Mesoscopic fluctuations of the ground-state spin of a small metal particle

Abstract

We study the statistical distribution of the ground state spin for an ensemble of small metallic grains, using a random-matrix toy model. Using the Hartree Fock approximation, we find that already for interaction strengths well below the Stoner criterion there is an appreciable probability that the ground state has a finite, nonzero spin. Possible relations to experiments are discussed. PACS numbers 71.24.+q, 75.10.Lp According to Hund's rule, 1 electrons in a partially filled shell in an atom form a many-body ground state with maximum possible spin. The maximum spin is preferred because it allows a maximally antisymmetric coordinate wavefunction in order to minimize the electrostatic repulsion between the electrons. In recent experiments, 2 Hund's rule was also observed in a cylindrically-shaped semiconductor quantum dot, or "artificial atom". The close similarity with real atoms is due to the degeneracy of single-particle levels, caused by the the high degree of symmetry of the device.In generic ultrasmall systems such as small metal grains, 3,4 semiconductor quantum dots, 5,6 or carbon nanotubes 7,8 there is no systematic degeneracy due to a spherical (or cylindrical) symmetric potential. However, even in the absence of degeneracies, a nonzero value of the ground state spin may occur, as long as the gain in electrostatic energy is larger than the loss in kinetic energy when an antisymmetric coordinate ground state wavefunction is formed. Such a ground state is most likely to be detected in ultrasmall metal and semiconductor devices, since in those systems, unlike in macroscopic samples, the spacing between single-particle energy levels and the typical interaction energies can be larger than the temperature. In fact, the possibility of such a "weakly ferromagnetic" ground state has been suggested as an explanation for some recent experiments, that could not be explained by simple noninteracting models. 4,7,9 In addition, a nonzero ground state spin from numerical simulations of a few particles in a chaotic dot, 10 and a theory of spin polarization in larger dots 11 were already mentioned in the literature. The stability of the zero spin ground state in a quantum dot was analyzed for weak interactions in Ref. 9.In this paper, we consider small metal grains in the mesoscopic regime, in which fluctuations of wavefunctions and energy levels, caused by, e. g., disorder or an irregular shape, control the behavior of kinetic and interaction energies at the vicinity of the Fermi energy. As a result, the ground state spin becomes subject to sampleto-sample fluctuations. Then, the relevant quantity to consider is the statistical distribution of the ground state spin for an ensemble of small metal grains or chaotic quantum dots, rather than the spin of a specific sample.Our starting point is a simple toy model that captures the essential mechanisms for mesoscopic fluctuations of the ground state spin. In second-quantized form, our model Hamiltonian H readswhere c † n,σ (c n,σ ) is the creation (annihilation) operator...

Spectroscopy of discrete energy levels in ultrasmall metallic grains

“…For example, already at the quite modest interaction strength of u/d ≃ 0.4, a ground-state spin of s = 1 was found to be more likely than s = 0. The probability to find non-minimal s is reduced if spin-orbit coupling is present, but may still be appreciable if the Coulomb interaction is strong [159]. More work is required to demonstrate definitively that the clustering observed in [16] is due to excitations within spin multiplet and not a non-equilibrium effect (of the sort described in Sec.…”

Interactions in chaotic nanoparticles: Fluctuations in Coulomb blockade peak spacings

“…Note that these requirements will yield a random matrix approach noticeably different from the ones discussed in Refs. [40,43,46].…”

“…(1), residual interactions make a contribution of order the mean level separation to the second difference Λ, comparable to the single-particle contribution in Eq. (3) [40,[42][43][44][45][46]. In the limit of a large nanoparticle-one whose typical dimension L is many times the electron wavelength, k F L ≫ 1-only the average effect of residual interactions needs to be added to the CI model [42][43][44][45].…”

“…Even if one neglects the fluctuations of the residual interaction terms, because of fluctuations of the single-particle levels, the "constant exchange" contribution can modify the total spin of the nanoparticle by favoring non-trivial occupation of the single-particle orbitals and, as a consequence, will significantly modify the fluctuations of the peak spacings [40,42]. The ground state spin of a large nanoparticle in this regime has been examined in detail [43,[45][46][47], but curiously the corresponding distribution of Coulomb blockade peak spacings has not appeared explicitly (see however Ref. [42]).…”

Semiclassical theory of Coulomb blockade peak heights in chaotic quantum dots

“…|ψ (r n )| 2 ψ , of the distribution are different from unity. Therefore, in its general from, the ensemble defined by (9) is not suitable for calculations which are sensitive to the n > 1 moments of the distribution P [ψ], such as for the description of the residual interactions in quantum dots [38][39][40].…”