Electron correlation in metal clusters, quantum dots and quantum rings
EElectron correlation in metal clusters, quantum dotsand quantum rings
M Manninen , S M Reimann NanoScience Center, Department of Physics, FIN-40014 University ofJyv¨askyl¨a, Finland Mathematical Physics, Lund Institute of Technology, SE-22100 Lund, Sweden
Abstract.
This short review presents a few case studies of finite electronsystems for which strong correlations play a dominant role. In simple metalclusters, the valence electrons determine stability and shape of the clusters.The ionic skeleton of alkali metals is soft, and cluster geometries are oftensolely determined by electron correlations. In quantum dots and rings, theelectrons may be confined by an external electrostatic potential, formed by agated heterostructure. In the low density limit, the electrons may form so-calledWigner molecules, for which the many-body quantum spectra reveal the classicalvibration modes. High rotational states increase the tendency for the electrons tolocalize. At low angular momenta, the electrons may form a quantum Hall liquidwith vortices. In this case, the vortices act as quasi-particles with long-rangeeffective interactions that localize in a vortex molecule, in much analogy to theelectron localization at strong rotation. a r X i v : . [ c ond - m a t . s t r- e l ] O c t etal cluster, quantum dot and quantum rings
1. Introduction
For a long time, the homogeneous electron gas has been the standard theoreticalmodel for a correlated, infinite Coulombic system where the fermionic character ofthe electrons plays the dominant role [1]. The so-called ’jellium’ model of a metal hasbeen a starting point for developing functionals for the density functional theory ofelectrons [2]. The long-ranged nature of the Coulomb interaction has posed a challengeto many-body theory since many decades. At high densities of the electron gas, theexchange interaction dominates, while in the low-density limit, the electrons may forma Wigner crystal [3].In real metals, the ions do not form a homogeneous charge background like in thesimple jellium model, but may rather be described by a lattice of pseudo-potentials.Nevertheless, many properties of alkali metals can be understood on the basis of thejellium model, as for example, the surface energy and work function [4], the vacancyformation energy [5], or collective plasmon excitations [1], etc. Since this works well forthe bulk, it is not surprising that the jellium model also applies well to the approximatedescription of the properties of small alkali metal clusters [6, 7].The intention of this article is to provide a brief survey of some of the fascinatingproperties of clusters or quantum dots: In Section 2, we discuss metal clusterson the basis of a two-component plasma and show that the overall shape and theplasmon excitations can be qualitatively explained within this simple model. Insemiconductor heterostructures, the valence electrons may be confined within a quasitwo-dimensional layer, forming two-dimensional electron gas (2DEG). Here, dependingon the material parameters, the electron density is small, and the electron wave lengthmuch larger than the lattice constant. Consequently, the model of the two-dimensionalhomogeneous electron gas is valid, if the electron mass is replaced by an effectivemass determined by the band structure, and the Coulomb interaction is replaced byan interaction screened by the dielectric constant of the semiconductor [8]. Withetching techniques and external gates, electrons can be localized in a nearly harmonicconfinement. In Section 3 we will study the fascinating many-particle physics of thisseemingly simple system: one example of its surprisingly rich physics is the occurrenceof internally broken symmetries such as spin density waves in the ground state. Thefollowing sections concentrate on the localization of electrons confined in quantumrings or 2d quantum dots. We will see how the reduction of the dimensionalityincreases the electron-electron correlation, and how the Pauli exclusion principle thencomes to play the dominating role. Using a simple model for a quantum ring, we showin Section 4 that the many-particle energy spectrum reveals the internal structure ofthe ground state, and its analysis provides information about the Wigner localization.The collective excitations of the particles are then the classical vibrational modes of theWigner molecule. Correspondingly, in section 5 we then show how all the low-energystates at high angular momenta can be described by a simple model Hamiltonian ofa vibrating molecule. In Section 6 we turn to a discussion of collective excitationsat low angular momenta. For large numbers of electrons, the low-energy excitationscorrespond to vortices in the quantum system. The localization of vortices in a ’vortexmolecule’ then determines the fine structure of the energy spectrum. We connect thediscussion to the physics of other, but intimately related many-body systems, such asfor example cold atoms in traps, pointing out the apparent similarities between finitefermionic or bosonic quantum systems. etal cluster, quantum dot and quantum rings
2. Metal clusters as electron-ion plasma
Mie [9] showed already 100 years ago that a metal sphere has an optical absorptionat a frequency which is independent of the size of the sphere. The absorption peakcorresponds to a surface plasmon related to the bulk plasmon as ω sp = ω p / √
3. Sincethe simplest model for a metal cluster is a conducting sphere, it is apparent thatsimilar plasmon peaks as those predicted by Mie, were observed [15]. As mentionedabove, the jellium model assumes the ions to be distributed in a homogenous, rigidcharge background. Modeling the cluster as a jellium sphere, the conduction electronsare then confided in the cluster by the Coulomb attraction of the sphere [10, 11, 12].The confining potential inside the sphere is harmonic, meaning that the center-of-massmotion of the electron cloud can be separated from the internal motion. The collectiveoscillation of the electrons against the background charge is the plasmon of the metalsphere. This result, in fact, emerged also from a shell-model type of calculation forthe correlated electrons in the cluster [13]. The phenomenon is analogous to the giantresonance in nuclei, where the protons oscillate against the neutrons [14]. A moredetailed calculation shows that the electrons spill out slightly from the region of theharmonic well. This causes a small red-shift of the plasmon peak (for a review, see [7]).Experiments with large spherical alkali-metal clusters have shown that the plasmonpeak occurs in fair agreement with this simple model [15].The spherical jellium model predicted a shell structure for small sodium clusterswhich was observed first by Knight et al. [16]. Immediately after this discovery,however, it was realized that only metal clusters with filled electron shell are nearlyspherical, while others should have strong (quadrupole) deformations, in much analogyto the physics of atomic nuclei [17]. Below, we shall discuss the origin of this shapedeformation by considering the sodium metal as a two-component plasma consistingof ions and electrons [18]. The ions are mimicked by positrons (but shall not allowannihilation with electrons). A cluster of such a fictive ’electron-positron’ plasma canbe studied using the density functional Kohn-Sham method, with the single particleequations − ¯ h m ∇ ψ i + v eff ψ i = (cid:15)ψ i , (1)where the effective potential is the functional derivative of the potential energyfunctional V [ n ]: v eff ( r ) = δV [ n ] /δn ( r ). In the case of our imaginary electron-positronsystem, the potential energy consists only of the exchange and correlation energiessince the total Coulomb potential (Hartree term) is zero due to the symmetry (theelectron and the positron densities are identical). Alternatively, we can imagine thepositive ions forming a completely deformable, ’floppy’ (classical) background chargewhich will always take the same density distribution as the electron density, but doesnot have its own correlation or exchange energy. This is the so-called ’ultimate jellium’model [19] where the potential energy is now only the exchange and correlation energyof the electron gas, V [ n ] = E xc [ n ]. The equilibrium density of the infinite ultimatejellium is close to that of the electron density of sodium (with a Wigner-Seitz parameterof r s ≈ . a , where a is the Bohr radius). Nevertheless, it is still surprising howwell the model can quantitatively describe shape-related properties of real sodiumclusters [20]. The obtained shapes are in agreement with those determined by thesplitting of the plasmon resonance to two separate peaks [21].The success of the simple plasma model for describing the overall shapes of sodiumclusters is based, on one hand, on the softness of the metal, and on the other hand, etal cluster, quantum dot and quantum rings Figure 1.
Shapes of clusters with 6 and 14 particles. The left panel shows theconstant-density surface of the universal model giving similar shapes irrespectiveof the interparticle interactions. The center panel shows the constant densitysurface and ion positions in sodium clusters calculated within DFT. The rightpanel shows stick-and-ball models of atom positions of TB clusters. Note that inthis case the constant density surface can not be defined, but the resulting atompositions are the same as in the DFT calculation. on the universality of the shapes of small fermion systems [22]. In small systems,there are only a small number of single particle states which contribute to the densitydistribution. The effective potential is a functional of this density distribution andwill thus have a similar shape determined by the few single-particle wave functions,irrespective to the details of the interparticle interactions binding the system together.Figure 1 illustrates the strength of this universality, showing the results for three apriori very different models for clusters containing 6 and 14 particles. The densitycontours of the two clusters on the left are results of the universal model (or so-called’ultimate jellium’ model), where the only essential parameter is the average densityof the bulk material. The clusters shapes shown in the center panel are the results ofab-initio density functional calculations using ab initio pseudo-potentials. The rightpanel shows the results of the simplest possible tight-binding model, which assumesconstant bond length and only nearest neighbor hopping. Note that this tight bindingmodel does not take into account any long-range Coulomb interactions.In semiconductors, is possible to make two-dimensional structures where electronsand holes are at different layers, the separation of the layers hindering theirrecombination. In the limit of vanishing interlayer distance the electrons and holesform a two-dimensional plasma. Reimann et al. [23] studied finite-size systems of sucha 2D plasma and observed that, like in 3D, the shapes of the plasma clusters are quiterobust. Interestingly, in 2D those clusters corresponding to filled electronic shells of2D circular traps do not have circular shapes but strong triangular deformations, asshown in Fig. 2. The reason is that in a triangular cavity the lowest energy shellsagree with those of the harmonic oscillator [24].It has also been suggested [25, 26] that the two-dimensional deformable jelliummodel could capture the main correlations of alkali metal clusters on a weaklyinteracting substrate, like oxide or graphite. In such systems computations basedon DFT and pseudo-potentials [22] indeed also result in a triangular shape for N = 12 etal cluster, quantum dot and quantum rings Figure 2.
Electrons densities in 2D plasma clusters calculated with the ultimatejellium model. Note that the sizes 12, 20, and 30 correspond to filled shells. in agreement with the simple model.
3. Two-dimensional quantum dots: DFT
In the previous sections, we studied free metal and plasma clusters and learned that,just like in nuclei, any open-shell system is deformed with respect to the sphericalshape. In 2D, even closed-shell clusters do not necessarily occur only for circularshape. The internal symmetry breaking is driven by the tendency to maximize theenergy gap between the highest occupied and lowest unoccupied single particle state,in accordance with the Jahn-Teller theorem. In the remaining parts of this briefsurvey, let us consider systems where the electrons (or ions and atoms, respectively)are trapped by a rigid harmonic confinement. Physical examples of such systems arethe conduction electrons in semiconductor quantum dots, as well as atoms and ionsconfined in magneto-optical traps.Generally speaking, with interactions between the particles, the degeneracy of anopen shell can be reduced by spin polarization driven by Hund’s first rule. However,this mechanism may be competing with other, internal, symmetry-breaking. Examplesare deformation effects (as discussed above for metal clusters and the jellium model),pairing (like in nuclei for interactions with an effectively attractive part), or theformation of a spin density wave (see below).Tunneling spectroscopy of small quantum dots [28] has clearly revealed the energy-lowering of the half-filled shells due to Hund’s first rule. The results are stronglysupported by the spin-density functional [29] and ab-initio many-particle calculationsfor electrons in a harmonic confinement (for a review see [8]).In the low-density electron gas, the electrons localize in a Wigner crystal. Closeto this limit, the difference of the total energy of the paramagnetic and ferromagneticelectron gas diminishes. When the electrons localize, the spin ordering is expected tofollow the Heisenberg model for the spin. Density functional theory in the local densityapproximation can not describe properly the Wigner crystal, since the Coulomb self-interaction of the localized electrons is not properly canceled by the local exchange.In order to study particle localization, despite of an a-priori lack of correlation, it wasargued that it is then more favorable to use the unrestricted Hartree-Fock method [30].At large densities, i.e. r s ≤ a ∗ (here a ∗ is the effective Bohr radius), the six-electron quantum dot has a closed-shell configuration. Its ground state is a statewith total spin S = S z = 0 in both the CI and SDFT methods, with circularlysymmetric particle densities. As r s increases, however, the total density remains etal cluster, quantum dot and quantum rings Figure 3.
DFT spin densities n ↑ and n ↓ (upper panel) and total density ( n ↑ + n ↓ )as well as (un-normalized) spin polarization ( n ↑ + n ↓ ) (lower panel) for a six-electron quantum dot at r s = 4 a ∗ , shown as 3D plots and their contours. FromRef. [39]. azimuthally symmetric, while the spin densities show a symmetry breaking, leadingto a pronounced spatial oscillation in the spin polarization (see Fig. 3), but stilltotal spin S z = 0. Such so-called ’spin density wave’ (SDW)-like states [29] havebeen much discussed in the literature, and it was claimed that such states are simpleartefacts of the broken spin symmetries in SDFT [31, 32]. To resolve this question,clearly one has to compare the SDFT results with the solutions of the full many-bodyHamiltonian. The many-particle state of a few electrons in a quantum dot can beobtained numerically nearly exactly, by diagonalizing the Hamiltonian in a properlyrestricted Hilbert space. However, density and spin densities of the exact solutionnecessarily have the same symmetry as the Hamiltonian. For a circular quantumdot, this means that the solutions also must have the azimuthal symmetry of theHamiltonian. Consequently, the possible localization of electrons, or other internalsymmetry breaking such as the above mentioned spin-density waves, can not be seendirectly in the total particle- or spin-densities. Instead, one then has to investigatethe pair correlation functions, i.e., conditional probabilities, to study the internalstructure [35, 36]. Indeed, the direct comparison to spin-dependent pair correlationfunctions in the configuration interaction method, clearly demonstrated that SDW- etal cluster, quantum dot and quantum rings
4. Energy spectra and localization: Quantum rings
In a strictly one-dimensional ring, the single-particle energy levels are solutions to theangular momentum-part of the Hamiltonian, H = − ¯ h mR ∂ ∂φ , (2)where R is the radius of the ring. The solutions are ψ (cid:96) ( φ ) = exp( i(cid:96)φ ) with energyeigenvalues (cid:15) (cid:96) = ¯ h (cid:96) / mR . For simplicity we will consider non-interacting electronswith the same spin. Then each electron occupies a different single-particle state, andthe energy is the sum of the energies of the single-particle energies. Figure 4 showsas an example the resulting (here non-interacting) many-particle energy spectrum forfour (spinless) electrons.More generally, we can consider particles interacting with an infinitely strongcontact interaction ( v ( r − r (cid:48) ) = v o δ ( r − r (cid:48) ), where v → ∞ ). Note that for polarizedelectrons, the contact interaction then does not play any role since the Pauli exclusionprinciple forbids the electrons to be in the same point. The many-particle problemwith spin can for contact interactions be solved exactly with the Bethe ansatz [40, 41].Figure 4 shows that for non-polarized electrons, the lowest energy of each angularmomentum (the so-called yrast state) is a smooth function of L , while for polarizedelectrons it oscillates with a period of four. The reason for this oscillation becomesobvious when one considers the configurations of each of these states, as illustrated inthe figure. If there is a non-occupied state between the occupied ones, the energy ishigher than for the compact states. The period of four is a result of the following fact:If Ψ L is a solution for a ring of N particles with angular momentum L , then alsoΨ L + νN = exp (cid:32) iν N (cid:88) k φ k (cid:33) Ψ L , (3)is a solution with angular momentum L + νN , that also has exactly the sameinternal structure. One can interpret the eigenenergies in Fig. 4 as a rotation-vibration spectrum of localized electrons. The states at the minima (lowest dashedline) correspond to rigid rotations of localized electrons while the states above havevibrational states accompanying the rigid rotation (two lowest vibrational states are etal cluster, quantum dot and quantum rings Figure 4.
Spectra of one-dimensional quantum rings. The right panel showsthe spectrum of four particles interacting with delta function interaction in astrictly 1D ring. Black bullets show results of the states with maximum spin( S = 2), circles have lower spin. The solid line connects the lowest states ofpolarized electrons. The dashed lines show the (lowest) rigidly rotating state, andthe lowest vibrational states. The left panel shows the configurations of lowest-energy states of polarized electrons for L = 0 · · ·
3, demonstrating how the lowestenergy state is obtained for L = 2. shown as dashed lines). The possibility of vibrational states of noninteracting electronsis a peculiarity of the 1D system: An electron is localized between the two neighboringelectrons since the Pauli exclusion principle prevents them to pass each other. Thisgives an effective 1 /r interaction between the electrons which in this case arises fromthe kinetic energy of the electrons. Indeed, quantizing the vibrational models of a ringformed of particles with 1 /r interactions gives exactly the spectrum shown in Fig. 4.In quasi-one-dimensional rings with electrons interacting by long-range Coulombinteractions, the localization may also be seen directly in the energy spectrum. Ifthe electrons are non-polarized, the charge and spin degrees of freedom separate. Thecharge excitations are the rigid rotations and vibrations, while the spin excitations canbe described by the anti-ferromagnetic Heisenberg model of localized spins [42] (for areview see [41]). The conclusion is that in narrow quantum rings the electrons localizein a ’necklace’ of electrons, and the collective low-energy excitations are vibrationsand spin excitations.
5. Rotation-vibration spectrum of electrons in quantum dots
Let us now return to electrons confined in a 2D harmonic potential, and considerfirst the rotational states of polarized electrons with large angular momentum.The electrons interact by their long-range Coulomb interaction. For not too largenumbers of electrons, the Hamiltonian can be solved numerically almost exactly.Maksym [43, 44] showed that the resulting energy spectrum can be quantitativelydescribed by quantizing the classical vibrational modes solved in a rotating frame.Figure 5 shows that for high angular momenta the whole energy spectrum can bequantitatively described by the vibration modes of the localized particles.Figure 5 shows that the lowest-energy state as a function of the angularmomentum has a similar periodicity of four as observed above for four electrons in etal cluster, quantum dot and quantum rings Figure 5.
Energy spectrum of four polarized electrons in a 2D quantum dot.The black bullets show the result of an exact quantum-mechanical diagonalizationcalculation, while the circles are results of the quantization of the vibrations androtations of classical electrons of a Wigner molecule. The numbers ( nm ) show theoccupancies of the two vibrational modes of the system. The lower panel shows thepair-correlation functions of lowest-energy states for three angular momenta. Theenergy is in atomic units and ∆ E = E i − L ¯ hω , where ω = 1 is the confinementfrequency. a ring. The figure also shows pair correlation functions for three cases. For L = 54,the pair correlation function shows that the electrons are clearly localized: Fixing theposition of one reference electron fixes the positions of the three other electrons. Inthe cases L = 56 and L = 57, the localization does not seem to be as strong due tothe fact that these states correspond to vibrational excitations (for a more detailedanalysis, see [45]).Let us consider the effects of the electron spin on the many-particle spectrumand on the electron localization. Note that the Hamiltonian of the system does notdepend on spin (since there is no magnetic field or spin-orbit interaction). However, asalready noted in connection with the discussion of particles on a ring, the spin-degreeof freedom has an important role in making the total wave function antisymmetric.Moreover, it turns out that in the case of a long-range Coulomb interaction, thelocalized electrons interact with an effective exchange interaction as in the Heisenberganti-ferromagnet. In the case of a one-dimensional ring, this can be understood onthe basis of the half-filled Hubbard model with nearest-neighbor hopping[51, 41]. Theeffective Hamiltonian for electrons localized in a ring can be written as H eff = ¯ h I M + (cid:88) ν ¯ hω ν n ν + J (cid:88) (cid:104) i,j (cid:105) S i · S j (4)where I is the moment of inertia, ω ν is the eigenfrequency of the vibrational mode ν , J is an effective exchange interaction, and S i is the spin operator. The value of J becomes smaller when the localization gets more pronounced. In the case of theinfinitely strong contact interaction discussed in the previous section, J = 0 and the etal cluster, quantum dot and quantum rings J is always finite, leadingto finite spin excitations. Figure 6.
Many-particle energy spectrum for four electrons in a 2D quantumdot and in a quantum ring. The numbers next to the lowest-energy levels showthe total spin of the state.
Figure 6 illustrates the effect of the spin in the cases of a 2D dot and a quasi-1D ring with four electrons. In both cases, the lowest band corresponds to a rigidrotation of the electrons localized in a square. An energy gap separates these statesfrom the vibrational excitations. In both cases, the low-energy spectrum is similarand consistent with the model Hamiltonian. The antisymmetry requirement dictateswhich spin-state is allowed at the given angular momentum. Group theory can be usedto resolve the allowed states. We note that the low-energy state for the maximum spin( S = 2) appears at angular momentum 2, in accordance with the simple model forquantum rings shown in Fig. 4 above. The energy differences between the differentspin states in the ring is consistent with the Heisenberg Hamiltonian (last term in Eq.(4)). For the dot, the order of the spin states for angular momentum 4 is different,most likely due to the fact that in this case there is also an exchange interactionbetween the opposite corners of the square of the four electrons, omitted in the simplemodel.
6. Localization of vortices
In the previous section we showed that the electron localization in a quantum dotshows characteristic features in the rotational spectrum. Especially, in the case of a fewparticles the geometry of the localized molecule, i.e. the symmetry group, determinesthe periodic features of the spectrum as a function of the angular momentum. Thesame method can be used to study localization of vortices in 2D quantum dots. Astrong magnetic field will polarize the 2D electron gas in the quantum dot and putit in a rotational state. At the angular momentum L = N ( N − /
2, the rotating etal cluster, quantum dot and quantum rings L is subtracted from the spectrum to emphasize the details of the yrast line. Thefigure shows that below L ≈
210 the yrast line is a smooth function. Beyond thatvalue, the yrast line oscillates with a period of two, three and four. These oscillationsresult from the existence of 2, 3 or 4 localized vortices in the system. The lowestpoints correspond to a rigid rotation of the ring of vortices, while the higher energiescorrespond to vibrational excitations of the vortex system.
Figure 7.
Energy spectrum of 20 polarized electrons in 2D a quantum dotshowing the periodic oscillations arising from localization of two, three and fourvortices, when the angular momentum is increased. The hole-hole pair-correlationcorresponding one of the states is shown on the right, showing the localization ofthe three vortices (the reference vortex is fixed at the point of the arrow).
If the single-particle basis is restricted to the lowest Landau level, i.e. to thosesingle-particle states of the 2D harmonic oscillator that have no radial oscillations, wecan use particle-hole duality to further analyze the localization of vortices [53]. In apolarized Fermi system, we can describe any quantum state in the occupation numberrepresentation equivalently by the particles, or the holes. In our present system, theholes correspond to the vortices and, consequently, the hole-hole correlation functiondescribes the spatial correlation between the vortices. Figure 7 shows the hole-hole correlation for one of the states in the three-vortex region, showing clearly thelocalization of the three vortices at the corners of an equilateral triangle. Let us finallyremark that the theories of quantum Hall liquids [48, 49, 50] can be used to discoverthe similarity of vortex formation in fermion and boson systems [54]. etal cluster, quantum dot and quantum rings
7. Conclusions
This article is not meant to be a comprehensive review of correlation effects in smallsystem of electrons. Rather, we have looked at selected case studies where the smallsize and low dimension of the systems emphasizes the correlation and the collectivemotion of the electrons.In the first section, we showed that in small clusters of simple metals it isthe optimal shape of the electron cloud that dictates the overall geometry of thecluster, making the ions the electrons’ slaves. The electron-ion system behaves likea structureless plasma, where the electron-electron exchange and correlation energydetermines the shape of the cluster. In small systems, this correlation is so strongthat, in fact, it does not matter what kind of internal interaction the particles have,or what kind of physical model is used for the system, as illustrated in Fig. 1.A strong external confinement hinders the spatial deformation. At low densities orhigh rotation, the electrons tend to localize in Wigner molecules. The energy spectrumis then dominated by the rigid rotation and the internal vibrations of the molecule. Thelocalization also separates the spin-excitations from the vibrational charge excitations,as most clearly seen in quasi-1D rings. A model Hamiltonian consisting of rigidrotations, quantized vibrations and an anti-ferromagnetic Heisenberg model, describeswell the low energy spectrum of the system. All low-energy excitations are thuscollective excitations of strongly correlated electrons.At high magnetic fields the electrons in a quantum dot form a ’miniature’ quantumHall liquid of polarized electrons. In this case, the elementary collective excitations arevortices. Interestingly, the vortices have a similar energy spectrum as the localizedelectrons. For example, the spectrum of three localized electrons shows the samevibrational modes as the spectrum of three vortices. The vortices can be interpretedas holes in the occupied Fermi sea, and the hole-hole correlation function can be usedto confirm the localization of the vortices. [1] A.L. Fetter and J.D. Walecka,
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