Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical
The behavior of the order parameter close to the superconductor-normal metal interface is researched. The thickness d of the normal layer is arbitrary and the temperature is close to critical. The method of quasiorthogonality to asymptotics is used to obtain boundary conditions. The expression t...
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| Cite this: | Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical / V.E. Sakhnyuk, A.V. Svidzinsky // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 159-167. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1206922017-06-13T03:06:07Z Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical Sakhnyuk, V.E. Svidzinsky, A.V. The behavior of the order parameter close to the superconductor-normal metal interface is researched. The thickness d of the normal layer is arbitrary and the temperature is close to critical. The method of quasiorthogonality to asymptotics is used to obtain boundary conditions. The expression that describes the dependence of the current states on the normal layer’s thickness is obtained. Asymptotic forms of solution at large and at small d (relatively to the coherence length) are considered. Досліджено поведінку параметра порядку поблизу границі розділу нормального металу та надпровідника. Товщина d нормального металу вважається довільною, а температура близькою до критичної. Для одержання граничних умов використано метод квазіортогональності до асимптотики. Отримано залежність виразу для струму від товщини нормального прошарку, а також розглянуті його асимптотичні форми при великих і малих d в порівнянні з довжиною когерентності £₀ . 2003 Article Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical / V.E. Sakhnyuk, A.V. Svidzinsky // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 159-167. — Бібліогр.: 6 назв. — англ. 1607-324X PACS: 74.50.+r DOI:10.5488/CMP.6.1.159 http://dspace.nbuv.gov.ua/handle/123456789/120692 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The behavior of the order parameter close to the superconductor-normal
metal interface is researched. The thickness d of the normal layer is arbitrary
and the temperature is close to critical. The method of quasiorthogonality
to asymptotics is used to obtain boundary conditions. The expression
that describes the dependence of the current states on the normal layer’s
thickness is obtained. Asymptotic forms of solution at large and at small d
(relatively to the coherence length) are considered. |
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Article |
| author |
Sakhnyuk, V.E. Svidzinsky, A.V. |
| spellingShingle |
Sakhnyuk, V.E. Svidzinsky, A.V. Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical Condensed Matter Physics |
| author_facet |
Sakhnyuk, V.E. Svidzinsky, A.V. |
| author_sort |
Sakhnyuk, V.E. |
| title |
Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical |
| title_short |
Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical |
| title_full |
Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical |
| title_fullStr |
Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical |
| title_full_unstemmed |
Current states in an SNS junction for arbitrary thickness of the normal layer at temperatures close to critical |
| title_sort |
current states in an sns junction for arbitrary thickness of the normal layer at temperatures close to critical |
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Інститут фізики конденсованих систем НАН України |
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2003 |
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http://dspace.nbuv.gov.ua/handle/123456789/120692 |
| citation_txt |
Current states in an SNS junction for
arbitrary thickness of the normal layer
at temperatures close to critical / V.E. Sakhnyuk, A.V. Svidzinsky // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 159-167. — Бібліогр.: 6 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT sakhnyukve currentstatesinansnsjunctionforarbitrarythicknessofthenormallayerattemperaturesclosetocritical AT svidzinskyav currentstatesinansnsjunctionforarbitrarythicknessofthenormallayerattemperaturesclosetocritical |
| first_indexed |
2025-07-08T18:24:58Z |
| last_indexed |
2025-07-08T18:24:58Z |
| _version_ |
1837104208507043840 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 159–167
Current states in an SNS junction for
arbitrary thickness of the normal layer
at temperatures close to critical
V.E.Sakhnyuk∗, A.V.Svidzinsky
Department of Theoretical and Mathematical Physics,
Lesya Ukrainka Volyn State University,
13 Voli Avenue, 43000 Lutsk, Ukraine
Received October 30, 2002
The behavior of the order parameter close to the superconductor-normal
metal interface is researched. The thickness d of the normal layer is arbi-
trary and the temperature is close to critical. The method of quasiorthogo-
nality to asymptotics is used to obtain boundary conditions. The expression
that describes the dependence of the current states on the normal layer’s
thickness is obtained. Asymptotic forms of solution at large and at small d
(relatively to the coherence length) are considered.
Key words: current states, the order parameter, the linear integral
equation, asymptote
PACS: 74.50.+r
1. Introduction
There are numerous papers in which the problem of constructing the current
states in an SNS junction (S – superconductor, N – normal metal) is considered.
However, the interest to the problem remains since some issues are still open or
their description leads to complicated theoretical calculations. Naturally, the model
approximations can be used here but they do not give an exact description of the
phenomena under consideration.
Papers [1,2] present the theory of the current states in an SNS junction in which
the thickness of the normal layer d � ξ0 and the temperature is close to critical.
The presence of the electron reflection from the NS interface is taken into account in
[3,4]. The case of a dirty superconductor has been investigated in [5]. The important
moment which permits to execute the calculations to the very end is the condition
d� ξ0 being assumed. Taking into account this condition one can find corresponding
asymptotics for the current.
∗E-mail: sve@lab.univer.lutsk.ua
c© V.E.Sakhnyuk, A.V.Svidzinsky 159
V.E.Sakhnyuk, A.V.Svidzinsky
There is no restriction on the thickness d of the normal layer in this paper, i.e.,
the consideration also includes the microscopic junction when d ∼ ξ0. The theory is
built for the temperature close to critical, since in this case one can use Ginzburg-
Landau theory. The complete mesoscopic theory is enlisted only for calculating the
parameter which determines the boundary condition.
2. The behavior of the order parameter in an SNS junction close
to NS interface
Let the surfaces, that separate normal and superconductive metals, be planar
and the axis OZ be perpendicular to them. The superconductor occupies the region
|z| > d/2 while the normal metal occupies |z| < d/2. The reflection of electrons from
NS interface is absent, i.e. the transition coefficient D = 1.
Near the NS interface, the superconductor is described by a linear integral equa-
tion for the order parameter ∆(z). The method of constructing this equation has
been presented in [2]. The equation has the following form:
∆(z) =
−d/2
∫
−∞
K(z − z′)∆(z′)dz′ +
∞
∫
d/2
K(z − z′)∆(z′)dz′. (1)
Its kernel is
K(z) =
πρTc
v0
∑
ωn
1
∫
0
dx
x
exp
(
−2|ωn|
v0x
|z|
)
, (2)
where ωn = πTc(2n + 1) is the odd Matsubara frequency (for Fermi statistics);
ρ = |g|N(0) is dimensionless coupling constant, N(0) is electron states density on
the surface of the Fermisphere; v0 is Fermi velocity. It is convenient to rewrite the
equation (1) in such a manner that integration can be performed on the half-axis
z > 0. To this end, we shall introduce the even ∆s(z) and the odd ∆a(z) parts of
the order parameter. Equations for them are as follows:
∆s,a(z) =
∞
∫
d/2
{K(z − z′) ±K(z + z′)}∆s,a(z
′)dz′. (3)
The sign plus corresponds to ∆s(z), minus corresponds to ∆a(z). Let us carry out
the shift of variables z and z′ on d/2 and denote the shifted functions by the same
letters, i.e. ∆s,a{z+d/2} → ∆s,a(z). Then, the integral equations take the following
form:
∆s,a(z) =
∞
∫
0
{K(z − z′) ±K(z + z′ + d)}∆s,a(z
′)dz′.
It is simpler to work with the equations which are written in dimensionless
variables. Let us put ζ = z/ξ0, a = d/ξ0, where ξ0 = v0/2πTc is the coherence
160
Current states in an SNS junction for arbitrary thickness. . .
length. Then,
∆s,a(ζ) =
∞
∫
0
{K(ζ − ζ ′) ±K(ζ + ζ ′ + a)}∆s,a(ζ
′)dζ ′,
K(ζ) =
ρ
2
∑
n
1
∫
0
dx
x
exp
(
−|2n+ 1|
x
|ζ|
)
. (4)
One can show from (1) that the asymptotics of the order parameter on the
infinity for both signs are linear, i.e.
∆(ζ)
as
=∆
′
+ζ + ∆+ , ζ → +∞,
∆(ζ)
as
= ∆
′
−
ζ + ∆− , ζ → −∞. (5)
Under the infinity we mean the region ζ � ξ0. For both the even and the odd
solutions we shall have, respectively:
∆s(ζ)
as
=C1(ζ + q1,∞), ∆a(ζ)
as
=C2(ζ + q2,∞), ζ → +∞. (6)
From (5) and (6) one can obtain the relations
C1 = ∆
′
+ − ∆
′
−
, C1q1,∞ = ∆+ + ∆− ,
C2 = ∆
′
+ + ∆
′
−
, C2q2,∞ = ∆+ − ∆− . (7)
Excluding the constants C1 and C2 we get
(∆
′
+ − ∆
′
−
)q1,∞ = ∆+ + ∆− ,
(∆
′
+ + ∆
′
−
)q2,∞ = ∆+ − ∆− .
It is known [1,2] that the constants q1,∞ and q2,∞ are not arbitrary. They are
uniquely fixed by the solution of the linear integral equation. We have to calculate
them, since they are involved in the expression for the density current.
There exist several methods which can be used for this purpose. In the paper
[1] the variational method of Rits was used. The authors have built a functional
minimization of which leads to the linear integral equation for the order parameter
∆(z). However, in the case of the junctions which have no left-right symmetry one
cannot build an appropriate functional. To avoid this the Gal’orkin’s method was
used in the further studies. In the method it is unnecessary to know any functional.
In the present work to calculate the required constants we use the so-called method
of quasiorthogonality to asymptotics [5] which is simpler than the above mentioned.
The starting point is the equation (4), which may be considered only for the even
part of the order parameter ∆s(ζ). Separating asymptotics on the infinity
∆s(ζ) = C1(ζ + q1,∞ + ψ(ζ))
161
V.E.Sakhnyuk, A.V.Svidzinsky
and substituting it into the equation (4) one obtains the following equation for a
function ψ(z)
ψ(ζ) −
∞
∫
0
{K(ζ − ζ ′) +K(ζ + ζ ′ + a)}ψ(ζ ′)dζ ′ =
=
∞
∫
0
{K(ζ + ζ ′) +K(ζ + ζ ′ + a)} ζ ′dζ ′
− q1,∞
∞
∫
0
{K(ζ + ζ ′) −K(ζ + ζ ′ + a)} dζ ′. (8)
Here we have used the formulas which are easy to check
∞
∫
0
K(ζ − ζ ′)dζ ′ = 1 −
∞
∫
0
K(ζ + ζ ′)dζ ′,
∞
∫
0
K(ζ − ζ ′)ζ ′dζ ′ = ζ +
∞
∫
0
K(ζ + ζ ′)ζ ′dζ ′. (9)
Now, let us integrate the equation (8) by ζ from zero to infinity. Using the first
formula of equation (9) we obtain a useful relation:
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ ′) −K(ζ + ζ ′ + a)}ψ(ζ ′)dζ ′ =
=
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ ′) +K(ζ + ζ ′ + a)} ζ ′dζ ′
− q1,∞
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ ′) −K(ζ + ζ ′ + a)} dζ ′. (10)
The second important relation is obtained multiplying (8) by ζ and integrating
by ζ from zero to infinity. Using the second formula of equation (9) one can find
∞
∫
0
dζ ζ
∞
∫
0
{K(ζ + ζ ′) +K(ζ + ζ ′ + a)}ψ(ζ ′)dζ ′ =
=
∞
∫
0
dζ ζ
∞
∫
0
{K(ζ + ζ ′) +K(ζ + ζ ′ + a)} ζ ′dζ ′
+ q1,∞
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ ′) −K(ζ + ζ ′ + a)}dζ ′. (11)
Let us introduce in (10) and (11) the function q(ζ) defined by a formula q(ζ) =
q1,∞ + ψ(ζ). Obviously, based on (6) one has lim
ζ→∞
q(ζ) = q1,∞. Substituting ψ(ζ) =
162
Current states in an SNS junction for arbitrary thickness. . .
q(ζ)− q1,∞ into equations (10) and (11) one has the final form of the relations, that
will be needed hereinafter:
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ ′) −K(ζ + ζ ′ + a)} q(ζ ′)dζ ′ = I1 + I1(a), (12)
∞
∫
0
dζ ζ
∞
∫
0
{K(ζ + ζ ′) +K(ζ + ζ ′ + a)} q(ζ ′)dζ ′ = 2q1,∞I1 − I2 − I2(a). (13)
Here we have introduced such notations:
I0(a) =
∞
∫
0
dζ
∞
∫
0
dζ ′K(ζ + ζ ′ + a), I1(a) =
∞
∫
0
dζ
∞
∫
0
dζ ′ ζ ′K(ζ + ζ ′ + a),
I2(a) =
∞
∫
0
dζ ζ
∞
∫
0
dζ ′ ζ ′K(ζ + ζ ′ + a), Ik = Ik(0), k = 0, 1, 2.
Let us clarify the sense of the above transformations. In the first case we multi-
plied the equation by unity, in the second one we multiplied by ζ and integrated by
ζ. Then, we did transformations to obtain zero in the left hand side of the equality.
We did not obtain zero because functions 1 and ζ are not the solutions of the homo-
geneous equation. They are only its asymptotics for large ζ. However, we neglected
the terms, that are limited or increasing on the infinity. Relations (12) and (13)
appear to be exact, and in [5] we called them the conditions of quasyorthogonality
to asymptotics.
Let us take a trial function Γ instead of q(ζ) in the relations (12) and (13). And
let Γ be an indefinite constant. We obtain
Γ(I0 − I0(a)) = I1 + I1(a), Γ(I1 + I1(a)) = −I2 − I2(a) + 2q1,∞I1 .
From here one can find both Γ and the expression for q1,∞
q1,∞ =
1
2I1
{
I2 + I2(a) +
(I1 + I1(a))
2
I0 − I0(a)
}
. (14)
The expression for q2,∞ differs from q1,∞ only in sign of Ik(a), which is easy to
understand from equation (4). Therefore,
q2,∞ =
1
2I1
{
I2 − I2(a) +
(I1 − I1(a))
2
I0 + I0(a)
}
. (15)
Certainly, formulas (14) and (15) give an approximate value of the coefficients
q1,∞ and q2,∞ because we have taken the constant Γ as a trial function instead
of the exact solution for q(ζ). Let us note that the same results are obtained by
other variation methods if the trial function is the same. However, the method of
quasyorthogonality is the simplest. From (14) and (15) it also follows that q1,∞ → ∞,
q2,∞ → 0 if a→ 0.
163
V.E.Sakhnyuk, A.V.Svidzinsky
3. The dependence of the current density on the thickness of
the normal layer in an SNS junction
Let us calculate the current, which can flow through an SNS junction. The thick-
ness d of the normal layer is assumed to be arbitrary. An expression for the density
of the current at a temperature close to critical in the Ginzburg-Landau theory [1,6]
is given by a formula:
j(ζ) = i
7ζ(3)
16π2
env0
p0ξ0T
2
c
∆
d
∗
∆
dζ
−
∗
∆
d∆
dζ
. (16)
We shall calculate the density of the current in the region, where both the asymp-
totics of the solution of the linear integral equation on the infinity (ζ � ξ0) and the
asymptotics of the solution of the Ginzburg-Landau’s equation for a small ζ,
ζ � ξ(T ), ξ(T ) =
√
7ζ(3)
12
ξ0
√
1 − T/Tc
,
overlap.
Therefore, substituting the expression for the order parameter into (16) we use
its asymptotics form (5). It makes it possible to rewrite (16) in the form:
j(ζ) = i
7ζ(3)
16π2
env0
p0ξ0T
2
c
(
∆+
∗
∆′
+ −
∗
∆+∆′
+
)
. (17)
Of course, we could take the asymptotics on a minus infinity. Then, instead of the
form ∆+
∗
∆′
+ −
∗
∆+∆′
+ we would have ∆−
∗
∆′
−
−
∗
∆−∆′
−
. However, these both forms
are equivalent. Really, the expressions (7) can be written as follows:
(
∆+
∆′
+
)
= M̂
(
∆−
∆′
−
)
, M̂ =
1
q1,∞ − q2,∞
(
q1,∞ + q2,∞ 2q1,∞q2,∞
2 q1,∞ + q2,∞
)
. (18)
Expressing one of the forms by another we can see that they differ by a multiplier
DetM̂ which is equal to unity. Therefore, we shall further neglect the argument in
the density of the current. For further calculations it is convenient to exclude the
derivatives of ∆, using the expression (7). Thus, one obtains
j = i
7ζ(3)
32π2
env0
p0ξ0T
2
c
(
1
q1,∞
− 1
q2,∞
)
(
∆+
∗
∆− −
∗
∆+∆−
)
. (19)
Quantities ∆+ and ∆− cannot be found from the solution of the linear integral
equation because this solution is determined up to a multiplication constant. In
order to determine this constant, we consider the condition of matching the solution
of the linear integral equation with the solution of the Ginzburg-Landau equation.
164
Current states in an SNS junction for arbitrary thickness. . .
In the presence of the current states, the solution of the Ginzburg-Landau equation
has the following form:
∆(ζ) = exp{±iϕ/2}∆∞f(ζ) exp{2imχ(ζ)}, (20)
where
f(ζ) = th
(
|ζ|+ γ − a/2√
2ξ(T )
ξ0
)
.
A phase of the order parameter is introduced here. It is bound up with a super-
fluid velocity: ∇χ = vs. This function is considered to be continuous throughout.
Specifically, we assume χ(a/2) = χ(−a/2) = 0.
To describe the current states we should assume the presence of a phase differ-
ence between the banks of the junction . To this end, we introduce the phase factor
exp{±iϕ/2}, which is independent of the coordinate within each of the supercon-
ductors, but differs in both of them. We shall assume that the phase ϕ/2 is for the
right hand superconductor and the phase −ϕ/2 is for the left hand one. The repre-
sentation (20) makes it possible to write down the formula for the density current
as follows:
j =
7ζ(3)
16π2
env0∆
2
∞
p0ξ0T
2
c
(
1
q2,∞
− 1
q1,∞
)
f+f− sinϕ, (21)
where f+ and f− are values of the function f(ζ) at ζ = a/2 and ζ = −a/2, respec-
tively, i.e. f+ = f− = th(γξ0/
√
2ξ(T )).
Before obtaining a result for the current, let us consider the condition which
follows from the relations (7). We write them down using the function f(ζ) which
was introduced by the (20). After comparing both the real and the imaginary parts,
the relation (7) takes the following form:
2q1,∞q2,∞f
′
+ − (q1,∞ + q2,∞)f+ + (q1,∞ − q2,∞)f− cosϕ = 0,
4mvsξ0q1,∞q2,∞f+ + (q2,∞ − q1,∞)f− sinϕ = 0,
2q1,∞q2,∞f
′
−
+ (q1,∞ + q2,∞)f− + (q2,∞ − q1,∞)f+ cosϕ = 0,
4mvsξ0q1,∞q2,∞f− + (q2,∞ − q1,∞)f+ sinϕ = 0.
(22)
These equations have a solution that is not equal to zero, at the condition
4mvsξ0 =
(
1
q2,∞
− 1
q1,∞
)
sinϕ.
Taking into account the last relation, it is easy to find from (22) that
f+ = f−, f ′
+ = −f ′
−
,
f ′
+
f+
=
1
2
(
1
q1,∞
+
1
q2,∞
)
+
1
2
(
1
q1,∞
− 1
q2,∞
)
cosϕ.
(23)
165
V.E.Sakhnyuk, A.V.Svidzinsky
Thus, we really have that f+ = f− since the system is symmetrical, the relation
f ′
+ = −f ′
−
follows from the symmetry as well. To find the unknown constant γ,
which is involved in the expressions for f+ and f−, one uses the relations:
∆′
+ =
∆∞ξ0√
2ξ(T )
eiϕ/2
/
ch
γξ0√
2ξ(T)
, ∆′
+q∞ = ∆∞ th
γξ0√
2ξ(T )
eiϕ/2, q∞ =
∆+
∆′
+
.
(24)
These relations are obtained from the matching condition of the solution of the linear
integral equation and the solution of the Ginzburg-Landau equation. We note that
from (23) and (24) there follows an identity q−1
∞
= cos2(ϕ/2)/q1,∞ +sin2(ϕ/2)/q2,∞ .
Based on (24) one has for f+ and f−
f+ = f− = th
(
γξ0√
2ξ(T )
)
=
q∞ξ0√
2ξ(T )
1
2
+
√
√
√
√
1
4
+
q2
∞
ξ2
0
2ξ2(T )
−1
. (25)
Thus, the expression for the density of the current can be written as follows:
j =
7ζ(3)
16π2
env0∆
2
∞
p0ξ0T
2
c
q1,∞ − q2,∞
q1,∞q2,∞
q2
∞
ξ2
0
2ξ2(T )
sinϕ
1
2
+
√
√
√
√
1
4
+
q2
∞
ξ2
0
2ξ2(T )
−2
. (26)
Taking into account the well-known relations:
ξ2(T ) =
7ζ(3)
12
ξ2
0
1 − T/Tc
, ∆2
∞
=
8π2T 2
c
7ζ(3)
(1 − T/Tc) ,
one can obtain a final expression for the density of the current in which there is no
restriction on the thickness d of the normal layer
j =
3
7ζ(3)
env0
p0ξ0
(1 − T/Tc)
2 q1,∞ − q2,∞
q1,∞q2,∞
q2
∞
sinϕ
(
1
2
+
√
1
4
+ q2
∞
6
7ζ(3)
(1 − T/Tc)
)2
. (27)
From the latter expression it is easy to obtain the asymptotics forms for the
density current. At small d, (d� ξ0)
j =
6
7ζ(3)
env0
p0ξ0
(1 − T/Tc)
2 q2,∞
cos(ϕ/2)
sin3(ϕ/2)
. (28)
At large d, (d� ξ0)
j =
3
7ζ(3)
env0
p0ξ0
(1 − T/Tc)
2 q1,∞
q2,∞
(q1,∞ − q2,∞) sinϕ. (29)
The latter formula coincides with the result which was obtained in the paper [4],
where from the very start it was assumed that (d� ξ0).
166
Current states in an SNS junction for arbitrary thickness. . .
References
1. Galaiko V.P., Svidzinsky A.V., Slyusarev V.A. // Zh. Eksp. Teor. Fiz., 1969, vol. 56,
No. 3, p. 835–840.
2. Svidzinsky A.V. Space-inhomogeneous Problems of the Superconducting Theory.
Moscow, Nauka, 1982.
3. Golubev L.V., Rakov Y.A., Svidzinsky A.V. // Physics of many partical systems, 1985,
No. 7, p. 40–45.
4. Svidzinsky A.V., Golubev L.V. // Teor. Mat. Fiz., 1984, vol. 59, No. 1, p. 129–138 (in
Russian).
5. Svidzinsky A.V., Sakhnyuk V.E. // Condens. Matter Phys., 2000, vol. 3, No. 3(23),
p. 683–696.
6. Svidzinsky A.V. The Microscopic Theory of Superconductivity. Lutsk, Vezha, 2001.
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