Low energy spectra of isotropic quantum dots are calculated in the regime of
low electron densities where Coulomb interaction causes strong correlations.
The earlier developed pocket state method is generalized to allow for
continuous rotations. Detailed predictions are made for dots of shallow
confinements and small particle numbers, including the occurance of spin
blockades in transport.Comment: RevTeX, 10 pages, 2 figure

Chakravarty S.

Häusler W.

Jefferson J. H.

Kouwenhoven L. P.

Press W.

Steffens O.

W Häusler

Weinmann D.

Weis J.

English

arXiv

Häusler, Wolfgang

OPUS - Augsburg University Publication Server

Rotational levels in quantum dots

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Low-energy spectra of isotropic quantum dots are calculated in
the regime of low electron densities where Coulomb interaction
causes strong correlations. The earlier developed pocket state
method is generalized to allow for continuous rotations.
Detailed predictions are made for dots of shallow confinements
and small particle numbers, including the occurrence of spin
blockades in transport

W. Häusler

EDP Sciences OAI-PMH repository (1.2.0)

OPUS Augsburg

                                                                    Rotational levels in quantum dots(∗)W. HäuslerI. Institut für Theoretische Physik, Universität Hamburg - D-20355 Hamburg, Germany                                                                                                                                                                                                                                                 Abstract. – Low-energy spectra of isotropic quantum dots are calculated in the regime of lowelectron densities where Coulomb interaction causes strong correlations. The earlier developedpocket state method is generalized to allow for continuous rotations. Detailed predictions aremade for dots of shallow confinements and small particle numbers, including the occurrence ofspin blockades in transport.Much of our present understanding of small quantum dots, with observable discrete levelstructure [1,2], concentrates on the regime of relatively high carrier densities where the inter-action and charging energy is comparable to the kinetic (Fermi) energy in magnitude [3–6].Similar to real atoms effective single-particle orbitals establish a reasonable approximationto the electronic states. The spins follow from Hund’s rule [5, 6] which is a perturbative re-sult though it accords well with experimental findings in small quantum dots at high particledensities [1].At lower densities Coulomb interaction is expected to destroy this single-particle picture,leaving strongly correlated or even crystallized electrons with collective low-energy excita-tions. While in the homogeneous two-dimensional case rs should exceed rc = 37 to reachthis regime [7] (rs = (πns)−1/2 measures the ratio between Coulomb and kinetic energy andis regulated by the two-dimensional carrier density ns), disorder is predicted to reduce thisvalue considerably to rc = 7.5 [8]. An even more pronounced reduction of rc in comparisonwith the homogeneous value is found for the transition into the “Wigner regime” in quantumdots [9, 10]. Careful quantum Monte Carlo (QMC) studies based on the spin sensitivity ofthe density-density correlation function yielded rc = 4 for parabolic quantum dots [10]. Ex-perimentally, this regime has been addressed using capacitance spectroscopy [11] which onlyprobes ground state energies. Non-linear transport behaviour [1, 2, 12, 13] has not yet beeninvestigated to detect the interesting correlation effects for the low-energy excitations.Numerical investigations of the low-density regime, emphasizing the spin states of rotatingthree electron Wigner molecules, have been carried out for shallow parabolic dots [14]. Investi-gations for larger particle numbers have focussed on dots of low symmetry where corners in the(∗) Dedicated to Alfred Hüller on the occasion of his 60th birthday.             232                   confining potential or impurities suppress zero modes to delocalize the charges in the Wignerregime by so that “pocket states” can be introduced [15], which are well suited to describelocalized charges. The “pocket states” served as basis to map the spin-sensitive low-energyphysics to the one of lattice models of the Hubbard form [16] that account for quantum corre-lations by hopping between nearest places. The applicability of this archetype for correlationphenomena has been demonstrated, e.g., in quantum dots of polygonal geometry [9].This mapping to a lattice model cannot be carried out straightforwardly if zero modes causecharge delocalization which by symmetry actually happens in most experimental quantumdots. They are fairly well described by an isotropic and in fact parabolic model [3, 17]H =N∑i=1p2i2m∗+ V , (1)whereV =m∗2ω20N∑i=1x2i +∑i<je2κ|xi − xj | . (2)Here, the effective mass m∗ and the dielectric constant κ are material parameters, and xj (pj)are electron positions (momenta) in two dimensions. This model does not explicitly involvespin (as opposed to real atoms spin-orbit coupling is negligible in quantum dots) so that all ofits eigenstates are simultaneously eigenstates to the square of the total spin Ŝ2 to eigenvaluesS(S + 1). The present work extends the pocket state method (PSM) to allow for rotationalsymmetry and compares with results obtained by QMC studies [10]. Being based on a recentlydeveloped multilevel blocking algorithm [18] to circumvent the infamous fermion sign problem,this QMC allows for high accuracy to resolve reliably even the low-energy spin structure atparticle numbers significantly larger than those treatable by diagonalizations.At low densities the charge carriers form a finite piece of an electron crystal [14], a Wignermolecule (WM), that might, classically [19], be arbitrarily oriented. Superposition of all of theazimuthal degeneracies leads to an isotropic charge density distribution, as required by thesymmetry of (2) [20]. For analytical progress it is tempting to separate out the normal coordi-nate related with the overall rotation and with total angular momentum quantum numbers `(in strictly harmonic confinements ` refers to the relative part of the Hamiltonian since thecenter-of-mass motion just adds integer multiples of ω0 to all of the eigenvalues and does notaffect the spin of any of the states [21,22]). However, the remaining normal coordinates thenwould in general no longer describe identical quantum particles obeying Pauli’s principle andFermi (or Bose) statistics but they would correspond to linear combinations of such particles.Within the PSM it is crucial to know the result of particle permutations in order to assigneventually the correct total spins S to the eigenstates and eigenenergies [23].Therefore, we treat all of the possible particle exchanges on equal footing, including discreteoverall rotations of the WM if they correspond to particle permutations. It depends on thegeometry of the WM whether rotations by 2π/p with p > 1 leave the electron places invariantso that the Pauli principle relates ` with S. Such a relationship is well known, for instancefrom the example of solid hydrogen H2, where the even ` are necessarily S = 0 singlet stateswhile the odd ` are S = 1 triplets (in this example the spins refer to the protons), the reasonbeing the equivalence of rotations by 180 degrees with the exchange of two identical spin-halffermions. Other examples are discussed in [24].The validity of the PSM requires that the spin-sensitive excitation energies ∆, to be calcu-lated by this method, should be smaller than charge (plasmon) excitations [23]. In the absenceof continuous symmetries this condition is easily fulfilled at small densities due to the almost                                       233exponential decay of ∆ ∼ exp[−√rs]. Plasmon energies decrease only according to a power law∼ r−3/2s for Coulomb repulsions. With their faster decay 1/2I = (2πm∗∫ ∞0dr r3n(r))−1 ∼r−2s (depending on the radial charge density distribution n(r), I is the moment of inertia)the total angular momentum excitations, however, still decay faster than the plasmons sothat eventually the low-energy levels will follow only from electron interchanges among theplaces defining the WM [25], including overall rotations by 2π/p, i.e. by processes permutingidentical quantum particles [26].From classical [19] as well as from quantum [10] Monte Carlo studies it is known that upto N ≤ 8 Wigner molecules in the parabolic quantum dots are very symmetric: the electronsform one spatial shell (N ≤ 5) so that p = N , or one electron occupies the center (i.e.p = N − 1). Here we focus on N ≤ 6. The method can be generalized straightforwardly tolarger N and more complicated geometries of the WM.The transition amplitudes for all possible particle permutations constitute the entries tof the pocket state matrix [15]. In the classically forbidden cases t can be estimated withinthe WKB approximation as discussed in [16, 23]. The complete potential (2), including theinteraction, goes into this estimate. Often the most important entries involve only two orthree adjacent particles, as in quantum dots of polygonal shapes [9], which then determine thehopping terms in the equivalent Hubbard model. This is different for the zero modes: therea much larger number of particles can be involved into a certain permutational transition,such as a rotation by 2π/p in isotropic quantum dots. Corresponding entries tR to the pocketstate matrix are not of tunneling type and, therefore, not exponentially small. In those casestR = −p2/8π2I is fixed by the energy constant 1/2I for rotational excitations (I follows fromn(r)).This way all of the relevant entries to the pocket state matrix can be estimated. Itsdiagonalization yields eventually the complete set of low-energy eigenvalues. Advantage canbe taken from the fact that pocket states constitute a faithful representation of the symmetricgroup SN , so that diagonalization can be carried out analytically for small systems, N ≤ 4,otherwise numerical help is required. Only irreducible representations [N/2+S, N/2−S] arecompatible with Pauli’s principle for spin-half fermions [23,27]. This fixes the spin S for eacheigenvalue.The entries |t| ∼ e−√rs and |tR| ∼ r−2s vary differently with the strength of the Coulombinteraction, so that the ratio t/tR is a measure for the interaction strength. We usey :=11 + t/tR> 0ranging from 1/(1+(π2/4)p), since |2t| cannot exceed the Fermi energy in the non-interactinglimit, up to unity at strong interactions, y → 1.Figure 1 shows the low-energy spectrum vs. y for N = 3. Our description is designedfor evaluating excitation energies, i.e. the differences between the energies of different spinstates. As expected for weak interactions (y < 0.5), the ground state is unpolarized [22]. Atransition into the spin-polarized ground state S = 3/2, not found in earlier diagonalizationstudies, is seen above a certain interaction strength which for Coulomb interactions and GaAsparameters can be estimated to happen when ω0 < 0.5meV [10]. This result complies withthe QMC studies and can also be seen when carefully examining fig. 1 of the study [14] ofa large quantum dot. We would like to emphasize that this spin polarization is an exactconsequence of correlations and not the result of the mean-field approximation or a magneticfield. In transport experiments, when contacting quantum dots with electron reservoirs, itshould show up as a “spin blockade” [28], since the ground states of N = 2 and N = 3 in234                   -40480.2 0.4 0.6 0.8 1E/t_Ry3/21/2Fig. 1-40480.4 0.6 0.8 1E/t_Ry5/21/23/23/21/21/2Fig. 2Fig. 1 – Low energy levels using pocket states vs. y for N = 3 in units of tR.Fig. 2 – Low energy levels using pocket states vs. y for N = 5 in units of tR.sufficiently large quantum dots differ then in spin by more than ∆S = 1/2 (by which enteringor escaping single electrons can change spin) since the N = 2 ground state (with time reversalsymmetry) is always a singlet [29].For N = 4 (not shown here) we confirm the Hund’s rule result of a S = 1 ground state, asobtained already in density functional calculations [5,6,20]. New is its persistence up to stronginteractions. The lowest singlet level S = 0 approaches this ground level ∼ exp[−ω−1/30 ] as ω0decreases. The rotationally first excited state ` = 1 consists only of triplet S = 1 levels whilethe spin polarized level S = 2 belongs to the doubly excited rotational state, ` = 2, togetherwith another singlet S = 0 level.For N = 5 (fig. 2), on the other hand, the polarized state S = 5/2 joins the unpolarizedground state S = 1/2 in the lowest rotational level at strong interactions. This low-energyhigh spin state makes negative differential conductances in the non-linear transport likely, dueto the spin blockade [28]. Rotationally excited levels consist of S = 1/2 as well as of S = 3/2spin states.The sixth electron is predicted [10], also classically [19], to occupy the center of a 5-foldring. This complicates the pocket state analysis since new types of pair exchanges appear(exchange with the central electron) and also the triple exchange t3 (cyclic permutations ofthree adjacent electrons, including the central one) turns out as important, in accordance withWKB estimates [30]. Indeed, the PSM spectra do not compare with the low energy levelsobtained from QMC unless t3 is included with a similar magnitude as the pair exchanges.This demonstrates how our approach complements most favorably the QMC simulationsfor quantum dots which yields abolute values for the many particle energies to high accuracy,contrary to the method based on pocket states. Very reliable estimates for the t-parameters canbe achieved which otherwise would have to be guessed by less trusty approximative means. Onthe other hand, QMC is incomplete for the low energy levels since only the lowest eigenenergiesto given z-component can be simulated.For N = 6 and confining energies ω0 ≈ 0.13meV (GaAs) we find, with increasing energy,the spin sequence 1-0-3-2-1-0-2-1-2-1-1-0. The spin S = 1 indicates another interaction-induced change in the ground state spin since from the non-interacting levels the point ofview N = 6 corresponds to a “noble gas” configuration implying an unpolarized ground state                                       235spin S = 0 [6]. This result also has to be contrasted with the conjecture S = 2 following froma static antiferromagnetic WM [16] of pentagonal symmetry. The rotational ground state` = 0 includes all possible spin states S = 1, 0, 3, 2, with the fully polarized state, S = 3, beinglower in energy than the lowest S = 2 state, in accordance with QMC. This again suggeststhe possible occurrence of negative differential conductances for the transition to N = 5.In conclusion, generalizing the pocket state method, we have developed a description for thelow-density regime in isotropic such as parabolic quantum dots. Low energy levels, includingspin quantum numbers were determined for N ≤ 6. Detailed predictions are made for spinblockades as they should be detectable in linear and non-linear transport through shallowquantum dots of confinement energies below ω0 . 0.4meV (GaAs).∗ ∗ ∗I am particularly indebted to my teacher A. Hüller for long-standing support and en-couragement. Numerous very fruitful discussions with Ch. Creffield, R. Egger, H.Grabert and J. Jefferson are acknowledged. This work has been carried out during staysat the University of Freiburg, the University of Jyväskylä, and the King’s College London.Support has been received from the DFG (through SFB 276) and the EPSRC (UK).REFERENCES[1] Tarucha S., Austing D. G., Honda T., van der Hage R. J. and Kouwenhoven L. P.,Phys. Rev. Lett., 77 (1996) 3613.[2] Kouwenhoven L. P., Marcus C. M., McEuen P. L., Tarucha S., Westervelt R. M. andWingreen N.S., in Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhovenand G. Schön, NATO ASI Ser. E, Vol. 345 (Kluwer Academic Publishers) 1997; Kouwen-hoven L. P., Oosterkamp T. H., Danoesastro M. W. S., Eto M. E., Austin D. G.,Honda T. and Tarucha S., Science, 278 (1997) 1788.[3] Merkt U., Physica B, 189 (1993) 165.[4] Pfannkuche D., Gudmundsson V. and Maksym P. A., Phys. Rev. B, 47 (1993) 2244.[5] Stopa M., Phys. Rev. B, 54 (1996) 13767; Koskinen M., Manninen M. and Reimann S. M.,Phys. Rev. Lett., 79 (1997) 1389.[6] Steffens O., Rössler U. and Suhrke M., Europhys. Lett., 42 (1998) 529.[7] Tanatar B. and Ceperley D. M., Phys. Rev. B, 39 (1989) 5005.[8] Chui S. T. and Tanatar B., Phys. Rev. Lett., 74 (1995) 458.[9] Creffield C. E., Häusler W., Jefferson J. H., Sarkar S., Phys. Rev. B, 59 (1999) 10719.[10] Egger R., Häusler W., Mak C. H. and Grabert H., Phys. Rev. Lett., 82 (1999) 3320.[11] Ashoori R. C., Stormer H. L., Weiner J. S., Pfeiffer L. N., Pearton S. J., BaldwinK. W. and West K. W., Phys. Rev. Lett., 68 (1992) 3088; Ashoori R. C., Nature, 379 (1996)413.[12] Johnson A. 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PACS. 73.50.-h { Electronic transport phenomena in thin ¯lms and low-dimensional struc-tures. PACS. 73.61.-r { Electrical properties of speci¯c thin ¯lms and layer structures. Abstract. { Low-energy spectra of isotropic quantum dots are calculated in the regime of low electron densities where Coulomb interaction causes strong correlations. The earlier developed pocket state method is generalized to allow for continuous rotations. Detailed predictions are made for dots of shallow con¯nements and small particle numbers, including the occurrence of spin blockades in transport. Much of our present understanding of small quantum dots, with observable discrete level structure [1,2], concentrates on the regime of relatively high carrier densities where the inter-action and charging energy is comparable to the kinetic (Fermi) energy in magnitude [3{6]. Similar to real atoms e®ective single-particle orbitals establish a reasonable approximation to the electronic states. The spins follow from Hund&apos;s rule [5, 6] which is a perturbative re-sult though it accords well with experimental ¯ndings in small quantum dots at high particle densities [1]

W. Häausler

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