Wave-vector dependence of spin and density multipole excitations in quantum dots

We have employed time-dependent local-spin density functional theory to analyze the multipole spin and charge density excitations in GaAs-AlGaAs quantum dots. The on-plane transferred momentum degree of freedom has been taken into account, and the wave-vector dependence of the excitations is discussed. In agreement with previous experiments, we have found that the energies of these modes do not depend on the transferred wave-vector, although their intensities do. Comparison with a recent resonant Raman scattering experiment [C. Sch\"uller et al, Phys. Rev. Lett {\bf 80}, 2673 (1998)] is made. This allows to identify the angular momentum of several of the observed modes as well as to reproduce their energies.


I. INTRODUCTION
The characteristic single particle and collective excitations of typical quantum dots (QD) are known to lie in the far-infrared (FIR) energy region, i.e., they have energies that, depending on the size of the dot, span the range from a few tens of meV to a fraction of meV. Experimental information about FIR spectra was first obtained from photon absorption experiments on InSb and on GaAs quantum dots 1,2 . Since the confining potential for small dots is parabolic to a good approximation, and in the FIR regime the dipole approximation works well, the absorption spectrum is rather insensitive to the number of electrons in the dot, measuring to a large extent only the center-of-mass excitations, which at non zero magnetic fields (B) correspond to the two allowed dipole transitions arising from each of the two possible circular polarizations of the absorbed light. Two limitations of the absorption process, namely that it is dominated by the L = 1 multipole of the incoming electromagnetic wave, and its insensitivity to the electronic spin degree of freedom, have motivated that theorists have been mostly concerned with the study of dipole charge density excitations (CDE). Yet, higher multipolarity CDE's have been discussed using a classical model 3 , a Hartree-random phase approximation method 4 , a classical hydrodynamical model 5 , and an equation of motion method 6 . Quadrupole L = 2 CDE's have also been addressed for the quantum-dot helium 7 .
The situation is changing with the use of inelastic light scattering to study QD excitations. This experimental technique is nowadays recognized as one of the more powerful tools to study the elementary excitations of low dimensional electronic nanostructures [8][9][10][11][12][13] , and it is contributing to a deeper understanding of the two dimensional electron gas [14][15][16][17][18]. Using polarization selection rules, it allows to disentangle CDE from spin density (SDE) and single particle excitations (SPE), and to observe them all in the same sample. Moreover, it offers the possibility of studying the wave-vector dispersion dependence of the excitations.
Measurements of Raman scattering on high-quality GaAs-AlGaAs quantum dots have been reported 11 . What make this experiment especially appealing is that sharp spin and charge density excitations have been measured in conventional backwards geometry as a function of the applied magnetic field B and of the transferred lateral wave-vector q. Previous studies were carried out at zero magnetic field 8,10 , or the experimental conditions were such that the spectra did not show a wave-vector conservation nor a clear polarization dependence 9 , hence it was not possible to resolve SDE and CDE from SPE, nor to record the spectra at predetermined q values. We attempt here a theoretical interpretation of these results based on the time-dependent local-spin density functional theory (TDLSDFT), addressing the description of high multipolarity spin and charge density modes of a QD, and incorporating in a realistic way the on-plane wave-vector dependence of these collective excitations.

II. TDLSDFT DESCRIPTION OF COLLECTIVE MODES IN QD
The dipole longitudinal response of quantum dots has been recently addressed in detail 19,20 . We sketch here how the method can be generalized to deal with other multi-polarities and the wave-vector degree of freedom.
The first task is to obtain the ground state (gs) of the dot solving the appropriate Kohn-Sham (KS) equations. The exchange-correlation energy density E xc (n, m), where n is the electron density and m the spin magnetization, constitutes a key ingredient of the method. It has been constructed from the results of Ref. 21 on the nonpolarized and fully polarized 2dEG using the two dimensional von Barth and Hedin prescription to interpolate between both regimes 22 .
Once the KS gs has been worked out, we have determined the induced densities originated by an external excitation field employing linear-response theory. For independent electrons in the KS mean field, the variation δn (0) σ induced in the spin density n σ (σ ≡↑, ↓) by an external spin-dependent field F , whose non-temporal dependence we denote as F = σ f σ ( r) |σ σ|, can be written as 23 where σσ ′ is the KS spin density correlation function. In this limit, the frequency ω corresponds to the harmonic time dependence of the external field F and of the induced δn (0) σ . Eq. (1) is a 2×2 matrix equation in the two-component Pauli space. In longitudinal response theory, F is diagonal in this space, and its diagonal components are written as a vector We consider first the external L-pole fields which cause, respectively, the charge and spin density L modes. For the monopole L = 0 mode, these fields are simply taken proportional to r 2 (see below). To distinguish the induced densities in each excitation channel they will be labelled with an additional superscript as δn (0,n) σ or δn (0,m) σ . The TDLSDFT induced densities are obtained from the integral equations where either A = n or A = m, and the kernel K σσ ′ ( r, r ′ ) is the electron-hole interaction. Equations (3) have been solved as a generalized matrix equation in coordinate space. Taking into account angular decompositions of χ σσ ′ and K σσ ′ of the kind , it is enough to solve them for each multipole separately because only modes with ℓ = ±L couple to the external L-pole field. One has where K n (x) is given by the hypergeometric function 24 π 2 F (1/2, n + 1/2; n + 1; x 2 ), and r > (r < ) is the greater (smaller) of r, r ′ . For a polarized system having a non zero magnetization in the gs, the ±L modes are not degenerate and give rise to two excitation branches with ∆L z = ±L, where L z is the gs orbital angular momentum. The induced charge or magnetization densities corresponding to density and spin responses are given by δn (A) = δn From them, the dynamical polarizabilities in the density and spin channels are respectively given by For each L value, taking into account both ±L possibilities we define α σ , correspond to the SPE. An analysis based on the use of the above multipole excitation operators r L e ±iLθ implies that no appreciable on-plane momentum q is transferred to the system, i.e., q ≈ 0. This will become apparent below. Even in this limit, some interesting features of the experimental spectra are reproduced. Moreover, it allows one to make contact with FIR photo-absorption spectroscopy. Yet, a more detailed analysis of Raman spectra calls for introducing the q dependence in a realistic way. A first attempt has been made in Ref. 9, although the analysis of the measured Raman spectra was carried out using a Hartree model which cannot address the spin degree of freedom on the one hand, nor take into account the contribution of charge and spin density collective modes to the scattering cross section on the other hand.
Hamilton and McWhorter 25 were the first in pointing out the important role played by spin density modes in the Raman scattering in GaAs. Their original formulation has been further elaborated by Blum 26 , and more recently the inelastic charge and spin density scattering cross sections have been discussed in terms of the charge S nn (q, ω) and spin S mm (q, ω) strength functions 27 (often called dynamic structure functions) where ω is the energy difference of the incoming and scattered photon ω i − ω s , andê i,s are the polarization vectors. We refer the reader to the review articles of Refs. 18,28 for a thorough discussion. The above expressions are deceptively simple, but this is somehow misleading, as simplicity arises from the approximations made to arrive at them 26,27 . Yet, they are often used to describe resonant Raman scattering in GaAs heterostructures 9,27,29,30 . These approximations might obscure the comparison of the calculated modes with these detected by Raman spectroscopy. We believe, however, that rather than testing the TDLSDFT description of charge and spin density excitations, it may manifest the limitations of theoretical schemes based on Eq. (6) to analyze resonant Raman scattering. It is worth to mention the application made by Wendler et al 31 to resonant Raman scattering in two electron quantum rings using a more general expression for the cross sections, of hopeless applicability to the N = 200 quantum dot described in Ref. 11.
To obtain S nn (q, ω) and S mm (q, ω) within TDLSDFT, instead of considering the response to multipole operators, one has to consider the plane wave operator e i q r involved in the inelastic scattering process. It is convenient to expand it into Bessel functions 24 : Depending on the q value, the number of terms in the expansion may be large, but the method is of direct applicability because the different L terms in the expansion do not couple. Physically, it is also sound to make the expansion, since the experimental results display quite distinct peaks whose multipolar character can, in some cases, be identified even at a transferred momentum as large as 0.8 × 10 5 cm −1 (see Fig. 2 of Ref. 11). Moreover, in the small q limit, the expansion of the Bessel functions leads to the multipole excitation operators we have previously considered. In particular, the r 2 operator used in the monopole case arises from the first non trivial term in the expansion of J 0 (qr). An r 2 term is also present in the quadrupole case, this time multiplied by the angular operators e ±2iθ . The TDLSDFT response to the plane wave operator can thus be obtained as in the multipole case substituting in Eq. (2) r L by J L (qr) and r 2 by J 0 (qr), and keeping as many terms in the expansion Eq. (7) as needed. A criterion to determine the number of terms to be considered is provided by the f-sum rule 20 . For a given q value, the f-sum rules of the plane wave operator and of each L component in Eq. (7) read (in effective atomic units): where n 0 ( r ) is the gs electron density. The maximum L value in the expansion has been fixed so as to fulfill the plane wave f-sum rule within 95 % or better. As a further numerical test, the second Eq. (8) has been used to check the accuracy in the calculation of the strength functions multipole by multipole.

III. RESULTS
As a case of study we present a theoretical interpretation of the results obtained in Ref. 11 for an N = 200 electron quantum dot of radius R = 120 nm in GaAs-AlGaAs. We have modeled the confining potential by the Coulomb potential created by a positively charged jellium disk of the same radius 32 . The only free parameter in the calculation is the number of positive charges in the disk, which has been set to N + = 404 to reproduce as many spin and density modes as possible at B = 0, with a special emphasis in the dipole SDE. We want to stress that this particular jellium disk plays no other role that creating a confining potential easy to generate and vary in a controlled way by simply changing N + . The question of whether the system is charged or not is misleading; after all, it could not be more 'charged' than any N electron dot confined by a parabolic potential. Image charges representing the gates and usually not considered in QD structure calculations will eventually make neutral the whole system 33 . Discarding a parabolic potential because of the large number of electrons in the dot, other confining potentials 9,34,35 and fitting parameter strategies might have been considered. However, a thorough testing of the confining potential for such a large dot would imply to obtain the charge and spin responses at B = 0 for several multipoles. Obviously, this is a very demanding task. A more elaborated search could have improved the results we are going to discuss, which in some cases are not in full agreement with experiment. Figure  1 shows the electron densities at B = 0, 3 and 6 T.
The choice of the spin dipole mode at B = 0 as the experimental quantity to be better reproduced in the fit is motivated by the emphasis we want to put in the spin channel results, and because for this mode two distinct branches with positive and negative B dispersions are seen in the experiment. The dipole CDE at B = 0 would have been a more conventional choice, but unfortunately its experimental value has not been reported 10,11 . It is worth to point out that even if there seems to exist a common belief that CDE's are well understood, for multipolarities different from the thoroughly studied dipole mode this belief does not stem from having confronted so far theory with real experiments. It is still an open question how quantitative is the agreement between theory and experiment when several CDE's have to be simultaneously described for the same QD.
The range of B values investigated in this work corresponds to filling factors larger than 3. Consequently, the use of other density functional approaches such as current density functional theory (CDFT) better suited at high magnetic fields 32,36,37 can be avoided. For a discussion of the difficulties one has to face to obtain the longitudinal response within time-dependent CDFT, we refer the reader to Ref. 38.
We first present the results obtained at q ≈ 0. The interest in studying this limit lies in the experimental observation 8,10 , thoroughly discussed at B = 0, that in QD's the energies of the excited modes do not depend on the transferred wave-vector q. This is at variance with the situation in nanowires and in the 2dEG, constituting a clear signature of the 'zero dimension' character of QD's. What changes with increasing q is the total strength (see the first Eq. (8)), and how it is distributed among the different peaks. We shall discuss these matters in the next Subsection. (2). They correspond to the two possible circular polarizations of the light absorbed or emitted in the excitation or deexcitation process. We have found that the spin peaks are rather fragmented, especially in the monopole case. However, they still are collective modes, with energies redshifted from the single particle ones due to the attractive character of the exchange-correlation vertex corrections.
We would like to draw the attention to the − type, low energy octupole SDE which is seen in Fig. 5 to carry an appreciable strength at ω ∼ 2.5 meV for B = 2 T. When a magnetic field is perpendicularly applied to a QD, it is well known that low energy modes in the density channel are dipole edge CDE's arising from intraband transitions, while bulk interband transitions lie at higher energy. That may change with increasing L, and it is easy to see that this is indeed the case for SDE's. An inspection to the KS single electron energies shown in Fig. 6 reveals that at high L's, interband electron-hole excitations are at lower energies than intraband ones. Since the electron-hole interaction is weak in the spin channel (only the exchange-correlation energy contributes to it), we have found that at B = 2 T the lowest energy octupole SDE is a mode built from interband electron-hole excitations. Still, one might consider it as an edge mode, as its existence is only possible because of the finite size of the system. When B increases further, the spin density edge mode has again a + polarization. In the L = 3, this happens at B = 3T. We have found that the low energy CDE is always a + type excitation, whereas the high energy CDE's are − type excitations arising from the corresponding component of F (n) . Figures 7 and 8 display the B dispersion of the more intense CDE's and SDE's, respectively. The cyclotron frequency appears as a peak in the calculated SPE (KS) dipole response, and we have not plotted it in Fig. 7. The solid symbols represent the experimental data 11 . We have connected with lines the more intense peaks obtained in the calculation of the strength, which displays some fragmentation, especially for high L and B values (see also Ref. 4). We recall that only for a pure parabolic confinement of frequency ω 0 and for the L = 1 mode in the dipole approximation, generalized Kohn's theorem 40 ensures that CDE's are distributed according to the classical dispersion laws Ω ± ω c /2, with Ω 2 = ω 2 0 + ω 2 c /4 and ω c being the cyclotron frequency. We also recall that the adiabatic TDLSDFT we are employing fulfills generalized Kohn's theorem 20 .
It can be seen from these figures that the experimental data are only partly explained, as not all the experimental modes are quantitatively described. In both spin and charge density channels, TDLSDFT reproduces the weak B dependence of the L = 0 mode found in the experiment at small B values. Our calculation confirms the L = 0, 1, and 2 multipolarity assigned in the experiment to the lower SDE's, but cannot identify the origin of the higher SDE, whose signal is weak and broad, as mentioned in Ref. 11. We will see in the next Subsection that including finite momentum transfer, as in actual experiments, does not greatly clarify the situation.
At B = 0, the energies of the L > 0 spin density excitations follow the simple rule E L ∼ LE 1 . We attribute this to the weakness of the electron-hole interaction in the spin channel. The prominent role played by the strong electron-hole interaction in the charge density channel causes that rule to fail for CDE's.
As a general trend, the strength carried by the positive B dispersion branch corresponding to the high L spin density excitations diminishes as B increases. We have also found that the spin strength becomes more fragmented with increasing L, whereas bulk and edge magnetoplasmons associated with the ±L excitations are better defined modes.
The positive B dispersion branches of the CDE's reveal a complicated pattern at intermediate B values, quite different from the expected classical one holding up to B ∼ 2-3 T, but that however fits a large set of the experimental modes. The behavior of these branches has an interesting quantal origin, namely the formation of well defined Landau bands for magnetic fields larger than a critical value. Above it, the more intense high energy collective peaks mostly arise from transitions between Landau bands whose index M differs in one unit, ∆M = 1. Since these bands are made of many single electron states with different ℓ values and energies rather ℓ independent if B is high enough 32 , this explains the otherwise striking quasi L-degeneracy of the plasmon energies, only broken by finite size effects and the L dependence of the electron-hole interaction. Other modes with ∆M = 2 build branches satellite of those formed by the more intense L-peaks, and are clearly seen in the calculation. Satellite branches of this kind appear even in the dipole case 2,41 , and are a clear signature of nonparabolic confinement 4,5,20 . We will see below how these branches emerge at high q and B values.
In contradistinction with the positive B dispersion branches of the CDE's, the negative B dispersion ones do not manifest the quasi L-degeneracy. While interband electron-hole excitations at the bulk of the dot are rather L independent as we have just mentioned, the negative B dispersion branches are built from intraband electron-hole excitations at the dot edge, and these are quite distinct for different L values (see Fig. 6, and Fig. 5 of Ref. 32 for instance).

B. Finite q results
The linear response to the multipole fields described before cannot tell what is the relative intensity of the different charge or spin density excitations. This limitation is circumvented using the plane wave operator for which S nn (q, ω) and S mm (q, ω) display the charge or spin density excitations with nonarbitrary relative intensity, allowing one to ascertain in each channel which L modes are more probably excited at given B and q values. This is clearly seen in Figs. 9 to 14. Figure 9 shows the CDE's and SDE's at B = 0 for selected q values used in Refs. 9 and 11 (we shall give q in 10 5 cm −1 ). Several interesting features show up in this figure. We see that for small q values the dipole mode takes most of the strength, and that for the q values employed in Ref. 11, the strength is exhausted by the modes with L ≤ 3. Another interesting observation, in full agreement with experiments, is that the peaks have no appreciable wave-vector dispersion 8,10 .
For a given L, Fig. 9 also reveals the mechanism by which the strength evolves with increasing q. Up to the q values of Ref. 11 only the lowest energy peak of each multipolarity is sizably excited, and with increasing q strength is transferred from dipole to quadrupole, monopole and octupole, successively. For larger q values, as those employed in Ref. 9, higher energy peaks of each multipolarity get predominantly excited. Conspicuous peaks corresponding to the second dipole and quadrupole modes, respectively, can be clearly seen at q = 5 and ω ∼ 16 meV and ∼ 17 meV. The same happens at finite B values, as it is shown in Fig. 10 for B = 1T. Figures 11 and 12 show the evolution with B of the spectra corresponding to the largest q value of Ref. 11. Figures 13 and 14 show the same for the largest q used in Ref. 9. As anticipated, Fig. 12 does not help identify the nature of the high energy SDE detected in the experiment. However, the results at higher q (Fig. 14) show at low B a broad distribution of the SDE strength centered around the energy of that experimental mode. One is tempted to speculate that the higher SDE seen in the experiment is just the envelope corresponding to our higher q spectrum, which is centered around the second dipole SDE. In this sense, it is worth to point out that our LSDA is essentially equivalent to a contact interaction in the spin channel, and thus it may underestimate finite momentum effects. Of course, a similar effect could also contribute to the broad features observed in the higher CDE's. The B = 6 T panels in Figs. 11 and 13 show that CDE's have a tendency to bundle, the energy spacing between bundles roughly being ω c . We have already discussed this effect at q = 0.
Finally, we have used our results at q = 1.32 to estimate the ratio r = (ω SP E − ω SDE )/(ω CDE − ω SP E ) for the more intense peaks. This ratio is a quantitative measure of the many-electron interactions in the dot 10 . At B = 0 we have obtained r ∼ 0.11, in good agreement with the experimental value 10 . This ratio decreases with increasing B; we have found that r ∼ 0.08 at B = 6T.

IV. SUMMARY
In this work we have thoroughly discussed spin and charge density modes of different multipolarity in GaAs-AlGaAs quantum dots, as well as their wave-vector dependence. This has allowed us to make a detailed comparison with experimental data obtained from resonant Raman scattering. In particular, our calculations reproduce the experimental finding that the excitation energies of the modes do not depend on the transferred wave-vector, although their intensities do. The ratio (ω SP E − ω SDE )/(ω CDE − ω SP E ) is also reproduced.
We have been able to compare the energies of several spin and density modes arising in the same dot. After fitting the value of the spin density dipole mode at zero magnetic field, the energies of the spin density modes up to L = 2 haven been quantitatively reproduced as a function of B.
The origin of the high energy spin density mode at q = 1.32 has not been elucidated by our calculations, although our results for larger q's predict a very broad distribution of strength centered around this experimental value. The analysis of the strength function at the experimental wave-vector q seems to indicate that no appreciable strength is carried by modes with L > 3, and that a broad structure consisting of peaks of different multipolarity L ≤ 3 and polarization appears between 1 and 3 T (see the appropriate panels in Fig. 12).
The top panel of Fig. 1 gives a hint about the difficulty to properly describe high L modes with rather simple confining potential models. While for L = 0 to 2 the excitation operator is probing the bulk region and part of the edge of the dot, for higher multipolarities it is only sensitive to its outermost edge structure 6 . Obviously, for a large N dot this region is very much influenced by the actual structure of the confining potential, and one should expect the larger disagreements between theory and experiment to appear for these modes.
In the charge density channel, the agreement between theory and experiment is more qualitative. At B = 0 one of the measured CDE's is between our calculated L = 2 and L = 3 modes and, as in the spin density channel, only the L = 0 mode is not appreciably dispersed with B. Yet, we have given an interpretation, and a fair quantitative description, of the peaks measured at intermediate B values which lie between the ω c and 2ω c lines. As indicated, we have given more weight in the fitting procedure to reproducing the SDE's.
Finally, we would like to point out that in spite of the difficulties in interpreting resonant Raman scattering in terms of spin and density modes arising only from excitations of the conduction band electrons, several features of the spectra are well described within timedependent local-spin density functional theory. A more quantitative description of some aspects of the experimental spectra would require to take fully into account the underlying structure of the system beyond the simple, idealized semiconductor model currently used to describe quantum dots, and likely a more realistic confining potential in the case of high multipolarities. FIG. 6. Single electron energies as a function of orbital angular momentum for B = 2 T. Our choice of B pointing towards +z favors that single particle states of negative angular momentum and upwards spin be occupied. To avoid dealing with single particle angular momentum quantum numbers which are mostly negative, the angular dependence of the single particle wave functions is written as e −iℓθ and hence, ℓ represents the orbital angular momentum changed of sign. The horizontal line represents the electron chemical potential. Full, upright triangles correspond to σ =↑ states, and the empty, downright triangles to σ =↓ states. Interband and intraband transitions with ∆ℓ = 2, 3 and 4 are represented to illustrate the energy crossing discussed in the text.