Collapse of the Zeeman gap in quantum dots due to electronic correlations Pawel Hawrylak, Charles Gould,* Andy Sachrajda, Yan Feng, and Zbig Wasilewski Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada K1A 0R6 ~Received 7 August 1998! We present results of calculations and ‘‘lateral’’ magnetotunneling experiments on small quantum dots in the few-electron regime. In the transition from a spin-unpolarized to a spin-polarized droplet, a systematic oscillation of the chemical potential of the droplet as a function of the magnetic field correlates with oscilla- tions in the current amplitude. The temperature dependence of the current minima indicates that the minima are associated with the collapse of the Zeeman gap in the quantum Hall droplet. The collapse of the Zeeman gap is associated with the nontrivial magnetic field dependence of the spin polarization of a quantum Hall droplet due to electronic correlations. @S0163-1829~99!05304-7# The Zeeman gap is the energy difference between the two spin orientations in a magnetic field. As the magnetic field is increased one would intuitively expect that an increasing number of free electrons would align their spins. This simple behavior, however, may be drastically altered by correlations and interactions. The properties of solids, for example, are determined by the interplay of the spin and orbital degrees of freedom of many correlated electrons. Our understanding of these effects is limited because of the large number of de- grees of freedom. 1 Semiconductor quantum dots with few electrons are artificial atoms that share many features com- mon to atoms and solids. 2 As a consequence of their semi- conductor host, magnetic fields typically available in the laboratory can drastically modify their electronic properties. These artificial atoms open up the possibility of studying the interplay between spin and orbital degrees of freedom of a sufficiently small number of electrons that the computation of their properties is within current capabilities. In this paper we present both experimental and theoretical results of such an investigation. In particular, we focus on the magnetic field induced transition of the spin in a few-electron droplet that can be understood in terms of the collapse of the Zeeman gap due to electron correlations. The most promising few-electron quantum dots in terms of control of electrons are vertical structures. 3,4 The interest- ing spectroscopic observation 3 of Hund’s rules in these struc- tures suggests that the electron-electron interactions are weak 5 and electron correlations may not play a significant role in these devices. A second difference between vertical and lateral devices is that while in vertical devices electrons tunnel primarily into the center of the quantum dot, in lateral devices they tunnel primarily to the edges from edge states in a bulk two-dimensional electron gas ~2DEG!. As we will show, this results in the tunneling of spin polarized electrons in lateral devices. In lateral devices 6–13 electron-electron interactions are strong, but devices fabricated so far have contained many electrons. McEuen et al. 6 pioneered the investigation of the Coulomb blockade ~CB! in lateral quantum dots in a mag- netic field. The random oscillations of the position of the Coulomb blockade peaks with magnetic field at low mag- netic fields became systematic at magnetic fields sufficient to dominate the quantization of energy. These regular oscilla- tions were initially associated with changes in occupation of the two lowest Landau levels. A subsequent Hartree calculation 7 suggested that the oscillations were rather asso- ciated with ‘‘spin flips’’ i.e., the redistribution of spin up and spin down electrons on orbitals of the lowest Landau level. Since the estimated number of electrons was N ’60, only a comparison with mean field calculations was possible. The subsequent reduction of the number of electrons to N ’30 allowed Klein et al. 9 to investigate the role of exchange in spin transitions and they compared experimental results with model Hartree-Fock ~HF! calculations. The HF calculations were required to reproduce some of the experimental features such as the observed magnetic field range and spacing of the oscillations of the CB positions in the regime of filling fac- tors 1 ,n ,2. However, the observed temperature depen- dence of the CB amplitude at the spin flips remained a puzzle. The energy scale involved, of the order of bare Zee- man energy, was orders of magnitude smaller than that pre- dicted by HF calculations. In HF calculations the Zeeman energy is significantly enhanced by exchange and the dis- agreement with experiment was severe. This suggested that effects beyond Hartree-Fock calculations, i.e., correlations, play a significant role. The role of spin and correlations is present in the generic model of correlated electrons, i.e., the Hubbard model. 1 When vacancies ~holes! or extra electrons are introduced into the half filled Hubbard model, they lead to massive changes in the magnetization. A 2DEG in a strong magnetic field at filling factor one is a simple realization of the half filled Hubbard model. 14 Rezayi 15 pointed out early on that the in- troduction of a single unpaired spin electron causes a large spin depolarization. Sondhi et al. 16 and MacDonald 17 related these spin instabilities to topological excitations ~skyrmions!. The competition between spin and angular momenta in controlling the ground state of few-electron quantum dots was pointed out previously by one of us 18 and by Yang et al. 19 Spin excitations 20,21 in quantum dots were also stud- ied by Oaknin et al., 22 who applied the idea of spin magne- toexciton condensation, while Imamura et al. 23 discussed the role of spin in the language of resonating valence bonds. To investigate these correlation effects we have fabricated a lateral quantum dot defined by electrostatic gates on a Al x Ga 1 2x As/GaAs wafer containing a high mobility 2DEG. PHYSICAL REVIEW B 15 JANUARY 1999-II VOLUME 59, NUMBER 4 PRB 59 0163-1829/99/59~4!/2801~6!/$15.00 2801 ©1999 The American Physical Society