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The entrainment effect of vortices by longitudinal ultrasonic waves in high-temperature superconductors
Physics C 235-240 (1994) 2080-2081 North-Holland
PHYSlCA@
'PHI,'ENTItAINMF, N T B F F E C T
Of,' V O R T I C E S
BY LONGITUDINAL
UI,TRASON]C
WAVES
IN
HIGH-TEMPERAq_'U P ~ S U P E R C O N D U C T O R S B.D.Gutli~nsky a
~Instltui,efor Physics, St~tky Ave. t94, Rostov. on-Don 344104, Russia It is shown that in superconductors in art external magnetic ~idd the. longitudinal u]trasonic waves cause a directed displacement of a vortex liq,id along the propagation direction of the waves and give rise to appearance g constant component of electricfidd iR superconductors balk. This electriclidd is directed perpendicular to the wave vectors of the ultrasonic waves and to the extentd m~netie #idd, sJ~d it has the temperature maximum.
One basic peculiarity of high-temperature superconductors is re.lativdyhigh thermoactivated vortex mobility [I]. This phenomena, arises due to the small coherence length, whidl le~ds to a small ~ctiwtlon energy of pinned vortcx cores, and due to the high "Pc values of these oxides. The spontao-mous depinniRg procvsse~ lea~ito gi~ t flux creep. This paper shows that a directed magnetic flux creep can be i~itiated by a Io~gitndin~l ultrasonic w,,ves (LUW). Physical rne~ing of tkls c~ect lic~ ;n the 1",~t that .l,lJW~ propaga.ti.g in s superconductor, indaces the ~ternating supercurrent in the bulk of the supcrconductor in the direction that is perperdiculax to the external magnetic tieJd said to thc wave vector of LUW. This current interacts with the vortices and causes a Loreattz force act, ing on each vortex along the wa.ve vector dlrection.Thc above force leads to the vortex oscill~ tlon. In the vortex liquid this motion leads to the osculation of the vortex density, according to the conthudty equation. Since the bomntz fmve acting on the vortex liquid is cqusl to the product of the vortex density by tttesupercurrent teen cousin.hi,component is ~i~en in this force in this expression, '.['hisforce ca.uses thc motion of the vortex structure as the whole. For the sake of simplicity a~sume that in sa isotropic superconductor p l ~ e d in sat external magnetic fidd Bn([~o [IZ) L U W propagates per pendicalax to the magnetic field in the OF dire(:-
tion. In this case the equation of the dynamic theory of dust|city ha~ the form Ont.t
:,OzO
P-d'#" = s-d-
+ 1,,
O)
where U in the component of the displacement vector of the superconductor, p is the density, E = pe2, c is the velocity of L U W said f i r is the fric. tion force between the vortex cores and the underlying ionic lattice. This force c ~ t be expressed in terms of the vortex velocity wltk respect to of the superconductor f f , =/~(l&' -- ['l)
(9.)
where W is a local velocity of the vortex structure in thc laboratory frame of reference)/z is the viscosity coefficient (per unit volume). AI,. though in thla paper for definltemess we consider the thermal ~.ctivatloni]ux IIow (TAFF) regime with /~TAFF=Ba/r [2], ( r----re exp ( u / k B T), however for flux flow regime all expressions given below axe valid. taJned by balancing the forces acting on a unit volume of the flux llne lattice, lax the continuum limit considered here it has the form
f~, = C,~ -gF" + [j.a,l~
(3)
where j, is the super current that is induced by thc L U W ~nd B, = ~ouZ, ~0 = 2~ ]~c/2c is the
0921-4534/94/507 00 © 1994- ElsevierScience B V. All nghts reserved SSDI 0921-4534(94)01606-2
2081
E D Gutliansky/Physica C 235-240 (1994) 2080-2081
flux quantum a~td n is the axe~l vortex density and Ct~ is the balk m o d u h m of the vortex llqu;d. It is shown [3], [4] that in situation considered here the supercurreRt hao the form up to the see. ond order of aanplltude j. =
k~Bo [UZ]
m
~
/~
l+At.
(~)
/~
where A~ is London's penetration depth. Tke continuity equation for vortices should be tzken into account to complete the system of equations corresponding to our problem, it has the followlag form:
One can verify that the electric~eld h~s the m ~ . ifcsted temperature m a x i m u m in T A F F ~ion. As numerical example we e~Jculate the electric field caused by a L U W inYBf~2Cu307 crystal Let us a~sume that external magnetic field is directed perpendlcu|zr to ab plane of the cry~ta]. The velocity of L U W is 4.1 • I0 '~m/s [51 and let w be I00 M H g sad A ---10 -9 m . In T A F F region the viscosity coettldent has the form y/-----H;t/r. 'lb calculate the resistaatee we take Tinkhaazt formula r ---- rJ~":~(To/2), 7o = J.2* lOa(l . - T / T v ) ~ . The results of e~leulations of electric fields axe shown ;n Fig.l.
(s)
=o
1000 YBazCusOv
The solation of the li~ea~ized ~ttta~ion with respect to U (1)-(5) causes the relativevelocity -~rlation a~id the sdditiona[ attenuation I , U W due to the interaction L U W with the vortex structure. In tlds work we are considering the second order effect with re~pect to U. After averaging over time of Eq.(3) and ignoring of the irregularity in the LUW amplitude vaxiatlon, a~sociated with its at. tenuatlon, one can ~ind the expression for mean velocity of vortex liquid
k~Bo ~o
< W >= - -
< ,~o >
X~ X"~ • A ~ 1+
~0~o
<
w
> +~o
<
~(w
-
t'I)>
(s)
where the angular brackets denote the average over the time. As a conseqaence ~or oue we get expression
E
=
k~A ~
X~ t +X 2
5T I,I
400 200
0 080
A--#..
~
0.85
.
•
0.90 0.95 T/To
1.00
Figure I. The electric rich dc~)endence induced by the longltudln~l ultrasonic wave in Y R a ~ C ~ s O r on temperature a,t the different values of e x t e ~ a l ms~gRetic field JR the T A F F region.
(z)
here X ffi ~c'xx ~ and A is the amplitude of LUW. The thneconstzat component of the electric field induced by this flux motion has fo~m =
E c> 600
(s)
where ~ = r#- ~o. Substituting ~z~ ~td U in the expression (6)~ foaRd from tinezri~ed set equz. lions (1).(5), we get < W > ~ k~o
ST
800
X~ ] ~4-A*
~2 ~
(9)
REFERENCES
I. Y.Yeshurun ~td A.P. Ma.lozemo]F Phys. Rev. Lett. 60 (1988), 2202, _2. ~.H. Br~ndt Z. Phys. BS0 (1990), 167, 3. E.D. Gutllxusky Soy. J. Low Temp. Phys., 1992, ;8 (1992), 290, 4. I'~.D. Gtttliaztsky J~-tph Lettem, 59 (1994), 7 5. IL bemmcns, IF'.Stellmach sad ors,, ,l.I~a~Comm. Mot., 151 (1989), 153.
PHYSlCA@
'PHI,'ENTItAINMF, N T B F F E C T
Of,' V O R T I C E S
BY LONGITUDINAL
UI,TRASON]C
WAVES
IN
HIGH-TEMPERAq_'U P ~ S U P E R C O N D U C T O R S B.D.Gutli~nsky a
~Instltui,efor Physics, St~tky Ave. t94, Rostov. on-Don 344104, Russia It is shown that in superconductors in art external magnetic ~idd the. longitudinal u]trasonic waves cause a directed displacement of a vortex liq,id along the propagation direction of the waves and give rise to appearance g constant component of electricfidd iR superconductors balk. This electriclidd is directed perpendicular to the wave vectors of the ultrasonic waves and to the extentd m~netie #idd, sJ~d it has the temperature maximum.
One basic peculiarity of high-temperature superconductors is re.lativdyhigh thermoactivated vortex mobility [I]. This phenomena, arises due to the small coherence length, whidl le~ds to a small ~ctiwtlon energy of pinned vortcx cores, and due to the high "Pc values of these oxides. The spontao-mous depinniRg procvsse~ lea~ito gi~ t flux creep. This paper shows that a directed magnetic flux creep can be i~itiated by a Io~gitndin~l ultrasonic w,,ves (LUW). Physical rne~ing of tkls c~ect lic~ ;n the 1",~t that .l,lJW~ propaga.ti.g in s superconductor, indaces the ~ternating supercurrent in the bulk of the supcrconductor in the direction that is perperdiculax to the external magnetic tieJd said to thc wave vector of LUW. This current interacts with the vortices and causes a Loreattz force act, ing on each vortex along the wa.ve vector dlrection.Thc above force leads to the vortex oscill~ tlon. In the vortex liquid this motion leads to the osculation of the vortex density, according to the conthudty equation. Since the bomntz fmve acting on the vortex liquid is cqusl to the product of the vortex density by tttesupercurrent teen cousin.hi,component is ~i~en in this force in this expression, '.['hisforce ca.uses thc motion of the vortex structure as the whole. For the sake of simplicity a~sume that in sa isotropic superconductor p l ~ e d in sat external magnetic fidd Bn([~o [IZ) L U W propagates per pendicalax to the magnetic field in the OF dire(:-
tion. In this case the equation of the dynamic theory of dust|city ha~ the form Ont.t
:,OzO
P-d'#" = s-d-
+ 1,,
O)
where U in the component of the displacement vector of the superconductor, p is the density, E = pe2, c is the velocity of L U W said f i r is the fric. tion force between the vortex cores and the underlying ionic lattice. This force c ~ t be expressed in terms of the vortex velocity wltk respect to of the superconductor f f , =/~(l&' -- ['l)
(9.)
where W is a local velocity of the vortex structure in thc laboratory frame of reference)/z is the viscosity coefficient (per unit volume). AI,. though in thla paper for definltemess we consider the thermal ~.ctivatloni]ux IIow (TAFF) regime with /~TAFF=Ba/r [2], ( r----re exp ( u / k B T), however for flux flow regime all expressions given below axe valid. taJned by balancing the forces acting on a unit volume of the flux llne lattice, lax the continuum limit considered here it has the form
f~, = C,~ -gF" + [j.a,l~
(3)
where j, is the super current that is induced by thc L U W ~nd B, = ~ouZ, ~0 = 2~ ]~c/2c is the
0921-4534/94/507 00 © 1994- ElsevierScience B V. All nghts reserved SSDI 0921-4534(94)01606-2
2081
E D Gutliansky/Physica C 235-240 (1994) 2080-2081
flux quantum a~td n is the axe~l vortex density and Ct~ is the balk m o d u h m of the vortex llqu;d. It is shown [3], [4] that in situation considered here the supercurreRt hao the form up to the see. ond order of aanplltude j. =
k~Bo [UZ]
m
~
/~
l+At.
(~)
/~
where A~ is London's penetration depth. Tke continuity equation for vortices should be tzken into account to complete the system of equations corresponding to our problem, it has the followlag form:
One can verify that the electric~eld h~s the m ~ . ifcsted temperature m a x i m u m in T A F F ~ion. As numerical example we e~Jculate the electric field caused by a L U W inYBf~2Cu307 crystal Let us a~sume that external magnetic field is directed perpendlcu|zr to ab plane of the cry~ta]. The velocity of L U W is 4.1 • I0 '~m/s [51 and let w be I00 M H g sad A ---10 -9 m . In T A F F region the viscosity coettldent has the form y/-----H;t/r. 'lb calculate the resistaatee we take Tinkhaazt formula r ---- rJ~":~(To/2), 7o = J.2* lOa(l . - T / T v ) ~ . The results of e~leulations of electric fields axe shown ;n Fig.l.
(s)
=o
1000 YBazCusOv
The solation of the li~ea~ized ~ttta~ion with respect to U (1)-(5) causes the relativevelocity -~rlation a~id the sdditiona[ attenuation I , U W due to the interaction L U W with the vortex structure. In tlds work we are considering the second order effect with re~pect to U. After averaging over time of Eq.(3) and ignoring of the irregularity in the LUW amplitude vaxiatlon, a~sociated with its at. tenuatlon, one can ~ind the expression for mean velocity of vortex liquid
k~Bo ~o
< W >= - -
< ,~o >
X~ X"~ • A ~ 1+
~0~o
<
w
> +~o
<
~(w
-
t'I)>
(s)
where the angular brackets denote the average over the time. As a conseqaence ~or oue we get expression
E
=
k~A ~
X~ t +X 2
5T I,I
400 200
0 080
A--#..
~
0.85
.
•
0.90 0.95 T/To
1.00
Figure I. The electric rich dc~)endence induced by the longltudln~l ultrasonic wave in Y R a ~ C ~ s O r on temperature a,t the different values of e x t e ~ a l ms~gRetic field JR the T A F F region.
(z)
here X ffi ~c'xx ~ and A is the amplitude of LUW. The thneconstzat component of the electric field induced by this flux motion has fo~m =
E c> 600
(s)
where ~ = r#- ~o. Substituting ~z~ ~td U in the expression (6)~ foaRd from tinezri~ed set equz. lions (1).(5), we get < W > ~ k~o
ST
800
X~ ] ~4-A*
~2 ~
(9)
REFERENCES
I. Y.Yeshurun ~td A.P. Ma.lozemo]F Phys. Rev. Lett. 60 (1988), 2202, _2. ~.H. Br~ndt Z. Phys. BS0 (1990), 167, 3. E.D. Gutllxusky Soy. J. Low Temp. Phys., 1992, ;8 (1992), 290, 4. I'~.D. Gtttliaztsky J~-tph Lettem, 59 (1994), 7 5. IL bemmcns, IF'.Stellmach sad ors,, ,l.I~a~Comm. Mot., 151 (1989), 153.