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Interaction-induced strong localization in quantum dotsA. D. Güçlü,1Amit Ghosal,2C. J. Umrigar,3and Harold U. Baranger11Department of Physics, Duke University, Box 90305, Durham, North Carolina 27708-03052Physics Department, University of California Los Angeles, Los Angeles, California 90095-1547, USA3Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA共Received 5 December 2007; published 2 January 2008兲We argue that Coulomb blockade phenomena are a useful probe of the crossover to strong correlation inquantum dots. Through calculations at low density using variational and diffusion quantum Monte Carlo 共up tors⬃55兲, we find that the addition energy shows a clear progression from features associated with shell structureto those caused by commensurability of a Wigner crystal. This crossover 共which occurs near rs⬃20 forspin-polarized electrons兲 is, then, a signature of interaction-driven localization. As the addition energy isdirectly measurable in Coulomb blockade conductance experiments, this provides a direct probe of localizationin the low density electron gas.DOI: 10.1103/PhysRevB.77.041301 PACS number共s兲: 73.23.Hk, 02.70.Ss, 73.63.KvLocalization of electrons induced by electron-electron in-teractions is a key issue in strongly interacting systems1which dates from the dawn of solid state physics whenWigner introduced the notion of an electron crystal.2Therecent focus on localization in inhomogeneous systems isstimulated in part by experiments which apparently seea metal-insulator transition in two dimensions.3This occursin low-density electron gas samples in which disorder ispresumably important. Recall that in a degenerate electrongas, low density implies strong interactions. Here we showthat an approach from nanoscale physics, in which thepotential confining the electrons plays a key role analogousto that of disorder, provides new information on this criticalproblem.The Coulomb blockade effect has been a valuable tool forprobing a variety of interaction effects in quantum dots.4–7By “quantum dot” we mean a confined region of electrongas containing between N =1 and ⬃1000 electrons; experi-mentally, they have proven remarkably tunable through theuse of gate voltages on nearby electrodes. We treat N ⱕ 20here. The large electrostatic charging energy usually forcesthe dot to have a fixed number of electrons, preventingthe flow of current. The blockade is lifted by using a gateto tune the energy for N electrons to be the same as thatfor N +1, inducing a finite conductance. The conductancethrough the dot as a function of gate voltage is thereforea series of sharp peaks. The height and position of thesepeaks encode information about the dot’s ground state; forinstance, the spacing between the peaks is proportional tothe second difference of the ground-state energy with respectto electron number N, a quantity known as the additionenergy. Quantum many-body physics probed in quantumdots includes, for instance, the atomiclike effect of exchangeand correlation in altering the filling of single-particleshells,6several kinds of Kondo effects,7aspects of thefractional quantum Hall effect,5and the entanglement ofspin and orbital degrees of freedom for quantum informationpurposes.4We show that the Coulomb blockade can be a valuableprobe of localization in quantum dots. In particular, the char-acteristic pattern of the addition energy changes as the dotcrosses over from the extended states of the high density-Fermi liquid regime to the localized states characterizing thelow density Wigner crystal. Interaction strength is oftencharacterized by rs=1/ aB*冑n 共in two dimensions兲 whereaB*is the effective Bohr radius and n is the electron den-sity. In our quantum dots, the crossover occurs for rssub-stantially smaller than the value8–10in bulk. This decreasein interaction strength needed for localization is connectedto the density inhomogeneity necessarily present in thisconfined system; density inhomogeneity produced in otherways, such as by disorder, may similarly enhance local-ization. Our main point is that such a crossover to localiza-tion can be directly measured in Coulomb blockade experi-ments.The crossover from Fermi liquid to “Wigner molecule”in quantum dots was studied previously using variousmany-body methods. Exact diagonalization,5,11while themost robust and direct approach, is limited to small elec-tron number and small rsdue to convergence problems.Path-integral quantum Monte Carlo12–14is well suited forfinite temperature properties, but preserves only Szsymmetryand has large statistical fluctuations at low temperatures.Variational and diffusion Monte Carlo15preserve Sz, S2,and Lzsymmetry; though limited by the “fixed node” error,which depends on the quality of the trial wave function,15they were successfully used to study quantum dots forrsup to ⬃ 416–19and then up to rs⬃15 in our recentwork.20,21Here, we apply recently developed energy minimi-zation methods22,23to floating Gaussian-based trial wavefunctions, enabling us to decrease the fixed-node error andinvestigate the strongly correlated regime up to as high asrs⬃55.Our model quantum dot consists of N interacting elec-trons in a two-dimensional circular quadratic potential. TheHamiltonian, expressed in effective atomic units 共electroniccharge e, dielectric constant⑀, effective mass m*, and ប areset to 1兲, is given byH =−12兺iN䉮i2+12兺iN2ri2+兺i⬍ jN1rij, 共1兲PHYSICAL REVIEW B 77, 041301共R兲共2008兲RAPID COMMUNICATIONS1098-0121/2008/77共4兲/041301共4兲 ©2008 The American Physical Society041301-1whereis the spring constant of the quadratic potentialwhich provides control of the strength of the Coulomb inter-action with respect to the kinetic energy. In analogy with 2Dbulk systems, we characterize the interaction strength usingrs=共n¯兲−1/2, where n¯⬅兰n2共r兲dr/ N is the mean density ofelectrons.Variational 共VMC兲 and diffusion 共DMC兲 Monte Carlotechniques15were used to calculate the properties of ourmodel quantum dots. One starts with a set of single-particleorbitals—simple Gaussian functions or from a self-consistentcalculation 共Hartree or Kohn-Sham兲. We then perform aVMC calculation using a trial wave function ⌿T, which is alinear combination of products of up- and down-spin Slaterdeterminants of these orbitals multiplied by a Jastrow factor.共The detailed form