Download Disoder of the vortex lattices in the type II superconductor How a

Ginzburg-Landau-Abrikosov Theory of
Type II Superconductors
- Phase diagram of vortex matter
李定平
北京大学物理学院理论物理研究所
2011-12-30, 台湾,台北
Main Collaborator: Prof. Rosenstein, National Chiao
Tung University, Hsinchu, Taiwan
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Dingping Li and Baruch Rosenstein, Phys. Rev. Lett. 86, 3618 (2001).
Dingping Li and Baruch Rosenstein, Phys. Rev. B 65, 024513 (2002).
Dingping Li, Rosenstein, Phys.Rev.Lett . 90,167004 (2003).
Dingping Li, Rosenstein, Phys. Rev. B 65, 220504 (2002), Rapid
Communication
Dingping Li,Rosenstein,Phy. Rev. B70,144521 (2004)
Dingping Li, Rosenstein, Vinokur, Journal of Superconductivity: Incorporating
Novel Magnetism, vol 19, 369 (2006).
H. Beibdenkopf, Myasoedov, Shtrikman, Zeldov, Rosenstein, Dingping Li,
Tamegai, PRL 98, 167004(2007)
Tinh, Li, Rosenstein, PRB81,224521(2010)
Review of Modern Physics, “The Ginzburg-Landau Theory of
Type II superconductors in magnetic field”,
By Rosenstein and Dingping Li,
VOLUME 82, JANUARY–MARCH 2010, Page 109.
Properties of a superconductor:
1. Zero resistivity
2. Perfect diamagnetism
H
Kamerlingh Onnes, (Mercury, 1911)
H
Hc
Persistent current
Normal
Meissner
Meissner and Ochsenfeld, (1933)
Tc T
Magnetic (Abrikosov) vortices in a type II superconductor
l x
B
l  x
H
Js
hc
0  *
e
H
Most new materials
Including high Tc are
Type II superconductors,
Most part of phase diagram
Is the mixed state.
Normal
Hc2
The perfect diamagnetism is lost
Mixed
Hc1
Meissner
Tc
T
STM (scanning tunneling microscope)Topographic
Image of vortex lattice of a superconductor NbSe2
Temperature: 400mK, Field: 0.5T
Magnetic Materials Laboratory, RIKEN, Dr. Hanaguri
Vortex line repel each other
forming highly ordered
structures like flux line
lattice ( STM and neutron
scattering)
Pan et al (2002)
H
Normal
Hc2
Hc1
Abrikosov
lattice
Meissner
Tc
T
Park et al (2000)
Flux flow
Fluxons can move. Electric current “induces” the flux flow if there
is no pinning force, and the flux flow will cause voltage.
B
J
F
zero resistivity, is also lost in
magnetic field above Hc1 if
there is no pinning
Field driven flux motion
probed by STM on NbSe2
Troyanovsky et al (04)
Pinning of vortices
Intrinsic random disorder
Artificial
STM of both
the pinning
centers (top)
and the
vortices
(bottom)
Schuler et al, PRL79, 1930 (1996)
Pan et al,
PRL 85, 1536 (2000)
For the materials to be
superconducting, we must pin
vortices (vortices can not move)
Due to huge thermal fluctuation in high Tc,
Large part of Vortex lattice can melt to vortex liquid
Vortex lattice
Vortex liquid
Phase Behaviour in the Mixed Phase in high
Tc
The conventional picture
An alternative view
Melting of the flux line lattice
local
magnetisation
measurements
on BSCCO
E. Zeldov et al.
Nature 1995
1st – order melting transition:revealed by jump
in magnetisation at Bm
E.Zeldov
Melting of the flux line
lattice
B = 0 T
Crabtree
B = 9 T
heat capacity
measurements on YBCO
A. Schilling et al.
PRL 1997
Entropy jumps at
1st order flux
lattice melting
• The electric field is induced in
a metal under
Nernst
effect
magnetic field by the temperature gradient
perpendicular to the magnetic field ,
T
phenomenon known as Nernst effect.
• The Nernst signal is defined as
H
E /ET
eN 
y
(T ) x
• In the vortex liquid phase of superconductor
the Nernst effect is large due to vortex
motion, while in the normal state and in the
vortex lattice or glass states it is typically
smaller.
The vortex-Nernst effect in a type-II
superconductor. Concentric circles
represent vorties
Xu, nature 406,
YBaZA,
2 Cu3O y
486(2000)
N. P. Ong’s group
(YBCO)
PRL88, 257003
(2002)
Bi2 SrYu
La y CuO
2  y Yang
et al 6
(BSCCO)
PRB 73,
024510 (2006)
N.P.Ong
Two complementary theoretical approaches to the mixed state
For H  H c1 vortex cores almost overlap.
Instead of lines one just sees array of
superconducting “islands”
H
Normal
Hc2
The Landau level
description for
constant B
Mixed
Hc1
Meissner
Tc
For H  H c 2 vortices are well
separated and have very thin cores
London appr.
for infinitely
T
thin lines
Outline
1. The Ginzburg – Landau Theory,
2. GL model of superconductivity.
3. Type I and Type II superconductor
4. Thermal Fluctutation
5. Phase diagram
6. Dynamics of vortex liquids
7. conclusion
Laudau Theory For Phase Transition
Example:Ising model can be mapped to a model with
order parameter m by using hubbard transformation
1
H  i     J i , j i   j    i hi ,J i , j  J , if i, jareneigboring
2 i, j
i
Z   exp    H  i     dmi exp    H  mi , h   ,  i  mi
 i 
Near mean Tc, we can expand H  m , h  in mi
i
as it is small and we can expand H  mi , h  as field M r

Magnetization density field
As coherence is long, we can omit higher derivative term.
  

H M r   d rC1 M r
C2 T    (T  Tcm ),
Near Tc,
2

 C2  T  M r
2

C3

M r
2
4
 
 M r h r ....;
C1 , C3 can be considered as constant, C2
Is linear dpendent on T and change sign at meanfield transition temperature
Above theory can correctly describe the physics
near Tc (including thermal fluctuation) for Ising
Model.
Landau Theory of
Second Order Phase Transitions
L.D. Landau ( 1937 )
Order parameter field and spontaneous
symmetry breaking
Any second order phase transition is described by:
a) Order parameter field
i (x)
b) Symmetry group G.
The Laudau theory is a lower energy effective theory
In contrast to original microscopic theory
GL Free energy for superconductors.
Without external magnetic
field the free energy near transition is:
V.Ginzburg
2


3
2
F [ ]   d x 
 *    (T  Tc ) *   ( *  ) 
2
 2m *

m*  2me
Ginzburg and Landau (1950) generalized
this to the

case of arbitrary magnetic field B (x )
using gauge invariance of electrodynamics.
The GL free energy with magnetic field is:
1
e*
 4 B2 3
2
F  [
(i   A)   (T  Tc)     ]d x
2m *
c
2
8
2
In equilibrium under fixed external magnetic field
the relevant thermodynamic quantity is the Gibbs
free energy:
1
G  F   dx
BH
4
3
The theory is generally believed to be a
correct theory describing the physics near
Tc, even the microscopic origin of order
parameters are unknown!
Historically GL theory of superconductor was
formulated at 1950 without knowing any paring
mechanism. Gorkov, later on based on BCS,
derived GL theory for superconductor.
Generally we believe GL theory is valid also for
high Tc even though its microscopic theory is
unknown.
Mean field equation (ignoring the thermal
fluctuation)-Ginzburg – Landau equations
Minimizing the free energy with covariant derivatives
one arrives at GL equations. The nonlinear
Schrodinger equation (variation of  ):
1
e* 2
2
(i   A)    (T  Tc)     0
2m *
c
And the super-current equation (variation of A):
2
c
ie *
e
*
2
*
*
 B  J  
(    ) 
 A
4
2m *
m*c
For finite sample, the equations should be
supplemented by the boundary conditions.
The covariant gradient is perpendicular to the
surface
ˆ  ( i  
n
e*
A)   0
c
While magnetization is parallel to it:


ˆ  (B  H )  0
n
Time dependence and disorder
The GL energy for inhomogeneos superconductor
F

r
2
*
2m
D
2
  T  Tc  r   
2


4
2
The friction dominated dynamics is described by the TDGL
 
2
 
2m  t
*
ie*




    x, t 
  r, t    x, t   
F

,



  x, t 

  x, t   0,   x, t    x ', t '  

2
*
m
  x  x '  t  t '
 the thermal noise,  the normal state inverse diffusion
constant.
The electric field is also minimally coupled to the order parameter:
Disorder can be introduced by adding a white noise term to Tc
Two characteristic scales
GL equations possess two scales. Coherence length x
characterizes variations of  (x) , while the
penetration depth l characterizes variations of B (x )
x (T ) 

2m * (T  Tc)
1
,
2
c  m* 
l (T ) 


e *  4 T  Tc 
Both diverge at T=Tc.
1
2
Ginzburg – Landau parameter
The only dimensionless parameter one can construct from
the two lengths is temperature independent:
l (T ) m * c 


x (T ) e * 2
Properties of solutions crucially depend on GL
parameter. If
1
 
2
(so called type II superconductivity ) there exist
topological solution –
the Abrikosov vortices.
Abrikosov (1957)
1
  l /x 
2
Type-I
Meissner Phase, where no magnetic field in the bulk
1
  l /x 
2
Type-II
In addition of Meissner phase, there is also a mix state where
magnetic field can penetrate the superconductor
Near Hc2, it is convenient to do follow scaling: length
in unit of
Unit of
x
, magnetic field in unit of Hc2.
2 Tc


2
in
Boltzmann factor is
2
2
 G 1 3 1

1
1

t
1

(
b

h
)
2
2
2
4
,
 g    d x  D   z     

2
2
2
4 
 T 
2

 D    iA, A  (by,0), b  B .

Hc2
With following parameters:
  2Gi t ;

2
2 2
2

1  32 e l Tc 
Gi  
 .
2 2
2
c h x


Gi is the so called Ginzburg parameters which
characterize the thermal fluctuation strength.
In high Tc, Gi is so large and thermal
fluctuation is very significant! Mean field the
solution (Ginzburg-Landau eq.) is not enough.
Contrast to low Tc superconductor, one must
include the thermal fluctuation effect in high Tc!
Increased role of thermal
fluctuations in high Tc materials
2
1 Ginzburg number is much
Metals::
High Tc:
1  32 e2l 2Tc 
larger Gi   2 2  ,
2 c h x 
Gi  10
 2  mc / mab
6
Gi  .0001(YBCO )  .5( BISCCO )
Remark: The thermal fluctuation (TF) is so small in low
Tc sup., melting and other TF effect can not be observed.
2 Magnetic field effectively reduces dimensionality of
fluctuations from D to D-2, and it will enhance the fluct..
Thermal fluctuations of vortices in high Tc
superconductors will lead to significant effects, melt the
Abrikosov vortex lattice into “vortex liquid”.
The thermal fluctuation changes the second order phase
transition of mean field solution into a first order
melting transition.
The disorder will change part of phase diagram to glass
phase (therefore becoming superconducting states),
including vortex glass and Bragg glass (Giamarchi,Le
Doussal
Thermal fluctuation
To calculate the thermal effect
G
Z   D D exp  
,
T 
*
How to calculate this very complicated functional integral.
Various method-Perturbative and Non Perturbative.
Phenomenological approach based on
elastic theory
Elastic theory of vortices (lower energy approximation of the GL
model) in the vortex lattice, E.H. Brandt, 1977.
Based on the elastic theory, and using Lindermann criterion of
theory of tradition melting (solid to liquid) , when the dislocation
exceeds to some extent, the vortex lattice melts to vortex liquid
(normal state).
The Lindermann criterion is a pure phenomenological criterion
without a sound theoretical basis. Also it can not predict the
latent heat et. al. along the melting line.
Furthermore they are qualitative, not quantitative.
Key theoretical developments
Second order transition between the vortex lattice and vortex
liquid based on the mean field solution of GL theory of type-II
superconductor (Abrikosov,1957).
GL model in the lowest Landau level modes with thermal
fluctuation included, studied G.J. Ruggeri and D.J. Thouless,
1976. The phase transition was speculated as the second order
phase transition.
The model studied by G.J. Ruggeri et. al. was reinvestigated
using RG. The phase transition was speculated as the first order
phase transition (melting). E. Brezin, D.R. Nelson and A.
Thiaville, 1985.
Larkin-Ovchinnikov collective-pinning theory (1979)
Brag glass theory by Giamarchi and Le Doussal (1995)
Review by Blatter etal.,RMP 1994; Brandt, Rep. Prog. Phys
(1995);Nattermann and Scheidl, Adv.Phys.,200; Larkin,
Varlamov, book by Oxford (2005)
Our approach:
Various method (perturbative and nonperturbative)
of field theory are used in calculating the thermal
fluctuation from the GL model . The main
approximation is the LLL mode approximation
which is valid near Hc2 line.
Using the theory, the melting line location can be
determined and magnetization and specific heat
jumps can be calculated , which agree well with
experimental results on YBCO and Monte Carlo
simulations. Considering disorder effects, we also
locate the glass transition line or superconducting
transition line
Phase diagram of vortex matter
-Spontaneous symmetry breaking
symmetry
TS (translation
invariant)
pattern
RS (replica
symmetric)
RSB (replica
symmetry broken)
liquid
Vortex glass
TSB (translational solid
symmetry broken)
Bragg glass
Different translation symmetry breaking patterns leads
square or hexagonal lattice-structure phase transition .
Generic Phase diagrams in high
Tc
Dynamics of vortex liquid
The friction dominated dynamics is described by the TDGL
 
2
 
2m  t
*
ie*




    x, t 
  r, t    x, t   
F

,



  x, t 

  x, t   0,   x, t    x ', t '  

2
*
m
  x  x '  t  t '
for a constant electric field   r,t    Ey
If E=0, the state in the long time limit will go to
Equilibrium state described by free energy F.
Large portion of mix state in phase diagram is liquid due to very
strong thermal fluctuations in high Tc. Vortex Lattice melts to liquid
Vortex liquid flow (response to electric field)
Tinh, Li, Rosenstein, PRB81,224521(2010)
   2 
2
2
 Gaussian approximation in TDGL is approximating the
nonlinear term
 The quadratic TDGL can be then solved exactly, and
conductivity or Nernst coefficient can be calculated. In
particular, Lawrence-Doniach model can be solved too
considering all Landau level.
As layered superconductor free energy
Lawrence Doniach model
In rescaled unit, TDGL can be written as
'
'
Gaussian variation approximation, the nonlinear
Term is approximated by       2  
2
n
2
n
n
2
*
n
n
n
The factor 2 is due to two contractions between

and 
. The nonlinear equation is replaced
By the linear one
     2  
*
n
n
2
n
2
n
 can be solved consistently.
n
2
*
n
n
n
Then we can solve the gap equation completely
And the final theory is cut off independent contrast
previous theory!
Nonlinear conductivity
Can be obtained and
compared to experiments
Nernst effect also can be
Calculated.
Noticing Tc
field Tcmean
in  (T  Tc ) 
2
actually is the mean
Actually Tc is related to (details can be found in paper)
Tc  T
mean
c

2Gi   c
1
 
 
,  E  0.577( Eulaconst.)
 ln 2   E   , c 
E
s  d
2 F e
 

The energy cutoff is Fermi energy as in high Tc,
the coherence length is of order atomic distance
Comment: 2d limit, d is very large, the above eq. shows
there is no superconductivity which is consistent with
Mermin Wagner theorem-in 2d, there is no U(1)
Symmetry breaking, therefore no superconductivity
After using the renormalized Tc (experimental
observed Tc!)in the gap eq., one can show that the gap
eq. becomes Cutoff independent, as it should be for a
renormalizable theory.
For a sample YBCO, real Tc is 87K, mean
field Tc is 101K. Between two Tc, there are
Local orderings, but lost global phase coherence,
This might be the pesudogap region.
One can also use the theory to explain the Nernst
Effect.
Conclusion
The GL model is systematically studied and used
to explain the experimental phase diagrams and
calculate thermodynamic quantities and
dynamical quantities.
.