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Influence of the electron-scattering mechanism on the critical current given by flux pinning at grain or twin
boundary in high T_c superconductors
M.-T. Lehoucq, R.-J. Tarento
To cite this version:
M.-T. Lehoucq, R.-J. Tarento. Influence of the electron-scattering mechanism on the critical current given by flux pinning at grain or twin boundary in highT_c superconductors. Journal de Physique III, EDP Sciences, 1994, 4 (2), pp.235-251. �10.1051/jp3:1994125�. �jpa-00249099�
Classification Physics Abstracts
74.60G
Influence of the electron-scattering mechanism on the critical current given by flux pinning at grain or twin boundary in high- l~ superconductors
M.-T. Lehoucq and R.-J. Tarento
L-P-M- CNRS Bellevue, I place Aristide Briand, 92195 Meudon Bellevue Cedex, France (Received 5 February 1993, revised 2 Noi>ember J993, accepted 22 November1993)
Abstract. The scattering caused by presence of grain or twin boundaries changes the local electronic properties of the material both in the normal state (mean free path, ...) and in the
superconducting state (coherence length, ...). The consequences of electron-scattering mechanism
on the critical current density given by ideal planar defects have been investigated for zero and large magnetic fields. For the case of large magnetic field, the critical current density dependences
versus the magnetic field and the defect distance have been carried out. For the case of a lo T
applied magnetic field strength and for defect spacing largely smaller than ? 000 A collective pinning is occurring, but for spacing largely larger than 2 000 h it is the strong pinning. Finally the
temperature behavior of the critical current density has been treated within the thermal flux creep model modified by the collective effects: I-e- the thermal activation energy is due to the
contributions of the pinning well depth and of the bundle hopping energies.
1. Introduction.
One of the most important industrial interest for the use of the high-T~ superconductors is linked with the property of lossless electrical transport if the current density is below a critical
value J~. In spite of the tremendous work on magnetic properties, the understanding of
J~ has not yet achieved a universal agreement. The reason is due to relationship existing
between J~, the defects and the microstructure of the high-T~ superconductors (twins, grain boundaries, local oxygen deficiency, secondary phases, amorphous strips, cracks, ). Indeed
experiments in sintered ceramics iii have established that J~ is limited by the weak-link behavior of grain boundaries. Furthermore, in a weak-magnetic field case, Dimos et al. [2]
have shown that all clean grain boundaries in YBa~CU~O~, except for very low-angle ones, are SNS-type Josephson junctions. But the situation is not so clear, in particular Larbalestier [3]
has reported results on high-angle grain boundaries which are reverse. Moreover materials such as epitaxial thin films grown on (100) SrTiO~ substrate do not indicate the existence of weak links with quasi Josephson-like properties [4]. In thin films, J~ has nearly the depaifing J~ value (J~ depairing1 5 x 10~ A/cm~) for non applied magnetic field and is still quite large
for large magnetic fields (I.e. for H
=
10-20 T, J~
=
10~-lU" A/cm~). This later behavior can be explained by the vortex pinning. As it has been mentioned in the previous ARC proceeding (I.e. Obrador and Senateur) [5] twins are effective for the pinning and has been intensively
studied using the high-resolution Bitter-pattem technique [6] and other methods (neutron scattering [7]). Moreover some works have been reponed on the vortex pinning at granular
boundaries [8] chiefly for low-angle ones.
The present article deals with the contribution of ideal planar defects to the critical current
and to the vortex pinning I-e- the study of planar defects with thickness largely inferior to the
coherence length such as clean grain boundaries or twins. Such clean grain boundaries have been observed at the atomic scale in high resolution transmission electron microscopy [3, 9].
The consequence of our choice is that the planar defect does not behave like a Josephson
junction (weak link) but like a barrier. The presence of this ideal planar defect, involving
electron scattering at the boundary, changes the properties of the material both in the normal
state (mean free path, ...) and in the superconducting state (coherence length, critical
temperature, .). So all the properties depend on the twin or clean grain-boundary distance.
Pinning in high-T~ superconductors by electron-scattering mechanism has been yet examined by Thuneberg [10] and Zandbergen et al. ill ]. In their approaches, they assumed a
planar defect which has a volume and which perturbe strongly the superconductivity in the
neighborhood of this one. In our calculation, we have considered a smaller perturbation given by the defect what it is true for twins and small misorientation clean grain boundaries.
In a first part, a phenomenological dependence of the mean free path and of the coherence
length is derived. The first investigated consequence is the temperature influence on the critical
current density given by a barrier (I.e. ideal planar defects) for no applied magnetic field. Then
the applied magnetic field case is treated. The formalism of the calculation of the elementary pinning force given by a single planar defect associated with the electron-scattering
mechanism has been developed. In part 4.3 the case of a set of planar defects has been
examined : I-e. the elastic deformation energy of the flux line lattice (FLL), the pinning energy
and the volume pinning force. The dependences of the critical current density both with the
magnetic flux density and with the defect distance have been derived. Finally the temperature influence on the critical current density has been investigated with the thermal flux creep model modified by the collective effects the thermal activation energy is the sum both of the
pinning well depth energy and the energy due to the bundle hopping.
2. Calculation of the mean free path and of the coherence length near the planar defect.
The high-T~ superconductors are compounds with a highly-anisotropic structure. They can be described as a stacking of planes (CUO~~ BaO~ Y and CUO for YBa~CU~O~). Like its structure~
their electronic properties are highly anisotropic and it is well established that their transport properties can be described by a 2D system (I,e, the electronic carriers are confined in the CUO~ planes). So the mean free path i~ is essentially two-dimensional. For the sake of
simplicity, the planar defect is assumed to be perpendicular to the CUO~ planes (the following part 4 gives an insight into our choice). Let x be the distance from the planar defect. The mean free path is obtained from
l~ P (r, o)r r)
~b" ~
~
~ (~)
P (r) dr
0
where P (r) is the probability of the electron surviving without scattering up to a distance r. In
the boundary free medium P (r) has the classical form
P (r)
= exp ) (2)
bo
where i~~ is the bulk mean free path. ( )
~
is the average over o (I.e. the angle between the
perpendicular to the planar defect and the electron path). It has been assumed that at the grain boundary or twin, the electron is scattered. So the dependence of the mean free path has the
following form :
ib(x) iboIi i
)l'~ exP ~~~li~~ d~ (3)
The two-dimensional evolution of i~(,t) is plotted in figure I and its change is spread over 200A. The distance dependence of the coherence length f and of the Ginzburg-Landau
parameter « are obtained within a framework developed by Yetter et al. [12]. So the variation of « is written as a function of Sand is given by :
Ax (x) i~(ibo)
~ f~(ib(x)) (4)
1,1
~ o.9
/m 0.8 w>
0.7
o.6
o.5
0 2 3 4 5
Xfl~~
Fig. I. Mean free path (in f~ unit) as a function of reduced distance between the grain boundary and
the vortex chosen as coordinate origin (x/f~~). Dashed line 3-D medium. Solid line 2-D medium.
The classical coherence length trends >.emus i~, at the temperature of 0 K, are the following I) in the clean limit :
f(T 0 K) 0.7~ So ~ ~
~~~
it) in the dirty limit :
f(T
=
OK
=
~°
/~ (6)
~ ~'~~
)
iii) in the intermediate zone :
7 i~
I(T
=
0 K)
= So z 'lj log ~ (7)
where So is the BCS coherence length and the coefficients 1~, are
l~o = 0.56551 1~1
=
0.17241 1~~
=
0.10153 1~~
=
0.05747 1~4 =
0.02518 1~~
=
0.02991
1~~ =
0.00237 1~~
=
0.00375
These behaviors are based on experimental ii 2] and theoretical (I,e. clean and dirty limits) [13]
considerations which are confirmed quite well with the high-T~ superconductors (I,e, for the intermediate zone).
3. Study of the critical current density in the case of no applied magnetic field : influence of the temperature.
The determination of the critical current density given by an ideal defect (cf. introduction) for
non applied magnetic field is a classical Ginzburg-Landau calculation and has been developed
by many authors. However their choice of the change of the spatial variation parameters
(«, p has been arbitrary. In the present article the parameter dependences have been derived within the phenomenological framework described in part 2. Let us summarize the idea of the
calculation which is based on the model of a planar barrier region across which the Ginzburg-
Landau parameters vary smoothly with position over several coherence lengths. It is assumed that the problem is one-dimensional. The order parameter of the Ginzburg-Landau theory is
written as ~ e'~ ~ and o
are fixed at large distance from the defect and their behavior are obtained by minimizing the Ginzburg-Landau free energy [14]. So they are the solutions of the coupled differential equations :
d h~ d~ h~ do
~ j2
~ 2 ~~ ~~ 2 ~~ ~ =2«~+2p~ (8)
J= ~~~~~~~° (9)
where «, p, m * are the Ginzburg-Landau parameters of the free energy.
In the following we have considered the effective mass m * constant. « and p are distance
dependent and are related with the changes of Sand « by :
f~= ~2
~
(10j
8w m*
«
K~=~ "
~~ (ll)
~
e
The behavior of f, «, A i>emus the defect distance are reported in figures 2, 3 and 4. The
0.705 0.7 o.695 0.69
i~ 0.685
iS
0.68 0.675 0.67 o.66s
0 2 3 4 5 6
~~b0
Fig. 2. Coherence length (in So unit) as a function of reduced distance between the grain boundary and
the vortex chosen as coordinate origin (x/fbo).
255
250
245
fl 240 235
230
225
0 2 3 4 5 6
~~b0
Fig. 3. Ginzburg-Landau parameter as a function of reduced distance between the grain boundary and
the vortex chosen as coordinate origin (,t/f~~).
equations (8) and (9) can be recasted into the following differential equation :
y"(x') £ J~
+ h(x') y(x') y3(x')
=
o (12)
27 y3(x')
168
166
ql' ~~
3
~
162
160
158
0 2 3 4 5 6
~b0
Fig. 4. London penetration depth (in fo unit) as a function of reduced distance between the grain boundary and the vortex chosen as coordinate origin (x/f~~),
fl 1>2
~ H t')
cy 2
With Y
- ~ l- t
,
x - n, h (x) - ii~ i-e- Hi - i
j
1(
32 e~ "~ ~'~j~ll~ J
[ ~'~'~~ 27 m * p (
The subscript w means the value at large distance from the defect. J is the critical current
density and is obtained by searching the greatest value which does not give oscillating or
unbounded solutions. A typical variation of the order parameter with distance is displayed in
figure 5 and shows a change with x over the range of 5 f~. J~ obtained by this model has to be
-fiv/"
i.o
no
o-o
0.7
o-o
o.5
~'~
0 I 2 3 4
4~- Fig. 5. Normalized order parameter ~ (- p ~la
~
)as a function of reduced distance between the grain boundary and the vortex chosen as coordinate origin (x/f~).
compared to J~ (depairing current) which is here the critical current density in the grain.
However the comparison with experiments is difficult : Dimos et al, have found a ratio
J~~z~~~~/J~
~~~~~
nearly of 0. I but their J~
~~~~~
is about 10~ A/cm~ [15] which is largely smaller than
the depairing J~ probably due to a small concentration of vortices in the grain,
The dependence with the temperature of the critical current density has been investigated with the following dependence for the coherence length
T 1'2
f(T)
= f(T
=
OK
~
~
~
(13)
C
The « parameter has been considered independent with the temperature, The barrier critical
current density behavior with the temperature is given in figure 6 and agree quite well with
Dimos experiments.
J~IT) /J~lo)
0 20 40 60 80 1O0
T(K)
Fig, 6, Normalized critical current density (J~/J~(0 K)) as a function of reduced temperature t in YBa~CU~O~.
4. Study of the critical current density in the case of large applied magnetic field.
4, j CALCULATION OF THE ELEMENTARY PINNING FORCE DUE TO THE ELECTRON-SCATTERING
MECHANISM. In type II superconductors (I,e. high T~ superconductors) large magnetic flux
density B penetrates into the material by creating an Abrikosov vortex lattice. The interaction between the defects and the flux line lattice FLL is classicaly expressed in the Ginzburg- Landau penurbational approach [16, 17]. It means that the presence of the defects does not
dramatically modify the order parameter ~.
In this paragraph a rigid FLL is assumed and as Yetter et al. did the interaction between the FLL and the strain field defect has not been investigated. The high-T~ superconductors built
with a stacking of CUO~ planes are anisotropic compounds and the electrons are confined in the Cu02 Planes. Moreover let us assume the case where the magnetic field strength H is parallel to the c-axis. So the FLL structure has to be considered as a « 2D pancake » stacking along the c-axis (Fig. 7). Consequently the 2D pancake structure in the Cu02 Plane is
isotropic, Let us investigate the situation in which the defect presence does not break the 2D
pancake-stacking along the c-axis : such situations are occuring for planar defect perpendicular
to the CUO~ planes. So all previous assumptions lead to c-independent physical quantities in the following calculation. The Ginzburg-Landau energy &E(r) of the interaction between the FLL and the planar defect for the electron-scattering mechanism is the sum over the interaction
energies between a CUO~ plane and the defect [10]
~~~~/ = La ~- ~~ ~~~~ ~(r, rj) j~ +
~~ ~~~~
~ (r, rj) j~ dry. (14)
Ho Hc r) ~ ~
z
grain
x
,4~~~
plane
' ' '
Fig, 7. Geometrical structure of the planar defect pinning in High-T~ Superconductor.
The integral is carried out over the a-b plane and the summation is running over the different
stacking planes. The validity of the assumption that the vonices are rigid can be examined by
the evaluation of the thermal fluctuations on the elementary pinning force f(~ (see pans 4,2 and 4.3. I : I.e. the better the hypothesis the smaller the thermal effects on f(~). Funher calculations force us to work in the reciprocal space (I,e. g~ vectors). So ~ ~ and ~ ~ have the following
classical forms
~(r,)j~ = z a~(I cos (g,, r, )) (15)
v#o
~(r, )(~ = z b~ cos (g,, r,) (16)
v
where a~ and b~ depend on the reduced magnetic induction B/B~~.
For B
~ 0, I B~~, the expansion could be limited to the first shells in the reciprocal space. In the case of H parallel to the c-axis, Brandt [18] has determined the a~, b~ dependence on B/B~~. The electron-scattering mechanism is essentially efficient when the motion of the vortex
is perpendicular to the twin or the grain boundary, so the pinning force is maximum for the two orientations of the FLL (Figs. 8a and 8b) and only these two FLL orientations have been
considered in this work. This fact can be understood by symmetry arguments. Hence the
elementary pinning force f(~ limited to the main contribution is :
f(~ = m ax ~f(((x ), f)2(x ) (17
with
f[[(x)
=
2 five Hj gi ((ai + bi y(gi sin (gi x) +
+ (a3 + b3) 2 y (2 gl Sin (2 gl X)) (18)
and
f([(x)
=
2 /qo H) gj (a~ + b~ / y(/
gj sin (/
gj x) (19)
where f((, ~(x) are the elementary pinning forces for the orientations land 2 which depend on
the distance x between the grain boundary and the vonex chosen as coordinate origin,
gj the modulus of one of the reciprocal FLL basis vectors gj =
~ "
j'~
~ andy the one
'fi ~ ~°
dimensional Fourier transform of A«/«.
a) b)
Fig. 8. a) The relative orientations of the FLL and the grain boundary in orientation I. b) The relative orientations of the FLL and the grain boundary in orientation 2.
4.2 INFLUENCE OF THE TEMPERATURE ON THE ELEMENTARY PINNING FORCE. Several
authors have pointed out that the thermal fluctuations on the FLL are quite imponant, some of them have proposed a melting transition of the FLL with the possibility of the vonex
entanglement. So it is interesting to examine the effect of the thermal fluctuations on the elementary pinning force ~f(~). In the following pan, we develop the framework where defects (vacancies, precipitates...) between planar defects have not been taken into account. These defects could pin the vonices and deform the flux lines. So our 2D discussion of thermal fluctuations neglects the depinning of these segments of votices. However our aim in this temperature dependence calculation is only to compare both f(~ (with and without thermal fluctuations).
So what we have to modify from the previous described calculation of the pinning force is the density of the superconducting electrons. We have to renormalize it by the thermal
fluctuations : I-e- to replace it by its thermal average ( ~ ~)~. Let ~o(~ be the density of the
superconducting electrons at OK, so
l~o1~
= i f (r R,) (20)
Rj e la",ce
at the temperature T, the density of the superconducting electrons is
l~ 1~ = z f (r R, u, (21)
R, e lat'ice
where u, the two-dimensional displacement is small.
So the second order expansion of (21) and the thermal average lead to
(j~j~)~
=
(~o(~ + u~)~( ~~~~~
+
~~~~~
(22)
4 ~x ~y
where (u~)
~
the thermal fluctuation of the flux line positions is given by (23) in the continuum
approximation and non local hypothesis
k~ J' kBz m
(u~)~ = j k~ dk~ dk~ x
2 w
o o
x
~
+ (23)
C~~k~ + C44(k)kj C,j(k)kj + C44(k)k)
where the coefficient Cjj referred to as the bulk modulus, C~~ to the shear modulus,
C~~ to the tilt modulus, k~z = (4 wB/4
o)~'~ m wlao to the radius of the first Brillouin zone (BZ) and ao to the flux line lattice parameter [19].
So, after taking into account of relations (15), (16) and (22), the a~ and b~ sets are modified by the thermal fluctuations and by the upper critical field H~~ temperature dependence. The
following equations give the temperature dependence adopted for H~ and H~~ in the calculation of fj'
H~ = H~(T
=
0 K )(I t~) (24)
H~2 ~ Hc2(T
" ° K )(' t) (25)
with t
=
~ Tc
Moreover, the Ginzburg-Landau parameter variaiion A«/« has a weak temperature dependence which can be neglected. After taking account of these changes, the calculation of
f(~ is similar to the calculation developed in part 4. I. The computation off(~ with(out) thermal
fluctuations is reported in the following part.
4,3 ELECTRON-SCATTERING MECHANISM PINNING FORCE AND CRITICAL CURRENT DENSITY IN
YBa~CU~O~.
4.3. I Calculation of the elementary pinning force. We have applied the theoretical results of the electron-scattering mechanism to the case of YBa~CU~O~. The parameters adopted in the
determination off(~ are reponed on the table I. The evaluation of the mean free path is derived
