Excitation of ''rotation'' collective modes in vortex lattice of clean type II superconductors

Ž .

Physica C 332 2000 199–202

www.elsevier.nlrlocaterphysc

Excitation of ‘‘rotation’’ collective modes in vortex lattice of

clean type II superconductors

A. Kasatkin

a,b

_{, B. Rosenstein}

c,)

a

Electrophysics Department, National Chiao Tung UniÕersity, Hsinchu 30043, Taiwan b

Institute of Metal Physics, Ukrainian Academy of Sciences, KieÕ, Ukraine c

National Center for Theoretical Sciences and Electrophysics Department, National Chiao Tung UniÕersity, Hsinchu 30043, Taiwan

Abstract

In superclean limit, the Magnus force on Abrikosov vortices is stronger than friction. Due to this nondissipative force, vortex segments rotate around pinning centers. Waves of such rotations under certain conditions are only weakly damped

Žnot overdamped as is usually the case and lead to resonances in ac response. Excitation of such waves by applied ac field.

near the surface is considered. Surface impedance, ac resistivity and magnetic permeability are calculated using elasticity theory of the vortex lattice. q 2000 Published by Elsevier Science B.V. All rights reserved.

Keywords: Excitation; Rotation; Vortex lattice

1. Introduction

Abrikosov vortex dynamics in type II supercon-ductors under magnetic field is usually thought to be overdamped. Due to large vortex viscosity, the dis-placement waves in vortex lattice do not propagate. In high-T superconductors, the situation under cer-_c tain conditions might be different. The dissipation during the vortex motion is, at least, to large extent due to excitation of quasiparticles inside the vortex core. At small temperatures, this process is frozen and instead of the usual Bardeen–Stephen friction force, h z, one only has a non-dissipative Magnus force h z = z perpendicular to the vortex velocity,

ˆ

where z is the direction of external magnetic field. As evidence to the increasing role of the Magnus

w x

force is the famous Hall anomaly 1,2 . In a series of

)

Corresponding author.

w x

direct experiments 3 , it was shown that in YBCO single crystals at low temperatures, the Hall angle

Ž . X y1

tan u '_{h rh diverges as T and clearly exceeds}

H

1 below 4 K reaching 2.5 at 3 K. This regime was

w x

termed by authors of Ref. 3 ‘‘superclean limit’’. Theoretically, such a behavior was predicted in Refs.

w_{4–8 . In such a superclean regime, vortex dynamics}x

might be non-overdamped and, for example, dis-placement waves in the vortex lattice can propagate.

w x

This type of phenomenon was used recently 9,10 to

w x

explain the magneto-absorption in BSCCO 11–14 , although alternative explanations based on the

w x

Josephson plasma oscillations exist 15,16 .

In this paper, we consider dynamics of vortices in ‘‘superclean’’ superconductors under applied ac field. We argue existence of weakly damped ‘‘rotation’’ waves in the pinned vortex lattice, calculate its dis-persion law, and consider linear vortex response on applied ac field. We show that excitation of the

0921-4534r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.

Ž .

( ) A. Kasatkin, B. Rosenstein r Physica C 332 2000 199–202

200

non-overdamped waves by applied ac field modify in an essential way the theory of the linear response

w x

developed by Brandt 17,18 and Coffey and Clem

w_{19,20 and point out possible resonance effects in}x

the surface impedance and ac resistivity.

2. Dispersion relation for waves in vortex lattice

For small magnetic induction values, BrF -0

ly2

, one can neglect exponentially small interactions between vortices and consider single vortex

dynam-Ž .

ics. Assuming that the vortex is pinned Fig. 1 , we describe it by equation of motion for displacement u:

mu q h u q h

¨

˙

Xz = u q a u s 0.

˙

Ž .

1

Here, a is the Labousch parameter describing restoring pinning force in the x–y plane. Neglecting

Ž .

Fig. 1. a Positions and displacements of vortices caused by

Ž .

external Lorentz force. b Displacement of a vortex segment

Ž .

under influence of ac field in the superclean limit. c Displace-ment of a vortex segDisplace-ment under influence of ac field in the conventional overdamped case.

Ž .

the vortex mass in Eq. 1 for single vortex

dynam-Ž .

ics, one obtains the following periodic solution: u ti

s_ei V"t_{u where:} i ih " hX V s a_{" .}

_{Ž .}

₂ X 2 2 h q h

When the friction coefficient, h, is small, one

ob-Ž .

tains clockwise and counterclockwise circular

vor-Ž .

tex motion around the pinning center see Fig. 1b with the frequency v s arh_M X.

Contribution of interactions between vortices to the vortex dynamics can be taken into account within harmonic approximation: u Ra _hX e u Ra _{qa u R}a

h Ž

˙

i

.

i j

˙

j

Ž

.

i

Ž

.

q F _Ra_y Rb u Rb.s_{0. 3}

Ž

.

Ž

Ž .

Ý

i j j b

Here F is the dynamical matrix and Ra _are

loca-i j

tions of vortices usually arranged in the lattice and

e_{i j} is the totally antisymmetric tensor. Since we are using elasticity theory, the detailed nature of the vortex matter is not very important as long as correct elastic moduli are used and most of the considera-tions are valid in vortex liquid or glass. We will consider only external forces homogeneous in y and

z directions; therefore, the only nonzero component

of momentum is k ' k. When external force isx

absent displacement vector for frequency, V , satis-fies: 2 X Ž . u k Ži Vh q a q c. 11Ž .k k i Vh x X 2

ž

u Ž .k

/

ž

yi Vh Ži Vh q a q c. 66Ž .k k

/

y ' A u s 0,i j j

Ž .

4 Ž .

where c11 and c66 are possibly dispersive elastic

w x

moduli of the vortex matter. In London limit 21 :

B2 1 c11 c

_{Ž .}

k s ' _; 11 2 2 2 2 4p 1 q l k 1 q l k BF0 c s66 2.

Ž .

5 4 4pl

Ž

.

( )

A. Kasatkin, B. Rosenstein r Physica C 332 2000 199–202 201

Ž .

The eigenfrequencies in Eq. 2 now become branches. In the superclean limit h s 0, one has non-damped waves with dispersion law:

2 2 4

(

a q a k c q k c11

Ž .

k c66

V"

Ž .

k s " X ,

Ž .

6

h

Ž .

where c ' c11 k q c . In the general case, when66

both h and hX are nonzero, the eigenfrequencies

Ž .

V_{pm k are complex values and dispersion law}

be-w x

comes rather complicated 22 .

Ž

Polarization of the waves which follows from

Ž .. Eq. 4 is as follows: u k a q c k2

Ž .

y y₁ 11 s y_{tan u q iH X.}

_{Ž .}

₇ u_x

_{Ž .}

k V_"

_{Ž .}

_{k h}

The fact that the ratio is imaginary means that vortices move on elliptic trajectories.

3. Linear response under applied ac field

In this section, we consider the pinned vortex system response to surface ac current caused by alternating field h eiv t _{in direction parallel to dc}

ac

field, H, and to the surface of the superconducting half space, see Fig. 1a. Linear response for such geometry for the case hXs_{0 was considered by}

w x w x

Brandt 17,18 and Coffey and Clem 19,20 , also taking into account pinning, viscosity and creep. Since we are interested mostly in the low tempera-ture regime, flux creep can be neglected while the

Ž

Magnus force term is important creep can be taken

w _x.

into account in a similar manner as in Refs. 17–20 . When one performs similar calculation for hX) 0,

new resonant phenomena are readily seen. We im-pose proper boundary conditions using the ‘‘bulk

w x

concept’’ methods of Refs. 17,18 , which allow referring the problem to an equivalent problem in whole space.

The external force is:

Bhac y_{< x < _ l i v t} Fext

Ž

x ,t s

.

e e ,

Ž .

8 4pl Bhac Fext

Ž

k , v s

.

_{2p 1 q l k}2 2 .

Ž .

9

Ž

.

The displacement in momentum space is obtained

Ž . Ž Ž ..

from Eq. 4 with external force Eq. 9 . Very often both c and k are ‘‘small’’. If c k2 _{is small}

66 66

Ž .

compared to vh q a , one can readily obtain dis-placements in the form:

Bhac ux

Ž

k , v s

.

_{2p a v 1 q k}2 _{c ra v q l}2

Ž

.

Ž

11

Ž

.

.

2 h l2 _v

Ž

.

ac C s _,

_Ž

₁₀

_.

2 2 B 1 q k l_ac

_Ž

v

_.

i vhX uy

Ž

k , v s

.

ux

Ž

k , v ,

.

Ž

11

.

i vh q a Ž . wŽ 2 X 2. Ž .x where a v ' i vh q a y v h r _{i vh q a and} 2_{Ž .}

the modified Campbell penetration depth l_C v is:

c11 B2

Ž

i vh q a

.

2 l_C

_Ž

v '

_.

s 2 2 X 2 a v

_Ž

_.

_4p

_Ž

_{i vh q a}

_.

yv h B2 _{i v q tanu v cos}2_u

Ž

H M

.

H s y _. 4ph v y Vq

Ž .

0 v y Vq

Ž .

0 12

Ž

.

The frequency-dependent complex ac penetration

Ž .

depth was introduced in Eq. 8 :

l2 _{v ' l}2_ql2 _{v .}

13

Ž

.

Ž

.

Ž

.

ac C

w x X

As is in the usual case 17–20 , h s 0, this quantity determines both the surface impedance:

Z v s 4p irc_s

_Ž

_.

_Ž

2

_.

vl_ac

_Ž

v

_.

_Ž

₁₄

_.

Ž . Ž . Ž .

and the ac resistivity r_ac v ' E x rJ x s Ž4p irc2.vl2_acŽv . These two quantities exhibit res-.

onance in the clean limit. On Fig. 2, real and imagi-nary parts of surface impedance for various values of cosu_H and b ' BrH s 10 are shown._c1

The general case, when k2_{c is not negligible,} 66

w x

was also studied in our work 22 . Qualitative

behav-Ž .

( ) A. Kasatkin, B. Rosenstein r Physica C 332 2000 199–202

202

Ž

Fig. 2. Frequency dependence of the surface impedance real and

.

imaginary parts for cosu s 0.1, 0.3, 0.6, 1. Magnetic inductionH B s10 H .c1

not change, while resonance peak becomes even more pronounced in this case.

4. Discussion

In this work, we determined conditions under

Ž

which a non-overdamped ‘‘rotation’’ around

pin-.

ning centers waves exist in clean type II supercon-ductors. There are clear indications that these condi-tions can be met in non-twinned YBCO single

crys-w x

tals 2 , and some resonance effects due to vortex motion were really observed in ac experiments on

w x

HTS materials 23,24 . Excitation of such waves by applied ac field near the surface is considered. The simplest realistic geometry is the superconducting half space with the dc magnetic field creating vor-tices parallel to the surface. We considered the direc-tion of the surface ac field parallel to the dc mag-netic field. In this case, linear response character-istics, such as surface impedance and ac resistivity, were calculated using the elasticity theory of the vortex lattice. The most pronounced effect of the rotation waves is resonance at characteristic fre-quency of order V s arh_s X. It is comparable or

larger than the depinning frequency, V_depinsarh, w x

which is of order 10–100 GHZ 25 .

Acknowledgements

The authors thank Professor Y.S. Gou and Dr. A. Knigavko for helpful discussions.

This work was supported by National Science Council, Republic of China through contract

aNSC88-2112-M009-026. A. Kasatkin, one of the

authors, is grateful to the Ministry of Education and National Science Council for their support on this work during his stay in Taiwan.

References

w x_{1 S.J. Hagen, A.W. Smith, M. Rajeswari, J.L. Peng, Z.Y. Li,}

R.L. Greene, S.N. Mao, X.X. Xi, S. Bhattacharya, Q. Li, C.J.

Ž .

Lobb, Phys. Rev. B 47 1993 1064.

w x2 J.M. Graybeal, J. Luo, W.R. White, Phys. Rev. B 49 1994Ž .

12923.

w x_{3 J.M. Harris, Y.F. Yan, O.K.C. Tsui, Y. Matsuda, N.P. Ong,} Ž .

Phys. Rev. Lett. 73 1994 1711.

w x_{4 N.B. Kopnin, V.E. Kravtsov, Pis’ma Zh. Eksp. Teor. Fiz. 23} Ž1976 631..

w x5 N.B. Kopnin, V.E. Kravtsov, JETP Lett. 23 1996 578.Ž . w x6 N.B. Kopnin, A.V. Lopatin, Phys. Rev. B 51 1995 15291.Ž . w x7 P. Ao, D.J. Thouless, Phys. Rev. Lett. 70 1993 2158.Ž . w x8 E.B. Sonin, Phys. Rev. B 55 1997 485.Ž .

w x_{9 N.B. Kopnin, A.V. Lopatin, E.B. Sonin, K.B. Traito, Phys.} Ž .

Rev. Lett. 74 1995 4527.

w10 E.B. Sonin, Phys. Rev. Lett. 79 1997 3732.x Ž .

w_{11 O.K.C. Tsui, N.P. Ong, Y. Matsuda, Y.F. Yan, B. Peterson,}x Ž .

Phys. Rev. Lett. 73 1994 724.

w12 O.K.C. Tsui, N.P. Ong, Phys. Rev. Lett. 76 1996 819.x Ž . w_{13 Y. Matsuda, M.B. Gaifullin, K. Kumagai, K. Kadowaki, T.}x

Ž .

Mochiku, Phys. Rev. Lett. 75 1995 4512.

w_{14 Y. Matsuda, M.B. Gaifullin, K. Kumagai, M. Kosugi, K.}x Ž .

Hirada, Phys. Rev. Lett. 78 1997 1972.

w_{15 L.N. Bulaevskii, M. Maley, H. Safar, D. Dominguez, Phys.}x Ž .

Rev. B 53 1996 6634.

w_{16 L.N. Bulaevskii, V.L. Pokrovski, M. Maley, Phys. Rev. Lett.}x Ž .

76 1996 1719.

w17 E.H. Brandt, Phys. Rev. Lett. 67 1991 2219.x Ž . w18 E.H. Brandt, Physica C 195 1992 1.x Ž .

w19 M.W. Coffey, J.R. Clem, Phys. Rev. Lett. 67 1991 386.x Ž . w20 M.W. Coffey, J.R. Clem, Phys. Rev. B 46 1992 11757.x Ž . w21 E.H. Brandt, Rep. Prog. Phys. 58 1995 1465.x Ž .

w22 A. Kasatkin, B. Rosenstein, Phys. Rev. B 60 1999 14907.x Ž . w_{23 E.J. Choi, S. Kaplan, S. Wu, H.D. Drew, Q. Li, D.B. Ferrer,}x

Ž .

J.M. Phillips, S.Y. Hoi, Physica C 254 1995 258.

w24 T. Ichiguchi, Phys. Rev. 57 1998 638.x Ž .

w_{25 M. Golosovsky, M. Tsindiekht, D. Davidov, Supercond. Sci.}x Ž .