Cold Melting of Solid Electron Phases in Quantum Dots

Cold Melting of Solid Electron Phases in Quantum Dots

M. Rontani, G. GoldoniINFM-S3, Modena, Italy

phase diagram

correlation in quantum dots

configuration interaction

spin polarization

Wignermolecule

Fermi liquid -

like

high density low density

Why quantum dots?

potential for new devicessingle-electron transistor, laser, single-photon emitter

quantum control of charge and spin degrees of freedom

laboratory to explore fundamentals of few-body physics

easy access to different correlation regimes

Energy scales in artificial atoms

experimental control: N, density, / e2/(l)

low density n high B fieldnT

Tuning electron phases à la Wigner

H = T + Vkineticenergy

e-einteraction

2/1nV T quenched

rs = l / aB n = 1 / l2

= lQD / aB

2DEG:

QD:2/1*

QD )/( ml

Open questions in correlated regimes

crystal

liquid

ferromagnet

Tanatar and Ceperley 1989

2D:spin-polarized phase?disorder favors crystal

0D:crystallization?spin polarization?melting?

controversy for N = 6

QMC: R. Egger et al., PRL 82, 3320 (1999)

CI: S. M. Reimann et al., PRB 62, 8108 (2000)

ji jir

ii

N

i rr

er

m

mH

||

1

22*2

22

20

*2

1

2

envelope function approximation, envelope function approximation, semiconductor effective parameters semiconductor effective parameters

dcbaabcd

abcdbaa

ab ccccVccH '†

'†

'

†

, second quantization formalismsecond quantization formalism

1) Compute 1) Compute HH parameters from the chosen single-particle basis parameters from the chosen single-particle basis

rdrrHr baab

)()()( 0

* 'dd)()'('

)'()(2

** rrrrrr

errV dc

rbaabcd

2) Compute the wavefunction as a superposition of Slater 2) Compute the wavefunction as a superposition of Slater

determinantsdeterminants ij

iii HHc 0|| †

'† mli cc

Configuration interaction

sp

d

Monitoring crystallization

example:N = 5

tota

l den

sity

con

ditio

nal p

rob

ab

ility

Rontani et al., Computer Phys. Commun. 2005

Classical geometrical phases

•crystallization around (agreement with QMC)•N = 6 ?

con

dit

ion

al pro

bab

ility

No spin polarization!N = 6

•single-particle basis: 36 orbitals•maximum linear matrix size ≈ 1.1 106 for S = 1•about 600 hours of CPU time on IBM-SP4 with 40 CPUs, for each value of and M

= 2 = 3.5

= 6

Fine structure of transition

conditional probability= fixed electron

N = 6

“Normal modes” at low density

N = 6 = 8

(mod 5) - replicas

rotational bands

cf. Koskinen et al. PRB 2001

Monitoring crystallization

= 2

Monitoring crystallization

= 2.5

Monitoring crystallization

= 3

Monitoring crystallization

= 3.5

Monitoring crystallization

= 4

Monitoring crystallization

= 5

Monitoring crystallization

= 6

The six-electron double-dot system

top view top-dot electron

bottom-dot electron

phase I phase II phase III

t

Numerical results

t tRontani et al., EPL 2002

Cold meltingI and III classical configurations

II novel quantum phase, liquid-like

I III

(rad)

same dot

different dots

Ronta

ni et

al.,

EPL

2002

Conclusion

phase diagram of low-density quantum dots

spin-unpolarized N = 6 ground state

classically metastable phase close to melting

How to measure?inelastic light scattering

[EPL 58, 555 (2002); cond-mat/0506143]

tunneling spectroscopies

[cond-mat/0408454]

FIRB, COFIN-2003, MAE, INFM I.T. Calcolo Parallelohttp://www.s3.infm.it