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Magnetic inclusion in the Josephson junction
Solid State Communications, Printed in Great Britain.
Vol. 73, No. 4, pp. 301-304,
MAGNETIC
INCLUSION
IN THE JOSEPHSON
A.T. Il’inikh Institute
of Inorganic
Chemistry,
Academy
(Received
0038-1098/90 $3.00 + .OO Pergamon Press plc
1990.
JUNCTION
and B.Ya. Shapiro
of Sciences,
Siberian
Division,
Novosibirsk,
630090, USSR
24 July 1989 by V.M. Agranovich)
The fluxon dynamic in the Josephson chains with magnetic inclusions is investigated. In the framework of the McLathlin and Scott perturbation theory it has been shown that fluxons and, antifluxons have different interaction energies with magnetic inclusions. The fluxons may be in the bound state with the magnetic inclusion and there is a distance between the centre of fluxon and the magnetic inclusion. The critical current (through the junction) at which the fluxon ‘unbinding takes place is determined. The frequency of the small oscillations of the fluxon in the equilibrium state is calculated. It is shown that there are conditions for the oscillation mode softening. A new type of electronic device is discussed.
1. INTRODUCTION IT IS WELL known that the fluxon motion along the Josephson chains is strongly influenced by different imperfections of the Josephson contact. As usual the point defects are divided into the fluxon attractive or fluxon repulsion types. (The microresistance and microshort, respectively [l-7]). In the present paper we will consider a new type of the point defects in the Josephson contact, namelymagnetic point defects. An example of such a defect may be a ferromagnetic conduction inclusion in the weak link of the Josephson structure. Of course such systems can exist only under conditions that do not suppress the superconductivity in the Josephson junction. Therefore these defects may play an important role in the Josephson structures consisting from high temperature superconductors. These substances possess a high value of the critical magnetic field. In our case the magnetic moment is perpendicular to the junction plane. The Josephson junction with a localized magnetic moment has been investigated in [8, 91. It has been shown that there are different signs of the interaction between this moment and either fluxons or antifluxons: the attractive interaction for antifluxons and repulsive interaction for fluxons. However, the authors of [8,9] have not taken into account the local change of the junction properties because they considered the Abrikosov vortex in the junction. Meanwhile, in order to take into account the influence of the magnetic defect on the fluxon (anti301
fluxon) motion one has to take into account both the fluxon interaction with magnetic moment of the defect and the local change of the critical current through the contact. As will be shown later on there are some new effects in these systems. 2. BASIC
EQUATIONS
The energy of the fluxon (antifluxon) interacting with the magnetic inclusion has the form: E
=
(dx
{+(&#$
+ +(&$+’
+ 1 -
cos $J (1)
CPO
Here 4 and x are the phase and the coordinate along the junction respectively, t is the time, p is the local current change (p > 0 is the microshort case, p < 0 is the microresistance) cp is the flux through the magnetic inclusion, ‘p. is the quantum of flux, y is the external current (see Fig. 1). The equation for the phase $ can be obtained from equation (1) in the usual way af,4 -
@4
=
sin 4 -
&S’(x) + p sin &5(x) + y. (2)
Applying to magnetic inclusion the perturbation theory [l] we obtain the equations for the soliton (fluxon) velocity u and coordinate x:
MAGNETIC
302
x
x sch
JI
-
INCLUSION
IN THE JOSEPHSON
JUNCTION
Vol. 73. No. 4
x
th ____ uz Jl -
z?
Fig. I. The Josephson junction transmission line with magnetic inclusion (4 is the magnetic flux of the inclusion y is the current).
(4) The soliton dynamics is completely determined by these equations. It is clear from (3)-(4) that fluxons and antifluxons in different ways interact with magnetic inclusion. Indeed, p is the same function both for a fluxon and for an antifluxon, while E > 0 for fluxons and c < 0 for antifluxons. (The fluxon magnetic field is parallel to the magnetic moment of the magnetic inclusion.) 3. PHASE
PLANE
ANALYSIS
Let us consider equations (3)-(4) by the phase plane method. It is easy to look for a focus (ti = 0) for these equations. In this point we have from (3)-(4):
However, the foci out of the defect are now the points of nonequilibrium state. It means that the bound state is absent. The focus X,, = 0 is the equilibrium point in this case. The fluxon may be trapped by the magnetic defect due to attractive forces of the microresistance. Under condition IpI < I (“weak” microresistance) the point X,, = 0 is nonequilibrium. The bound state in this case is absent. (b) If the current is flowing through the junction then the constant force acting on solitons appears (7 # 0). This force tends to destroy the bound solitons state in the junctions. To take into account the current one should introduce the effective potential Gel,-:
(5) sch X th X
x sch
x $-IT
sch X + L 2p
(9)
(a) Under condition y = 0 (the current through the junction is absent) we have for foci positions ch X,, = X,, =
-
2@ : = fl
In [/I + (/I’ -
(6) l)“‘].
(7)
With c < 0 (antifluxon) and c > 0 (microshort) there are two foci. In this case the soliton centres are outside the magnetic inclusion. Under condition Ifi < 1 equation (7) is not valid. In this case we have from (3) a new sohton x0
=
0.
It is clear from (9) that the term -y leads to the decreasing of the pinning forces for the bound solitons. The critical value y, of the depinning current depends on the relations between ~1and E. The critical current
(10)
(8)
The antifluxon in this case is located at the magnetic inclusion. Under condition E > 0, ~1 < 0 (microresistance) the soliton of equation (5) is equivalent to (6)-(7).
x >
0.
Under condition X < 0 one has to exchange F,., in all formulae.
F,,,,, and
Vol. 73. No. 4
MAGNETIC
INCLUSION
IN THE JOSEPHSON
303
JUNCTION
4. THE FREQUENCY OF THE SMALL SOLITON OSCILLATION The trapped solitons may oscillate at the equilibrium points. The oscillation frequencies depend on the parameters y, E, p. For zero current (y = 0) one can expend (5) near the focus point. Under condition I/?1 < 1 (V,, has two minimums) we obtain:
(11)
B Fig. 2. The critical current of the antifluxon trapping by the magnetic inclusion in the junction vs magnetic inclusion properties.
In the opposite case (I/31 > 1, y = 0) there is one focus at the point X0 = 0. We have from equation (5): G
The y region for the trapped solitons is shown in the Fig. 2. In this figure there are three different regions. In region I with increasing current (fl = const) the soliton to the right from the magnetic inclusion becomes free (y = yc2, Fig. 3b). Under further current increasing the second minimum of the potential U, may disappear too. In this case (y = y,, , Fig. 3c) the soliton to the left from magnetic inclusion is transformed into a free soliton, too. In the region II with increasing y the left minimum disappears first and soliton passes to the right minimum (y = yc,, Fig. 3d). Under condition y >, yr2 the right minimum disappears and soliton becomes free. In the region III (I/?1 3 1) the function V,, changes and transforms into the function with a single minimum. Changing the current direction it is possible to obtain a symmetrical picture.
(a)
(b)
Fig. 3. The effective energy of the antifluxon in the junction with magnetic inclusion at the different values of current y@ < 0.6), (a) y = 0 (b) y = ycz (c) y = y,., and (d) y = yc, (0.6 < fi < 1, jl = ~P/E).
=
5 (IBI -
(12)
1).
The general dependence G(b) is shown in Fig. 4. One should notice the softening of the frequency at 1flI = 1. This is caused by the changing of the topology of the phase plane (two poles are merged into one). The characteristic frequencies of the small oscillations depend on the current flowing through the junction. We have in this case: (13)
2 =
u.
The frequency
(14) of small oscillation
is
(15) (X0 may be obtained from equation (5)). R, is the function of y and p. Under conditions y + yc,, y -+ yc2(b = const) G -+ 0. Indeed, under these values of current one of the minimums disappears. The numerical results for h(y)@ = const) are shown in Fig. 5.
B Fig. 4. The frequency lation vs flux through
of the small antifluxon the magnetic inclusion.
oscil-
304
MAGNETIC
INCLUSION
IN THE JOSEPHSON
JUNCTION
Vol. 73, No. 4
netic field one can switch off the shuttle (because of trapping the antifluxons by the magnetic inclusions). - We would like to thank V.B. Pavlenko for numerical calculations. This paper was supported by Grant No 198 of the HTS program of the Soviet Academy of Sciences.
Acknowledgement
REFERENCES Fig. 5. The frequency of the small antifluxon oscillations vs current y.
1. 2.
5. CONCLUSIONS A new class of the electronic devices may be constructed using the phenomenon described in this paper. In these devices one can change the power of the Josephson radiation from the junction by changing the external magnetic field (the magnetic moment of the magnetic inclusions, if there are many inclusions in the contact, may be varied by the external magnetic field). Changing the current flowing through the junction one can change the frequency of the radiation. These junctions may be used as a “shuttle of fluxons”. Changing the direction of the external mag-
3. 4. 5. 6. 7. 8.
K. Lonngren & A. Scott, Solitons in Action, Academic Press (1978). D.W. McLaughlin & A.C. Scott, Appl. Phys. Lett. 30, 545 (1977). D.W. McLaughlin & A.C. Scott, Phys. Rev. 18A, 1652 (1978). I.L. Serpuchenko & A.V. Ustinov, JETF Lett. 46, 435 (1987). Yu.S. Galperin & A.T. Fillipov, JETF Lett. 35. 470 (1982).
B.A. Malomed, IL. Serpuchenko, M.I. Tribe]‘sky & A.V. Ustinov, JETF Lett. 47, 505 (1988). O.A. Chubikalo, A.M. Kosevich & YuS. Kivshar, Fiz. nizk. temp. 13, 800 (1987). L.G. Aslomazov & E.V. Gurovich, JETF Lett. 40, 22 (1984).
9.
E.V. Gurovich & V.G. Mihalev, JETF 93, 1293 (1987).
Vol. 73, No. 4, pp. 301-304,
MAGNETIC
INCLUSION
IN THE JOSEPHSON
A.T. Il’inikh Institute
of Inorganic
Chemistry,
Academy
(Received
0038-1098/90 $3.00 + .OO Pergamon Press plc
1990.
JUNCTION
and B.Ya. Shapiro
of Sciences,
Siberian
Division,
Novosibirsk,
630090, USSR
24 July 1989 by V.M. Agranovich)
The fluxon dynamic in the Josephson chains with magnetic inclusions is investigated. In the framework of the McLathlin and Scott perturbation theory it has been shown that fluxons and, antifluxons have different interaction energies with magnetic inclusions. The fluxons may be in the bound state with the magnetic inclusion and there is a distance between the centre of fluxon and the magnetic inclusion. The critical current (through the junction) at which the fluxon ‘unbinding takes place is determined. The frequency of the small oscillations of the fluxon in the equilibrium state is calculated. It is shown that there are conditions for the oscillation mode softening. A new type of electronic device is discussed.
1. INTRODUCTION IT IS WELL known that the fluxon motion along the Josephson chains is strongly influenced by different imperfections of the Josephson contact. As usual the point defects are divided into the fluxon attractive or fluxon repulsion types. (The microresistance and microshort, respectively [l-7]). In the present paper we will consider a new type of the point defects in the Josephson contact, namelymagnetic point defects. An example of such a defect may be a ferromagnetic conduction inclusion in the weak link of the Josephson structure. Of course such systems can exist only under conditions that do not suppress the superconductivity in the Josephson junction. Therefore these defects may play an important role in the Josephson structures consisting from high temperature superconductors. These substances possess a high value of the critical magnetic field. In our case the magnetic moment is perpendicular to the junction plane. The Josephson junction with a localized magnetic moment has been investigated in [8, 91. It has been shown that there are different signs of the interaction between this moment and either fluxons or antifluxons: the attractive interaction for antifluxons and repulsive interaction for fluxons. However, the authors of [8,9] have not taken into account the local change of the junction properties because they considered the Abrikosov vortex in the junction. Meanwhile, in order to take into account the influence of the magnetic defect on the fluxon (anti301
fluxon) motion one has to take into account both the fluxon interaction with magnetic moment of the defect and the local change of the critical current through the contact. As will be shown later on there are some new effects in these systems. 2. BASIC
EQUATIONS
The energy of the fluxon (antifluxon) interacting with the magnetic inclusion has the form: E
=
(dx
{+(&#$
+ +(&$+’
+ 1 -
cos $J (1)
CPO
Here 4 and x are the phase and the coordinate along the junction respectively, t is the time, p is the local current change (p > 0 is the microshort case, p < 0 is the microresistance) cp is the flux through the magnetic inclusion, ‘p. is the quantum of flux, y is the external current (see Fig. 1). The equation for the phase $ can be obtained from equation (1) in the usual way af,4 -
@4
=
sin 4 -
&S’(x) + p sin &5(x) + y. (2)
Applying to magnetic inclusion the perturbation theory [l] we obtain the equations for the soliton (fluxon) velocity u and coordinate x:
MAGNETIC
302
x
x sch
JI
-
INCLUSION
IN THE JOSEPHSON
JUNCTION
Vol. 73. No. 4
x
th ____ uz Jl -
z?
Fig. I. The Josephson junction transmission line with magnetic inclusion (4 is the magnetic flux of the inclusion y is the current).
(4) The soliton dynamics is completely determined by these equations. It is clear from (3)-(4) that fluxons and antifluxons in different ways interact with magnetic inclusion. Indeed, p is the same function both for a fluxon and for an antifluxon, while E > 0 for fluxons and c < 0 for antifluxons. (The fluxon magnetic field is parallel to the magnetic moment of the magnetic inclusion.) 3. PHASE
PLANE
ANALYSIS
Let us consider equations (3)-(4) by the phase plane method. It is easy to look for a focus (ti = 0) for these equations. In this point we have from (3)-(4):
However, the foci out of the defect are now the points of nonequilibrium state. It means that the bound state is absent. The focus X,, = 0 is the equilibrium point in this case. The fluxon may be trapped by the magnetic defect due to attractive forces of the microresistance. Under condition IpI < I (“weak” microresistance) the point X,, = 0 is nonequilibrium. The bound state in this case is absent. (b) If the current is flowing through the junction then the constant force acting on solitons appears (7 # 0). This force tends to destroy the bound solitons state in the junctions. To take into account the current one should introduce the effective potential Gel,-:
(5) sch X th X
x sch
x $-IT
sch X + L 2p
(9)
(a) Under condition y = 0 (the current through the junction is absent) we have for foci positions ch X,, = X,, =
-
2@ : = fl
In [/I + (/I’ -
(6) l)“‘].
(7)
With c < 0 (antifluxon) and c > 0 (microshort) there are two foci. In this case the soliton centres are outside the magnetic inclusion. Under condition Ifi < 1 equation (7) is not valid. In this case we have from (3) a new sohton x0
=
0.
It is clear from (9) that the term -y leads to the decreasing of the pinning forces for the bound solitons. The critical value y, of the depinning current depends on the relations between ~1and E. The critical current
(10)
(8)
The antifluxon in this case is located at the magnetic inclusion. Under condition E > 0, ~1 < 0 (microresistance) the soliton of equation (5) is equivalent to (6)-(7).
x >
0.
Under condition X < 0 one has to exchange F,., in all formulae.
F,,,,, and
Vol. 73. No. 4
MAGNETIC
INCLUSION
IN THE JOSEPHSON
303
JUNCTION
4. THE FREQUENCY OF THE SMALL SOLITON OSCILLATION The trapped solitons may oscillate at the equilibrium points. The oscillation frequencies depend on the parameters y, E, p. For zero current (y = 0) one can expend (5) near the focus point. Under condition I/?1 < 1 (V,, has two minimums) we obtain:
(11)
B Fig. 2. The critical current of the antifluxon trapping by the magnetic inclusion in the junction vs magnetic inclusion properties.
In the opposite case (I/31 > 1, y = 0) there is one focus at the point X0 = 0. We have from equation (5): G
The y region for the trapped solitons is shown in the Fig. 2. In this figure there are three different regions. In region I with increasing current (fl = const) the soliton to the right from the magnetic inclusion becomes free (y = yc2, Fig. 3b). Under further current increasing the second minimum of the potential U, may disappear too. In this case (y = y,, , Fig. 3c) the soliton to the left from magnetic inclusion is transformed into a free soliton, too. In the region II with increasing y the left minimum disappears first and soliton passes to the right minimum (y = yc,, Fig. 3d). Under condition y >, yr2 the right minimum disappears and soliton becomes free. In the region III (I/?1 3 1) the function V,, changes and transforms into the function with a single minimum. Changing the current direction it is possible to obtain a symmetrical picture.
(a)
(b)
Fig. 3. The effective energy of the antifluxon in the junction with magnetic inclusion at the different values of current y@ < 0.6), (a) y = 0 (b) y = ycz (c) y = y,., and (d) y = yc, (0.6 < fi < 1, jl = ~P/E).
=
5 (IBI -
(12)
1).
The general dependence G(b) is shown in Fig. 4. One should notice the softening of the frequency at 1flI = 1. This is caused by the changing of the topology of the phase plane (two poles are merged into one). The characteristic frequencies of the small oscillations depend on the current flowing through the junction. We have in this case: (13)
2 =
u.
The frequency
(14) of small oscillation
is
(15) (X0 may be obtained from equation (5)). R, is the function of y and p. Under conditions y + yc,, y -+ yc2(b = const) G -+ 0. Indeed, under these values of current one of the minimums disappears. The numerical results for h(y)@ = const) are shown in Fig. 5.
B Fig. 4. The frequency lation vs flux through
of the small antifluxon the magnetic inclusion.
oscil-
304
MAGNETIC
INCLUSION
IN THE JOSEPHSON
JUNCTION
Vol. 73, No. 4
netic field one can switch off the shuttle (because of trapping the antifluxons by the magnetic inclusions). - We would like to thank V.B. Pavlenko for numerical calculations. This paper was supported by Grant No 198 of the HTS program of the Soviet Academy of Sciences.
Acknowledgement
REFERENCES Fig. 5. The frequency of the small antifluxon oscillations vs current y.
1. 2.
5. CONCLUSIONS A new class of the electronic devices may be constructed using the phenomenon described in this paper. In these devices one can change the power of the Josephson radiation from the junction by changing the external magnetic field (the magnetic moment of the magnetic inclusions, if there are many inclusions in the contact, may be varied by the external magnetic field). Changing the current flowing through the junction one can change the frequency of the radiation. These junctions may be used as a “shuttle of fluxons”. Changing the direction of the external mag-
3. 4. 5. 6. 7. 8.
K. Lonngren & A. Scott, Solitons in Action, Academic Press (1978). D.W. McLaughlin & A.C. Scott, Appl. Phys. Lett. 30, 545 (1977). D.W. McLaughlin & A.C. Scott, Phys. Rev. 18A, 1652 (1978). I.L. Serpuchenko & A.V. Ustinov, JETF Lett. 46, 435 (1987). Yu.S. Galperin & A.T. Fillipov, JETF Lett. 35. 470 (1982).
B.A. Malomed, IL. Serpuchenko, M.I. Tribe]‘sky & A.V. Ustinov, JETF Lett. 47, 505 (1988). O.A. Chubikalo, A.M. Kosevich & YuS. Kivshar, Fiz. nizk. temp. 13, 800 (1987). L.G. Aslomazov & E.V. Gurovich, JETF Lett. 40, 22 (1984).
9.
E.V. Gurovich & V.G. Mihalev, JETF 93, 1293 (1987).