VOLUME 82, NUMBER 19 PHYSICAL REVIEW LETTERS 10 MAY 1999 Novel Spin Features in Raman Spectra of Few-Electron Quantum Dots Oliver Steffens and Michael Suhrke Institut f ür Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany (Received 3 November 1998) We employ a self-consistent, dynamic spin-density functional concept beyond the dipole approxi- mation in order to calculate the collective modes of few-electron quantum dots with partial spin polarization. Because of Hund’s rule in a magnetic field the mode energies exhibit characteristic discontinuities in their magnetic-field dependence, which is a manifestation of ground-state transitions between different spin configurations. A finite ground-state spin density couples collective charge- and spin-density excitations. We predict their observability in state-of-the-art Raman experiments. [S0031-9007(99)09101-2] PACS numbers: 73.20.Mf, 71.15.Mb, 73.23. – b, 78.30.Fs Mesoscopic quantum wires and dots made from later- ally structured two-dimensional electron systems (2DES) at semiconductor heterojunctions (e.g., GaAs Ga x Al 12x As) have attracted much attention regarding their collective properties: With an effective Bohr radius comparable with the interelectron spacing, inter- action effects are important and can be detected using far-infrared (FIR) spectroscopy [1] or inelastic light (Raman) scattering [2]. Especially quantum dots, with their potential applicability in novel electronic devices, have been a key object of Raman experiments for the past few years [3]. Individual quantum dots containing less than ten electrons have been studied by means of capacitance and transport spectroscopy [4,5]. Their en- ergy spectra exhibit characteristic features in dependence on particle number and magnetic field which are related to crossings of effective single-particle (SP) levels at particular magnetic field values. It has been observed in experiment [5] and confirmed by many-particle ground- state calculations [6] that level crossings at the Fermi energy can be considered as “quasi-energy-shells” that are occupied according to what is known as “Hund’s first rule” in atomic physics: If two pairs of spin- degenerate effective SP levels become sufficiently close in energy, it is favorable to occupy as many identical spin states as possible under the constraint of the Pauli principle, thus creating an exchange gap between the different spin states in partially occupied shells. Technological advances offer the prospect that these exchange effects could be identified also in Raman ex- periments on self-assembled few-electron quantum dot arrays [7]. While FIR absorption spectroscopy with pho- ton energies of only a few meV is restricted to dipole excitations, inelastic light scattering allows one to ex- cite many multipole components by a large momentum transfer from absorption and emission of photons with energies close to the band gap (eV); thus it contains much more information. In order to address this timely issue theoretically, we have developed a self-consistent treatment of multipole collective excitations using spin- density functional theory (SDF T) and a time-dependent local spin-density approximation (LSDA). A similar ap- proach (limited to dipole excitations) has recently been reported in Ref. [8]. In this Letter we predict novel spin features in the Raman cross section of few-electron quantum dots due to the transition from an unpolarized ground-state configuration with vanishing spin density to a partially polarized one, which occurs in the magnetic-field region around SP level crossings at the Fermi energy due to the above-mentioned Hund’s rule [6]. They manifest themselves as a twofold discontinuity of the peaks in the Raman spectra at the crossing point. Selection rules connected with the scattering geometry [2] usually allow one to excite charge-density excitations (CDE) and spin-density excitations (SDE) separately. A finite ground-state spin density, however, couples both types of excitations. Therefore, in contrast with unpolarized sys- tems, simultaneous CDE and SDE may appear. A high momentum transfer well beyond the regime of the usually applied dipole approximation leads to sig- nificant contributions from monopole (“breathing”) and quadrupole modes. We start with a 2DES in the x -y plane subject to a homogeneous magnetic field B in the z direction. The lateral confinement is modeled by a harmonic potential V 0 r   1 2 m  v 2 0 r 2 . Without interaction, this yields the (spin-degenerate) SP spectrum ´ nm  ¯ hv h 2n 1 jmj 1 1 1 ¯ hv c m2 with the cyclotron frequency v c  eBm  c and v h  v 2 0 1v 2 c 4 12 ; m is the angular- momentum quantum number. In the framework of SDFT, the particle and spin densities of the N -electron interacting ground state are calculated from the self-consistent solutions of the Kohn- Sham (KS) equations [9], H KS c as r   ´ as c as r  , n s r   X a f as jc as r j 2 , X s Z d 2 rn s r   N , (1) 0031-9007 99  82(19)  3891(4)$15.00 © 1999 The American Physical Society 3891