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The interaction of ultrasonic waves with vortex structure in the taff region of high temperature superconductors
Phystca C 235-240 (1994) 2078-2079 North-Holland
PHYSICA
THI~ INTERACTION OF ULTRASONIC WAVES WITH VORTEX STRUCTURE IN 'PHE TAFF RBGION OF HIGH TEMPERATURE SUPERCONDUCTORS
E.D.Gutlitmsky, T.V.Kolesnikov~~ ~Instltute for Physics, Staeltky Ave. 194, Roetov-on-Don 844104, Russia It is shown that in a superconductor, positioned in the external magnetic field, the passing ultrasonic wave causes the induction superconductive currents, which leads to the additional interaction, apart from the p h u d ~ forces, of the ultrasonic wave with the superconductor vortex structure. It is demonstrated how this interaction results in a modified behavior of ultrasonic attenuation and dispersion, if the vortex dynamics is assumed to be described by thermally assisted flux flow. The attenuation coefficient and the shift of ultrasonic wave velocity are expressed in terms of DC resistance and elasticity moduU of f u x lattice.
Investigation of the mixed state in the high temperature su- perconductors has culminated in the considerable interest to novel phases of vortex matter. $o a number of new phases of vortex lattice were considered, namely: entan$1ed vortex lattice, flux solid , flux liquid, vortex glass and a hexatic vortex glass. Information about vortex structure is usually obtained from study of superconductor response on small alternating magnet. ic fieldor transport current from cxtcrn~l source. The applied field amd tile tr~usport current interact with the vortex structure only at superconductor surface. Ultr~onic method investigation of superconductors have no tkls defect. In the work [1,2] we have shown that in superconductors in the external magnetic ~eld, the passing ultra. sonic wave (UW) causes the inducted superconductive currents inside whole of superconductive volume, and this effect allow to probe the vortex structure inside a superconductor. Below we considered the physical effect of the first order of ultrasound amplitude which result from existence of this current in the T A F F re&~ne. In this work for the simplicity we suppose that a superconductor is isotropic one and consider only two type of ultrasound waves: y axiallydirected longitudinal wave (LW) and z axiallydirected transverse wave (TW) with polarization tdong the y sxis. The vortex l~ttice equ~ttlon of motion is oh-
tained by balandng the forces acting on the unit volume of the fux llne lattice. In the conthnuum limit considered here R has the form
where B , =B0 Z, B is a compoltent of magnetic induction in the xy plane,
'~'~ = ~ \ ax~ + a z , /
(2)
aatd ~: are the componentv of the two- dimensional vortex lattice displacemeut vector, ]
l'>L =
\ av/
+
k Oz/
(3)
and Cll,Ca4 are the bulk and tilt moduli correspondingly [3]. As shown in [I] B must obey generalized London's equation B - ;~,XB = c~,,'~ ~ R z ]
(4)
Here AL is L<)ndon's penetration depth and U is the superconductor displacement vector. To close the system of the equations (3) and (4) we must write the equ~tlons of the dynamlc theory of elasticity P
0921-4534/94/S07O0© 1994- ElsevierSocnce B V All rights reserved. SSDI 0921-4534(94)01605-4
-
÷
2079
E.D. Guthans~., Z. If. Kolesntkoua/Pkystca C 235-240 (1994) 2078-2079
Here p is the mass density. It can be shown that ia the T A F F regime the mea~ pinning force, has form
(f,,,,,), =
- ud
where ~A~'~' -
S=/r in], r =
~'o~P
(-U/ksT).
Solving the system of equations (1),(4),(~) and (6) we find the variation velocity and the aAdL fional attenuation of ultreson~c. These effects axe resulted from taldng into account the interaction between UW m-td the vortex structure. Expressions for the relative deviation velocities of lon. gitudJnal A ~ / ~ and t r u s v e r s e A ~ / ~ of the ultrasound w~ves and for the additional attenuation of longitudinal ~l and transverse ~t of UW n~c written out below
art(t___/)=
t
b,,(.)
= 2pv,,)
i+
I
the tramsveraal gltrasonic velocity Ct = 2, 5 * 10 s m / c [7]. Inserting these value~ into cqs. (6)- (7) we have calculated the shift in ultrasound velocity and the additlonal attenuation. The results axe shown in Fig,I.
(z)
(s)
(9) Here Ct, Ct axe the velocities ofLW ~ad T W corPeflpondentiy, CII = ~I1 +D+ (~44 = C 4 4 + D , D = B~/(po (1 + A~ka)) • The compaxison of F.qs. (7)- (I0) with the amalogous results of Pa~tkert [4] shows that for our c u e , A C / C a n d ot axe approximately twice as large as these analogous obtained in [4]. The dhTerence is connected with the fact that the induced superconductive currents were not taken into consideration. It causes the renormalizatlon of the coefficients in the expressions (7) - (10). As a numerical "£11ustration consider • crystal of the Bi2.2 S r ~ C a o . s C ~ 2 O s + s • For example we take the case of the traasverse u!trazonic ,waves, propagatin$ ~.long the direction of m~gnetic field oriented perpendicular to the ~,b basal pl~,~e. Fo]lowh-tg [5] the resistivity for our case cs~t be represented by r = ~'oeXp -- 642/T at B = 3,21 T; r = r 0 e ~ - 523/T at B = 7,14 T; r = roe.zp - 459/T ~t B = 8,75 T with r0 = 10-a~m. For the ala~tic modulus C ~ we assume for sufficiently Idglx field, Oft = B~/~, sad for the density p = S • i03 k g / m ~ [6] and for
Bi~SrCauCuz0,m, -
0.8
2.0
8,7~ ,@..
0.0 O @
0.4 0.2
• 7.|
8.5
~
°'°3d 4O 5. 0.. eo f o 06 9 o , o- ~"B o
Figure 1, The temperature dependence of the sh{ft in the velocity and the additional attenuation of the transverse ultrasonic waves in a Bi~.2Sr2Oao.sOu2Os+ crystal at the several val. ues of a extern~l m,.gnetlc field directed perpendieulax to the a,b basal plane and parallel to the propagation direction. The frequency is 10 MI-h and the fieJdz ~Lre8,75T and 7J4T.
REFERENCES
I. E.D. Gudisasky, Soy. 3. Low Temp. Phys., 18 (1992), 290 2. E.D.Gut]iaasky, Jatph Letter% 59(1994), 7 3. E.IL Br~ad, Int. Jour. M o d e m Physics B., 5
0991), 751 4. 3. Paakert, Physic~ C168 (1990), 335 5. T . T . ".["-~.il~15_v~l. ' ...... =__n~.._ A 1 / ,\n~ovnw'j ~~ {lJ.V,,Jt r,,,J,.t~,/,.11. l t y u . D^..z, a~r---,v, a.~., "xz 6621 6. A. Gupta and ors., Phys. Rev. Left, 63 (1989), 1869 7', P. L,emmenB ~t.RdorB, J,L,css-Comm. Met.., 151 0989),
is3.
PHYSICA
THI~ INTERACTION OF ULTRASONIC WAVES WITH VORTEX STRUCTURE IN 'PHE TAFF RBGION OF HIGH TEMPERATURE SUPERCONDUCTORS
E.D.Gutlitmsky, T.V.Kolesnikov~~ ~Instltute for Physics, Staeltky Ave. 194, Roetov-on-Don 844104, Russia It is shown that in a superconductor, positioned in the external magnetic field, the passing ultrasonic wave causes the induction superconductive currents, which leads to the additional interaction, apart from the p h u d ~ forces, of the ultrasonic wave with the superconductor vortex structure. It is demonstrated how this interaction results in a modified behavior of ultrasonic attenuation and dispersion, if the vortex dynamics is assumed to be described by thermally assisted flux flow. The attenuation coefficient and the shift of ultrasonic wave velocity are expressed in terms of DC resistance and elasticity moduU of f u x lattice.
Investigation of the mixed state in the high temperature su- perconductors has culminated in the considerable interest to novel phases of vortex matter. $o a number of new phases of vortex lattice were considered, namely: entan$1ed vortex lattice, flux solid , flux liquid, vortex glass and a hexatic vortex glass. Information about vortex structure is usually obtained from study of superconductor response on small alternating magnet. ic fieldor transport current from cxtcrn~l source. The applied field amd tile tr~usport current interact with the vortex structure only at superconductor surface. Ultr~onic method investigation of superconductors have no tkls defect. In the work [1,2] we have shown that in superconductors in the external magnetic ~eld, the passing ultra. sonic wave (UW) causes the inducted superconductive currents inside whole of superconductive volume, and this effect allow to probe the vortex structure inside a superconductor. Below we considered the physical effect of the first order of ultrasound amplitude which result from existence of this current in the T A F F re&~ne. In this work for the simplicity we suppose that a superconductor is isotropic one and consider only two type of ultrasound waves: y axiallydirected longitudinal wave (LW) and z axiallydirected transverse wave (TW) with polarization tdong the y sxis. The vortex l~ttice equ~ttlon of motion is oh-
tained by balandng the forces acting on the unit volume of the fux llne lattice. In the conthnuum limit considered here R has the form
where B , =B0 Z, B is a compoltent of magnetic induction in the xy plane,
'~'~ = ~ \ ax~ + a z , /
(2)
aatd ~: are the componentv of the two- dimensional vortex lattice displacemeut vector, ]
l'>L =
\ av/
+
k Oz/
(3)
and Cll,Ca4 are the bulk and tilt moduli correspondingly [3]. As shown in [I] B must obey generalized London's equation B - ;~,XB = c~,,'~ ~ R z ]
(4)
Here AL is L<)ndon's penetration depth and U is the superconductor displacement vector. To close the system of the equations (3) and (4) we must write the equ~tlons of the dynamlc theory of elasticity P
0921-4534/94/S07O0© 1994- ElsevierSocnce B V All rights reserved. SSDI 0921-4534(94)01605-4
-
÷
2079
E.D. Guthans~., Z. If. Kolesntkoua/Pkystca C 235-240 (1994) 2078-2079
Here p is the mass density. It can be shown that ia the T A F F regime the mea~ pinning force, has form
(f,,,,,), =
- ud
where ~A~'~' -
S=/r in], r =
~'o~P
(-U/ksT).
Solving the system of equations (1),(4),(~) and (6) we find the variation velocity and the aAdL fional attenuation of ultreson~c. These effects axe resulted from taldng into account the interaction between UW m-td the vortex structure. Expressions for the relative deviation velocities of lon. gitudJnal A ~ / ~ and t r u s v e r s e A ~ / ~ of the ultrasound w~ves and for the additional attenuation of longitudinal ~l and transverse ~t of UW n~c written out below
art(t___/)=
t
b,,(.)
= 2pv,,)
i+
I
the tramsveraal gltrasonic velocity Ct = 2, 5 * 10 s m / c [7]. Inserting these value~ into cqs. (6)- (7) we have calculated the shift in ultrasound velocity and the additlonal attenuation. The results axe shown in Fig,I.
(z)
(s)
(9) Here Ct, Ct axe the velocities ofLW ~ad T W corPeflpondentiy, CII = ~I1 +D+ (~44 = C 4 4 + D , D = B~/(po (1 + A~ka)) • The compaxison of F.qs. (7)- (I0) with the amalogous results of Pa~tkert [4] shows that for our c u e , A C / C a n d ot axe approximately twice as large as these analogous obtained in [4]. The dhTerence is connected with the fact that the induced superconductive currents were not taken into consideration. It causes the renormalizatlon of the coefficients in the expressions (7) - (10). As a numerical "£11ustration consider • crystal of the Bi2.2 S r ~ C a o . s C ~ 2 O s + s • For example we take the case of the traasverse u!trazonic ,waves, propagatin$ ~.long the direction of m~gnetic field oriented perpendicular to the ~,b basal pl~,~e. Fo]lowh-tg [5] the resistivity for our case cs~t be represented by r = ~'oeXp -- 642/T at B = 3,21 T; r = r 0 e ~ - 523/T at B = 7,14 T; r = roe.zp - 459/T ~t B = 8,75 T with r0 = 10-a~m. For the ala~tic modulus C ~ we assume for sufficiently Idglx field, Oft = B~/~, sad for the density p = S • i03 k g / m ~ [6] and for
Bi~SrCauCuz0,m, -
0.8
2.0
8,7~ ,@..
0.0 O @
0.4 0.2
• 7.|
8.5
~
°'°3d 4O 5. 0.. eo f o 06 9 o , o- ~"B o
Figure 1, The temperature dependence of the sh{ft in the velocity and the additional attenuation of the transverse ultrasonic waves in a Bi~.2Sr2Oao.sOu2Os+ crystal at the several val. ues of a extern~l m,.gnetlc field directed perpendieulax to the a,b basal plane and parallel to the propagation direction. The frequency is 10 MI-h and the fieJdz ~Lre8,75T and 7J4T.
REFERENCES
I. E.D. Gudisasky, Soy. 3. Low Temp. Phys., 18 (1992), 290 2. E.D.Gut]iaasky, Jatph Letter% 59(1994), 7 3. E.IL Br~ad, Int. Jour. M o d e m Physics B., 5
0991), 751 4. 3. Paakert, Physic~ C168 (1990), 335 5. T . T . ".["-~.il~15_v~l. ' ...... =__n~.._ A 1 / ,\n~ovnw'j ~~ {lJ.V,,Jt r,,,J,.t~,/,.11. l t y u . D^..z, a~r---,v, a.~., "xz 6621 6. A. Gupta and ors., Phys. Rev. Left, 63 (1989), 1869 7', P. L,emmenB ~t.RdorB, J,L,css-Comm. Met.., 151 0989),
is3.