Flux pinning in general

Adrian CrisanSchool of Metallurgy and Materials, University of Birmingham, UK

andNational 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 IIType 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 normalwith an increasing magnetic field- Typically they superconductat higher temperatures and fieldsthan 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 fluxconfined 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. VORTICESVortex (mixed) state- Normal regions thread through superconductor- Ratio between surface andvolume of the normal phaseis maximised- Cylinders of normal material parallel to the applied field (normal cores)- Cores arranged in regular patternto 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. PINNINGLorenz 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.

)3

1(

42

babda

mJ c

(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

Geometry

of pin

Type of

centre

Pinning

function

p, q Position of

maximum

Max.

Const.

Volume Normal A(1-h)2 p=0; q=2 - A=1

Δκ Bh(1-h) p=1; q=1 h=0.5 B=4

Surface 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

Point 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

0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8

Volume normal; (1-h)2, no max; F=1

0

0.05

0.10

0.15

0.20

0.25

0.2 0.4 0.6 0.8

Volume Dk; h(1-h) , max at 0.5, Fm=0.25, A=4

0

0.1

0.2

0.3

0.2 0.4 0.6 0.8

Surface n; h1/2(1-h)2 ; max at 0.2, Fm=0.286, B= 3.5

0

0.05

0.10

0.15

0.20

0.2 0.4 0.6 0.8

Surface Dk; h3/2(1-h) ; max at 0.6, Fmax=0.186, C=5.37

0

0.05

0.10

0.15

0.2 0.4 0.6 0.8

Point n; h(1-h)2; max at 0.33, Fmax=0.148, D=6.76

0

0.05

0.10

0.15

0.2 0.4 0.6 0.8

Point Dk; h2(1-h) ; max at 0.67, Fmax=0.148; E=6.76

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)

(1)(1)

(3)

(2)

(2)

(3)

(1): I=0(2): 0<I<Io

(3): I=Io

)(

)/1()/1(

/

0

00

00

0000

00

00

kTU

kTU

b

f

ktBJvxU

kTBJvxU

bf

bf

ee

JJUUJJU

JvxJBJUBJvxUU

ee

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:

)ln(ln

)1(ln

0

0

0

0

)/1(

0

00

tJ

JJ

kTU

e

c

c

kTJJU c

Logarithmic decay, magnetic relaxationJ; 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

1;)( 0

m

m

JJUJU c

The above dependence leads to a non-ohmic current-voltage characteristic of the form:

m

JJ

kTUV c0exp

In an inductive circuit, V is proportional to dJ/dt

)/1ln()/(1)( 00 ttUkTJtJ c T010-6 s

mm

/1

0

0

/1

00

lnln)(

UkTJ

tt

UkTJtJ cc

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

JJUU eff

*

0 ln

E. Zeldov et al, PRL 62, (1989) 3093 , PRL . 56, (1990) 680A.C. et al, SuST 22, 045014, 2009

kTU

V effexp

kTU

kTU

JJCt

JJCt

JJ

kTUCtV

00***

0 .lnexp.lnexp.