Investigation of relaxation phenomena in high-temperature
  superconductors HoBa2Cu3O7-d at the action of pulsed magnetic fields
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Investigation of of of of relaxation phenomena in high-temperature superconductors HoBa2Cu3O7-δ at 
the action of pulsed magnetic fields  
J.G. Chigvinadze*, J.V. Acrivos**, S.M. Ashimov*, A.A. Iashvili*, T. V. Machaidze*,   Th. Wolf*** 
* E. Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia 
** San Jose’ State University, San Jose’ CA 95192-0101, USA 
              *** Forschungszentrum Karlsruhe, Institut für Festkörperphysik, 76021 Karlsruhe, Germany 
Summary 
It is used the mechanical method of Abrikosov vortex 
stimulated dynamics investigation in superconductors. 
With its help it was studied relaxation phenomena in 
vortex matter of high-temperature superconductors. It 
established that pulsed magnetic fields change the course 
of relaxation processes taking place in vortex matter. The 
study of the influence of magnetic pulses differing by 
their durations and amplitudes on vortex system of 
isotropic high-temperature superconductors system 
HoBa2Cu3O7-δ showed the presence of threshold 
phenomena. The small duration pulses doesn’t change 
the course of relaxation processes taking place in vortex 
matter. When the duration of pulses exceeds some 
critical value (threshold), then their influence change the 
course of relaxation process which is revealed by 
stepwise change of relaxing mechanical moment
relτ . 
These investigations showed that the time for 
formatting of Abrikosov vortex lattice in HoBa2Cu3O7-δ 
is of the order of 20 �s which on the order of value 
exceeds the time necessary for formation of a single 
vortex observed in type II superconductors.  
1. Introduction 
The present communication is devoted to the 
experimental investigation of relaxation phenomena in 
high-temperature superconductors of HoBa2Cu3O7-δ 
system. 
High-temperature superconductors are characterized 
by such high critical transition temperatures Тс in the 
superconducting state, they remain superconductors at 
temperatures when their thermal fluctuations energy 
becomes compared with the elastic energy, and also with 
the pinning energy [1]. It creates prerequisites for phase 
transitions.  Due to the layered crystal structure and 
anisotropy, which is a characteristic high-temperature 
superconductors, they reveal conditions for the 
appearance of different phases on B-T diagram.( B is 
magnetic induction, T-is temperature)[2-13]. As 
example, Abrikosov vortex lattice begin melting near the 
critical Тс temperature what is followed by the essential 
change of vortex continuum flow dynamics along with 
sharp change of character (dynamics) of relaxation 
phenomena. In high-temperature superconductors it is 
observed such relaxation processes as a slow logarithmic 
decrease of captured flux with time at temperatures much 
below their superconductive critical transition 
temperature Тс [14-16]. The logarithmic character of 
relaxation is explained by the Anderson [17]. Near Тс, in 
the range of Abrikosov vortex lattice melting, the 
logarithmic character of relaxation is changed by the 
power one with 2/3 exponent [18]. 
Consequently, the study of relaxation processes in 
high-temperature superconductors is an important 
problem.  
2. Experimental 
For Investigation it was used currentless mechanical 
method of Abrikosov vortex stimulated dynamics study 
by magnetic pulses revealing relaxation phenomena in 
vortex matter described in work [19]. This method is a 
development of currentless mechanical method of 
pinning investigations [20,21] and is based on pinning  
forces   countermoments measurements and viscous 
friction, acting on a axially symmetrical superconducting 
sample in an outer  (transverse) magnetic field.   
Countermoments of pinning forces and of viscous 
friction, acting on a superconductive sample from  
quantized vortex lines side (Abrikosov vortices) are 
defined the way as it was described in work [22,23].  The 
sensitivity of the method accordingly works [24], is 
equivalent to 10-8 V×cm-1   in the method of V-A 
characteristics. 
The high-temperature superconducting samples of 
HoBa2Cu3O7-δ system were prepared by the standard 
solid state reaction method. Samples were made 
cylindrical with height L=13mm and diameter d=6mm. 
Their critical temperature was Tc=92 K. The investigated 
samples were isotropic what was established by 
mechanical moment τ measurements appearing 
H > 1cH   with the penetration of Abrikosov vortices 
into a freely suspended on a thin elastic thread 
superconducting sample. The appearance of such 
moment ατ sinMH= , characteristic for anisotropic 
superconductors, is related with penetrating Abrikosov 
vortices and the mean magnetic moment M
 of a sample 
which could deviate on angle α  from the direction of 
outer magnetic field H
. In superconducting anisotropic 
samples it is presented energetically favorable directions 
for the arrangement of emerging (penetrating) vortex 
lines which in their turn are fastened by pinning centers 
creating aforementioned moment τ . The lack of τ  
moment is characteristic for isotropic and investigated by 
us samples, no matter magnetic field value and its 
previous orientation in respect to H
 in the axial 
symmetry plane. Pulsed magnetic fields were created by 
Helmholtz coils. The value of pulsed magnetic fields was 
changed in Oeh 2002 ÷=∆   limits. 
In experiments it was used both single and continuous 
pulsed with repetition frequency ν   from 2.5 s-1  to 500s-1 
.  The duration 
x of pulses was changed from 0,5 до 
500 �s. Magnetic pulse could be directed both parallel 
h||H) and perpendicularly (
h⊥H) to applied steady 
magnetic field H
, creating mixed state of 
superconducting sample. The standard pulsed generator 
and amplifier were used to feed Helmholtz coils. The 
current strength in coils reached up to  40÷50 A.  
Samples were high-temperature superconductors of 
HoBa2Cu3O7-δ system placed in the center between 
Helmholtz coils. 
The principal set-up of experiment is shown in fig.1 
[19,20]. In experiments it is measured the rotation angle 
2ϕ  of sample depending on the angle of rotation of a 
torsion head 1ϕ , transmitting the rotation to a sample by 
means of suspension having the torsion stiffness 
K ≈4·10-1 [dyn•cm], which can be replaced when 
necessary by a less stiff or stiffer one.  
The measurements were carried out at a constant speed 
of rotation of the torsion head, making ω1=1,8·10
-2 rad/s . 
Angles of rotation поворота   2ϕ   and 1ϕ  were 
determined with an accuracy of ±4,6·10-3 and  ±2,3·10-3 
rad, respectively. The uniformity of the magnetic field’s 
strength along a sample was below H
∆ = 10-3.
                  Fig. 1.  The schematic diagram and the geometry of the experiment. 1-sample, 2-upper elastic filament, 3-lower filament, 4 - leading head,  
                              5 - glass  road. φ is angle between Mr  and Hr  
To avoid effects, connected with the frozen magnetic 
fluxes, the lower part of the cryostat with the sample was 
put into a special cylindrical Permalloy screen, reducing 
the Earth magnetic field by the factor of 1200. After a 
sample was cooled by liquid nitrogen to the 
superconducting state, the screen was removed, a 
magnetic field of necessary intensity H  was applied and 
the 2 1( )ϕ ϕ  dependences were measured. To carry out 
measurements at different values of H , the sample was 
brought to the normal state by heating it to до T > cT   at 
H =0,  and only after returning sample and torsion head 
to the initial state 1 2 0ϕ ϕ= = , the experiment was 
repeated.  
                                            
3. Results and discussions 
During rotation of the sample both of normal and 
superconducting states in the 
absence external magnetic field ( H =0) the 2ϕ  
dependence versus 1ϕ  is linear and the condition is 
satisfied. 
       tωϕϕ == 21  
    The character of the   2 1( )ϕ ϕ   dependence is changed 
significantly, when the sample is in magnetic fields 
H > 1cH  at  T < cT . Typical 2 1( )ϕ ϕ dependences at 
T=77K and various magnetic fields for HoBa2Cu3O7-δ  
sample  ( length of a cylindrical sample L=13mm and 
diameter d=6mm ) is shown in Fig.2.   
                              
Fig.2. Dependence of the rotation angle of the sample HoBa2Cu3O7-δ   
2ϕ  on the rotation angle of the leading head 1ϕ  in magnetic field 
H=1000 Oe at T=77K. 
Three distinct regions are observed in Fig.2.  In the 
first (initial) region, the sample does not respond to the 
increase in  1ϕ  ,  i.e. to the applied and increase with 
time torsion torque as  1ϕ ~ )( 21 ϕϕτ −= K  or 
responds weakly. Such behavior of the sample can be 
explained by fact that Abrikosov vortices are not 
detached from pinning centers at small values of 1ϕ ~τ , 
but if the sample is still turned slightly, this can be 
caused by elastic deformation of magnetic force lines 
beyond it or, possibly, by separation of the most weakly 
fixed vortices. As it is seen from fig.2 , as soon as a 
certain critical value  φсmin depending on H is reached , 
the first region under goes a transition to the second 
region in which the velocity of the sample increases 
gradually with  1ϕ  increasing resulting from the 
progressive process of detachment of vortices from their 
corresponding pinning centers. One should expect that 
just in this region, in the rotating sample “the vortices 
fan” begins to unfold, in with the vortices are distributed 
according to the instantaneous angles of orientations with 
respect to the fixed external magnetic field.   In this case 
the of orientation angles of separate vortex filaments are 
limited from frϕ  to pinfr ϕϕ + , where  frϕ  is the angle 
on which the vortex filament can be turned with respect 
to  H
 by forces of viscous friction with the matrix of 
superconductor, and  pinϕ  is the angle on with the 
vortex filament can be turned by the most strong pinning 
center, studied for the first time in [25]. 
The gradual transition (at high 1ϕ  values)  to the third 
region where the linear 2 1( )ϕ ϕ dependence was 
observed, allows one to define the countermoments of 
pinning forces  pτ and  frτ , independently. Just in this 
region, when  21 ωω =  the torque τ , appeared to the 
uniformly rotating sample, is balanced by the 
countermoment  pτ  and  frτ . In particular, in the case 
of continuously rotating sample with frequency  
21 ωω =    one could find similarly to   [26,27]  the 
expression for the total braking torque  τ  [19] . 
Indeed, if we consider in this case a vortex element 
  moving with velocity ⊥υ
 perpendicular to sd
then the average force acting on this elements is 
dsFdsfd l
⊥ += υ
and the associated braking torque, exerted on the rotating 
specimen becomes:                    
υτ fdrd
 where r
is the vector pointing from the rotational axis 
to the vortex elements, lF  is the pinning force per flux 
thread per unit length, and η  is the viscosity coefficient. 
For a cylindrical specimen of radius R   and height L  
integrating over the individual contribution of all vortex 
gives a total braking torque τ  
 ωτττ 0+= p                                                        (1) 
   with      
=τ  ,                
 and 
B ηπτ
= ,               
Where B is the inductivity averaged over the sample, 
0Φ  is the flux quantum , L  is the height and R  is the 
radius of the sample. 
As it is shown in Fig.2, starting with the point   
(a), where 21 ωω = , to the superconducting sample 
uniformly rotating in the homogeneous stationary 
magnetic field  H=1000 Oe, is applied stationary 
dynamic torsion moment  fr
p τττ += .  
If in this region the torsion head is stopped, then at the 
expense of relaxation processes connected with the 
presence of viscous forces acting on vortex filaments, the 
sample will continue the rotation in the same direction 
(with decreasing velocity)   until it reaches a certain 
equilibrium position, depending on the H  value. The  
Fig.3 shows curves of  
2ϕ∆ time dependences at the 
stopped leading head for HoBa2Cu3O7-δ  sample at 
T=77K and  H=1000 Oe. 
Fig.3. Dependence of momentum 
relτ  on time  t  after the stopping of 
rotating head for HoBa2Cu3O7-δ      sample at  T=77K  and  H=1000 Oe.  
If during the relaxation after rotation of sample one 
applies the pulsed magnetic field in parallel to the outer 
magnetic field H
, then additional vortices, created as 
result of magnetic pulse, influence the structure already 
existing in the sample as “the vortex fan” what could 
result in the decrease of the angle of its unfolding or to 
its folding. The letter in its turn, would cause the 
additional change in the relaxation process taking place 
in the sample, and, correspondingly, results in the 
stepwise decrease of moment related with viscous 
forces frτ . 
But the change of relaxation process character and, 
correspondently, the stepwise decrease of moment could 
happen if the duration of magnetic pulse is larger as 
compared with the time necessary for creation of a new 
vortex structure, which will influence the 
superconducting sample relaxing in magnetic field. If it 
is the case, then at the small durations of magnetic pulses 
the relaxation curve, presented in Fig.3, doesn’t  change, 
but when this duration becomes the order of a time for 
penetration of vortices into the sample and the creation 
of vortex structure, then the aforementioned change of 
relaxation processes could principally appear. Namely, 
this situation when the duration �x of magnetic pulses 
was larger then the time for  Abrikosov vortex lattice 
creation �xс,  have been described by us our previous 
work [19], when it was shown that the influence of one 
magnetic pulse �h≈400 Oe (�h||H)  with duration 
30�сек>�xс   was stepwisely decreased the relτ  moment 
and the relaxation process continued with the reduction 
relτ on a level as far as a new magnetic pulse similar 
the first one is not applied. 
In the presented work it was studied the influence of 
different duration and amplitude pulses on relaxation 
processes in vortex matter. The results shown in  Fig.4 
on action of single pulses of different durations on 
relaxation processes in vortex matter and, consequently, 
on mechanical moment  
relτ revealed that at small pulses 
durations up to  15 �s  the relτ doesn’t change, but at 
duration of applied pulse  >15 �s  it is observed the 
stepwise change  
relτ , what speaks on the existence of 
the  �xс   threshold. 
Fig.4. Dependence of momentum � relτ on  the duration �x of 
magnetic field single pulse    � h=172 Oe applied in parallel to the 
main magnetic field  H=1000 Oe  at  T=77K  for  HoBa2Cu3O7-δ  
sample. 
This way one could say that the Abrikosov vortex 
lattice creation time in high-temperature isotropic 
superconductor of HoBa2Cu3O7-δ makes value on the 
order of 20�s. This value approximately on the order of 
value higher then time for the single-vortex creation for 
the first time measured by G. Boato, G.Gallinaro and C. 
Rizzuto [28],  who showed that this time is less than 10-5 
sec. 
In work [19]  it was also shown that continuous action  
of aforementioned pulses with the train frequency equal 
to 2,5 s-1  more sharply reveals their influence on 
relaxation processes in vortex matter and in these 
conditions the processes of penetration of vortices into 
superconductors bulk are made more sharply expressed. 
In fig.5 it is presented the clear picture of magnetic 
pulses continuous action with �h=172 Oe (�h||H)  , and 
the duration  20�sec, what  is larger than the �xс with the 
train frequency  ν  = 2,5 s-1. As it is seen from picture the 
pulses of 5, 10 and 15 �sec durations doesn’t  change 
)(tfrel =τ  which is observed at absence of magnetic 
pulses. 
The results presented in Fig.5 show that at durations of 
pulses in 20�sec, 30�sec and 40�sec the Abrikosov 
vortices penetrate into the superconductor. This way the 
threshold value on the magnetic pulses duration observed 
at the action of single pulses  (Fig.4)   coinside with the 
threshold observed when their repetition frequency is  ν  
= 2.5 s-1. 
Fig.5. Dependence of momentum  
relτ on time  t  after the stopping of 
rotating head with the influence since  t=5 min  on the relaxation 
process of  HoBa2Cu3O7-δ sample of the continuous magnetic field 
pulses      h=172 Oe  with  ν=2,5 s-1  frequency and different                            
durations     x=5;  10;  15;  20;  30;  and 40 �s.  Pulsed magnetic field 
was parallel to the main magnetic field  H=1000 Oe  at  T=77K.   
 In Fig.6 it is presented the curve of  relτ =f(t) 
dependence on time at the influence of magnetic pulses  �h=172  (�h||H)  the duration of which is below the time 
of Abrikosov vortex system creation �x=5�s<�xс 
(�xс≥15�s for the investigated HoBa2Cu3O7-δ). As it is 
seen from the picture when �x<�xс, the relaxation curve 
doesn’t change in spite the increase of the repetition 
frequency of magnetic pulses ν  from 2.5 up to 500 s-1. As 
soon as the duration of pulses exceeds the critical value 
and becomes  �x=30�s, the relaxation curve undergoes 
the essential  (stepwise)  change. For example in Fig.6 it 
is presented measurement for ν=5s-1  и ν= 500s-1. 
Fig.6. Dependence of momentum  
relτ on the time  t  after the 
stopping of rotating head with the influence since  t=10 min  on the 
HoBa2Cu3O7-δ  sample relaxation process of  the continuous                       
pulses magnetic field  with frequency ν=2,5 ÷500s-1  at �x=5�s< �xс , 
and also at �x= 30�s > �xс. The pulsed magnetic field �h=172  Oe was 
parallel to the main magnetic field  H=1000 Oe   at  T=77K.   
And finally, we have observed the threshold on the 
value of applied pulses. In Fig.7 it is shown that in spite 
the fact that we applied magnetic pulses of the  large 
duration 300�s>>�xс, much longer as compared with the 
time of Abrikosov vortex creation at small amplitudes of 
pulsed field  �h ~7, 11, 14 Oe  relτ =f(t) doesn’t change. 
The stepwise change of the relaxing moment 
relτ is 
revealed only at �h ~18 Oe  and higher. 
       
Fig.6. Dependence of momentum  
relτ on the time  t  after the 
stopping of rotating head with the application after 5 minutes on the 
HoBa2Cu3O7-δ  sample relaxation process of  the single magnetic                    
field  pulses �h=(7÷36)  Oe  with duration �x= 300�s >>�xс.                             
The pulsed magnetic field was parallel to the main magnetic field  
H=400 Oe at T=77K. 
The further investigations of relaxation phenomena are 
anticipated for anisotropic high-temperature 
superconductors among them in strongly anisotropic 
high-temperature superconductors of  Bi-Pb-Sr-Ca-Cu-O  
system.   
4. Conclusion 
      
The simple mechanical method of Abrikosov vortex 
stimulated dynamics investigations it was applied for the 
study of pulsed magnetic fields influence on relaxation 
phenomena in vortex matter of high- temperature 
superconductors. It was observed the change of 
relaxation processes in vortex matter as a result of pulsed 
magnetic field influence on it.   
    The study of influence of different duration and 
amplitude pulsed magnetic fields influence was revealed 
the existence of threshold phenomena. A small duration 
pulse doesn’t change the course of relaxation processes 
in vortex matter of isotropic high- temperature 
superconductor HoBa2Cu3O7-δ. When the duration of 
pulses exceeds some critical value (threshold), then their 
influence change the course of relaxation processes. The 
latter is revealed in a stepwise decrease of relaxing 
mechanical momentum
relτ , apparently, related with a 
sharp change of pinning and the rearrange of vortex 
system of superconducting sample as a result of 
penetration into its bulk of a new portion of vortices at 
application of pulsed field on the outer magnetic field 
creating the main vortex structure in the investigated 
HoBa2Cu3O7-δ sample. A new portion of vortices 
“shakes” the vortex lattice existing in a sample causing 
the detachment of vortices from a weak pinning centers 
what, apparently, is the reason for the stepwise decrease 
of mechanical momentum
relτ . 
 All these made it possible to define the Abrikosov 
vortex lattice creation time in HoBa2Cu3O7-δ which 
turned out to be on the order of value higher as compared 
with the time of single- vortex creation observed in type 
II superconductors. 
  Acknowledgements 
     
The work was supported by the grants of International 
Science and Technology Center (ISTC) G-389 and G-
593. 
                             
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