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DissipationlessDissipationless currentcurrent carryingcarrying states in states in
type II type II superconductorssuperconductors in in magneticmagnetic fieldfield
NTHU, January 5, 2011
D. P. Li Peking University, Beijing, China
B. Shapiro Bar Ilan University, Ramat Gan, Israel
P. J. Lin Florida State University, Tallahassee, USA
R. F. Hung, D. Berco Nat. Chiao Tung University
Baruch RosensteinNat. Chiao Tung University
H
Meissner
Normal
Tc T
H
Hc
Two defining electromagnetic properties of a superconductor:
1. Zero resistivity 2. Perfect diamagnetism
Kamerlingh Onnes, (1911)
Persistent current flows in
superconductors for years
Electron tomography
Tonomura et al, PRL66, 2519 (1993)
B
Magnetic (Abrikosov) vortices in a type II superconductor
Normal
Mixed
Meissner
H
Hc2
Hc1
Tc T
The second defining property,
perfect diamagnetism, is lost
H
x x
sJ0 *
hc
e
Vortex line repel each other Vortex line repel each other
forming highly ordered forming highly ordered
structures like flux line structures like flux line
lattice (as seen by STM and lattice (as seen by STM and
neutron scatteringneutron scattering)
Pan et al PRL (2002)
Park et al,PRL (2000)
Normal
A. Vortex lattice
Meissner
H
Hc2
Hc1
Tc T
Field driven flux motion
probed by STM on NbSe2
Flux flowFlux flow
FluxonsFluxons are light and move. The motion is generally a friction are light and move. The motion is generally a friction dominated one with energy dissipated in the vortex cores. Electric dominated one with energy dissipated in the vortex cores. Electric current “induces” the flux flow, causing voltage via phase slips.current “induces” the flux flow, causing voltage via phase slips.
The first defining property, zero
resistivity, is also lost in
magnetic field above Hc1
This however is not the end of the storyThis however is not the end of the storyTroyanovsky et al, Nature (04)
B
J
F
Pinning of vortices
STM of both
the pinning
centers (top)
and the
vortices
(bottom)
Schuler et al, PRL79, 1930 (1996)
Intrinsic random disorderArtificial
Recently techniques were developed to
effectively pin the vortices on the scale of
coherence length. The most effective
pinning is achieved at the matching field –
one vortex per pin. Fortunately this case is
also the simplest to treat theoretically.
Pan et al
PRL 85, 1536 (2000)
A pinning center acts as an attractive force on the vortex since
the energy loss due to the necessity to create a vortex core is
reduced.
A single vortex description of the pinned state
Jc
FL
fpin
V r
When the pinning force is able to
oppose the Lorentz force, the flux
motion stops, electric field cannot
penetrate the material and the
superconductivity is restored.
Schuler, PRL79, 1930 (1996)
London appr. for infinitely
thin lines
For vortices are well
separated and have very thin cores
For vortex cores almost overlap.
Instead of lines one just sees array of
superconducting “islands”
The Landau level description for
constant B
Normal
Mixed
Meissner
H
Hc2
Hc1
Tc T
Two complementary theoretical approaches to the mixed stateTwo complementary theoretical approaches to the mixed state
1cH H
2cH H
B
Homogeneity of magnetic induction B
for is a result of overlap of
magnetic fields of roughly
magnetic fields of individual vortices
Magnetization (although
inhomogeneous) is small ( ) and
one replaces B(r)=H
Homogeneity of magnetic fieldHomogeneity of magnetic field
2
2/ cB H
2
2cH
1/ 2 1/ 2
2 2
c 0 0t=T/T 1; 1 ; 1 ; / 1t t x x x
a
Kemmler et al, PRB79, 184509 (2009)
Outline
1. The Ginzburg – Landau description of the charged BEC
in magnetic field.
2. Basic mathematical tool – perturbation theory around a
bifurcation point. Abrikosov lattice as an example.
3. How current carrying states look like in the Landau level
basis?
4. Calculation of the critical (depining) current.
5. Solutions of the time dependent GL equations. Electric
field and dissipation in the moving vortex matter.
6. Excitations in the pinned vortex core. How to increase the
energy gap?
* *
, ; ,t
ie iet D t
c t
D A r r
The friction dominated dynamics is described by the TDGL
2
*, ,
2 ,t F
m t t
rr
The electromagnetic field is minimally coupled to the order
parameter:
the normal state
inverse diffusion
constant
2
2 2 4
*2 2
cF T Tm
r D r
The GL energy for inhomogeneous superconductor
The current density is
**
2n
i e
m
J D D E
Disintegration of the magnetic flux at the normal line into Disintegration of the magnetic flux at the normal line into
vortices at type II SCvortices at type II SC
The vortex dynamics then can be simulated
I. Shapiro, B. Shapiro (2006)
GL energy (using as a unit of length, and neglecting
pinning) is
21
2 2
bH D
2 4*F hH a
x 2 2 2
0/ 2
kN kNH Nb
Is a nonnegative definite operator – QM Hamiltonian of a particle in homogeneous magnetic field:
2/ cb B H
2
1/ 2
1/ 2
;
2 2exp 2
2 2
kN Nk
N N
l
T
l l b lH b y i l x y
b b
Landau levels basis and the bifurcation point
t
1
b
1
normal
lattice
2 ha
2/ cb B H
/ ct T T
1
12
ha t b
Is the distance from the bifurcation point in which the nonzero solution disappears.
Below this point it is reasonable to assume that order parameter is small. The basic idea is to guess the critical exponent and expand the rest in
1/ 2 2
0 1 2 ..h h ha a a
Lascher, PR A140, 523 (65)
B.R., PR B60, 4268 (99)
B.R., Li, Rev. Mod. Phys. 82 , 109 (2010)
ha
Dimensionless parameter
2
*0h
FH a
t
1
b
1
normal
lattice
2 ha
The leading ( ) order equation gives the LLL restriction:
0 0 0 00H C
1/ 2
ha
haPerturbation theory in
-4 -2 0 2 4
-4
-2
0
2
4
The normalization is determined by higher order
The next to leading order, , the equation is:3/ 2
ha
22 *
1 0 0 0 0 0 0 0H C C C
Making a scalar product with one obtains 0
22 *
0 1 0 0 0 0 0 0H C C C
2
0 1.16AC 2 4
0 01 0C
Higher order correction will include the higher Landau level
contributions
1 1
0
N
N
N
C
3/220
1 1 1
N N
N N
CH N bC C
N b
(6)
1
(12)
1
0.279;
6
0.025
12
Cb
Cb
20hH a
haTherefore the perturbation theory in is useful up to surprisingly
LLL is by far the leading contribution above this line.
low fields and temperatures, roughly above the line 1
115
b t
2
/ 0.5
/ 0.1
0.2
c
c
h
T T
B H
a
The LLL component is found from the order etc.5/ 2
ha
Li, B.R. PRB60, 9704 (1999)2 3 4
2 3 2
0.044 0.056
2 6 6
h h h
A
a a af
b b
Currents in an equilibrium state (when pinning is absent) are
purely diamagnetic and overall current of the equilibrium state is
zero in accordance with the generalized Bloch theorem.
tr diamagj r j j r
Pinning creates a stationary nonequilibrium state supporting
transport current.
Bohm, PR141 (1959)
Before considering a more complicated problem of pinning of the
vortex lattice by a stress created by the current (via Lorentz
force) one would like to imagine how the current carrying state
looks like in the Landau level basis.
The order parameter configuration cannot
belong to LLL, since for a general LLL
configuration 2
i ij jJ
Affleck, Brezin, NPB257, 451 (1985)
A small 1LL correction like
Produces an appreciable net current of one percent (the
unit of current is the depairing current of superconductor)
When all the vortices are pinned there is current without dissipation.
0 10.02
LLL is not enoughLLL is not enough
The force balance equation for a periodic pinning potentialThe force balance equation for a periodic pinning potential
2
L pin
Bf z J V f
c
The force balance equation
J
fL
fpin
Qualitatively the best pinning is achieved when the gradient of the
pinning potential is proportional to the Abrikosov vortex
superfluid density
*
DerivationDerivation
Multiplying a covariant derivative of GL equation
** 210
21i i i it V DDD V
2 *1
2iD D
211 0;
2D t V
By one obtains
2* 21 1
02 2
i i i iD D DD V
Using the commutator and integrating over
the sample, one gets:
2, i ij jD D i bD
With full derivatives dropped due to periodicity, leads to
*1
2ij j i ij j ii bD V bj V
Bz j V
c
This relation is making a calculation of the persistent
supercurrent in a lattice pinned by an arbitrary periodic potential
at matching field very simple.
One can systematically expand solutions of GL eqs. around the “new” bifurcation point for the inhomogeneous case to first order in pinning potential
We consider a periodic hexagonally symmetric potential. In this case a conflict between interactions of vortices and pinning potential is avoided and quasimomentum is conserved
21
2 2k
bH D
1
2h h
t ba
k
Solution of GL with a periodic pinning potential
To first order in potential and the order parameter is
02
hx y kN k
A
aj j ij i V
b
and it carries a current
ha
Current vs displacement of vortices
It is important to note that in addition to the displacement, the shape of vortices changes in the current carrying states: shape degrees of freedom that can be used and manipulated
Transition to the flux flow state as current is
increased passed is always at same
quasimomentum (same place within the unit
cell)
1.5 hc
A
aj v
0.46 0.46h hx x
A A
a aj k y
b
cj
1/ 20.695,0ck b
Beyond certain pinning strength the perturbation theory breaks
down. It is quite enough to consider a variational method in which
the configuration is restricted two the lowest two LL.
Beyond perturbation theory in potential
0 0 1 1k kc c
Above the critical current Lorentz force becomes larger than the
pinning force, vortices start moving and electric field enters the
superconductor. Since electric field is inhomogeneous, Maxwell
equations should be solved. Simplicity is lost.
Beyond certain potential the
critical current stops rising.
B.R., B. Shapiro, I. Shapiro, PRB81,
064507 (2010)
Maxwell –GL equations simulation
It turns out that at the depinning current large inhomogeneous
electric fields are generated.
J=0.007
J=0.0075
J=0.01
J =0.0073c
When current significantly exceeds critical, electric field is present electric field is present and, due to superposition between vortices, is also homogeneous in and, due to superposition between vortices, is also homogeneous in sufficiently dense vortex matter sufficiently dense vortex matter
Troyanovsky et al, Nature (04)
Hu, Thompson, PRL27, 1352 (75)
*
;t
ieD r r Ey
t
The friction dominated dynamics is
described by the TDGL
2
*,
2tD F
m
Here is constant and this makes it
possible to apply the bifurcation
perturbation theory again
E
Bifurcation perturbation theory in constant electric fieldBifurcation perturbation theory in constant electric field
2| | 0hL a
2 211 /
2ha t b b
*
4 GL
GL GL
EE
e t Ex
is not Hermitean.
2
GL xUsing the natural units
Time dependent GL equation can be written as
Electric field therefore is an additional pair breaker. The critical line beyond which just a trivial normal solution exists is
2 21 / 0t b b 2 2
2 2, 1 /c cH T E H t b
Hu, Thompson, PRL27, 1352 (75)
2
22tL D H
b
The adaptations to the method are the following. One first looks for eigenfunctions of the linear part of the equation
Np Np NpL
2( ) 1/ 2e / exp /
2
i kx t
Np N
bH b y k b iv y k b iv
The right eigenfunctions are:
Np Nb i vk
Note the “wave” exponential despite absence of Galileo
invariance (due to microscopic disorder tied to the rest frame)
3/ 2v b
Within the bifurcation method one uses scalar products. In the present case these should be formed with the left eigenfunctions:
2( ) 1/ 2 *e ( / ) exp /
2
i kx t
Nk N Nk
bH b y k b iv y k b iv
Li, Malkin, B.R., PRB70, 214529 (04)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
dissipation superfluid density2
2
*2tp E J D
m
The moving lattice solution
The lattice is no longer hexagonal, but is slightly deformed.
In the presence of periodic pinning the corrections and the AC conductivity can be obtained.
Maniv, B.R., Shapiro, PRB80, 134512 (2009)
DeGennes found that normal quasiparticles in the vortex core
have a spectrum
Vortex core excitations
2
FE
However solving Bogoliubov – DeGennes equations with a
dielectric core, one finds that the low angular momentum
excitations are pushed up: very hard to dissipate energy.
Conclusions
1. Bifircation point perturbation theory is a convenient
systematic universal method which can be applied to vortex
matter in type II superconductors when electromagnetic
field is essentially homogeneous.
2. It was applied to describe quantitatively nonequilibrium
supercurrent carrying states supported by a periodic array
of pins of arbitrary shape and the flux flow at sufficiently
large flux velocities.
3. Pins on the scale of coherence length can manipulate the
distribution of the order parameter. The critical current is
maximized when gradient of potential is proportional to the
Abrikosov lattice superfluid density. Core excitations have
large energy gap.
B.R., Li, Rev. Mod. Phys. 82 , 109 (2010)
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