We study atomiclike properties of artificial atoms by measuring Coulomb oscillations in vertical quantum dots containing a tunable number of electrons starting from zero. At zero magnetic field the energy needed to add electrons to a dot reveals a shell structure for a two-dimensional harmonic potential. As a function of magnetic field the current peaks shift in pairs, due to the filling of electrons into spin-degenerate single-particle states. When the magnetic field is sufficiently small, however, the pairing is modified, as predicted by Hund's rule, to favor the filling of parallel spins. [S0031-9007(96) PACS numbers: 73.20.Dx, 72.20.My, 73.40.Gk The "addition energy" needed to place an extra electron in a semiconductor quantum dot is analogous to the electron affinity for a real atom [1]. For a fixed number of electrons, small energy excitations can take these electrons to a higher single-particle state. However, due to Coulomb interactions between the electrons, the addition energy is greater than the energy associated with these excitations. Both the addition energy spectrum and the excitation energy spectrum are discrete when the Fermi wavelength and the dot size are comparable. Until now a direct mapping of the observed addition energy, and the single-particle excitation energy, to a calculated spectrum has been hampered, probably due to sample specific inhomogeneities [2].The three-dimensional spherically symmetric potential around atoms gives rise to the shell structure 1s, 2s, 2p, 3s, 3p, . . . . The ionization energy has a large maximum for atomic numbers 2, 10, 18, . . . . Up to atomic number 23 these shells are filled sequentially, and Hund's rule determines whether a spin-down or a spin-up electron is added [3]. Vertical quantum dots have the shape of a disk with a diameter roughly 10 times the thickness [2,4]. The lateral potential has a cylindrical symmetry with a rather soft boundary profile, which can be approximated by a harmonic potential. The symmetry of this twodimensional (2D) harmonic potential leads to a complete filling of shells for 2, 6, 12, . . . electrons. The numbers in this sequence can be regarded as "magic numbers" for a 2D harmonic dot. The shell filling in this manner is previously predicted by self-consistent calculations of a circular dot [5]. In this Letter we report the observation of atomiclike properties in the conductance characteristics of a vertical quantum dot. We find an unusually large addition energy when the electron number coincides with a magic number. We can identify the quantum numbers of the single-particle states by studying the magnetic field dependence. At a sufficiently small magnetic field ͑B , 0.4 T) we see that spin filling obeys Hund's rule. At higher magnetic fields ͑B . 0.4 T) we observe the consecutive filling of states by spin-up and spin-down electrons, which arises from spin degeneracy.The gated vertical quantum dot shown schematically in Fig. 1 is made from a double-barrier heterostructure (DBH). The use of well-defined heterostructure tu...

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Abstract. We study ground states and excited states in semiconductor quantum dots containing 1 to 12 electrons. For the first time, it is possible to identify the quantum numbers of the states in the excitation spectra and make a direct comparison to exact calculations. A magnetic field induces transitions between excited states and ground state. These transitions are discussed in terms of crossings between single-particle states, singlet-triplet transitions, spin polarization, and Hund's rule. Our impurity-free quantum dots allow for atomic physics experiments in magnetic field regimes not accessible for atoms. 