We have investigated edge modes of different multipolarity sustained by
quantum antidots at zero magnetic field. The ground state of the antidot is
described within a local density functional formalism. Two sum rules, which are
exact within this formalism, have been derived and used to evaluate the energy
of edge collective modes as a function of the surface density and the size of
the antidot.Comment: Typeset using Revtex, 8 pages and 6 Postscript figure

A. Emperador

A. Lorke

B. Tanatar

E. Lipparini

G. Y. Wu

I. S. Gradshteyn

K. Bollweg

K. Kern

Ll. Serra

M. Abramowitz

M. Barranco

M. Ferconi

M. Pi

O. Bohigas

S. Giovanazzi

Y. Zhao

English

arXiv

We have investigated edge modes of different multipolarity sustained by quantum antidots at zero magnetic field. The ground state of the antidot is described within a local-density-functional formalism. Two sum rules, which are exact within this formalism, have been derived and used to evaluate the energy of edge collective modes as a function of the surface density and the size of the antidot

Emperador, Agustí

Pi Pericay, Martí

Barranco Gómez, Manuel

Lipparini, Enrico

Serra, Llorenç

Diposit Digital de la Universitat de Barcelona

PHYSICAL REVIEW B 15 SEPTEMBER 1998-IVOLUME 58, NUMBER 11Structure and far-infrared edge modes of quantum antidots at zero magnetic fieldA. Emperador, M. Pi, and M. BarrancoDepartament d’Estructura i Constituents de la Mate`ria, Facultat de Fı´sica, Universitat de Barcelona, E-08028 Barcelona, SpainE. LippariniDipartimento di Fisica, Universita` di Trento, 38050 Povo, ItalyLl. SerraDepartament de Fı´sica, Facultat de Cie`ncies, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain~Received 10 February 1998!We have investigated edge modes of different multipolarity sustained by quantum antidots at zero magneticfield. The ground state of the antidot is described within a local-density-functional formalism. Two sum rules,which are exact within this formalism, have been derived and used to evaluate the energy of edge collectivemodes as a function of the surface density and the size of the antidot. @S0163-1829~98!00332-4#With the progress of microstructure technology, the studyof the two-dimensional electron gas ~2DEG! has evolved tothat of laterally confined superlattices made of either electronislands ~dots! or holes surrounded by electrons ~antidots!.Much effort has been devoted in the past to the study ofquantum dots, as compared to that put in the study of quan-tum antidots. One of the goals of their study has been todetermine the far-infrared response of these semiconductormicrostructures, and the formation of compressible and in-compressible states when a magnetic field B is perpendicu-larly applied. In the case of antidots, which is the subject ofthe present work, experimental evidence of collective exci-tations sustained by these structures has been presented inRefs. 1–3. A theoretical description based on magnetoplas-mons in two-dimensional antidot structures has been given inRef. 4 that compares well with the experimental data of Ref.2. Recently, the existence of compressible and incompress-ible strips at the edge of antidots has been determined byfar-infrared spectroscopy.5We have started a systematic study of the structure andcollective far-infrared response of antidots, whose aim is toachieve a level of sophistication in the description of thesesystems similar to that attained for quantum dots. As a firststep, we present here results at zero magnetic field obtainedwithin density-functional theory ~DFT!. To some extent, thepresent study is similar in scope to that carried out in Ref. 6on the surface excitations of cavities in 3D metals. TheBÞ0 case, which requires a rather different and far morecomplex approach, will be presented in a forthcoming paper.We have modeled an antidot of radius R in a 2DEG ofsurface density ns by a positive jellium background of den-sity nsQ(r2R) to which we have added a parabolic poten-tial barrier of the type Vp(r)5mv02(R22r2)/2 acting on theelectrons for r<R . A potential barrier of one kind or anotheris needed to prevent the electrons from spilling in the antidottoo much, producing an unphysical representation of the ex-perimental device. Actually, we have found that if v0 is setto zero, high-density antidots such as those studied in Ref. 5would have nonzero electron densities even at r50. We shallcall Vext the sum of the jellium and Vp potentials.PRB 580163-1829/98/58~11!/6732~4!/$15.00The ground state ~gs! of the antidot is obtained solving theKohn-Sham ~KS! equations as indicated, for example, inRef. 7. The problem is simplified by the imposed circularsymmetry, and only the radial KS equations have to be con-sidered to determine the electronic radial wave functionsRl(r). We have used a dielectric constant e and an electroneffective mass m5m*me , which are characteristics ofGaAs: e512.4 and m*50.067, which we have chosen forthe numerical applications in view of the existing experimen-tal data of Refs. 2 and 5. The single electron potential V(r)entering the KS equations is made of Vext , of the Hartreeelectron-electron potential and of the exchange-correlationpotential. The correlation potential has been obtained fromthe correlation energy of Ref. 8 in a local-density approxi-mation. Sometimes we shall use effective atomic units. Inthis system of units, the length unit is the Bohr radius timese/m*, and the energy unit is the Hartree times m*/e2, de-noted here as a0* and H*, respectively. For GaAs we havea0*;97.9 Å and H*;11.9 meV;95.6 cm21.Physically acceptable solutions to the radial KS equationshave to be regular at r50 and behave asymptotically asRl(r ,k);Jl(kr)1tan(dl)Yl(kr), where Jl and Y l are theinteger Bessel functions of the first and second kind,9 andk5A2m(E2V`)/\2. The wave number k has to bek<kF5A2pns. Taking into consideration the spin degen-eracy, the electron density r(r) is obtained asr~r !5 (l52`l5` 2~2p!2EukW u<kFRl2~r ,k !dkW . ~1!We have checked that the number of points used to carry outa Gaussian integration over k , and the maximum ulu em-ployed in Eq. ~1! lead to stable results.We show in Fig. 1 the electronic densities correspondingto antidots of R57.5 a0* and ns50.05, 0.1, 0.2, 0.3, and 0.4(a0*)22. They span the size and density range of those fab-ricated and experimentally studied in Ref. 2, and have beenobtained using v050.3H*.6732 © 1998 The American Physical SocietyPRB 58 6733BRIEF REPORTSThese results can be employed to determine the B50far-infrared multipole response of antidots. To do so, we relyon the formalism described in detail in Ref. 10. For thepresent purpose, it consists in obtaining the so-called m1 andm3 sum rules ~SR! for an excitation operator QL . We havethat11FIG. 1. Electronic densities as a function of r for antidots ofR57.5 a0* and ns50.05, 0.1, 0.2, 0.3, and 0.4 (a0*)22. Also shownare the jellium densities ~dotted lines!.m1~QL!5 12 ^0u@QL1 ,@H ,QL##u0&,~2!m3~QL!5 12 ^0u@@H ,@H ,QL1## ,@H ,QL##u0&,where u0& is the gs of the system. These SR have been ex-tensively studied in the literature.11,12 As indicated in thesereferences, if only a collective state is contributing to thestrength function, E3(QL)[(m3 /m1)1/2 represents the aver-age excitation energy. This is the situation experimentallyfound for antidots at zero magnetic field.The operator QL is taken to beQL5(j51N 1r jL eiLu j. ~3!This choice is inspired in that (qr)2LeiLu is the small qexpansion of the function Y L(qr)eiLu, which is the restric-tion to the z50 plane of the irregular solution of the Laplaceequation in cylindrical coordinates. Following Ref. 10, alengthy but straightforward calculation yieldsm1~QL!52pL2E0`dr1r2L11r~r !, ~4!m3~QL!5m3~T !1m3~ee !1m3~Vexte !, ~5!wherem3~T !52pL2~L11 !E0`dr1r2L13@Lt~r !12~L12 !l~r !# , ~6!m3~ee !54pL2~2L21 !!!2LL!E0`r8~r !drH 1r2L11E0r@2r8~r8!1r8r9~r8!#ELS r8r D dr81Er` 1r8~2L11 !@~2L11 !r8~r8!2r8r9~r8!#ELS rr8D dr82 2L11L!~2L11 !!! 1r2L r8~r !J14pL2E0`dr1r2L12@rr9~r !2~2L11 !r8~r !#H 1r E0r@3r8r~r8!1r8 2r8~r8!#ES r8r D dr82Er`r8r8~r8!ES rr8D dr822rr~r !1 limR`!`R`nsES rR`D J , ~7!m3~Vexte !5pL2E0`drVext~r !Fr9~r !2 2L11r r8G 1r2L11 . ~8!The definition of the densities t(r) and l(r) and of the function EL can be found in Ref. 10, and the primes on the densitydenote r derivatives. The jellium potential V1(r) entering in Vext isV1~r !54nsH R`E~r/R`!2RE~r/R !, r,RR`E~r/R`!2rE~R/r !1r12~R/r !2K~R/r !, r.R . ~9!6734 PRB 58BRIEF REPORTSIn the above equations, K and E are the complete ellipticintegrals of the first and second kind, respectively,13 and R`represents a large r value. In practice, it is the largest r usedin the structure calculation, which we have also taken as thepoint where the asymptotic behavior of Rl(r) holds. Wewant to point out that the two Coulomb diverging terms inm3(ee) and m3(Vexte) cancel each other.The present formalism can be applied to antidots with therestriction that r(r) vanishes in a small disk around the ori-gin. In practice, this is not a limitation, as can be inferredfrom Fig. 1 ~see also Fig. 3!. Some technical details abouthow the above integrals are handled can be found in Sec. IVof Ref. 6.For a large antidot, the electronic density is constanteverywhere apart from a narrow region along the border ofthe hole. Following the method outlined in Ref. 10, it iseasy to show that E3 yields the classical hydrodynamicsdispersion relation for edge waves, namely, E35v(q);A2nsq ln(q0 /q). It is also worth to notice that the induced~or transition! density associated to the operator QL has theform10r tr~rW !}L1rL11r8~r ! ~10!that manifests the edge character of the excitation.Figure 2 shows the frequency of the L51 mode as afunction of ns compared with the experimental points of Ref.2. For completeness, we have also plotted the results corre-sponding to L52. We have checked that similar results areobtained using as potential barrier the parabola Vp(r)5mv02(R2r)2/2 for r<R with v051H*. One can see thatthe agreement with experiment is good. Furthermore, ourcalculation yields a frequency of ;68 cm21 for the R5180 nm, ns5931011 cm22 antidot, in good agreementwith the experimental findings of Ref. 5.If the electronic density r(r) is approximated by a qua-sistep function, an analytical expression can be obtained forE32. Proceeding as in Ref. 10 one getsFIG. 2. Mode frequency for L51 and 2 as a function of theelectron surface density corresponding to R57.5 a0* . The crossesare experimental data from Ref. 2, and the lines are drawn to guidethe eye.E3252pnsL~L11 !R214nsLRF12 lnS g Ra D112 (m51L 12m21G ,~11!where a represents the width of the quasistep function, andthe precise value of g depends on the way the electronicdensity goes to zero.14 This equation tells one that the fre-quencies have a 1/AR linear dependence if the Coulomb en-ergy contribution dominates. For reasonable values of g/a , ithappens for any realistic value of R .We have also studied the size dependence of the modeenergy. Figure 3 represents the electronic densities for anti-dots of R510, 15, and 20 a0* , and a surface densityns50.2 (a0*)22. The frequencies of the L51 and 2 modesare shown in Fig. 4 as a function of 1/AR . They exhibit adistinct R dependence, indicating a clear departure fromparabolicity of the confining potential, i.e., a physical situa-tion where the generalized Kohn theorem does not apply.FIG. 3. Electronic densities as a function of r for antidots ofR510, 15, and 20 a0* , and ns50.2 (a0*)22. Also shown are thejellium densities ~dotted lines!.FIG. 4. L51 and 2 mode frequencies as a function of R21/2for ns50.2 (a0*)22. From right to left, the dots correspond toR57.5, 10, 12.5, 15, 17.5, and 20 a0* . The lines are drawn to guidethe eye.PRB 58 6735BRIEF REPORTSIn conclusion, we have shown that density-functionaltheory is able to reproduce the zero-magnetic-field frequencyof antidot edge modes in a way that is quite similar to thatfor quantum dots. Although a satisfactory description of thecollective spectrum of antidots can be achieved using a mag-netoplasmon approach,4 an alternative method based on amore microscopic approach such as DFT is needed to discussother interesting problems, such as edge reconstruction andthe formation of compressible and incompressible strips atthe antidot edge.5It is a pleasure to thank Ricardo Mayol and Francesc Sal-vat for useful discussions. This work has been performedunder Grant Nos. PB95-1249 and PB95-0492 from CICYT,Spain, and GRQ94-1022 from Generalitat of Catalunya. A.E.acknowledges support from the Direccio´n General deEnsen˜anza Superior ~Spain!.1 K. Kern, D. Heitmann, P. Grambow, Y. H. Zhang, and K. Ploog,Phys. Rev. Lett. 66, 1618 ~1991!.2 Y. Zhao, D. C. Tsui, M. Santos, M. Shayegan, R. A. Ghanbari, D.A. Antoniadis, and H. I. Smith, Appl. Phys. Lett. 60, 1510~1992!.3 A. Lorke, I. Jejina, and J. P. Kotthaus, Phys. Rev. B 46, 12 845~1992!.4 G. Y. Wu and Y. Zhao, Phys. Rev. Lett. 71, 2114 ~1993!.5 K. Bollweg, T. Kurth, D. Heitmann, V. Gudmundsson, E. Vasili-adou, P. Grambow, and K. Eberl, Phys. Rev. Lett. 76, 2774~1996!.6 Ll. Serra, F. Garcias, J. Navarro, N. Barbera´n, M. Barranco, andM. Pi, Phys. Rev. B 46, 9369 ~1992!.7 M. Ferconi and G. Vignale, Phys. Rev. B 50, 14 722 ~1994!.8 B. Tanatar and D. M. Ceperly, Phys. Rev. B 39, 5005 ~1989!.9 M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions ~Dover, New York, 1970!.10 E. Lipparini, N. Barbera´n, M. Barranco, M. Pi, and Ll. Serra,Phys. Rev. B 56, 12 375 ~1997!.11 E. Lipparini and S. Stringari, Phys. Rep. 175, 103 ~1989!.12 O. Bohigas, A. M. Lane, and J. Martorell, Phys. Rep. 51, 267~1979!.13 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series andProducts ~Academic Press, New York, 1980!.14 S. Giovanazzi, L. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 72,3230 ~1994!.

Structure and far-infrared edge modes of quantum antidots at zero magnetic field

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Structure and far-infrared edge modes of quantum antidots at zero
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Structure and far-infrared edge modes of quantum antidots at zero magnetic field

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