Symmetry breaking phenomena in mesoscopic systems: quantum dots and rotating nuclei Rashid Nazmitdinov UIB, Palma de Mallorca, Spain and JINR, Dubna Russia Kazimerz, September 2008
Outline : • Quantum dot: basics • Models: mean-field results • Mechanisms of symmetry breaking : RPA analysis • Summary
QDs created in thin film semiconductor heterostructures With the use ofepitaxial deposition techniques(like molecular beam epitaxy) it is possible to grow semiconductor crystals in coherent layers, a few lattice constant thick, and createmultilayer semiconductor heterostructures. By sandwiching a AlGaAsbetween GaAs (insulator) layers, one confines electrons in a AlGaAs'quantum well'. The electrons in the 2DEG result fromSi donors in the n- AlGaAs layer. (The thickness of the diferent layers is not toscale.) Reduction of the remaining 2D ‘infinite’ extension of the quantum well, i.e. lateral confinement, leads tocarrier confinement in all three dimensionsand creation of aQD.
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. Electrostatic DQs M.A.Reed et al, PRL 60, 535 (1988)
Basics 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 ofsize quantization. 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.
Basics • 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 aparabolic 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
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-calledFock-Darwin energy levels are where ωL = eB/2m* is the Larmor frequency and
The Kohn theorem The Kohn theorem - In a parabolic confining potential the centre-of-mass (CM) and relative (rel) motion decouple. For a N-electron QD: Since Q/M =e/m*, the CM energy is identical to the single-electron energyEnm. The generalized Kohn theorem – The far-infrared (FIR) absorption spectra are independent on the number of electrons.
Shell effects S.Tarucha et al, Phys. Rev. Lett. 77, 3613 (1996).
Models 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: where e, m*, ε0 and εr and the unit charge, effective electron mass, vacuum and relative dielectric constant of a semiconductor, respectively.
Single-electron capacitance spectroscopy Reimann&Manninen, RMP 74,1283 (2002) Oosterkamp et al, PRL 82,2931(1999)
Wigner molecule in Quantum Dot Serra, Nazmitdinov, Puente, PRB68 (2003) 035341;PRB69(2004)125315 • Hamiltonian • Units The length The energy The strength • Result
Spurious modes 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
The Quasi-Boson approximation 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 Spurious mode: Rotational Symmetry
Spurious mode: Rotational Symmetry Canonical operators Lz and an angle operator (Thouless, 61; Marshalek and Weneser, 69)
Spurious mode: Rotational Symmetry Two additional RPA vectors For the C operator one needs to solve the linear system of equations
Spurious mode: Rotational Symmetry • 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 Spurious mode: Rotational Symmetry
Spurious mode: Rotational Symmetry RPA UHF
Spurious mode: Rotational Symmetry RPA
Spurious mode: Rotational Symmetry Nazmitdinov&Simonovic,PRB76 (2007)193306 RPA
Rotational behavior of the experimental, kinematical¶(1)=I/Ωand dynamical¶(2) ≈ 4/∆Eγmoments of inertia
RPA analysis: Moments of inertia Nazmitdinov,Almehed,Donau, PRC65,041307(2002)
Rigid Rotor in Principal Axes Frame Do triaxial nuclei exist ? Bohr&Mottelson,1975
Wobbling excitations Marshalek (1979),Kvasil&Nazmitdinov(2007).
Criteria for wobbling excitations: Kvasil&Nazmitdinov, Phys.Lett. B650 (2007) 33.
Criteria for wobbling excitations: Nazmitdinov&Kvasil, JETP105 (2007) 962
Reflection asymmetric shape Nazmitdinov, Kvasil,Tcvetkov, Phys.Lett.B657 (2007) 159
Summary • 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.
Thanks to Collaborators: • 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 Thank you for your attention !
Summary • 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 degeneracyof 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.
The Hartree-Fock equations • 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.
E Shell structure in quantum dot Heiss&Nazmitdinov,Phys.Lett.A222(1996) 309
E Shell structure in quantum dot Heiss&Nazmitdinov, Phys.Rev.B55 (1997)16310 • B = 0 the magic numbers (including spin) turn out to to bethe usual sequence of thetwo-dimensional isotropic oscillator, ωx=ωy, that is 2, 6, 12, 20, . . .. • B ≈ 1.23 we find a newshell structure as if the confining potential would be a deformedharmonic oscillator withoutmagnetic field. The magic numbers are 2, 4, 8, 12, 18, 24, . . . which are just the numbersobtained from the two-dimensional oscillator with ω+ = 2 ω-. • B ≈ 2.01 the magic numbers2, 4, 6, 10, 14, 18, 24, . . . which corresponds to ω+ = 3 ω- .
The Wigner molecule in 2eQD Nazmitdinov&Simonovic,PRB76 (2007)193306
Model Hartree-Bogoliubov equations: