Ladungen in der Falle
Michael Bonitz
Institut für Theoretische Physik und Astrophysik
Christian-Albrechts-Universität zu Kiel
Stuttgarter Physikalisches Kolloquium, 30. Mai 2006
Co-workers:
A. Filinov, V. Golubnychiy, V.S. Filinov (guest)
C. Henning, H. Baumgartner, P. Ludwig
J. Böning, K. Balzer, A. Fromm
Cooperations:
A. Piel, H. Kersten (Kiel)
Yu. Lozovik (Moscow)
A. Melzer, H. Fehske (Greifswald)
H. Stolz, W.D. Kraeft (Rostock)
©Michael Bonitz, Universität Kiel, 2006
Charged particles
Electrons, ions, holes, positrons, quarks ...
From few (N=2...100) to very many particles
Behavior dominated by Coulomb interaction U (r )  q 2 / r
This interaction is strong and long range!
 Expect: coordinated, correlated behavior of all particles
Many similarities with correlation effects in solids, nuclei...
©Michael Bonitz, Universität Kiel, 2006
Plasma
I. Langmuir/L. Tonks (1929): ionized gas - „plasma“
„4th state of matter“: solid  fluid  gas  plasma
ideal hot classical gas made of electrons and ions
BUT: there exist unusual „non-ideal“ plasmas
©Michael Bonitz, Universität Kiel, 2006
Coulomb Systems (Plasmas)
1eV  10 4 K
Magnetic Fusion
Magnet fusion
Sun
Sun
core
Dwarf
stars
Lightning
Neutron
stars
10,000K
Jupiter
Planet
cores
Semiconductors
Dusty
Plasmas
Lightning
Metals
300K
Electron density, 1/ccm
Trapped
ions
Strongly correlated (nonideal) plasmas
©Michael Bonitz, Universität Kiel, 2006
Contents
1. Strong Coulomb correlations. Wigner crystals in traps
2. Mesoscopic 2D Coulomb crystals
3. Quantum crystals: electrons in quantum dots
4. Excitons: from gas to fluid and superfluid/supersolid
5. Coulomb balls: spherical 3D crystals in a dusty plasma
6. Neutral plasmas. From stars to hole crystals in semiconductors
7. Outlook
©Michael Bonitz, Universität Kiel, 2006
Correlation effects in classical plasmas
Coulomb Interaction:
U (r )  q 2 / r
Coupling parameter:
n1/ dim
  U  / kBT  q
kBT
2
Strong
Coulomb
correlations
 1
  175
Coulomb-Kristall
©Michael Bonitz, Universität Kiel, 2006
Strong correlations: Wigner crystal
Ground state of the electron gas in metals
E. Wigner, Physical Review 46, 1002 (1934):
 exchange and correlation energy of the electron gas
„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....“
©Michael Bonitz, Universität Kiel, 2006
Experimentally observed
Wigner crystal
Electrons on Helium droplets (Grimes 1979) 2D crystal, T=4K
(Leiderer 1982)
Strongly correlated (cooperative) behavior of all charges
Realization in plasma in the laboratory???
©Michael Bonitz, Universität Kiel, 2006
Correlation effects in classical plasmas
Coulomb Interaction:
U (r )  q 2 / r
Coupling parameter:
n1/ dim
  U  / kBT  q
kBT
2
Bound
state
formation
 1
  175
Coulomb-Kristall
©Michael Bonitz, Universität Kiel, 2006
Classical and Quantum
charged particles in traps
2-component plasma: strong coupling inhibited by recombination (formation of neutrals)
Idea: non-neutral systems with only single charge species
External confinement potential  stability plus arbitrarily strong coupling!
n (x)
Single classicle particle
Quantum particle
Quantum diffraction and spin effects
Interacting particles
Real atoms (molecules, clusters etc.): central Coulomb potential
| U C |
rs 
 1...5
 moderate coupling:
©Michael Bonitz, Universität Kiel, 2006
T 
Mesoscopic Coulomb clusters
in traps („artificial atoms“)
Mesoscopic: small particle number N
quasi-2-dimensional system
-
0

classical ground state: potential minima (cf. Wigner)
©Michael Bonitz, Universität Kiel, 2006
2D finite
Coulomb
Clusters:
„Mendeleyev
table“
universal!
Dusty plasma
experiments by
A. Melzer et al.
©Michael Bonitz, Universität Kiel, 2006
Strongly correlated Coulomb systems in traps
„Artificial Atoms“
Dusty Plasmas, Coulomb crystal
A. Piel, A. Melzer
Ions in Paul-/Penning traps
G. Werth, Uni Mainz
Electron crystal
(Quantum dots)
A.Filinov, MB, Yu. Lozovik
R. Blatt, Uni Innsbruck
Ions in storage rings, U. Schramm, LMU München ©Michael Bonitz, Universität Kiel, 2006
Contents
1. Strong Coulomb correlations. Wigner crystals in traps
2. Mesoscopic 2D Coulomb crystals
3. Quantum crystals: electrons in quantum dots
4. Excitons: from gas to fluid and superfluid/supersolid
5. Coulomb balls: spherical 3D crystals in a dusty plasma
6. Neutral plasmas. From stars to hole crystals
7. Outlook
©Michael Bonitz, Universität Kiel, 2006
Correlation and Quantum effects
Coulomb Interaction:
  U  / kBT
U ab (r )  ea eb / r
rs  U  / EF  r / aB
EF
- Fermi Energy a B - Bohr Radius
  n3  1
r

Strong
Coulomb
correlations
 r
 1
  175
Quantum effects
Overlap
of wave functions,
Spin effects
DeBroglie
wave length
  h / 2mkBT
crystal
rs  1
rs  100
©Michael Bonitz, Universität Kiel, 2006
Mesoscopic quantum Coulomb
clusters in quantum dots
PIMC
-
quasi-2-dimensional system
Kinetic energy finite  quantum ground state
Density increase  quantum („cold“) melting of „crystal“
rs  200
rs  100
rs  20
Phys. Rev. Focus (April 2001), Sciences et Avenir, Scientific American, FAZ 1.8. 2001
©Michael Bonitz, Universität Kiel, 2006
Phase diagram of the mesoscopic
Wigner crystal
Classical
Particle Number
Liquid
Quantum Liquid
(„Wigner-Molecule“)
Wigner Crystal
strongest correlations
4 phase transitions
Confinement Strength
RM - Radial Melting,
OM – Rotational Melting
  U  / kBT
rs  U  / EF  r / aB
A. Filinov, M. Bonitz, Yu. Lozovik, Phys. Rev. Lett. 86, 3851 (2001)
©Michael Bonitz, Universität Kiel, 2006
Phase diagram of e-h bilayers
density
electrons
d
holes
Bose-Einstein condensation,
Suprafluidity of excitons?
Radial confinement
Variation of distance d: switch
- from Coulomb to dipole interaction
- from plasma to excitons
- between liquid and crystal
- from fermions to bosons
d 
Distance d
A. Filinov, P. Ludwig, V. Golubnychiy, M. Bonitz, Yu. Lozovik, phys. stat. sol. (c)No.5, 1518 (2003)
©Michael Bonitz, Universität Kiel, 2006
Ez
QW
CB
CB
AlGaAs
Ee
e-
AlGaAs
electron energy
electron energy
Band structure
Electric field induced spatially separated excitons
Eg GaAs hw
VB
Eh
h+
L
z
AlGaAs
e
Ee
Eg GaAs hw
-
VB
Eh h+
L
AlGaAs
Probability density
z
QMC results, A. Filinov, P. Ludwig, Yu. Lozovik, M. Bonitz and H. Stolz, J. Phys. Conf. Ser., (2006)
©Michael Bonitz, Universität Kiel, 2006
Electrostatic trap by quantum confined Stark effect
F [kV]
Single point electrode
GaAs/AlGaAs quantum well
zBufferlayer= 300nm
U = 100V
RElektrode = 50µm
LQW = 30nm
U
z
Ez,Er [kV/cm]
substrate
0
r
x2 y2
W (r )  PEX (r ) Ez
Ez(r=0) = 20kV/cm
 Total energy minimum at r=0
A. Filinov, P. Ludwig, M. Bonitz and H. Stolz, phys. stat. sol. (2006)
©Michael Bonitz, Universität Kiel, 2006
First principle Monte-Carlo results
N=300
N=3000
T=0.01mK
T=1K
N=30
20 m
©Michael Bonitz, Universität Kiel, 2006
Coupling controlled by excitation intensity
U corr
k BT
crystal
fluid
gas
Condensate fraction from exchange permutations
N identical bosons: wave function symmetric under particle permutation
Zn
ZnSe
rs  18
N cond
Pcr  1 / N x
Frequency of exchange permutation cycles from Monte Carlo simulation
©Michael Bonitz, Universität Kiel, 2006
Suprafluidity
 Loss of viscosity below critical temperature
- discovered in liquid He by P.L. Kapitza 1938
- L.D. Landau: two-fluid (normal+suprafluid) model
Andronikashvili experiment
Superfluid density (fraction) from
Classical/quantum moment of inertia I
I quant
s
 1

I class
First principle result for interacting
bosons from path area A:
 Quantum Monte Carlo simulations
(D. Ceperley, 1995 )
©Michael Bonitz, Universität Kiel, 2006
Suprafluid fraction of excitons
Path integral Monte Carlo results
for strongly correlated excitons
ZnSe 30nm quantum well, E=20kV/cm
Coupling parameter
e 2 / l0


, l0 
w
mrw
A.Filinov, M. Bonitz, P. Ludwig, and
Yu. Lozovik, phys. stat. sol. (2006)
©Michael Bonitz, Universität Kiel, 2006
Exciton supersolid?
Coexistence of solid and superfluid behavior
Prediction: Andreev/Lifshitz (1969), Leggett (1970), ...
Discovery in He claimed: Kim/Chan (2004)
 Requires imperfect crystal (defects)...
  14
Our finite exciton „crystals“
are imperfect!
ZnSe, 30nm Qwell
6 excitons
e
h
A.Filinov, M. Bonitz, P. Ludwig,
and Yu. Lozovik,
phys. stat. sol. (2006)
Expect: finite superfluid fraction in crystal  exciton supersolid
©Michael Bonitz, Universität Kiel, 2006
Open questions
1. Theory of strongly correlated Coulomb/exciton systems
- excitation spectrum,
- optical and transport properties
a) analytical approach (normal modes)
b) simulation: Nonequilibrium Green‘s functions,
quantum Molecular dynamics
2.
Experimental realization and verification of
- electron and exciton crystal
- Bose condensation and suprafluidity
Problems: low density, clean materials, finite lifetime ...
©Michael Bonitz, Universität Kiel, 2006
Contents
1. Strong Coulomb correlations. Wigner crystals in traps
2. Mesoscopic 2D Coulomb crystals
3. Quantum crystals: electrons in quantum dots
4. Excitons: from gas to fluid and superfluid/supersolid
5. Coulomb balls: spherical 3D crystals in a dusty plasma
6. Neutral plasmas. From stars to hole crystals
7. Outlook
©Michael Bonitz, Universität Kiel, 2006
Dusty plasmas. Setup
Discharge chamber, micrometer-size particles, Z=5,000...100,000
Dust particles surrounded by electrons, ions and neutrals
©Michael Bonitz, Universität Kiel, 2006
Spherical 3D Coulomb clusters
1. Ion plasmas:
Bollinger et al., Walter et al. (1987), N>10,000
2. Dusty plasmas:
Arp, Block, Piel, Melzer, Phys. Rev. Lett. 93, 165004 (2004)
N=190
N~6000
Video image, room temperature
 Problem: unknown interaction between dust particles
©Michael Bonitz, Universität Kiel, 2006
Mesoscopic 3D crystals
Simulation with Coulomb potential
Concentric spherical shells
Mendeleyev table
Closed shells, „magic“ numbers:
N=12 (12,0)
N=13 (12,1) [Rafac et al. 91]
....
N=57 (45,12)
N=58 (45,12,1)
N=59 (46,12,1)
N=60 (48,12) [Tsuruta/Ichimaru 93]
....
N=154 (98,44,12)
N=155 (98,44,12,1)
Ludwig, Kosse, Bonitz, PRE 71, 046403 (2005)
©Michael Bonitz, Universität Kiel, 2006
Experiment vs. Simulation
N=190
First principle simulation
2
q
(Molecular dyamics) with
Vdd (r )  e r
r
Coulomb and Yukawa-potential
©Michael Bonitz, Universität Kiel, 2006
Experiment vs. Simulation
Shell radii in units of mean interparticle distance
Experiment (symbols):
43 clusters,
N=100...500
Molecular dynamics with
Coulomb and
Debye potential
q 2 r
Vdd (r )  e
r
Excellent agreement without free parameters!
means nothing... (if the wrong quantity is chosen)
Bonitz, Block, Piel et al.,
Phys. Rev. Lett. 96, 075001 (2006)
©Michael Bonitz, Universität Kiel, 2006
Experiment vs. Simulation
Shell populations
Experiment (symbols):
43 clusters,
N=100...500
Molecular dynamics with
Coulomb and
Debye potential
q 2 r
Vdd (r )  e
r
Bonitz, Block, Piel et al.,
Phys. Rev. Lett. 96, 075001 (2006)
Systematic screening dependence
predict screening in experiment:
  0.6  0.3
©Michael Bonitz, Universität Kiel, 2006
Density distribution in spherical trap
Apparent equidistant shells, constant average density, but...
q 2 r
Vdd (r )  e
r
Harmonic confinement
Finite particle number N
Exact analytical results
from fluid model
Henning, Ludwig, Bonitz, Block, Piel, Melzer, to be published
©Michael Bonitz, Universität Kiel, 2006
Density distribution in spherical trap
q 2 r
Vdd (r )  e
r
Harmonic confinement
N=2000
Lines: fluid model
(mean field theory)
Crosses: 3D crystal
(simulation, exact)
One to one correspondence between pair interaction
and macroscopic density profile
Goals: - pair correlation function at very strong correlations
- excitation spectrum, wave properties
- response to external E/B fields etc.
Henning, Ludwig, Bonitz, Block, Piel, Melzer, to be published
©Michael Bonitz, Universität Kiel, 2006
Contents
1. Strong Coulomb correlations. Wigner crystals in traps
2. Mesoscopic 2D Coulomb crystals
3. Quantum crystals: electrons in quantum dots
4. Excitons: from gas to fluid and superfluid/supersolid
5. Coulomb balls: spherical 3D crystals in a dusty plasma
6. Neutral plasmas. From stars to hole crystals
7. Outlook
©Michael Bonitz, Universität Kiel, 2006
Proton crystallization in dense Hydrogen
T = 10,000 K, n = 31025 сm-3,  = 50.2 g/сm3
- proton
- electron
- electron
1st-principle
Path integral
Monte Carlo
simulation
Filinov, Bonitz, Fortov, JETP Letters 72, 245 (2000)
©Michael Bonitz, Universität Kiel, 2006
Correlations in 2-component plasmas
Coulomb Interaction:
Atoms, molecules
(neutrals)
  U  / kBT
U ab (r )  ea eb / r
rs  U  / EF  r / aB
EF
- Fermi Energy a B - Bohr Radius
 e  nee 3  1
r

e  re
Strong
Coulomb
correlations
 p  rp
rsp  100
 1
  175
Proton crystal
&
Electron gas
a  h / 2ma k BT
crystal
rs  1
rs  100 Pressure ionization
©Michael Bonitz, Universität Kiel, 2006
White dwarf star
classical
fluid and crystal
in „quantum sea“
of electrons
Size ~ our Earth
Mass ~ our Sun
density:
  106  ERDE
D. Schneider, LLNL
©Michael Bonitz, Universität Kiel, 2006
Neutron star
crystal and
quantum fluid
of Fe-nuclei
in „quantum sea“
of electrons
Radius ~ 10km
Mass ~ our Sun
  1015 g cm3
Source: Coleman, UMD
©Michael Bonitz, Universität Kiel, 2006
Conditions for TCP Coulomb crystals
I. strong „hole“ coupling (OCP):
II. no Coulomb bound states:
h  cr  175, rsh  rs  100 (3D)
Mott
3
kT

E
,
r

r
/
a

r
 1.2
e
B
se
e
B
s
2
cr
Additional parameters in TCP: mass, charge and temperature asymmetry
mh
qh
Te
M
, Z  , 
me
qe
Th
Analytical Results
1. Classical system:
 finite temperature
range for crystal or
minimum charge Z
2 kTe 2Z 2 / 3

 cr
3 EB
 rse
Te 
3 kT
2 ER
Density 
©Michael Bonitz, Universität Kiel, 2006
Conditions for TCP Coulomb crystals
I. strong hole coupling (OCP):
II. no Coulomb bound states:
h  cr  175, rsh  rs  100 (3D)
Mott
3
kT

E
,
r

r
/
a

r
 1.2
e
B
se
e
B
s
2
cr
Additional parameters in TCP: mass, charge and temperature asymmetry
mh
qh
Te
M
, Z  , 
me
qe
Th
Analytical Results (2): Quantum system
 M  1  Mott
 ne   cr
 n
 M 1
3
• Finite density range:
n Mott
• Critical mass ratio:
r
M  M cr (Te )  4 / 3 sMott
1
Z rs (Te )
cr
• Maximum temperature:
k BTe
Z 2  ( M  1)
4
cr
EB
 cr rs
Phase diagram of
TCP Coulomb crystal
K
M 1
M cr  1

Te 
3 kT
2 ER
• for Z=1 (e.g. electron-hole plasma):
1
Z 2
1 aB

 n1/ 3
rse re
M cr  83 (3D), 60 (2D)
Bonitz, Filinov, Fortov, Levashov, and Fehske, Phys. Rev. Lett. 95, 235006 (2005)
©Michael Bonitz, Universität Kiel, 2006
Predicted parameters of
TCP Coulomb crystals
nmin [cm3 ]
O 6  Ions
26
nmax [cm 3 ]
3.6 10
33
Tmax [ K ]
10 9
(white dwarfs)
210
Protons
(hydrogen)
51024
1.0 10 28
66,000
Semiconductors
(M=100)
1.2 10 20
2.110 20
9.0
©Michael Bonitz, Universität Kiel, 2006
Verification using
path integral Monte Carlo
simulations*
First principle treatment of all many-body effects:
- Coulomb correlations,
- bound state formation,
- quantum and spin effects
* See: “Introduction to Computational Methods for Many-Body systems“,
M. Bonitz and D. Semkat (eds.), Rinton Press, Princeton 2006
©Michael Bonitz, Universität Kiel, 2006
Partially ionized electron-hole plasma
Electrons with different spin are shown by clouds of yellow and blue dots.
Holes with different spin are shown by clouds of red and pink dots.
M=40
rs=13, T/Ry=0.13
rs =13, T/Ry=0.6
©Michael Bonitz, Universität Kiel, 2006
Partially ionized electron-hole plasma
Electrons with different spin are shown by clouds of yellow and blue red dots.
Holes with different spin are shown by clouds of red and pink dots.
M=40
rs = 5, T/Ry = 0.13
rs = 5, T/Ry = 0.6
©Michael Bonitz, Universität Kiel, 2006
Fully ionized electron-hole plasma
M=40
rs = 1.3, T/Ry = 0.13
rs = 1.3, T/Ry = 0.6
©Michael Bonitz, Universität Kiel, 2006
Pressure ionization of excitons
Mott density (10% bound states):
rs
Mott
 1.2
©Michael Bonitz, Universität Kiel, 2006
Typical snapshots of the Coulomb liquid
T / Ry  0.06, rs  0.5
M = 50
M = 12
©Michael Bonitz, Universität Kiel, 2006
Typical snapshots of the Coulomb crystal
T / Ry  0.06, rs  0.5
M  800
M  100
©Michael Bonitz, Universität Kiel, 2006
Pair distribution functions
h-h
e-h
Fermi gas
e-e
Hole crystal
Hole liquid
electron density
modulation,
spin ordering
©Michael Bonitz, Universität Kiel, 2006
Mass ratios of M=40 observed in TmSe0.45Te0.55
(Wachter et al., ETH Zürich)
EXPERIMENTAL PHASE DIAGRAM
©Michael Bonitz, Universität Kiel, 2006
Schematic semiconductor band structure
Conduction
band
electron
Excitation
momentum
hole
valence
band
C
B
A
Hole mass increase
©Michael Bonitz, Universität Kiel, 2006
Band structure
and
collective hole
behavior
Increase of M:
Increasing hole localization
Hole Fermi gas  Fermi liquid
 Wigner crystal
Early predictions of hole crystal:
Halperin/Rice, Abrikosov, ...
Experimental verification?
High temp. Superconductivity?
©Michael Bonitz, Universität Kiel, 2006
Animation: G. Schubert,
M=5,12,25,50,100
Our results in the news
Dec 2 2005
Deutschlandfunk, NDR
New Scientist
BBC Focus
Superconductor Week
ca. 100 Wissenschafts-Seiten
.......
©Michael Bonitz, Universität Kiel, 2006
Outlook: theory challenges
Theory missing (non-perturbative regime!)
I.) develop thermodynamic and kinetic theory,
classical and quantum normal mode analysis
II.) First principle classical/quantum computer simulations
for strongly correlated Coulomb systems:
1. Classical systems: Monte Carlo, Molecular dynamics,
Particle in cell
2. Quantum systems: Path integral Monte Carlo,
quantum Molecular dynamics,
Nonequilibrium Green‘s functions
Text book:
„Introduction to computational methods for many-body systems“,
M. Bonitz and D. Semkat (eds.), Rinton Press, Princeton (2006)
©Michael Bonitz, Universität Kiel, 2006
Summary
Charged particles in traps: electrons, ions, holes, dust...
highly organized, collective behavior, universal phase diagram
Mesoscopic electron/exciton clusters: fermion/bosonnano-crystals; exciton suprafluid, supersolid (?)
3D dust Coulomb crystals: unique test system for correlations
(room temperature, real-time correlation dynamics...)
Coulomb crystal in neutral/Two-component plasmas:
hole crystal in semiconductors (for M>80), superconductivity?
Information, animations etc.: http://www.theo-physik.uni-kiel.de/~bonitz
Supported by DFG via
Transregio-SFB
Greifswald/Kiel „Grundlagen Komplexer Plasmen“
Many thanks for your attention!
http://www.theo-physik.uni-kiel.de/~bonitz