Flux pinning in general
Adrian Crisan
School of Metallurgy and Materials, University of Birmingham, UK
and
National Institute of Materials Physics, Bucharest, Romania
CONTENTS
• Introduction: type I vs. type II
• Vortices
• Pinning
• Bulk Pinning Force Density
• Pinning Potential
INTRODUCTION: Type I vs. Type II
Type I superconductors
- They cannot be penetrated by
magnetic flux lines
(complete Meissner effect)
- They have only a single critical field
at which the material ceases to
superconduct, becoming resistive
- They are usually elementary metals,
such as aluminium, mercury, lead
Type II superconductors
- Gradual transition from
superconducting to normal
with an increasing magnetic field
- Typically they superconduct
at higher temperatures and fields
than Type I
- Between Meissner and normal state
there is a large “mixed” or “vortex” state
- They have two critical fields
(upper and lower)
- They are ussually metal alloys,
intermetallic compounds, complex oxides (e.g., Cu-based HTSC)
and, recently discovered, pnictides and chalcogenides
Phase diagram of “classical” superconductors
Penetration depth (l)
- Diamagnetic material
(no internal flux)
- Currents to repel external flux
confined to surface
- Surface currents must flow in
finite thickness (penetration depth l)
Coherence length (x)
- characterises the distance over which the superconducting wave
function y(r) can vary without undue energy increase
- the distance over which the superconducting carriers
concentration decreases by Euler’s number e
- GL parameter k=l/x; if k<1/21/2 then Type I; k>1/21/2 then Type II
II. VORTICES
Vortex (mixed) state
- Normal regions thread
through superconductor
- Ratio between surface and
volume of the normal phase
is maximised
- Cylinders of normal
material parallel to the
applied field (normal cores)
- Cores arranged in regular pattern
to minimize repulsion between cores
(close-packed hexagonal lattice) – flux lattice
Flux quanta - vortex
Phase diagram of High-Tc superconductors and
Vortex Melting Lines
The vortex lattice undergoes a first-order melting transition transforming the vortex solid into
a vortex liquid [Fisher et al, PRB 43,130, 1991].
For high anisotropy, at low magnetic fields (approx 1 Oe in BSCCO [A.C. et al, SuST 24, 115001,
2011), there is a reentrance of the melting line [Blatter et al, PRB 54, 72, 1996].
The flux lines in the vortex -liquid are entangled resulting in an ohmic longitudinal response,
hence the vortex liquid and normal metallic phases are separated by a crossover at Hc2.
For low enough currents
-VL- linear dissipation: E ≈ J
-VS (VGlass)- strongly nonlinear dissipation: E ≈ exp[-(JT/J)m]
III. PINNING
Lorenz force (FL) and pinning force (Fp)
In the presence of a magnetic field perpendicular to the current
direction, a Lorentz force FL = j ×f0, where j is the current and f0
is the magnetic flux quantum, acts on the vortices
• If FL is smaller than the pinning force Fp, vortices do not move.
Defect-free sample
Point defects
Columnar defects
Dimensionality and strength of PCs
IV. BULK PINNING FORCE DENSITY
• FP determined from magnetization loops M(Ha)
Fp=BxJc
Jc=Ct.DM
Jc 
4m
a
a bd (1  )
3b
.
2
(thin films; m=DM/2;
d-thickness; a,b-rectangle dim.)
Dew-Hughes model
F = Fp/Fpmax = hp(1-hq)
;
h = B/Birr
p and q depend on the types of pinning centres.
- Classified by the number of dimensions that are large
compared with the inter-flux-line spacing; and
- by the type of the core: “Dk pinning” and “normal pinning”
Ususlly there are several types of pinning centres.
F = Ahp1(1-hq1)+Bhp2(1-hq2)+Chp3(1-hq3)+.......
D. Dew-Hughes, Philosophical Magazine 30 (1974) 293
Max.
maximum
Const.
p=0; q=2
-
A=1
Bh(1-h)
p=1; q=1
h=0.5
B=4
Normal
Ch1/2(1-h)2
p=1/2; q=2
h=0.2
C=3.5
Δκ
Dh3/2(1-h)
p=3/2; q=1
h=0.6
D=5.37
Normal
Eh(1-h)2
p=1; q=2
h=0.33
E=6.76
Δκ
Fh2(1-h)
p=2; q=1
h=0.67
F=6.76
Type of
Pinning
of pin
centre
function
Volume
Normal
A(1-h)2
Δκ
Surface
Point
p, q
Position of
Geometry
0.3
1.0
Volume normal; (1-h)2, no max; F=1
0.8
0.2
0.6
0.4
0.1
Surface n; h1/2(1-h)2 ; max at 0.2, Fm=0.286, B= 3.5
0.2
0
0
0.2
0.4
0.6
0.8
0.25
0.2
0.4
0.6
0.8
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
Volume Dk; h(1-h) , max at 0.5, Fm=0.25, A=4
Surface Dk; h3/2(1-h) ; max at 0.6, Fmax=0.186, C=5.37
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
0.15
0.15
0.10
0.10
Point n; h(1-h)2; max at 0.33, Fmax=0.148, D=6.76
0.05
0.05
0
0
0.2
0.4
0.6
0.8
Point Dk; h2(1-h) ; max at 0.67, Fmax=0.148; E=6.76
0.2
0.4
0.6
0.8
V. PINNING POTENTIAL
• Energy needed by the flux line to escape from
the potential well crated by the pinning centre
• Shape and influence on superconducting
properties modelled in several ways,
depending on material, strength and
distribution of pinning centres
• In 1962 Anderson predicted that movement of
vortices with a drift velocity v will create
dissipation (electric field) E=Bxv
Dissipation occurs through two mechanisms:
• 1. Dipolar currents which surround each moving
flux line (eddy currents) and which have to pass
through “normal conducting vortex core”
• 2. Retarded relaxation of the order parameter
when vortex core moves
• Anderson and Kim predicted that thermal
depinning of flux lines can occur at finite
temperatures T (“flux creep”).
Anderson-Kim model
• Model assumes that flux creep occurs due to
thermally-activated jump of isolated bundles of
flux lines between two adjacent pinning centres.
• The jump is correlated for a bundle of vortices of
volume (correlated volume), Vc due to the
interactions between them
• In the absence of transport current (i.e., no
Lorentz force) the bundle is placed in a
rectangular potential well of height Uo.
• Due to thermal energy, there are jumps over the
barier with a frequency = oexp(-Uo/kT)
   f  b   0 e
(3)
(2)
(1)

U 0  BJvx
kT
 0 e

U 0  BJvx
kt
U f  U 0  BJvx  U 0  BJ0vxJ0 / J
(1)
(2)
 U 0 (1  J / J 0 )
U b  U 0 (1  J / J 0 )
(3)
   0 (e
(1): I=0
(2): 0<I<Io
(3): I=Io
Second term is ussualy neglected, since
current densities of interest are smaller
than J0
Critical current density is defined arbitrarily
at a certain electric field, e.g., 10-6 V/cm. It
follows:

Uf
kT
U


e
0
e

0 (1 J c
Ub
kT
)
/ J0 )
kT
 
U0
Jc


ln   
(1  )
kT
J0
 0 
 
J c  ln    ln(t )
 0 
Logarithmic decay, magnetic relaxation
J; M (a.u.)
ln (t)
K-A model: -pinning potential decreases linearly with current
-remnant magnetization and persistent current (or critical
current density) decay logarithmically with time
Modified Anderson-Kim model
• Tilted-washboard cosine potential, which
leads to U=U0[1-(J/Jc)]3/2
• The two forms can be generalized as
U=U0[1-(J/Jc)]
• Such forms focus on the detailed behavior
near Jc, which is appropriate for the classic
superconductors where fluctuation effects
cause only slight degradation of Jc
Larkin-Ovchinnikov collective pinning model
• Cooperative aspects of vortex dynamics
• Formation of vortex lattice will be a result of a
competition between:
-vortex-vortex interaction, which tends to
place a vortex on a lattice point of a periodic
hexagonal/triangular lattice; and
- vortex-pin interaction, which tends to place
a vortex on the local minimum of the pinning
potential
• v-v interaction promotes global translational
invariant order
• v-p interaction tend to suppress such long-range
order, if pinning potential varies randomly
• Long-range order of an Abrikosov lattice is
destroyed by a random pinning potential, no
matter how weak it is.
• Periodic arrangement is preserved only in a small
corellated volume vc which depends on the
strength of the pinning potential and the elasticity
of the vortex lines
• Correlated volume vc increase strongly with
decreasing current density J, which leads to a
power-law dependence of effective pinning
potential on the current density
m
 Jc 
U ( J )  U0   ; m  1
J 
The above dependence leads to a non-ohmic current-voltage characteristic of the form:
 U 0  J c m 
V  exp
  
 kT  J  
In an inductive circuit,
V is proportional to dJ/dt
J (t )  J c 1  (kT / U 0 ) ln(1  t / t0 )
 kT t 
J (t )  J c 
ln 
 U 0 t0 
1 / m
T010-6 s
 kT  0 
 J c 
ln 
 U0  
1 / m
Zeldov effective pinning
• Magneto-resistivity and I-V curves of YBCO films
• Potential well having a cone-like structure
exhibiting a cusp at its minimum and a broad
logarithmic decay with the distance
U eff
 J* 
 U 0 ln 
 J 
 U eff
V  exp 
 kT



U0
U0






*
 U 0  J * 
 J *  kT
  J  kT  
V  Ct. exp
ln   Ct. expln  
 Ct. 



 J 
 kT  J 
  J   

 
E. Zeldov et al, PRL 62, (1989) 3093 , PRL . 56, (1990) 680
A.C. et al, SuST 22, 045014, 2009