stark effect in parabolic quantum dot

7/29/2019 Stark Effect in Parabolic Quantum Dot

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JOURNAL DE PHYSIQUE IVColloque C.5, supplBment au Journal de Physique 11, Volume 3, octobre 1993

Stark effect in parabolic quantum dotS. JAZIRI, G. BASTARD*an d R. BENNACEUR*Dkpartement de Physique de 1'E.N S. de Bizerte, 7021 Jarzouna Bizerte, Tunisia* Laboratoire de Physique de la Mati2re Condenske de 1'E.N.S. de Pans, 24 rue Lhomond, 75231 Parisceda 05, France** Dkpartement de Physique de la Facultk des Sciences de Tunis, Campus Universitaire Le Belvkdere Tunis,Tunisia

Abstract: We theoretically investigate the optical properties of theexciton confined in parabolic quantum-dot , with and without electricfield, by means of perturbative-variational method. The quantum-dotsize enhances the 1s eigenvalue ahd oscillator strength . In smaller dotthe relative extension of the exciton wave function is equal t o the sizeof the dot . The 1s exciton bihding energy is found t o be almost 2-3times that in the quantum-well of the same thickness. In the presenceof an external electric field, we calculate the quantum-confined Starkeffect . The energy split is found about the same as in quantum-wellwith the same size .The wider quantum-dot has a larger Stark shift. Wealso analyse the special case of high electric field . In this case theCoulombic interaction can be approximated by parabolic potential.

1 INTRODUCTIONIn recent years excitons states in quantum dots have been studied in a numberof papers 11-41 and have been observed by photoluminescence experiments [5-61. The study of electronic states in quantum dots depends on either theconfining potential and the interacting force between the particles. Followingthe" Generalized Kohn's Theorem "; heoretical studies show that the confinedpotential for electrons [7-91 and holes [2,9] in quantum dots is nearlyparabolic, so the center-of-mass motion can be solved exactly. The effect ofan electrostatic field on the electron-hole states and on the confined excitonicstates is referred t o the quantum confined Stark effect (Q.C.S.E.) has receivedintensive discussions in quantum well structures [12,15]and few studies inquantum wire and in quantum dot systems [13]. In this work , using ir simpleand efficient approximation , we propose to study the exciton properties in aparabolic quantum dot structure with and without the presence of anelectrostatic field. In sect.2-, we present the formalism of the perturbative-variational method [14]. We investigate the Stark shifts. We also analyse thespecial case of high electric field. The results for the exciton ground stateproperties in parabolic quantum dot , and the energy level split under anelectric field , are presented and discussed in sect.3- .2- THEORYWithin the effective-mass approximation and neglecting the band-structureeffects , the Hamiltonian of an exciton in a parabolic quantum dot with the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1993577
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same quantization energy MSZ ( for the electron and hole)[Z],and subjected to anexternal ele~ tr ic ield , can be expressed as :

p, 1H z- 4 1 e2+- m a2re2- - m ~~r~~ - eFze - eFZh (1 )2 m e 2 e 2 m h 2 h ~rwhere me-( mh ) are the single-particle Hamiltonian and the effective-massfor the electron (hole),, E is the background dielectric constant . .Using therelative coordinate r=(re-rh) and the corresponding momenta p with reducedmdss p=m,mh/M and center-of-mass coordinate R =mere+mhrh/M and thecorresponding momenta P with the total-mass M=me+mh , the Hamiltonian H isrepresented as :

p2 1 1 e2H = =+ I ~ n 2 ~ 2 ++ zp&r2 - - eFz2P ~r ( 2 )In Eq. (2) the part which depends only on the center-of-mass coordinatecorresponds t o the Hamiltonian of a well-known three dimensional harmonicoscillator. The exciton properties is essentially determined by the relativeHamiltonian . As under the influence of the electric field the potential energyis z-axial symmetric,we use conventional cylindric coordinates . The fieldterm added to the z-direction confinement describes a displaced harmoniceFoscillator centred in -zo=- with the frequency a, inferior ro n.ln order to

PQsolve the Hamiltonian H we introduce an interaction potential which obeysto the Hooke's force with the parameter h by adding and substracting the

1potential V(r)= A (T pn2r2-L( to be able t o split the Hamiltonian H into twoterms , with the one term being~xactly olvable while the other can be treatedas a perturbation . This potential is similar t o the interaction potentialbetween electron-electron used by Johson et al [ I 11. This approximation is notcorrect for all electron-hole separation but the interaction parameter h can beadjusted t o give the best f it of the true interaction which is the Coulombinteracion, grid for the dominant range of separation r . The attractionpotential V(r) must have negative value with positive h , this yields a-reasonable f i t t o the exact interaction for electron-hole separation r-42 Ro?where Ro is the quantum dot radius defined by R We determine thebest choice of has the one wich ensures the fasted convergence of theperturbation series . We rewrite H as :

H,= H,+H, (3)1+hin which :Ho = Pf_+ pa2r2 - SZ - e2 F'2v ~ $ 2 ~3a)


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HI is taken as a perturbation by choosing an applicable value of h in t h equantum dot ; and Ho as the unperturbed part of the Hamiltonian H The totalenergy corresponding to exciton ground state is obtained as :

3 3E T = z H Q + z H Q , - h H Q - e2 F~ where Q, = 0 fib (4)2pQ2The parameterh may be determined[4,141 by: Hl (h)==O ( 5 )with q(r,h) is the exciton enveloppe wave function . The field-induced energyshifts can be expressed as: AE = ET(F=O)-ET(F), E(F=O)is the correspondingenergy in the zero-field state.We calculate the exciton binding energy definedby EB= Ee+Eh-ET, where Ee,Eh are the energies corresponding t o the single-particle Hamiltonian. The extension of the exciton relative motion iscalculated by the expectation value in the ground state of r, We also calculatethe oscillator strength of the exciton ground state in the dot normalised t othat of a free exciton in a bulk material with volumeV=4/3n R which given by

whereEex=EG-R*andE = EG+Ef are the exciton energies in the bulk and in thequantum dot respectively, and a* is the effective Bohr radius,R*is theeffective Rydberg energy and EG is the gap energy and x(R,r,h)is the totalexciton enveloppe wave functionEffect of strong electric fieldDue t o the confinement effects the enhancement of the exciton binding energyin quantum dot is found to be significantly large. I t is interesting t oinvestigate the particular large field effects on the confined exciton i nparabolic quantum dot. One pos'sible approach is the perturbgtive-variationalmethod as mentioned iJ1 the last subsection . A better way is to analyse in thiscase the expanding of the Coulomb interaction . Large electric field induces aspatial shift of the particles along the field direction , leading to a reductionof the Coulomb interaction which can be approximated by :

We only consider the first and the second order of the expansion in the relativeHamiltonian H The longitudinal relative motion of the exciton is a linearharmonic oscillator pushed on the left by quantity (2,-z,) , and displayed withthe frequency Q/, inferior to 52, due to combined effects of the electric fieldand Coulombic potential . The transverse relative-motion is represented by aflater harmonic oscillator with the frequency 52,superior to 52.Th.e energy ofthe exciton in the ground state is given by :

3 1E T ( F ) = ~ Q + b ( q + ~ H Q ~ / - hQ - e2 F' e2 1- -2 p ~ 2 EZO 'Z (8)- 2p9 / /5with Q =Q.\I-, n = and zl = I-ZyZn where 5 =A~ ~ 2 ~ ~ 5 2Obviously the expansion of the Coulomb interaction is not correct foyal l fieldstrength . In order for the model t~ be applicable ,we must determine a


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condition which makes the approximation of the Coulomb attraction physicallyreasonable . The longitudinal relative coordinate z should be greater than therelative position coordinates particulary than the mean value of the displacedharmonic oscillator in the z-direction which is equal t o zl . Then the validitycriterion of this method is found for z,


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The electron-hole separationbecomes insensitive t o the wide quantum dots andconverge to the bulk value Separation.The size dependence of the normalizedoscillator strength (fig.2) is determined essentially by the integral in eq.(6),while the energy dependent term displays negligible variations [2-41. As thequantum dot size is increased the envelope function becomes more and moreflat, the integral term decreases rapidely and converges towards constant .This point is already been noticed by [2-4. This result should be generalindependent of the confining and interaction potentials.The electric field effects on exciton in GaAs parabolic quantum dot are plottedin Fig 3,4. In fig. 3 we plot the calculated heavy-hole exciton resonance energyshift AE versus the electric field strength for several quantum-dot radius.

Fig.4The exciton binding energ as afunction of electric field for dfferentquantum dots.The split variation of the exciton energy is found about the same as inparabolic quantum-wet1 [I 1 and is a few larger than the excitonic shift in thesquarg quantum-well case bith the same thickness [I 1. The 1S binding energyof the heavy-hole exciton is shown in fig.4 as a function of an applied electricfield for three different quantum-dot radius. Finally, we examine the casewhere the Coulombic potential is approximated by an interaction potentialwhich is composed essentially by quadratic terms . We investigate the energystates of the exciton in parabolic quantum-dot subjected to an applied electricfield when the condition (9) is satisfied : high field strength for narrowerquantum-dot and relatively weak field strength for wider quantum-dot. Wehave compared these energies with those determined by the variational-perturbathe method. The corresponding accuracy is of the order of 1%. Itshould be noted that the actual method is much simpler but cannot enviseagedfor any strength electric field and/nor any quantum dot size.REFERENCES[I]- Y.Z. Hu ,M.Lindberg and S.W.Koch ,Phy.Rev. B 42 ,1713 (1990).G.W. Bryant,Phys. Rev.B 37,8763 (1988) andT. Takagahara ,Surf. Sci. 267 , 310 (1992).[2]- Weiming Que ,Solid Stat. Comm. ,81, 721 (1992).[3]- Y.Kayanuma ,Phys.Rev B 44 , 13085 (1991) .


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141-S.Jaziri,G.Bastard and R.Bennaceur,Semicond.Sci.andTechnol.(tobe published)151- K. Kash ,A.Scherer,J.M. Worlock ,H.G. Craighead and M. C. Tamargo,Appl.Phys. Lett.49,1043 (1986).[6]M. A. Reed, R.T. Bate, K. Bradshaw, W.M. Duncan, W. R.Frensley, J.W. Lee andH.D. Shih ,J. Vac. Sci Technol. B4,358 (1986).[7]- L. Brey ,N.F. Johnson and B.1 Halperin ,Phys. Rev. B 40,10647 (1989).[8]- P.A. Maksym and T. Chakraborty ,Phys. Rev. Lett. 65 ,108 (1990)[9]- F.M.Peeters ,Phys. Rev. B 42 ,1486 (1990).[10]-V. Halonen, T. Chakraborty and P. Pietilainen Phys. Rev. B 45 ,5986 (1992).{I11-N.F. Johnson and M.C. Payne ,Phys. Rev.Lett ,67 ,1157 (1991).[12]- G. Bastard ,"Wave Mechanics Applied to Semiconductor heterostructures ",Les Editions de Physique ,Les Ulis (1988) ;DAB. Miller, D.S. Chemla, T.C. Dqmen,A.C. Gossard, W. Wiegmann, T.H. Wood and C.A. Burrus , Phys. Rev. B 32,1043(1985).-G. Bastard and J.Brum , EEE J. Quantum Electron. Q E 22,1625 (1986).[13]-Y. Chiba and S.Ohnishi, Phy.Rev.B, 38, 12988 ( 1988).[14)-Y.C. Lee ,W.N. Mei and K.C.Liu ,J.Phys. C15, L469 (1982).[IS]- Chen and T.G. Andersson ,Semicond. Sci. Technd. 7828 (1 92).

stark effect in parabolic quantum dot

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