PLASMA

Artificial classical atoms and molecules:from electrons to colloids and to superconducting vortices

François Peeters

V. Bedanov, V. Schweigert, M. Kong, B. PartoensG. Piacente, J. BetourasS. Apolinario

Wigner crystalWigner crystal

Ground state of the electron gas in metalsE. Wigner, Physical Review 46, 1002 (1934)

„If the electrons had no kinetic energy, they would settle in configurations which correspond to the absolute minima of the potential energy. These are close-packed lattice configurations, with energies very near to that of the body-centered lattice....“

- 2D electrons on liquid heliumC.C. Grimes and G. Adams, PRL 42, 795 (1979)

- Colloidal particles on surfaces or interfaces- Dusty plasmas- Charged metallic balls, ….

NbSe2 measured by STM

- Superconductors Abrikosov lattice (1957)Nobel prize in 2003

Theoretical research on colloids in India

• Bangalore (Indian Institute of Science): - A.K. Sood- H.R. Krishnamurthy- J. Chakrabarti

• Kolkata (S.N. Bose National Research Centre for Basic Sciences):- S. Sengupta

Confinement

• Geometrical constrainted motion- 1D: microchannels- 0D: artificial atoms

• Self-organization in reduced dimensions• Reduced phase space

- Diffusion (e.g. anisotropic diffusion)- Non-linear dynamics

Hamiltonian (2D)

Coupling constant

Hamiltonian: artificial atom

Kinetic energy Confinement Interaction

Energy unit

Length unit

2

203 ,

mea

a

aB

B

0

00 / ma

23 For vertical quantum dot:

2 23

1 1

1

2 2

N Ni i

i i i j i j

rH

r r

1. Confinement potential

2. Typical energy scale

3. The considered artificial atoms are two-dimensional4. The number of electrons, the size and the geometry of

artificial atoms can be changed arbitrarily

Differences with real atoms

Real atom Artificial atom

Parabolic potential

2 20

1

2m rr/1

Coulomb potential

0

0Real atom: Ry = 13.6 eV

Vertical quantum dot: = 3 meV

Classical artificial atoms ()

• Potential energy

• Ground state Energy minimalization

22 20

1

1 1

2

N

ii i j i j

eH m r

r r

New units:

2 1/3 1/3

2 2/3 1/3

20

' ( / )

' ( / )

/ 2

r e

E e

m

11 2

2 ( , ,.. ,1

. )N

N

ii i j i j

E E r rrr

rHr

,

0 , ; i=1,...,Ni

Ex y

r

Ground state configurations

Classical configurations

N N ConfigurConfiguration ation

1 1 2 2

3 3 4 45 56 1, 57 1, 68 1, 79 2, 710 2, 811 3, 812 3, 913 4, 914 4, 1015 5, 1016 1, 5, 10

(1,7) (1,7,12)

V.M. Bedanov and F.M. Peeters, Phys. Rev. B 49, 2667 (1994)Classical atoms (J.J. Thomson (1904))

W. T. Juan, et al, Phys. Rev. E 58, 6947 (1998)

Dusty plasma

Leiderer et al., Surf. Sci. 113, 405 (1982)

(1,7) (1,7.12)Electrons on He surface

Vortices in helium, imaged by injecting electrons that become trapped at the vortex core. (R.E. Packard)

Superfluid helium

Bose-Einstein condensate

Con

dens

ate

dens

ity

MIT, Ketterle group, 2001

Superconducting vortices in a disk

I.V. Grigorieva et al, Phys. Rev. Lett. 96, 077005 (2006)

Generic model (2D)for different systems, energy and length scales

Dusty plasma Ions in traps Colloids Metallic balls Vortices in SC

Diameter: 1 - 10 m Å 1 - 10 m 0.1-1 mm ~ nm - m

Potential Z ~ 104

YukawaZ ~ 1Coulomb

ParamagneticDipole

K0 (r/)

ln(r/)

ln(r/)

a/ ~ 50 >> 1 ~ 2 - 10 ~ 2 - 10 10 - 1000

m (kg) 10-13 – 10-14 10-26 10-13 – 10-14 10-4 – 10-3 0

Environment Plasma Vacuum Liquid Air (surface friction)

Electron gas

Relax. time (s) 0.1 10-4 0.1 10-12 – 10-13

Dynamics Damped ~ 1 - 10

Undamped >> 1

Overdamped < 1

Damped ~ 10

Overdampd < 1

( ) ( )conf i inter particle i ji i j

U V r V r r

LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL Superparamagnetic colloidal spheres

2

, 3,

;i ji j

MV M B

r

Metallic balls

10 mm

M. Saint Jean et al, Europhys. Lett. 55, 45 (2001)

1kV~

I.V. Grigorieva et al, Phys. Rev. Lett. 96, 077005 (2006)

vortex

Decoration exp.

Nb: d = 150 nm; D

1µm; T 3.5 K

Saddle point

Saddle point

M. S. Jean et al ( Europhys. Lett. 55, 45 2001))

N=6

(1,5) (6)

V.A. Schweigert and F. Peeters, Phys. Rev. B 51, 7700 (1995)

2 6

0

Normal modesEigenfrequencies and eigenvectors

2

6

Exp. on dusty plasma:

A. Melzer, Phys. Rev. E 67, 016411 (2003)

Magic numbers

exp.

N=19 (1,6,12) N=20 (1,7,12)

V.A. Schweigert and F.M. Peeters, Phys. Rev. B 51, 7700 (1995)A. Melzer, A. Piel (Kiel University) => dusty plasma (Phys. Rev. Lett. 87, 115002 (2001))

Magic number clusters

Melting: small clusters

Radial fluctuations

Relative angularintershell fluctuations

Anisotropic melting Two-step melting process

Experiments on paramagnetic colloids:R. Bubeck et al, Phys. Rev. B 82, 3364 (1999).V.M. Bedanov and F.M. Peeters,

Phys. Rev. B 49, 2667 (1994)

1/ = kBT/<V>

Artificial molecules

Competition between electron correlationsin the single dots and correlations betweenelectrons in the different dots.

d

Classical artificial molecules

1 1 2 2 1 22 2

1 1

1 1 1+

N N N N N N

i ii i j i i j i ji j i j i j

H r rr r r r r r d

N=10

B. Partoens and F.M. Peeters, Phys. Rev. Lett. 79, 3990 (1997)

Competition between particle correlationsin the single atoms and correlations betweenparticles in the different atoms.

Molecule2x(3 particles)

2x(5 particles)

2x(5 particles)

One dimensional:Microchannels

0 1 2 3 4 50

2

4

6

8others

ñe

6 c.s 5 chains

4 chains

3 chains

4 chains

2 chains

1 chain

Phase diagramPhase diagram

G. Piacente, B. Betouras and F.M. Peeters., PRB 69, 045324 (2004)

2exp( )i j

ij i ii j

r ry

r rH

=r0/r0=(2q/m0

2)2/3

E0=(m 02q4/22)1/3

zig-zagtransition

Continuous transition 2nd order.

Q1D Q1D channels channels

Institut für Physik, Universität Mainz,

Ion chains

M Block, A Drakoudis, H Leuthner, P Seibert and G Werth

Crystalline ion structures in a Paul trap

Experimental evidence for the “zig-zag” transition

lowerelectrode

groove mg

QE

y

x

Complex plasma

(B. Liu and J. Goree, Phys. Rev. Lett. 71, 046410 (2005)

A. Melzer, Phys. Rev. E 73, 056404 (2006)

0 1 2 3 4 50

2

4

6

8others

ñe

6 c.s 5 chains

4 chains

3 chains

4 chains

2 chains

1 chain

2 4 transition

Discontinuous transition 1st order

zig-zag shift over a/4

Lorentian shapedLorentian shapedconstriction constriction

V0’

1/

Driving forceDriving force

Pinning and de-pinning of a Q1D system

G. Piacente and F.M. Peeters, PRB 72, 205208 (2005)

Non-linear physics

Increase of the density(no driving force)

Lane reductionat the constriction

W=60m L=2mm=4.55 m 2.5

Phys. Rev. Lett, 97, 208302 (2006)

G. Piacente and F.M. Peeters, PRB 72, 205208 (2005)

7 lanes 6 lanes

Elastic Depinning (small values of V0’)

Quasi-elastic Depinning (large

values of V0’)

f < fc Pinning

f > fc Depinning

v ( f – fc ) β

== 2/3 2/3 as for as for Infinite 2D SystemsInfinite 2D Systems

Elastic depinningElastic depinning

0.000 0.004 0.008 0.012 0.0160.00

0.01

0.02

0.03

0.04

(b)

T'=0 T'=0.002 power fit for T'=0

vx

f

=1

Vl

0=1

l=1

N=400

= 0.68 ± 0.04

Quasi-elastic depinningQuasi-elastic depinning

0.00 0.02 0.04 0.06 0.08 0.100.00

0.05

0.10

0.15

0.20

T'=0 T'=0.002 power fit for T'=0

vx

f

Vl

0=5

l=1

= 0.89 ± 0.03

N=400

(b)

=1

0 1 2 3 4 50.6

0.7

0.8

0.9

1.0

V0'

1

T'=0.002

l=1

N=300

Crossover Crossover from thefrom the elastic elastic toto quasi-elastic quasi-elastic flowflow

Tuning of the critical exponent

Conclusions

• 0D systems artificial atoms- Ground state: ring structures Thomson model- Lowest normal mode: intershell rotation (small N)

vortex / antivortex rotation- Melting: anisotropic (radial / angular)

• Artificial molecules- ‘Structural’ phase transitions

• 1D systems microchannels- Chains: 1 2: zig-zag transition (continuous)

2 4: first-order transition- Constriction: tuning of the critical exponent: from

elastic to quasi-elastic (no plastic depinning)

The end

Scheme of the experimental setup.Pictures are taken from the website of Lin I and A. Piel’s group

Dusty Plasma (Complex Plasma)

14Mhz

Newton optimization technique

( ) ( ), , , , ,

( ) ( ), , , , ,

, ,

1

2

n ni i i i i

i

n nij i i j j

i j

H r H r H r r

H r r r r

position of the particle after n iteration steps

=x,yi=1,

…,N

The potential energy in the vicinity of this configuration can be expanded in a Taylor series:

( ), ,/ ni iH r Force:

2

, ( ) ( ), ,

ij n ni j

HH

r r

Dynamic matrix:

Eigenfrequencies normal modes

Metallic balls

10 mm

M. Saint Jean et al, Europhys. Lett. 55, 45 (2001)

1kV~

Real atoms versus artificial atoms

3 *0 / Ba a

18 nm5 nm

InAs/GaAs