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PhysicaC 170 (1990) 481-485North-Holland

Dynamics of a permanent magnet levitating above a high-TCsuperconductorV.V. Nemoshkalenko a, E.H. Brandt b, A.A. Kordyuk aand B.G. N&tin aa Inst it ute of M etal Physics, 252680 Ki ev 142, USSRb Max-Planck-Instituttir Festkt ir perfor schung, D- 7000 Stut tgart 80. FRGReceived 31 July 1990

The oscillation modes of a small spherical permanent magnet ( 1.7 mm diameter) above a flat high-T, superconductor areinvestigated. From the attenuation and nonlinearity of the vertical oscillation we obtain the viscosity of the vortex lattice and theonset of hysteretic damping, which indicates depinning of flux lines. The shape of the potential well in which the magnet floats,rotates, and oscillates is found to depend on the magnetic history of the superconductor.

In view of the new physical phenomenon - the freesuspension of a permanent magnet (PM) in a finiteinterval of stability above and below a high-temper-ature superconductor (HTSC) [ l-31, the investi-gation of the macroscopic dynamics of this levita-tion is now being carried out [ 3-5 1. In this paper weinvestigate the free oscillations of a small PM abovea HTSC, in particular, the resonance frequency v.and the damping rate y=Au as a function of the am-plitude of the driving force J: We define Au as thewidth of the resonance curve at the height A,,,/J 2 where A,,, is the maximum amplitude of theforced oscillations of the PM when the frequency isscanned at a given amplitude of the driving forcef:Various technical characteristics of HTSCs are easilyobtained in this way, namely the viscosity q of thevortex motion, the pinning force, and the energy lossW, in the system in an alternating magnetic field.These characteristics are mainly determined by ir-reversible processes during the motion of vortex Ill-aments of magnetic flux in the superconductor,namely, either by the stripping of vortices fromthe pinning centers (this depinning occurs at largeamplitudes A& for the applied AC field and is char-acterized by the area of a hysteresis loop) or, at loweramplitudes, by the viscous flow of the vortex struc-ture (flux flow) or by thermally activated flux mo-tion (flux creep and thermally assisted flux flow [ 6-8] ). Low-amplitude processes occur in the quasi-

elastic sections of the hysteretic magnetizationcurve, where Mx - AB, since the magnetic flux isnearly ideally pinned and thus B x constant. (If de-magnetization effects may be disregarded one hasM=B-B,).

This work proposes a method for studying thecharacteristics of HTSC-materials in an alternatingmagnetic field in both the hysteresis and quasi-elas-tic areas. The investigations were carried out at atemperature of 80 K. Our PM of SmCo, had theshape of a spheroid with excentricity z 0.6 and withmass mx 0.021 g and magnetic moment ,UUX.2 Gcm3. It levitated above the HTSC at a height ofx0x0.3 cm with fi orientated parallel to the HTSCsurface. Our HTSC had the shape of discs with 2 cmdiameter and 0.3 cm thickness made from the castuniphase Y ,B~ ZU~O~.~~with T, z 93 K and AT, z 2K. From X-ray data this Y ,Ba2Cu306.95 has a perfectorthorhombic structure with the parametersao=3.840f0.004 A, bo=3.860k0.004 A,co= 11.669+0.004 A and an orthorhombicity ratiobo/aoz 1.005. The oxygen deficiency S inY,Ba2Cu30,_6 was determined by a mass-spectrom-eter with oxygen-calibrated volume [ 9 1.

Figure 1 shows the scheme of the experiment. Theforced oscillations of the PM with frequency v=o/2x were excited by the AC coil C located above themagnet coaxially with the x-axis. These oscillationsare a sensitive detector of the vortex lattice in the

092 l -4534/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

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482 V. V Nemoshkdenko et al . /D ynami cs of a permanent magnet above a HTSC

Fig. 1. Experimental scheme for the investigation of the free os-cillation of a permanent magnet (PM) freely levitating above anY, BazCu,06.95 high-T, superconductor (HTSC).

HTSC. The oscillating PM is the source of an alter-nating magnetic field which is registered by two coilsC,, C, with an identical number of turns connectedin opposition and positioned at the ends of the coilC coaxially with the x-axis. A double-beam oscillo-graph (OS) shows simultaneously the free oscilla-tion of the magnet and the phase difference Aq be-tween the driving field and this oscillation. Theresonance point Y= v. is defined by the phase shiftAp= KC/~. n our experiment three oscillations modeswith different resonance frequencies were observed:(a) translation along the y-axis at v{ z 17 Hz; (b)translation along the x-axis at uij225 Hz; (c) ro-tation about the y-axis at a frequency of ~70 Hz.These resonance frequencies belong to low ampli-tudes of these in general nonlinear oscillations. Theanalysis of the amplitude-frequency relationshipA(V), for convenience of experimental observationand simplicity of the further considerations, was car-ried out in the vicinity of the frequencies v$ of theoscillations along the vertical x-axis over an ampli-tuderangeAz10-3cmtoA~10-cm.Atlargeam-plitudes the registration of the PM oscillations couldbe carried out both electronically by using an oscil-loscope (OS) and optically by a microscope (M).The microscope allowed to calibrate the oscillo-scope. Small amplitudes were only measuredelectronically.

Figure 2 shows the amplitude A ( vx) of the PMlevitating at the height ~~~0.3 cm above a disc ofY, BazCu306.9, for various values of the external force

EU.

-7x

T

$4 o.15dyn

\

&=o.S Qn4, = 0.6 dyn$4= 1.3.&n

\,= 3.0 dyn

Fig. 2. Several amplitude-frequency characteristics for the trans-lational mode ti of the magnet above a disk of Y, Ba2Cu306,9, atvarious values of the driving force/: The amplitude is obtainedfrom the current in the coil C, fig. 1.

fdetermined by the current in the coil C. In this case(see area A in fig. 3 (I ) ) the damping factor y, de-fined as the width of the resonance curve (fig. 2) atthe height A,,,/$, is practically constant. The mo-tion of the vortex lattice in this region can be de-scribed as motion in a viscous medium by the for-mula F,= -rpVo, where F, is the viscous frictionalforce exerted on a volume V. of the vortex latticemoving with velocity v and q the (volume) viscositycoefficient, which may be estimated as follows. Ac-cording to the definition the quality factor of the os-cillating system can be written as

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V. Nemoshknlenko et al. /D ynami cs of a permanent magnet above a HTSC 483

where W= 4 mA&,, (27~~5)~ is the energy stored inthe PM-HTSC system and W, = $ FJ.x is the energyloss during one period. Averaging over all vorticeswe obtain W,=r,+( v2) Vo/v=2n2qv~a2Vo, where a2is the mean square oscillation amplitude of the vor-tex lattice and V, is the volume of the superconduc-tor in which the main energy dissipation occurs. Dueto the steeply decreasing dipolar magnetic fieldH- 1 x3 the volume V, z xi, and the correspondingsurface area on the HTSC &XX$ [cf. eq. (4) be-low]. Substituting W and W, in eq. ( 1) we obtain

27cmyI=v,crzy (2)where (Y= a/A,,,. Estimating the viscosity coeffi-cient in the area A of fig. 3 by using (2 ) and the ex-perimental data from this work, y= 1.6 Hz, B,x 15G, V,zx; x2.7x lO-2 cm3, ~~~0.3 cm, m=0.021g we obtain qz (800/a) kg s-i mW3. A quantitativeestimation of q is difficult since a is not well known.Note that in this experiment within stable equilib-rium of the PM, a cannot be larger than ao, the maindistance between the vortices, otherwise the vorticesare depinned and the damping is no longer of the vis-cous type but becomes amplitude dependent (hys-teretic damping) [ lo]. At B= 15 G one has ao= 3.9urn whereas A,,, 10 to 30 pm, hence (Y lo6 G is the upper critical field [ 81.The magnified section of the curve ( yversus A,,,)(fig. 3 (II) ) shows that within the area A there oc-curs a small (about 6%) but sharp change of y atfrequencies V$near 24.5 Hz and 23 Hz. As in this casethere are no peculiarities of the e versus A,,, de-pendence, variations in y apparently do not influ-ence the elastic properties of the vortex-pin systembut the depinning process changes slightly at theseamplitudes. This conclusion is supported by the factthat such peaks in y are not observed in the ampli-tude interval A even when this is passed repeatedly.

Taking into account that in the area A in fig. 3-Ione has AB, - 0.1 G < B,,, (B,, is the lower criticalfield) and that after switching off the alternating forcefthe PM returns into its initial equilibrium state, wemay assume that in the area A weak flux creep oc-curs in a quasi-elastic area. This thermally activateddepinning from weak pinning centres results in a de-crease of the averaging pinning force and explainsthe anharmonicity observed in this case in the sys-tem PM-HTSC within the area A of fig. 3 (I ), wherey is constant. The role of of thermal depinning is ex-perimentally supported by additional data obtainedwith powder-compacted samples of Y, Ba2Cu307.0

lt3456759l .A,,,.,~Io~, mFig. 3. (I) The driving forcef; the resonance frequency 6, and the damping factory= Av of the translational mode along the x-axis, fig.I, plotted vs. the resonance amplitude of the oscillations A,= A( ~6). Av is the width of the resonance curve (fig. 2) at I/@ timesthe maximum amplitude&. =A( vi;). (II) The same ~=AL*, fand V$vs. A,,=A( v;) on an enlarged scale.

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484 V. V. Nemoshkalenko et al. /Dynam ics of a permanent magnet above a HTSC(the size of grains in the powder is 3 to 10 urn ) , wherecompared to the cast ones, the relative portion ofweak pinning centres is decreased at the expense ofan increase of strong surface pinning. Here, underanalogous experimental conditions an anharmonic-ity in the system PM-HTSC was not observed up toamplitudes of A,,, N 1Oe2 cm.

With the increase of the amplitude of the drivingforcefin the area B in fig. 3 (I), the damping y beginsto increase and the PM oscillation becomes anhar-manic. In this case a decrease of ti with increasingamplitude A,,, is caused both by stripping off somevortices from the pinning centres, which results in asoftening of the vortex-pin system (similar as in thearea A in fig. 3 (I) ) and by the increase of y accord-ing to the basic relationship v. = ,/m ( vR isthe resonance frequency of frictionless oscillations[ 111). An increase of A,,, enhances the depinning.This conclusion is supported by the fact that an in-crease of A,,, above the values given in fig. 2 (curve5), where the amplitude of changing magnetic fieldis A&x 20 G > B,, results in complete depinning ofthe vortex lattice and the PM occupies a new equi-librium state with height xb > x0 (floating up ofthe magnet ) . The hysteresis found in the system PM-HTSC indicates that the shape of the potential well,unlike with the usual nonlinear oscillator, dependson the magnetic history.

In fig. 4 the energy W stored in the damped os-cillation is plotted as a function off where the forceamplitude f= mA max~O*2ny) [ 111 is determined

I

2

JO2

from the experimental data A,,, wo=2xv,,, and y(fig. 2). Clearly, two types of behavior can bedistinguished:

k,f 2, ffc (hysteretic damping) (3)

corresponding to the areas A and B in fig. 3(I)(k, x 0.4s2/g, k2 x 0.45 cm). Taking into account thatW=f */my2 [ 111, we may conclude that in the areaA one has k, = l/my*=0.4 s2/g and y=const. In thearea B, k,=f/my*=0.45 cm, and it should bey w Jx From the relationship for the loss per unit area(cf. W, from eq. ( 1) with W inserted),W 47c3m . yA, &,,-=SO so

(4)with So z xl, we can estimate the losses WI/So in boththe quasi-elastic (A) and hysteresis (B) regions.Under the experimental conditions v$ N 20 Hz thelosses causes by air resistance may be neglected sincethey are of only a few percent. We obtain for thequasi-elastic or viscous losses (area A) W,/So= 5.3~ lo- J/cm2 and for the hysteretic losses(area B) W,/S,- - 1.34x 10m6 J/cm2. The above con-siderations show that the investigation of the mac-roscopic dynamics of a magnet levitating above atype-II superconductor gives valuable insight intopinning and depinning of the flux lines inside thesuperconductor.

eferences

[ ] V.V. Nemoshkalenko, V.V. Klimenko and B.C. Nikitin,Metallofiyika 11 (1989) 124.E.H. Brandt, Science 243 (1989) 249;ibid., Am. J. Phys. 58 ( 1990);ibid., Appl. Phys. Lett. 53 ( 1988) 1554.V.V. Nemoshkalenko, M.A. Ivanov, B.C. Nikitin, Yu.G.Pogorelov and G.A. Klimenko, Solid State Commun. 7( 1990) 637;V.V. Nemoshkalenko, M.A. Ivanov, B.C. Nikitin,Vyveshennoye sostoyaniye magnita nad i podvysokotemperatumym sverkhprovodnikom (Suspendedstate of a magnet above and below a high-temperaturesuperconductor). Preprint 6.89, Institute of Metal Physics,Kiev, 1989) (in Russian).Fig. 4. The stored energy Wof the oscillations versus the squaredexciting force f 2. [4] AI. Spitsyn and E.A Lichman, Zhum. Tekhn. Fiz. 59 ( 1989)193 (in Russian).

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[ 5 ] F.C. Moon, M.M. Yanoviak and R. Ware, Appl. Phys. Lett.52 (1988) 1534.

[6] D. Dew-Hughes, Cryogenics 29 ( 1988) 674.[7 ] P.H. Kes, J. Aarts, J. van den Berg, C.J. van der Beek andJ.A. Mydosh, Supercond. Sci. Technol. 1 ( 1989) 242.

[ 81 E.H. Brand& 2. Physik B 80 (1990) 167.[9] V.V. Nemoshkalenko, Y.V. Komyushin, N.S. Kobzenko,B.G. Nikitin et al., Ukr. Fiz. Zhum., 34 (1989) 1367 (in

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[lo] E.H. Brandt, P. Esquinazi and H. Neckel, J. Low Temp.Phys. 64 (1986) 187;E.H. Brandt, J. de Physique Colloque C8 ( 1987) 3 I.[ I1 ] L.D. Landau and E.M. Lifshits, Mechanics (Course ofTheoretical Physics, vol. I), (Pergamon, Oxford, 1966) p.58.