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Cold Melting of Solid Electron Phases in
Quantum Dots
M. Rontani, G. Goldoni
INFM-S3, Modena, Italy
correlation in
quantum dots
Fermi
liquid like
Wigner
molecule
configuration
interaction
spin polarization
phase diagram
high density
low density
Why quantum dots?
potential for new devices
single-electron transistor, laser, single-photon emitter
laboratory to explore fundamentals of few-body
physics
quantum control of charge and spin degrees of
freedom
easy access to different correlation regimes
Energy scales in artificial atoms
experimental control: N, density, De
De / e2/(le)
Tuning electron phases à la Wigner
H = T + V
kinetic
energy
low density n
T n
2DEG:
QD:
e-e
interaction
high B field
V  n1/ 2
T quenched
rs = l / aB
n = 1 / pl2
l = lQD / aB
lQD  ( / m* )1/ 2
Open questions in correlated regimes
ferromagnet
crystal
liquid
2D:
spin-polarized phase?
disorder favors crystal
0D:
crystallization?
spin polarization?
melting?
Tanatar and Ceperley 1989
controversy for N = 6
QMC: R. Egger et al., PRL 82, 3320 (1999)
CI: S. M. Reimann et al., PRB 62, 8108 (2000)
Configuration interaction
 2 2 m*02 2 e2
H  
i 
ri 
2m *
2
2 r
i 1
N
d
1
 

|
r
i j
i  rj |
p
s
envelope function approximation,
semiconductor effective parameters
H  e ab ca† cb 
a ,
V
†
†
abcd a b ' c ' d
abcd
c c c c
second quantization formalism
'
1) Compute H parameters from the chosen single-particle basis
e ab


 
   (r ) H 0 (r ) b (r )dr
*
a
Vabcd


   (r )b* (r ' )
*
a

  
e2

(
r
'
)

(
r
)dr dr '
  c
d
r r  r'
2) Compute the wavefunction as a superposition of Slater determinants
| i   cl† cm†  ' | 0
   ci i
i

H  H ij
Monitoring crystallization
example:
N=5

l2
l  10
l  10

Rontani et al., Computer Phys. Commun. 2005
conditional probability
l2
conditional probability
Classical geometrical phases






•crystallization around l  4 (agreement with QMC)
•N = 6 ?
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
l and M
Fine structure of transition
l=2
N=6
conditional probability
= fixed electron
l = 3.5
l=6
“Normal modes” at low density
rotational
bands
cf. Koskinen et al.
PRB 2001
N=6
l=8
(mod 5) - replicas
Monitoring crystallization
l=2
Monitoring crystallization
l = 2.5
Monitoring crystallization
l=3
Monitoring crystallization
l = 3.5
Monitoring crystallization
l=4
Monitoring crystallization
l=5
Monitoring crystallization
l=6
The six-electron double-dot system
Numerical results
top
view
top-dot electron
bottom-dot electron
phase I
phase II
phase III
De  t
De  t
De  t
Rontani et al., EPL 2002
Cold melting
I and III classical
configurations
I
same dot

different dots
III
II novel quantum phase,
liquid-like
 (rad)
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 Parallelo
http://www.s3.infm.it