Far-infrared edge modes in quantum dots

We have investigated edge modes of different multipolarity sustained by quantum dots submitted to external magnetic fields. We present a microscopic description based on a variational solution of the equation of motion for any axially symmetric confining potential and multipole mode. Numerical results for dots with different number of electrons whose ground-state is described within a local Current Density Functional Theory are discussed. Two sum rules, which are exact within this theory, are derived. In the limit of a large neutral dot at B=0, we have shown that the classical hydrodynamic dispersion law for edge waves \omega(q) \sim \sqrt{q \ln (q_0/q)} holds when quantum and finite size effects are taken into account.


I. INTRODUCTION
Collective excitations induced in finite fermion systems by external probes have been extensively studied in the last years. Particular effort has been devoted to an understanding of the giant dipole resonance in nuclei, 1 and of the plasmon mode in metallic clusters. [2][3][4][5] Recently, a strong collective state has also been observed in quantum dot structures. 6,7 These collective modes all have in common the feature of being Lϭ1, Sϭ0 excited states induced by a dipolar external radiation which is the dominant component of the electromagnetic field when its wavelength is much larger than the size of the system. Dipole modes correspond to oscillations of protons against neutrons in nuclei, and of electrons against the positive background in metal clusters and dot structures. Whereas in nuclei the restoring force of the collective motion is the symmetry potential which acts differently on protons than on neutrons, in metal clusters and dots it arises from the Coulomb interaction between ions and electrons.
Collective states with multipolarity LϾ1 were predicted to exist, and have been observed in nuclei. At present, we have a complete systematics of quadrupole (Lϭ2) and octupole (Lϭ3) collective excitations in nuclei. 1,8,9 Multipole collective states in 3 He droplets have also been studied, [10][11][12] but have not been experimentally detected so far. In metal clusters, the predicted multipole states 13,14 also have not been observed. The basic reason is the experimental difficulties arising from the fact that free clusters are produced and analyzed on-fly, with the added difficulty in the case of 3 He drops of being electrically neutral systems.
Far-infrared absorption spectroscopy experiments on large radius quantum dots submitted to a static external magnetic field B have likely evidenced quadrupole excitations, 7 and an anticrossing between the Lϭ1 and 2 resonances, each of them split into two branches, one with negative and another with positive B dispersion. We recall that, for these systems, L has to be understood as the angular momentum about an axis perpendicular to the dot plane.
The collective spectrum of quantum dots has been ad-dressed in different ways. An explanation based on classical arguments can be found in Ref. 7 In Refs. 15-18, use has been made of an edge-magnetoplasmon model, whereas in Refs. 19 and 20 a Hartree random-phase approximation ͑RPA͒ method has been employed, and a Hartree-Fock RPA one in Ref. 21. Multipole modes have been considered in Refs. 15 and 20. All these approaches neglect the electron correlation energy, and therefore, no microscopic calculation of Lу1 modes exists which incorporates exchange correlation energy as well as quantum effects arising from finite size and shell structure of dots in a magnetic field. A workable, yet reliable formalism that takes into account all these effects is called for to describe these excitations. In this paper we present one of such possible formalisms. It is based on the equation of motion method within the framework of a local current density-functional theory ͑CDFT͒. 22 This functional theory is well suited to study electronic systems in presence of a magnetic field, and has been successfully employed to study ground-state ͑g.s.͒ properties of quantum dots. 23 This paper is organized as follows. We describe the equation of motion method in Sec. II, as well as the basics of the strength function and its moments ͑sum rules͒ needed to interpret the experimental and theoretical results. In Sec. III we present a rigorous solution of the dipole mode in the case of the full Pauli many-electron Hamiltonian with a parabolic lateral confining potential. The exactness of the dipole solution in this case, irrespective of the value of the magnetic field, has been discussed in Refs. 24 and 25. Here we have obtained this result in a way that yields an explicit expresion not only for the spectrum, but also for the eigenstates. In Sec. IV we present a variational approach to the description of multipole excitations in which exchange and correlation effects are taken into account within CDFT. We show in Sec. V that for neutral large dots at Bϭ0, these excitacions build an edge wave with dispersion relation of the type (q)ϳͱq ln(q 0 /q). Detailed numerical results are presented in Sec. VI for dots with different number of electrons, and the concluding remarks are presented in Sec. VII. Finally, an appendix contains some technical details.

II. EQUATION-OF-MOTION METHOD
We consider N electrons moving in the zϭ0 plane where they are confined by the dot potential V ϩ ͑r͒ with rϭͱx 2 ϩy 2 . On this system it may act a constant magnetic field in the z direction described by the vector potential A ជ ϭ1/2(Ϫy,x,0)B, and we suppose that the system can be described by the N-electron Pauli Hamiltonian: where m* is the electron effective mass which together with a dielectric constant ⑀ and gyromagnetic factor g* are characteristics of the semiconductor ͑for example, g*ϭϪ0.44, ⑀ϭ12.4, and m*ϭ0.067m e in GaAs͒, B * is the effective Bohr magneton B *ϭបe/(2m*c), ជ is the Pauli matrix vector, and V c is the electron-electron (e-e) interaction can be rewritten as where c ϭeB/m*c is the cyclotron frequency and l z i is the angular momentum operator about the z axis: Given the exact g.s. ͉0͘ of the N-electron system, it is possible to obtain the exact spectrum corresponding to a broad class of collective vibrations if one is able to find an operator O ϩ such that the following equation of motion is fulfilled: The state O ϩ ͉0͘ has an excitation energy ប, and the g.s. fulfills O͉0͘ϭ0. As the excited states have a well-defined angular momentum, so the operators O ϩ must have. Consequently, one has to solve Eq. ͑5͒ for each L value.
When a magnetic field acts perpendicularly on the dot, it causes a splitting of the excited Bϭ0 states into two branches of energy ប ϮL , each carrying an angular momemtum ϮបL over that of the g.s. It implies that, besides Eq. ͑5͒, the physically acceptable O ϮL ϩ operators have to fulfill with Lϭ1,2,3, . . . and L z ϭ͚ iϭ1 N l z i . If បL 0 is the angular momentum of ͉0͘, the states ͉ϮL͘ϵ O ϮL ϩ ͉0͘ have an angular momentum ប͑L 0 ϮL͒.
To ease the formulas that otherwise would be very cumbersome, from now on we shall be using effective atomic units, defined by បϭe 2 /⑀ϭm*ϭ 1. In this system of units, the length unit is the Bohr radius a 0 times ⑀/m*, and the energy unit is the Hartree times m*/⑀ 2 , which we call a 0 * and E H * , respectively. For GaAs we have a 0 *ϳ 97.94 Å and E H *ϳ11.86 meV.
It is obvious that to find these operators is in general as difficult a task as to solve the Schrödinger equation corresponding to the Hamiltonian Eq. ͑1͒ for the vibrational states, and one is led to solve Eq. ͑5͒ in an approximate way, one of these being, for example, the random-phase approximation. Another fruitful approximation, originally proposed by Feynman to describe density excitations of superfluid 4 He, 26 consists of making an ansatz on O ϩ ͉0͘. Acting upon ͉0͘ with Eq. ͑5͒ and projecting onto O ϩ ͉0͘, one obtains ͑7͒ Equations ͑5͒ and ͑7͒ are completely equivalent if ͉0͘ is the exact ground state, and O ϩ is the sought operator. The advantage of Eq. ͑7͒ is that we may look for approximate solutions of variational type, guessing O ϩ and obtaining ͉0͘ within a workable, yet accurate scheme, such as the localdensity approximation ͑LDA͒ at Bϭ0, or CDFT at B 0. To find these approximate solutions is the subject of Sec. IV. We shall also see that, remarkably, Eqs. ͑1͒ and ͑7͒ have exact solutions in the dipole case when the confining potential has a parabolic form. The excitation spectrum of the system is usually probed by different external fields, or given the well-defined angular momentum of the excited states, by a selected L-polar component of the field. For an excitation operator F representing it, a useful, often experimentally accessible quantity, is the so-called strength function where E n and ͉n͘ are, respectively, the excitation energy and the excited state, and the sum or integral in the case of continuum spectrum extends over all excited states of the system. Of special interest are some energy moments of the strength function which we shall call sum rules ͑SR's͒. They are the m 1 and m 3 moments, which can be also written as These SR's have been extensively studied in the literature. 1,8 For the present purposes it is enough to recall that, if only one excited state is contributing to S(E), E 3 ϵ(m 3 /m 1 ) 12 coincides with the corresponding excitation energy. In a more physical situation, whenever the strength is concentrated in a narrow, ''high''-energy region, E 3 is a fair approximation to the resonant energy. We call E 3 the scaling energy because m 3 can also be obtained by scaling the ͉0͘ wave function as and then carrying out the following derivative:

͑12͒
The m 3 sum rule can be obtained either from Eq. ͑10͒ or ͑12͒. The latter allows one to identify m 3 as the restoring force that causes the collective vibration, and, consequently, to identify m 1 with the collective mass parameter. We want to stress that not all possible approximation schemes fulfill the m 1 and m 3 sum rules in the sense that a direct evaluation of Eq. ͑9͒ yields the same result as Eqs. ͑10͒. A test on the consistency of the approximations made to get the g.s. and excitation spectrum, is the fulfillment of these SR's, especially of m 1 , which is very model independent.

III. EXACT SOLUTION OF THE DIPOLE MODE FOR THE PARABOLIC CONFINING POTENTIAL
It has been frequently argued in the literature that V ϩ (r)ϭ1/2 0 2 r 2 can be a good approximation of the confining potential. That is the case, for example, when the number of electrons in the dot is small as compared with the number of positive ions N ϩ that produce the confining potential. 27 If that approximation holds, it is easy to check that the solutions of Eq. ͑7͒ for the Hamiltonian Eq. ͑3͒ and Lϭ1 are The corresponding frequencies are and it is easy to verify that The states O Ϯ ϩ ͉0͘ are normalized to unity and orthogonal, carrying an angular momentum (L 0 Ϯ1͒. This exact result stems from the translational invariance of the electronelectron interaction for which ͓ ͚ iϽ j N V c (͉r ជ i Ϫr ជ j ͉),P͔ ϭ0, and consequently it is independent of the e-e interaction, provided it is local. 25 It is a simple matter to check that the states ͉Ϯ1͘ exhaust the m 1 and m 3 SR's for the dipole operator Dϭ ͚ iϭ1 N x i , and that the dipole strength is equally distributed between them. One obtains and, for example, It is interesting to notice that in the Bϭ0 limit, ϩ ϭ Ϫ ϭ E 3 (D) ϭ 0 . This result is independent of the number of electrons in the dot, in agreement with the generalized Kohn theorem. 28 The parabolicity of the potential is expected to break down when the number of electrons in the dot increases, and the electronic density extends up to the edge of the dot. Departure of V ϩ (r) from the parabolic law originates an N dependence in Ϯ and E 3 (D). The exact solution to Eq. ͑5͒ can be used to obtain the static dipole polarizability of the dot ␣(D), which is twice the m Ϫ1 (D) sum rule: 1,8 Consequently, in the parabolic potential approximation the dipole polarizability is independent of the magnetic field. We expect ␣(D) to be B dependent in the case of a more general confining potential. Using that 27 we obtain The R dependence could have been anticipated from a dimensional analysis.
Let us finally discuss the nature of the dipole modes with regard to their geometrical shape. If we consider the transition density associated with the excited states ͉Ϯ1͘ where is the N-electron density operator it is easy to obtain where (r) is the g.s. electronic density, and the prime denotes the r derivative. This transition density is characteristic of an edge excitation.

IV. VARIATIONAL SOLUTION FOR GENERAL AXIALLY SYMMETRIC POTENTIALS
When the confining potential is no longer parabolic, irrespective of the value of the magnetic field and of the multipolarity of the mode, the equation of motion ͑5͒ cannot be solved exactly, and we resort to an approximate method based on Eq. ͑7͒. In the dipole case, the natural guess for the where b Ϯ are variational parameters and a Ϯ have to be determined from the normalization condition. This process can indeed be carried out, and one obtains the dipole spectrum of the system. At this point, we consider it more convenient to generalize the Q and P operators in such a way that the calculation can be done for any L value. To this end, we have first taken the following Q L : This choice is inspired in that (kr) L e iL is the small k expansion of the function J L (kr)e iL , which is the restriction to the zϭ0 plane of the general solution of the Laplace equation in cylindrical coordinates. J L is the L Bessel function of the first kind. 29 Next, we have taken as partner of Q L in the O ϩ expression the operator where the i-particle index is implicit in the particle and momentum coordinates. This choice is again guided by the exact dipole case, since the conmutator ͓H,Q L ͔ yields a combination of Q L and R L , which reduces to the one we have found in Sec. III when Lϭ1.
We have thus considered as O ϮL ϩ operators the following combinations: Equations ͑31͒ reduce to Eqs. ͑28͒ for Lϭ1, and the proposed operators fulfill Eq. ͑6͒. These modes are edge modes whose transition densities read To minimize Eq. ͑7͒, it is necessary to have a description of the g.s. ͉0͘. We have taken for it the Kohn-Sham sate built from single-particle ͑s.p.͒ wave functions obtained in the framework of the CDFT of Ref. 23. We refer the reader to that work for the details. For the present purposes, it is enough to recall that the s.p. wave functions nl (r,) are separable in r and variables as nl ͑r,͒ϭu nl ͑r͒e Ϫil , ͑33͒ with lϭ0,Ϯ1,Ϯ2, . . . being the orbital angular momentum of the s.p. state. Upon minimization, one obtains Besides the particle and kinetic-energy densities as well as the E L function where F is the hypergeometrical function. 29 The density (r) represents a sort of centrifugal kinetic-energy density. It can be shown that in the zeroth-order Thomas-Fermi ͑TF͒ approximation (r)ϭ 1 2 (r), similar to the (r)ϭ 2 3 (r) relation holding in three dimensions. 30 The function j p (r) is the paramagnetic current density. 23 It is worthwhile to see that, at Bϭ0, ␥(r) vanishes due to time-reversal invariance. Its contribution at high magnetic fields is crucial to have wellbehaved B-dependent collective energies. Equations ͑34͒-͑42͒ constitute the main outcome of the present section, and will be used in Sec. VI to obtain numerical results within CDFT.
The goal of describing the multipole modes in a situation as general as possible makes these expressions look very cumbersome at first sight. However, the following can be noted.
͑i͒ For Lϭ1 and the parabolic V ϩ , one recovers the exact solution.
͑ii͒ For any L and zero magnetic field, b Ϯ ϭ1/ 3 and O ϮL ϩ ͉0͘ merge into a degenerate state of energy This is precisely the Bϭ0 scaling energy E 3 (Q L ), since m 1 is actually the general m 1 (Q L ) sum rule, and, as is outlined in the Appendix, m 3 reduces to the zero magnetic field m 3 (Q L ) sum rule making ␥(r)ϭ0. This result is the variational analog of the case we have discussed for the dipole mode in a parabolic V ϩ . ͑iii͒ In the Lϭ1 case, irrespective of B, all terms in m 3 but m 3 (ϩe) vanish, and for any axially symmetric ionic potential V ϩ (r) we find which reduces again to the exact case if V ϩ ϭ 1 2 0 2 r 2 . ͑iv͒ For any B, the states ͉ϮL͘ and their energies satisfy the m 1 sum rule, i.e., This implies that, although O ϮL ϩ ͉0͘ are only approximate solutions to the L-polar excitation spectrum, within our method there are no other excited collective states corresponding to that multipolarity built as coherent superposition of one electron-hole states. The sum rule is also fulfilled. Moreover, the square transition amplitudes to the ͉ϮL͘ states are equal, The fulfillment of m 1 and m 3 constitutes a rather stringent test on our variational method. ͑v͒ It is worth noticing that in all cases, there is no explicit contribution to these formulas arising from the exchange correlation energy terms in the CDFT Hamiltonian. This is because Q L is a solution of the two-dimensional Laplace equation. A similar result holds in three dimensions within the LDA if one takes for Q L a solution of the corresponding Laplace equation. 31

V. EDGE WAVES IN NEUTRAL LARGE DOTS AT B‫0؍‬
It is possible to use the scaling energy E 3 , which is a good approximation of the collective excitation energy at Bϭ0, to obtain the dispersion relation of the edge waves in the case of a neutral large dot. Under these conditions, the g.s. electronic density is constant everywhere apart from a narrow region along the border of the disk. Let 0 ϭN/(R 2 ) be that density, and let a be the width of the edge region. Using the zeroth-order TF approximation to the kinetic and centrifugal kinetic-energy densities ϭ 2 , ͑53͒ ϭ 1 2 , one obtains, from Eqs. ͑38͒ and ͑40͒-͑42͒, with zϭ1ϪO(a/R) at least, and which diverges at zϭ1. Physically, this divergency is associated with the known divergency of the electric field at the edge of the disk in the case of a constant electron density; 32 see below. Thus F L (z) can be written as a function of F 0 (z), which is the elliptic function K(z). 29 . ͑59͒ Using this, we obtain

͑60͒
On the other hand, F 1 (z) can be related to the electric field E(r) generated by the electrons at the edge of the disk: The divergency of E(r) at rϭR is removed when one considers the existence of the edge width: 32

͑62͒
where ␥ is a constant that depends on the precise way the electronic density goes to zero at R. Thus

͑63͒
If Lӷ1 but still NӷL, that expression can be further elaborated since where C is the Euler constant 0.5772 . . . . Introducing the wave vector qϭL/R, and keeping only the leading q term, we obtain where q 0 ϭ␥/(a␤) and ␤ϭ0.964. Result ͑65͒ can also be obtained using classical hydrodynamics. 32 It is worthwhile to notice that the above-(q)model differs from that obtained within the edgemagnetoplasmon model ͑Refs. 15-17͒: which is ͱ2/3 times that of the two-dimensional plasma frequency. We would like to stress that, to obtain the ͱqln(q 0 /q) dispersion relation, it is crucial to take into account width effects in the electronic density. These effects are important in quantum dots, where the number of electrons usually is much less than the number of ions, and in any case the electron density has a non-negligible edge region.

VI. NUMERICAL RESULTS
We have applied the method of Sec. IV to dots made of Nϭ6, 20, 30, 42, and 56 electrons. We have taken the values of g*,⑀ and m* indicated after Eq. ͑1͒, and N ϩ ϭ125, i.e., a positive density of ϳ4ϫ10 11 ions/cm 2 . This corresponds to a GaAs disk of about 1000-Å radius modeling the positive background.
The g.s. of the dots has been obtained using the CDFT of Ref. 23. We checked that we reproduce their results when we use the same V ϩ . For large dots and intense magnetic fields, sometimes one has to face severe convergence problems in the solution of the Kohn-Sham equations. Rather than a deficiency of the numerical algorithm, we consider it as a consequence of an inherent characteristic of the system under study, namely the existence of a very dense s.p. energy spectrum. To overcome it, on the one hand, and to carry out the calculations under conditions closer to the experimental ones, on the other hand, we have found it convenient to approach the description of the collective spectrum as the low-temperature limit of the results obtained from the finite temperature generalization of the formalism of Sec. II ͑see Ref. 33͒, and of the Kohn-Sham equations ͑see, for example, Ref. 34͒. Consequently, the numerical results we discuss below have been obtained at a temperature Tϳ 1-2 K. A comparison with several cases in which the Tϭ0 calculation is easy to converge, allows us to state that the small temperature we use does not influence the results here presented. Thermal effects on the collective spectrum of quantum dots will be described in detail elsewhere. Figures 1 and 2 show the electronic densities corresponding to dots with Nϭ6, 30, and 56 electrons for Bϭ0, and for Bϭ5 T, respectively. Figures 3-5 display the B-dependent, multipolar spectrum of the same dots up to Lϭ4. This is the interesting region where the crossing between ϩL and ϪL Ј branches may occur and has been experimentally observed. The energies are drawn in meV, and the magnetic field in T. These figures show that as the ϪL energies go to zero, they may reach a value comparable to the electron-hole s.p. energy difference at a rather moderate B value. When this happens, the collective state lies within the s.p. excitations region, and experiences a strong Landau damping, losing its collectivity and eventually being washed out. This is one of the reasons why the ϩL branch has been experimentally observed up to higher B values than the ϪL branch. 7 The structures appearing along the LϾ1 curves are due to drastic changes in the particle, kinetic and paramagnetic current densities of the dot arising from the effect of B through the exchage-correlation energy, which has a profound influence on them ͑see Fig. 3 of Ref. 23͒. The structures in the ϮL branches roughly correspond to values of B at which the total g.s. spin has a minimum. For example, for Nϭ6 we have that 2S is equal to 2 at Bϭ1 T, to zero at Bϭ2 T, and to 2 at Bϳ3 T. For Nϭ56, one has that 2Sϭ13 at Bϭ4 T, 7 at Bϳ5 T, and 8 at Bϭ6 T. For the Nϭ6 dot, the rising of the ϩLϾ1 curves at Bϳ5 T is due to the full alignment of the electron spins. No similar risings show up for Nϭ30 and 56 because for them, the alignment occurs at B values higher than those displayed in the figures.
It can be seen from Figs. 3-5 that the crossing between ϩ1 and ϪL branches does not follow a clear N systematics. We have also plotted in Fig. 4 the E 3 (Q L ) energies for Lϭ1 and 4 ͑dashed lines͒. The scaling energy is reproducing ϩL to within 10-20 %, since the ϪL branch is contributing to m 3 ͑Q L ) with the same weight as the ϩL one, and the negative B-dispersion energies are going to zero rather slowly.
We also display in Fig. 5 ͑dashed lines͒, the collective energies obtained for a parabolic potential whose 0 has been fixed to 5.6 meV in order to reproduce the 1 energy at  Bϭ0. It is worth noticing that this value does not equal the one which fits the Coulomb potential generated by the 1000-Å-radius disk charged with N ϩ ϭ125 ions near the origin, which is 4.4 meV, and consequently, 0 has to be interpreted here as an effective parameter to reproduce the dipole energy at zero magnetic field. For Lϭ1, to the scale of the figure both calculations coincide. It may be seen that the B slopes of the ϮL branches for Lϭ2 -4 are roughly the same for the parabolic and disk confining potentials.
At Bϭ0, Figs. 6 and 7 show, for Nϭ6 and 56, respectively, the different relative contributions to m 3 coming from kinetic and Coulomb energies, as a function of L. The Coulomb energy is also decomposed into e-e and dot-electron (ϩe) energies. These figures show that, for small L values, m 3 is dominated by the (ϩe) contribution, the kinetic and (e-e) ones being of minor importance. However, for a fixed N the kinetic contribution eventually takes over the Coulomb energy contribution. This occurs at an angular momentum L cr which increases when N increases. Since Q L behaves as ϳr L , the higher the L, the more external its influence on the electronic density. This means that for large enough L, it just acts on the s.p. wave functions having the larger l angular momentum and radial quantum number n. Consequently, it no longer generates collective but s.p. excitations. As collectivity has its origin in interparticle effects, it was argued in Ref. 14 that L cr roughly represents the largest L collective mode the system can sustain, since for LϾL cr the restoring force represented by m 3 is basically determined by an independent particle property like the kinetic energy. This criterion yields L cr ϳ 4, 6, and 8 for Nϭ6, 30, and 56, respectively.
In Fig. 8 we represent the energy of modes with Lϭ1 -4 at Bϭ0, as a function of the number of electrons in the dot. The ratios 3 / 1 and 2 / 1 have, for NϾ20, average values 1.68 and 1.37, respectively, instead of 3 and 2 as corresponds to the harmonic-oscillator sequence. It is apparent from that figure the N dependence of L . As we already discussed, only for the dipole mode in the case of a parabolic confining potential does one have 1 ϭ 0 . The deviation FIG. 5. Same as Fig. 3 for Nϭ56. The dashed lines represent ϮLϾ1 obtained using a parabolic potential instead of the one generated by the disk.  from that rule becomes more and more important as the number of electrons in the dot increases. This was already discussed in Ref. 19. The N dependence of LϾ1 is more complex, and this is due to the interplay between kinetic and Coulomb energy contributions to the excitation energy. It can be easily understood from Eq. ͑58͒, which shows that the contribution of the kinetic energy term is important for small N and increases with increasing L. For fixed L, the Coulomb energy eventually dominates, and LϾ1 depends on N in a way similar to the dipole mode.
Finally, we have also studied the collective spectrum of the Nϭ6 dot in the parabolic potential with 0 ϭ5.6 meV, and have found that, as expected, the LϾ1 energies at Bϭ0 depend on N. For example, 2 decreases from 8.2 to 7.5 meV, 3 decreases from 10.9 to 9.0 meV, and 4 decreases from 14.3 to 10.4 meV when one goes from Nϭ6 to 56.
So far, we presented a systematic study carried out under well-defined conditions which could render a comparison with the results of a given experiment difficult, since there are several variables that have to be fixed at the experimental values to permit a sensible comparison. To end this section, we compare some results obtained within our formalism with the experimental data of Ref. 7. This is possible only in part because, in that work, the Lϭ1 and 2 well-defined branch were obtained for a dot made of a large number of electrons, Nϭ210. At present, this is too large a value for us to deal with microscopically. Let us remind the reader that the s.p. wave functions behave near the origin as r ͉l͉ ͑see, for example, Ref. 24͒, and since the s.p. levels are nondegenerate when B 0, large N's imply large s.p. angular momenta. Consequently, we have only attempted to describe the Nϭ25, Rϭ1000-Å dot. This can be inferred from the value of the dipole energy Demel et al. found at Bϭ0, that N ϩ ϳ 28. For this dot, the ratio 2 / 1 they obtained is around 2, a value we are unable to reproduce, whereas for the Nϭ210 dot it is ϳ1.5, in better agreement with our systematics. Nevertheless, it can be seen from Fig. 9 that ϩ1 and E 3 (Q 1 ) nicely reproduce the positive B-dispersion branch, which is the only one for which a detailed comparison is possible.
We notice that the influence of the dot-dot interaction has not been considered in the comparison with the experiment. That effect was estimated in Ref. 25, and it can be shown to have a negligible influence on the frequency of the collective modes for the experimental dot array of Ref. 7 because the dot-dot distance is much larger ͑about ten times larger͒ than the dot radius.

VII. SUMMARY
In this work we used a variational approach similar to the one proposed by Feynmann in the case of liquid 4 He, to describe the multipole spectrum of quantum dots. One of the merits of the method is that quantum and finite-size effects can be taken into account. It may be easily applied to dots hosting several tens of electrons. Rather than in the method itself, this limitation has its origin in technical difficulties inherent to current microscopic approaches to handle a large number of electrons in intense magnetic fields.
The ground state of the dots has been described within current density functional theory. This has permitted us to incorporate, in a description of the spectrum, an exchangecorrelation energy contribution that takes naturally into account current effects appearing at nonzero magnetic fields.
We presented a systematic description of edge modes up to Lϭ4 in the region of interest to describe level crossing at nonzero magnetic fields. We gave explicit formulas for the m 1 and m 3 sum rules corresponding to the general multipole operator Eq. ͑29͒, which reduce to very simple expressions in the dipole case, Eqs. ͑A23͒. These sum rules are exact within CDFT, and may be of interest to check the accuracy of any detailed calculation of the L-mode strength, similarly to the way they are currently used within time-dependent local-density approximation. 35 Besides this practical application, it is worth noticing that there are few studies in the literature of an m 3 sum rule corresponding to a physical situation where time-reversal invariance is violated ͑see, for example, Refs. 36 and 37 for the three-dimensional polarized electron gas͒.
For large neutral dots at zero magnetic field, we showed that the classical hydrodynamic dispersion law for edge waves (q) ϳͱq ln (q 0 /q) holds when quantum and finitesize effects are taken into account. Finally, we also showed that in the case of a parabolic potential, the dipole mode can be exactly solved, yielding the well-known classical formula for Ϯ1 . The exactness of the dipole collective spectrum was stressed in Refs. 24 and 25. Here we obtained it in a different way, and went a step further explaining the structure of the ͉Ϯ1͘ collective states. This allowed us to obtain a variational solution valid for any axially symmetric lateral confining potential.

ACKNOWLEDGMENTS
It is a pleasure to thank Luis Brey for a useful correspondence. This work was performed under Grant Nos. PB95-1249 and PB95-0492 from CICYT, SAB95-0388 from DGID, Spain, and GRQ94-1022 from Generalitat of Catalunya.

APPENDIX
In this appendix we give some hints about how to derive Eqs. ͑34͒-͑42͒, and the sum rules m 1 and m 3 corresponding to the external one-body operator Q L of Eq. ͑29͒, which are given in Eqs. ͑38͒ and ͑51͒. The details of the method we use here can be found in Refs. 1,8,14  Splitting the Hamiltonian Eq. ͑3͒ into a one-body term H 0 and a two-body term Vϭ͚ iϽ j V (͉r ជ i Ϫr ជ j ͉), and using that Q L is a local operator which commutates with V, one obtains ͓H,Q L ͔ϭ͓H 0 ,Q L ͔ϭϪiLR L ϩ 1 2 L c Q L , ͑A2͒ where we used the conmutation relations ͓T,Q L ͔ϭϪiLR L , ͓L z ,Q L ͔ϭLQ L . ͑A3͒ From Eq. ͑A2͒ and it is then inmediate to derive, for the first two terms of Eq. ͑A1͒, the results ͗0͉ †Q L ϩ ,͓H,Q L ͔ ‡͉0͘ϭ2L 2 ͵ r 2LϪ2 ͑r ͒dr ជ ϭ2m 1 ,

͑A11͒
The remaining two terms in Eq. ͑A9͒ give It is seen that the exchange-correlation energy does not give any explicit contribution to m 3 (ee). However, it affects g.s. magnitudes like (r) and the other densities, thus implicitly influencing all these quantities. The commutator ͓O ϩL ϩ ,O ϩL ͔ of the denominator of Eq. ͑7͒ is easily evaluated to be ͗0͉͓O ϩL ϩ ,O ϩL ͔͉0͘ϭa ϩ 2 ͗0͉͓Q L Ϫib ϩ LR L ,Q L ϩ ϩib ϩ LR L ϩ ͔͉0͘ ϭ4a ϩ 2 ͑b ϩ m 1 ϩb ϩ 2 L 2 ⌫͒. ͑A21͒ The cubic energy weighted sum rule Eq. ͑10͒ for the external operator Q L is given by where we employed Eq. ͑A2͒. Using the previous result it is easy to recover the expression Eq. ͑51͒, which at Bϭ0 reduces to m 3 ͓Eq. ͑39͔͒ with ␥(r)ϭ0.
Let us finally indicate that, for the dipole operator, the m 1 and m 3 sum rules for any value of B and an axially symmetric confining potential V ϩ have the simple expressions