Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities
The boundary conditions are obtained for NS interface at the presence of arbitrary concentrations of nonmagnetic impurities. The current states in SNS junction are considered. The solution obtained for the current is analysed, in particular it was found out how the characteristic length of the c...
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| Zitieren: | Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities / A.V. Svidzinsky, V.E. Sakhnyuk // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 683-696. — Бібліогр.: 2 назв. — англ. |
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irk-123456789-1210052017-06-14T03:06:02Z Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities Svidzinsky, A.V. Sakhnyuk, V.E. The boundary conditions are obtained for NS interface at the presence of arbitrary concentrations of nonmagnetic impurities. The current states in SNS junction are considered. The solution obtained for the current is analysed, in particular it was found out how the characteristic length of the current decreasing ξ₀ for pure superconductor turns to ξd for dirty one. A variant of the variational method, called the method of quasiorthogonality to asymptotics, is used to obtain the boundary conditions. Отримано граничні умови на NS-границі за наявності немагнітних домішок довільної концентрації. Розглянуто струмові стани в SNS- контакті. Зроблено аналіз отриманого виразу для струму, зокрема з’ясовано, як відбувається перехід від характерної довжини спадання струму ξ₀ для чистого надпровідника до ξd у випадку забрудненого. Для отримання граничних умов використаний варіант варіаційного методу, названий методом квазіортогональності до асимптотики. 2000 Article Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities / A.V. Svidzinsky, V.E. Sakhnyuk // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 683-696. — Бібліогр.: 2 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.683 PACS: 74.50.+r http://dspace.nbuv.gov.ua/handle/123456789/121005 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The boundary conditions are obtained for NS interface at the presence
of arbitrary concentrations of nonmagnetic impurities. The current states
in SNS junction are considered. The solution obtained for the current is
analysed, in particular it was found out how the characteristic length of the
current decreasing ξ₀ for pure superconductor turns to ξd for dirty one. A
variant of the variational method, called the method of quasiorthogonality
to asymptotics, is used to obtain the boundary conditions. |
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Article |
| author |
Svidzinsky, A.V. Sakhnyuk, V.E. |
| spellingShingle |
Svidzinsky, A.V. Sakhnyuk, V.E. Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities Condensed Matter Physics |
| author_facet |
Svidzinsky, A.V. Sakhnyuk, V.E. |
| author_sort |
Svidzinsky, A.V. |
| title |
Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities |
| title_short |
Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities |
| title_full |
Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities |
| title_fullStr |
Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities |
| title_full_unstemmed |
Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities |
| title_sort |
current states in sns junction for arbitrary concentrations of nonmagnetic impurities |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2000 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121005 |
| citation_txt |
Current states in SNS junction for arbitrary concentrations of nonmagnetic impurities / A.V. Svidzinsky, V.E. Sakhnyuk // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 683-696. — Бібліогр.: 2 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT svidzinskyav currentstatesinsnsjunctionforarbitraryconcentrationsofnonmagneticimpurities AT sakhnyukve currentstatesinsnsjunctionforarbitraryconcentrationsofnonmagneticimpurities |
| first_indexed |
2025-07-08T19:00:30Z |
| last_indexed |
2025-07-08T19:00:30Z |
| _version_ |
1837106444035424256 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 683–696
Current states in SNS junction for
arbitrary concentrations of
nonmagnetic impurities
A.V.Svidzinsky, V.E.Sakhnyuk
Department of Theoretical and Mathematical Physics,
Lesya Ukrainka Volyn State University,
13 Voli Avenue, 43000 Lutsk, Ukraine
Received April 18, 2000
The boundary conditions are obtained for NS interface at the presence
of arbitrary concentrations of nonmagnetic impurities. The current states
in SNS junction are considered. The solution obtained for the current is
analysed, in particular it was found out how the characteristic length of the
current decreasing ξ0 for pure superconductor turns to ξd for dirty one. A
variant of the variational method, called the method of quasiorthogonality
to asymptotics, is used to obtain the boundary conditions.
Key words: superconductor, order parameter, impurities, Green function
PACS: 74.50.+r
1. Introduction
To describe the current states in the superconducting junctions like SNS (super-
conductor – normal metal – superconductor) at the temperature not too close to
the critical the model with a constant modulus of the order parameter is used. Here
the availability of the coherent phase difference is assumed. It makes possible to
describe the current states. In reality the order parameter modulus decreases near
the border, but this change has a characteristic length ξ0 – the coherence length –
and it effects only the numerical coefficient in the current for wide junctions, when
the thickness d is much larger than ξ0. The simplified model cannot be applied at a
temperature close to the critical, where the coherence length increases (1−T/Tc)
−1/2
times. Therefore in this region of temperature one has to take into account spatial
distribution of the order parameter. In this work we are interested in the current
states at the temperature close to the critical. Both pure junction and the junction
with nonmagnetic impurities are considered. It’s natural to expect that the magni-
tude of such currents decreases, when d increases. One can raise a question: what
is a characteristic length of this decreasing? Since the order parameter close to the
c© A.V.Svidzinsky, V.E.Sakhnyuk 683
A.V.Svidzinsky, V.E.Sakhnyuk
critical temperature is small, physical quantities of the linear by ∆ approximation
will be expressed by the Green function of the normal metal. This Green function
has the characteristic length ξN(T ) of the order v0/T ; because T ∼ Tc, the charac-
teristic length is v0/T , i. e. it is of the order ξ0. If the free path length is finite, then
the characteristic length of the normal metal Green function equals (1/ξ0 + 1/l)−1.
In other words, the characteristic length must be equal to min(ξ0, l). Hence, just
this characteristic length will determine the current decreasing when d increases. In
reality, as it’s well known [1], if l ≪ ξ0 , then ξd =
√
ξ0l/3 will be the characteristic
length. Thus the effect is associated not with a harmonic mean of ξ0 and l, but with
their geometric one. It’s also known, that for l ≫ ξ0, ξ0 becomes the characteristic
length. Obviously, the appearance of new characteristic length is associated with the
induction of weak superconductivity in the normal layer, but it’s less obvious which
way the smooth transition from
√
ξ0l/3 to ξ0 takes place when the free path length
increases. This question will be considered in the first part of the work. Another
goal of the work is to build a new form of variation method that is more efficacious
than Rits method and Gal’orkin method. This will be done in the second part of the
work. Let’s recall the scheme of building the superconductivity theory equations for
the temperatures close to the critical. Since in this region of temperatures the order
parameter is small, the set of the Gor’kov’s equations can be solved by expanding
the Gor’kov’s Green functions by the degrees of the order parameter. As a result,
we’ll obtain the integral equation for the order parameter. Thus we shall obtain
the integral equation for the order parameter. We assume additionally, that the or-
der parameter changes slowly, then the obtained equation can be turned into the
differential. Here the linear by ∆(~r ) terms will vanish and the expansion must be
continued to cubic terms, which may be taken in a local form by ∆(~r ). The obtained
nonlinear differential equation is the Ginzburg-Landau equation. So for the order
parameter near the border we have the linear integral equation and at the distances
much larger than the coherence length we obtain a nonlinear differential equation.
We know its solution if the external magnetic field is absent. The Ginzburg-Landau
equation is just for the distances from the border of the order ξ0. It can be used
to obtain the boundary condition for the Ginzburg-Landau equation. Such a pos-
sibility arises because there exists a spatial region where both the linear integral
equation and the Ginzburg-Landau equation are true. For solutions of the integral
equation this is the region of the large distances from the border (comparing to
ξ(T )). Asymptotics of the linear integral equation proves to be the linear function.
It’s important, that the relation of two coefficients of this function is determined
from the equation. In case of SN interface it will be denoted as q∞ and it has to
be calculated explicitly. The parameter q∞ determines a boundary condition for the
Ginzburg-Landau equation – the tangent of a slope angle of the dimensionless order
parameter to the abscissa axis in zero. For SNS junction two linear equations arise,
their asymptotics being the linear functions as well, and such two parameters take
place. Since the current is expressed by these parameters they should be calculated
as well.
684
Current states in SNS junction for . . .
2. SNS junction in the presence of impurities
It’s known that the set of the linear integral equations, that arise for the order
parameter of a superconductor in the presence of impurities, has the form
∆(z) =
πρTc
v0
∑
ωn
1
∫
0
dx
x
∞
∫
−∞
exp
(
−2|ω′
n|
v0
|z − z′|
)
∆ωn
(z′)dz′, (1)
∆ωn
(z) = ∆(z) +
1
2l
1
∫
0
dx
x
∞
∫
−∞
exp
(
−2|ω′
n|
v0
|z − z′|
)
∆ωn
(z′)dz′. (2)
Here |ω′
n| = |ωn|+ 1/τ, ωn = πTc(2n+ 1) – Matsubara’s frequency, τ – mean free
time; τ = l/v0, v0 – Fermi-velocity; ρ = |g|N(0) – dimensionless attraction constant,
N(0) – electron state density on the surface of the Fermi-sphere. Let the surfaces,
that separate normal and superconductive metals, be planar and the axis OZ be
perpendicular to them. In such a geometry the spatial homogeneity is broken in the
direction of the axis OZ that is clear from the equations (1) and (2). To simplify our
dealing with equations (1) and (2) it’s convenient to write them in the dimensionless
form. Supposing z ≡ ζξ0 we obtain
∆n(ζ) = ∆(ζ) +
1
2λ
1
∫
0
dx
x
∞
∫
−∞
exp
(
−|ζ − ζ ′|
x
|2n′ + 1|
)
∆n(ζ
′)dζ ′, (3)
∆(ζ) =
ρ
2
∑
n
1
∫
0
dx
x
∞
∫
−∞
exp
(
−|ζ − ζ ′|
x
|2n′ + 1|
)
∆n(ζ
′)dζ ′. (4)
Here |2n′ + 1| = |2n+ 1|+ 1/λ, λ = l/ξ0 – dimensionless electron free path.
Equation (3) can be easily solved by Fourier transformation relative to ∆̃n(k).
Using this solution we can exclude ∆n and write a closed linear integral equation
for the order parameter
∆(ζ) =
∞
∫
−∞
K(ζ − ζ ′)∆(ζ ′)dζ ′. (5)
Fourier-component of the kernel of this equation has the form
K̃(k) = ρ
∑
n
[
1
k
arctan
k
|2n′ + 1|
] / [
1− 1
λk
arctan
k
|2n′ + 1|
]
. (6)
Using variational methods for equation (5) one can obtain the boundary condition
for the Ginzburg-Landau equation in the case of NS boundary. Omitting this (see
[1,§40]) we shall investigate the current states in SNS junction. The linear integral
equation (5) for the order parameter in the case of SNS junction has the form
∆(ζ) =
−a/2
∫
−∞
K(ζ − ζ ′)∆(ζ ′)dζ ′ +
∞
∫
a/2
K(ζ − ζ ′)∆(ζ ′)dζ ′, (7)
685
A.V.Svidzinsky, V.E.Sakhnyuk
because ∆ vanishes in a normal layer, i. e. in the region |z| < d/2, or |ζ | < a/2,
a = d/ξ0. The equation can be reduced to the equation on the half-axis ζ > 0 if we
introduce the even ∆s(ζ) and the odd ∆a(ζ) parts of the parameter ∆(ζ) and carry
out the shift on a/2. After the simple transformation we obtain (shifted functions
are denoted by the same letters)
∆s,a(ζ) =
∞
∫
0
{K(ζ − ζ ′)±K(ζ + ζ ′ + a)}∆s,a(ζ
′)dζ ′. (8)
Sign (+) corresponds to the even part of the function ∆(ζ) and (−) – to the odd
part of it. The explicit form of the kernel K(ζ − ζ ′) in the configuration space can
be obtained in the limit cases λ ∼ 0 and λ ≫ 1, i. e. in the cases of pure and dirty
superconductors. We shall consider the common scheme of description for whole the
interval of lengths λ.
Proceeding to the solution of equation (8), let us separate the asymptotics on
the infinity
∆(ζ) = ζ + q∞ + ψ(ζ).
We still do not put indices s and a near ∆(ζ) and q∞, because of the likeness of
the equation for them which is enough to consider the equation only for one of the
functions, for example, for ∆s(ζ). After the substitution shown we obtain
ψ(ζ)−
∞
∫
0
K(ζ − ζ ′)ψ(ζ ′)dζ ′ =
=
∞
∫
0
K(ζ + ζ ′)ζ ′dζ ′ − q∞
∞
∫
0
K(ζ + ζ ′)dζ ′ +
∞
∫
0
K(ζ + ζ ′ + a)∆(ζ ′)dζ ′.
We shall find the solution using the perturbation theory, taking into account only
the first order correction
ψ(ζ) = ψ(0)(ζ) + ψ(1)(ζ), q∞ = q(0)
∞
+ δq∞ .
The equation for the zero approximation has the form
ψ(0)(ζ)−
∞
∫
0
K(ζ − ζ ′)ψ(0)(ζ ′)dζ ′ =
∞
∫
0
K(ζ + ζ ′)ζ ′dζ ′ − q(0)
∞
∞
∫
0
K(ζ + ζ ′)dζ ′.
In the first approximation
ψ(1)(ζ)−
∞
∫
0
K(ζ − ζ ′)ψ(1)(ζ ′)dζ ′ =
= −δq∞
∞
∫
0
K(ζ + ζ ′)dζ ′ +
∞
∫
0
K(ζ + ζ ′ + a)∆(0)(ζ ′)dζ ′. (9)
686
Current states in SNS junction for . . .
Solution of equation (9) can be found from the condition of orthogonality of the
right side to the solution of the corresponding uniform equation, i. e. to ∆ (0)(ζ).
Thus, we have
δq∞ =
1
2I1
∞
∫
0
dζ∆(0)(ζ)
∞
∫
0
K(ζ + ζ ′ + a)∆(0)(ζ ′)dζ ′.
Here the identity proved in [2] was taken into account:
∞
∫
0
dζ∆(0)(ζ)
∞
∫
0
K(ζ + ζ ′)dζ ′ = 2I1,
Let us use the solution of the equation of zero approximation obtained by the
variational method: ∆(0)(ζ) ∼= ζ +Γ, where Γ = I1/I0 [1]. Thus, for δq∞ one obtains
δq∞ =
1
2
(
I2(a)
I1
+ 2
I1(a)
I0
+
I1
I20
I0(a)
)
, (10)
where
I0(a) =
∞
∫
0
dζ
∞
∫
0
dζ ′K(ζ + ζ ′ + a),
I1(a) =
∞
∫
0
dζ
∞
∫
0
dζ ′ζ ′K(ζ + ζ ′ + a),
I2(a) =
∞
∫
0
dζζ
∞
∫
0
dζ ′ζ ′K(ζ + ζ ′ + a). (11)
An expression for the current is given by a common formula of the Ginzburg-Landau
theory [2] (that should be calculated)
j =
6
7ζ(3)
env0
p0ξ0
(
1− T
Tc
)2
δq∞ sinϕ. (12)
This formula, however, contains δq∞, that should be found. Thus, describing
the current states in SNS junction we shall deal with the expression for current
(12), where δq∞ is represented by formula (10) with yet unknown magnitudes of the
integrals I1(a), I2(a), I3(a). Since for the wide SNS junction (d ≫ ξ0) the current
decreases exponentionally if the normal metal layer’s width increases, a characteristic
length of such a decrease appears. It equals ξ0 for the pure superconductor and ξd for
the extremely dirty one. This dependence has the origin in the mentioned integrals.
Thus, to investigate the transition from one characteristic length to another, one
has to transform these integrals to a convenient form.
We shall carry out the calculation in detail for one of them, e. g. for I0(a), the
scheme is the same for all of the three.
687
A.V.Svidzinsky, V.E.Sakhnyuk
Let us consider
I0(a) =
∞
∫
−∞
dζ
∞
∫
−∞
dζ ′K(ζ + ζ ′ + a).
First of all, let’s express K(ζ) by Fourier component K̃(k), an expression for the
latter is given by equation (6). Let
K(ζ) =
1
2π
∞
∫
−∞
K̃(k)e−ikζdk.
Then
I0(a) =
1
2π
∞
∫
−∞
dkK̃(k)e−ika
∞
∫
0
dζ ′
∞
∫
0
dζe−ik(ζ+ζ′)
= − 1
2π
∞
∫
−∞
K̃(k)e−ika
(k − iε)2
dk, ε→ 0. (13)
In the same manner
I1(a) =
i
2π
∞
∫
−∞
K̃(k)e−ika
(k − iε)3
dk, I2(a) =
1
2π
∞
∫
−∞
K̃(k)e−ika
(k − iε)4
dk, ε→ 0.
Let us use the relation
1
(k − iε)n
=
Pf
kn
+ iπ
(−1)n−1
(n− 1)!
δ(n−1)(k),
and integration rule of the pseudo function Pf/kn with function ϕ(k):
∞
∫
−∞
Pf
kn
ϕ(k)dk =
∞
∫
−∞
[
ϕ(k)− ϕ(0)− kϕ′(0)− ...− kn−2
(n− 2)!
ϕ(n−2)(0)
]
dk
kn
.
For the integral I0(a) we have
I0(a) = − 1
2π
∞
∫
−∞
K̃(k)e−ika − 1
k2
dk − i
2
lim
k→0
(K̃(k)e−ika)′
=
1
2π
∞
∫
−∞
1− K̃(k) cos ka
k2
dk − a
2
=
1
2π
∞
∫
−∞
(1− K̃(k)) cos ka
k2
dk. (14)
Here we used evenness of function K̃(k), equality K̃ ′(0) = 0 and the elementary
integral
∫
∞
−∞
[(1 − cos ka)/k2]dk = πa . The analogous expression for integrals I1(a)
and I2(a) are given by representations
I1(a) =
1
2π
∞
∫
−∞
(
K̃(k)− 1− 1
2
k2K̃ ′′(0)
)
sin ka
dk
k3
, (15)
I2(a) =
1
2π
∞
∫
−∞
(
K̃(k)− 1− 1
2
k2K̃ ′′(0)
)
cos ka
dk
k4
. (16)
688
Current states in SNS junction for . . .
To calculate K̃ ′′(0) we use an asymptotical expression for a small k
K̃(k)− 1
as
=
ρ
3
∑
n
k2
(2n+ 1)2|2n′ + 1| , k ∼ 0.
From here
K̃ ′′(0) =
2
3
ρ
∑
n
1
(2n+ 1)2|2n′ + 1| .
Let us replace the kernel K̃(k) by its implicit expression and change the variable in
the integrals I0(a), I1(a), I2(a) so that k = |2n′ + 1|x. Taking into account, that
1 = ρ
∑
n
1/|2n+ 1|, we obtain
I0(a) =
ρ
π
∞
∫
−∞
x− arctanx
1 + (x− arctanx)/λx
cos
[(
1 +
1
λ
)
ax
]
dx
x3
, (17)
I1(a) =
ρ
π
λ
1 + λ
∞
∫
−∞
(
arctan x− x
1 + (x− arctan x)/λx
+
x3
3
)
sin
[(
1 +
1
λ
)
ax
]
dx
x4
, (18)
I2(a) =
ρ
π
λ2
(1 + λ)2
∞
∫
−∞
(
arctan x− x
1 + (x− arctanx)/λx
+
x3
3
)
cos
[(
1 +
1
λ
)
ax
]
dx
x5
. (19)
Here we used the fact, that for d≫ ξ0 the main contribution to the sum is given
by n = 0 and n = −1. For the next consideration of these integrals we proceed to the
complex variables and use the logarithmic representation of arctangent Arctan z =
Ln[(1 + iz)/(1− iz)]/2i.
Let us integrate by contour closed in the upper half-plane, a cut being done by
an imaginary axis and the point of branching z = i being by-passed as it’s shown in
figure 1.
Figure 1. Path of integration for estimation of the integrals Is(a).
The contour contains a single pole, which is determined by the condition of equal-
ity of the function 1 + 1/λ− 1/λz arctanz to zero. The pole lies on the imaginary
689
A.V.Svidzinsky, V.E.Sakhnyuk
axis (z = iy0, y0 < 1). When integration by the imaginary axis we define a logarithm
so that on the left side (γ+) Ln(1 − y) = ln(y − 1) + iπ and on the right side (γ−)
Ln(1− y) = ln(y − 1)− iπ.
Using the residue theorem we obtain
Is(a) = ρ
λs+2
(1 + λ)s
∞
∫
1
exp{−a(1 + 1/λ)y}
(1 + λ+ ln[(y − 1)/(y + 1)]/2y)2 + π2/4y2
dy
y3+s
+ ρ
λs+2
(1 + λ)s
2(1− y20)
y1+s
0
exp{−a(1 + 1/λ)y0}
(1 + λ)y20 − λ
, s = 0, 1, 2. (20)
From this formula one can see that the integrals and, consequently, the magnitude
of the current through the junction decrease exponentially. To understand how the
characteristic length of decreasing is formed, let us consider both addends in (20).
In the pole one we have an exponent
exp
{
−a
(
1 +
1
λ
)
y0
}
= exp
{
−d
(
1
ξ0
+
1
l
)
y0
(
l
ξ0
)}
,
from where we see that the magnitude of the pole has an effect upon the forming
of the characteristic length. The root of the corresponding equation depends on the
concentration of impurities and can be found numerically (see figure 2).
Figure 2. Pole of integrand in the integrals Is(a).
As for the integral addend, it can be evaluated by Laplace method, and the
expression that determines the dependence in d can be expressed in the form
exp
{
−d
(
1
ξ0
+
1
l
)}
y1
(
l
ξ0
)
.
Here y1(l/ξ0) should be found from the condition of the minimum of the function
a
(
1 +
1
λ
)
y + ln
(
1 +
1
λ
+
1
2yλ
ln
y − 1
y + 1
)2
+
π2
4y2λ2
,
690
Current states in SNS junction for . . .
i.e. from equation
a+
a
λ
=
1
λ
[(
1 +
1
λ
+
1
2y1λ
ln
y1 − 1
y1 + 1
)(
1
y21
ln
y1 − 1
y1 + 1
− 2
y1
1
y21 − 1
)
+
π2
2y31λ1
]
×
(
1 +
1
λ
+
1
2y1λ
ln
y1 − 1
y1 + 1
)2
+
π2
4y21λ
2
−1
(21)
Thus the characteristic length of the current decreasing is inversed to the expres-
sion (1/ξ0 + 1/l)f(l/ξ0), where f(l/ξ0) is either y0(l/ξ0) or y1(l/ξ0).
Let us consider the extreme cases λ ∼ 0 and λ ≫ 1. If λ ∼ 0 then the main
contribution to (20) is made by the pole. Asymptotics of the contribution has the
form
2ρ
λ2+s
(1 + λ)s
1− y20
y1+s
0
exp{−a(1 + 1/λ)y0}
(1 + λ)y20 − λ
as
=
as
=2ρ
λ2+s
y1+s
0
exp
{
−a
λ
y0
}
y20 − λ
= ρ
(
λ
3
)
s+ 1
2
exp
−a
√
3
λ
. (22)
From this for the integrals Is(a), s = 0, 1, 2, we have
Is(a) ∼= ρ
(
λ
3
)
s+ 1
2
exp
−a
√
3
λ
, s = 0, 1, 2, λ≪ 1.
And accordingly, for the current we obtain
j ∼= 0, 5
env0
p0ξ0
√
l
ξ0
e−d
√
3/ξ0l
(
1− T
Tc
)2
sinϕ, d≫
√
ξ0l
3
. (23)
In the limit λ ≫ 1 the main contribution to (20) is made by the integral term,
that asymptotically for the integrals Is(a), s = 0, 1, 2 gives:
Is(a) ∼= ρ
ξ0
d
exp
(
− d
ξ0
)
, λ≫ 1.
and, thus, currently we obtain the expression
j ∼= 2, 5
env0
p0d
e−d/ξ0
(
1− T
Tc
)2
sinϕ, d ≫ ξ0. (24)
3. The common equation for NIS-junction with nonmagnetic
impurities
In a common case, when the reflecting of electrons from an interface is available,
a set of linear integral equations, written in the dimensionless quantities, has the
691
A.V.Svidzinsky, V.E.Sakhnyuk
form [1, §40]:
∆(ζ) =
ρ
2
θ(ζ)
∞
∫
−∞
dζ ′
∑
n
1
∫
0
dx
x
{
exp
(
−|ζ − ζ ′|
x
|2n′ + 1|
)
+R(x) sign ζζ ′ exp
(
−|ζ |+ |ζ ′|
x
|2n′ + 1|
)}
∆n(ζ
′), (25)
∆n(ζ) = ∆(ζ) +
1
2λ
∞
∫
−∞
dζ ′
1
∫
0
dx
x
{
exp
(
−|ζ − ζ ′|
x
|2n′ + 1|
)
+R(x) sign ζζ ′ exp
(
−|ζ |+ |ζ ′|
x
|2n′ + 1|
)}
∆n(ζ
′). (26)
Although now one cannot exclude ∆n(ζ) and obtain a closed equation for ∆(ζ),
the asymptotical properties of the equation solutions in +∞ remain. Really, if
ζ → ∞, then the additional terms, associated with R, disappear from the equa-
tion kernels, and the asymptotical form of both equations does not contain R any
more. If we denote ζ − ζ ′ = u, and take into account that in the integrand ζ ≫ u,
we can write
∆n(ζ)
as
=∆(ζ) +
1
2λ
∞
∫
−∞
du(∆n(ζ) +
1
2
∆′′
n(ζ)u
2)
1
∫
0
dx
x
exp
(
−|u|
x
|2n′ + 1|
)
,
i.e.
∆n(ζ)
as
=∆(ζ) +
1
λ
∆n(ζ)
|2n′ + 1| +
1
3λ
∆′′
n(ζ)
|2n′ + 1|3 . (27)
The first equation will have the asymptotical form
∆(ζ)
as
= ρ
∑
n
∆n(ζ)
|2n′ + 1| +
ρ
3
∑
n
∆′′
n(ζ)
|2n′ + 1|3 . (28)
Expressing ∆n(ζ) from equation (27) and substituting it to equation (28) we obtain
∆(ζ) = ρ
∑
n
1
|2n+ 1|∆(ζ) +
ρ
3λ
∑
n
∆′′
n(ζ)
|2n+ 1||2n′ + 1|3 +
ρ
3
∑
n
∆′′
n(ζ)
|2n′ + 1|3 .
∆(ζ) drops out from this equation and the terms with the second derivative are
reduced. As a result
ρ
3
∑
n
∆′′
n(ζ)
|2n+ 1||2n′ + 1|3 = 0,
from where ∆′′ = 0. And taking into account (27)
∆n(ζ)
as
=
∣
∣
∣
∣
2n′ + 1
2n+ 1
∣
∣
∣
∣
∆(ζ). (29)
692
Current states in SNS junction for . . .
From the asymptotical equality of the derivatives ∆ ′′(ζ) and ∆′′
n(ζ) to zero one
can conclude, that the asymptotics of the solutions of the integral equation on +∞
are linear:
∆(ζ)
as
= ζ + q∞,
∆n(ζ) =
∣
∣
∣
∣
2n′ + 1
2n+ 1
∣
∣
∣
∣
(ζ + q∞), ζ → +∞. (30)
Obviously, at ζ → −∞ asymptotics is zero, because ∆(ζ) = 0 at ζ < 0, and
∆n(ζ) goes to zero if ζ → −∞, as one can see from the equation. The asymptotics
is extracted as usually, and here we shall write it in the form Aζ + B, that gives
more possibilities afterwards, for example, we can include a case A = 0 (q∞ = ∞).
So we write
∆(ζ) = (Aζ +B)θ(ζ) + ψ(ζ),
∆n(ζ) =
∣
∣
∣
∣
2n′ + 1
2n+ 1
∣
∣
∣
∣
(Aζ +B)θ(ζ) + ψn(ζ).
After extracting the asymptotics, the terms that do not go to zero on the infinity
are cancelled and the equations get homogeneous.
ψn(ζ)− ψ(ζ)− 1
2λ
∞
∫
−∞
dζ ′ψn(ζ
′)
1
∫
0
dx
x
×
{
exp
(
−|ζ − ζ ′|
x
|2n′ + 1|
)
+R(x) sign ζζ ′ exp
(
−|ζ |+ |ζ ′|
x
|2n′ + 1|
)}
=
=
A
2λ|2n+ 1||2n′ + 1|
1
∫
0
dxx(1 +R(x) sign ζ) exp
(
−|2n′ + 1|
x
|ζ |
)
− B sign ζ
2λ|2n+ 1|
1
∫
0
dxD(x) exp
(
−|2n′ + 1|
x
|ζ |
)
. (31)
ψ(ζ)− ρ
2
θ(ζ)
∞
∫
−∞
dζ ′
∑
n
ψn(ζ
′)
1
∫
0
dx
x
×
{
exp
(
−|ζ − ζ ′|
x
|2n′ + 1|
)
+R(x) sign ζζ ′ exp
(
−|ζ |+ |ζ ′|
x
|2n′ + 1|
)}
=
= A
ρ
2
θ(ζ)
∑
n
1
|2n+ 1||2n′ + 1|
1
∫
0
dxx(1 +R(x)) exp
(
−|2n′ + 1|
x
ζ
)
−B
ρ
2
θ(ζ)
∑
n
1
|2n+ 1|
1
∫
0
dxD(x) exp
(
−|2n′ + 1|
x
ζ
)
. (32)
We shall give here a form of the variation method that can be called a method of
quasiorthogonality to asymptotics. Let us integrate both equations by ζ from zero
693
A.V.Svidzinsky, V.E.Sakhnyuk
to infinity. We obtain
∞
∫
0
ψn(ζ)dζ =
|2n′ + 1|
|2n+ 1|
∞
∫
0
ψ(ζ)dζ
− 1
2λ|2n+ 1|
∞
∫
−∞
dζψn(ζ) sign ζ
1
∫
0
dxD(x) exp
(
−|2n′ + 1|
x
|ζ |
)
+
A
2λ|2n′ + 1|(2n′ + 1)2
1
∫
0
dxx2(1 +R(x))− B
2λ(2n+ 1)2
1
∫
0
dxxD(x), (33)
∞
∫
0
ψ(ζ)dζ − ρ
∑
n
1
|2n′ + 1|
∞
∫
0
ψn(ζ)dζ
+
ρ
2
∑
n
1
|2n′ + 1|
∞
∫
−∞
dζψn(ζ) sign ζ
1
∫
0
dxD(x) exp
(
−|2n′ + 1|
x
|ζ |
)
=
= A
ρ
2
∑
n
1
|2n+ 1|(2n′ + 1)2
1
∫
0
x2(1 +R(x))dx
−B
ρ
2
∑
n
1
|2n+ 1||2n′ + 1|
1
∫
0
xD(x)dx. (34)
Now let us substitute the expression (33) into equation (34). After simplification
we have an equality
∑
n
1
|2n+ 1|
∞
∫
−∞
dζψn(ζ) sign ζ
1
∫
0
dxD(x) exp
(
−|2n′ + 1|
x
|ζ |
)
=
= A
∑
n
1
|2n′ + 1|(2n+ 1)2
1
∫
0
x2(1 +R(x))dx− B
∑
n
1
(2n+ 1)2
1
∫
0
xD(x)dx. (35)
Similar manipulations should be performed with equations (31), (32), multiplied
by ζ and integrated by ζ . Thus, in the same way we substitute an expression for
∫
∞
0 ψn(ζ)ζdζ into the relation for
∫
∞
0 ψ(ζ)ζdζ and after all simplifications we obtain
∑
n
1
|2n+ 1||2n′ + 1|
∞
∫
−∞
dζψn(ζ)
1
∫
0
dxx(1 +R(x) sign ζ) exp
(
−|2n′ + 1|
x
|ζ |
)
=
= B
∑
n
1
(2n+ 1)2|2n′ + 1|
1
∫
0
x2D(x)dx
− A
∑
n
1
(2n+ 1)2(2n′ + 1)2
1
∫
0
x3(1 +R(x))dx. (36)
We shall show how to use the method of quasiorthogonality for the approximate
calculation of the coefficient q∞ in case when on the NS interface we have a reflection
694
Current states in SNS junction for . . .
and the dimensionless free path length λ≫ 1. Let us rewrite equations (35) and (36)
in the form of a relation that contains a function qn(ζ) = ψn(ζ) + |(2n′ + 1)/(2n+
1)|q∞θ(ζ). Here we introduce the notation
χ(λ−1) =
∑
n
1
(2n+ 1)2|2n′ + 1| , χ1(λ
−1) =
∑
n
1
(2n+ 1)2(2n′ + 1)2
.
This way, instead of equations (35) and (36) we obtain the following:
∑
n
1
|2n+ 1|
∞
∫
−∞
dζqn(ζ) sign ζ
1
∫
0
dxD(x) exp
(
−|2n′ + 1|
x
|ζ |
)
=
= χ(λ−1)
1
∫
0
(1 +R(x))x2dx, (37)
∑
n
1
|2n+ 1||2n′ + 1|
∞
∫
−∞
dζqn(ζ)
1
∫
0
dxx(1 +R(x) sign ζ) exp
(
−|2n′ + 1|
x
|ζ |
)
=
=
2
3
q∞χ(λ
−1)− χ1(λ
−1)
1
∫
0
x3(1 +R(x))dx. (38)
Writing a trial function in the form qn(ζ) ≡ θ(ζ)|(2n′ + 1)/(2n + 1)|Γ, where
Γ = const, we get from the relation (37) and (38) the following:
q∞ =
3
2
χ1(λ
−1)
χ(λ−1)
1
∫
0
x2(1 +R(x))dx+
3
2
χ(λ−1)
[∫ 1
0
x2(1 +R(x))dx
]2
S2
∫ 1
0
xD(x)dx
.
This formula contains a result for NIS surface at the presence of impurities, including
(l ≫ ξ0). But the formula
q∞
as
=
2
3
χ(λ−1)
S2
(∫ 1
0
xD(x)dx
)−1
(39)
is asymptotically exact at small D for arbitrary concentrations of impurities, includ-
ing l ≪ ξ0, when it gets the form
q∞
as
=
2
3
λ
(∫ 1
0
xD(x)dx
)−1
. (40)
To obtain this solution, we have to note, that when R = 1 the set of equations (25),
(26) is satisfied by the constant ∆n = |(2n′ + 1)/(2n + 1)|∆, ∆ = const. It means,
that in this case A = 0 and B is any constant that can be taken as a unit.
If D is small, one can apply the perturbation theory by D, taking for zero ap-
proximation A(0) = 0, B(0) = 1 and ψ(0)
n (ζ) = 0, ψ(0)(ζ) = 0. Considering now
(35) we see that the left part of this relation is a quantity of the second order by
695
A.V.Svidzinsky, V.E.Sakhnyuk
D and thus can be neglected in the first approximation. Then equation (35) gives
A(1) 2
3
χ(λ−1) = S2
∫ 1
0 xD(x)dx, because B is equal to unit in the zero approximation.
Thus, for q∞ = 1/A we obtain (39). In particular, for λ ≪ 1 in the expression for
χ(λ−1) we can let
χ(λ−1) =
∑
n
1
(2n+ 1)2|2n′ + 1| =
∑
n
1
(2n + 1)2[|2n′ + 1|+ 1/λ]
∼= λS2.
Thus we obtain an asymptotically exact expression (40).
References
1. Svidzinsky A.V. Space-Inhomogeneous Problems of the Superconducting Theory.
Moscow, Nauka, 1982.
2. Svidzinsky A.V. The theory of the Josephson’s currents in superconducting junctions
with normal layer. – In: Problems of the Modern Statistical Physics. Kyiv, Naukova
dumka, 1985, p. 245–256.
Струмовi стани в SNS-контактi за наявностi
немагнiтних домiшок довiльної концентрацiї
А.В.Свідзинський, В.Є.Сахнюк
Кафедра теоретичної і математичної фізики,
Волинський державний університет ім. Лесі Українки,
43000 Луцьк, проспект Волі, 13
Отримано 18 квітня 2000 р.
Отримано граничні умови на NS-границі за наявності немагнітних
домішок довільної концентрації. Розглянуто струмові стани в SNS-
контакті. Зроблено аналіз отриманого виразу для струму, зокрема
з’ясовано, як відбувається перехід від характерної довжини спадан-
ня струму ξ0 для чистого надпровідника до ξd у випадку забруднено-
го. Для отримання граничних умов використаний варіант варіаційно-
го методу, названий методом квазіортогональності до асимптотики.
Ключові слова: надпровідник, параметр впорядкування, домiшки,
функція Гріна
PACS: 74.50.+r
696
|