Dynamics of Vortex Structures

Valerii VinokurArgonne National Laboratory

L. IoffeM. FeigelmanD. GeshkenbeinA. KoshelevA. LarkinA. LopatinD. Nelson

G. CrabtreeP. KesM. KonczykowskiL. Krusin-ElbaumW. Kwok A.MalozemoffA-C. MotaM. OcioC. RosselC. van der BeekU. WelpY. YeshurunE. Zeldov

Outline

1. Superconductors and vortices 2. Vortices and point disorder3. Critical currents and creep4. Quantum mechanical mapping and

Bose glass5. Competing localization

Superconductors

The Meissner Effect

TYPE I AND TYPE II SUPERCONDUCTORS

VORTEX FORMATION

Metal

Superconductor

x

ψ

H

Js

ξ

λJs

quantum of flux hc/2e

20.7 Oe-µm 2

H

λ ξ

Abrikosov vortices

… vortices…

Union City tornado

Cordell tornado, in a sinuous stage

Js

H

Js

H

Vortex lattice

Nb at T= 1.2 K H = 985 GU. Essmann, H. Träuble, Phys. Lett. 24A, 526 (1967)

decoration by magnetic smokeimage in electron microscope

Disorder

Dislocations in crystals CDW and SDW

Domain walls Interacting electrons

Wigner crystals on disordered substratesSpin- and other glasses…

Exemplary system: elastic medium in random environment

Models a wealth of physical systems and phenomena:

Collective pinning

L

recovers pinning

Critical current is defined by the relation:

and reads:Lorentz force

Pinningforce

Three regimes of vortex motion:

T. Nattermann and S. Scheidl, Vortex glass phases in type II superconductors,Advances in Physics, 2000, vol. 49, No. 5, 607, 704

vortex dynamics

- On n'est pas sérieux, quand on a dix-sept ansEt qu'on a des tilleuls verts sur la promenade.

(One isn’t serious when one is seventeen,And when there are green lime-trees on one’s

promenade.)

Arthur Rimbaud

We consider creep regime

Linear response

For vortices: v=µFfind µ ?

L

u

( )

c

barrier pc

LwL

L x xE E fL w L

ς

χ

ξ⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ′⎛ ⎞−= ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

∼

0On the intermediate scales where u a<

20

Dp cE a L −= ∆

2 2Dχ ζ= − +

Roughness:

Being placed in disordered medium elastic object adjusts itself to rugged potential relief

Thermally activated vortex dynamics in random media

=

Activated vortex motion:

Creation a vortex loop

F

ℓ

Nucleation of new phasein first order transitions

1/(2 )p

f c

fL L

f

ς−⎛ ⎞

= ⎜ ⎟⎝ ⎠

( , )F x L

x

x

barrierE

LfL

fL

BE

2

( ) 1Dbarrier p p

c c f

L L LE E w L fL EL L L

ζχ χ −⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎢ ⎥− = − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

fL

Meaning of :

At distances

pinning is not effective

f

f

L

L L>

Driving force f :

( ) pbarrier f B p

fE L E E

f

µ⎛ ⎞

≡ = ⎜ ⎟⎝ ⎠

(0)barrier p

c

LE EL

χ⎛ ⎞

= ⎜ ⎟⎝ ⎠

Thermally activated dynamics of random systems:

Governed by the wide distribution of barriers

0ln( / )E T τ τ

0

time to overcome barrier :

exp

EET

τ τ ⎛ ⎞⎜ ⎟⎝ ⎠

Basic law of relaxation in random (glassy) systems:

CREEP

O Snail, Climb Fuji slope, slowly, slowly Up to the top…

Matsuo Bashō

1v τ −∼( )

exp p pE f Tv

T f

µ⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦∼

2χµ

ς=

−

Consequences of general law of relaxation:

0ln( / )pB p

fE E T

f

µ

τ τ⎛ ⎞

= ⋅⎜ ⎟⎝ ⎠

Motion under Constant force:

0vnf JcΦ

= [ ]0

1/0

( )ln( / )

JJ tt µτ

∼Current (magnetizationrelaxation:

No linear response!

et al

Experiment Unambiguous proof: divergent barriers: E J µ−∝

BOSE GLASS:VORTICES + COLUMNAR DEFECTS

A PROBLEM:B

A

CD

sail over the sea. The paths are straight lines and all the velocities are different. Ship A ‘collided’ with ships B, C, and D; B collided with C and D (triple collisonsexcluded).

Four ships,

Prove that C collided (or will collide) with D

AB

D

C X

X X

X

X

?

X

Y

X

Y

t

C

D

B

A

Z

r Z

r

t

exp 1T

− ∫2

2drdzdz

ε ⎛ ⎞⎜ ⎟⎝ ⎠

exp 1T

− ∫2

2drdzdz

ε ⎛ ⎞⎜ ⎟⎝ ⎠

mt ti

21 ( ) ( )

2i

N i iji i j

drF dz U r V rdz

ε<

⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

∑ ∑∫

Vortex trajectories can be mapped onto world lines of the 2D Bosons

Quantum mechanical mapping

3/ 21/ 24 2exp3

m Uw

e E

⎛ ⎞⎜ − ⎟⎜ ⎟⎝ ⎠

1~ exp kE JwT J

⎛ ⎞−⎜ ⎟⎝ ⎠

1/ 2 3/ 20

0

4 2exp3

lc UJ

ε⎛ ⎞≡ −⎜ ⎟⎜ ⎟Φ⎝ ⎠

HALF-LOOP EXCITATIONS: v∝ exp(- const / J)

CURRENT

1/30exp kE J

T Jρ

⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

∼

Variable range vortex hopping

Variable range hopping(Shklovskii formula for semiconductors)

VARIABLE RANGE FLUX HOPPING: v∝ exp(-const/J1/3)

HALF-LOOP EXCITATIONS: v∝ exp(- const / J)

CURRENT

et al

BOSE GLASS DYNAMICS

Creep is a general phenomenon

Other random systems?All of them?

Vortex lattice melting

melting

266C u L

Mean-field (cage) approach

- elastic energy to deform vortex in a cage

T - energy of thermal fluctuations

22

66 ,uC u LL

ε 266 0/C aε≈ 0L a≈

0a

u

L2

66 0C u a TEnergy balance:

366 0C a TMelting condition: u 0aLc

0.1 0.2Lc ≈ ÷

melting and disorder

cLWeak disorder does not influence melting much since disorder becomes relevanton the scales of the order , and melting characteristic distances are ,and

0a

0cL a ?

Bose glass transition and vortex localization

:

0th

Ta

ε 2 0th

Tauε

Lindemann criterion:2 2 2

0th Lu c a=

εthε

20m LT c aε=

Energy criterion:2( )th m LT cε ε= 2

0m LT c aε=

Energy Lindemann criterion

In the first order with respect to disorder the correction to the energyis zero (after averaging with respectto disorder). Thus the first correctionappears in the second order of perturbation theory. But the secondorder correction is always negative.Therefore, the bound state in the rippled well is more deep.

εε

Bound state in the parabolic well

Bound state in the rippled parabolic well

Shift of the melting line due to columnar defects

mBT

B

BGBMelting line shifts upwards

Is that all?

Bose glass melting vs. delocalization process

Formation of the Bose glass phase is equivalent to localization of 2D quantum particles in the random field of point defects. Melting into a liquid phase corresponds to the delocalization effect

Then little Gerda shed burning tears; and they … thawed the lumps of ice, and consumed the splinters of the Bose-glass…

Hans Christian Andersen, The Snow Queen

44 (tilt modulus) is related to the superfluid density of bosonsc

( )2 144 / 4 1 (4 )sc B nπ π λ −⎡ ⎤= +⎣ ⎦

Bose glass transition takes place at ns = 0

Consider vortex liquid state

Disorder-induced depletion of superfluid density

Stiffening of the tilt modulus

Any disorder reduces ns. May be interpreted as the fact that somefraction of the vortices is always localized

(partially pinned)

What happens to pinned vortices as we raise temperature?

Intermediate vortex state liquid+pinned?

B

T

Intermediate phase?Bm

Bose glass

→ equivalent to superfluid state of 2D bosons

There are always bound states in 2D, therefore one vortex is always pinned by one columnar defect.

What happens if there are many interacting vortices (take high enoughtemperatures where binding is exponentially weak)?

Naïve picture:Vortices wander freely and screen each other out from columnar defect. Thus, if the defect potential is not sufficiently strong, vortices may get depinned.

Increasing temperature effectivelysuppresses pinning potential. Therefore,by increasing temperature, one can depinvortices from columnar defects

B

Tmelting line

Boseglass

vortex liquid

delocalization line

Hamiltonian of bosons:2 2 2 2

1 2 1 1 12 2 2ˆ / 2 ( ) ( ) ( ) ( ) ( ) ( )H d r p m U r d r d r r r V r r rψ µ ψ ψ ψ ψ ψ+ + +⎡ ⎤= − + +⎣ ⎦∫ ∫

Low defect concentration:defects can be considered separately ( ) ( )i

iU r u r r= −∑

Single defect pinning energy:

20

0 2 2( )2

ru rr

εξ

≅+ ( )

20

0 24φεπλ

=

21/

1 21

TE e β

ε ξ−∼

2 20 1 0 / 4r Tβ ε ε=

1 1

TE ε⊥ ∼Localization length:

Calculations: finding the occupation number n for the columnar defects

In case of strong vortex interaction double occupation is prohibited

0 0 1 1ˆ ˆ ˆ

k kk

b b bψ ϕ ϕ ϕ= + + ∑

0 1ˆ ˆ ˆcondensate, localized state, excitationskb b b− − −

ˆ ˆ 0b b+ + =

Now we use the basis of the exact eigenstates of the noninteracting problem:

Effective model that describes occupation of the localized sates:

1 1 1 1 1ˆ ˆ ˆ ˆˆ ( ) ( )effH b E b b bα µ β+ += + − + +

α : +L L

C C

L C

C L

β :L C

C C

C L

C C

+

0 10 1a aψ = +

221 2 2 2 2

2Occupation of the lowest state: 4 4

n aE E E

αα α

= =+ + +

( )2 21 42

E E E α± = ± +

1E E α µ= + −

1 when / 1n E α→ − 0 when / 1n E α→

At 0 localization-delocalization crossover occursE =

2 21 1 2 1 1 1 2 2 2

0

( ) ( ) ( )E E d rd r r v r r rvµ φ φ= + −∫

20 ( )v d rv r= ∫

20 ln( / )Vn ε λ⊥ ⊥

Localization-delocalization crossover:2 4 2

4 4 *0 0 1 0

expln( / ) ( / 4 ) ln( / )V

B T T TnT

κε ε λ π λ⊥ ⊥ ⊥

⎛ ⎞≡ −⎜ ⎟Φ Φ ⎝ ⎠

*0 1 0(1/8)T r ε ε=

VORTEX PHASE DIAGRAM

4 20

4 *0

exp( / 4 ) ln( / )

T TBT

κπ λ ⊥

Φ ⎛ ⎞= −⎜ ⎟Φ ⎝ ⎠

T

BTransition occurs due tolines wanderings

In a liquidvortices fluctuatemuch easier,thus we can expect thetransition to shift tohigher temperatures,if we go from the solidside!

FIRST ORDER TRANSITION

(0)MB

MB

( )exp p pE f T

vT f

µ⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦∼

2χµ

ς=

−

Competing localization