phase diagram of a point disordered model type-ii superconductor
DESCRIPTION
Phase Diagram of a Point Disordered Model Type-II Superconductor. Peter Olsson Stephen Teitel Umeå University University of Rochester. IVW-10 Mumbai, India. What is the equilibrium phase diagram of a strongly fluctuating type-II superconductor?. Experiments:. - PowerPoint PPT PresentationTRANSCRIPT
Phase Diagram of a Point Disordered Model Type-II
Superconductor
Peter Olsson Stephen TeitelUmeå University University of Rochester
IVW-10 Mumbai, India
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Tg
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Tm
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TL
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Hmcp
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Hcep Vortex liquid
Bragg glass
Vortex glass
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H*
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33Y3H1
Shiba et al., PRB 2002
Optimally doped untwinned YBCO
Pal et al., Super. Sci. Tech 2002
What is the equilibrium phase diagram of a strongly fluctuating type-II superconductor?
Bragg glass vortex liquidvortex glass? vortex slush?critical end point? multicritical point?
Experiments:
point disorder
Hu and Nonomura, PRL 2001 Kierfeld and Vinokur, PRB 2004
Lindemann criterionXY model simulations
Phase Diagram
Theoretical:
Outline• Introduction
• 3D XY model and parametersthermodynamic observables and order parameters
• Low disordervortex lattice melting
• Large disordervortex glass transitiongauge glass and screening
• Intermediate disordervortex slush?
• Conclusionsthe phase diagram!
3D Frustrated XY Model
phase of superconducting wavefunction
magnetic vector potential
kinetic energy of flowing supercurrentson a discretized cubic grid
coupling on bond i
density of magnetic flux quanta = vortex line densitypiercing plaquette of the cubic grid
uniform magnetic field along z directionmagnetic field is quenched
weakly coupled xy planes
constant couplings between xy planes || magnetic field
random uncorrelated couplings within xy planes disorder strength p
Parameters
anisotropy
system size ~ 80 vortex lines
disorder strength varies
vortex line density fixed
ground state vortex configuration for disorder-free system
increasing disorder strength p at fixed magnetic field f
€
↔ increasing magnetic field f at fixed disorder strength p
exchange Monte Carlo method (parallel tempering)
or
systematically vary p to go from weak to strong disorder limit
Thermodynamic observables
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
free energy F
= 1/kBT
Structure function vortex lattice ordering parameter
nz is vortex density in xy plane
or
K1K2
or
real-space
k-spacekx
ky
vortex liquid
vortex solid
Helicity Modulus phase coherence order parameter
twisted boundary conditions
twist dependent free energy
phase coherent: F[] varies with free energy sensitive to boundary
phase incoherent: F[] independent of free energy insensitive to boundary
helicity modulus:(phase stiffness)
evaluate at the twist 0 that minimizes the free energy F, ormaximizes the histogram P
twist histogram: measure by simulating in fluctuating twist ensemble
0 = 0 for a disorder-free system, but not necessarily with disorder p > 0
Low disorder the vortex lattice melting transition
Structure function
p = 0.16
T = 0.1985solid
T = 0.2210liquid
liquidsolid
Structure function indicates vortex solid to liquid transition
0.0
0.2
0.4
0.6
0.8
1.0
0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24T
S(K1)p = 0.16
S(K2)
S = S(K1) − S(K2)
Tm ~ 0.21
p = 0.16
Helicity modulustwist histograms
normalsuperconducting
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24T
Uz/J z
Ux
Uy
p = 0.16Tm ~ 0.21
0.000
0.005
0.010
0.015
0.020
0.025
P(Dz)-0.193P(Dz)-0.204P(Dz)-0.215P(Dz)-0.221
z
T p = 0.16Tm ~ 0.21
0 π−π π/2−π/2
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
P(Dx)-0.193P(Dx)-0.204P(Dx)-0.215P(Dx)-0.221
x
T p = 0.16Tm ~ 0.21
0 π−π π/2−π/2
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
P(Dy)-0.193P(Dy)-0.204P(Dy)-0.215P(Dy)-0.221
y
T p = 0.16Tm ~ 0.21
0 π−π π/2−π/2-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24T
z0
p = 0.16
x0
y0
Tm ~ 0.21
Plots of S, U, E, or Q vs. T do not directly indicate the order of the melting transition. Need to look at histograms!
Bimodal histogram indicates coexisting solid and liquid phases!1st order melting transition
vortex lattice ordering parameter S
10-4
10-3
10-2
10-1
100
0.0 0.2 0.4 0.6 0.8 1.0
0.1930.19850.2040.20950.2150.2210.2275
S
Tp = 0.16Liquid
Solid
Use peaks in P(S) histogram to deconvolve solid configurations from liquid configurations.
Construct separate E and Q histograms for each phase to compute the jumps E and Q at the melting transition.
10-4
10-3
10-2
10-1
0.0 0.2 0.4 0.6 0.8S
p = 0.16Liquid Solid
T = 0.2095
0.000
0.002
0.004
0.006
0.008
0.010
0.012
-1.38 -1.36 -1.34 -1.32E
LiquidSolid
p = 0.16T = 0.2095
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.20 0.24 0.28 0.32Q
LiquidSolid
p = 0.16T = 0.2095
Melting phase diagram
As disorder strength p increases, E decreases to zero, but Q remains finite.Transition remains 1st order, without weakening, along melting line.
P. Olsson and S. Teitel, Phys. Rev. Lett. 87, 137001 (2001)
f = 1/5
0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.260.00
0.05
0.10
0.15
0.20
0.25
T Temperature
p disorder strength
lattice
liquid
Tm(p)1st order melting
16 independent realizations of disorder
???
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.00 0.05 0.10 0.15 0.20 p disorder
E
Q
Large disorder the vortex glass transition
well above the melting transition linep = 0.40
No longer any vortex solidhistograms of lattice ordering parameter P(S)
T = 0.90 below Tg
T = 0.221 above Tg
Phase coherenceLooking for a 2nd order vortex glass transition with critical scaling.In principal, scaling can be anisotropic since magnetic field singlesout a particular direction.
If anisotropic scaling, situation very difficult; need to simulate many aspectratios Lz/L. So assume scaling is isotropic, = 1, and see if it works! (it does!)
Use constant aspect ratio Lz = L.
P. Olsson, Phys. Rev. Lett. 91, 077002 (2003)
Curves for different L all cross at t = 0, i.e. T = Tg
0.000
0.001
0.002
0.003
0.004
0.005
0.006
P(Dz) 0.090P(Dz) 0.125P(Dz) 0.183P(Dz) 0.2335
Dz
T
p = 0.4Tg = 0.11
0 p- p p/2- p/2 0.000
0.005
0.010
0.015
0.020
P(Dx) 0.090P(Dx) 0.125P(Dx) 0.183P(Dx) 0.2335
Dx
Tp = 0.4
Tg = 0.11
0 p- p p/2- p/2
0.000
0.005
0.010
0.015
0.020
P(Dy) 0.090P(Dy) 0.125P(Dy) 0.183P(Dy) 0.2335
Dy
T p = 0.4Tg = 0.11
0 p- p p/2- p/2 0.0
1.0
2.0
3.0
4.0
5.0
0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17T
z0
p = 0.40
x0
y0Tm ~ 0.11
Histograms of twist for a particular realization of disorder
twist histogram develops several local maxima as enter the vortex glass phase
p = 0.40 well above the melting transition line
0.0
0.5
1.0
1.5
2.0
2.5
0.09 0.11 0.13 0.15
L = 15L = 20L = 25L = 10L = 15L = 20L = 10L = 15L = 20
T
p = 0.55p = 0.40p = 0.30
Tg
Tg
Tg
0.0
0.5
1.0
1.5
2.0
2.5
-1 0 1 2 3 4
L = 15L = 20L = 25L = 10L = 15L = 20L = 10L = 15L = 20
tL1/n
p = 0.55p = 0.40p = 0.30
n = 1.56n = 1.48n = 1.26
Helicity modulus p = 0.30, 0.40, 0.55
curves for a particular p cross at single Tg
averaged over 200 600 disorder realizations
scaling collapse of data
0.05 0.10 0.15 0.20 0.25 0.300.00
0.10
0.20
0.30
0.40
0.50
0.60
T Temperature
p disorder strengthlattice
liquid
Tm(p)1st order melting
vortex glass Tg(p)
2nd order glass
???
Phase diagram for melting and glass transitions
How do glass and melting transitions meet???
Vortex glass vs. gauge glass
uniform random distribution
gauge glass model:(Huse & Seung, 1990)
gauge glass is intrinsically isotropic average magnetic field vanishes
(Katzgraber and Campbell, 2004)
vortex glass model:
magnetic field breaks isotropy
although vortex glass is not isotropic, critical scaling is isotropic
(Olsson, 2003)
gauge glass and vortex glass are in the same universality class(also, Kawamura, 2003, Lidmar, 2003)
Screening
(Bokil and Young, 1995, Wengel and Young, 1996)When include magnetic field fluctuations due to a finite , the gauge glasstransition in 3D disappears,
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.
Critical scaling will break down when one probes length scales
Resistance in vortex glass will be linear at all T, for sufficiently small currents.
how small?
(Kawamura, 2003)
0.05 0.10 0.15 0.20 0.25 0.300.00
0.10
0.20
0.30
0.40
0.50
0.60
T Temperature
p disorder strengthlattice
liquid
Tm(p)1st order melting
vortex glass Tg(p)
2nd order glass
???
Phase diagram for melting and glass transitions
How do glass and melting transitions meet???
Simulations get very slow and hard to equilibrate.
Intermediate disorder still a vortex solid, but now two!
p = 0.22
0.0
0.2
0.4
0.6
0.8
1.0
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22T
S(K1)p = 0.22
S(K2)
S = S(K1) − S(K2)
T 1m ~ 0.16
T 2m ~ 0.12
0.008
0.016
0.024
0.032
0.040
0.048
0.056
0.0 0.2 0.4 0.6 0.8 1.0
T = 0.115T = 0.118T = 0.1215
S
p = 0.22
Liquid
2Solid
1Solid
0.000
0.005
0.010
0.015
0.020
0.025
0.030
-1.440 -1.435 -1.430 -1.425 -1.420E
Liquid
Solid 1
p = 0.22T = 0.118
Solid 2
0.000
0.001
0.002
0.003
0.004
0.25 0.30 0.35 0.40Q
LiquidSolid 1
p = 0.22T = 0.118
Solid 2
E and Qconsistentwith valuesfrom lower p
Phase coherencep = 0.22
0.0
0.2
0.4
0.6
0.8
1.0
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22T
Uz/J z
Ux
Uy
p = 0.22
Tm1 ~ 0.16
Tm2 ~ 0.12
0.000
0.005
0.010
0.015
0.020
0.025
Dz
T p = 0.22
Tm1 ~ 0.16
Tm2 ~ 0.12
0 p- p p/2- p/2
0.1150.12150.12850.1550.1830.204
0.000
0.010
0.020
0.030
0.040
0.050
Dx
T p = 0.22
Tm1 ~ 0.16
Tm2 ~ 0.12
0 p- p p/2- p/2
0.1150.12150.12850.1550.1830.204
0.000
0.010
0.020
0.030
0.040
0.050
-1 -0.5 0 0.5 1
Dy
T p = 0.22
Tm1 ~ 0.16
Tm2 ~ 0.12
0 p- p p/2- p/2
0.1150.12150.12850.1550.1830.204
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22T
z0
p = 0.22
x0
y0
T 1m ~ 0.16
T 2m ~ 0.12
Intermediate solid phase “solid 1”
p = 0.22
snapshot of vortex configurations for 4 successive layers
Intermediate solid consists of coexisting regions of ordered and disordered vortices.
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?
Only the orientationgiven by K1 is coherentthroughtout the thicknessof the sample.
Orientation K2 mayexist locally in individual layers, butwithout coherence from layer to layer.
---
0.05 0.10 0.15 0.20 0.25 0.300.00
0.10
0.20
0.30
0.40
0.50
0.60
T Temperature
p disorder strengthlattice
liquid
Tm(p)1st order melting
vortex glass Tg(p)
2nd order glass
Tm1(p)
Tm2(p)
???
Phase diagram of point disordered f = 1/5 3D XY model
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?
Conclusions