Theory of Phase Transitions Theory of Phase Transitions

in Type II superconductors in Type II superconductors

������

������� �������

OutlineOutline

�� IntroductionIntroduction aa,,bb,,cc

�� MotivationMotivation

�� ModelModel

�� Theoretical resultTheoretical result

�� Comparison with experimentsComparison with experiments

�� ConclusionConclusion

�� Vortex Phase diagramVortex Phase diagram——melting, glass melting, glass transition, structure phase transition etc, transition, structure phase transition etc, very rich phase diagramvery rich phase diagram

�� Traditional theoryTraditional theory--using using LindermannLindermanncriterion to obtain the transition line, not criterion to obtain the transition line, not really a theory.really a theory.

�� There are structures in vortices near There are structures in vortices near phase transition.phase transition.

Phase Behaviour in the Mixed Phase Behaviour in the Mixed PhasePhase

The conventional picture An alternative view

Melting of the flux line Melting of the flux line latticelattice

local magnetisationmeasurements

on BSCCO

E. Zeldov et al.

Nature 1995

1st – order melting transition:-revealed by jump in magnetisation at Bm

Melting of the flux line Melting of the flux line latticelattice

heat capacitymeasurements

on YBCO

A. Schilling et

al.

PRL 1997

B = 0 T

B = 9 T Entropy jumps at 1st order flux lattice melting

001

010

Results in niobium: B // Results in niobium: B // (100)(100)

“isosceles phase”

B = 0.16 T, T = 4 K

010

001

001

010

Results in niobium: B // Results in niobium: B // (100)(100)

“scalene phase”

B = 0.26 T, T = 2 K

010

001

ActualActual B B --TT Phase DiagramPhase Diagram

B

(mT)

T (K)

μ o

H

app

2nd

order

1st

order

Phase diagram of vortex matterPhase diagram of vortex matter

--Spontaneous symmetry breaking Spontaneous symmetry breaking

patternpattern

Bragg glassBragg glasssolidsolidTSB TSB (translational (translational symmetry broken)symmetry broken)

Vortex glassVortex glassliquidliquidTS (translation TS (translation invariant)invariant)

RSB (replica RSB (replica symmetry broken)symmetry broken)

RS (replica RS (replica symmetric)symmetric)

symmetrysymmetry

Different translation symmetry breaking patterns leads square or hexagonal lattice-structure phase transition .

Generic Phase TransitionGeneric Phase Transition

3 1

4G F d x B H

π= − ⋅∫

� �

In equilibrium under fixed external In equilibrium under fixed external magnetic field, themagnetic field, the relevant thermorelevant thermo--dynamical quantity is the Gibbs free dynamical quantity is the Gibbs free energy:energy:

22 2 ' 22 2 43 2

( ) ( )2 2 2 8

z

ab c

c

ie b BF d x T T

m c mψ ψ α ψ ψ

π= − + ∂ + − + +∫ A

� �

�

∇

Ginzburg-Landau theory for superconductiv

Mean field phase diagramMean field phase diagram

Near Hc2, it is convenient to do follow scaling: length

in unit of , magnetic field in unit of Hc2. in

Unit of

With following parameters:

ξ

2 22 2 2 43

2

1 1 1 1 1 ( ),

2 2 2 2 4

, ( ,0), .

z

c

G t b hg d x D

T

BD iA A by b

H

κψ ψ ψ ψω

⎧ ⎡ ⎤− −= = + ∂ − + +⎪ ⎢ ⎥⎪ ⎣ ⎦⎨⎪ =∇− = =⎪⎩

∫�� �� �� ��

2ψ2 cTα

β

( )

2

22 2 4 2 2

2 22

2 ;

3 21.

2 0c c

c

G i t

e T TG i

c h H

ω π

π λ γ κ γξ

⎧ ≡⎪⎪⎨ ⎛ ⎞

≡ ∝⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

Gi is the so called Ginzburg parameters which characterize the thermal fluctuation strength.

ω is a better number which can be used to judge the fluctuation effect near Hc2(T) line. If this number is large enough, the fluctuation effect could be observed.

-50.08 3 10Giω > → = ⋅Vortex melting (magnetization jumps) is observable.

Near Hc2(T), magnetic field is nearly constant and nonfluctuating when is large

Near Hc2(T), the lowest Landau level (LLL) approximation also can be used(Abrikosov,57).

κ2 2

2 2

1 ( ) 1

4

b hb h

κκ κ

−− ∝ → ∝

We will study the phase diagram of the superconductors with not too big thermal fluctuation (where the LLL approximation can be justified ).

Near Hc2, we can use the LLL approximation

2 2

2 22 2 2 43

2 22 2 43

1

2 2

1 1 1 1 1 ( )

2 2 2 2 4

1 1 1 1 ( )

2 2 2 4

z

z

bD

t b hg d x D

t b b hd x

ψ ψ

κψ ψ ψ ψω

κψ ψ ψω

≈ →

⎡ ⎤− −= + ∂ − + +⎢ ⎥⎣ ⎦

⎡ ⎤− − −≈ ∂ − + +⎢ ⎥⎣ ⎦

∫ ∫

∫

∫

LLL modelLLL model and LLL scalingand LLL scaling

In the LLL limit, the model simplifies after In the LLL limit, the model simplifies after further further rescalingrescaling and large kappa and large kappa approximationapproximation::

With the only parameter being LLL scaled temperature:

2/ ,cb H H= / ct T T=

2 2 431 1 1

2 24 2z T

Fg d x a

Tψ ψ ψ

π⎡ ⎤= = ∂ + +⎢ ⎥⎣ ⎦

∫

2/3

1,

4 2T

tb Gi t ba

π−

⎛ ⎞ − −≡−⎜ ⎟⎜ ⎟⎝ ⎠

Vortex Melting TheoryVortex Melting Theory

1 Requires accurate vortex solid free energy. There isInfrared problem due to soft phonon modes (goldstone modes).

2 Requires liquid free energy. Usually Gaussian or Hatree-Fock is not enough for the precision of the free energy. Nonperturbative method is needed in calculating the liquid free energy

Traditional melting theory using Lindermann criterion based on elasticity theory is purely phenomenogical.

Theory of Vortex glassTheory of Vortex glass

Replica symmetry breaking will be used to determine the transition as Mezard and Parisi did for other models usingvariational method.

Thermal fluctuations are taken into account

via statistical sum

Thermal fluctuationsThermal fluctuations

[ ]

( ) ( )G

TZ D x D x eψ

ψ ψ−∗= ∫

Gaussian Gaussian VariationalVariational Calculation for Calculation for

Vortex latticeVortex lattice

Gaussian Gaussian variationalvariational method, method, 2D as an example:2D as an example:

( )2

. ,

1( , ) ( , ) e ( , )( )

2 2

k

z

k k k

k B Z k

x y v x y x y O iAθ

ψ ϕ ϕπ

−

∈

= + +∫

( ) ( ) ( ) ( )1 1 1 1

;

1k OO k k AA k k AO k k OA k

g K V

K O G k O A G k A A G k O O G k Aω

− − − −− − − −

= +

⎡ ⎤= + + +⎣ ⎦

One found after analyzing gap equations that:

( )( )

1 0

0k

k

E kG

E k

γγ

− ⎛ + Δ ⎞= ⎜ ⎟− Δ⎝ ⎠

Finally the equations:

( ) ( ) ( )

( ) ( )

2

2

1 12 2

1 1

T k k lk k

A kk k

E k a vE k E k

vE k E k

β βγ γ

β γγ γ

−

⎛ ⎞= + + +⎜ ⎟⎜ ⎟+ Δ − Δ⎝ ⎠

⎛ ⎞Δ = + −⎜ ⎟⎜ ⎟− Δ + Δ⎝ ⎠

The solution can be obtained using mode Expansion and it converges rapidly:

with

( )

( ) ( )

( ) ( )

2 2

0

0

e x p

nk n

n

n

X n a

n nn

k

k i k X

E k E k

β χ β

β

β

Δ

∞

=

=

∞

=

=

= •

=

∑

∑

∑

2

e x p2

2e x p

3

aχ

π

Δ⎡ ⎤= −⎢ ⎥

⎣ ⎦

⎡ ⎤ = −⎢ ⎥⎣ ⎦

So we have an algebraic equation for !!!

becomes smaller with bigger n. We can solve theequation very easily by iterations along with shiftequation

, nEΔ

nE

3D calculationsDingpingLi,B.Rosenstein

PRL90,167004 (2003).

5.5Ta ≈ −

SpinodalSpinodal PointPoint

Our Our guassianguassian calculation showed that calculation showed that there is a spin nodal point (or line) for there is a spin nodal point (or line) for vortex solid. There exists the solid above vortex solid. There exists the solid above melting temperature down tomelting temperature down to

The experiments done by Z.L. Xiao, O. Dogru, E.Y. Andrei, P. Shuk, M. Greenblatt also confirmed that there is superheated solid to , and then stop at this line.

Z.L. Xiao et.al,PRL(2004)

Dingping Li,B.Rosenstein PRB65,220504 (2002).

5.5Ta ≈ −

Perturbation Theory of LLL GL Perturbation Theory of LLL GL

Model Gaussian ExpansionModel Gaussian Expansion for liquidfor liquid

Ruggeri and Ruggeri and ThoulessThouless developed high temperature developed high temperature perturbation theory around perturbation theory around gaussiangaussian liquid stateliquid state

where excitation energy satisfies the cubic gap eq. optimizing gaussian (mean field) energy

4 [1 ( )],liqf g x= Ω + ( ) ,nng x c x=∑ 31

2x −= Ω

Ω

3 4 0TaΩ − Ω − =

The series are divergent.

BorelBorel--PadePade SummationSummation

BorelBorel--PadePade summation of summation of

converge so

( ) ,nng x c x=∑

4,5.k =

0

2 1

( , 1 )1

( ) ( ) e x p ( ) ,

( ) [ ]!

k k

knn

k k kn

g x d t g x t t

cg x P a d e x

n

∞

−

−=

′= −

′ =

∫

∑

( )kg x

1/ 24 [1 ( )]liqf g xε= + converges. The result is consistent with MC and OPT.

Overcooled liquid stateOvercooled liquid state

BP calculation shows that there is a meta stable liquid state down to very lower temperature

We speculate that there is a meta stable liquid state down to very lower temperature for any repulsive interaction system.

Recent experiments done by Z.L. Xiao et.al. (PRL2004) confirmed that there is a meta stable liquid state down to very lower temperature (NbSe2).

Plotting ,one found that for 3D

Similar calculation in 2D, one found that.Magnetization curves etc. are

consistent with MC (Kato, Nagaosa, Phy. Rev. B31,7336 (1992))and OPT in 2D too.

,sol liqf f 9.5mTa = −

13.2mTa = −

Application of the Application of the metastablemetastable liquid and liquid and

solid theorysolid theory--Melting of Vortex LatticeMelting of Vortex Lattice

One can use the above theory toOne can use the above theory tocalculate various quantitiescalculate various quantities

�� Magnetization JumpMagnetization Jump

�� Entropy JumpEntropy Jump

�� Parameter FittingParameter Fitting

�� LLL Scaling FunctionLLL Scaling Function

( )2 2 22 * *

*2 2z

bG L dx D a

mψ ψ ψ ψ ψ′⎡ ⎤′= + +⎢ ⎥

⎣ ⎦∫

��

Disorders are introduced via:

Disorder effects in typeDisorder effects in type--II superconductorsII superconductors

( ) ( ) ( )( )1 1* * 1 ,m m U x− −

→ + ( ) ( ) ( )U x U y P x yδ= −

( )( ) ( ) ( ) ( )1 ,a a W x W x W y R x yδ′ ′→ + = −

( )( ) ( ) ( ) ( )1 ;b b V x V x V y Q x yδ′ ′→ + = −

[ ]{ }log , , , /G T G U W V Tψ= −

Disorder average can be done using replica trick:

( )( )

( ) ( )2 22 2 * *2

,

1exp

42 4a

na a b a a b b

a a b

qZ g r

ψ

ψ ψ ψ ψ ψ ψ ψπ

⎡ ⎤′⎡ ⎤′= − + +⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦∑ ∑∫

There is a replica symmetry breaking solution in partof phase diagram!! It is a glass phase region!

1 storder

2 ndorder

380 Oe 0.1G

B –

Lin(T

) (G)

420 Oe0.2G

350 Oe

0.2G

Glass transitionmeasured by Beidenkopf, Zeldov, etc. PRL,2005

2 ndorder

1 storder

AbrikosovAbrikosovLatticeLattice

BraggBraggGlassGlass

Amorphous GlassAmorphous Glass LiquidLiquid

Second-order Glass transition

Disorder effectsDisorder effects to the melting lineto the melting line

A universal melting line, smoothly interpolating from low temperature where disorder is important, to high temperature where disorder is negligible.

Observation of a FieldObservation of a Field--Driven Structural Driven Structural Phase Transition in the FLLPhase Transition in the FLL

ErNi2B2C: triangular to square!

Eskildsen et al. PRL 78,1968(1997)

Small-angle neutron scattering

magnetic decoration

R=Dy, Ho, Er: SC and AntiferroR=Y, Lu :Sc

B

Dewhurst et al PRB 72, 014542 (2005)

YNi2B2C

MechanismMechanism

Symmetry and reorientation of FLL relative to Symmetry and reorientation of FLL relative to the crystallographic axes is determined by: the crystallographic axes is determined by:

�� Anisotropy of gap parameter Anisotropy of gap parameter ((ObstObst 1971)1971)

It can be analyzed phenomenologically

Ginzburg-Landau (GL) approach

�� Anisotropy of Fermi surface Anisotropy of Fermi surface ((UllmaierUllmaier 1973)1973)

Theory

�� Affleck et al. PRB Affleck et al. PRB 55,55, R704 (1997)R704 (1997)

Mixture of s-wave and d-wave coupling

�� Rosenstein et al. PRL Rosenstein et al. PRL 80,80, 145 (1998)145 (1998)Anisotropic contribution to GL free energy

( ) ( )2 22 2[ ]Aniso x y x y y xF D D D D D Dψ η ψ ψ⎡ ⎤

⎢ ⎥⎣ ⎦= − − +

Thermal fluctuationThermal fluctuation

�� With weak thermal With weak thermal fluctuation,fluctuation,perturbation method has perturbation method has negativenegative slopeslope..

� Near melting line, the thermal fluctuation became stronger, Gaussian method tells that the slope became positive.

� However, the thermal fluctuations is not important for LTSC.

Liquid

�� Without thermal fluctuation, the SPT line is Without thermal fluctuation, the SPT line is parallel to the x axis.parallel to the x axis.

�� It is a 2nd order phase transitionIt is a 2nd order phase transition..

Quenched Disorder effectQuenched Disorder effect--resultresult

�� Slope of SPT is always Slope of SPT is always positivepositive

�� The SPT is 1The SPT is 1stst

order phase order phase transition.transition.

�� Disorder strongly Disorder strongly influences the influences the system atsystem at low low temperaturetemperature

Comments on other theoriesComments on other theories

Dorsey et al,PRL91,097002(2003)

Those theories can only obtain experimentalPhase diagrams by using unjustified assumption of some parameters dependence on temperature

The LLL GL model is solved and used to The LLL GL model is solved and used to explain the experimental results.explain the experimental results.

Our theoryOur theory is the quantitative is the quantitative theory. theory. The theory can explain The theory can explain the experimental results well.the experimental results well.

Conclusion

�� �������� ������ ������� ����������������������� �� �� ��������� ���������������������

�� �������� ������ ����������������� ���� ��������� ����������������� ���� ��������� ��!"�#�$�%%&�'!��������� ��!"�#�$�%%&�'!���

�� ����������������������������������������������������� ������ ���((��������))��

�� ������ �������� � �������� �*��+&,*��+&, � �Journal ofJournal ofSuperconductivity: Incorporating Novel Magnetism, Superconductivity: Incorporating Novel Magnetism, volvol 19, 369 (2006); 19, 369 (2006); ,��-�!,��'.��������,��/����,��-�!,��'.��������,��/����

/����!.��/������0�,��#!���/��1�0.��!&,!��/����!.��/������0�,��#!���/��1�0.��!&,!���,�/���,�*���,�/���,�*��2�30&,42�30&,4��

�� ������������������ ���������������������� ����������������������

������������������������������������Phys. Rev. B 74, 174518 Phys. Rev. B 74, 174518 (2006) .(2006) .

�� �������������������������������������� �������� ���������!�������!������

"�#�� "�#�� �� ����������� ���������������������$�!�%�$�!�%�����������

����&�#��������&�#���� �� � � 0707