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Reduced fluxon viscosity in a Josephson junction
# ' ~ Solld State Colmunlcatlons, Vol. 68, No. 12, pp.1097-1100, 1988. ~ i ~ P r l n t e d in Great Brlta~n.
0038-1098/88 $3.00 + .00 Pergamon Press plc
REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION A. M. Portis* and K. W. Blazey IBM Research Division, Zurich Research Laboratory, 8803 R0schlikon, Switzerland (Received
August 4'oh , 1988 by P. Wachter)
Flu~on viscosities within w e a k Josephson junctions are orders of magnitude smaller than in bulk. Highly mobile fluxons within weak junctions, driven by surface m i c r o w a v e currents, are responsible for the large low-field m i c r o w a v e absorption observed in the granular high-temperature superconductors. While typical fluxon displacements in bulk are in the Angstrom range, displacements in weak Josephson junctions are in the micron range.
1. Introduction Since the discovery of superconductivity in the planar cuprates 1 a number of investigators have reported intense m i c r o w a v e absorption signals from these materials at low magnetic fields. 2-9 These signals have been analyzed in terms of an effective-medium t h e o r y 1° with absorption arising from the damped n o t i o n of fluxons driven by m i c r o w a v e currents. 11 Recent studies of fluxon nucleation by m i c r o w a v e currents 12'~3 indicates that low-field m i c r o w a v e absorption takes place within Josephson junctions and not in the superconducting bulk. As shown in this communication, the viscosity of fluxons in Josephson junctions can be orders of magnitude lower than in bulk, accounting for the intense microwave absorption observed from the relatively small volum~ fraction of weak junctions in the granular high-temperature superconductors.
Fig. 1. View of a domain boundary of extension d in a single crystal of YBa2Cu307. 6. The domain boundary is in the xz-plane and is parallel to the {110} planes of the crystal. The domain boundary forms a Josephson junction of length t and depth #'. A variable static magnetic field is applied in z-direction and flux penetrates at distance s = d + 2 t L in the vicinity of the junction. A m i c r o w a v e magnetic field is applied in x-direction and m i c r o w a v e currents flow across the junction in the y-direction. A geometric area S associated with the junction intercepts the static magnetic field.
2. Static', Magnetic Fields Before considering the excitation of fluxons by microwaves, we must first discuss the structure of quantized flux states in an intergranular junction. 14 When a static magnetic field is applied to an intergranular junction, tunnel currents flow across the junction and cause a variation of the magnetic field along the junction with a period of J-B. A junction parallel to the xz-plane as shown in Fig. 1 with the static field along the z-direction has tunneling currents in the y-direction. Such a situation exists for a domain boundary parallel to {110} twin planes of a YBa2Cu307_6 crystal with the x-direction parallel 1o the crystal [001] axis.
the gradient junction is a&/ax =
of the
phase
difference
- (2~/(#0) aq~/ax =
across
- (2~s/q~o) g
,
the (3)
The tunneling current density is is J := J o s i n J
,
w h e r e q~ is the entire flux that penetrates the junction and the adjoining superconducting regions with s = d + 2iLl I. The distance d is the physical width of the juriction and J.LII~8400 ~, is the penetration depth 16'17 of a field p,~'rallel to the conducting planes. Differentiating (3) yields
(1)
w h e r e Jo is the critical current density and (f the quantum phase difference across the junction. From the Josephson relation for the voltage across a junction 15
¢32(~/~x2 = a&/at =
- (2~c/~P0) V
- (2~zs/~Po)~B/ax =
- (8~t2s/(Po c) J . (4)
(2) Using (1) for the tunneling current leads finally to the FerrelI-Prange equation t8
*Permanent address: Department of Physics. University of California, Berkeley CA 94720, USA.
a2(~/ax2 = (1/J.j) 2 sin 1097
(5)
1098
REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION
with the penetration depth into the junction given by
~j = (~oc/8~sJo) ~/~ From (2) and (5) the phase weakly coupled junction a p p r o x i m a t e l y as
(6)
difference across a may be written
&(x) = - (2~sB/~0) x - (2J0~o0/scB 2) sin (2~sBx/~o). (7) The flux enclosed in a period of the modulation ~B = ~P0/sB is equal to the quantum of flux ~o0 = hc/2e. Fluxon states are obtained by equating the junction length t to an integral multiple p of the period JIB, ensuring t h a t vortex currents are parallel to the crystal surfaces. The weak coupling condition is then t << p~j. Taking t ~ 100 ~.m as the upper limit of the YBa2Cu307.~ crystal of Ref. 1, the tunneling current density is (10/c)J 0 = 10q)o/8~2st 2 ~. 1.5A/cm 2. Such currents are substantially smaller than usually expected and suggest that the pair potential is considerably w e a k e n e d by the short coherence length ~ in the high T c materialsJ g 3. Microwave Excitation of Fluxons Having established a model of the quantization of distributed flux we now consider the effect of m i c r o w a v e currents in the vicinity of the junction. The m i c r o w a v e field H 1 is applied in the x-direction of Fig. 1. At an incident m i c r o w a v e p o w e r P0 the m i c r o w a v e field within an e m p t y cavity is H 1 = (32~PoQ/~Vc) 1/2 ,
(8)
w h e r e V c ~ 10 cm 3 is the cavity v o l u m e and Q ~ 8000 is the quality factor of the unloaded m i c r o w a v e cavity. For PO = 200 mW at e ) / 2 ~ 9 4 0 0 M H z we estimate H 1 ~ 1 . 7 0 e . M i c r o w a v e magnetic fields normal to the conducting planes of YBa2CU3OT. ~ penetrate the crystal only to a depth ls,17 of ~L.I. ~ 9 0 0 A. For H 1 normal to the plane of a thin crystal, the m i c r o w a v e field near the edge is enhanced by a factor of =. The computed surface m i c r o w a v e current density at this ~ o w e r level is (10/c)J 1 = (IO/4~)
= (1/C) J l ~ 0
•
(9)
Here /~ is the mass per unit distance, ~/ is the viscosity per unit distance and k is a restoringforce constant. The mass # arises primarily from the d e v e l o p m e n t of electric fields by the moving fluxon and only secondarily from carrier motio n . The ratio = /~/~/ is the m o m e n t u m relaxation time. From the fact that the fluxon contribution to tile resistivity is observed to increase linearly with field, p = (Pn/Hc2) B ,
(10)
Strnad, Hempstead and Kim 2° have suggested tha! the viscosity of fluxons in a bulk superconductor is
Vol.
tt ~ ~00Hc2/PnC2 = (1/2/~ 2) ((p2/PnC2)
68, No. 12 ,
(11)
w h e r e 2 ~ 2 is the area of the quasi-normal fluxon core and Pn is the resistivity of the normal phase above the upper critical field Hc2 ~ (P0/2~2. Stephen and Bardeen 21 have shown that (11) is a consequence of electrostatic screening, which concentrates current through the core of a moving fluxon. Suh122 has calculated the mass arising primarily from the e n e r g y in the electric field of a moving fluxon /~c2 = (1/3 ~) (~o0/AD)2 ,
(12)
w h e r e ~.p is the Debye screening length within the superconductor. Taking the ratio of (12) to (11) yields for the m o m e n t u m relaxation time = /~hl = (1/3 ~) (~/~.D) 2 Pn -
(13)
Taking the ratio of the coherence length ~ to the Debye screening length ,~1) for the oxide superconductors 19 (~/~.O) 2 ~ 10 and using, 23 c2p,, x 10 -9 = 10 -4 Q-cm leads to z = 10-16 sec for the'"mornentum relaxation time. Obtaining expressions for the viscosity and mass of fluxons within weak Josephson junctions requires a separate but similar treatment. The electric field d e v e l o p e d by a fluxon moving with velocity v within the plane of a Josephson junction is largely screened from the superconductor and establishes a potential V = (v/c) ~0/,~B ,
(14)
which as expected from (2) appears largely across the junction. As for fluxons in bulk we assume quasi-normal current flow with ~/v2 = ~BV2/Rn w h e r e R n is the normal phase tunneling resistance across the junction. The viscosity within a tunnel junction is then //j = (p02/~.BRnC2
(15)
which is smaller than the bulk viscosity (11) by a factor of f = lz~2pn/Rn~.B . Similarly, (14) leads to a mass per unit distance
#j = (~./4.J) ~2/~Bd
(18)
w h e r e ¢ ~ 1 0 is the dielectric constant within the junction. The estimated relaxation time ~j is longer than (13) by Rn/Pnd for ~ ~ , t D. The additional m i c r o w a v e resistivity over an area S produced by the motion of a fluxon within a junction is obtained by equating the loss rates ~/jv2 = S j 2 3 p , which gives an expression for the incremental resistivity of the form of (10) with Hc2 replaced by q~OPn/Rn~B. For a single additional fluxon, the incremental resistivity averaged over an area S is (~p = ~.BRn/S .
(17)
4. Comparison with Experiment Expressions are obtained in this section for typical bulk fluxon velocities, displacements and energy relaxation rates. Corresponding expressions are then obtained for fluxons in weak Josephson junctions. Velocities, displacements and relaxation rates are
Vol. 68, No. 12
seen to be junctions.
orders
1099
.REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION
of
magnitude
larger
in
the
From (9) and (11) the velocity of a bulk fluxon driven by microwave current is
Hcl J ~ (4~/c) J0~.j =
~P0/2~s).j
where ~j is given by (6). Taking for the Josephson current density 24 J0 = (=/4)(2A/eRn) tanh (A/2kBT)
V/C = Jl(P0/~/c 2 =
(PnC/4=,~L)(Ht/Hc2).
(1/32~K2) Pnc2H;2 (19)
where 2A(T) is the BCS energy gap, the velocity of a fluxon at temperatures we!l below Tc is v/c =
(l/u) dw/dt =
(1/4 ~ 2) (PnC2/)2)
(~B/~.j) (RnC/4~(P0)H 1
= (1/8s)(2A/e)(~B/~j)(H1/H21j).
(23)
With 300 x (2A/e) 27 mV and taking Hcl J ~ 0 . 5 0 e , (23) gives v/c ~ 0.27 H 1, which is five orders larger than bulk fluxon velocities. Fluxon displacement may be as large as x -'= vh) ~ 140 /~m. The energy relaxation rate is
with K = ~L/~ the Ginzburg-Landau parameter. An energy relaxation rate r =
(22)
(18)
With a magnetic field applied parallel to the ab-plane of YBa2Cu307_.~ , the upper critical field 17 is Hc211 ~ 2.4 x 10~ Oe and the resistivity perpendicular to the ab-plane 23 is pn c2 x 10 -8 ~ 10 -4 ~-cm. Substituting these values into (18) gives v / c ~ 1.3 x 10 -6 H 1 . At a typical microwave power of 40 dB below 200 roW, the computed displacement is x=v/~) ~ 7 ,~. The mean rate at which energy is absorbed is dw/dt = (v/2c)JIcp 0 =
(21)
r =
(l/u)dw/dt
=
(~2/2)(c/q~0)(2A/e)
(24)
(20)
is defined with u = (1/8~) (~,~L2) H 2 , the microwave magnetic energy stored in the region of the fluxon. Taking K ~ 225, the computed energy relaxation rate is r ~ 2 x 106 . Microwave absorption is observed in ceramic YBa2Cu307_ 6 at fields well below an Oersted in a sample cooled in zero field. The onset of this absorption at fields far below the bulk Hcl arises from weak Josephson junctions, which have critical fields
with dw/dt=(v/8~).j)Hlq~ 0 and u =(1/8~)(S2L)H12. The relaxation rate r ~ 6 x 1017 is independent of J0 and ten orders larger than the bulk rate. The driven motion of fluxons in junctions is by far the dominant mechanism for microwave loss in a magnetic field.
A c k n o w l e d g e m e n t - W e acknowledge helpful conversations with A. Baratoff, M. H. Devoret, D. Esteve, F. Guinea, P. Martinoli, K. A. M011er, Ch. Rossel, C. Urbina and F. Waldner.
5. References 1j. G. Bednorz and K. A. M011er, Z. Phys. B 64, 189 (1986). 2S. V. Bhat, P. Ganguly, T. V. Ramakrishnan, C. N. R. Rao, J. Phys. C 20, L559 (1987). 3R. Durn~', J. Hautala, S. Ducharme, B. Lee, O. G. Symko, P. C. Taylor, D. J. Zheng and J. A. Xu, Phys. Rev. B 36, 2361 (1987). 4K. W. Blazey, K. A. M011er. J. G. Bednorz, W. Berlinger, G. Amoretti, I'. Buluggiu, A. Vera and F. C. Matacotta, Phys. Rev. B 36, 7241 (1987). SM" Per=c, .v B. Rakvin, M. Prester, N. BrmcevJc . .v and A. .V ,v Dulclc. Phys. Rev. B 37, 522 (1988). A. Dulclc, B. .V .V Leonhc, M. Pertic and B. Rakvin, Europhys. Lett. 4, 1493 (1987). 6K. Khachaturyan, E. R. Weber, P. Tejedor, A. M. Stacy and A. M. Portis, Phys. Rev. B 36, 8309 (1987). 7E. J. Pakulis and E. J. Osada, Phys. Rev. B 37, 5940 (1988). sS. H. Glarum, J. H. Marshall and L. F. Schneemeyer, Phys. Rev. B 37, 7491 (1988). ~N. J. Tomasch, H. A. Blackstead, S. T. Ruggiero, P. J. McGinn, J. R. Clem, K. Shen, J. W. Weber and D. Boyne, Phys. Rev. B 37, 9864 (1988).
10A. M. Portis, K. W. Blazey, K. A. M011er and J. G. Bednorz, Europhys. Lett. 5, 467 (1988). 11y. B. Kim and M. J. Stephen, in Superconductivity, Volume II, R. D. Parks, ed., (Marcel Dekker, Inc., New York, 1969), Chapter 19. 12K. W. Blazey, A. M. Portis, K. A. M011er and F. H. Holtzberg, Europhys. Lett., 6 457 (1988). 13K. W. Blazey, A. M. Portis and F. H. Holtzberg, submitted to Physica C. 14j. R. Clem, Physica C, in print. lSB. D. Josephson, Phys. Lett., 1, 251 (1962); Rev. Mod. Phys., 36, 216 (1964); Adv. Phys., 14, 419 (1965). I~/V. J. Gallagher, T. K. Worthington, T. R. Dinger, F. H. Holtzberg, D. L. Kaiser and R. L. Sandstrom, Physica, 148B, 228 (1987). 17A. Umezawa, G. W. Crabtree, J. Z. Liu, T. J. Moran, S. K. Malik, L. H. Nunez, W. L. Kwok and C. H. Sowers, submitted to Phys. Rev. Lett. 18R. A. Ferrell and R. E. Prange, Phys. Rev. Lett., 10, 479 (1963). IgG. Deutscher and K. A. M011er, Phys. Rev. Lett., 59, 1745 (1987).
ii00
REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION
2°A. R. Strnad, C. F. Hempstead and Y. B. Kim, Phys. Rev. Left., 13, 794 (1964). 21M. J. Stephen and J. Bardeen, Phys. Rev. Lett., 14, 112 (1964); John Bardeen and M. J. Stephen, Phys. Rev., 140, Al197 (1965). 22H. Suhl, Phys. Rev. Lett., 14, 226 (1965).
Vol. 66, No. 12
23S. W. Tozer, A. W. Kleinsasser, T. Penney, D. L. Kaiser and F. H. Holtzberg, Phys. Rev. Lett., 59, 1786 (1987). 24V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486 (1963) and 11, 104 (1963).
0038-1098/88 $3.00 + .00 Pergamon Press plc
REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION A. M. Portis* and K. W. Blazey IBM Research Division, Zurich Research Laboratory, 8803 R0schlikon, Switzerland (Received
August 4'oh , 1988 by P. Wachter)
Flu~on viscosities within w e a k Josephson junctions are orders of magnitude smaller than in bulk. Highly mobile fluxons within weak junctions, driven by surface m i c r o w a v e currents, are responsible for the large low-field m i c r o w a v e absorption observed in the granular high-temperature superconductors. While typical fluxon displacements in bulk are in the Angstrom range, displacements in weak Josephson junctions are in the micron range.
1. Introduction Since the discovery of superconductivity in the planar cuprates 1 a number of investigators have reported intense m i c r o w a v e absorption signals from these materials at low magnetic fields. 2-9 These signals have been analyzed in terms of an effective-medium t h e o r y 1° with absorption arising from the damped n o t i o n of fluxons driven by m i c r o w a v e currents. 11 Recent studies of fluxon nucleation by m i c r o w a v e currents 12'~3 indicates that low-field m i c r o w a v e absorption takes place within Josephson junctions and not in the superconducting bulk. As shown in this communication, the viscosity of fluxons in Josephson junctions can be orders of magnitude lower than in bulk, accounting for the intense microwave absorption observed from the relatively small volum~ fraction of weak junctions in the granular high-temperature superconductors.
Fig. 1. View of a domain boundary of extension d in a single crystal of YBa2Cu307. 6. The domain boundary is in the xz-plane and is parallel to the {110} planes of the crystal. The domain boundary forms a Josephson junction of length t and depth #'. A variable static magnetic field is applied in z-direction and flux penetrates at distance s = d + 2 t L in the vicinity of the junction. A m i c r o w a v e magnetic field is applied in x-direction and m i c r o w a v e currents flow across the junction in the y-direction. A geometric area S associated with the junction intercepts the static magnetic field.
2. Static', Magnetic Fields Before considering the excitation of fluxons by microwaves, we must first discuss the structure of quantized flux states in an intergranular junction. 14 When a static magnetic field is applied to an intergranular junction, tunnel currents flow across the junction and cause a variation of the magnetic field along the junction with a period of J-B. A junction parallel to the xz-plane as shown in Fig. 1 with the static field along the z-direction has tunneling currents in the y-direction. Such a situation exists for a domain boundary parallel to {110} twin planes of a YBa2Cu307_6 crystal with the x-direction parallel 1o the crystal [001] axis.
the gradient junction is a&/ax =
of the
phase
difference
- (2~/(#0) aq~/ax =
across
- (2~s/q~o) g
,
the (3)
The tunneling current density is is J := J o s i n J
,
w h e r e q~ is the entire flux that penetrates the junction and the adjoining superconducting regions with s = d + 2iLl I. The distance d is the physical width of the juriction and J.LII~8400 ~, is the penetration depth 16'17 of a field p,~'rallel to the conducting planes. Differentiating (3) yields
(1)
w h e r e Jo is the critical current density and (f the quantum phase difference across the junction. From the Josephson relation for the voltage across a junction 15
¢32(~/~x2 = a&/at =
- (2~c/~P0) V
- (2~zs/~Po)~B/ax =
- (8~t2s/(Po c) J . (4)
(2) Using (1) for the tunneling current leads finally to the FerrelI-Prange equation t8
*Permanent address: Department of Physics. University of California, Berkeley CA 94720, USA.
a2(~/ax2 = (1/J.j) 2 sin 1097
(5)
1098
REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION
with the penetration depth into the junction given by
~j = (~oc/8~sJo) ~/~ From (2) and (5) the phase weakly coupled junction a p p r o x i m a t e l y as
(6)
difference across a may be written
&(x) = - (2~sB/~0) x - (2J0~o0/scB 2) sin (2~sBx/~o). (7) The flux enclosed in a period of the modulation ~B = ~P0/sB is equal to the quantum of flux ~o0 = hc/2e. Fluxon states are obtained by equating the junction length t to an integral multiple p of the period JIB, ensuring t h a t vortex currents are parallel to the crystal surfaces. The weak coupling condition is then t << p~j. Taking t ~ 100 ~.m as the upper limit of the YBa2Cu307.~ crystal of Ref. 1, the tunneling current density is (10/c)J 0 = 10q)o/8~2st 2 ~. 1.5A/cm 2. Such currents are substantially smaller than usually expected and suggest that the pair potential is considerably w e a k e n e d by the short coherence length ~ in the high T c materialsJ g 3. Microwave Excitation of Fluxons Having established a model of the quantization of distributed flux we now consider the effect of m i c r o w a v e currents in the vicinity of the junction. The m i c r o w a v e field H 1 is applied in the x-direction of Fig. 1. At an incident m i c r o w a v e p o w e r P0 the m i c r o w a v e field within an e m p t y cavity is H 1 = (32~PoQ/~Vc) 1/2 ,
(8)
w h e r e V c ~ 10 cm 3 is the cavity v o l u m e and Q ~ 8000 is the quality factor of the unloaded m i c r o w a v e cavity. For PO = 200 mW at e ) / 2 ~ 9 4 0 0 M H z we estimate H 1 ~ 1 . 7 0 e . M i c r o w a v e magnetic fields normal to the conducting planes of YBa2CU3OT. ~ penetrate the crystal only to a depth ls,17 of ~L.I. ~ 9 0 0 A. For H 1 normal to the plane of a thin crystal, the m i c r o w a v e field near the edge is enhanced by a factor of =. The computed surface m i c r o w a v e current density at this ~ o w e r level is (10/c)J 1 = (IO/4~)
= (1/C) J l ~ 0
•
(9)
Here /~ is the mass per unit distance, ~/ is the viscosity per unit distance and k is a restoringforce constant. The mass # arises primarily from the d e v e l o p m e n t of electric fields by the moving fluxon and only secondarily from carrier motio n . The ratio = /~/~/ is the m o m e n t u m relaxation time. From the fact that the fluxon contribution to tile resistivity is observed to increase linearly with field, p = (Pn/Hc2) B ,
(10)
Strnad, Hempstead and Kim 2° have suggested tha! the viscosity of fluxons in a bulk superconductor is
Vol.
tt ~ ~00Hc2/PnC2 = (1/2/~ 2) ((p2/PnC2)
68, No. 12 ,
(11)
w h e r e 2 ~ 2 is the area of the quasi-normal fluxon core and Pn is the resistivity of the normal phase above the upper critical field Hc2 ~ (P0/2~2. Stephen and Bardeen 21 have shown that (11) is a consequence of electrostatic screening, which concentrates current through the core of a moving fluxon. Suh122 has calculated the mass arising primarily from the e n e r g y in the electric field of a moving fluxon /~c2 = (1/3 ~) (~o0/AD)2 ,
(12)
w h e r e ~.p is the Debye screening length within the superconductor. Taking the ratio of (12) to (11) yields for the m o m e n t u m relaxation time = /~hl = (1/3 ~) (~/~.D) 2 Pn -
(13)
Taking the ratio of the coherence length ~ to the Debye screening length ,~1) for the oxide superconductors 19 (~/~.O) 2 ~ 10 and using, 23 c2p,, x 10 -9 = 10 -4 Q-cm leads to z = 10-16 sec for the'"mornentum relaxation time. Obtaining expressions for the viscosity and mass of fluxons within weak Josephson junctions requires a separate but similar treatment. The electric field d e v e l o p e d by a fluxon moving with velocity v within the plane of a Josephson junction is largely screened from the superconductor and establishes a potential V = (v/c) ~0/,~B ,
(14)
which as expected from (2) appears largely across the junction. As for fluxons in bulk we assume quasi-normal current flow with ~/v2 = ~BV2/Rn w h e r e R n is the normal phase tunneling resistance across the junction. The viscosity within a tunnel junction is then //j = (p02/~.BRnC2
(15)
which is smaller than the bulk viscosity (11) by a factor of f = lz~2pn/Rn~.B . Similarly, (14) leads to a mass per unit distance
#j = (~./4.J) ~2/~Bd
(18)
w h e r e ¢ ~ 1 0 is the dielectric constant within the junction. The estimated relaxation time ~j is longer than (13) by Rn/Pnd for ~ ~ , t D. The additional m i c r o w a v e resistivity over an area S produced by the motion of a fluxon within a junction is obtained by equating the loss rates ~/jv2 = S j 2 3 p , which gives an expression for the incremental resistivity of the form of (10) with Hc2 replaced by q~OPn/Rn~B. For a single additional fluxon, the incremental resistivity averaged over an area S is (~p = ~.BRn/S .
(17)
4. Comparison with Experiment Expressions are obtained in this section for typical bulk fluxon velocities, displacements and energy relaxation rates. Corresponding expressions are then obtained for fluxons in weak Josephson junctions. Velocities, displacements and relaxation rates are
Vol. 68, No. 12
seen to be junctions.
orders
1099
.REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION
of
magnitude
larger
in
the
From (9) and (11) the velocity of a bulk fluxon driven by microwave current is
Hcl J ~ (4~/c) J0~.j =
~P0/2~s).j
where ~j is given by (6). Taking for the Josephson current density 24 J0 = (=/4)(2A/eRn) tanh (A/2kBT)
V/C = Jl(P0/~/c 2 =
(PnC/4=,~L)(Ht/Hc2).
(1/32~K2) Pnc2H;2 (19)
where 2A(T) is the BCS energy gap, the velocity of a fluxon at temperatures we!l below Tc is v/c =
(l/u) dw/dt =
(1/4 ~ 2) (PnC2/)2)
(~B/~.j) (RnC/4~(P0)H 1
= (1/8s)(2A/e)(~B/~j)(H1/H21j).
(23)
With 300 x (2A/e) 27 mV and taking Hcl J ~ 0 . 5 0 e , (23) gives v/c ~ 0.27 H 1, which is five orders larger than bulk fluxon velocities. Fluxon displacement may be as large as x -'= vh) ~ 140 /~m. The energy relaxation rate is
with K = ~L/~ the Ginzburg-Landau parameter. An energy relaxation rate r =
(22)
(18)
With a magnetic field applied parallel to the ab-plane of YBa2Cu307_.~ , the upper critical field 17 is Hc211 ~ 2.4 x 10~ Oe and the resistivity perpendicular to the ab-plane 23 is pn c2 x 10 -8 ~ 10 -4 ~-cm. Substituting these values into (18) gives v / c ~ 1.3 x 10 -6 H 1 . At a typical microwave power of 40 dB below 200 roW, the computed displacement is x=v/~) ~ 7 ,~. The mean rate at which energy is absorbed is dw/dt = (v/2c)JIcp 0 =
(21)
r =
(l/u)dw/dt
=
(~2/2)(c/q~0)(2A/e)
(24)
(20)
is defined with u = (1/8~) (~,~L2) H 2 , the microwave magnetic energy stored in the region of the fluxon. Taking K ~ 225, the computed energy relaxation rate is r ~ 2 x 106 . Microwave absorption is observed in ceramic YBa2Cu307_ 6 at fields well below an Oersted in a sample cooled in zero field. The onset of this absorption at fields far below the bulk Hcl arises from weak Josephson junctions, which have critical fields
with dw/dt=(v/8~).j)Hlq~ 0 and u =(1/8~)(S2L)H12. The relaxation rate r ~ 6 x 1017 is independent of J0 and ten orders larger than the bulk rate. The driven motion of fluxons in junctions is by far the dominant mechanism for microwave loss in a magnetic field.
A c k n o w l e d g e m e n t - W e acknowledge helpful conversations with A. Baratoff, M. H. Devoret, D. Esteve, F. Guinea, P. Martinoli, K. A. M011er, Ch. Rossel, C. Urbina and F. Waldner.
5. References 1j. G. Bednorz and K. A. M011er, Z. Phys. B 64, 189 (1986). 2S. V. Bhat, P. Ganguly, T. V. Ramakrishnan, C. N. R. Rao, J. Phys. C 20, L559 (1987). 3R. Durn~', J. Hautala, S. Ducharme, B. Lee, O. G. Symko, P. C. Taylor, D. J. Zheng and J. A. Xu, Phys. Rev. B 36, 2361 (1987). 4K. W. Blazey, K. A. M011er. J. G. Bednorz, W. Berlinger, G. Amoretti, I'. Buluggiu, A. Vera and F. C. Matacotta, Phys. Rev. B 36, 7241 (1987). SM" Per=c, .v B. Rakvin, M. Prester, N. BrmcevJc . .v and A. .V ,v Dulclc. Phys. Rev. B 37, 522 (1988). A. Dulclc, B. .V .V Leonhc, M. Pertic and B. Rakvin, Europhys. Lett. 4, 1493 (1987). 6K. Khachaturyan, E. R. Weber, P. Tejedor, A. M. Stacy and A. M. Portis, Phys. Rev. B 36, 8309 (1987). 7E. J. Pakulis and E. J. Osada, Phys. Rev. B 37, 5940 (1988). sS. H. Glarum, J. H. Marshall and L. F. Schneemeyer, Phys. Rev. B 37, 7491 (1988). ~N. J. Tomasch, H. A. Blackstead, S. T. Ruggiero, P. J. McGinn, J. R. Clem, K. Shen, J. W. Weber and D. Boyne, Phys. Rev. B 37, 9864 (1988).
10A. M. Portis, K. W. Blazey, K. A. M011er and J. G. Bednorz, Europhys. Lett. 5, 467 (1988). 11y. B. Kim and M. J. Stephen, in Superconductivity, Volume II, R. D. Parks, ed., (Marcel Dekker, Inc., New York, 1969), Chapter 19. 12K. W. Blazey, A. M. Portis, K. A. M011er and F. H. Holtzberg, Europhys. Lett., 6 457 (1988). 13K. W. Blazey, A. M. Portis and F. H. Holtzberg, submitted to Physica C. 14j. R. Clem, Physica C, in print. lSB. D. Josephson, Phys. Lett., 1, 251 (1962); Rev. Mod. Phys., 36, 216 (1964); Adv. Phys., 14, 419 (1965). I~/V. J. Gallagher, T. K. Worthington, T. R. Dinger, F. H. Holtzberg, D. L. Kaiser and R. L. Sandstrom, Physica, 148B, 228 (1987). 17A. Umezawa, G. W. Crabtree, J. Z. Liu, T. J. Moran, S. K. Malik, L. H. Nunez, W. L. Kwok and C. H. Sowers, submitted to Phys. Rev. Lett. 18R. A. Ferrell and R. E. Prange, Phys. Rev. Lett., 10, 479 (1963). IgG. Deutscher and K. A. M011er, Phys. Rev. Lett., 59, 1745 (1987).
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REDUCED FLUXON VISCOSITY IN A JOSEPHSON JUNCTION
2°A. R. Strnad, C. F. Hempstead and Y. B. Kim, Phys. Rev. Left., 13, 794 (1964). 21M. J. Stephen and J. Bardeen, Phys. Rev. Lett., 14, 112 (1964); John Bardeen and M. J. Stephen, Phys. Rev., 140, Al197 (1965). 22H. Suhl, Phys. Rev. Lett., 14, 226 (1965).
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23S. W. Tozer, A. W. Kleinsasser, T. Penney, D. L. Kaiser and F. H. Holtzberg, Phys. Rev. Lett., 59, 1786 (1987). 24V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486 (1963) and 11, 104 (1963).