Random-signal interaction with flowing lattice of Abrikosov vortices

PHYSICA Physica C 224 (1994) 377-383

ELSEVIER

Random-signal interaction with flowing lattice of Abrikosov vortices V.D. Ashkenazy a, G. Jung b,c, I.B. Khalfin a, B.Ya. Shapiro ~'* ° Jack and Pearl Reznik Institute of Advanced Technology and Department o f Physics, Bar-llan University, 52900, Ramat Gan, Israel b Department of Physics, Ben Gurion University of the Negev, 84105 Beer-Sheva, Israel c Instytut Fizuki PAN, 02668 Warszawa, Poland

Received 15 January 1994; revised manuscript received 3 March 1994

Abstract

The interaction of a random-noise signal with a flowing lattice of Abrikosov vortices leads to strong modifications of the original noise spectrum. The detailed shape of the spectrum depends on the lattice-flow velocity. At low velocities of the lattice flow the spectrum is attenuated at low frequencies similarly to the case of a motionless lattice. For the velocities above the crossover velocity the spectrum magnitude oscillates with a characteristic frequency depending on the time of flight of the vortices across the strip dimension perpendicular to the direction of current flow. For sufficiently high magnetic fields the bias current determining the crossover to the oscillatory regime depends only on the temperature and film thickness.

1. Introduction

High critical t e m p e r a t u r e superconductors ( H T S C ' s ) operate at t e m p e r a t u r e s where the effects o f thermal activation can be clearly pronounced. This fact, together with a strong anisotropy and low pinning energies o f the oxide superconductors, results in easy r a n d o m m o v e m e n t s o f the flux vortices. M o v i n g vortices p r o d u c e a high level o f intrinsic low-frequency noise as frequently observed in H T S C materials a n d devices in the form o f strong 1/f-like fluctuations [ 1,2 ] a n d characteristic r a n d o m telegraph noise signals, extending even to M H z frequencies [3,4]. R a n d o m voltages developing across a H T S C thin film cause r a n d o m Lorentz forces to act u p o n the flux vortices in the sample. The m o t i o n o f vortices caused by an interaction with r a n d o m compo* Corresponding author.

nents o f the Lorentz force generates additional voltages across the sample that change the power spectrum o f the original r a n d o m signal. I f the vortices are pinned, or strongly interact, the viscous vortex relaxation leads to the attenuation o f the low-frequency part o f the power spectra o f r a n d o m voltages. In a recent p a p e r we have presented a detailed theoretical m o d e l o f this p h e n o m e n o n a n d its experimental verification [ 5 ]. In particular we have shown that the extent a n d frequency b o u n d a r i e s o f the d u m p i n g effect d e p e n d on the elastic properties o f the vortex lattice and on the structure a n d strength o f the pinning potential. In the m o d e l we have assumed that all vortices are p i n n e d a n d thus the dissipative state in the H T S C film is not due to the correlated movements o f flux vortices under the driving Lorentz force o f the bias current. The dissipative voltage state is due to switching o f intrinsic Josephson j u n c t i o n s into the voltage state by a current flow exceeding the Jo-

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14D. A s h k e n a z y et al. / P h y s i c a C 224 (1994) 3 7 7 - 3 8 3

sephson critical current. Such a mechanism is typical for granular samples and indeed, the experiments performed with a granular YBaCuO thin film fully confirmed the predictions of our model [ 5 ]. Nevertheless, the mechanism of the dissipation in high-quality HTSC thin films is almost exclusively due to the flux-creep and flux-flow mechanisms. When the external bias current exceeds the critical value the Lorentz force exceeds the pinning force and the Abrikosov lattice starts to flow. In this paper we demonstrate that the flowing lattice can transform the noise spectra even more drastically than the static pinned lattice does. In particular, we predict the appearance of a set of periodically distributed bumps in the power spectra when the bias current exceeds a temperature- and film thickness-dependent threshold value.

2. Basic equations The voltage due to the motion of flux vortices in a superconducting film can be expressed as a superposition of elementary voltages V, contributed by individual vortices:

8 t'(t) = t ' ( t ) - ( t ' ( t ) i~,

= ~ d2pg(p)SJ(p, t) .

(5)

where 8g(p, t)=J(p, t ) - ( J ( p , t)), and . . . ? , denotes the time average. The voltage autocorrelation function is: 7J,.(r) = ( 8V(t)SV(t+ r.) .>:

---fdZpfd2p'~g,~(p)glAP')K~(p,p',r,. where

K~(p,p',r)=(SJ,~(p,t)aJ~(p',t+r)).,.

(7)

is the vortex-flow correlation function; c~ and/J are the Cartesian coordinates. In the following we shall consider a thin superconducting film filled with rigid vortices. We will assume that the vortices undergo small deviations u(aio, t) from their equilibrium positions a,o in the Abrikosov lattice frame (ALF). The positions of the vortices in the film are thus given by two-dimensional vectors Pi [ 7 ], with (8 )

(1)

The relation between the elementary voltage V, and the ith vortex velocity, for uniform thickness film t,,, is fully determined by a resolution function g(p,), p~ is the ith vortex radius-vector, that depends on the geometry of the sample and on the layout of the voltage measuring circuit [ 6 ]. We have [bmt(p)-bmb(P)]

•

(2)

In the harmonic approximation the energy of interaction between the vortices can be expressed in the terms of the elastic matrix G( i,j):

U(u(aio, t))=½ ~ G(i,j)u(a,o,t)u(ajo, t).

u(a,o, t)= f~ ~(q2)exp(iqa,o)Qq~ .

J(p, t) = ~ v~(t)~2(p-p~(t) ).

D(q)= ~ G(h)exp(iqh),

(3)

qa

(10)

Qca(t) are the normal-mode amplitudes, q is the wave vector and 2 stands for the polarization; here D(q)~(q2 ) =D.~(q2

(11)

) .

h=a~o-ajo.

(12)

h

i

In the long-wave approximation

The total voltage due to the vortex motion,

V(t) = [ d2pg(p)J(p, t),

(9)

Expanding the vortex deviations from their equilibriurn positions in the basis of the polarization vectors ~(q, 2) diagonalizing the dynamic matrix D(q2) we have

Here ~o is the flux quantum, bmt and bmb are the values of the magnetic induction due to the current flow Im in the measuring circuit on the top and on the bottom of a film, respectively. All information about the vortex dynamics is contained in the time-dependent vortex-flow density J(p, t) [ 7 ],

d

(6,

~fl

pi = ai0 + u (a,o, t ) .

V, =g(p~)v~.

g(p)= ~~o

may possess a fluctuating random component V(t):

(4)

Dqx -~

( OoC6o/B)q2:Dq 2 .

( 12a )

V.D.Ashkenazy et al. / Physica C 224 (1994) 377-383 where B is the magnetic induction in the sample and C66 is the shear modulus of the Abrikosov lattice. According to Eq. (6) the voltage noise can be expressed in the terms of the vortex-flow correlation function. If the dimensions of the measuring circuit are large with respect to the intervortex spacing, the vortex lattice can be treated as a continuum. Within the first-order approximation, assuming small displacements of the vortices from their equilibrium positions, the change in the vortex-flow density is

• l(p, t)=no~v(p-vot, t)+vo~n(p-Vot, t),

(13)

where no is the equilibrium vortex density, ~ / a n d p are vectors in the laboratory reference frame, and 8v and 8n are measured in the ALF system moving with a velocity Vo. Identifying ~v with du/dt and ~n with ( - noVu), we get from Eqs. ( 10-13 )

5J(p,t)=no~[~(q2)~-~-ivoq.e(q2)Qca(t)] Xexp[iq(p-vot) ] .

(14)

Substituting &lfrom Eq. (14) into Eq. (5) we obtain the noise voltage: 8 V ( t ) = E 8Vqa(t), qa

(15)

with

8Vqa( t)= (FqadQ,a( t) /dt +Gq~Q~a( t) ) exp( -iqvot) ,

(16)

where Fq~ =no f d2pg(p)E(q2) exp(iqp) ,

(17)

and

Gq~= -no ~ d2pg(p)voexp(iqp)(iq~(q2) ).

(18)

The noise defined by Eq. (16) consists of two components. The first one is proportional to vortexvelocity fluctuations, i.e., is proportional to dQq~/dt, while the second term is proportional to vortex-density fluctuations, i.e., to Qq~. The overall shape of the noise power spectrum depends on the peculiarities of the voltage measuring configuration. Let us consider a practical experimental configuration in which a long thin film strip has voltage measuring leads applied perpendicular to the

379

Fig. 1. The schematicof the consideredgeometricalarrangement of the thin film superconducting strip and the voltagemeasuring setup. Current is flowingalong the strip length and vortices are directed perpendicular to the strip surface. film surface at two contact spots. The voltage leads are kept far away from the surface elsewhere. The detailed scheme of the considered arrangement is shown in Fig. 1. The superconducting strip possesses length L in the x-direction, width W in the y-direction, and thickness d in the z-direction, such that L >> W>> d. The voltage measuring leads are attached at points with coordinates [xa, 0] and [Xb, 0] such that x a - xb = R >> W. The resolution function g(p) for this arrangement takes the form [ 7 ]

Ffw

g(a)= Lk7 )

2

21-' 2 , -y J r

(19)

where ey is the unit vector in the y-direction. Eq. (19) holds for vortices moving between the leads far away from the contracts. A more complicated expression would arise if the vortex position would fall within a distance less than Wfrom the contacts. However, the condition R >> W allows us to neglect contributions of such vortices. Using Eqs. ( 17-19) we obtain for Fqa and Gqa

BR r fqyW~x x rqz = ~c JOk---~j,,qx;o,,a;to ,

(20)

..

Gqa = -~c (q'vo)Jo

6q,;oaa;to

(21)

where Jo is the Bessel function of the zeroth order.

3. Vortex motion

The equation of motion for an ith vortex reads

V.D. Ashkenazy et al. /Physica C 224 (1994) 377-383

380

dPi

rl~ 7 = - Y. G(j, i)u(j, t ) + L x t ( P , t ) , J

(22)

du( aio, t) + ~j G(i,j)u(ajo, t ) = 5f(aio, t ) , dt

i

Z(q, 0)) = 4~ 2

where r/is the vortex viscosity per unit length,f~xp (p,, t) is the linear density of the external force. The intervortex interaction term is accounted for according to Eq. (9). Taking the time average of Eq. (22) we obtain for vortex displacements from the equilibrium positions

~1

where 0)'=0)-qvo, and 14"/2 I./2 -- g ' / 2

w

×exp(i0)t+i(q.,x+q),y) ) ctx-dt.

(30)

Putting in Z ( t ) from Eq. (28) and taking into account Eqs. (20), (21) and (12a) we obtain for P,.(0))

P,.( 0) )

(23)

where 6f=fext- (f'). Expanding Eq. (23) by the normal displacements, we obtain

- 1/2

r~BeRZ7o ~ -

Wc 2

)

sin e

j2

(?)

0) 2 d q

[4qe(D2qa+q2(0)+qvo)e)] (31)

dQ~t(t) +Vq~Qq~(t) = 6fqa(t) ,

(24)

where 1 8fq~ = ~ ~ exp(-iqajo)~(qA)ff(ajo, t ) ,

(25)

Let us rewrite Eq. (31 ) introducing the characteristic frequency of the vortex system 0)c, the time of flight of a vortex across the strip r, and the wave vector normalized to the strip width ~. The new variables are defined as

H" eq,

and N is the number of vortices. The solution of Eq. (24) is the following:

03c =

Oqa=

in the new variables Eq. ( 31 ) reads

~ - aa(q, 0))Sfqa(0)) exp ( - i 0 ) t ) ,

(26)

(~0('66/B)/Weq=D/

r = W/t,'o.

~=ql~'.

(32)

2( ()()

e~ aa(q, 0)) = - [i0)*l-Dqa] -~ ,

(27)

Therefore, knowledge of the fluctuating component of the external force, ~f enables us to calculate Q¢a and its time derivative dQqa/dt, and thus enables us to calculate the resulting noise voltage.

4. Voltage noise power spectrum

Let us assume that the vortex system interacts with a spatially uniform white noise. The randomness of the noise becomes manifest in the force-force correlator:

Z(t, t') = ( 8 f ( t ) 6 f ( t ' ) ) = Z o ~ ( t - t ' )

.

(28)

The spectral density of fluctuations, following Eqs. (6) and ( 16-18 ), can be represented in the form p~(0)) = ~] [Fq~.O.) 2 ,2 + 2iFqa Gqa0)'- G~a ]

× Z ( q , 0)')la(q, 0)') [2

(29)

rEB2RZZo .? sm P,,(0)) =

?12C2

j

~ Jo ~ (co'r)ed(

(2[(0)cT)2(4+(0)T+()2]

'

(33) The solutions of our problem clearly depend on the relation between the relaxation frequency of the vortex lattice 0)° and the time of flight of the vortices across the strip r. Let us consider different regimes of flux flow.

4. 1. Rapidly flowing lattice (~ocr<< 1) The main characteristic frequency of the problem is set by the inverse of the time of flight across the sample l/r. For a rapidly flowing lattice this frequency substantially exceeds the relaxation frequency of the Abrikosov lattice 0)c- Therefore, the vortex lattice propagates across the sample as a rigid structure stressed by the external signal. Lattice-density fluctuations, the second term in Eq. (13), as a result are frozen in the lattice during the time of flight.

V.D. Ashkenazy et al. /Physica C 224 (1994) 377-383

Their role in determining the spectrum shape is set by an interplay between the characteristic frequencies of the problem and the signal frequency co. The kernel of integral (33) reaches the giant maxima at ( = - t o z and at ( = 0 . The integral may be calculated asymptotically for frequencies much lower and much higher than the certain characteristic value O9o>> 1/ 3; 1

too-- r(tocZ)t/3 .

(34)

For frequencies o2<< o20 the value of integral (33) is determined by the first kernel maximum ~= - toz and the spectral density Pv (o2) becomes

•B2R P~(tO) ~

sin 2(toz/2 )j2 (tOz/2) c2r/2 (toz)2 (tour)

duced random distribution of vortices flowing across the strip. This is physically equivalent to the rigidly distorted vortex lattice as formally described by us. For extremely large frequencies, o2>> O9o the second maximum at ( = 0 becomes predominant. The external signal is changing fast, and the lattice-density fluctuations are changed many times by the driving force during the time of flight. Therefore, their average effect is null and only the velocity fluctuations, first term in Eq. ( 13 ), produce the spectrum. At very high frequencies the vortex lattice adiabatically follows the external random signal. Thus the spectral density measured at high frequencies will correspond simply to the white-noise spectrum ItB2R2Zo

2Z 0

(35)

Observe that the spectral density oscillates with a characteristic frequency 1/r determined by the time of flight of vortices across the strip. Eq. (3 5) was derived in the approximation that excludes points in the vicinity of frequencies at which P~ (o2) = 0. The shape of the power spectrum calculated according to Eq. (3 5 ) is presented in Fig. 2. Physically, these oscillations are due to propagating frozen vortex-lattice density fluctuations. In this sense they are similar to the oscillations discussed by Oojen and van Garp [8] and explained by them as manifestations of the phenomenologically intro-

Pv(to) ~ cEt/~---T -

- const.

(36)

Although the characteristic frequency o20 has mainly a formal meaning, one can ascribe a physical meaning to tOo: the frequency separating two different asymptotic regimes of the power-spectra behavior; oscillating with frequency below too and white spectrum above o20.

4.2. Slowly flowing lattice (toot >> 1) In this case the time of flight is long and the dynamics of the system, excluding extremely low frequencies, is controlled by the vortex-lattice relaxation processes. In the following we shall estimate asymptotically the integral (33) in different frequency ranges with respect to a characteristic frequency co, << 1/z. We have o21 -

¢.¢1)

381

1

1 (toot)

"

(37)

( 1 ) For frequencies much lower than a characteristic frequency o9<< to1 we get for the power spectrum nB2R 2Z o

Pv(to)~ 4c2/~2toc,t.=Avo, where A=const.

t.IJo

frequency Fig. 2. The shape of the power spectrum o f the while-noise signal interacting with an Abrikosov vortex lattice flowing at very high velocities.

(38)

For extremely low frequencies of the driving signal one can neglect vortex-velocity fluctuations and consider only static density fluctuations in the vortex lattice. Namely, this effect brings about the DC-like term for the frequencies up to the frequency of the order of o21when velocity fluctuations start to play a significant role.

V.D. Ashkenazy et al. /Physica C 224 (1994) 377-383

382

(2) For frequencies 091 <• 09<• (-t)cthe motion of vortices can be considered as their oscillations in the nonflowing lattice and the power spectrum will not be influenced by the vortex-lattice motion at all. In this frequency range we obtain from Eq. (33)

nB2R2Zo(09/09c)J/2

Pv(09) ~

c2q2

(39)

Observe that the spectral density is suppressed at low frequencies and increases with increasing frequency as 09 1/2. This corresponds to the static case of a motionless, pinned vortex lattice which we have recently discussed in ref. [ 5 ]. (3) For extremely large frequencies 09>> 09c we obtain again a white-noise spectrum with the same magnitude as in the case of a rapidly flowing lattice, see Eq. (36). The behavior of the power spectrum for a slowly flowing vortex lattice is shown schematically in Fig. 3. Observe that o91, similarly to the previously discussed 090 separates two asymptotically different behaviors of the power spectra; a white spectrum below the characteristic frequency and a quasi static regime of low-frequency attenuation above it. In practice the parameter controlling the crossover between the regime of a slowly and rapidly flowing lattice is the current flow determining the lattice-flow velocity. The crossover velocity v~= W09c depends on the temperature, vortex viscosity, strip geometry, and through the quantity D, see Eq. (12a), on the external magnetic field. For a sufficiently dense Abrikosoy lattice, where C66=¢oB/(8n2)2 [9] and 2 is the

(0c

C

-(3

}

*d (9 C).. O)

' i F----

/

frequency

Fig. 3. Schematic o f the white-noise power spectrum resulting from a n i n t e r a c t i o n with a slowly flowing A b r i k o s o v vortex lattice.

penetration length, the crossover velocity becomes field independent: vc =0o

2/(8~2)2qW.

(40)

The corresponding crossover current flow depends only on the film thickness d and the temperature. We have

Oocd

( 41 )

lo= ( 8 ~ ) 2 .

For temperatures close to the critical temperature I'~

Oocd(

Io = ~(8~2o)

T) 1-~

,

(42)

where 2 0 = 2 ( T = O ) .

5. Conclusions In conclusion we have evaluated the interaction between a flowing Abrikosov lattice and a randomnoise signal applied to the thin film superconducting strip. The behavior of the power spectra depends on the velocity of flux lattice. There is a crossover velocity v~r separating two regimes of interaction between the flowing Abrikosov lattice and the random noise. The crossover velocity is determined by the characteristic frequency of the vortex lattice. For low fluxflow velocities, below the crossover, the resulting power spectrum is suppressed at low frequencies. This effect is analogous to the one predicted by us for the case of a pinned vortex lattice. However, in a marked difference to the motionless-lattice case, for a moving lattice the power spectrum is not suppressed to zero at low frequencies. For the flux flow with velocities above the crossover velocity the power spectrum oscillates with a characteristic frequency corresponding to the time of flight of vortices across the strip width. The oscillating regime should be easily reached experimentally. For a HTSC superconducting thin film strip with a thickness d ~ 1 0 - 4 cm, width W~ 10 - 2 c m and typical viscosity q ~ 10 -8 CGS units and 2 ~ 10- 5 cm, the crossover occurs at a current flow of 1~ 10 -3 A. The characteristic modulation frequency corresponds to a time of flight of the order of z~lO

-6.

V.D. Ashkenazy et al. /Physica C 224 (1994) 377-383

Acknowledgement T h i s w o r k was s u p p o r t e d by t h e M i n i s t r y o f Science a n d T e c h n o l o g y o f Israel a n d by the R a s c h i Foundation.

References [ 1 ] For an excellent review see: L.B. Kiss and P. Svendlingh, IEEE Trans. Electron Devices, to be published, and references therein.

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[2 ] H.K. Ohlsson, R.H. Koch, P.-A. Nilsson and E.A. Stepantsov, IEEE Trans. Appl. Supercond. 3 (1993) 559. [ 3 ] G. Jung, B.Savo and A. Vecchione, Europhys. Lett. 21 ( 1993 ) 947. [4] G. Jung, S. Vitale, J. Konopka and M. Bonaldi, J. Appl. Phys. 70 (1991) 5440. [5] V.D. Ashkenazy, M. Bonaldi, G. Jung, I.B. Khalfin, B.Ya. Shapiro and S. Vitale, Solid State Commun., to be published. [6] J.R. Clem, J. Phys. (Paris) 39 (1978) C6-619. [7] J.R. Clem, Phys. Rep. 75 ( 1981 ) 1. [8] D.J. van Oojen and G.J. van Garp, Phys. Lett. 17 (1965) 230. [9] G. Blatter, M.V. Feigelman, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev. Mod. Phys., to be published.