Download RPA - Department of Theoretical Physics UMCS

Rashid Nazmitdinov
• Quantum dot: basics
• Models: mean-field results
• Mechanisms of symmetry breaking :
RPA analysis
• Summary
QDs created in thin film
semiconductor heterostructures
With the use of epitaxial deposition techniques (like molecular beam epitaxy) it is
possible to grow semiconductor crystals in coherent layers, a few lattice constant
thick, and create multilayer semiconductor heterostructures.
AlGaAs between GaAs (insulator) layers, one confines
electrons in a AlGaAs 'quantum well'. The electrons in the 2DEG result from Si
donors in the n- AlGaAs layer. (The thickness of the diferent layers is not to scale.)
By sandwiching a
Reduction of the remaining 2D ‘infinite’ extension of the quantum well, i.e. lateral
confinement, leads to carrier confinement in all three dimensions and creation of a QD.
Electrostatic DQs
M.A.Reed et al, PRL 60, 535 (1988)
A scanning electron micrograph of various size GaAs nanostructures containing quantum dots. The
dark regions on top of the column is the electron-beam defined Ohmic contact and etch mask. The
horizontal bars are 0.5 μm.
One can consider the QD as a tiny laboratories in which quantum
mechanics and the effects of electron-electron interaction can be
studied.
If the carrier motion in a
solid is limited in a layer
of a thickness of the
order of the carrier de
Broglie wavelength (λ),
one will observe effects
of size quantization.

h
h
1.22 nm


p
3meff kT
Ekin /[ eV]
Quantum dots (QD) are small boxes (2 – 10 nm on a side,
corresponding to 10 to 50 atoms in diameter), contained in
semiconductor, and holding a number of electrons.
At 10 nm in diameter, nearly 100000 quantum dots can be fit within the width of a human thumb.
•
By sandwiching a 10 nm thickness of GaAs
between AlGaAs (insulator) layers, one
confines electrons in a GaAs 'quantum well'.
By placing electrostatic gates on the surface
of the wafer, we can laterally confine this 2D
electron gas and create a quantum dot
•
For the typical voltage  1V applied to the
gate (top plate), the confining potential is
some eV deep which is large compared to
the few meV of the confining frequency.
Hence, the electron wave function is
localized close to the minimum of the well
which always can be approximated by a
parabolic potential.
FIR spectroscopy
Sikorski and Merkt, PRL 62 (1989) 2164 - The first direct observation of resonance transitions
between discrete states of QDs on InSb.
Left: Scanning electron micrograph of arrays of QDs on InSb
and the schematic sketch of the band structure across the dots.
Right: Experimental resonance positions (bullets) together with
theoretical curves calculated from    ( 02   L2 )1 / 2   L .
FIR spectroscopy
Sikorski and Merkt, PRL 62 (1989) 2164 - The first direct observation of resonance transitions
between discrete states of QDs on InSb.
The problem was solved more than 70 years ago
(Fock 1928, Darwin 1930). The so-called FockDarwin energy levels are
H


1
p  eA 2  1 m *02 x 2  y 2 .
2m *
2
A
1
B  r, B  0,0, B ,
2
p2
l z2
1
H


m * 2  2  Ll z ,
2
2m * 2m * 
2
where ωL = eB/2m* is the Larmor frequency and
Enm   2n  m  1  L m ,
 2  02  L2 .
   ( 02   L2 )1 / 2   L .
The Kohn theorem
The Kohn theorem - In a parabolic confining potential the centre-ofmass (CM) and relative (rel) motion decouple.
For a N-electron QD:
H
1
P  QA 2  1 M02 R 2  H rel ,
2M
2
N
N
i 1
i 1
where P   p i , R   ri N , Q  N e, M  N m * .
Since Q/M = e/m*, the CM energy is identical
to the single-electron energy Enm.
The generalized Kohn theorem –
The far-infrared (FIR) absorption
spectra are independent on the
number of electrons.
d nm,nm  nm r e i  nm ,
m  1, n  0,1.
E      L ,
S.Tarucha et al, Phys. Rev. Lett. 77, 3613 (1996).
The standard theoretical model is based on a number of approximations:
•
•
•
The underlying lattice structure is taken into account in effective
mass approximation
The confining potential is parabolic
The electrons interact via a pure Coulomb interaction
The Hamiltonian for N electrons interacting in a QD in a magnetic field B, perpendicular to the
dot plane reads:
1
e2
 1
2
2 2
p i  eA i   m *0 ri  
H  
2
 40 r
n 1  2m *
N
N 1, N
1
i 1 j i
ij
 r
 g * B B S z ,
where e, m*, ε0 and εr and the unit charge, effective electron mass, vacuum and relative
dielectric constant of a semiconductor, respectively.
Oosterkamp et al, PRL 82,2931(1999)
Reimann&Manninen, RMP 74,1283 (2002)
Serra, Nazmitdinov, Puente, PRB68 (2003) 035341;PRB69(2004)125315
• Hamiltonian
• Units
The length
The energy
The strength
• Result
Thouless (60,61):
Stability
Symmetry
Orthognality
The generators of symmetries broken on the mean field level create
eigenstates with zero energy in RPA
Rotational symmetry
Equations of motion
lead the generalized eigenvalue problem
Conditions to construct the RPA ground state
One seeks solution in the form (Thouless theorem)
The matrix Zmi,ni is complex and symmetric in the boson indexes, i.e,
Zmi,ni = Zni, mi
Canonical operators Lz and an angle operator 
(Thouless, 61; Marshalek and Weneser, 69)
Two additional RPA vectors
For the C operator one needs to solve the linear system of equations
• Once these two sets of coefficients are determined,we can
calculate the Thouless-Valatin moment of inertia
And determine the complete set
Serra, Nazmitdinov, Puente, PRB68 (2003) 035341, PRB69 (2004) 125315
The particle density
The RPA ground state density
where
RPA
UHF
RPA
Nazmitdinov&Simonovic, PRB76 (2007)193306
RPA
Rotational behavior of the experimental, kinematical
¶(1)=I/Ω and dynamical ¶(2) ≈ 4/∆Eγ moments of inertia
Nazmitdinov,Almehed,Donau, PRC65,041307(2002)
Do triaxial nuclei exist ?
Bohr&Mottelson,1975
Marshalek (1979), Kvasil&Nazmitdinov (2007).
S  
jz jy
E2  2
Kvasil&Nazmitdinov, Phys.Lett. B650 (2007) 33.
Nazmitdinov&Kvasil, JETP105 (2007) 962
Nazmitdinov, Kvasil,Tcvetkov, Phys.Lett.B657 (2007) 159
J , O   KO
z

K

K
H  H  J Z
• We suggest to consider the vanishing of one of the
vibrational modes in the rotating frame
as an indicator of the possible shape transitions in fast
rotating nuclei.
• We propose the selection rules for electromagnetic
transitions from excited vibrational states of the negative
signature that can be used to identify wobbling
excitations and the sign of -deformation.
•
•
•
•
•
•
Jan Kvasil , Charles University, Prague, Czech Republic
Daniel Almehed, Notre Dame University, USA
Fritz Donau , Rossendorf, Dresden, Germany
Llorens Serra , University Illes Balears, Palma, Spain
Nenad Simonovic , UInstitute of Physics, Belgrade, Serbia
Antonio Puente , University Illes Balears, Palma, Spain
• Shell effects play important role in small quantum dots. At specific
values of the magnetic field the interplay between the Coulomb
interaction and shell structure may lead to degeneracy of the
quantum spectrum.
• We suggest to consider the ratio
as an indicator of the possible shape transitions in fast rotating
nuclei.
• We propose the selection rules for electromagnetic transitions from
excited vibrational states of the negative signature that can be used
to identify wobbling excitations and the sign of -deformation.
•
oscillator basis
•
An arbitrary single-particle orbital |i is expanded as
The oscillator states |a are characterized by radial (na) and angular
momentum (ma) quantum numbers ( Fock-Darwin states; Fock, 28)
•
We assume a good spin : each orbital i has non zero components only
for a given spin orientation i=or i=, i.e., B(i)a=iB(i) a.
Heiss&Nazmitdinov, Phys.Lett.A222 (1996) 309
E
  H   x y  0
Heiss&Nazmitdinov, Phys.Rev.B55 (1997) 16310
•
B = 0 the magic numbers (including
spin) turn out to to be the usual
sequence of the two-dimensional
isotropic oscillator, ωx=ωy, that is 2,
6, 12, 20, . . ..
•
B ≈ 1.23 we find a new shell
structure as if the confining
potential would be a deformed
harmonic oscillator without
magnetic field. The magic numbers
are 2, 4, 8, 12, 18, 24, . . . which
are just the numbers obtained from
the two-dimensional oscillator with
ω+ = 2 ω-.
•
B ≈ 2.01 the magic numbers 2, 4,
6, 10, 14, 18, 24, . . . which
corresponds to ω+ = 3 ω- .
E
Nazmitdinov&Simonovic, PRB76 (2007)193306
Hartree-Bogoliubov equations:
h(1) kl  ( H 0 ) kl  jklx
h(2) kl  ( H 0 ) kl  jklx
Kvasil&Nazmitdinov, PRC73,014312(2006)
Kvasil&Nazmitdinov, PRC73 (2006) 014312
J.A.Krumhansl and R.J. Gooding,
Phys.Rev.B39 (1989) 3047
  1/ 2
Analog of Landau phase transition of 2 order !
Serra, Nazmitdinov, Puente, PRB68 (2003) 035341
•
It is assumed that the system's excitations are created by operators of the
type
whose action on the as yet unknown ground state |0 yields the excited
vibrational states
•
The quasi-boson approximation (QBA) amounts to treat each particle-hole
pair (mi) as an elementary boson