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Travelling 4π-kink in nonlocal Josephson electrodynamics
Physics Letters A 177 (1993) North-Holland
259-262
PHYSICS
LETTERS
A
Travelling 4x-kink in nonlocal Josephson electrodynamics Yu.M. Aliev and V.P. Silin P.N. Lebedev Physical Institute, Leninsky Prospect 53, 1 I7924 Moscow, Russian Federation Received 20 March 1993; accepted for publication Communicated by V.M. Agranovich
26 April 1993
Basic statements of the nonlocal electrodynamics of Josephson junctions between two different superconductors are formulated. In the limit of strong nonlocality when the Josephson penetration depth is much smaller than the London length, the solution describing a travelling 4x-kink corresponding to the propagation of an Abrikosov-like vortex with a regular core carrying two magnetic flux quanta is obtained.
The development of nonlocal Josephson electrodynamics under the conditions where the Josephson length AJ is smaller than the London depth II of magnetic field penetration into the superconductor has started recently [ I-31. Such a situation occurs for large enough Josephson critical currents and smaller depairing currents, which is typical for high temperature superconductors, when the Ginzburg-Landau parameter K is large enough. In ref. [ 11 for the contact between identical superconductors an integrodifferential equation for the phase difference was formulated that generalizes the well-known sineGordon equation. The static solution of this equation was obtained in ref. [ 21 for the case A>> A,. This solution like the known solution in the local theory [ 4 ] represents a static 2x-kink. It describes the static magnetic field vortex of Abrikosov type with regular core. In ref. [ 31 for the resistive limit in the framework of the nonlocal Josephson electrodynamics the relaxation of nonstationary vortices is presented and it was shown how the static 2rr-kink is established. In this communication we present, first, the basic equation of nonlocal Josephson electrodynamics in the case of contact between two different superconductors and analyze its general features. Second, we give the asymptotic solution of this integro-differential equation, that describes a 4x-kink, travelling with constant velocity of propagation. The main difference between this vortex and the known Josephson vortices is that our 4n-kink carries two magnetic Elsevier Science Publishers
B.V,
flux quanta, but not one quantum as in the local theory. Let us consider a plane uniform tunnel layer with width 2d limited along the x axis by infinite in-plane y, z boundaries of two superconductors x= -d, x= d. Then for the phase difference p we have the equation [41
(1) Here j, is the density of the critical current through the junction, HY(z, t) the magnetic field intensity independent of x inside the contact and the usual definitions
(2) are used, where e= ]el is the electron charge, C,= c/&d the capacity and R the resistivity per unit area of the contact. In order to connect the right hand side of eq. ( 1) with the phase difference v we use the Maxwell equation aE aE Z_X=_Y ax aZ
1 aH at .
c
(3)
Let us insert into eq. (3) the approximate relation for the z component of the electric field valid inside the contact, 259
Volume
177, number
3
PHYSICS
EAx, z, t) = t [Ed4 z, t) +E,( -4 z, t) I +
5
LEA4 z, t) -EA -d, z, t) 1
and use the Josephson
relation
41eld
$
]&(d,
--
z, t) -&(
fi
41eld
-d,
A
14 June 1993
hc m dz’ Q(z-z’) t)= - s 214 --co
(5)
z, t) 1
azat
1 aH,(z, 0
7
(affiat)
hc - -21el
z, t)
at
.
,
_12
dH,(x,
ax
x=d
z> t)
x=-d
ap az
=2dH,,(z, t) .
(10)
00
X
dz’ Hy( z’, t) s --oD
CO:at
of
at2
m dk sexp[ik(z-z’)5 -co
aw,
___
t)
(11)
azf ’
-cc
where ;1i=Ac2/8rrle(jc. In the local limit, the smooth changing of a, at the scale lengths A+ and A_, the approximation
d(z) Q(z) = A++d_+2d
(12)
is valid, where 6(z) is the delta function. Using the usual definition for the Josephson length A: = Az(A+ +A_ +2d)-‘,weobtainfromeq. (1) thewellknown sine-Gordon equation with dissipation. The limit A+ =A _ =a>> 2d analyzed in ref. [ 1 ] is obtained in the case when (13) This expression is also valid for the case A+ =IZ P+ I _, 2d. Finally, let us analyze the asymptotic limit of eq. ( 1) when rpsharply changes at lengths less than A + and ;1_. Then, using the asymptotic expression
Q(z) =
yac)[ln(2/lzl)-C+f(a+,a-)l
x(a2
z
+
(14) (kx-d)dw]
,
(8) where + and - correspond to the half-spaces x>,d and x< -d, respectively. Substituting (8) into the left hand side of eq. (7) and solving the integral equation obtained, we have 260
eq. ( 1) in the equation for
(7)
In connection the equations a’, AH,, -H, = 0 for the magnetic fields inside the superconductors are expressed by means of the boundary value H,,( z, t) with the help of the relation
Hy(x, z, t) =
(9)
Q(z)=-cc f,+@$;P::;m+2d’
(6)
where ,? f are the London lengths for the x2 d and x=$ -d superconductor regions, respectively. Equation (6) takes the form
ax
1
where
dz’ Q(z-z’)
Az dH,(x, +
t)
sin~+-!T2Y+L!Y!?
awz,0
rot
azl
Expression ( 9 ) enables us to represent closed form of an integro-differential the phase difference ~1,
eq. (3) takes the form
Taking into account that the electric field inside the superconductor can be found from the Maxwell equation by means of the magnetic field intensity, E= ;a.:
aa’, ~
7
[4]
at ’
Then inside the junction
HJz, (4)
E,=-~=_- fi ap 2d
LETTERS
where C,= 0.577 is the Euler constant
and
/(a+,a_)=(a:_a2)-1[,a:ha+-aTha_ -a+A_
tan-l($$-)].
(15)
Volume
177, number
PHYSICS
3
LETTERS
14 June 1993
A
Then eq. ( 11) takes the form
A;
=m;
dz’ ap(2’, f)
l”O
-___ z’-z
f
(16)
azf ’
-cv
where the integral in the right hand side means the principal value of the Cauchy integral and corresponds to the Hilbert transformation. Formulas (8 ), (9) permit one to connect the magnetic fields in the superconductors with the phase difference, HY(X>z, t)
?ic
=--
214
cvdz’ Q, (z-z’, f -m
2x-d)
aw, T,
The finite conductivity energy dissipation d% -=dt
=-=
f f ‘d_/=
-d A2
rs of the junction
dz uE;
-co
m dz !?!! 2 . f 0 at --co
t)
m, 2)=v
(y
(z-vt)
(22)
(23)
Qk (z, x) O” dk
f
-cc
exp(ikz-xJ_)
%cn+Jm
t-L,/-
+2d’ (18)
Here the signs + and - correspond to x>,d and x< -d, respectively. In the asymptotic limit, when p is a function of z and changes sharply at lengths less than 2 + and A_-) it is necessary to use the asymptotic expression Q, (z, X) = Q(Jm) where for the Q function expression (14) should be used. Equation ( 11) for /I=0 corresponds to the Hamiltonian function
c%=$${_~dz[&($+l-cosv] m
in; f
Cm
dz’
--m
f
dz” Q( z’- z”)
-c0 (19)
The
SW(<) >
satisfies the equation
where
+
(21)
As an application of the general expressions of the present theory let us consider the solution of eq. ( 16 ) for a travelling wave with constant velocity of propagation v, where
(17)
=
gives the
latter
formula
H= W+ T, where
Josephson
vortices,
can
be written
in the
where cr2= [v(L$ +A?)/c+ni]‘, and the integral means the principal value of the Cauchy integral. Here we present the following solution of eq. (23 ), w(t)=4
tan-‘(
+c) .
(24)
The velocity of propagation of the kink (antikink) is defined by the relation &= 1. In accordance with the relation [ 41 ~=~~OIC(Z=-“‘t)-P(Z=+co,t)]
(25)
connecting the magnetic vortex flux with the phase difference and the flux quantum &, = KAC/1el we find that our 4x-kink (eq. (24) ) carries two magnetic flux quanta. This qualitatively differentiates the travelling kink (24) from the moving kinks of the local Josephson theory, and so from the static 2n-kink, obtained in ref. [2] in the framework of the nonlocal Josephson theory, carrying only one flux quantum. The magnetic field corresponding to the 4x-kink (24) has the form
form
W is the energy of the interacting
and the kinetic energy T is 261
Volume
177, number
HY(X, z,t)= -ln[
3
rr(a:
( IxTdl
PHYSICS
+1.2> {ln2-C+f(A+,i_) +a2_))2+
(z-ut>2]}
A
14 June 1993
Finally let us determine the mass of the kink by means of the relation T= f mu2 (compare ref. [ 21) and we obtain
2@0
+;i$(A:
LETTERS
. (26)
The total energy of the 4rc-kink is @’ 2rc2(a: +A?)
#=
1 ___ + n: -2: -L+il_
{1-C+ln((n:+ii)e)
$(A: +A?)
ln(;l+//Z_)
[ tan-‘($$$)]~.
(27)
It should be noted that this energy exceeds exactly four times the energy of the static (v= 0) 2n-kink, corresponding to the value a2 = 0 in eq. (23 ) and to the solution w( 5) = K+ 2 tan-‘< [ 21. The latter solution corresponds to the zero kinetic energy ( T= 0)) whereas for the travelling kink the kinetic energy is
(28)
(29) that is twice larger than the estimate of the mass presented in ref. [ 21 for the case I + =a _ =a. So, the present results may be considered as the foundations of nonlocal Josephson electrodynamics for a tunnel junction between two different superconductors. The description of the travelling 4n-kink for the phase difference of the Cooper pairs is presented. The magnetic field of the kink, carrying two magnetic flux quanta, the core of which essentially defines its energy, corresponds to the Abrikosov-type vortex. Due to the nonlinearity of the tunnel current and nonlocality the core of the moving vortex obtained regularizes at a length Lz(L: +a: )-I, instead of the superconductivity coherence length.
References [ 1] Yu.M. Aliev, V.P. Silin and S.A. Uryupin, Superconductivity
The magnetic field in the frame system moving with velocity v is functionally the same as the magnetic field of the static 27c-kink and is twice larger than value (26). The magnetic field of the 4n-kink is formed by Josephson and superconductor currents as well as by displacement currents, in contrast to the static 27t-kink for which the displacement current is absent.
262
5 (1992)
230.
[ 21 A. Gurevich, Phys. Rev. B 46 ( 1992) 3 187. [ 31 Yu.M. Aliev, V.P. Silin and S.A. Uryupin, Pis’ma Zh. Eksp. Teor. Fiz. 57 (1993)
187.
[ 41 A. Barone and G. Paterno, Physics and applications Josephson
effect (Wiley, New York, 1982).
of the
259-262
PHYSICS
LETTERS
A
Travelling 4x-kink in nonlocal Josephson electrodynamics Yu.M. Aliev and V.P. Silin P.N. Lebedev Physical Institute, Leninsky Prospect 53, 1 I7924 Moscow, Russian Federation Received 20 March 1993; accepted for publication Communicated by V.M. Agranovich
26 April 1993
Basic statements of the nonlocal electrodynamics of Josephson junctions between two different superconductors are formulated. In the limit of strong nonlocality when the Josephson penetration depth is much smaller than the London length, the solution describing a travelling 4x-kink corresponding to the propagation of an Abrikosov-like vortex with a regular core carrying two magnetic flux quanta is obtained.
The development of nonlocal Josephson electrodynamics under the conditions where the Josephson length AJ is smaller than the London depth II of magnetic field penetration into the superconductor has started recently [ I-31. Such a situation occurs for large enough Josephson critical currents and smaller depairing currents, which is typical for high temperature superconductors, when the Ginzburg-Landau parameter K is large enough. In ref. [ 11 for the contact between identical superconductors an integrodifferential equation for the phase difference was formulated that generalizes the well-known sineGordon equation. The static solution of this equation was obtained in ref. [ 21 for the case A>> A,. This solution like the known solution in the local theory [ 4 ] represents a static 2x-kink. It describes the static magnetic field vortex of Abrikosov type with regular core. In ref. [ 31 for the resistive limit in the framework of the nonlocal Josephson electrodynamics the relaxation of nonstationary vortices is presented and it was shown how the static 2rr-kink is established. In this communication we present, first, the basic equation of nonlocal Josephson electrodynamics in the case of contact between two different superconductors and analyze its general features. Second, we give the asymptotic solution of this integro-differential equation, that describes a 4x-kink, travelling with constant velocity of propagation. The main difference between this vortex and the known Josephson vortices is that our 4n-kink carries two magnetic Elsevier Science Publishers
B.V,
flux quanta, but not one quantum as in the local theory. Let us consider a plane uniform tunnel layer with width 2d limited along the x axis by infinite in-plane y, z boundaries of two superconductors x= -d, x= d. Then for the phase difference p we have the equation [41
(1) Here j, is the density of the critical current through the junction, HY(z, t) the magnetic field intensity independent of x inside the contact and the usual definitions
(2) are used, where e= ]el is the electron charge, C,= c/&d the capacity and R the resistivity per unit area of the contact. In order to connect the right hand side of eq. ( 1) with the phase difference v we use the Maxwell equation aE aE Z_X=_Y ax aZ
1 aH at .
c
(3)
Let us insert into eq. (3) the approximate relation for the z component of the electric field valid inside the contact, 259
Volume
177, number
3
PHYSICS
EAx, z, t) = t [Ed4 z, t) +E,( -4 z, t) I +
5
LEA4 z, t) -EA -d, z, t) 1
and use the Josephson
relation
41eld
$
]&(d,
--
z, t) -&(
fi
41eld
-d,
A
14 June 1993
hc m dz’ Q(z-z’) t)= - s 214 --co
(5)
z, t) 1
azat
1 aH,(z, 0
7
(affiat)
hc - -21el
z, t)
at
.
,
_12
dH,(x,
ax
x=d
z> t)
x=-d
ap az
=2dH,,(z, t) .
(10)
00
X
dz’ Hy( z’, t) s --oD
CO:at
of
at2
m dk sexp[ik(z-z’)5 -co
aw,
___
t)
(11)
azf ’
-cc
where ;1i=Ac2/8rrle(jc. In the local limit, the smooth changing of a, at the scale lengths A+ and A_, the approximation
d(z) Q(z) = A++d_+2d
(12)
is valid, where 6(z) is the delta function. Using the usual definition for the Josephson length A: = Az(A+ +A_ +2d)-‘,weobtainfromeq. (1) thewellknown sine-Gordon equation with dissipation. The limit A+ =A _ =a>> 2d analyzed in ref. [ 1 ] is obtained in the case when (13) This expression is also valid for the case A+ =IZ P+ I _, 2d. Finally, let us analyze the asymptotic limit of eq. ( 1) when rpsharply changes at lengths less than A + and ;1_. Then, using the asymptotic expression
Q(z) =
yac)[ln(2/lzl)-C+f(a+,a-)l
x(a2
z
+
(14) (kx-d)dw]
,
(8) where + and - correspond to the half-spaces x>,d and x< -d, respectively. Substituting (8) into the left hand side of eq. (7) and solving the integral equation obtained, we have 260
eq. ( 1) in the equation for
(7)
In connection the equations a’, AH,, -H, = 0 for the magnetic fields inside the superconductors are expressed by means of the boundary value H,,( z, t) with the help of the relation
Hy(x, z, t) =
(9)
Q(z)=-cc f,+@$;P::;m+2d’
(6)
where ,? f are the London lengths for the x2 d and x=$ -d superconductor regions, respectively. Equation (6) takes the form
ax
1
where
dz’ Q(z-z’)
Az dH,(x, +
t)
sin~+-!T2Y+L!Y!?
awz,0
rot
azl
Expression ( 9 ) enables us to represent closed form of an integro-differential the phase difference ~1,
eq. (3) takes the form
Taking into account that the electric field inside the superconductor can be found from the Maxwell equation by means of the magnetic field intensity, E= ;a.:
aa’, ~
7
[4]
at ’
Then inside the junction
HJz, (4)
E,=-~=_- fi ap 2d
LETTERS
where C,= 0.577 is the Euler constant
and
/(a+,a_)=(a:_a2)-1[,a:ha+-aTha_ -a+A_
tan-l($$-)].
(15)
Volume
177, number
PHYSICS
3
LETTERS
14 June 1993
A
Then eq. ( 11) takes the form
A;
=m;
dz’ ap(2’, f)
l”O
-___ z’-z
f
(16)
azf ’
-cv
where the integral in the right hand side means the principal value of the Cauchy integral and corresponds to the Hilbert transformation. Formulas (8 ), (9) permit one to connect the magnetic fields in the superconductors with the phase difference, HY(X>z, t)
?ic
=--
214
cvdz’ Q, (z-z’, f -m
2x-d)
aw, T,
The finite conductivity energy dissipation d% -=dt
=-=
f f ‘d_/=
-d A2
rs of the junction
dz uE;
-co
m dz !?!! 2 . f 0 at --co
t)
m, 2)=v
(y
(z-vt)
(22)
(23)
Qk (z, x) O” dk
f
-cc
exp(ikz-xJ_)
%cn+Jm
t-L,/-
+2d’ (18)
Here the signs + and - correspond to x>,d and x< -d, respectively. In the asymptotic limit, when p is a function of z and changes sharply at lengths less than 2 + and A_-) it is necessary to use the asymptotic expression Q, (z, X) = Q(Jm) where for the Q function expression (14) should be used. Equation ( 11) for /I=0 corresponds to the Hamiltonian function
c%=$${_~dz[&($+l-cosv] m
in; f
Cm
dz’
--m
f
dz” Q( z’- z”)
-c0 (19)
The
SW(<) >
satisfies the equation
where
+
(21)
As an application of the general expressions of the present theory let us consider the solution of eq. ( 16 ) for a travelling wave with constant velocity of propagation v, where
(17)
=
gives the
latter
formula
H= W+ T, where
Josephson
vortices,
can
be written
in the
where cr2= [v(L$ +A?)/c+ni]‘, and the integral means the principal value of the Cauchy integral. Here we present the following solution of eq. (23 ), w(t)=4
tan-‘(
+c) .
(24)
The velocity of propagation of the kink (antikink) is defined by the relation &= 1. In accordance with the relation [ 41 ~=~~OIC(Z=-“‘t)-P(Z=+co,t)]
(25)
connecting the magnetic vortex flux with the phase difference and the flux quantum &, = KAC/1el we find that our 4x-kink (eq. (24) ) carries two magnetic flux quanta. This qualitatively differentiates the travelling kink (24) from the moving kinks of the local Josephson theory, and so from the static 2n-kink, obtained in ref. [2] in the framework of the nonlocal Josephson theory, carrying only one flux quantum. The magnetic field corresponding to the 4x-kink (24) has the form
form
W is the energy of the interacting
and the kinetic energy T is 261
Volume
177, number
HY(X, z,t)= -ln[
3
rr(a:
( IxTdl
PHYSICS
+1.2> {ln2-C+f(A+,i_) +a2_))2+
(z-ut>2]}
A
14 June 1993
Finally let us determine the mass of the kink by means of the relation T= f mu2 (compare ref. [ 21) and we obtain
2@0
+;i$(A:
LETTERS
. (26)
The total energy of the 4rc-kink is @’ 2rc2(a: +A?)
#=
1 ___ + n: -2: -L+il_
{1-C+ln((n:+ii)e)
$(A: +A?)
ln(;l+//Z_)
[ tan-‘($$$)]~.
(27)
It should be noted that this energy exceeds exactly four times the energy of the static (v= 0) 2n-kink, corresponding to the value a2 = 0 in eq. (23 ) and to the solution w( 5) = K+ 2 tan-‘< [ 21. The latter solution corresponds to the zero kinetic energy ( T= 0)) whereas for the travelling kink the kinetic energy is
(28)
(29) that is twice larger than the estimate of the mass presented in ref. [ 21 for the case I + =a _ =a. So, the present results may be considered as the foundations of nonlocal Josephson electrodynamics for a tunnel junction between two different superconductors. The description of the travelling 4n-kink for the phase difference of the Cooper pairs is presented. The magnetic field of the kink, carrying two magnetic flux quanta, the core of which essentially defines its energy, corresponds to the Abrikosov-type vortex. Due to the nonlinearity of the tunnel current and nonlocality the core of the moving vortex obtained regularizes at a length Lz(L: +a: )-I, instead of the superconductivity coherence length.
References [ 1] Yu.M. Aliev, V.P. Silin and S.A. Uryupin, Superconductivity
The magnetic field in the frame system moving with velocity v is functionally the same as the magnetic field of the static 27c-kink and is twice larger than value (26). The magnetic field of the 4n-kink is formed by Josephson and superconductor currents as well as by displacement currents, in contrast to the static 27t-kink for which the displacement current is absent.
262
5 (1992)
230.
[ 21 A. Gurevich, Phys. Rev. B 46 ( 1992) 3 187. [ 31 Yu.M. Aliev, V.P. Silin and S.A. Uryupin, Pis’ma Zh. Eksp. Teor. Fiz. 57 (1993)
187.
[ 41 A. Barone and G. Paterno, Physics and applications Josephson
effect (Wiley, New York, 1982).
of the