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)