Table Of ContentVortex motion in superconducting YBa Cu O inferred from the damping of the
2 3 7−δ
oscillations of a levitating magnetic microsphere.
R. Grosser, P. Ho¨cherl, A. Martin, M. Niemetz, and W. Schoepe
Institut fu¨r Experimentelle und Angewandte Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
9 (February 1, 2008)
9
The damping of the oscillations of a small permanent magnet (spherical shape, radius 0.1 mm)
9
levitating between two parallel YBCO surfaces is measured as a function of oscillation amplitude
1
and temperature. The losses in the samples (epitaxial thin films, bulk granular and bulk melt-
n
textured) are analyzed in terms of oscillating shielding currents flowing through trapped flux lines
a
whosemotiongivesrisetoelectricfields. Wefinddissipationtooriginatefromdifferentmechanisms
J
of flux dynamics. At small amplitudes there is a linear regime described by a surface resistance
1 varying from 10−9Ω for the bulk samples down to 10−13Ω for the thin films at low temperatures.
1
With increasing amplitude various nonlinear regimes are observed, firstly collective pinning with
diverging energy barriers, secondly in bulk samples above 85K hysteretic damping, and finally in
]
n thin films exponentially large losses which can be described by pinning energies vanishing linearly
o at large currents.
c
-
PACS numbers: 74.60.Ge;74.76.Bz;74.80.Bj.
r
p
u
s I. INTRODUCTION further increase of the drive and at temperatures above
.
t 85K we find that the dissipated power in the bulk sam-
a
ples varies with the cube of the amplitude. We discuss
m
A permanent magnet levitating above, or suspending this regime in terms of hysteretic dissipation having a
d- below, a superconducting surface has an equilibrium po- coefficient which divergesnear Tc. In the films, however,
n sition which is determined by magnetic forces due to aquitedifferentbehaviorisobservedinstead,namely,an
o screening currents and trapped flux.1 Except for a small exponential increase of the dissipation which ultimately
c logarithmic long-termrelaxationof the magnetizationof exceeds even the bulk values. In this regime we can de-
[
the superconductorthis positionis stable. The systemis scribe the data by a Kim-Anderson type model of flux
1 in a metastable state with stationary flux lines and con- motion with linearly vanishing pinning energies at large
v sequentlywithoutanydecayofthedcsupercurrents,i.e., currents. In summary, by varying the drive of the oscil-
5 without any energy dissipation. However, oscillations of lating magnet by five orders of magnitude we can study
8 the levitatingmagnetaboutits equilibriumpositiongive differentregimesofvortexdynamicsinYBCOatagiven
0
rise to ac magnetic fields at the surface of the supercon- temperature. Someofourpreliminaryresultsongranular
1
ductorwhichmodulatetheshieldingcurrents. Fluxlines samples2 andonthin films3 havealreadybeen published
0
9 are now moved back and forth periodically as a result inshortform. Inthisarticleamoredetailedanalysisand
9 of the oscillating component of the Lorentz force. This a more complete account of our work is presented.
/ leads to energy dissipationwhichwe candetermine from This article is organizedas follows. In the next Chap-
t
a measurementsofthe oscillationamplitude ofthe magnet ter we describe our experimental method, and in Ch.III
m as a function of an externaldriving force. The measured themodelusedfordataanalysisisoutlined. InCh.IVwe
- lossescanbeattributedtovariousregimesoffluxmotion present the results of our experiments for the thin films
d
in the superconductor. andforthebulksamples. ChapterVconcludesourwork.
n
In our present work we have investigated the energy
o
c dissipation when the magnet is levitating between two
v: horizontalYBa2Cu3O7−δ (YBCO)surfaceseitherofbulk II. EXPERIMENTAL METHOD
material(bothsinteredandmelt-textured)andofepitax-
i
X ial thin films. In all samples and at all temperatures we Our experimental method consists of placing a mag-
r findthatatsmallamplitudesofthe acshieldingcurrents netic microsphere made of SmCo5 (radius 0.1mm, mass
a ( 10A/m)thedissipationcanbedescribedbyasurface m=5.2 10−8kg)betweentwohorizontalsurfaces(spac-
r≃esistance which varies from 10−9Ω for the bulk sam- ing 1mm·, diameter 4mm) of a parallel-plate capacitor
ples down to below 10−12Ω for the thin films at 4K. At made of YBCO. This is an extension of earlier work,
larger amplitudes nonlinear behavior is observed which where the capacitor was made of niobium4 and the lev-
can be attributed to thermally activated flux flow with itating sphere was used for hydrodynamic studies in su-
pinning energies whose diverging current dependence is perfluid helium.5 In Fig. 1 a schematic of our electronics
typical for collective pinning. In addition, for a given is shown. Before the capacitor is cooled through T we
c
current, the pinning energies have a linear temperature apply a dc voltage of about 800V to the bottom elec-
dependence which is an indication of flux creep. Upon trode. Therefore, the magnet carries an electric charge
1
q of about 2pC when levitating. The dc voltage is then a = v/ω from 50nm to 30µm. The dissipated power
switchedoffandoscillationsofthemagnetcanbeexcited varies from 10−9W down to below 10−17W.
with an ac voltage Uac ranging typically from 0.1mV to The YBCO we used for the superconducting capaci-
20V and having a frequency at the resonance of the os- tor was either home-made sintered material2 (thickness
cillations ( 300Hz). These vertical oscillations induce 2mm) or melt-textured samples6 of the same geome-
a current q≈v/d in the electrodes, where v is the veloc- try, and epitaxial thin films3 having a thickness of ei-
ity of the magnet, which is measuredby an electrometer ther 450nm or 190nm laser deposited oninsulating sub-
connected to a lock-in amplifier. The capacitively cou- strates of SrTiO3 (lower electrode) and Y2O3-stabilized
pledsignalduetothedrivingvoltageUac whichissuper- ZrO2 (upper electrode) both cut with an inclination of
imposed upon the signal from the oscillating magnet is 2 degrees off the (001) orientation. The inclination im-
electronicallynulledoffresonancebyaddingtheinverted proves both flux pinning and, due to initiating a terrace
pickup of appropriate amplitude. For a given amplitude growth, crystallinity of the films. The lower electrode is
of the driving force F = qUac/d we measure the maxi- protected by a 20 nm thick PrBa2Cu3O7 epitaxial layer.
mumsignalwhenslowlysweepingthe frequencytowards TheYBCOfilmshaveanextremelysharpsuperconduct-
resonance. Because at large oscillation amplitudes the ing transition at T = 89.5 K measured by an ac sus-
c
restoring forces become nonlinear, the resonance curves ceptibility method, the complete transition width being
areusuallyhystereticandcarehastobetakentoachieve 0.1 K. Further details of the thin film production can be
resonance. We therefore also monitor the phase of the found in Ref. 7.
signalwithrespectto thatofthe drive. Inthe Appendix
ashortdiscussionofsomeoftheelasticpropertiesofthis
oscillator is presented. Further data are taken either by
III. DIPOLE MODEL
varying the driving force at constant temperature or by
sweeping the temperature at a fixed driving force while
keeping the oscillator at resonance by adjusting the fre- The oscillations of the magnet give rise to ac surface
quency of the drive. currents which exert an oscillating Lorentz force on the
flux lines. This causes energy dissipation which can be
determined from the damping of the oscillations either
by measuring the amplitude as a function of the driving
force or by recording a free decay. In this section a brief
discussion of a model used for a quantitative analysis of
our data is given.
FIG. 1. Schematic of the experimental setup. The elec-
trically chargedmagneticmicrosphereislevitatinginsidethe
superconducting YBCO capacitor.
FIG.2. Geometry of the dipole moment above the super-
conducting surface and its image below.
In order to infer the velocity amplitude from the mea-
sured current amplitude and the driving force from the
acvoltagewedeterminethechargeqbyrecordingareso- In our experiment the superconductors are cooled in
nance curve at very small drive (which is not hysteretic) the field of the magnet and therefore flux is trapped in
and by analyzingits width (which is given by the damp- the superconductingelectrodes. We estimate thedensity
ing coefficient γ) and its height which then determines of the trapped flux to be nΦ0 1mT corresponding to
the charge. Alternatively, in particular at low damp- a flux line density n 5 101∼1m−2. After cooling we
∼ ·
ing, we measure the time constant τ of the freely decay- find that the magnet always levitates near the middle
ing oscillations from which we obtain γ = 2m/τ. Com- between the superconductors, i.e., at a levitation height
paring this result with the driven response at the same h0 = 0.5mm.9 In this position the field of the magnet
temperature gives the charge. The driving forces range at the surface of the superconductors is below Bc1 and
−13 −7
from 10 N to 10 N, the velocity amplitudes from for a quantitative analysis we approximate the surface
0.1 mm/s to 50 mm/s, and the oscillation amplitudes shielding currents induced by the magnet by an image
2
dipole model. In this model the magnetic field on either density j =∆J/δ inside the samples, where δ is either
h i
surface is given by the dipolar fields of the magnet and the penetrationdepth incaseofbulk samplesorthe film
the first image dipole (Fig. 2), neglecting images further thickness. Wemaythendescribethedissipationperunit
away and also modifications of the field due to trapped area by ∆J E, where E(j) depends on j .
· h i
flux lines. The magnetic moment of the magnet calcu-
lated from the magnetic remanence of the material and
the radius is M0 = 4 10−6Am2. It is simple to calcu- a)
late the dc sheet curre·nt on the surface for a particular α=0°
orientation of the dipole with respect to the surface.10 0.5
ThedccurrentdistributionJ~=(J (x,y,h),J (x,y,h))
x y
(x and y are coordinates on the surface) is given by
J = 2H and J =2H with
x y y x
−
3M0
H = (xycosα hysinα), (1)
x 4πr5 −
3M0 2 r2
Hy = 4πr5 y − 3 cosα−hysinα (2) y/h 0
(cid:26)(cid:18) (cid:19) (cid:27)
where r2 = x2 + y2 + h2. The parallel orientation of
the dipole (α = 0) has the lowest energy but because
of trapped flux we cannot rule out that some other ori-
entation applies. Although the variation of the surface
currentisquitecomplexandchangesdrasticallywiththe
10
orientation of the dipole , the maximum value Jmax is
fairly insensitive to the orientation: Jmax = 4400A/m −0.5
−0.5 0 0.5
when the dipole is perpendicular and Jmax = 5100A/m x/h
for the parallel orientation at h = h0. In either case
b)
the magnetic field is below Hc1, but some flux remains
trapped because of the field-cooled situation. α=90°
The ac sheet current ∆J~ due to the oscillations of the 0.5
magnet about its position at h0 is
∂J ∂J
∆J~(x,y,t)= x , y acos(ωt), (3)
∂h ∂h
(cid:18) (cid:19)h0 (cid:18) (cid:19)h0!
where a is the oscillation amplitude. Again the cur-
rent distribution (shown in Fig. 3) is complicated but
the maximum amplitude of ∆J is insensitive to the ori- y/h 0
entation of the dipole. For an oscillation amplitude
a = 0.5µm corresponding to a velocity amplitude of
v 1mm/s(at300Hzoscillationfrequency)amaximum
≈
current amplitude of 14A/m is found for the perpendic-
ular orientation and 15A/m for the parallel one. The
current is localized in a small area of the superconduct-
ing surfaces below and above the sphere, i.e., dissipation
−0.5
occurs only here.
−0.5 0 0.5
This ac sheet current is actually an oscillating current x/h
density j in the samples. Flux lines will be set into mo-
FIG.3. Vector plots of the ac surface currents calculated
tion thereby creating an electric field E =vfnΦ0, where from Eq. 3: a) for parallel and b) for perpendicular orienta-
v is the flux line velocity. Because j and E are par-
f tion of thedipole. In our case h=0.5mm and therefore the
allel the dissipation per unit volume is given by j E. displayed surface area is 0.5×0.5mm2 each.
·
In bulk samples the shielding current is flowing within
the London penetration depth where it decays exponen-
tially, whereas in thin films the penetration depth can We calculate the stationary velocity amplitude of the
exceed the sample thickness. (Note that the Campbell magnet v(F) = ωa(F) by employing the energy balance
penetration depth is an order of magnitude smaller be- between the gain from the driving force (which at reso-
cause of the low trapped flux density and therefore can nanceisinphasewiththevelocity)andthelosspercycle
be neglected.) For simplicity, we use an average current (period τ) on both surfaces:
3
τ τ
2 Fvπ Here we assume that Rs is a constant on the surface of
Fv sin (ωt)dt= =2 ∆J Edxdydt (4) both superconductors. The surface integral S is evalu-
ω ·
Z0 Z0 xZ,y ated numerically. In Fig. 6 the Rs values are plotted on
therightscalefortheparallelorientationofthedipole.11
Because∆J aitisevidentfromEq.4thatalinearde-
∝
pendence of E(j) corresponds to a linear response v(F) 60
while a nonlinear response implies a nonlinear current- 81K
voltage relation. Secondly, it is obvious that the driving 50
82K
force is a measure of the electric field, i.e., of the vortex
s]
velocity. In the following chapter we present our experi- m/ 40
m
mental results and analyze them on the basis of Eq. 4. e [ 83K
plitud 30 bulk 85K
m 84.5K
a
IV. RESULTS AND DISCUSSION ocity 20 85.5K
el
v 10
We first describe and discuss the data obtained with
the thin film samples whicharevery similarfor both the 0
190nm and the 450nm thick films. Then, the measure- 0 10 20 30 40 50 60
ments on the bulk samples are presented which are very driving force amplitude [nN]
similar for both granular and melt-textured YBCO. FIG. 4. Resonant velocity amplitude of the oscillating
magnet as a function of the driving force at various temper-
atures. Note the steep increase at small driving forces and a
logarithmic dependence at large drives. The lines are fits of
A. Thin films
ourmodeltothedata,seetext. Analmostlineardependence
observed with melt-textured bulk samples (from Fig. 11) is
Asummaryofourdataobtainedwiththe450nmfilms shown for comparison.
above77KisshowninFig.4. Atsmalldrivingforceswe
find a steep initial increase of the velocity amplitude of
the magnet, whereas at driving forces above ca. 10nN 40
there is only a logarithmic dependence. This implies an 4K 20K 30K 45K
exponentialincreaseofthedissipation. Atlowertemper-
aturesthe velocityamplitudes increaseveryrapidlywith s] 30
driveasshowninFig.5withalargelyexpandedabscissa. mm/
Further increase of the driving force amplitude gave un- e [ 60K
d
stable oscillations. Only at very small amplitudes we plitu 20
do find a linear regime between the driving force ampli- m
a
tude F and the velocity amplitude v, i.e., F = γv with y
cit
a strongly temperature dependent coefficient γ(T), see o 10
el
Fig. 6. While γ decreasesconsiderablybelow the critical v
temperature T , it obviously levels off at 4K. Here the
c 81K
Q-factor mω/γ which we attribute exclusively to losses 0
8 6 0 5 10 15
in the superconductor exceeds 10 . The steep drop be-
low T follows approximatelya (T T)−2 law as can be driving force amplitude [pN]
c c
seen in the insert of Fig. 6. − FIG.5. Velocityamplitudeatverysmalldrivingforcesin-
dicating the linear initial increase at various temperatures.
In the linear regime F = γv the induced electric field
The lines are fitsof ourmodel tothe data, see text.
inthesuperconductorsisproportionaltothesurfacecur-
rentamplitude,E =R ∆J,andthe loss(r.h.s. ofEq.4)
s
can be described by a surface resistance R γ, namely Fromthesurfaceresistancewecaninfertheimaginary
s
∝
part of the penetration depth of the ac field being given
γω2 by λ′′ = Rs/µ0ω. In our case (ω/2π 300 Hz) we find
Rs = , (5) λ′′ to drop from 84 nm at 85 K dow≈n to only 0.2 nm
S
at 4 K. A steep decrease of the surface resistance with
where temperature has been observed at radio and microwave
frequencies12 and is discussed in terms of various mod-
2
els of flux dynamics. In the limit of very low frequencies
∂J ∂J
S =2 x , y dxdy. (6) it has been calculated13 that the complex resistivity ρ
xZ,y (cid:12)(cid:12) (cid:18) ∂h (cid:19)h0 (cid:18) ∂h (cid:19)h0!(cid:12)(cid:12) due to vortex motion is greatly altered when flux creepv
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
is included. Instead of a purely imaginary resistivity the Eq. 4 must have a nonlinear current dependence. This
result has a real part given by Re(ρ ) = ρ ǫ where is usually described by thermally activated flux motion
v f
·
ρf = BΦ0/η is the usual resistivity for free vortex flow E(j) exp( U(j)/kT) with current dependent pinning
(η is the vortex viscosity)and ǫ(ν)=I−2(ν) 1, where energi∝es U(j−), for a review see Ref. 16. Essentially two
0
≪
I0 is a modified Besselfunction and ν =U/2kT depends different dependences are being considered, firstly barri-
on the pinning energy U(T). There is ample evidence14 ers diverging for j 0:
→
thatbelow70Kν isatemperatureindependentquantity µ
j0
rangingfrom3to9. Infact,forν =9.5ourresultswould U(j)=U0 , (7)
j
agree very well with Rs = Re(ρv)/t (t is the film thick- (cid:18) (cid:19)
ness)ifatrappedflux densityof1mT andtypicalvalues where 0<µ<1. Vortex glass and collective creep mod-
for η(T) are assumed, see Ref. 14 for a compilation of els yield this behavior which implies zero dissipation in
viscositydata. This value for ν appears quite reasonable the limit j 0. Secondly, for large currents, one con-
→
because,aswillbeshownbelow,inthenonlinearregime, siders barriers vanishing at a critical current density j
c
where ν begins to drop with increasing current, we find as
ν = 8 at a current amplitude of 50A/m. It should be j
mentioned that recent work on the surface impedance of U(j)=U0 1 , (8)
− j
conventional superconductors15 has shown that a more (cid:18) c(cid:19)
complicated electrodynamical description is required. If which leads to an exponential dissipation for large cur-
this theoryweredirectlyappliedto ourexperiment(thin rents (Kim-Anderson model), E sinh(j/j1) with j1 =
∝
films, low frequency, low vortex density) one would ob- jckT/U0.
−2
tainthesimple relationR B (T)whichwouldagree
s ∝ c1
both in temperature dependence (including the region 60
near Tc) and in order of magnitude with our data. How- 450 nm YBCO 2
ever,thisremainstobejustifiedandthereforeatpresent 50 1
81K
is only a speculation.
40 82K
3
10-7 10-9 m/s] 30 83K
m
slope=-2 v [
10-8 10-10 20 84.5K
10 85.5K
γ [kg/s]10-9 10 (Tc-T) [K] 100 10-11ΩR []s 0 0.01 0.1 1 10
F [nN]
10-10 10-12
FIG. 7. Data of Fig. 4 plotted on a logarithmic scale for
thedrivingforce. ThesolidlinesarefitsofEq.4tothedata,
see text. Forthe 81K curvethecontributions of threediffer-
10-11 10-13 ent mechanisms of dissipation are indicated separately: 1 is
0 10 20 30 40 50 60 70 80 90
the linear one (which is an exponential curve because of the
T [K]
logarithmicabscissa), 2isthecontributionfromEq.7,and3
FIG.6. Linearcoefficientγ =F/v fromFig.5andsurface
is from Eq. 8.
resistance Rs from Eq. 5. Open symbols are data obtained
with the thinner film δ =190nm. The solid line is the same
as in the insert where thedata are plotted vsTc−T. Wefitthecompletev(F)curvesbyincludingbothnon-
linear mechanisms (Eqs. 7 and 8) and a linear term ad-
ditively in the energy balance, see Eq. 4. An example
In Fig. 7 the data of Fig. 4 are plotted on a logarith- is shown in Fig. 7, where the separate contributions and
mic scale for the driving force F. Above 10nN we find theirsumarecomparedwithanexperimentalcurve. The
a logarithmic dependence v ln(F/F0) which implies linear term (surface resistance) contributes only at very
∝
an exponential increase of the losses. For driving forces low amplitudes. In the intermediate range the glass (or
between 0.1nN and 10nN the amplitudes appear to be creep) term (Eq. 7 with µ = 0.2) is dominant. Only at
reduced indicating an additional loss mechanism in this the largest drives the Kim-Anderson term (Eq. 8) be-
regime. Furthermore,itisnoteworthythatobviouslythe comes relevant. It is this mechanism which leads to the
drivingforceandhencetheelectricfield,orvortexveloc- observedslow logarithmic increase of the oscillation am-
ity, determines which mechanism dominates and not the plitudes. From these fits we can determine the barriers
oscillation amplitude, or shielding current. U(j)atvarioustemperatures,seeFig.8. Itisapeculiar-
In the nonlinear regime of v(F) the electric field in ityofEq.8thatonlytheconstantslope dU/dj =U0/jc
−
5
can be obtained reliably from a fit to the data because The values for ν are determinedby the logarithmofvar-
the quantity U0 determines only the prefactor which we ious properties of the pinning centers and of the inverse
donotevaluateforitcontainsotherunknownquantities, frequency. For this reasonour values obtained at 300Hz
e.g. the density of the trapped flux n or the prefactor arelargerthanthoseobtainedatmicrowavefrequencies12
of the vortex velocity v . From Fig. 8 it is evident that but lower than those obtained from magnetic relaxation
f
at large currents and high temperatures the barriers of measurements. Approaching T the effective barrier fi-
c
Eq. 8 ultimately drop below those of Eq. 7. For the ex- nally decreases (see Fig. 8) because the quantity j0 in
ponent µ determining the barrierswe find µ=0.2which Eq. 7 which is related to the critical current density16
is closeto the theoreticalvalue 1/7 for single vortexpin- goes to zero.
ning which is resonable for the small vortex densities in
our experiment.17 We should mention that the exponen-
tial fitting parameters are not affected by the particular
∆J = 50 A/m
currentdistributionused(parallelorperpendicularorien-
2 ∆J = 500 A/m
tationofthedipole),onlyprefactorsaredifferentbecause
the paralleldipole leads to a lowertotaldissipationthan
the perpendicular one. K]
30
1
4 T) [ 1
U(
3
0
30 K]2 0 20 40 60 80
U [1 T [K]
FIG. 9. Temperature dependence of the effective barriers
1 (Eq. 7) for two different current amplitudes. Below 60K the
dependenceislinearforbothcurrentswithslopes10and16,
respectively. The solid lines are merely a guide to theeye.
0
0 5 10 15 20 25
< j > [108 A/m2]
B. Bulk samples
FIG.8. Dependenceof thepinningpotential U on theav-
erage current density in the film obtained from the fits of
our model to the data: curved lines are Eq. 7, straight lines Wewillnowgiveanoverviewofthemeasurementswith
areEq.8atthefollowingtemperatures(fromtoptobottom): YBCO bulk samples and compare these results with the
78K,81K,83K,84.5K,85.5K.NotethatfromEq.8wecan thin films. We have found that the dissipative behav-
determine only the slopes U0/jc; in the figure we arbitrarily ior of melt-textured and sintered bulk samples is quite
set U0 =4·103K similar. Measurements on the sintered samples have al-
readybeenpublished2 andthereforewewillfocusonour
It is interesting to note that all three mechanisms of melt-textured results. We emphasize, however, that our
dissipationarefoundto coexistatthe sametemperature present interpretation of the data is more complete and
but dominate in different regimes of the driving force. to some extent makes our earlier analysis obsolete.
Furthermore, we find that the current j1 which sets the Figures 10 and 11 show the velocity amplitude of the
scale for the exponential dissipation extrapolates to zero magnet as a function of driving force at different tem-
at 86.5 K (well below T = 89.5 K) which might indi- peratures for melt-textured and sintered samples. At
c
cate anonset of free flux flow where the magnetloses its 4K we find a nonlinear increase of the amplitude even
lateral stability. at small amplitudes followed by a nearly linear depen-
Towards lower temperatures only the linear term and dence for large drives. This initial nonlinear increase is
the glass term contribute as no data could be obtained less pronounced at higher temperatures where we find
atlargerdrivesbecausetheoscillationsbecameunstable. approximately linear curves up to 85K. For higher tem-
For temperatures below 60K we find that the evaluated peratures additional dissipation arises and above 88K
barriers decrease nearly linearly with temperature, see the dependence is simply v √F indicating a quadratic
∝
Fig. 9. We find U = 16T and U = 10T (i.e., ν = 8 and damping, i.e., losses proportional to the cube of the am-
5) for current amplitudes of 50A/m and 500A/m, re- plitude. Becauseofthevariousmechanismsofdissipation
spectively. The ratio 1.6 is given by the factor 100.2, see (to be discussed below) dominating at diffent tempera-
Eq. 7. The linearity of U(T) is again a signature of flux tures and amplitudes, there exists a temperature range
creep, but now with a current dependent creep factor. below T where the oscillation amplitudes surprisingly
c
6
increasewithtemperatureatconstantdrive,correspond- rising to 10−8Ω at 85K. Comparing these results with
ing to a minimum of dissipation. This can be seen more our data on YBCO thin films we note that at 78K R
s
clearlyinFig.12wherethevelocityamplitudesareshown has increased by three orders of magnitude.
as a function of temperature for various constant drives. At 4K the barrier in the nonlinear regime is much
We note that for sintered YBCO the maximum is much smaller than at high temperatures (there are no experi-
more pronounced, see Ref. 2. mentaldatabetween4Kand77K).Above77Kthebar-
riersdecreasewithincreasingtemperature. Theseresults
aresimilartothoseobtainedwiththefilmsexceptforthe
melt-textured factthatthebarriersinthebulkareafactorof2smaller
60 granular 4 K than in the films.
F [nN] melt-textured
s] 40 25 57
m/
m
v [ 77 K
20 20
0
0 20 40 60 15
F [nN] m/s] 29
m
FIG. 10. Velocity amplitudes as a function of the driving v [
forceforsinteredandmelt-texturedsamplesat4Kand77K. 10 20
The lines are fitsof ourmodel to thedata.
11
5
85 K 5.7
melt-textured
87 K
0
20 75 80 85 90
88.5 K T [K]
s] FIG. 12. Temperature dependence of the velocity ampli-
m/
m tude at various constant driving forces. Note the maximum
v [10 at about 88K (i.e., minimum of dissipation).
90 K
For temperatures above 85K quadratic damping
rapidly increases and eventually becomes to be the only
relevant dissipation above 88K. We attribute this to a
0
0 20 40 60 mechanism proposed by Melville18 for conventional su-
F [nN] perconductorsinwhichsinglevorticesareconsideredbe-
FIG.11. Velocity amplitudes versus driving force at tem- ing trapped perpendicular to the surface of the super-
peraturesabove85Kwheredampingbecomesquadraticlead- conductor. When an external field smaller than Hc1 is
ing to a square-root dependence. The lines are fits of our appliedshieldingcurrentsareinducedatthesurfacepro-
model to thedata. ducing a Lorentz force on the trapped vortices. If this
force is strong enough to depin the vortices they will
Considering the initial nonlinear increase of the am- bend in the direction of the applied force, while they
plitude at 4K followed by a nearly linear rise for larger stay pinned in the interior of the bulk where no current
drives we find that this behavior can be well fitted by is flowing. The amount of bending is given by the equi-
adding a linear dissipation and the collective flux creep librium between the Lorentz force and the elastic force
(Eq. 7), using an exponent µ = 0.14. Because Eq. 7 of the vortex. When the applied field is reduced and fi-
describes the data over almost the entire range the ex- nally reversed the vortex moves back in an irreversible
ponent µ can be determined with much better accuracy way, so that the flux component parallel to the surface
thanwiththethin-filmsamples: increasingµto0.2,e.g., passes through a hysteresis. This leads to dissipation
leadstoaclearlypoorerfit. Fromthelinearterm(domi- where the loss per cycle and surface area is given by
3
natingonlyatthesmallestamplitudes)weinferasurface BH /Hc1jc, where H is the amplitude of the external
resistance which is of the order of R 10−10Ω at 4K field and j the depinning current density. In our exper-
s c
∼
7
iment, H a and therefore, with Eq.4, this mechanism theflux linesareexposedtolargecurrentdensitiesalong
leads to qu∝adratic damping F v2 having a coefficient theirentirelength,whereasinthebulktheLorentzforces
∝
whichdivergesatT aswasfoundexperimentallyalready decay with increasing distance from the surface.
c
for sintered samples.2 Whencomparingourexperimentalmethod(whichwas
It should be noted that the well known Bean critical motivated by an interest in the physics of superconduct-
state also leads to quadratic damping and is used suc- ing levitation) with more standard techniques for inves-
cessfully in the interpretation of most other levitation tigating vortex dynamics in superconductors, like mea-
experiments involving larger magnets. However, in our surements of the ac susceptibility, of the current volt-
experiment the magnetic field is low and a critical state age characteristics ,of the magnetic relaxation, or me-
is not reached. Also, in the model of Melville the dis- chanical oscillators containing superconducting samples,
sipation is proportional to the density of trapped flux we note several obvious differences some of which are
B which makes it plausible that the losses in the melt- disadvantageous while others are of advantage. Firstly,
textured samples are larger than those in the sintered ourmethodisnotaffectedbydemagnetizationfactorsor
samples. edge effects, the geometry is just a magnetic dipole in
Besides the much larger damping found in bulk sam- themiddlebetweenessentiallyinfiniteandflatsupercon-
ples, there are two main features distinguishing the bulk ductingsurfaces. Secondly,itisaresonancemethodwith
results from the thin film ones: Firstly, for bulk samples highsensitivityandwide dynamic range: The dissipated
the damping becomes quadratic approaching T so that powervariesfrom10−9Wdownto10−17Wandtheelec-
c
−4
the amplitudes exceedthosefor thinfilms (Fig.4)where tricfields causedby vortexmotionrangefrom10 V/m
the dissipation grows exponentially with amplitude. We down to 10−11V/m. Because of the resonance method,
believe that the reason for this is the sample thickness. though, the frequency cannot be varied. The magnetic
In bulk samples the vortices stay pinned in the interior field is inhomogeneous and rather small, and the distri-
andmovejustwithinthepenetrationdepth. Inthinfilms bution of trapped flux lines is unknown. Nevertheless,
the vorticesareexposedto the drivingcurrentovertheir the damping of the oscillating magnet can be described
entirelength. Thisleadstohighflowratesofthevortices quantitatively by various mechanisms of flux dynamics
in the films whereas in bulk samples the flow rates are with flux creep being an important issue.
limited due to pinning deeper inside the samples.
Secondly, for bulk samples there is a minimum in dis-
sipationbelowT while forthin films the dampinggrows ACKNOWLEDGMENTS
c
monotonically with temperature. This minimum might
be due to a softening of the flux line lattice leading to
We are gratefulto K.F. Renk for support of our work,
improved pinning because the vortices can adjust more
toE.V.PechenforprovidingthefilmsandtoH.J.Borne-
effectivelyto astate oflowertotalenergy. Atstillhigher
mann for the melt-textured material. We had help-
temperatures pinning becomes less effective, hysteretic
ful discussions with N. Lu¨tke-Entrup and B. Plac¸ais on
dampingoccursanddissipationdivergesapproachingT .
c the interpretation of surface impedance measurements.
In thin films the exponential growth of dissipation dom-
Financial support by the Deutsche Forschungsgemein-
inates and no minimum in damping occurs.
schaft is gratefully acknowledged by R.G. and M.N.
(Graduiertenkolleg “Komplexit¨at in Festko¨rpern”).
V. CONCLUSION
APPENDIX: ELASTIC FORCES ON THE
OSCILLATING MICROSPHERE
Insummary,thedampingoftheoscillationsofthelev-
itatingmagnetisduetoenergylossescausedbyfluxmo-
tionofvariousdynamicregimes. Atsufficientlysmallam- The elastic forces on the levitating microsphere were
plitudeslinearfrictionisobservedwhichcanbedescribed investigated by measuring the amplitude dependence of
byasurfaceimpedancehavingparticularlylowvaluesfor the resonance frequency ω(a) andalso of the secondand
thin films at low temperatures. The temperature depen- third harmonic amplitude. In addition, the effect of a
dence of the surface impedance is discussed in terms of large dc voltage (up to 1000V) across the capacitor
±
a vortex resistivity caused by flux creep. Towards larger on the above quantities was determined. By and large,
amplitudeswefindlossestooriginatefromthemotionof theoscillatordisplaystypicalnonlinearproperties.19The
collectively pinned vortices. From our analysis pinning “backbonecurve”ω(a),e.g.,istheusualparabolaforthe
barriers are obtained having the typical inverse current thin-film samples below 81K. However, at higher tem-
dependence. The effective barriers increase linearly with peraturesandforallbulk samplesinparticular,stronger
temperature up to 60K as expected from creep effects. dependences are found which indicate that a third order
Finally, at evenlarger amplitudes, the losses in the films Taylor expansion of the return forces at the equilibrium
grow exponentially and eventually exceed those in the position is no longer adequate. On the other hand, the
bulk samples. This seems plausible because in the films coefficients of the second and third harmonic are found
8
to be independent oftemperature. Application of the dc 6H.J.Bornemann,C.Urban,P.Boegler,H.Kpfer,H.Riet-
voltage changes the equilibrium position by few tens of schel, Advances in Superconductivity VI, edited by T. Fu-
micrometers as canbe estimated fromthe stiffness. Fur- jitaandY.Shiohara(Springer-Verlag,Tokyo,1994),Vol2,
thermore, the resonance frequency changes because of p.1311.
the changing tension, in analogy with a vibrating string: 7E. V. Pechen, A. V. Varlashkin, S. I. Krasnosvobodtsev,
ω2 is proportional to the static force as observed earlier B. Brunner, and K. F. Renk, Appl. Phys. Lett. 66, 2292
with a niobium capacitor.4 The coefficient of the second (1995).
harmonic is stronglyaffected andcanbe reducedto zero 8The Q-factor is still low enough to neglect dissipation by
theinputimpedanceoftheelectrometerorbyresidualgas
ataparticularvoltage,whichimpliesasymmetricreturn
(the cell was evacuated and contained charcoal for cryop-
force at the new equilibrium position.
umping). Eddy current losses in normal conducting metal
These elastic properties may be discussed in terms of
partsofthemeasuringcellaredifficulttoestimatebutap-
the image-dipole model presented in Ch. III. Neglecting
pear to betoo small to limit theQ value at 4 K.
higherorderimagestheresonancefrequencyiseasilycal- 9At the end of the experiment the capacitor was heated
culated to be ω/2π 200Hz if the equilibrium position
≈ above Tc and the magnet fell to the lower electrode. The
is in the middle between the two surfaces. Gravitation changeof theinduced chargeq·h0/d was measured. From
and especially trapped flux lead to unsymmetric forces this we determine the levitation height to be h0 = (0.4±
andthereforetoanunsymmetricpositionandhencetoa
0.1) mm.
second harmonic amplitude. The force between a single 10S.B. Haley and H.J. Fink, Phys. Rev. B53, 3506 (1996).
vortex and a magnetic dipole perpendicular above a su- 11FortheperpendicularorientationRswouldbesmallerbya
perconducting thin film has been considered in Ref. 20. factorof2.Becausethesurfacecurrentscalesproportional
Because we have no quantitative information on the dis- tothemagneticmomentofthesphere,theabsolutevalues
tribution and the orientation of the trapped vortex lines of Rs are uncertain to within a factor 2 to4.
in the samples we cannot calculate their contribution to 12N. Belk, D. E. Oates, D. A. Feld, G. Dresselhaus, M. S.
the elastic forces on the levitating magnet. Therefore, Dresselhaus, Phys. Rev. B53, 3459 (1996); J. R. Powell,
the image-dipole model remains a rather crude descrip- A.Porch,R.G.Humphreys,F.Wellh¨ofer,M.J.Lancaster,
tionoftheelasticforces. Forlateralstabilityofthemag- and C. E. Gough, Phys.Rev. B57, 5474 (1998).
net, which is a prerequisite of our method, trapped flux 13M. W. Coffey and J. R. Clem, Phys. Rev. B45, 10527
is clearly necessary. (1992).
14M.Golosovsky,M.TsindlekhtandD.Davidov,Supercond.
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15N.Lu¨tke-Entrup,B. Pla¸cais, P. Mathieu, Y.Simon, Phys.
Rev.Lett. 79, 2538 (1997); Physica B, to bepublished.
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2R.Grosser,J.J¨ager,J.Betz,andW.Schoepe,Appl.Phys. valuesofµupto0.6.This,however,leadstobarrierscom-
parable to kT for our measuring currents, which appears
Lett. 67, 2400 (1995).
3R. Grosser, A. Martin, M. Niemetz, E.V. Pechen, and W. 18uPn.rHea.sMonealvbillele., J. Phys.C: Solid St.Phys. 4, 2833 (1971).
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(John Wiley & Sons, NewYork,1979), Chap. 4.
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9
