Phase Diagram of a Point Disordered Model Type-II Superconductor

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Phase Diagram of a Point Disordered Model Type-II
Superconductor

• Peter Olsson Stephen Teitel
• Umeå University University of Rochester

IVW-10 Mumbai, India
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What is the equilibrium phase diagram of a
strongly fluctuating type-II superconductor?
Experiments
Optimally doped untwinned YBCO
Vortex liquid
Vortex glass
Bragg glass
Shiba et al., PRB 2002
Bragg glass -- vortex liquid vortex glass?
vortex slush? critical end point? multicritical
point?
Pal et al., Super. Sci. Tech 2002
point disorder
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Phase Diagram Theoretical
Hu and Nonomura, PRL 2001
Kierfeld and Vinokur, PRB 2004
Lindemann criterion
XY model simulations
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Outline

• Introduction
• 3D XY model and parameters
• thermodynamic observables and order parameters
• Low disorder
• vortex lattice melting
• Large disorder
• vortex glass transition
• gauge glass and screening
• Intermediate disorder
• vortex slush?
• Conclusions
• the phase diagram!

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3D Frustrated XY Model
kinetic energy of flowing supercurrents on a
discretized cubic grid
phase of superconducting wavefunction
magnetic vector potential
coupling on bond im
density of magnetic flux quanta vortex line
density piercing plaquette a of the cubic grid
uniform magnetic field along z direction magnetic
field is quenched
constant couplings between xy planes magnetic
field
random uncorrelated couplings within xy planes
disorder strength p
weakly coupled xy planes
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Parameters
anisotropy
system size 80 vortex lines
disorder strength varies
vortex line density fixed
ground state vortex configuration for
disorder-free system
systematically vary p to go from weak to strong
disorder limit
exchange Monte Carlo method (parallel tempering)
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Thermodynamic observables
free energy F
b 1/kBT
E - energy density
Q - variable conjugate to the disorder strength p
E and Q should in general change discontinuously
at a 1st order phase transition
E and Q must both be continuous at a 2nd order
phase transition
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Structure function - vortex lattice ordering
parameter
nz is vortex density in xy plane
real-space
K2
K1
vortex liquid
or
vortex solid
ky
kx
k-space
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Helicity Modulus - phase coherence order parameter
twisted boundary conditions
twist dependent free energy
Dm 0 0 for a disorder-free system, but not
necessarily with disorder p gt 0
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Low disorder - the vortex lattice melting
transition
Structure function p 0.16
T 0.2210 liquid
liquid
solid
T 0.1985 solid
Structure function indicates vortex solid to
liquid transition
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twist histograms
Helicity modulus
p 0.16
normal
superconducting
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Plots of DS, U, E, or Q vs. T do not directly
indicate the order of the melting transition.
Need to look at histograms!
vortex lattice ordering parameter DS
Bimodal histogram indicates coexisting solid and
liquid phases! 1st order melting transition
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Use peaks in P(DS) histogram to deconvolve solid
configurations from liquid configurations.
Construct separate E and Q histograms for each
phase to compute the jumps DE and DQ at the
melting transition.
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Melting phase diagram
f 1/5
As disorder strength p increases, DE decreases to
zero, but DQ remains finite. Transition remains
1st order, without weakening, along melting line.
P. Olsson and S. Teitel, Phys. Rev. Lett. 87,
137001 (2001)
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Large disorder - the vortex glass transition
well above the melting transition line
p 0.40
No longer any vortex solid
histograms of lattice ordering parameter P(DS)
T 0.90 below Tg
T 0.221 above Tg
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Phase coherence
Looking for a 2nd order vortex glass transition
with critical scaling. In principal, scaling can
be anisotropic since magnetic field singles out a
particular direction.
If anisotropic scaling, situation very difficult
need to simulate many aspect ratios Lz/L. So
assume scaling is isotropic, z 1, and see if it
works! (it does!)
Use constant aspect ratio Lz L.
Curves for different L all cross at t 0, i.e. T
Tg
P. Olsson, Phys. Rev. Lett. 91, 077002 (2003)
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Histograms of twist Dm for a particular
realization of disorder
p 0.40
well above the melting transition line
twist histogram develops several local maxima as
enter the vortex glass phase
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Helicity modulus p 0.30, 0.40, 0.55
averaged over 200 - 600 disorder realizations
curves for a particular p cross at single Tg
scaling collapse of data
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Phase diagram for melting and glass transitions
How do glass and melting transitions meet???
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Vortex glass vs. gauge glass
gauge glass model (Huse Seung, 1990)
gauge glass is intrinsically isotropic - average
magnetic field vanishes
although vortex glass is not isotropic, critical
scaling is isotropic
gauge glass and vortex glass are in the same
universality class
(also, Kawamura, 2003, Lidmar, 2003)
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Screening
(Kawamura, 2003)
(Bokil and Young, 1995, Wengel and Young, 1996)
If gauge glass and vortex glass are in the same
universality class, expect the same. Vortex glass
transition will survive only as a cross-over
effect.
Resistance in vortex glass will be linear at all
T, for sufficiently small currents.
how small?
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Phase diagram for melting and glass transitions
How do glass and melting transitions meet???
Simulations get very slow and hard to equilibrate.
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Intermediate disorder - still a vortex solid,
but now two!
p 0.22
DE and DQ consistent with values from lower p
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Phase coherence
p 0.22
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Intermediate solid phase solid 1
p 0.22
Intermediate solid consists of coexisting regions
of ordered and disordered vortices.
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Intermediate solid phase solid 1
p 0.22
Some similarities to vortex slush of Nonomura
and Hu. Does it survive as a distinct phase in
larger systems?
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Phase diagram of point disordered f 1/5 3D XY
model
Conclusions

1. Melting transition remains 1st order even where
   it meets glass transition
2. Glass transition becomes cross-over on large
   enough length scales
3. Possible intermediate solid? Needs more
   investigation
4. Lattice to glass transition at low T?