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Vortex Glass, Dislocation Glass, Stripe Glass: Long Range Interactions at Work
1. Vortex Glass: Long vs. Short Range Interactions2. Dislocation Structures in 2D Vortex Matter3. Stripe Glasses in Magnetic Films & 2DEG
M. Chandran, C. Pike, R. Scalettar M. Winklhofer & G.T. Zimanyi U.C. DavisB. Bako, G. Gyorgyi & I. Groma Budapest
Long Range Interactions Form Slow Structures in Cuprates
Competing Energies:Kinetic energyShort range magnetic
Long range Coulomb
- Phase separation (Emery, Kivelson)
- Stripe formation (Littlewood, ZaanenEmery, Kivelson, …)
Experiment (Davis, Yazdani, …)J.C. Davis,Physics Today, September 2004
1. Vortex Glass: The Original Proposition
2~ scJ
Vortex Glass with Long Range Interactions: the Gauge Glass
No Screening: Glass Transition(Young 91)
Expt.: No Extended Defects - No Vortex Glass
Yeh (1997)Lopez, Kwok (1997)Lobb (2001)
Foglietti, Koch (1989)
Screening: Short Range Interactions: No Gauge Glass
Young (95)
Vortex Glass Transition Arrested by Screening: Vortex Molasses
Jc does not vanish as a power law:levels off around
Langevin dynamics for vortices:
1~/ renBCS
ren
1. Interacting elastic lines
2. In random potential
3. Overdamped dynamics
Resistivity in Vortex Molasses
Resistivity finite below “Jsc”:Vortex Molasses
Resistivity can be fitted by a - power law; or the- Vogel-Fulcher law
Finite Size Scaling
Long Range Interaction Short Range Interaction
log (T-TG)
)/log(
)log(
)log(
Vortex GlassVortex Molasses
Interaction Crossover from Long Range to Short Range Causes Criticality Crossover from Scaling to Structural Glasses
Vortex Molasses
~
short rangeinteractions
long rangeinteractions
2. Dislocation Glass
In 2D Disordered Vortex Matterdislocations were supposed to:
• Distributed homogeneously• Characterized by single
length scaleD
Giamarchi-Le Doussal ’00Inspired by KT-Halperin-Nelson-Young theory of 2D melting
Magnetic Field Sweep
B/Bc2 = 0.1 (a)0.4 (b)0.5 (c)0.6 (d)0.8 (e)0.9 (f)
v
•Blue & Red dots: 5 & 7 coordinated vortices: disclinations
• Come in pairs: dislocations
Dislocations form
domain walls at
intermediate fields
What is the physics?
Dislocations are dipoles of disclinations, with anisotropic logarithmic interaction.
Theory averages anisotropy and applies pair unbinding picture ~ KTHNY melting.
However: - The dipole-dipole interaction is strongly anisotropic:
- parallel dipoles attract when aligned;- energy is minimized by wall formation;- energetics different from KTHNY.
Dislocation structures formed by anisotropic interactions
“Absence of Amorphous Vortex Matter”
Fasano, Menghini, de La Cruz, Paltiel, Myasoedov, Zeldov, Higgins, Bhattacharya, PRB, 66, 020512 (2002)
• NbSe2
• T= 3-7K • H= 36-72 Oe
Sim
ula
tion
sN
bS
e 2
Low DisorderMedium DisorderN
bS
e 2S
imu
lati
on
Domain Configurations
We
acce
ssed
low
est
disl
ocat
ion
dens
ities
Dislocation Domain Structures in Crystals
Pattern formation is typical
Rudolph (2005)
Dislocation Simulations
)()(g rnBrnBv PKccc
PKgg
1. Overdamped dynamics
2.is the glide/climb component of the stress-related Peach-Kohler force
3. Dislocation interaction is in-plane dipole-dipole type
4. No disorder
Novelty:
1. Dislocations move in 2D: Bg- glide mobility, Bc - climb mobility;
2. Dislocations rotate: through antisymmetric part of the displacement tensor
3. Advanced acceleration technique
Glide
Climb
Computational Details
Kleinert formalism
1. Separate elastic and inelastic displacement
2. Isolate the antisymmetric component of displacement tensor
3. Rotate Burgers vector
Observation I: Separation of Time Scales
Fast fluctuations: from near dislocationsSlow fluctuations: large scale dynamics from far dislocations
Observation II: Stress Distribution Modeling
2/32 )]()[(
)()(
rC
rCP
ave
Stochastic Coarse Graining
• 1. Divide simulation space into boxes
• 2. Calculate mean (coarse grained) dislocation density for each box
• 3. Slow interactions (AX): Approximate stress from box A in box X by using coarse grained density.
• 4. Fast interactions (BX): Generate random stress t from distribution P(t) with average stress tave.
• 5. Move dislocations by eq. of motion.
• 6. Repeat from 2.
• 1-10 million dislocations simulated in 128x128 boxes
X
A
B
Stochastic Coarse Graining: No Climb, No Rotation, Shearing
Full simulations:
-1 million dislocations-(~20 million vortices)
-Profound structure formation
-Sensitive to boundary, history
-Work/current hardening
Stochastic Coarse Graining:
No Climb, No Rotation, Shearing
Box counting:
- Domains have fractal dimension
-D=1.86
- No single characteristic length scale
Number of domains N(L) of size L with no dislocations
Stochastic Coarse Graining:Climb, No Rotation, No Shearing
Climb promotes structure formation, even without shearing
Stochastic Coarse Graining:Climb, Rotation, No Shearing
log(
time)
Bc/Bg=1.0 Bc/Bg=0.1
1. Domain structure formation without shear
2. Climb makes domain structures possible
3. Domain distribution:not fractal
4. Effective diffusion constgoes to zero:Domain structure freezes:Dislocation Glass
Andrei group PRL 81, 2354 (1998)
Expt.: Shearing Increases Ic
Rudolph et al
Expt.: GaAs: Increasing Climb Induces Domain Structure Formation
Climb
3. Stripe Glass
Co/Pt magnetic easy axis: out of plane
Potential perpendicular recording media
[Co(4Å)/Pt(7Å)]N: Hellwig, Denbeaux, Kortright, Fullerton, Physica B 336, 136 (2003).
Co
Pt
Happ
N=50
Transmission X-ray Microscopy
3m
Stage 1: Sudden propagation of reversal domains.
Stage 2: Expansion/contraction of domains, domain
topology preserved.
Stage 3: Annihilation of reversaldomains.
Modeling Magnetic Films
• Classical spins, pointing out of the plane• Spins correspond to total spin of individual domains:
spin length is continuous variable• Competing interactions:
– Exchange interaction: nearest neighbor ferromagnetic
– Dipolar interaction: long range antiferromagnetic (perpendicular media)
• Finite temperature Metropolis algorithm (length updated)
• Spivak-Kivelson: Hamiltonian same as 2DEG & Coulomb systems
• Tom Rosenbaum: Glassy phases in dipolar LiHoYF
T
C(T)
Equilibrium Phases
Expt.: Two Phases Observed in FeSiBCuNb Films
Henninger
Non-equilibrium Anneal: Supercooled Stripe Liquid Stripe Glass
Protocol:
1. Cool at a finite rate to T
2. Study relaxation at T
Typically configuration is far from equilibrium:
Supercooled Stripe LiquidStripe Glass
~ Schmalian-Wolynes
T 1/T
~ Fragile Glass ~ Strong Glass
Relaxation of Persistence
)(T )](log[ T
])/(exp[~)( 0tPtP
)/exp()( 0 TT
Aging
P(t, tw)
tw=104
tw=105
tw=106
tw=107
Good fit: P(t, tw) = P[(t-tw)/tw]
t
Blue regions: frozen
Summary1. Vortex Glass:
- Crossover of range of interaction from long to short changes Glass transition from Scaling to Molasses transition
2. Dislocation Glass: - In 2D in-plane dipoles form frozen domain structures: Dislocations, Vortex matter - Climb, rotation, shearing, disorder- Stochastic Coarse Gaining, ~ 10 million vortices
3. Stripe Glass: - In 2D out-of plane dipoles form Stripe Glass: Magnetic films, 2DEG, Coulomb systems - Persistence, aging - Strong and Fragile Glass aspects observed
How to see your glass? Low frequency spectrum of noise is large (Popovic), slow dynamics, imaging