On the theory of current states in superconducting junctions of SNINS type
The behavior of the order parameter close to the NS interface in an SNINS junction is considered. To this end, a linear integral equation, which is valid near the superconductor-normal metal interface, is obtained and researched. The Ginzburg-Landau equation is solved taking into account the effec...
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| Дата: | 2006 |
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | On the theory of current states in superconducting junctions of SNINS type / V.E. Sakhnyuk, A.V. Svidzynsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 169–177. — Бібліогр.: 12 назв. — англ. |
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Sakhnyuk, V.E. Svidzynskyj, A.V. 2017-06-14T04:44:29Z 2017-06-14T04:44:29Z 2006 On the theory of current states in superconducting junctions of SNINS type / V.E. Sakhnyuk, A.V. Svidzynsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 169–177. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 74.50.+r DOI:10.5488/CMP.9.1.169 https://nasplib.isofts.kiev.ua/handle/123456789/121311 The behavior of the order parameter close to the NS interface in an SNINS junction is considered. To this end, a linear integral equation, which is valid near the superconductor-normal metal interface, is obtained and researched. The Ginzburg-Landau equation is solved taking into account the effect of the current on the space behaviour of the order parameter. Applying the method of quasiorthogonality to asymptotics, the boundary condition for the Ginzburg-Landau equation is obtained. The calculation of the current states, which can exist in an SNINS junction is carried out. We assume the normal layer thickness to be arbitrary and the temperature to be close to critical. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the theory of current states in superconducting junctions of SNINS type До теорії струмових станів в SNINS контакті Article published earlier |
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On the theory of current states in superconducting junctions of SNINS type |
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On the theory of current states in superconducting junctions of SNINS type Sakhnyuk, V.E. Svidzynskyj, A.V. |
| title_short |
On the theory of current states in superconducting junctions of SNINS type |
| title_full |
On the theory of current states in superconducting junctions of SNINS type |
| title_fullStr |
On the theory of current states in superconducting junctions of SNINS type |
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On the theory of current states in superconducting junctions of SNINS type |
| title_sort |
on the theory of current states in superconducting junctions of snins type |
| author |
Sakhnyuk, V.E. Svidzynskyj, A.V. |
| author_facet |
Sakhnyuk, V.E. Svidzynskyj, A.V. |
| publishDate |
2006 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
До теорії струмових станів в SNINS контакті |
| description |
The behavior of the order parameter close to the NS interface in an SNINS junction is considered. To this
end, a linear integral equation, which is valid near the superconductor-normal metal interface, is obtained
and researched. The Ginzburg-Landau equation is solved taking into account the effect of the current on
the space behaviour of the order parameter. Applying the method of quasiorthogonality to asymptotics, the
boundary condition for the Ginzburg-Landau equation is obtained. The calculation of the current states, which
can exist in an SNINS junction is carried out. We assume the normal layer thickness to be arbitrary and the
temperature to be close to critical.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121311 |
| citation_txt |
On the theory of current states in superconducting junctions of SNINS type / V.E. Sakhnyuk, A.V. Svidzynsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 169–177. — Бібліогр.: 12 назв. — англ. |
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| first_indexed |
2025-11-25T23:28:33Z |
| last_indexed |
2025-11-25T23:28:33Z |
| _version_ |
1850581087567216640 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 169–177
On the theory of current states in superconducting
junctions of SNINS type
V.E.Sakhnyuk∗, A.V.Svidzynskyj
Department of Theoretical and Mathematical Physics,
Lesya Ukrainka Volyn States University,
13 Voli Avenue, 43000 Lutsk, Ukraine
Received August 3, 2005, in final form November 8, 2005
The behavior of the order parameter close to the NS interface in an SNINS junction is considered. To this
end, a linear integral equation, which is valid near the superconductor-normal metal interface, is obtained
and researched. The Ginzburg-Landau equation is solved taking into account the effect of the current on
the space behaviour of the order parameter. Applying the method of quasiorthogonality to asymptotics, the
boundary condition for the Ginzburg-Landau equation is obtained. The calculation of the current states, which
can exist in an SNINS junction is carried out. We assume the normal layer thickness to be arbitrary and the
temperature to be close to critical.
Key words: current, critical temperature, Ginzburg-Landau equation, the order parameter
PACS: 74.50.+r
1. Introduction
To calculate the current in the superconducting junctions of SNS, SINS, SNINS type (where
S – superconductor, N – normal metal, I – insulator) at the temperature not too close to critical,
the model with the piecewise constant order parameter ∆(~r) has been used. In this model a modulus
of the order parameter is assumed to be a constant in the superconducting region and zero in the
normal metal. Such an approach for the investigation of the current states in pure SNS junction
was realized in paper [1]. The calculation of the current taking into account the reflection of
electrons on the junction (junctions of SINS, SINIS, SNINS type) has been made in [3–5]. The
numerous results in the model with the piecewise constant order parameter are contained in [6].
However, for the temperature close to critical such a reduced model cannot be used. In this
region of temperatures the spatial nonhomogeneity of the order parameter near NS interface should
be taken into account. In order to investigate the spatial behaviour of the order parameter at a
temperature close to critical the Ginzburg-Landau equation has been used. However, the boundary
condition has to be taken into account here. One can find the boundary condition from the linear
integral equation which is valid near the interface for the distances of the order ξ0 (ξ0 – the
length coherence). Taking into account the spatial behaviour of the order parameter the current
in SNS, SINS junctions has been calculated in papers [3,9], while in [7] the SNINS junction is
considered. However, in these papers we have investigated the asymptotic case, when the thickness
of the normal layer d� ξ0.
The purpose of this paper is to investigate the current states in SNINS junction for arbitrary
thickness of the normal layer at the temperature close to critical.
∗
E-mail: sve@lab.univer.lutsk.ua
c© V.E.Sakhnyuk, A.V.Svidzynskyj 169
V.E.Sakhnyuk, A.V.Svidzynskyj
2. The common scheme for the investigation of the behavior of the order
parameter near NS interface.
It is well-known that the information about the spacial behavior of the order parameter can be
found from the self-consistent condition [3,8]:
∆(~r) = |g|T
∑
ωn
F ∗
ωn
(~r, ~r), (1)
where g is the coupling constant, T is temperature, ωn = πT (2n+1) is the odd Matsubara frequen-
cy, Fωn
(~r, ~r) is the so-called anomalous Matsubara Green’s function. Both the anomalous Matsub-
ara Green’s function and the usual Matsubara Green’s function Gωn
(~r, ~r′) satisfy the Gorkov’s
system of equations:
(iωn − ξ̂)Gωn
(~r, ~r′) + ∆(~r)Fωn
(~r, ~r′) = δ(~r − ~r′),
(iωn + ξ̂)Fωn
(~r, ~r′)+
∗
∆ (~r)Gωn
(~r, ~r′) = 0.
(2)
Here ξ̂ = ~̂p
2
/(2m)−µ+V (~r), where V (~r) is the potential, which we will later take as δ – functional.
Let us suppose that the magnetic field is absent. Thus, in order to obtain the equation for ∆(~r),
it is necessary to express the function Fωn
(~r, ~r′) from system (2) and substitute it into the self-
consistent condition (1). In the general case this problem is very complicated. However, for the
temperature close to critical the order parameter ∆(~r) is small, so that Green’s functions can be
expanded into a series in terms of the ∆(~r). For Fωn
(~r, ~r′) in the linear approximation one has
Fωn
(~r, ~r) =
∫
G
(0)
−ωn
(~r, ~r′)G(0)
ωn
(~r′, ~r)∆∗(~r′)d~r′. (3)
Inserting (3) into (1) we obtain the linear integral equation
∆(~r) =
∫
K(~r, ~r′)∆(~r′)d~r′ (4)
with the kernel
K(~r, ~r′) = |g|Tc
∑
ωn
G
(0)
−ωn
(~r, ~r′)G(0)
ωn
(~r′, ~r),
where G
(0)
ωn
(~r, ~r′) is the Green’s function of the structure under investigation in the normal states.
It satisfies the equation
(iωn − ξ̂)G(0)
ωn
(~r, ~r′) = δ(~r′ − ~r). (5)
In order to obtain the explicit form of the linear integral equation (4) describing the behaviour of
the order parameter near NS-interface in a SNINS junction, it is necessary to have the expression
for the Green’s function G
(0)
ωn
(~r, ~r′). This function can be found from equation (5) in which ξ̂ =
~̂p
2
/(2m) − µ + U0δ (z), where the last term simulates the thin layer of an insulator placed in the
middle of the normal region (plane X0Y ). The best way of constructing a Green’s function is
to solve the corresponding one-particle problem (see e.g. [10]). After substituting the expressions
for the Green’s function in the formula for K(~r, ~r′) one can carry out the averaging on the atom
distance [2]. Thus, we obtain the linear integral equation for the order parameter
∆(ζ) =
ρ
2
∑
n
−d/2
∫
−∞
dζ′∆(ζ′)
1
∫
0
dx
x
exp
(
−|2n+ 1|
x
|ζ − ζ′|
)
+
ρ
2
∑
n
∞
∫
d/2
dζ′∆(ζ′)
1
∫
0
dx
x
signζζ′R(x) exp
(
−|2n+ 1|
x
(|ζ| + |ζ′|)
)
. (6)
170
On the theory of current states
Here we take into account that the spatial homogeneity is broken only in the direction of 0Z
axis. Therefore, we introduced the dimensionless quantities: ρ = |g|N(0) is dimensionless coupling
constant,N(0) is electron state density on the surface of the Fermisphere; ζ = z/ξ0 is dimensionless
variable; R(x) is coefficient of the reflection, a = d/ξ0 is dimensionless thickness of the normal
layer. Hereinafter, it is convenient to consider the equation (6) separately in the regions ζ > a/2
and ζ < −a/2, i. e. in both the right superconductor and the left one. Then, introducing the even
∆s(ζ) = 1/2(∆(ζ)+∆(−ζ)), and the odd ∆a(ζ) = 1/2(∆(ζ)−∆(−ζ)) parts of the order parameter
∆(ζ), one can obtain the following equations
∆s(ζ) =
∞
∫
0
dζ′∆s(ζ
′) {K(ζ − ζ′) +K(ζ + ζ′ + a)} , (7)
∆a(ζ) =
∞
∫
0
dζ′∆a(ζ′) {K(ζ−ζ′) +KD(ζ + ζ + a)} . (8)
Here the shift of variables was performed: ζ → ζ + a/2, ζ′ → ζ′ + a/2 and the shifted functions are
denoted by the same letters, i. e. ∆s (ζ + a/2) → ∆s(ζ), ∆a (ζ + a/2) → ∆a(ζ). Furthermore, the
following notations are introduced
K(ζ) =
ρ
2
∑
n
1
∫
0
dx
x
exp
(
−|2n+ 1|
x
|ζ|
)
,
KD(ζ) =
ρ
2
∑
n
1
∫
0
dx
x
τ(x) exp
(
−|2n+ 1|
x
|ζ|
)
,
where τ(x) = 1 − 2D(x). D(x) is the coefficient of transition across the insulator.
From equations (7) and (8) it follows that the asymptotes of ∆s(ζ) and ∆a(ζ) are linear at
infinity (ζ → ∞).
Let us substitute the expressions for ∆s(ζ) and ∆a(ζ) with separated asymptote
∆s(ζ) = C1(ζ + q1,∞ + ψs(ζ)),
∆a(ζ) = C2(ζ + q2,∞ + ψa(ζ)),
ζ → ∞, lim
ζ→∞
ψs,a(ζ) = 0 (9)
into (7) and (8), respectively.
It is important to calculate the coefficients q1,∞ and q2,∞, since they are included into the
expression for the current. To calculate them we used the method of quasiorthogonality to asymp-
totics [11]. As a result we obtain two systems
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ′) −K(ζ + ζ′ + a)} qs(ζ′)dζ′ = I1 + I1(a),
∞
∫
0
dζ ζ
∞
∫
0
{K(ζ + ζ′) +K(ζ + ζ′ + a)} qs(ζ′)dζ′ = 2q1,∞I1 − I2 − I2(a),
(10)
∞
∫
0
dζ
∞
∫
0
{K(ζ + ζ′) −KD(ζ + ζ′ + a)} qa(ζ′)dζ′ = I1 + I1(D, a),
∞
∫
0
dζ ζ
∞
∫
0
{K(ζ + ζ′) +KD(ζ + ζ′ + a)} qa(ζ′)dζ′ = 2q2,∞I1 − I2 − I2(D, a).
(11)
Here new functions: qs(ζ) = q1,∞ + ψs(ζ), qa(ζ) = q2,∞ + ψa(ζ) are introduced, and the following
notation is used:
I0(D, a) =
∞
∫
0
dζ
∞
∫
0
dζ′KD(ζ + ζ′ + a), I1(D, a) =
∞
∫
0
dζ
∞
∫
0
dζ′ ζ′KD(ζ + ζ′ + a),
171
V.E.Sakhnyuk, A.V.Svidzynskyj
I2(D, a) =
∞
∫
0
dζ ζ
∞
∫
0
dζ′ ζ′KD(ζ + ζ′ + a), Ik(a) = Ik(0, a), Ik = Ik(0, 0),
k = 0, 1, 2.
Note also that
I0 =
ρ
4
S2 , I1 =
ρ
6
S3 , I2 =
ρ
8
S4 , Sα =
∞
∑
n=−∞
1
|2n+ 1|α . (12)
Let us take trial functions Γs, Γa instead of qs(ζ
′), qa(ζ′), which are put to be indefinite
constants. Then from (10) and (11) one has the final form expressions for q1,∞, q2,∞
q1,∞ =
1
2I1
{
I2 + I2(a) +
(I1 + I1(a))
2
I0 − I0(a)
}
, (13)
q2,∞ =
1
2I1
{
I2 + I2(D, a) +
(I1 + I1(D, a))
2
I0 − I0(D, a)
}
. (14)
0 1 2 3 4 5 6
0,6
0,8
1,0
1,2
1,4
1,6
0 1 2 3 4 5 6
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
q 1(
d
/ξ
0)
d/ξ
0
D=0.8
D=1
D=0.2
D=0.01
q 2(
d
/ξ
0)
d/ξ
0
k=991/2p
k=2p
k=0.5p
k=0
D=p 2x2/(p2x2+k2)
Figure 1. q1 and q2 as functions on the normal layer thickness d/ξ0.
From figure 1 one can see that the difference between q1 and q2 increases at the small thickness
of the normal layer. It is the consequence of a marked increase in coupling of the supercoductors.
In the opposite case q1 and q2 are close to 0.64, which is characteristic of NS – junction.
Assuming the transition coefficient of electrons D in (14) to be equal to unity, we can reproduce
the well-known result for an SNS junction [12].
3. Solution of the Ginzburg-Landau equation for the order pa rameter which
takes into account a superfluid velocity
Let the order parameter be small and let us assume that ∆(~r) is a slowly changing function;
such a region must exist, since the order parameter tends to a constant in the depth of a super-
conductor. Then, it is necessary to continue the expansion (3) to the third power of the order
parameter ∆(~r). However, the last term can be taken in the local form. As a result we come to the
nonlinear second order differential equation, which is called the Ginzburg-Landau equation. Let us
write down this equation
ξ2(T )
ξ20
d2∆(ζ)
dζ2
− 1
∆2
∞
|∆(ζ)|2∆(ζ) + ∆(ζ) = 0, (15)
172
On the theory of current states
where
∆2
∞ =
8π2T 2
c
7ζ(3)
(
1 − T
Tc
)
, ξ(T ) =
√
7ζ(3)
12
ξ0
√
1 − T/Tc
are the magnitude of both the order parameter at infinity and a characteristic length of the change
of the order parameter close to critical temperature, respectively; ξ0 = v0/(2πTc) – a coherence
length, ζ = z/ξ0 – a dimensionless variable.
Since we investigate the junctions at the presence of the current states, the solution of equation
(15) has to be presented in the form
∆(ζ) = e±iϕ/2∆∞f(ζ)e2imχ(ζ). (16)
A phase χ(ζ) of the order parameter is introduced here. It is bound up with a superfluid velocity:
∇χ = vs. This function is considered to be continuous throughout. Specifically, we assume that
χ(d/2) = χ(−d/2) = 0. To describe the current states we should assume the presence of a phase dif-
ference between the banks of the junction. To this end, we introduce the phase factor exp{±iϕ/2},
which is independent of the coordinates within each of the superconductors, but differs in both of
them. We shall assume that the phase −ϕ/2 is for the left hand superconductor and the phase ϕ/2
is for right hand one. After substituting (16) into (15) and separating both the real part and the
imaginary one, we obtain
ξ2(T )
ξ20
f ′′(ζ) − ξ2(T )4m2v2
s (ζ)f(ζ) + f(ζ) − f3(ζ) = 0, (17)
vs(ζ)f
2(ζ) = const. (18)
At infinity f is a constant and one gets from equation (17)
f2
∞ = 1 − 4m2v2
s (∞)ξ2(T ), f∞ ≡ f(∞). (19)
Based on both the last relation and the equation (18) we have
v2
s (ζ) =
1 − f2
∞
4m2ξ2(T )
f4
∞
f4(ζ)
. (20)
Substituting (20) into (17) one can obtain the equation for f(ζ)
ξ2(T )
ξ20
f ′′(ζ) − (1 − f2
∞)
f4
∞
f3(ζ)
+ f(ζ) − f3(ζ) = 0.
The last equation has the first integral
ξ2 (T )
ξ20
(f ′ (ζ))
2
+
(
1 − f2
∞
)
f2
∞
(
f2
∞
f2 (ζ)
− 1
)
+ f2 (ζ) − f2
∞ +
1
2
f4
∞ − 1
2
f4 (ζ) = 0. (21)
Finally, carrying out the integration, we obtain
f (ζ) =
{
2
(
1 − f2
∞
)
+
(
3f2
∞ − 2
)
th2 ζ + C√
2ξ (T )
ξ0
}1/2
. (22)
Another form of the solution which includes the superfluid velocity is
f(ζ) =
1 − 4m2v2
s (∞)ξ2(T ) +
12m2v2
s (∞)ξ2(T ) − 1
ch2 ζ+C√
2ξ(T )
ξ0
1/2
. (23)
Comparing the obtained result (formula (22)) with the one which was known before ([8], section 3,
equation (4.5)) it is easy to see that taking into account the term with the superfluid velocity in
the Ginzburg-Landau equation brings a new component in the expression for f(ζ).
173
V.E.Sakhnyuk, A.V.Svidzynskyj
4. Boundary value of the order parameter
The obtained solution of the Ginzburg-Landau equation contains an unknown constant C. In
order to fix it, it is necessary to have the boundary condition for the function f(ζ) at ζ = a/2.
Let us put ζ = a/2 in equation (21) then, introducing the notation f(a/2) ≡ f+, one can obtain
the relation between f ′
+ and f+
ξ(T )
ξ0
f ′
+ − f2
∞ − f2
+
f+
√
f2
∞ − 1 +
1
2
f2
+ = 0. (24)
Another relation between f ′
+ and f+ we have from conditions on the NS interface (see [9])
f ′
+
f+
=
cos2
ϕ
2
q1,∞
+
sin2 ϕ
2
q2,∞
≡ 1
q∞
. (25)
Using (25) let us exclude f ′
+ from (24), as a result we come to the closed equation with respect to f+:
f6
+ − 2
(
1 +
ξ2(T )
ξ20q
2
∞
)
f4
+ + f2
∞(4 − 3f2
∞)f2
+ + 2f4
∞
(
f2
∞ − 1
)
= 0. (26)
f∞ satisfies such an equation
f6
∞ − f4
∞ +
(
ξ(T )
2ξ0
)2(
1
q2,∞
− 1
q1,∞
)2
sin2ϕf4
+ = 0. (27)
Indeed, using the condition (6.21) in [9] from equation (18) one has
vs(∞) =
sinϕ
4mξ0
(
1
q2,∞
− 1
q1,∞
)
f4
+
f2
∞
. (28)
Then substituting (28) into (19) we come to (27).
Let us return to equation (26). It is a cube equation with respect to f2
+ ≡ x and if we introduce
the following notation:
a = −2
(
1 +
ξ2(T )
ξ20q
2
∞
)
, b = f2
∞(4 − 3f2
∞), c = 2f4
∞(f2
∞ − 1), (29)
then, equation (26) gets the form
x3 + ax2 + bx+ c = 0. (30)
It is well-known that the solution of the cube equation has been given by the Kardano-Tartalia’s
formula. Among three roots of the cubic equation we must take the one which coincides with the
result as f∞ = 1. Thus, for f2
+ one finds
f2
+ = ∓2
3
√
a2 − 3 b sin
(
π
6
− 1
3
arctan
√
4(a2 − 3b)3
(9 ba− 27 c− 2 a3)2
− 1
)
− a
3
, (31)
here sign (−), at 9 ba− 27 c− 2 a3 > 0 and (+) at 9 ba− 27 c− 2 a3 < 0. The obtained expression
(31) is the boundary value of the function f(ζ) at ζ = a/2. It allows us to find a constant C which
is included in the solution of the Ginzburg-Landau equation.
174
On the theory of current states
5. The current density in an SNINS junction
Now let us consider the equilibrium current states, which can exist in an SNINS junction. As
a starting point the common expression for current in the Ginzburg-Landau theory [3,8] has been
taken
j(ζ) = i
7ζ(3)
16π2
env0
p0ξ0T 2
c
(
∆
d
∗
∆
dζ
−
∗
∆
d∆
dζ
)
. (32)
We shall calculate the density of the current in the region ξ0 � z � ξ(T ), where both the linear
integral equation and the Ginzburg-Landau equation are valid. Then one can use the asymptote
of the solution of the linear integral equation for ∆(z) at |z| � ξ0 which is linear
∆(ζ) = ∆
′
+ζ + ∆+, ζ → +∞,
∆(ζ) = ∆
′
−ζ + ∆−, ζ → −∞. (33)
-4 -3 -2 -1 0 1 2 3 4
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
-4 -3 -2 -1 0 1 2 3 4
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
-4 -3 -2 -1 0 1 2 3 4
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
-4 -3 -2 -1 0 1 2 3 4
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
a)
D(1)=0.8
D(1)=0.2
j( ∆
ϕ)
T=0.98T
c
d=0,01ξ
0
∆ϕ
b) D(1)=0.8
D(1)=0.2
j(∆
ϕ)
T=0.98T
c
d=0,1ξ
0
∆ϕ
c)
D(1)=0.8
D(1)=0.2
j( ∆
ϕ)
T=0.98T
c
d=ξ
0
∆ϕ
d) D(1)=0.8
D(1)=0.2
j( ∆
ϕ)
T=0.98T
c
d=10ξ
0
∆ϕ
Figure 2. Current-phase relation in SNINS junction at T/Tc = 0.98 for different values of the
thickness of the normal layer d and for different barrier transparencies D(1). The current density
is divided by the maximum value for D(1) = 0.8.
Let us substitute (33) into (32) and use representation (16). Thus, one gets
j =
1
2
env0
p0ξ0
(
1 − T
Tc
)(
1
q2,∞
− 1
q1,∞
)
f2
+ sinϕ. (34)
Here the symmetry of the junction is used. It gives f+ = f−, f ′
+ = −f ′
− (see [9]). Taking into
account (31) finally we have
j =
1
3
env0
p0ξ0
(
1 − T
Tc
)(
1
q2,∞
− 1
q1,∞
)
{
∓
√
a2 − 3f2
∞(4 − 3f2
∞)
175
V.E.Sakhnyuk, A.V.Svidzynskyj
× sin
(
π
6
− 1
3
arctg
√
4(a2 − 3f2
∞(4 − 3f2
∞))3
(9f2
∞(4 − 3f2
∞)a− 54f4
∞(f2
∞ − 1) − 2 a3)2
− 1
)
− a
2
}
sinϕ, (35)
here
a = −2
1 +
7ζ(3)
12(1 − T/Tc)
(
cos2 ϕ
2
q1,∞
+
sin2 ϕ
2
q2,∞
)2
.
From figure 2 it follows that the dependence of the current density on the phase difference
between the banks of the junction is strongly nonsinusoidal in Josephson junction SNINS type
with the thin normal metal layer at the high barrier transparency and the temperature close to
critical.
Both the current-phase and the current-thickness relations d� 1/p0 are taken for analysis. The
reason is that in the Green’s function G
(0)
ωn
(~r, ~r′) we ignore the terms, which are the fast oscillation
(on the distance order 1/p0). In fact we used a quasiclassical approximation.
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1
0,0
2,0x10-4
4,0x10-4
6,0x10-4
8,0x10-4
1,0x10-3
1,2x10-3
1,4x10-3
1,6x10-3
1,8x10-3
2,0x10-3
2,2x10-3
D(1)=0.2
D(1)=0.8
j C
/j 0
j
0
=env
0
/2p
0
ξ
0
T=0.98T
c
d/ξ
0
Figure 3. The critical current as a function of the mean normal layer thickness d for different
values of the barrier transparency D(1).
6. Conclusion
The main results of this study are as follows.
(I) There is a common scheme for building the linear integral equation which describers the
behaviour of the order parameter near NS interface.
(II) The boundary condition for the Ginzburg-Landau equation is obtained using the linear inte-
gral equation and the method of quasiorthogonality to asymptotics. The boundary condition
involves: a barrier transparency, an interlayer thickness, a temperature, a phase difference.
(III) The Ginzburg-Landau equation is solved taking into account the effect of the current on the
spatial behaviour of the order parameter. It is shown that the presence of the current in a
junction reduces the value of the order parameter in comparison with the currentless case.
(IV) The main result of this work is formula (35) for the current states in a SNINS junction for
an arbitrary thickness of the normal layer. Both current-phase and current-thickness relations
are shown in figures 2, 3. As a result, one can see that the dependence of the current density
on the phase difference between the banks of the junction is strongly nonsinusoidal for the
thin (d� ξ0) normal metal layer at a high barrier transparency and at the temperature close
to critical.
176
On the theory of current states
References
1. Svidzinsky A.V., Antsygina T.N, Bratus’ E.N., Zh. Eksp. Teor. Fiz., 1971, 61, No. 4, 1612 (in Russian);
Sov. Phys. JETP., 1971, 34, 860.
2. Svidzinsky A.V. The theory of Josephson current in the supercoducting junction whith nomal layer.
– In: Modern Problems of Statistical Physics. Kiev, 1985, 245–255.
3. Svidzinsky A.V. Space-inhomogeneous Problem of the Superconducting Theory. Moscow, 1982.
4. Golubev L.V., Rakov Y.A., Svidzinsky A.V., Fiz. Mnogochast. Sys., 1985, 7, 40 (in Russian).
5. Akhramovich L.N., Svidzinsky A.V., Fiz. Nizk. Temp., 1988, 14, 815 (in Russian).
6. Svidzynsky A.V., Viligursky O.M., Biruk O.M., Rakutsky A., Journal Physical Studies, 1999, 3, No. 3,
359.
7. Akhramovich L.N., Rakov Y.A., Svidzinsky A.V., Teor. Mat. Fiz., 1988, 77, 450 (in Russian).
8. Svidzinsky A.V. The Microscopic Theory of Superconductivity. Part 1. Lutsk, 2001.
9. Svidzinsky A.V. The Microscopic Theory of Superconductivity. Part 2. Lutsk, 2003.
10. Svidzinsky A.V. Mathematical Methods of The Theoretical Physics. Kyiv, 1998.
11. Svidzinsky A.V., Sakhnyuk V.E., Condens. Matter Phys., 2000, 3, 683.
12. Sakhnyuk V.E., Svidzinsky A.V., Condens. Matter Phys., 2003, 6, 159.
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PACS: 74.50.+r
177
178
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