PHYSICAL REVIEW B, VOLUME 63, 125334

Ground state and far-infrared absorption of two-electron rings in a magnetic field

Antonio Puente and Llorenc¸ SerraDepartament de Fı´sica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain

~Received 29 June 2000; published 13 March 2001!

Motivated by recent experiments@A. Lorke et al., Phys. Rev. Lett.84, 2223 ~2000!# an analysis of theground state and far-infrared absorption of two electrons confined in a quantum ring is presented. The heightof the repulsive central barrier in the confining potential is shown to influence, in an important way, the ringproperties. The experiments are best explained assuming the presence of both high- and low-barrier quantumrings in the sample. The formation of a two-electron Wigner molecule in certain cases is discussed.

DOI: 10.1103/PhysRevB.63.125334 PACS number~s!: 73.21.2b, 72.15.Rn

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I. INTRODUCTION

The possibility of confining a controlled number of eletrons in artificial nanostructures at the interface of two seconductors, achieved in recent years,1 has attracted muchinterest. An obvious reason is the potential applicationthese systems to submicron electronics technology but, fthe theoretical point of view, much importance is attributto their character of novel quantum systems in which nphysics, as well as analogies with atomic clusters, atomsnuclei, may be found. The most widely studied systems hbeen the so-called ‘‘quantum dots,’’ in which a certain nuber of electrons form a two-dimensional compact islamodeled by a confining potential of parabolic type~for smalldots! plus the mutual electron-electron interactions. Telectronic islands have been also formed in the shaperings. In fact, mesoscopic rings were produced by Dahlet al.and their magnetoplasmon resonances measured for differing widths.2 Subsequent theoretical models, using semicsical methods3 and density functional theory,4 provided agood interpretation of the measured ring spectra. Thereexist approaches based on exact diagonalization methodvery small numbers of electrons that have addressed thecalled ‘‘persistent current’’5 and the optical excitations fovery narrow rings@approaching the one-dimensional~1D!limit # as well as for dots with a repulsive scatterer cente6,7

Besides, a description of the optical properties in twelectron rings based on an analytical treatment of the grostate was given by Wendleret al.8

Very recently, Lorke and co-workers reported the fabriction, using self-assembly techniques, of electron rings intrue quantum limit, not affected by random scatterers, ctaining two electrons.9 The far-infrared~FIR! spectra, as afunction of the magnetic field, has shown the appearancelectronic excitations peculiar for a ring geometry, as wella magnetic-field induced transition in the ground-state anlar momentum. The two-electron system is the smallestwith electron-electron interaction and its reduced size plait within the reach of essentially exact methods, such asHamiltonian diagonalization, in a basis of functions metioned above. In this paper we have followed an alternatalthough also an exact method, based on the solution ofSchrodinger equation by discretizing the coordinate spacea uniform grid of points. The method is therefore free frobasis selection and truncation although its limitation lies,viously, in the number of spatial points. This scheme c

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also be used to obtain the dynamical properties by solvithe time-dependent Schro¨dinger equation and monitoring theevolution of the relevant physical quantities with time. Ithis paper we have addressed the linear-response regimesmall amplitude oscillations. However, one of the potentiaties of the real-time approach is the study of nonlinear~largeamplitude! dynamical processes. By finding the exact soltion, one can easily quantify the effect of the electroelectron interaction, as well as check the validity of approxmate descriptions, such as density-functional theory. In faan analysis of the experiment based on the local spin-denapproximation~LSDA! in a symmetry-restricted approachhas already been presented in Ref. 10.

An important input to the theory is the external confininpotential acting on the electrons. Since we aim for a direcomparison with experiment, the strategy should be to intduce a reasonable model potential, depending only on aparameters, and to explore the exact solution as a functionthese parameters. Our approach has been to take the paraparameterv0 and ring radiusR0 as guessed in Ref. 9 andexplore the effect of the central repulsive barrier by solvinfor a low-, an intermediate-, and a high-barrier ring. Witthese barriers the system changes from a compact dotstructure to a well-developed ring one. We show that tbarrier height modifies the ring properties and that the eperiments are best explained assuming a mixed sample ctaining both low- and high-barrier rings.

In recent papers, Yannouleas and Landman,11 as well asKoskinenet al.12 have discussed the electronic localizatioin the relative frame, known as the formation ofWigner mol-eculesin two-electron quantum dots and in rings with sielectrons, respectively. Both groups have addressed the plem by calculating the low-lying states and discussing therotorlike structure. This crystallization is in good agreemewith the prediction of mean-field~MF! theories like theHartree-Fock or the local-density-functional approaches. Walso comment on the formation of a Wigner molecule in thhigh-barrier rings by comparing the conditional probabilitdistribution with the LSDA density. In this case, the LSDApredicts a symmetry-broken ground state that nicely agrewith the electronic distribution built from the conditionaprobability density of the exact solution to the Schro¨dingerequation.

The structure of the paper is as follows. Section IIdevoted to the exact solution for the ground state and SIII, to the corresponding exact solution for the dynamics.Sec. IV we discuss the comparison with the mean-field aproach and the formation of the Wigner molecule. Finallthe conclusions are presented in Sec. V.

©2001 The American Physical Society34-1

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ANTONIO PUENTE AND LLORENCSERRA PHYSICAL REVIEW B63 125334

II. GROUND STATE

A. Hamiltonian and resolution method

The Hamiltonian of the two-electron system, in a manetic field, using effective units,13 is

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whereA is the vector potential,g5e/c ~we assume Gaussian magnetic fields! and V(r ) is the confining external potential. The electronic positions are restricted to thexy plane,i.e., r[(x,y), and we consider a constant and perpendicumagnetic fieldB5Bez , described in the symmetric gaugwith a vector potentialA(r )[B/2(2y,x). The last piece isthe Zeeman term, depending on the total spinS5s11s2, theeffective gyromagnetic factorg* , and electronic effectivemassm* .14

The ring external potential is modeled by the followincircularly symmetric piecewise function,

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In this expressionR0 is the ring radius andv0 gives thecurvature around the minimum atR0. Fixing a barrier heightVb at the origin determines the parametersc andd, with theadditional requirement of smoothness atR0.15

Since Hamiltonian~1! depends on spin only through thZeeman term, the spin part of the wave function is giventhe well-known singlet and triplet combination of spinorOur method to obtain the spatial wave function is basedthe only assumption that its gradient and Laplacian canobtained by using finite difference formulas in a spatial gof uniformly spaced points. Associating a macroindexI[(x1 ,y1 ,x2 ,y2) to each point and discretizing the derivativoperators~we use five-point formulas!, the Schro¨dinger ei-genvalue equation transforms into the linear problem

HIJCJ5EC I . ~3!

It is worth to mention that the interaction and potential terare diagonal~local! and only the kinetic contribution givenondiagonal elements. Besides, since the derivatives invonly a finite number of points~we have typically used fivepoint formulas! the Hamiltonian matrix is very sparse. Whave solved Eq.~3! for the ground state using the iterativimaginary time-stepmethod.16 In principle, one can also obtain excited states by introducing the constraint of orthonalization to the lower eigenvectors, although the computional effort increases very rapidly. A very usefoptimization is introduced by considering only symmet~singlet! or antisymmetric~triplet! wave functions with re-spect to the exchange of the two-particle positions, sinceallows us to reduce, by a factor two, the wave-functionmension.

The above method has proved to be quite stable andconvergence with the number of points very fast. Typicawe have used fromN1'20 up to'50 points for the discreti-zation of a single dimension. Notice that the wave-functdimension (N1

4) would normally imply that the Hamiltonian

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in the ground-state energy is found.

B. Results

In this subsection we present the results for the groustate of rings havingR051.4 a0* and v051.1 H* . Thecorresponding physical values are, approximately, 14 nm13 meV, and are taken as reasonable values from the exment. Three different barrier heights are considered,Vb50.75, 3, and 12H* . As shown in Fig. 1, these three vaues correspond, going from low to high barrier, to decreascentral densities. In what follows, the three rings will breferred to asV1, V2, andV3 for increasing barrier heightrespectively. While ringV1 has a compact density becauof its rather low barrier,V3 has a real ring shape, withhollow central region.

Figure 2 displays the expectation value of the angumomentum Lz& as well as the energy of the lowest level fboth singlet and triplet states as a function of the magnfield. At B50, the ground state is a singlet, and with increaing B, there is an alternation of triplet and singlet groustates in the three rings. The first singlet-triplet transitioccurs approximately at 3, 2, and 1 T for V1, V2, andV3,respectively. The triplet-singlet energy differences remquite small after the first crossing, which hints at possitransitions induced by small nonspin conserving perturtions, like spin-orbit interactions or magnetic impurities. Walso notice how the ground state increases in steps its ang

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GROUND STATE AND FAR-INFRARED ABSORPTION OF . . . PHYSICAL REVIEW B63 125334

momentum~in absolute value! as a function ofB. This re-flects the exact property that angular momentum is a gquantum number since the Hamiltonian commutes withLz operator. The angular momentum gain with magnetic fiis faster for the higher barrier rings. In fact, forV3 the evo-lution is markedly linear, already indicating a rotorlike evlution for this ring. This is further discussed in Sec. IV.

FIG. 1. Electron density for the three rings~symbols! as a func-tion of r. Notice that the exact solution always yields circularsymmetric densities. In order of decreasing density atr 50, theresults correspond toV1, V2, andV3, respectively. Also shown inthis plot are the LSDA densities~solid lines!, circularly averaged inthe case of broken symmetry (V3). Inset: External potentialV(r )for rings V1 ~solid!, V2 ~dash!, andV3 ~dot-dash!.

FIG. 2. Evolution with magnetic field of the lowest singlet antriplet states for the three rings. Left panels show the state en~in H* ) and right ones, its corresponding total angular momentin the perpendicular directionLz&, in units of\.

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C. Noninteracting solution

When neglecting the Coulomb interaction, the Hamtonian ~1! separates for each particle and, therefore, a stion built from single-particle (sp) orbitals with radialn andangularl quantum numbers is adequate.1 Figure 3 shows thesp eigenvalues for all, low and high barrier, rings. In facfrom the lowest maximum of the capacitance-voltage sptrum Lorke et al. obtained the single-electron ground-stashift with magnetic field. This manifests a change in slopeB;8 T that was explained as the transition froml 50 to l521 in a model potential. Of course, this is in agreemewith the sp levels forV1, since the low-barrier ring essentially coincides with the model potential used in Ref. 9. Tfirst crossing forV2 is located around 4 T, while as shownFig. 3, the high-barrier case (V3) has again a change in slopat B;8 T, attributed to thel 521→ l 522 transition,which may indicate that the experimental sample could ctain both low- and high-barrier rings. This point will be futher discussed when presenting the FIR spectra.

The noninteracting ground state with two electrons is otained by filling the lowest orbitals of thesp scheme, withthe constraint of having a singlet or triplet spin. The corsponding plots for the evolution of^Lz& and the ground-stateenergy as a function of magnetic field are displayed in Fig

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FIG. 3. Dependence of the single-particle energy levels withBfor the three potentials. The different branches, taken in vertorder at B51 T, correspond for all three cases to (nl)5(10),(121),(11),(122),(123), wheren ~radial! and l (z componentof angular momentum! are the usual quantum numbers. Eabranch presents a small spin splitting for increasingB, with thespin-up states placed below the spin-down ones.

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ANTONIO PUENTE AND LLORENCSERRA PHYSICAL REVIEW B63 125334

By comparing it with Fig. 2, we notice the effects of thCoulomb interaction between the two electrons. Obviouthe noninteracting case gives lower ground-state enersince the Coulomb repulsion is neglected. Due to the increing effect of the confining potential, the difference is highfor V1 than forV3. Differences in angular momentum aalso higher for the low-barrier ring. This is indicating thatwell-developed rings, the interaction effects are less imptant than in more compact structures, in agreement withfindings of Ref. 5 and with the obvious expectation oflarger Vb keeping the electrons more apart from each otthan a lower barrier.

We also realize that the triplet solution is unfavored in tnoninteracting case. InV1, the ground state has alwaysS50, except forB58 T for which singlet and triplet areessentially degenerate. A similar thing happens in the orings, with the difference that the singlet-triplet degenerapoints appear at lowerB’s. These differences with respectthe exact results, in which clearer singlet-triplet oscillatioare seen, indicate the importance of the Coulomb interacfor a proper ground-state description.

III. TIME INTEGRATION

The time evolution of the two-electron wave functiongiven by the time-dependent Schro¨dinger equation

i]

]tC~r1 ,r2 ;t !5HC~r1 ,r2 ;t !, ~4!

whereH is the Hamiltonian operator~1!. We have solved Eq~4!, in the same spirit as for the ground state, by discretizthe coordinate space in a uniform grid of points. Time is adiscretized and the evolution from time~n! to (n11) is ob-tained through the unitary Crank-Nicholson algorithm

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This is an implicit scheme, in which the newC (n11) may beobtained by iteration. The above algorithm allows us to pform the dynamical simulation of Eq.~4! by using a smalltime stepDt. Of course, the feasibility of this approachthe solution of Eq.~4! relies on the very small number oconstituents~two electrons! of the system.

In this paper we discuss the FIR absorption of twelectron rings and therefore our interest will focus ondipole states corresponding to small amplitude charge olations. The ground-state wave function is perturbed byinitial ~small! rigid displacement in a certain directione, rep-resenting the electric field direction, and the dipole momevolution is then followed in time. The absorption cross stion is given by the frequency transform of the dipole sign

s~v!'v

2pU E dt^C~ t !uDuC~ t !&exp~ ivt !U, ~6!

where the dipole operator isD5e•(r11r2). As a test of themethod, developed from previous calculations within mefield approaches, we have checked that the oscillation

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quencies, when assuming a pure parabolic potentialgiven, as a consequence of the generalized Kohn theoremthe values17

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wherevc5eB/c is the cyclotron frequency.

A. FIR spectra

Figure 5 shows the absorption spectra for singlet and tlet states corresponding to different magnetic fields. Thetersection of the baseline for each spectrum with the horiztal axis indicates the magnetic field in each case. Noticethe horizontal scale for each spectrum is arbitrary, and thfore, it is not given by the horizontal axis. The spectra adrawn in linear scale and arbitrary units while the verticscale gives the energies in meV. Not surprisingly, theV1spectrum is very similar to that of compact dots, with twbranches of oppositeB dispersion that merge atB50. Thisis indeed evident by comparing the spectra with the analcal expressions~7! ~dashed lines!, where an empiricalv0value fixed to the lowestB50 peak energy has been useExcept for some fragmentation details, these two brancare rather similar for both singlet and triplet cases. Moimportantly, we want to stress that essentially no strengtcontained in the energy region 15 meV,v,25 meV forB,6 T. This fact does not imply that there are no excitions in this region. In fact, they can be seen using logarmic scale, as has been done in the LSDA approach of R10. However, we have stuck to the linear scale first becathe real-time approach requires extremely long simulattimes in order to extract so low-intense frequencies, and s

FIG. 4. Same as Fig. 2 for the noninteracting case.

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GROUND STATE AND FAR-INFRARED ABSORPTION OF . . . PHYSICAL REVIEW B63 125334

ond we do not think that the experiments can easily idensuch low-intensity peaks.

The V2 spectrum and, more clearly, theV3 one reflect abehavior qualitatively different from that ofV1. The high-energy branch does not merge with the low one for vanishB, deviating from the analytical prediction of Eq.~7!. Thisbehavior is well known for quantum rings and is a manifetation of the violation of Kohn’s theorem in these systems.2–8

For V2, a global gap'8214 meV persists at all magnetifields for both spins, while in the case ofV3, the correspond-ing gap lies approximately in the region 7217 meV. There-fore, this gap constitutes a direct manifestation of the barheight in the FIR response. TheB evolution of the spectra ischaracterized by small jumps in the peak positions for ebranch that must be attributed to changes in total angmomentum. SinceV3 is the one gaining more angular mmentum, its spectrum also displays more abundant disconuities. We also notice that when the ring barrier gro~from V2 to V3) an important amount of strength is placin the interval 15–25 meV at low-magnetic fields. Actualthe V3 peak in this region has a strength comparable toone at low energy'50%, as found experimentally,10 whileour calculation forV1, as well as the LSDA results of Re10, yield a much lower ratio.

FIG. 5. Evolution of the FIR spectrum for singlet and triplground states in the three rings. The results forV1, V2, andV3 aredisplayed in the three rows from top to bottom, while left and rigcolumns correspond to singlet and triplet states, respectively.comparison, the dashed lines show the analytical prediction fKohn’s theorem~7!, see Sec. III A.

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A comparison with experiment of the calculated specreproduces, in each single case, some of the experimefeatures, but misses others. A plausible explanation isthe sample contains a mixture of high- and low-barrier rinIn Fig. 6 we display a superposition of theV1 andV3 spec-tra in comparison with the available experimental points,suming that the system takes the theoretical ground staeach magnetic field, although as already noticed in mcases, the triplet-singlet energy difference is quite smWith this interpretation, the circles are attributed to higbarrier rings, while the triangles and rombuses would corspond to low-barrier ones. Although the positions of the eperimental peaks are not perfectly reproduced~this wouldinvolve a further optimization with respect tov0 andR0) thequalitative agreement is good. A richer mixture includinintermediate-barrier rings would induce a blurring of the dferent branches with the introduction of additional~interme-diate! peaks, but would also yield the above discussion acomparison with experiment more involved. The crossesFig. 6 are not reproduced by any of our ring calculations aas hinted by the experimental group, they could be duepure quantum dots in the sample. We also mention that inexperiment,9 an insufficient signal-to-noise ratio at low enegies does not allow the detection of the lower branch av,10 meV.

B. Noninteracting spectra

Figure 7 shows the singlet and triplet spectra whenCoulomb interaction is neglected as in Sec. II C. Therefothey correspond to pure particle-hole transitions insingle-particle level scheme. On a first look they seem qusimilar to the spectra of Fig. 5. However, after a secoinspection, important differences appear. A dramatic one

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FIG. 6. Superposition ofV1 ~solid lines! andV3 ~dashed! spec-tra at different magnetic fields. The experimental data are tafrom Lorke et al. ~Ref. 9!. See text~Sec. III A! for details.

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ANTONIO PUENTE AND LLORENCSERRA PHYSICAL REVIEW B63 125334

the softening, i.e., a decrease in energy, of the lowest brain all three cases. In fact, forB'8 T the lowest state has aalmost vanishing energy. This fact is explained as the formtion of quasi-Landau bands, even for such a small systwith the property that single-particle states of the same bare quasi-degenerate and therefore, particle-hole transiare nearly gapless.

There are also sizeable shifts of peak positions andgeneral, a higher fragmentation is present in the singparticle picture. This is easy to understand since the coltivity associated to the Coulomb interaction is absent insecond case, and therefore, the description of collective mnetoplasmon states is quite rough. Altogether, the corqualitative result must be understood as an indicationinteraction effects are not overwhelming for the spectrocofeatures of a two-electron system.

IV. MEAN-FIELD THEORY AND BROKEN-SYMMETRYSOLUTION

Usually, the exact solution of the many-body Schro¨dingerequation is not possible and one must contend with apprmate solutions. In this sense, the mean-field approximaallows in many cases the description of relevant physAlthough, in principle, the mean field provides an accurpicture for a big enough number of particles, it is interestto compare it with the exact solution in the present casetwo particles. This will provide a strong test of the mea

FIG. 7. Same as Fig. 5 for the noninteracting case.

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field result, that will allow us to quantify its validity. Weremark that for systems confined in an external potensuch as atomic or quantum dot electrons the restrictionhaving a sufficient number of particles in order to develthe mean field is not as strict as in self-bound systems, sas liquid drops or atomic nuclei. This is easy to understasince in confined systems, the fixed external potential isready an important contribution to the mean field and, thefore it is not solely based on particle-particle interactions

Imposing circular symmetry, the mean-field theory hindeed been applied to medium-large quantum dotsN'20–200), in the Hartree,18 Hartree-Fock,19 and LSDAschemes.20,21 Nevertheless, when using symmetrunrestricted approaches, even at the mean-field level,normally necessary to restrict to smaller sizes (N;10) be-cause of the computational requirements.22–25

A. LSDA ground state

We have applied the LSDA symmetry-unrestricted aproach developed by us in Ref. 23 to the present twelectron rings. Figure 8 summarizes the ground-state proties, for both singlet and triplet forV1, V2, andV3. Thecomparison with the energies for the exact solution~Fig. 2!prove that the mean field is providing a quantitatively godescription of the ground-state energy, with relative enedifferences around 2%. We notice that the MF descriptionthe triplet is slightly better than that of the singlet. In facafter the first singlet-triplet crossing, the MF fails to reprduce a singlet state more bound than the triplet one. Thisbe attributed to a more important effect of correlations duethe interaction in the singlet than in the triplet, which aonly approximately taken into account within LSDA. Thoverall evolution of the total angular momentum is also wreproduced in MF, especially for the triplet. Quite strikingl

FIG. 8. Same as Fig. 2 within the LSDA.

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GROUND STATE AND FAR-INFRARED ABSORPTION OF . . . PHYSICAL REVIEW B63 125334

however, the angular momentum of the singlet doesshow the steplike evolution peculiar of the exact solution.fact, it is taking in most cases clear noninteger values. Tmust be attributed to a spontaneous breaking of the rotatisymmetry in the frame of the mean field.

The symmetry of the MF solution is more easily appreated in Fig. 9, that shows the MF ground state densityr5r↑1r↓ and magnetizationm5r↑2r↓ , wherer↑ , r↓ arethe two spin densities, forV1 andV3 at B50. Both casescorrespond to singlet solutions but, whileV1 has a circularlysymmetric density and vanishing magnetization,V3 is show-ing a separation of spin-up and -down densities, leading tooscillation of the magnetization along the ring perimeter~aspin-density wave! as well as of the total density~a charge-density wave!. This behavior is in fact common to all singlestates atB.0 for which ^Lz& takes noninteger values. Thpeculiar states were predicted by Reimannet al.26 in narrowrings with a bigger number of electrons, and have beenstudied by us in Ref. 27.

The formation of broken-symmetry solutions within Mapproaches is a well-known phenomenon in nuclphysics.28 In this field, it has been shown that the MF isfact providing the system structure in anintrinsic referenceframe, given by properintrinsic coordinates, while there isdegeneracy with respect to the remainingcollectivecoordi-nates. In a simplifying picture, this corresponds to assumcertain structure of the system relative to its mean field, tin turn may be moving, thus leading to a restoration ofexact symmetry. In fact, as shown in Fig. 1, when perforing an angular average of the MF density forV3, an excel-lent agreement with the exact result is obtained. As wecuss further in the next section, the MF symmetry breakcan be interpreted as an electronic crystallization, i.e.,formation of a Wigner molecule.

B. The Wigner molecule

The symmetry breaking of the MF solution forV3 ispointing an incipient electron localization in the intrinsframe of this system. The formation of these localized str

FIG. 9. Results obtained within the LSDA forB50 T: Lowerplots display the densityr and magnetizationm for V3, while theupper one shows the density forV1. In the latter case the magnetization vanishes everywhere.

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tures has been recently discussed usingexact solutions byYannouleas and Landman for two-electron parabolic do11

and, for rings with six electrons by Koskinenet al.12 Theseauthors have computed the exact excitation spectrumshown that it adjusts to rotor bands, thus revealing the etence of the electron molecule.

Even when the exact wave function is known, it is notrivial task to notice the existence of a Wigner moleculethe intrinsic frame. It can be hinted at by means of intrindistribution functions, like the conditional probability densir(r 2 ,u21ur 15j), which gives the probability of finding anelectron at (r 2 ,u21) when the other one is fixed atr 15j.Therefore, it provides valuable insight on the intrinsic struture of the two-electron system. In Ref. 29, Berry reviewsuse of the conditional probability density to describe the fmation of electron molecules in the doubly excited statesthe helium atom. In what follows, we show that a similanalysis can be used for the two-electron rings.

Figure 10 shows the conditional probability density whthe first electron is placed at its most probable value forradial coordinate in bothV1 and V3 confining potentials.Since this position is taken as the origin for the angleu21,this implies that the first electron is placed in those graph(j,0), with j51.5, 1.8 a0* for V1 and V3, respectively.Notice that the two distributions yield a peak for the secoelectron that lies opposite to the first one, i.e., at (2j,0). Ona first look, both functions look rather similar althoughcloser inspection reveals that the inner hole is more pnounced in the case ofV3. This is in agreement with astronger localization of the second electron in this caseorder to manifest more clearly this result, it is helpfulconstruct the density in the intrinsic frame and comparewith the MF one. To this end, we assume that once the mprobable relative positions are determined, (x,y)5(j,0) and(2j,0), the intrinsic frame density can approximately

FIG. 10. Upper plots: conditional probability densitier(r 2 ,u21ur 15j) obtained from the exact wave function atB50 T. The first electron has been fixed at (j,0), indicated by asolid dot, wherej51.5,1.8 a0* is the most probable radius forsingle electron, inV1 andV3, respectively. Lower plots: Intrinsicframe density built from the upper conditional probability densitas explained in the text~Sec. IV B!.

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ANTONIO PUENTE AND LLORENCSERRA PHYSICAL REVIEW B63 125334

obtained by centering in each position the conditional prability density. This procedure leads to the two lower plotsFig. 10. Quite interestingly, theV3 result is clearly breakingthe circular symmetry and both are in excellent agreemwith the mean-field densities of Fig. 9.

Although qualitative, the preceding discussion suppothe interpretation that the MF describes the intrinsic structof the ring. Consequently, the existence of a high barrieV3 produces an incipient Wigner molecule. However, itmains an interesting task for the future, to obtain the fullof excited states and discuss its rotorlike character, followRefs. 11,12, as well as the analytical expressions of Ref

C. Unrestricted time-dependent LSDA„TDLSDA …

Based on the mean-field picture, the TDLSDA allowsto describe the collective dipole oscillations. This schehas been used to describe both spin and density excitatioquantum dots and rings in a symmetry-restrictapproach.30,10 In Ref. 23, we described the applicationTDLSDA relaxing the constraint of circular symmetry anusing a real-time approach similar to that of Sec. III. WithTDLSDA, this implies the solution of the time-dependeKohn-Sham equations, which are the analog of Eq.~4!,within the LSDA. We have applied this technique to tpresent two-electron rings and show the correspondingglet and triplet spectra forV1, V2, andV3 in Fig. 11.

The TDLSDA provides a rather good description of trings’ dipole spectra, as compared to the exact ones.

FIG. 11. Same as Fig. 5 within the LSDA.

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notice that the overall evolution withB and the location ofthe strength is quite well reproduced for each branch. Hoever, finer details of the spectra, such as the fragmentastructures, are not reproduced in many cases. Although bon a single-particle picture, the unrestricted TDLSDA corects the lower-branch instability found in Fig. 7. This ismanifestation of the interaction effects taken into accountthe TDLSDA. From the present results we conclude thdespite that the present rings only contain two electrons,TDLSDA provides a rather sensible picture of the dipooscillations.

V. CONCLUSION

In this paper we have theoretically analyzed two-electrings motivated by recent measurements of their FIR abstion. We have found that in order to explain the experimenspectra it is necessary to assume that the sample containmixture of rings with different barrier heights. As expectehigh-barrier rings are characterized by a clear violationKohn’s theorem, showing a sizeable amount of strengthenergiesv.15 meV forB,6 T. On the other hand, lowbarrier rings approximately fulfill Kohn’s theorem, with aeffective v0, and have lower-energy excitation branchmore similar to the measured ones. A mixture of these ttypes of rings has been used to explain the main featurethe experiment. A more detailed analysis would requirecluding intermediate barrier rings~ideally knowing the actualbarrier-height distribution in the sample!, for which the spec-tra lie in-between the preceding two extreme situations. Dspite the very small number of constituents, these systpossess a rich FIR spectrum with an intricateB dependence.

The evolution of angular momentum and total spin wbarrier height and magnetic field has been discussed andtrasted with independent particle predictions. In agreemwith previous studies, we find that interaction effects becoless important for increasingly narrow rings, obviously rflecting a bigger electronic separation than in a compstructure and a corresponding increase in angular momtum.

The comparison with the symmetry-unrestricted LSDhas allowed us to discuss the formation of an incipieWigner molecule for theV3 case and, in general, has showthe good quality of this approximation. We have proposemethod to obtain the density in the intrinsic reference frambased on the conditional probability density, that quanttively agrees with the LSDA result. The FIR absorptionTDLSDA provides a good overall strength distribution adispersion with the magnetic field, although it misses sofiner details of the fragmentation patterns.

ACKNOWLEDGMENT

This work has been performed under Grant No. PB90124 from DGESeIC, Spain.

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GROUND STATE AND FAR-INFRARED ABSORPTION OF . . . PHYSICAL REVIEW B63 125334

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14For bulk GaAs, the effective gyromagnetic factor isg* 5

12533

-

B

20.44.15For a barrier heightVb the condition of continuity up to the sec

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