Stark effect in parabolic quantum dot

We theoretically investigate the optical properties of the exciton confined in parabolic quantum-dot , with and without electric field, by means of perturbative-variational method. The quantum-dot size enhances the 1s eigenvalue ahd oscillator strength . In smaller dot the relative extension of the exciton wave function is equal t o the size of the dot . The 1s exciton bihding energy is found t o be almost 2-3 times that in the quantum-well of the same thickness. In the presence of an external electric field, we calculate the quantum-confined Stark effect . The energy split is found about the same as in quantum-well with the same size .The wider quantum-dot has a larger Stark shift. We also analyse the special case of high electric field . In this case the Coulombic interaction can be approximated by parabolic potential. 1 -INTRODUCTION In recent years excitons states in quantum dots have been studied in a number of papers 11-41 and have been observed by photoluminescence experiments [561. The study of electronic states in quantum dots depends on either the confining potential and the interacting force between the particles. Following the" Generalized Kohn's Theorem "; theoretical studies show that the confined potential for electrons [7-91 and holes [2,9] in quantum dots is nearly parabolic, so the center-of-mass motion can be solved exactly. The effect of an electrostatic field on the electron-hole states and on the confined excitonic states is referred t o the quantum confined Stark effect (Q.C.S.E.) has received intensive discussions in quantum well structures [12,15]and few studies in quantum wire and in quantum dot systems [13]. In this work , using ir simple and efficient approximation , we propose to study the exciton properties in a parabolic quantum dot structure with and without the presence of an electrostatic field. In sect.2-, we present the formalism of the perturbativevariational method [14]. We investigate the Stark shifts. We also analyse the special case of high electric field. The results for the exciton ground state properties in parabolic quantum dot , and the energy level split under an electric field , are presented and discussed in sect.3. 2THEORY Within the effective-mass approximation and neglecting the band-structure effects , 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 368 JOURNAL DE PHYSIQUE IV same quantization energy MSZ ( for the electron and hole)[Z],and subjected t o an external ele~tric field , can be expressed as : p', 1 H z 4 1 e2 +m a2re2 +-+m ~~r~~ + eFze eFZh (1 ) 2 m e 2 e 2 m h 2 h ~r where me-( mh ) are the single-particle Hamiltonian and the effective-mass for the electron (hole),, E is the background dielectric constant . .Using the relative coordinate r=(re-rh) and the corresponding momenta p with reduced mdss p=m,mh/M and center-of-mass coordinate R =mere+mhrh/M and the corresponding momenta P with the total-mass M=me+mh , the Hamiltonian H is represented as : p2 1 1 e2 H = =+ I~n2~2+ & + zp&r2 + eFz 2 P ~r ( 2 ) In Eq. (2) the part which depends only on the center-of-mass coordinate corresponds t o the Hamiltonian of a well-known three dimensional harmonic oscillator. The exciton properties is essentially determined by the relative Hamiltonian . As under the influence of the electric field the potential energy is z-axial symmetric,we use conventional cylindric coordinates . The field term added to the z-direction confinement describes a displaced harmonic eF oscillator centred in -zo=with the frequency a, inferior ro n.ln order to PQ solve the Hamiltonian H, we introduce an interaction potential which obeys to the Hooke's force with the parameter h by adding and substracting the 1 potential V(r)= A (T pn2r2 -L( to be able t o split the Hamiltonian H, into two terms , with the one term being~xactly solvable while the other can be treated as a perturbation . This potential is similar t o the interaction potential between electron-electron used by Johson et al [ I 11. This approximation is not correct for all electron-hole separation but the interaction parameter h can be adjusted t o give the best f it of the true interaction which is the Coulomb interacion, grid for the dominant range of separation r . The attraction potential 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 the best choice of has the one wich ensures the fasted convergence of the perturbation series . We rewrite H, as : H,= H,+H, (3) 1 +h in which : Ho = Pf_+ pa2r2 S Z e2 F'