PLASMA Artificial classical atoms and molecules: from electrons to colloids and to superconducting vortices François Peeters V. Bedanov, V. Schweigert, M. Kong, B. Partoens G. Piacente, J. Betouras S. Apolinario

Wigner crystal Ground state of the electron gas in metals E. 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 helium C.C. Grimes and G. Adams, PRL 42, 795 (1979) - Colloidal particles on surfaces or interfaces - Dusty plasmas - Charged metallic balls, …. NbSe 2 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 ConfinementInteraction Energy unit Length unit For vertical quantum dot:

1.Confinement potential 2.Typical energy scale 3.The considered artificial atoms are two-dimensional 4.The number of electrons, the size and the geometry of artificial atoms can be changed arbitrarily Differences with real atoms Real atomArtificial atom Parabolic potential Coulomb potential Real atom: Ry = 13.6 eV Vertical quantum dot: = 3 meV

Classical artificial atoms (  ) Potential energy Ground state  Energy minimalization New units:

Ground state configurations Classical configurations N Configuration , 5 7 1, 6 8 1, 7 9 2, , , , , , , , 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 Condensate density MIT, Ketterle group, 2001 Superconducting vortices in a disk I.V. Grigorieva et al, Phys. Rev. Lett. 96, (2006)

Generic model (2D) for different systems, energy and length scales Dusty plasmaIons in trapsColloidsMetallic ballsVortices in SC Diameter:  m Å mm  ~ nm - m PotentialZ ~ 10 4 Yukawa Z ~ 1 Coulomb Paramagnetic Dipole K 0 (r/)  ln(r/) a/ ~ 50>> 1~ m (kg) – – – EnvironmentPlasmaVacuumLiquidAir (surface friction) Electron gas Relax. time (s) – DynamicsDamped  ~ Undamped  >> 1 Overdamped  < 1 Damped  ~ 10 Overdampd  < 1 Superparamagnetic colloidal spheres Metallic balls 10 mm M. Saint Jean et al, Europhys. Lett. 55, 45 (2001) 1kV~ I.V. Grigorieva et al, Phys. Rev. Lett. 96, (2006) vortex Decoration exp. Nb: d = 150 nm; D  1µm; T  3.5 K

Saddle point M. S. Jean et al ( Europhys. Lett. 55, )) N=6 (1,5) (6)

V.A. Schweigert and F. Peeters, Phys. Rev. B 51, 7700 (1995) Normal modes Eigenfrequencies and eigenvectors  2  6 Exp. on dusty plasma: A. Melzer, Phys. Rev. E 67, (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, (2001)) Magic number clusters

Melting: small clusters Radial fluctuations Relative angular intershell 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/  = k B T/

Artificial molecules Competition between electron correlations in the single dots and correlations between electrons in the different dots. d

Classical artificial molecules N=10 B. Partoens and F.M. Peeters, Phys. Rev. Lett. 79, 3990 (1997) Competition between particle correlations in the single atoms and correlations between particles in the different atoms.

Molecule 2x(3 particles) 2x(5 particles)

One dimensional: Microchannels

Phase diagram G. Piacente, B. Betouras and F.M. Peeters., PRB 69, (2004)  =r 0 / r 0 =(2q/m  0 2 ) 2/3 E 0 =(m  0 2 q 4 /2  2 ) 1/3 zig-zag transition Continuous transition  2 nd order. Q1D  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 y x Complex plasma (B. Liu and J. Goree, Phys. Rev. Lett. 71, (2005) A. Melzer, Phys. Rev. E 73, (2006)

2  4 transition Discontinuous transition  1 st order zig-zag shift over a/4

Lorentian shaped constriction V0’V0’ 1/  Driving force Pinning and de-pinning of a Q1D system G. Piacente and F.M. Peeters, PRB 72, (2005)

Non-linear physics Increase of the density (no driving force) Lane reduction at the constriction W=60  m L=2mm  =4.55  m  2.5 Phys. Rev. Lett, 97, (2006) G. Piacente and F.M. Peeters, PRB 72, (2005) 7 lanes 6 lanes

Elastic Depinning (small values of V 0 ’) Quasi-elastic Depinning (large values of V 0 ’) f < f c  Pinningf > f c  Depinning

v  ( f – f c ) β   = 2/3 as for Infinite 2D Systems Elastic depinning Quasi-elastic depinning

Crossover from the elastic to quasi-elastic flow 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 position of the particle after n iteration steps  =x,y i=1,…,N The potential energy in the vicinity of this configuration can be expanded in a Taylor series: Force: 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 18 nm5 nm InAs/GaAs