Ground-state energy and spin in disordered quantum dots

We investigate the ground-state energy and spin of disordered quantum dots using spin-density-functional theory. Fluctuations of addition energies (Coulomb-blockade peak spacings) do not scale with average addition energy but remain proportional to level spacing. With increasing interaction strength, the even-odd alternation of addition energies disappears, and the probability of non-minimal spin increases, but never exceeds 50%. Within a two-orbital model, we show that the off-diagonal Coulomb matrix elements help stabilize a ground state of minimal spin. 73.61.-r, 71.15.Mb, 71.70.Ej, 75.75.+a Typeset using REVTEX 1 The control of spin in semiconductor nanostructures [1] is essential for a number of applications such as spintronics [2] and quantum bits [3], and for fundamental studies of the Kondo effect in quantum dots [4]. In clean quantum dots with circular symmetry, Hund’s rule is observed – high-spin states (i.e. states of non-minimal spin) appear for partly filled shells of degenerate single-particle levels [5]. In disordered or chaotic quantum dots [6], highspin states are suppressed by the rarity of degenerate or nearly degenerate levels. However, the observed absence of even-odd alternation of addition energies in quantum dots [7–10] is consistent with ground states of non-minimal spin. Recently, evidence for such highspin states has been obtained from studies of the magnetic dispersion of Coulomb-blockade peaks [11,12]. A number of theoretical treatments have considered the conditions under which high-spin ground states might occur [13–20]. Many of these works [16,18,19] address the regime of large dimensionless conductance g >> 1 [21]; other works rely on the validity of the Hartree-Fock approximation [14,17]. Simulations on small lattice models indicate a significant fraction of S = 1 ground states [15], but we infer g ∼ 0.1−0.3 in the simulations, compared to the experimental range g > 1 [7–10]. Density-functional theory enables in silico experiments on the ground-state spin of small, disordered quantum dots for which g > 1, over a wide range of interaction strengths. The results presented here challenge the view that high-spin states dominate in disordered dots for rs > 1 [17]: even-odd alternation of addition energies disappears by rs = 1.25, but the probability of a high-spin ground state never exceeds 50%. Using a two-orbital model, we show that off-diagonal Coulomb matrix elements help stabilize a ground states of minimum spin, as argued by Jacquod and Stone [20]. We also find that addition-energy fluctuations do not scale with the average addition energy, but remain proportional to the single-particle level spacing, consistent with our previous results for spin-polarized dots [25]. The ground-state energy and spin of disordered, two-dimensional quantum dots are obtained within spin-density-functional theory (SDFT). Specifically, we solve the following Kohn-Sham equations [22] numerically, and iterate until self-consistent solutions are obtained; 2