The basic characteristics of the thermoactivated vortex mobility in Y1Ba2Cu3O7 are determined by measurement of the kinetics of magnetization in two time regimes. The analysis of the kinetics of the approach of the equilibrium results in the activation energy, while the measurement of the log-creep rate allows determination of the activated moment. It is shown that the movement of vortices can be regarded as the diffusion process

Roytburd, A. L.

Turchinskaya, M. J.

English

NASA Technical Reports Server

N92-21633
VORTEX MOVEMENT AND MAGNETIZATION OF HIGH Tc SUPERCONDUCTORS
A.L. Roytburd and M.J. Turchinskaya*
Engineering Materials Program
Department of Chemical and Nuclear Engineering, University of Maryland
College Park, MD 20742-2111
*Metallurgical Division, NIST, Gaithersburg, MD 20899
ABSTRACT
The basic characteristics of the thermoactivated vortex mobility in YIBa2Cu307 are
determined by measurement of the kinetics of magnetization in two time regimes.
The analysis of the kinetics of the approach of the equilibrium results in the
activation energy, while the measurement of the log-creep rate allows
determination of the activated moment. It is shown that the movement of vortices
can be regarded as the diffusion process.
INTRODUCTION
The problem of vortex pinning is of great importance for the application of oxide
high Tc superconductors. In comparison with pinning of dislocations, interface
boundaries or domain walls, the pinning problem for vortices is much more complex
because the density of vortices is large and adequate to produce strong
interactions between the vortices in many practical important cases.
In accordance with the effects of vortex-vortex interactions on pinning
two different cases can be considered. The first one corresponds to strong
pinning centers with the spacing much larger than the distance between vortices.
In this case the system of vortices can be considered an equilibrium phase
(crystal, glass or melt), and the interaction of this phase with randomly
distributed pinning obstacles can be analyzed. A classic example of such an
approach is the Labusch theory of pinning for an elastic vortex lattice. I
Another example is Nelson's melt theory, 2 which pretends to the explanation of
the very weak pinning at temperatures above the reversibility temperature. Note
that before melting, a great number of dislocations must be formed in the vortex
crystal, and the real problem of the dynamics of the vortex medium in this case
is the problem of movement and pinning of dislocations in the vortex lattice.
In other words the trend to melting has to be manifested by a transition from a
laminar flow of the vortex medium to a turbulent one. This problem seems very
important in the case of pinning by interfaces or particles with spacing larger
than a distance between vortices. 3
However, the problem of intrinsic pinning in oxide superconductors may be
closer to reality in the other case, where the individual vortices are pinned by
the obstacles, and the distance between the obstacles is comparable with the
distance between vortices. Collective effects of the vortex-vortex interactions
are important in this case also, but we can expect that they manifest themselves
in the renormalization of parameter values rather than in the alteration of the
physical picture of the vortex motion. This case of small vortex density is under
consideration in the paper. Our goal is to obtain basic characteristics of vortex
mobility from the study of the magnetization of zero-field cooled samples.
203 l_'_(r']_,"r'4G _.U£ i_,._t_J_ _'_I(|7 F,t_D
https://ntrs.nasa.gov/search.jsp?R=19920012390 2020-03-17T13:07:40+00:00Z
THEORETICAL CONSIDERATION
The penetration of magnetic flow into a superconductor during its magnetization
in an external magnetic field, Fig. i, is similar to the flux flow diffusion, 4,5
and is described by the generalized diffusion equation:
aB/at - V(Def f VB), (i)
where D.ff-vB/VB is the effective flux diffusion coefficient, B is the local
induction, VB is the gradient of the induction, and v is the flux velocity.
The equation (i) is correct if there is no nucleation, multiplication
or annihilation of vortices inside the sample, i.e. there are no weak links
within a superconductor. The flux velocity for the thermoactivated movement of
a vortex through an array of pinning obstacles can be determined as
v - voexp(-U/kT)sinh(_VB/kT ) . (2)
This equation is obtained under the assumption that the energy barrier AF,
which has been overcome for a vortex displacement, is disturbed under the driving
force (VB), and it is an analytical function of VB at VB-O: 4,6
AF±(VB) - _F(0) • _VB, &F(0) - U, (3)
where + corresponds to displacements with VB, - to the opposite direction, U
is the activation energy, and a is the activated moment. _-m_1, m-_on_2, where
m is the magnetic moment of the vortex or vortex bundle, 4o-2,7xi0 -7 Gcm 2 is the
flux of one vortex, n is the number of the vortices in the bundle, _1 is the
displacement of the bundle or vortex at an activation event, _2 is the effective
length of a displaced section of the bundle or vortex, factually the distance
between the nearest pinning points along the vortex, (Fig. 2). v o in Eq.(2) is
the limiting velocity which equals _1 u, where v is the frequency of vibration of
the vortex section, _2. Assuming w-(_/22)v,_ , where _ is the coherence length or
vortex "radius" (-10-8cm), vat is the atomic vibration frequency -I013s -I, we get
for vo=_vat=105cm/s, a value on the order of the sound velocity.
If the driving force is larger than the critical force, VB>VBc=-U/a , the
movement of the vortex is overbarrier. The vortex movement is not described by
Eq. (2) because it is controlled not by the frequency of fluctuation depinning
events but by the rate of the dissipation of the vortex energy. In this case
v-(i/7)VB, where 7 is the viscosity. L8
The vortex velocity, v(VB), and the corresponding diffusion coefficient,
D.ff(VB), are given in Fig. 3. There are three regions for which the solution
of Eq. (I) can be obtained analytically.
The first one corresponds to the distribution of magnetic induction with
VB>VBc=. This is the so called flux flow region when Def_-Df-B/7 is independent
on VB.
The second region corresponds to the near critical state, when eVB/kT>>l,
and sinh(_VB/kT)=exp(_VB/kT). In this region the total magnetization of a
superconductor, or its average induction, is proportional to the logarithm of
204
time : 4,6
M(t) = PInt, P - dM/dlnt = kTR/(aAF/aVB) - (kTR/_(R)), (4)
where R is the radius of the sample, _(R) is the activated moment at its surface,
where the induction B(R) is closed to the equilibrium one, Bo.
The third
again:
region corresponds to the constant Def_-D at the small VB
D - (Bo_vo/kT)exp(-U/kT) - Doexp(-U/kT), DO u Bo_vo/kT (5)
This regime of ordinary diffusion flow was named the thermally assisted flux flow
(TAFF). 8
It is important to note, that all of these regimes can be observed by
measuring the magnetization as a function of time of the zero-field cooled
superconductor sample. At the first moment after applying the magnetic field, VB
is great and is larger than the critical value. Unfortunately, this stage is
difficult to study because of nonstationary effects while the field is turned
on. The sample then goes through the near critical state, where linear behaviour
of magnetization vs. logarithm of time has been expected. At last, in the final
stage, the magnetization approaches to the equilibrium saturation, Mo(or Bo) , at
t>t*, and the kinetics of the magnetization is described by the equation:
M(t) - Mo + M'exp(-t/r), (6)
where M' is a constant. The relaxation time, r , is connected with the diffusion
coefficient, D(Bo) , by the relation r - R2/D, where R is the sample radius. It
is necessary to stress that this near equilibrium asymptotic behaviour does not
depend on the distribution of the local induction at the moment t*, when the
exponential stage begins, provided the induction is closed to Bo (RVB/B o << i).
Thus the study of the kinetics of the magnetization at the different stages
gives the opportunity to find the basic characteristics of the vortex mobility,
and U, without any additional assumption about the distribution of the
induction within the sample. The results of such a kind of study are represented
below.
SAMPLES AND EXPERIMENTAL METHOD
Three different YIBa2Cu307 samples with sharp transitions at Tc-93K were under
investigation: a single crystal, a sintered polycrystal and a sample of
quench,melt and growth (QMG) material. I° The QMG sample is a bar of 0.9x0.9x5.8
mm, which was cut from a grain of a polycrystal. The sample has no weak links, 11
making it possible to determine its effective radius, R (R=0.45 mm). Our single
crystal sample contains some fraction of normal phase. So, the single crystal as
well as sintered samples were not good for quantative estimates.
The samples were cooled in zero magnetic field. The magnetic field was then
applied, and the magnetization was measured as a function of time using either
a vibrating sample magnetometer, (VSM), or a SQUID system. The magnetic field was
directed along c axes of the single crystal sample and perpendicularly to c axes
of the QMG sample.
205
EXPERIMENTAL RESULTS
Approach to Equilibrium Magnetization,Mo, Dif_usiqn Coefficient,D. Magnetization
versus time, M(t), curves were measured at rather high temperatures (70 and 80K)
and for long exposures (up to 340 ksec) at relatively weak magnetic fields (2 and
4 kOe), so the density of vortices was small up to approaching the equilibrium
magnetization. M(t) data for all the samples at 70K and 2 kOe are plotted in
curves (a) of Figures 4,5 and 6 for the QMG, single crystal and sintered
samples, respectively. M vs. log-tlme data are plotted in curves (b). Curves
obtained for the other temperature and field setting have the same shape as in
these Figures: for all the curves there are log-time dependences M(t), and at
much longer times, t>t,* there are stages of the approach to the equilibrium
magnetization, Mo. In these stages the magnetization shows the exponential
behaviour:
in[M(t)/M o - i] - in(M'/Mo) - -t/r, (7)
where In(M'/Mo) are constants. In curves (c) of Fig.4,5 and 6, where the
calculated data of In[M(t)/Mo-i ] vs. time for all the samples are plotted, it can
be seen that at t>t* there are linear dependences of in[M(t)/Mo-i ] on time. The
result of the determination of r depends strongly on the choosen value of Mo, and
a special asymptotic method was used 12 for the correction the values of Mo. For
the QMG sample we obtained the values ofM0--3.87 (emu/gr) at 70K and 2 kOe, and
Mo--2,5 (emu2gr) at 70K and 4 kOe.
The values of the relaxation time, f, were obtained as a slope of the
linear portion of the curves (c), and the diffusion coefficients, D-R2/r, where
R is the effective radius of the sample, were calculated. For the QMG sample at
70K we obtained D=6.7x10 -g cm2/s for 2 kOe, and D=7.8x10 -s cm2/s for 4 kOe.
Activated Moment, _, Activated Sengtb, 2, and Intervortex Spacing. a, The
logarithmic creep rate P-dM/dlnt for the QMG and sintered samples was measured
in the temperature interval 6-80K. I° The results are given in Fig. 7,a and in
Fig. 7,b, respectively. For the QMG sample we calculated the values of the
activated moment, a (using the relation a-kTR/P, Eq. (4)). Derived values of
_/40, where 40 is the flux of one vortex, for the QMG sample are plotted in Fig.
8.
Values of the average activated length, _-(a/4o) I/2, are displayed in Fig.
9 together with values of the distance between vortices, a-(Bo/4o) I/z. B o was
obtained from measuring the equilibrium magnetization, Mo: Bo-4_Mo+H.
Activation Energy., U, and Pre-exponential Factor, Do. Obtaining r and _ from the
kinetics of the magnetization, it is possible to estimate the activation energy
of the thermoactivated movement of the vortex bundle, U, from the Eq. (5):
U/kT = In(D/Do) = in(rvoBo_/kTR2). (8)
Assuming that vo-105 cm/s, we obtained, with log-accuracy by vo, U/kT=33, U=0.19
eV, and the pre-exponential factor Do-Bo_vo/kT=6xl05 cmZ/s for the QMG sample at
70K and 2 kOe. For 70K and 4 kOe U=0.2 eV, Do=4xl06 cm2/s.
2O6
DISCUSSION
The exponential approach to equilibrium is evidence of the linear dependence
between the vortex velocity and the driving force and supports the assumption of
Eq .(3). Taking into account the relation between D,z_ and resistance, p : p =
D/c 2, where c is the velocity of light, we have to admit that at 70 and 80K and
H-2 and 4 kOe the QMG sample of YBa2Cu307 manifests small but finite
resistivity. From the point of view of vortex mobility in this stage, the vortex
medium is rather "melt" than "glass" (in terms of M.Fisher's theory 13) or
"crystal" with collective pinning. 14 For both these models AF must be the
divergent function of VB at V_0: AF-I/VB #, _>0 for the vortex glass state, or
-inVB for the collective pinning model.
By "vortex melt", we imply that the effects of correlations between the
individual thermoactivated depinning events are small. 15 To prove this suggestion
it is necessary to consider the pre-exponential factor, DO, in Eq. (5) in more
detail, rewriting it through the activated length, _, and the intervortex
distance, a , as
DO = (4o2/kT) (_2/a2)Vo = LVo(22/a 2) . (9)
The term, 4o2/kT_L, has the dimension of length (e.g. L=4 cm at 7OK). From Fig.
I0 it is clear that up to 70K at 2 kOe and 60K at 4 kOe, _<a, i.e. a bundle
consists of only one vortex, and _ is the weak function of temperature. Then
rises rapidly, i.e. the number of vortices in the bundle increases. For n-l,
_-(_i_2) I12, that is, _ is close to an average distance between pinning points.
At 70K and higher temperatures, all the pinning centers are occupied by
vortices, and the pinning-depinning movements of vortices with the driving force
VB can be interpreted as back-directed diffusion of pinning points in vortex
medium. Indeed, the velocity of the point flow is determined by the Einstein
diffusion relation:
vp - (Dp/kT)V_ = (Dp/kT)(_VB/a), (I0)
where Dp-a2vexp(-U/kT) is the point diffusion coefficent, and V# is the gradient
of chemical potential of the pinning points in the magnetic field. The point
velocity, vp, is equal numerically to the vortex velocity, v:
-vp- v - DVB/B (11)
(the relation between D and v arises from the definition of D.ff in Eq. (I)). If
we take into account that av-vo, it follows from Eqs. (i0) and (Ii) that the
vortex diffusion coefficient, D, is determined by Eq. (5). At 2=a,
Do=LVo=10%m2/s: that is in accordance with the experimental determination of DO
through measuring _ and Bo.
It is clear that the penetration of the flux flow into a sample during
magnetization can be considered in terms of the diffusion theory for polycrystal
samples with weak grain boundaries too. For this case it would be necessary to
bring into consideration the vortex diffusion along the grain boundaries as well
as the bulk grain diffusion. Of course the diffusion description may be not
207
complete if the nucleation of vortices is possible inside "bad" samples.
CONCLUSION
I. It is shown that approach to equilibrium magnetization with time of the zero-
field cooled QMG, single crystal and sintered samples of YBa2Cu307 proceeds
according to the exponential law. The equilibrium values of the induction,
relaxation time and vortex diffusion coefficient are determined for the QMG
sample at 70 and 80K and 2 and 4 kOe.
2. Log-creep rate was measured at 2 and 4 kOe for a set of temperatures between
6 and 8OK. From these data the activated moment a was found. The analysis of
shows that an activated vortex bundle contains only one vortex at the
temperatures and fields under consideration. An activation length, _ - (_/_o) I/z,
is almost constant up to 70 or 80 K. Beyond these temperatures the distance
between vortices becomes smaller than _. Up to these temperatures _ is likely the
average distance between effective pinning points. At temperatures higher than
70K at field 2 kOe and temperatures -75K at 4 kOe the number of vortices in a
bundle grows, as does _.
3. Using Bo and D, which are obtained from the approach to the equilibrium , and
from the log-creep stage, the activation energy, U, has been determined for the
QMG sample.
4. The character of vortex movement during magnetization with time at 70 and 80K
and 2 and 4 kOe is in good agreement with the diffusion picture, which does not
take into account the effects due to the correlation between individual
thermoactivation events.
ACKNOWLEDGEMENT
We are very grateful to L.H. Bennet and L.J. Swartzendruber for their friendly
collaboration and to K. Sawano, D.L. Kaiser, J.E. Blendell, and C.K. Chiang for
the samples.
REFERENCES
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(1969).
D.R. Nelson and H.S. Seung, Phys. Rev. B39, 9153 (1989).
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M.R. Beasly, R. Labusch, W.W. Webb, Phys. Rev. 181, 681 (1969).
E.M. Brandt, Z. Physik B., in press.
P.W. Anderson, Phys. Rev. Lett., 9, 309 (1962).
M. Tinkham, Phys. Rev. Lett., 13, 804 (1964).
L. Bardeen and M.J. Stephen, Phys. Rev. 140, AI197 (1965).
P.H. Kes, J. Aarts, J. van der Berg, and J.A. Mydosh, Supercond. Sci.
Technol., !, 242, (1989).
i0. M.J. Turchinskaya, L.H. Bennet, L.J. Swartzendruber, A. Roytburd, C.K.
Chiang, M. Hill, J.E. Blendell, and K. Sawano, High-Temperature
Superconductors: Fundamental Properties and Novel Materials Processing. MRS
Symp. Proc. 169, in press, (1990).
ii. M. Murakami, M. Morita, and N. Koyama, Jpn. J. Appl. Phys., 28, LI125 (1989).
12. M.J. Turchinskaya, A.L. Roytburd, L.H. Bennett, L.J. Swartzendruber, and K.
208
Sawano, in preparation.
13. M.P.A. Fisher, Phys. Rev. Lett., 62, 1425 (1989).
14. M.V. Feigelman, V.B. Geshkenbain, A.L. Larkin, and V.M. Vinocur, Phys. Rev.
Lett., 63, 2303 (1989).
15. A.L. Roytburd, M.J. Turchinskaya, L.H. Bennett, and L.J. Swartzendruber,
submitted for publication.
2O9
-4,_ .... 4
(b)
# ,
Y oI , -
L "
o & , -].--
;I N N m
o 1-2 ,3 q
Fig. i. Kinetics of magnetization of a zero-field cooled sample.
a. Equilibrium magnetization dependence on magnetic field.
b. Time dependence of magnetization.
c. Distribution of magnetic induction.
O - the initial diamagnetic state, i-2- the near critical state (VB_<VBcr),
3 the near equilibrium state, 4 - the equilibrium state (VB-0, B-Bo).
(o)
(b)
>__l_2!o' v_
li
I
dvB
d_sp/_ romeo t
__V_o .....
/ I
Loq-creep .)_-_ It
iii / I
I I/ / II
i i _l" I
-/_-;_ - -,,
t ¢. 17
I I
__D.#_I.__ .
i
% &
Fig. 2. Fig. 3.
Fig. 2. Thermoactivated movement of a vortex, a. A vortex section, 12, pinned
by obstacles, b. Energy as function of a vortex displacement.
Fig. 3. Vortex velocity, v, and vortex diffusion coefficient, D,f z.
210
-41
_0
-4.5
0
(o,)
2oo
ti_ (kse¢)
la
-4.:
4OO
9 11
in (time, sec)
"21 c)
-_
0 2dO
time (ksec)
Fig. 4.
13
400
0.68_
_-o 7
°."I _
(a)
o loo 26o
time (kJec)
-°-n i
5 _; 13
in (time. ksec)
-3_
-42
G,
t__
0 2O0 400
time (ksec)
] "
_"-4ol
7 13
-2
v
-4
0
9 f,
In (tiRe. see)
i<
2OO
time (kae¢)
400
Fig. 5.
.-3
-5
{c)
tlme (ksec)
Fig. 6.
Fig. 4. Kinetics of magnetization
of the QMG sample, Ho--3.87(emu/gr).
Fig. 5. Kinetics of magnetization of the
single crystal sample, Mo--36.25 [emu/_] .
Fig. 6. Kinetics of magnetization of
the sintered sample, Mo--0.68 (emu/gr).
211
i0 20 40 60 80 100
Temperature, K o
Fig. 7, Fig. 9.
Fig. 7. Logarithmic rate of magnetization, P.
a. the QM0 sample, b. the sintered sample.
Fig. 8. Temperature dependence of a/$ o for the QMG sample at 2 and 4 kOe.
Fig. 9. Temperature dependence of average activated length, 2, and intervortex
spacing, a, for the QMG sample at 2 and 4 kOe.
212


Vortex movement and magnetization of high Tc superconductors

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