ELSEVIER Physica C 275 (1997) 322-326

PHYSICA

Magnetic levitation of thin superconducting disks

Alvaro Sanchez *, Caries Navau Grup d'Electromagnetisme, Departament de F[sica, Universitat Autbnoma Barcelona, 08193 Bellaterra (Barcelona), Spain

Received 20 September 1996; revised manuscript received 17 December 1996

Abstract

We derive the analytical expressions describing the magnetic force experienced by a thin superconducting disk when being displaced in the presence of the magnetic field of a permanent magnet. The important demagnetization effects associated with the film geometry are taken into account. The model explains the characteristic features observed in the experiments. In particular, the results for the levitation force and its dependence on the film thickness are compared with published experimental data, showing good agreement.

Keywords: Levitation; Applications of high-T c superconductors; Magnetization

There is recently much interest for understanding the magnetic properties of thin superconducting films, disks in particular. When subjected to transverse fields, the magnetic properties of thin superconduct- ing disks are different from those for samples with smaller aspect ratio. The works of Mikheenko and Kuzovlev [1], Zhu et al. [2], and Clem and Sanchez [3] have introduced a theoretical background that satisfactorily describes static (or low-frequency) magnetic properties of thin disks. The results of these models have been confirmed by several experi- mental works [4,5].

On the other hand, magnetic levitation of super- conductors is becoming an important topic of re- search. Although most of the works until now have been carried out for long samples, it is clear that samples with thin film geometry have an enormous

" Corresponding author. Fax: + 34 3 581 1350; e-mail: alvar@elema.uab.es.

potential in this field. For example, it was shown in Ref. [6] that for thin films a very small amount of superconductor can produce levitation forces compa- rable to a bulk conductor 10 2 times its thickness. Some levitation experiments for thin films have been performed. The measurements show that the general behavior (i.e., hysteresis in the levitation force be- tween ascending and descending curves) is similar to that for long samples. However, some unique impor- tant features appear in these experiments for thin films, which are presently not well understood: typi- cal measurements [6,8,9] give, for example, a sus- pension effect almost as large as the repulsion effect, while in the case of bulk samples the suspension force is often much smaller than the repulsion force. A closely related effect, the fact that the slope of the force versus distance curve when reversing the sense of motion of the superconductors is much larger for films than for bulk samples, has been also pointed out by several groups [7,8]. Sch/Snhuber and Moon [7] have introduced a model based on the force

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A. Sanchez, C. Navau / Physica C 275 (1997) 322-326 323

balance between fluxoids and the critical-state model that satisfactorily describes the general properties. However, as noted by these authors, the unique features observed for thin films arise from the impor- tant demagnetization effects associated to the shape of the films, which were not taken into account.

In this work, we derive the analytical expressions describing the levitation force experienced by a thin disk in the presence of a magnetic field created by a permanent magnet, incorporating the demagnetiza- tion effects on the superconductor. The results are general and can be easily adapted to other configura- tions. The model will enable both to qualitatively understand the features observed in the experiments for levitating thin disks and to quantitatively fit the experimental levitation force curves. Moreover, other issues such as the dependence of the vertical levita- tion force upon the thin disk thickness will be ana- lyzed in terms of the model, showing again a good agreement with the experiments.

We consider a type-II superconducting thin disk with thickness L and radius R (L << R) which has been zero-field cooled and is displaced over a cylin- drical permanent magnet along their common axis, in the z direction. We use cylindrical coordinates (r,th,z), with the origin located at the center of the upper side of the magnet. We consider in this work only quasi-static displacements disregarding the dy- namics associated with flux motion in the sample. The superconductor is assumed to be always in a critical state, with a constant critical current density

Jc. It can be demonstrated that the main equation for

describing the levitation force over a cylinder with thickness L and radius R in which there are az- imuthal currents circulating with a density J , in the presence of a vertical magnetic field Hz, is

where z is the distance between the origin and the lower side of the cylinder.

Eq. (1) is valid as long as the applied magnetic field is vertical and has a constant value all over a sample cross section, that is, if He× t = H z ( z , r = 0)~

for a given height z. In our case, this assumption is easily met in practice provided that the ratio between

the radii of the magnet and the superconductor is not too small [10].

In this work, we consider a permanent magnet with uniform magnetization .~v in the z direction; the z component of the field it creates at the axis is

-gg z + b z //-'=Y a fg ?-z ,)

where a and b are the magnet radius and thickness, respectively.

In the case of thin films Eq. (1) adopts the simpler expression

aH z F. = txoqrR2LM, ., , (3)

" o Z

since we assumed L << R and therefore the variation of the magnetic field created by the magnet over the thickness of the superconductor can be neglected. M z

is the magnetization induced in the superconductor. It is useful to work with non-dimensional quanti-

ties, which will allow a general description with a minimum number of parameters. For this purpose, we define

H, M: h = ----=" m

H o ' X0 Hdqr /4 '

Z F z (4) ~ ' = - ~ , f =

tx o x o H 2 RZ~'2 / 4 "

The field H d equals J c L / 2 ; it corresponds to the characteristic field at which the critical-state region (defined as that for which the supercurrent density J equals Jc) has penetrated a significant distance to- wards the center of the sample [3]. X0 is equal to 8 R / 3 7 r L and corresponds to the external magnetic susceptibility in the case of perfect screening [3]. M z

is normalized to the saturation magnetization (Msa t = ( T r X o / 4 ) H d = J c R / 3 for thin disks [3]). The nor- malization factor for F z corresponds to the force created by a field gradient H d / L over a thin super- conducting disk fully penetrated by supercurrents flowing with a density of Jc.

Eq. (3) then becomes

Oh f = mo--- ~ • (51

324 A. Sanchez, C. Navau / Physica C 275 (1997 ~ 322-326

The magnetization of thin disks has been calcu- lated in Refs. [1-3], assuming that Jc is independent of the magnetic field. Applying the results to our case we find that the normalized force on the super- conductor when descending from very far towards the permanent magnet (which corresponds to an increasing of the applied field) is

4 Oh f = - -~hS(h) -~z' (6)

where the function S(x) is defined as in Refs. [1-3] as

S(x ) = cos -l ~ + cosh2x 1. (7)

This force arises from the supercurrents induced in the superconductor due to the variation of the applied field when descending towards the magnet. Once the superconductor has been lowered down to a vertical position Zm~n with a corresponding mag- netic field hma×, when reversing its motion, super- currents with opposite sign are induced in the super- conductor giving rise to a (normalized) force

+ ( h r n a x - h ) s ( h m a 2 - h ) ] Oh 0 . (8)

In Fig. 1 we plot the results for the normalized force as a function of the normalized vertical posi- tion for the ascending and descending curve, ob- tained from Eqs. (6)-(8). It is interesting to distin- guish three characteristic cases depending on the ratio HmaJH d to be greater, smaller, or equal than 1. For all cases there is a hysteresis in the force when descending and then rising the superconductor. The hysteresis in the force gets larger with increasing H m a x / n d , because when Hma x < H d the magnetic field is not large enough for complete (or almost complete [3]) penetration of the critical-state region, and therefore the hysteresis, which results from vor- tex pinning in these current-penetrated zones, is small.

The three mentioned cases can be found in gen- eral, although a more careful inspection shows that in actual experiments the c a s e Hma x > H d is the one almost always met for thin films. This can be

B

5

4

3

c ~ 2

E- l

0

- 1

i i 1 t

- - - -Hd=2H m

Hd=H m

Hd=Hrn/2

" " , " .

20100 40100 60;0 I 10000 8000

z/L

Fig. 1. Normalized vertical force f as function of normalized vertical distance ~', for the cases H a = 2Hma x (solid line), Hj = Hma x (dashed line), and H a = Hma X/2 (dotted line); the descend- ing branch corresponds, for each case, to the entirely positive curve.

demonstrated by realizing that a typical thin film may have a thickness of several ixm, with a critical current density at 77 K (when experiments are most often performed) of typically of the order of 109 A / m 2, which gives a value for the characteristic field H d of the order of 103 A / m . Taking into account that typical values of a magnetic field cre- ated by a conventional powerful permanent magnet (such as F e - N d - B ) are of the order of 105 A / m , we see that Hma x is in most cases two or more orders of magnitude larger than H d. In this case (Hma ~ :~ Hd), the descending and ascending branches are almost symmetric, which is what is found in the experi- ments [6-8]. We next discuss a particular example in more detail in order to illustrate all these features.

In Ref. [8], Weinberger measured the levitation force of both a stacked array of Y - B a - C u - O thin films and a melt processed bulk Y - B a - C u - O sam- ple at 77 K, when they were lowered in the field of identical permanent magnets. Both the repulsive force branch (when descending the sample) and the attrac- tive branch (when the motion is reversed) were measured. The main difference between films and bulk samples was shown to be that the attractive force curve for the film was larger in magnitude, relatively to the repulsive force, making in this case the complete force curve almost symmetric when plotting the force versus the vertical distance. Fig. 2 shows Weinberger's experimental data points to- gether with a fit using Eqs. (6) and (8) with the

A. Sanchez, C. Navau / Physica C 275 (1997) 322-326 325

, i I 0 I 0 I 0 2 30 40

z (mm)

Fig. 2. Vertical force F as function of vertical distance z for a stack of Y-Ba-Cu-O films at 77 K, after Ref. [8]. Points are experimental data and the solid line is the theoretical fit.

6000

I 4000 i

2000

0

- 2000

-4000

values of parameters (magnet and superconductor dimensions) given in Ref. [8]. The value of Jc (not given in Ref. [8]) was our fitting parameter. We obtain a good adjust with a value Jc = 6.4 × 108 A / m 2, which is among the typical values for Y - B a - C u - O films at 77 K. By inputting this and the value of L of Ref. [8] into the expression for H a, we find that H a = 875 A / m in our case. Since it is said in Ref. [8] that the maximum field (at the surface of the magne0 is n m a x = 2 X 105 A / m , we are clearly in the situation Hma , >> H a. Therefore the critical- state region with J = Jc starts filling completely the whole superconductor at a small field (large dis- tance) when descending the sample. When reversing the motion, supercurrents induced with opposite sign fill completely the sample as soon as the supercon- ductor reaches an applied field Hma x - H a = Hmax ,

which explains the almost vertical slope of the attrac- tive curve in both Weinberger 's data and our model fitting. The overall agreement between experimental and fitting curve is rather satisfactory, although the thin films of Ref. [8] were not disks. The distinctive features observed for films (large attractive force and large slope at the reversing point) are well depicted by the model. The discrepancies between experimen- tal data and the model fit are slightly larger in the central region of both curves. This is understandable by realizing that in these regions, as mentioned above, the sample is practically fully penetrated by supercurrents and therefore, after Eq. (5), since the magnetization is saturated, the force is simply pro-

portional to the derivative of the applied field, which is modeled in an only approximated way [Eq. (2)].

We can further check our model by calculating the dependence of the levitation force upon the thickness of the superconductor. For the most com- mon case of Hma x > > H d this dependence has a simple form. From Eqs. (4) and (5), we find that when the superconductor is at a fixed height z (which corresponds to fixed field and gradient), the force results simply proportional to the thickness L, since the magnetization has a constant value ( M s a t =

----- 3Jc R). Weinberger measured the levitation force for several thin films with different thicknesses and at several heights [8]. When plotted in a linear graph (in Ref. [8] the plot was semi-logarithmic), his re- suits clearly agree with the predicted linear behavior, as can be seen in Fig. 3, where we have reproduced his experimental data together with straight lines obtained from our calculations. The values of Jc used in the fits coincide quite well with the value Jc = 6.4 X 108 A / m 2 used above; the discrepancies can be attributed to slightly different values of Jc for the different films forming the stacks. It is worth noticing that, although the dependence of the levita- tion force upon film thickness has a more complicate expression for the other cases (for example, when Hma ~ = H0), a general relation, valid for all cases, can be found out accurately from Eqs. (4), (6) and (7).

6000 I I I I

5000

4000

3oo0 U . 2 0 0 0

1 0 0 0

0 , I I I I i i

0,0 0,5 1,0 1,5 2.0 2,5 3,0

t (~m)

Fig. 3. Dependence of the force F on the total film thickness t for different stacks of Y-Ba-Cu-O films at 77 K, after Ref. [8]. The different curves correspond to different values of the applied field (from up to down): 1100, 815, 540, 370, 235 and 110 Gauss. Symbols are experimental data and lines are the theoretical fits.

326 A. Sanchez, C. Navau / Physica C 275 (1997) 322-326

In summary, we have introduced a model to calculate the levitation force experienced by a super- conducting thin disk when displaced along the axis of a cylindrical permanent magnet. The model satis- factorily explains the main characteristic features observed in the experiments. The levitation force as function of the superconducting position for both ascending and descending curves has been calculated and compared to experimentally measured data, showing a good agreement. The model also accu- rately describe the dependence of the levitation force upon the film thickness. Most of the previous ap- proaches to model levitation force have assumed that a constant magnetization can be considered for the whole superconductor. This is only a rough approxi- mation when the superconductor is a bulk sample, since the external magnetic field varies along the thickness. In the present case of thin films, assuming a mean magnetization for the whole sample is quite reasonable, because of their shape. Moreover, our model takes into account the important demagnetiza- tion effects. Therefore, this geometry provides an ideal test for comparing the theoretical models for levitation with the experimental data. The results presented here confirm that the recent theoretical advances in the understanding of the physics of superconducting films are indeed sound.

We thank John R. Clem for helpful discussions. Financial support from Spanish DGICYT (project number PB93-1249) and from Catalan CIRIT (pro- ject number 1995SGR 00039) is acknowledged. C.N. acknowledges a grant from CUR (Generalitat de Catalunya).

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Phys. Rev. B 49 (1994) 9222. [5] A.V. Kuznetsov, A.A. Ivanov, D.V. Emerenko and V.N.

Trofimov, Phys. Rev. B 52 (1995) 9637. [6] F.C. Moon, Superconducting Levitation (Wiley, New York,

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