. 10, No. PR3, 2000, pp. 69-84.
[8] D. Jerome, “2 (TM)X Organic Superconductors and
Related Physics,” Molecular Crystals and Liquid Crystals,
Vol. 380, No. 1, 2002, pp. 3-13.
[9] N. Dupuis, C. Bourbonnais and J. C. Nickel, “Supercon-
ductivity and Antiferromagnetism in Quasi-One-Dimen-
sional Organic Conductors,” Journal of Low Temperature
Physics, Vol. 32, No. 4-5, 2006, pp. 380-391.
[10] W. A. Little, “Possibility of Synthesizing an Organic
Superconductor,” Physical Review, Vol. 134, No. 6A,
1964, pp. A1416-A1424.
[11] Y. Fuseya and M. Ogata, “Increase of Superconducting
Correlation due to Dimensionality Change in Quasi-One-
Dimensional Conductors,” Journal of the Physical Soci-
ety of Japan, Vol. 76, No. 093071, 2007, p. 4.
[12] I. J. Lee, S. E. Brown and M. J. Naughton, “Unconven-
tional Superconductivity in a Quasi-One-Dimensional
System 2 (TMTSF)X ,” Journal of the Physical Society
of Japan, Vol. 75, No. 051011, 2006, p. 9.
[13] J. Friedel, “Quasi-Low-Dimensionality in a Weak Cou-
pling Limit,” Physica C, Vol. 153-155, 1988, pp. 1610-
1616.
[14] C. Bourbonnais and L. G. Caron, “New Mechanism of
Phase Transitions in Quasi-One-Dimensional Conduc-
tors,” Europhysics Letters, Vol. 5, No. 3, 1988, pp. 209-
215.
[15] H. Eyring, J. Walter and G. E. Kimball, “Quantum Chem-
istry,” Wiley, New York, 1957.
[16] C. C. J. Roothaan, “New Developments in Molecular
Orbital Theory,” Reviews of Modern Physics, Vol. 23, No.
2, 1951, pp. 69-89.
[17] S. Olszewski, “Hartree-Fock Approximation for the One-
Dimensional Electron Gas,” Zeitschrift für Physik B, Vol.
45, No. 4, 1982, pp. 297-306.
[18] C. Kittel, “Quantum Theory of Solids,” 2nd Edition,
Wiley, New York, 1987.
[19] M. Cyrot and R. Pavuna, “Introduction to Superconduc-
tivity and High-Tc Materials,” World Scientific, Singa-
pore, 1992.
[20] N. H. March, W. H. Young and S. Sampanthar, “Many-
Body Problem in Quantum Mechanics,” University Press,
Cambridge, 1967.
[21] E. A. Lynton, “Superconductivity,” Methuen, London,
1962.
[22] E. Perfetto and J. Gonzalez, “Electronic Correlations of
Small Diameter Carbon Nanotubes,” Journal of Physics:
Condensed Matter, Vol. 18, No. 33, 2006, pp. s2105-
s2114.
[23] J. M. Ziman, “Principles of the Theory of Solids,” Uni-
versity Press, Cambridge, 1972.
[24] M. R. Schafroth, “Theoretical Aspects of Superconduc-
tivity,” In: F. Seitz and D. Turnbull, Eds., Solid State
Physics, New York, 1960, pp. 293-498.
[25] C. Kittel, “Introduction to Solid State Physics,” 7th Edi-
tion, Wiley, New York, 1996.
[26] D. Ivanenko and A. Sokolov, “Classical Field Theory,”(in
Russian), GITTL, Moscow, 1949.