quantum tunneling of vortices through the pinning barriers. The theory of
quantum collective creep has been developed by Blatter, Geshkenbein, and
Vinokur ...
Research Collection
Doctoral Thesis
Quantum flux creep in high-Tc superconductors Author(s): Aupke, Klaus Publication Date: 1995 Permanent Link: https://doi.org/10.3929/ethz-a-001513585
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ETH Library
Diss. ETH No. 11259
Quantum Flux Creep in
High-Tc Superconductors
A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the
degree
of
Doctor of Natural Sciences
presented by
KLAUS AUPKE
Dipl. Phys. ETH bom
on
the 28th of citizen of
accepted
on
September
1964
Germany
the recommendation of
Prof. Dr. A. C. Mota, examiner Prof. Dr. J. Blatter, co-examiner
1995
meinen Eltem
Contents
Abstract
1
Kurzfassung
5
1
Introduction
2
Theory 2.1
3
4
Creep
9
13 in
2.1.1
Classical
2.1.2
Quantum Creep
16
Creep
19 24
Layered Materials
2.2
Creep
2.3
Hall Tunneling of Vortices
2.4
Quantum Creep
in
14
Anisotropic Materials
in
26
High-Tc Superconductors
with Columnar Defects
.
28 35
Experimental Arrangement 3.1
Low-Temperature Measuring System
35
3.2
High-Temperature Measuring System
37
41
Experimental Results 4.1
Sample Descriptions
41
4.2
Experimental Procedure
46
4.3
Magnetic Relaxation
4.4
in
a
Single Crystal Measurements
of
YiBa2Cu307
(T
> 4.2
48
K)
48
4.3.1
High-Temperature
4.3.2
Experimental
4.3.3
Low-Temperature Measurements (7mK
0
proportional
die
£
zu
und ho¬
den Defekten
ungefahr
einen Fak-
„Columnar
wird und der Radius gro¬
Smith, Caldeira
worden
„Columnar Defects" anwendbar
Defekten
jedoch, da8
wir
in dem ein
Quantenkriechrate
„Quantum Collective Creep" gefunden mit
um
Modell,
von
mit
einer Do-
Temperaturen
c-Achse und
0 in der bestrahlten Probe
Stromdichte jc. Andere Parameter wie pn und
Bestrahlung
bei hohen
Temperaturen beobachten
dafi die Resultate fiir die
Supraleitem
von
—
Kohaienzlange,
gezeigt worden, Theorie des
Pinnings
ist als in der unbestrahlten. In einem
Defect" durch ein scharfes
YjBa2Cu307-Kristall
580-MeV-Zinn-Ionenbestrahlung
mit dem au£eren Peld
Quantenkriechrate
in einem
erzeugt wurden. In Proben mit dieser Art
hen Feldern berichtet worden. Bei tiefen
die
Ubereinstimmung
Experiment.
Schlie81ich haben wir
in der
in zufriedenstellender
fur T
-*
sind, auch
und Blatter
0, die in der auf den Fall
sind. Gema£ diesen Resulta-
zur
Wurzel
aus
der kritischen
werden nicht wesentlich
von
der
der verwendeten Dosis beeinflugt. Da die kritische Stromdichte durch
Bestrahlung
um
ungefahr
einen Faktor vier erhoht
wurde, finden wir,
da£
unsere
experimentellen Resultate gut mit der in der „Quantum Collective Creep"-Theorie und
von
Morais Smith et aJ.
iibereinstimmen.
gefundenen v^-Abhangigkeit
der
Quantenkriechrate
Leer
-
Vide
-
Empty
Chapter
1
Introduction
Like in ture
other
no
almost the whole
superconductor,
can
sustain
a
moving by pinning
from vides
pinning, there
are
dissipation-free
current
centers. Besides the two
and quantum fluctuations.
only if the
vortices
leading
activation
[1,2],
to
One of the most
for T
—
has been
where vortices
creep.
made
by Mota
conductors
through
the
et a/,
[4-6].
in the
jump
over
the
pinning
barriers
by thermal
relaxation rates which vanish with decreasing temperature.
non-vanishing on
Chevrel
a
extrapolate
phase superconductor [3J.
to
The
flux creep at millikelvin temperatures have been
high-Tc superconductors, heavy fermion,
This has been
pinning
Creep
successfully described by considering only
0, has been made by Mitin in
first observations of
prevented which pro¬
important phenomena caused by the
The first experimental observation of relaxation rates, which do not zero
are
types of dynamical disorder in the system: thermal
type-II superconductors
thermal fluctuations
high-tempera¬
static, quenched disorder,
competition between quenched disorder and fluctuations is flux classical
of
to the vortex state. In the vortex state the super¬
superconductors corresponds
conductor
phase diagram
interpreted
barriers, and the
theory
satisfactory agreement with most of the
as
quantum tunneling
organic
super¬
of the vortices
of quantum collective creep, which is in
experimental findings,
9
and
has been
developed
10
Chapter
by Blatter, Geshkenbein,
and Vinokur
tum creep rate is enhanced in
normal state are
in classical
herence
type-II
length,
magnitude.
study
the
tization after
performed
H,
zero
2250 Oe.
of quantum creep in
anisotropy, quantum
superconductors YiBa2Cu30T
—
phenomenon
superconductors with their low normal state
and small
field
and
theory,
the quan¬
length, high
anisotropy. Therefore, high-Tc superconductors
We have therefore chosen to
We have
to this
with short coherence
superconductors
and strong
resistivity,
ideal candidates to
[7,8]. According
Introduction
1.
investigate quantum
resistivity, large by
creep is smaller
detail, whereas co¬
several orders of
creep in the
high-Xc
Bi2Sr2CaCu20I.
measurements of the relaxation of the remanent magne¬
and
cooling
the
cycling
At low temperatures, the
specimens
in
external field of
an
decays deviate only slightly from
arithmic-in-time law. The measured quantum creep rates
as
well
as
the
a
log¬
crossover
temperature separating the quantum regime from the regime of thermal activation are
in
satisfactory agreement
tum creep
theory.
anisotropy
in
In
with the values calculated with the results of the quan¬
particular,
Bi2Sr2CaCu20x
as
the
expectation
compared
has been confirmed that the strong
Y1Ba2Cu307
to
leads to
a
considerable
enhancement of quantum creep.
Regarding
technical
applications,
it is
high-rc superconductors. Very promising
important
results have been obtained
columnar defects into the material. Columnar defects of
damaged, non-superconducting
size of the vortex
irradiation. been
a
Particularly
reported to result
of the in
core.
in
YiBa2Cu307 crystal
3 T. In the
of
B$
to
the tracks,
of two
at a
irreversibility region
tin-ion irradiation at =
higher
we
a
material with
These tracks
optimize pinning
to
can
be
a
are
extended
cylindrical
and
fields,
We have
with columnar defects which
dose of 1.5
configuration
x
this kind of defects have an
enlargement
investigated quantum
were
introduced
10uions/cm2 corresponding
with the external field
find that the quantum creep rate for T
in the irradiated
tracks
diameter of approximately the
considerable enhancement of pinning and
plane.
by introducing
generated by high-energy heavy-ion
high temperatures
in the H-T
in the
to
parallel
—
0 is
a
by
matching
field
to the c-axis and
by roughly
specimen than in the unirradiated
creep
580-MeV
one.
a
factor
Chapter
1.
Introduction
This thesis is
U
organized
as
Chapter
follows:
2 summarizes the results of the
quantum collective creep theory for anisotropic and layered superconductors for low and moderate
the
superclean
Furthermore, defects
are
magnetic
recent results
described.
In
presented investigations measurements in
crystal chapter gives
a
of
fields in the limits of strong
dissipation
as
well
as
in
limit, where the Hall term of vortex motion becomes dominant. on
quantum
chapter
are
3 the
described. In
single crystals
of
creep in
superconductors
experimental arrangements chapter
YiBa2Cu307
4
are
discussed and
compared
summary and conclusions of the
we
and
Y1Ba2Cu3C>7 with columnar defects down
5 the data
with columnar used in the
present magnetic relaxation
Bi2Sr2CaCu20I and
to
single
millikelvin temperatures. In
with theoretical results.
presented
a
work.
Chapter
6
Leer
-
Vide
-
Empty
Chapter
2
Theory
A
theory
case
for
tunneling of vortices in bulk superconductors has
of strong
been
developed for the
dissipation by Blatter, Geshkenbein, and Vinokur [7]
within the frame¬
work of collective
pinning theory j9]. The effects
of
anisotropy, layering and finite
temperature have been studied by Blatter, and Geshkenbein in the low and interme¬ diate field regime, where
single
vortex
Geshkenbein, Larkin, and Levit ductors may
tunneling
belong
of motion of vortices. For the
by
Morais
mainly
[10] pointed
out that
to the class of very clean
of vortices is shown to be
quantum creep
collective pinning is relevant
rate has been
case
of
high-temperature
materials,
in which
governed by the Hall
superconductors
derived by Vinokur
[8]. Feigel'man,
case
term in the
with columnar
supercon¬
quantum
equation
defects,
[11], by Radzihovsky [12],
Smith, Caldeira, and Blatter [13]. In this theoretical overview,
follow the references
given above
as
13
well
as
we
the work of Blatter et aJ.
the and
will
[14].
14
Chapter
The ered
in
Creep
2.1
new
oxide
superconductors
between the directions
sentially isotropic behaviour description
in terms of
cable. On the other
is
Lawrence-Doniach model
of the ered
physics,
superconductors"
perpendicular
for very
[15].
applicable
strongly layered Bi
and Tl
in terms of
family
by the
mass
effective electronic
them, respectively.
superconductors, direction
A
over
are
a
magnetic field
H
an-
and
es¬
layers
theory
is
of
set of
weakly coupled
appli¬
supercon¬
provided by the discrete
is
an
anisotropic
continuous
accurate
description
an
"lay¬
term
materials, which have
to be described
general,
anisotropic descrip¬
a
continuous
to the class of
the
by
more
materials, for which
requiring
a
discrete
=
m/M
£sf/A.
to
zero
temperature gives
et al. conclude that HTSCs
In this limit
they
free
mean
indeed
might
obtain effective Euclidean
actions of
for the 2D
for the 3D
case
result
case.
and in
(2.15)
Here
£2
S/h
is the
£2LC
case
are
is of
It is
enclosed
by
the
superconducting
one
to notice that in the
interesting
is the volume enclosed
trajectory of the the
by
vortex
trajectory
of the
multiplied by the respective superfluid densities, cases
course
The estimation
course
per
are
given by
the number of
particles
the vortex.
important question is of
(2.14)
superfluid density
area
in the two
by the trajectory of
types of tunneling. tion
case
expressions
the Euclidean actions
An
(2.16)
superfluid density.
for the 3D
vortex. Since these
enclosed
?l~?:?Lc^nseLc denotes the
ns
for the 2D
(2.16)
(2.15)
and
and ns is the 3D
layer
5£~f^~,m«?
how to
distinguish
given by Feigel'man
between the different et ed.
based
rough, because the used parameters
very
gap, the Fermi energy, the coherence
length,
and the
mean
free
as
path
equa¬
on
the energy are
not
eas¬
ily accessible quantities and therefore rather poorly known. Choosing only slightly different parameters and i.e. that the viscous An
drag
combining them,
important qualitative difference between the
tum creep rate.
it is
equally
well show that
coefficient and the Hall coefficient
predicted concerning the low-temperature
T2,
one can
Whereas in the
exponentially small,
i.e.
two
case
proportional
of the
—»
1,
order.
0 limit of the quan¬
this correction is
to exp
same
~
types of tunneling however is
correction to the T
dissipative
are
wr
(—To/T),
proportional
in the Hall
case.
to
28
Chapter
Quantum Creep
2.4
in
High-Tc Superconductors
Theory
2.
with
Columnar Defects
The vortex
dynamics
in
high-Tc superconductors
twin boundaries and artificial columnar defects
sample
high-energetic heavy-ions,
with
Vinokur
[25]
and
by
modeled
by
and a
by
Morais
Smith, Caldeira,
cylindrical cavity with radius
the saddle point
where Vb represents the vortex. The liberation of
of
a
finite segment of
well, creating
a
half-loop
[11], by
Vinokur
by
[13].
rate is determined
-(|c% + e(?2)|u(q,u;)|2 + Vb(q,u)
L and
a
transverse
excitation of the
at
Radzi-
A columnar defect is
tunneling
ej
=
e2«o
vortex line from its rod takes
length
Nelson and
by
action
pinning potential and a
theoretically by
and Blatter
r^. The
trajectory of the Euclidean
27T27T /dojdq
as
Vortex creep from columnar defects
very low temperatures and low fields has been treated
hovsky [12],
disorder, such
from irradiation of the
resulting
has been treated
[14].
Blatter et al.
with correlated
(2-17)
is the line tension of the
place
via thermal activation
displacement
u
out of the
pinning
string. The energy barrier for creating
a
half loop of length L is given by
U
where
eT is
balancing
the
eT
L
depth of pinning potential. The tunneling
time tt is determined
the kinetic energy term against the elastic term in
The saddle
point
optimal
ratio between L and
energy and the
(2.17)
solution of the Euclidean action then follows
h
The
=
pinning
energy
h u
which
by
yields
as
hsi
is determined
by
the
equality
of the elastic
2.4
so
Quantum Creep
that the
in
with Columnar Defects
High-Tc Superconductors
half-loop
of the
geometrical shape
is determined
29
by
fir
u _
The Euclidean action then reads 5
v^tf
_
H where e(
=
e2s0
and r/i
=
0HC2/p„c2
e2pn \£j
~
ft2/e2pn£2
the Lorentz force and the
c
Using jo
a
-=-, one
that
In order to determine which a
=
length
i
\
Upin(x)
are
equal.
u
£jo
_
(*\" Ik
1* scale
J
=
we
have to
pinning
energy
p(x)
is
a
=
I
dx'
use
a
we
radius ro
begin by considering
>
f
> r°
a
vortex is
U^(x')p(x'
form function which for the
for u,
else
0
Ep^x) experienced by
Epin(z)
lxl
~5r
£
and
j
case
we
have ro
same as
potential for t/p;n,
now
r0 >
£
the relevant
2 double
A. The investigated YiBa2Cu307 single crystals
technique which
the
by
a
the lattice parameters
L. Krusin-Elbaum at IBM Yorktown
an
fully oxygenated YjI^CusOt
samples x
parallel
1.7
x
are
approximately
0.7
x
1.8
0.025 mm3 for the irradiated
to the shortest dimension.
41
x
0.03 mm3
crystal with
The latter
one
has
42
Chapter
Experimental Results
4.
Cu
•
4.1: Structure of
Figure
o
o
O
Ba
@
Y
fully oxygenated YjBa^CusCV
been irradiated with 580-MeV u6Sn30+ ions with the incident beam at 2° to the c-axis
[40
and references
therein].
This irradiation produces permanent
in form of linear tracks with diameters of
fects in
YiBaaCusOr
aligned
with the initial beam direction. The
tion dose of 1.5
x
which is defined columnar defect
10u as
ions/cm2.
This
sample
corresponds
has been
to
a
approximately
exposed
matching
occupied by
transition temperatures
are
one
vortex
approximately Tc
carrying c±
one
to
an
field of 5*
the magnetic field that would thread the the
were
damage
sample,
flux quantum
50
ef¬
A
irradia¬ =
3T,
if each
$0. The
94.5 K for the unirradiated
crystal
4.1
43
Sample Descriptions
0.2
t i
i
i
|
i
i
iT-rri"i
i
jttt
i
|
i
i
i
i
f
i
i
i
rj-1
0.0
'S
r
rrfi
o
o
o
o
o
o
o-
-0.2
I
J-
-0-4
-0.8
?*
1.0
-o
-1.2 90
o
o
91
o
o
o
94
93
92
96
95
97
98
100
99
T(K) Figure
4.2:
susceptibility
ac
YiBazCuzOj single crystal.
The
arbitrary
for the normal state and -1 for the
complete
of the
units have been chosen to be
zero
diamagnetic signal.
0.2
i
-|—i—r—.—|
I—p
|
i—i
i
|
i—i—i—|—i—i—i—r—'—'—"~
0.0
units) itrary (arb
ODOOOOO o o o
o
-0.2
-0.4
-0.6
-0.8
-1.0
OOOOOOOCOO'
-1.2 86
-J
I
I
88
I
I
1
1—I
I
90
92
1
'
L_J
94
'
I
L-J
96
I
I—I
1—
98
100
T(K)
Figure
4.3:
defects with
ac a
susceptibility
matching field
of the of
Bf
YiBajCusOr single crystal with =
3T.
columnar
44
Chapter
40
4.
Experimental
Results
50
T(K)
Figure for
an
4.4:
Temperature dependence
applied
crystals from the represent
an
6eldofB*
Figure
of the critical current density in the
Seld of H =1T with H same
batch
as
the
ones
=
same
in
the c-axis
in this work.
irradiated
an
ab-plane
[39] for Yj B32 Cuj O7
crystal
The squares
with
a
matching
3T.
4.5: Fieid
from tie
to
investigated
unirradiated crystal, the circles
dependence of the critical
temperature of T =5KforH parallel
same as
parallel
Fig.
batch 4.4.
as
the
ones
current
to the c-axis
investigated
density in the ab-plane
at
a
[40] for YiBazCusO7 crystals
in this work. The
symbojs
are
the
Sample Descriptions
4.1
and Tc
of the
~
ac
95 K for the irradiated
from
are
magnetization from the
samples
at
a
field of B$
at T
=
a
5 K is shown in
considerably 5 K and
Fig.
=
ones
et
al.
an
For all
[40]
and
Fig.
by
by
measurements
from the
seen
density jc
a
10 %
field,
the unirradiated
jc is about
a
factor of 4
higher
for
work. The temperature an
irradiated
crystal
dependence
and temperatures, jc is
enhanced due to the presence of the columnar defects. In
zero
[39]
et ai.
4.4 and the field
investigated fields
graphs
criterion,
has been determined
Thompson
unirradiated and
3 T is shown in
4.5.
be
investigated in this
1T for
=
can
transition width, defined
by Civale
the
field of H
dependence of jc with
matching
as
a
The critical current
measurements
batch
4.2 and 4.3. As
Figs.
specimens.
same
and have been determined
one
sharp with
very
0.5 K for both
>
-0.4
03
a 3
: -0.6
f
•5-
°
J
-0.8
-1.0
:
'-
o .
60
65
.
i
.
70
75
80
85
90
95
100
105
T(K)
Figure
4.8:
ac
susceptibility
of the
units have been chosen to be
diamagnetk signal
zero
The
arbitrary
for the normal state and -I for the
complete
BijS^CaC^Ox single crystal
Chapter
48
high-temperature cryostat TS90,
In the the
sample
up to
Tc.
decay
measurement
is thus
sample
measurement of the
plus
the flux
changes
flux
The total remanent
Experimental Results
4.
be recorded while
can
at the
magnetization
heating
beginning of the
given by the flux difference recorded during the
being
released from the
sample during
the warm-up
toTc.
Magnetic Relaxation in a Single Crystal of YxBa2Cu307
4.3
High-Temperature
4.3.1
Remanent
In
Figs.
Magnetization
field of Ht
given
=
Fig.
reaches
increasing temperature. Fig. constant
strongly
with
temperature of T
increasing cycling
This behaviour the critical state. balanced
by
gradient
which
density flux
the
the
so
implies
a
of
for
simplicity.
K, Mrem increases with
field
of MKm
dependence
density jc(T).
of the local
magnetic
The
a
field
pictures (A)
to
magnetization
acting
lossless
flux
density
macroscopic
current
Bean assumed field.
are
(C)
a
a
critical current
Qualitative pictures in
an
shown in
for
the vortices is
on
field leads to
magnetization
to the external
flux distribution of the remanent
[45]
described with the Bean model
sustaining
distribution of the remanent
parallel
function of
dependence of Mlem for
10
cycling
change of magnetic
capability
independent
a
field.
force and
the
as
The remanent magnetization increases
4.2 K.
qualitatively
be
is shown
In the critical state, the Lorentz force
pinning
slab of thickness d
neglect Hci
approximately
4.10 shows the
=
called critical current
which is
density
can
K)
maximum at around 30 K and decreases then
a
with
density,
magnetization Mrem
Above
at
a
> 4.2
4.9 shows the temperature
2250 Oe.
increasing temperature,
(T
Function of Temperature and Field
4.9 and 4.10 the remanent
temperature and field. a
as a
Measurements
infinitely
Fig.
4.11.
of the
extended
Here
we
in the first line represent the
for three different
cycling
fields Hi,
4.3
Magnetic Relaxation
1200
in
a
""Ill IT)
Single Crystal
1
1000
°
•
*a
ITTTI'I
of
1
'
49
YiBaiCu307
"
"
iT-i-n 111111 i-i-fTi
1
ii
o
8oo
i
3
0
.
,
3
600
o
_
^
_
o
B
o
400 -cPo
200
-
0
-
11111
0 0
1..,.
1
70
60
50
40
30
20
10
-
80
90
100
T(K) magnetization
Figure
4.9: Remanent
tion of
temperature for
a
cycling
of tie
field of Hi
YiBa2Cu$07 single crystal =
as a
func¬
2250 Oe.
700
O
600 :
:
500
•a 3
400
e
g
"
O
:
300
O
•
200
100
O
:
0 0
2000
1500
1000
500
2500
3000
Ht (Oe)
Figure
4.10:
Remanent
function of tie
magnetization
cycling Geld
at
a
of the
YiBaiCusOi single crystal
temperature ofT
=
4.2K.
as
a
50
Chapter
#.djc
4
Experimental
—
=
H=0
Results
H
^7 crystal.
right
order of
which is in
magnitude,
measured data. A model which takes flux
MnJt=to) ^"f
=
penetration
correction of
proposed
into account is
Relaxation of the Remanent
a
satisfactory qualitative agreement
demagnetization appendix
and discussed in
with the
effects due to
A.
Magnetization
The relaxation of the remanent
magnetization
has been
investigated
in the
high-
temperature measuring system TS90 for temperatures between 4.2 K and 18.5 K. In this temperature range, the in-time behaviour
as
described
Therefore, the slope —T
decay
by
on
the chosen
slope always
fitting
deviate
slightly
the classical flux creep
of the
omt
slightly
curves
decay
curve
in
a
from the
theory of
logarithmic
logarithmic-
Anderson scale
depends
interval. We have therefore chosen to determine the
in the time window from 10 to 100 seconds. The normalized rates
S
[1].
dMKm(t) =
MKm(t
=
to
dint
t0
=*
10s
4.3
are
Magnetic
Relaxation in
temperature with S
on
~
Experimental
4.3.2
As mentioned in section
result,
a
certain
and hence
at 4.2 K and
typically
1
% will
up to
is not in the
exposed
vortices is
for by
critical state at temper¬
to
sample
a
flux
the
a
density gradient
during
observed time window.
the relaxation is not detected
in fact counted in the total remanent
[46],
2250 Oe. As
=
and for relaxation rates
experimentally
sample and
magnetization,
therefore detected
relaxation from the
a
incomplete
in the normalization
are
by
during
relaxation rate is therefore smaller
Tc. The measured normalized
considering
0.1%/K.
to the inside instead of to the outside of the
not leave it within the
are
dependence
linear
a
field of Ht
trapped
force, pointing
Pollini et ai.
fully
cycling
than the rate Sf that would be obtained in
proposed by
55
Flux Penetration
40 K with the used
because all the vortices have to leave the
As
slope
4.3.1, the sample
measuring system, they
warming
of about
Incomplete
Whereas the movement of those vortices
the
a
Those vortices relax to the inside of the
sample. of
a
Y1Ba2Cu307
Error Due to
part of the
Lorentz
%
0.5
approximately
atures below
of
Single Crystal
4.14. In this temperature range, the data show
Fig.
shown in
a
fully
flux penetration
only those
vortices
can
subjected
critical state. be accounted
to
outward
an
pointing flux density gradient
For the
of
case
Fig. 4.11),
state.
For
a
calculate. It
2, with
r
=
and
Mtma*a(t
infinitely extended slab,
an
((a)
undercritical state in
d\nt{1
dint
sf\T)
,
the relaxation rates
>l-f(T)
the are
5.3
Quantum Creep
Thermal Enhancement of
10
T—T-TTTTTP|
1
jt-tvry,r-r,i
-r"T"l
II
[ttti |
i
79
1
ll|
i
i
i
.
.
1—rTTTIT|"
| iti-i"|
i
CO '
0
I
I
t
1
t
.
I
I
I
-CL¬
-©-
10°
io-i
IO"2
O
/
5
4
Tr
io-!
I
3
12
101
T(K)
Figure
5.2: Fits to the normalized relaxation rates for the
The continuous
experimental
curve
to the law for the
corresponds
data up to 4.2K. The dotted and the dashed
the law for strong dissipation, where the dotted 2.5K and the dashed a
In
curve
is
a
Figure
5.1
we
show
fits
at 7mK has been used for
is
a
to
the
Sts with
fit to the data up to
£t up to 4.2K. In the inset the Sts
of these
laws to the
are
shown in
the limit
——(T
—>
been used
as
the
points
only
—
400
500
T(mK)
Figure
5.9:
as a
function of temperature for relaxation
curves
of the remanent magnetization in
UPtg single crystal after cycling the
sample in
Time-stretch exponent
a
field of Hi
below 350 mK. In been
cycled
to the
up to
temperatures
as
-
in the first
typical
low
as
-
Mrem(oo)
=
curves are
can
Ht
=
-
33 Oe
a
105s.
of the initial remanent
{MKm(0)
-
stretched
Mrem(oo)] exp
x
sample
has
applied perpendicularly
10 mK, the remanent
half of its value in about 1.8
the fitted fraction
shown after the
be fitted with
[Mrem(0)
approximately
Mrem(104 s)] /Mrem(0) 104s,
relaxation
c-axis. The decays
MKm{t)
approximately
[Mrem(O)
5.7
[31].
33 Oe
maximum external field of
a
crystallographic
law of the form
to
Fig.
=
0
exponential
[(-t/r)"]
[31].
At
magnetization decays
Values of the
quantities
magnetization which decays
MTsm(oo)]/Mrem(0)
which decays
until the saturation of the relaxation process, and the fitted stretched exponent are
a
shown in
non-zero
Figs.
5.8 and 5.9.
logarithmic
The
decay
increasing cycling fields,
rate at short times
at very low fields and which
bulk.
With
can
(see Fig. 5.10)
be attributed to the
of the bulk vortices is
thermally
the
decays
/?
show
which is not observed
decay
activated.
of the vortices in the
On the other
hand,
Quantum Creep
5.5
in Other
Types of Superconductors
r-r-rT7Tnj•"^
'
1
'
-T-l
lUFf
1
|
nil[
|
91
—i—n
irmi
nrm
1—i
«
e°
°
~
"""
10
o
d
^^**^-^^
-i
"—
:l.
rH
-2
II *4
if
"a
rf-3 i
^
I
i
I
**•*%. x.
-2
\
\
f-6
-4
1
^
ST-5
10°
\
'.
101
102
104
103
t(a) ""1
'
1
-
'
Ill'
1
8
\
-
\
s »
—
i
3"4
•
^.t_
^\.
II "
—.
1_'
1
'
"'
'
101
100
''
1
102
"'
1
'
'"!
103
t
104
105
t(s)
Figure T a
5.10: Relaxation
450 mK after
=
a
curve
smaller cycling field Hi
in the
same
of remanent
cycling £eld Hi
arbitrary
=
3.4 Oe. The remanent
depth and £bcs
obtain
2±
jo
107A/cm2
=
action
0
[31]
and pn
Sg '\
~
~
4/iQcrn [69],
of the order of 105 ft
5
by several
orders of
the
of
theory
QCC.
x
we
a
origin
has been
suggested by Sigrist
curve
is
regime —
3600
is
by investigating
[71]
given
cannot be
A
for the
length [69],
density. Neglecting
(2.9)
an
we
the
density
effective Euclidean
creep rate of the order of
10~3 %. So this
temperature independent decay which
is
theoretically obtained from
creep of vortices close to the surface may
in the unconventional nature of the order parameter in et al.
at
for
sample).
for the critical current
than the value
anomalous, giant
have its
superconductor
regime
103A/cm2
current
calculate with
or a
magnitude higher
The
at the
theory. Using Al
depairing
Itheor.
material exhibits in the low-field
similar
200 A for the BCS-coherence
for the
anisotropy [70] and using jc
small at T
value of
=
a
rates in the low-field
with the quantum collective creep
London penetration a
decay
UPt3 crystal
the
magnetization M„m
decays (flux quanta 0q
units for both
the strong, temperature independent
explained
magnetization for
680 Oe. In the insert,
—
that
one
way of
surface effects.
probing
an
UPt3.
It
unconventional
A similar behavior
as
in
UPt3
92
Chapter
-1—I—I—r—H'l
0.020
'
1
i
1
i
-i—i—r
•
III'
7
materials
T
that in this
the influence of temperature,
as
lay¬
e2p„
study the dependence of quantum
anisotropy
so
of
_
h
In order to
d
case
is smaller than the
pinning length LI
replaced by
has to be
In the
density.
current
cooling
and
cycling
At low temperatures, the
the
specimens
decays
rithmic-in-time law. The measured quantum creep
deviate
in
an
external field of
only slightly from
rates at T
—>
0
are
a
loga¬
approximately
0.15% for YiBa2Cu3C>7 and 1.7% for Bi2Sr2CaCu20I. Considering the experimen¬ tal uncertainties and the
theory and experiment
approximations made
is
satisfactory.
In
in the
theory,
particular,
firmed that the strong anisotropy in Bi2Sr2CaCu20I leads to
a
the as
considerable enhancement of quantum creep.
the agreement between
expectation compared
to
has been
con¬
YiBa2Cu3C>7
Chapter
6.
Summary and Conclusions
A further
97
point of comparison between theory and experiment is the
temperature from the quantum regime result of the quantum collective creep
T ,c
where Uc is the
pinning potential, This
ature relaxation data.
Tqc
4K for
~
theory
~
regime of thermal activation. The
is
~V*
h
Sf(T
kB
which
expression
Bi2Sr2CaCu20j.
to the
crossover
be determined from the
can
leads to
This is also in
0)
=
Tqc
1.5 K for
~
high-temper¬
YiBa2Cu307 and
satisfactory agreement
with the exper¬
imental observations.
Regarding
technical
applications,
it is
important
high-Tc superconductors. Very promising results
damaged, non-superconducting material
the size of the vortex
core.
diation. Particularly at
reported the a
to result in
a
YlBa2Cu307 crystal a
They
in the H-T
dose of 1.5
x
plane.
to
the
tracks,
we
of two
higher
in the irradiated
fects,
columnar defect is modeled by
of the columnar defect is
larger
the quantum creep rate for T critical current
by
pinning
We have
and
—»
case
a
introduced to
parallel
—*
0 is
superconductors
—>
0 should be
to the c-axis and
one.
density jc. Other parameters
as
p„ and
£
a
factor
On the theo¬
and Blatter that
theory of quantum with columnar de¬
and the radius
length. According
proportional
creep in
matching field
a
sharp square-well potential
than the coherence
by
of
580-MeV
by roughly
0 obtained in the
of
enlargement
an
investigated quantum were
irra¬
of defects has been
by Morais Smith, Caldeira,
collective creep is also applicable to the a
fields, this kind
specimen than in the unirradiated
been shown
cylindrical
diameter of approximately
the external field
the result for the quantum creep rate at T
if
and
find that the quantum creep rate for T
recently
by introduc¬
extended
10nions/cm2 corresponding
configuration with
=
a
are
in the
generated by high-energy heavy-ion
with columnar defects which
B$
retical side it has
be
with
considerable enhancement of
of
3 T. In the
can
high temperatures
irreversibility region
tin-ion irradiation at
have been obtained
into the material. Columnar defects
ing columnar defects tracks of
optimize pinning
to
to this
result
to the square root of the are
the irradiation of the used dose. Since the critical current
not
strongly affected
density
is enhanced
by
98
roughly
Chapter
a
factor of four in the irradiated
one, the enhancement of the two in the irradiated
with the
quantum creep rate
specimen
y/Jl dependence
specimen
as
compared
to
as
6.
Summary
compared
at T
—
0
to the unirradiated
by roughly
the unirradiated
of the quantum creep rate found in the
collective creep and in the work of Morais Smith et al.
and Conclusions
one
a
factor of
is in agreement
theory of quantum
Appendix
A
A Model for
Demagnetization
Effects in
the Critical State
In order to interpret the
simple
magnetization
Bean model is often used.
extended slab
simple
results.
sample
in
an
or
Particularly
cylinder parallel
For
example,
to
up to
a
to the
the
perpendicularly
samples to the
are
the
externally applied field by
factor which is given by the correction is the
of revolution
following:
parallel
of
cases
a
factor
1/(1
to the
a
many
—
D),
or
so
simplest
geometrical shape For
after
as an
a
platelets
that strong
way of
virgin zero
or
demagnetization is to correct
demagnetization
sample. The reasoning for
externally applied field H",
is
and the external
doing this
where D is the
of the
gives
experiments with high-Tc
homogeneously magnetized ellipsoid
99
infinitely
the model
cycling
the
infinitely extended slab
an
flat disks
large surface,
effects have to be taken into account. The
applied field,
magnetization
externally applied field. In
superconductors however, field is applied
simple geometries
maximal value Hi and back to
given by the equations (4.1) and (4.2) for the
cylinder parallel
for
the externally
the remanent
externally applied field
type-II superconductors,
in irreversible
it
can
with
one
this axis
be shown that the
A.
Appendix
100
A Model for
effective field HeS in the
sample
state
BeS
have
we
magnetization 0,
=
H"
=
of the
that 0
so
|i0Beff
=
anymore because BeB
simply given by presence of
—
as
a
increasing external
an
simple
case a
into account.
calculated or
model for
HeS(H")
Once
as a
applied
obtains
case one
equation (A.l)
in
expression
is not
Furthermore in the
differently. is not
applied
cannot be
homogeneous
simple demagnetization
a
and the
factor any¬
for the effective field HeS
in the Bean critical state for the to the
demagnetization has been
Afeff
and
sample
an
applied field
field is
in the Meissner
found,
We propose for
virgin sample.
effects which takes flux the remanent
case
penetration
magnetization
can
be
(4.1)
function of the externally applied cycling field using equations
(4.2). We
use
the Bean model for
order to estimate the
approximated by
a
an
infinitely
extended slab
dependence
and any field
field, neglecting Hc\
demagnetization factor,
flat rotational
this geometry with the field is
by
is to find
appendix
Meff. In this
sample, this formula
in the
effects cannot be treated of this
+
determined
to be
magnetization
function of the externally
where this
goal
The
the
superconductor
-T^d
longer fulfilled
no
HeS but has
pinning,
demagnetization more.
0 is
=
(A.l)
a
HeS
=
where vortices enter the
case
MeS
D
-
sample. In
H However in the
Effects in the Critical State
given by
is
tfeff where M*e is the
Demagnetization
parallel
to
of the critical current we
assume
ellipsoid perpendicular
that the to
the
the effective
density jc. In
sample
applied
can
field.
along the shortest dimension, the demagnetization
be For
factor
approximately given by D
=
l-\\
(A.2)
where t is the thickness and d is the diameter of the
part of Fig. A.l, flux profiles
HeS
=
H*, and (3a) H*s
We first consider the The field
profile
can
>
are
shown for the three
H* with H*
case
of the not
be considered
as
:=
sample [72].
cases:
(la)
In the upper
HeS
1.»
..
0.1
i
0.2
.1...tit,
12
M
0.4
0.3
0.5
3
4
H°/H* Figure the
A.3:
Eniancement factor between the effective magnetic field HeS and
externally applied Held H"
ratio of d/t
as a
function of H" for
a
diameter to thickness
20.
=
H*s
=
Ha+(l-^\H*
In summary, the effective field in the sample
as
a
(A.4)
function of the applied field is
given by 1
It 2d
H*
+
SI
H"
B-
Fig.
d/t
=
20.
is Ha
~
case
of
The effective field H*s reaches H*
More precisely, H^ reaches H* when if"
example
H*>fa
H*
A. 2 this function is shown in units of H* for the
thickness ratio of
f tic
H*
H*+(l-4) In
