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