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Polaronic states in II–VI quantum dot M. Trikia,*, S. Jazirib a

Laboratoire de Physique de la Matie`re Condense´e, Faculte´ des Sciences de Tunis, Bizerte, Tunisia b De´partement de Physique, 7021 Bizerte, Tunisia

Abstract In self-assembled quantum dots, the strong electron–phonon interaction should lead to some polaron effects in optical spectra (absorption and Raman scattering). We study the influence of the longitudinal optical phonons on polaron formation in CdS/ZnS self-assembled quantum dots. # 2004 Published by Elsevier B.V. Keywords: Polaronic states; Self-assembled quantum dots; Electron–phonon interaction

Investigation of exciton in quantum dots have been attracted much attention in the last decade as it has been possible to produce well characterised quasizero-dimensional structures. The optical properties are influenced by electron and hole interaction with longitudinal optical LO phonons [1]. In low dimensional structures, the electron and the phonon become more correlated and give rise to polarons. So the knowledge of the polaron energy spectrum is important for understanding the optical spectra of quantum dots. In this work, we propose an analytic study to determine the lowest polaronic states in a self assembled II–VI quantum dot. These semiconductor quantum structures can be considered as a new class of materials of basic interest [2,3]. The shape of the dot (Fig. 1) is modelled by a cone lying on a wetting layer with a radius r, height h and a fixed ratio h/r ¼ 0.21.

* Corresponding author. E-mail address: [email protected] (M. Triki).

0169-4332/$ – see front matter # 2004 Published by Elsevier B.V. doi:10.1016/j.apsusc.2004.05.265

The approach is based on the strong coupling between the electron and the LO phonon. To be specified, we focus our analysis to the study of the anti-crossings between the electronic and the excitonic states which are separated by the LO phonon energy. So, this work is divided on two parts: firstly, we study the electron– phonon interaction and then we concentrate to the exciton-phonon one. Within the effective-mass approximation, the single particle Hamiltonian can be written: Hi ¼

P2ir P2iz þ þ Vdot ðzi ; ri Þ 2mi== 2mi?

(1)

where i ¼ e, h for electron and hole and // and ? stands for parallel and perpendicular to the growth axis (z), mi the effective mass of the electron or the hole and Vdot(zi, ri) is the band offset potential quantum dot. It is zero in the dot and Vi outside. It is assumed to have a cylindrical symmetry along the growth direction. The wave function is simplified to the form Cin ðri Þ ¼ einyi fn ðri ; zi Þ due to the commutation ½Lzi ; Hi ¼ 0 between the Hamiltonian and the angular momentum Lzi induced by the symmetry. The electron

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Fig. 1. The shape of the quantum dot.

and hole wave functions are calculated by numerical diagonalization on a Fourier–Bessel function basis. n l pm zi win;l;m ð~ ri Þ ¼ anl eln yi Jn l ri sin (2) Z R where Z and R are the high and the radius of a large cylinder. lnl is the lth root of the n-order Bessel function Jn. anl is the normalization Constance.

Fig. 2 shows the lowest energies of the electron in a CdS cone versus its radius. It is found that this semiconductor quantum dot displays discrete levels (Ee < 810 meV). Same results are obtained by most quantum dots [4], reason for what they are usually called artificial atoms. The two first states are with a symmetry n ¼ 0 and n ¼ 1. They will be denoted by |Sei and |Pei in the rest of our discussion. The two first states of the hole will be denoted |Shi and |Phi,

Fig. 2. The lowest energies of the electron in CdS/ZnS quantum dot as a function as its radius. The two first states are with a symmetry n ¼ 0 and n ¼ 1, respectively.

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respectively. These calculations have, also, pointed out that the energy gap between |Sei and |Pei states varies between 30 and 120 meV. But, what can happen after considering phonons contribution? After considering the phonon contribution, the polaronic Hamiltonian can be expressed:

diagonalize the polaron Hamiltonian on the subspace {|Se, 1qnin ¼ 1,2,. . ., |Pe, 0qi}. The matrix elements can be written:

Hpe ¼ He þ Hph þ He ph X Hph ¼ hoLO a~q a~þ q

hSe ; 1qn jHpe jPe ; 0qi

(3) (3-a)

~ q

He ph

iA X i~q~re ¼

ðe a~q e i~q~rei a~þ qÞ q ~q

(3-b)

3

A2 ¼ 4pðhoLO Þ

(4-a)

hPe ; 0qjHpe jPe ; 0qi

(4-b)

¼

Epe

¼ hSe ; 1qn jHe ph jPe ; 0qi ¼ Vqn

(4-c)

Such problem admits exact solutions:

with 1=V a F 2ðh2 =2me Þ2

hSe ; 1qn jHpe jSe ; 1qn0 i ¼ ðESe þ hoLO Þdnn0

(3-c)

Hph is the longitudinal optical (LO) phonons Hamiltonian. He–ph is the Fro¨ hlich Hamiltonian. aF and V are the Fro¨ hlich constant and the dot volume, respectively. a~þ q Þ is the creation (annihilation) operator of LO q ða~ phonon considered as dispersionless with frequency oLO. In the following, we restrict our consideration to the lower lying polaron states in which the number of phonon per mode is at most one and only the |Sei and |Pei electronic states included. So, we have to

e E ¼ 12 ðESe þ EPe þ hoLO Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R ð12 ðESe EPe þ hoLO ÞÞ2 þ dq3 jVq j2

(5)

These polaronic states will be estimated by a series over the basis set of the unperturbed Hamiltonian H ¼ He þ Hph given by the tensorial product between the two first carrier and the phonon eigenstates {|Se, 1qnin ¼ 1,2,. . ., |Pe, 0qi}. The polaron energies, measured from the ground electron state, are plotted versus the cone radius in Fig. 3. The material parameters [5] used are me ¼ 0.21mo, mo is the free electron mass, aF ¼ 0.526, hwLO ¼ 38:26 meV, Ve ¼ 897 meV. For comparison, we have also plotted, in this same figure, the energies of the non interacting states |Se, 1qi and |Pe, 0qi (dashed lines). These non interacting levels

Fig. 3. The polaron energies, measured from the ground state of the electron, plotted vs. the dot size. The dashed lines represent the unperturbed states |Se, 1qi and jP e ; 0qi.

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M. Triki, S. Jaziri / Applied Surface Science 238 (2004) 213–217

˚ . This cross means that the cross near r 145 A electronic levels separation is equal to hwLO . It is replaced by large anticrossing energy levels (25 meV). This is the rabi splitting of the electron levels caused by the electron phonon coupling. In such e anticrossing, many transitions: ESe ! E , ESe ! ESe þ e e hoLO , ES ! EP may take place because the wave functions of these levels are mixed in this region. In this case, electron–LO phonon interaction can never be shown as a weak coupling. In other hand, one can easily note that the rabi splitting obtained in CdS QDs is more significant than that obtained for GaAs ones [6] which seems as promising result. Considerable efforts are being devoted to the investigation of the effects due to the exciton–phonon interaction on the optical properties of quantum dots, reason for what we should outline the importance of the research of the excitonic polaron spectrum. The polaronic excitonic Hamiltonian can be expressed:

level spacing for electrons and holes is large compared to the temperatures T < 200 K explored experimentally. This allows us to restrict our attention to the lowest orbital levels in the conduction and valence band of the QD’s. After this discussion the excitonic Hamiltonian can be projected on the tensorial product of the two bases of these particles. It is noted that the Coulomb Hamiltonian do not couple two states with different Lz. The matrix element is given by:

Hpex ¼ Hex þ Hph þ Hex ph

(6-a)

Hex ¼ He þ Hh þ He h

(6-b)

e2 4pe0 er j~ re ~ rh j

(6-c)

hcne cnh jHex jcme cmh i ¼ Ene dne;me þ Enh dnh;mh Z 3 Z e2 d~ q d3~

re Þcme ð~ re Þei~q~re re cne ð~ 2 q e0 er Z d3~ rh Þcmh ð~ rh Þe i~qrh rh cnh ð~

He h ¼

He–h is the Coulomb Hamiltonian which couples the two particles. We should note that, the coupling between heavy hole states and light hole states in the valence band has been neglected on account of the strain effects and the large energy separation between the zone center states (sub-levels) in small dots. So the mixing of states from these two sub-bands is considerably reduced. Thus, the states of a hole in a dot can be described, in a way, similar to that of the electron states, only with different effective mass (which for a hole is no longer isotropic) and potential discontinuity at the interface. In the other hand, we consider small quantum dots, i.e. the strong confinement regime, which leads to well-separated electron and hole levels within the dot. This allows us to safely treat the Coulomb interaction as a perturbation [7,8]. In CdS/ ZnS II–VI semiconductor quantum dots, this energy spacing is about 100–350 meV, which is large compared to other III–V semiconductor quantum dots (for example it is about 50–150 meV for InAs/GaAs QDs). For CdS/ZnS II–VI semiconductor quantum ˚ , the single-particle dot with radii little than 150 A

hcne cnh jHex jcme cmh i e2 ¼ Ene dne;me þ Enh dnh;mh

4pe0 er Z Z c ð~ r Þc re Þcnh ð~ rh Þcmh ð~ rh Þ e me ð~ d3~ re d3~ rh ne j~ re ~ rj (7-a) Using Fourier transformation this matrix element is rewritten:

(7-b)

This expression can be easily deduced from the previous calculation of the Fro¨ hlich Hamiltonian. After diagonalizing such Hamiltonian, we obtain the excitonic states, which are slowly mixed with the Coulomb interaction. For example the excitonic ^ h i is mainly formed of |SePhi and slowly state jSe P mixed with |PeShi. Next, we consider the pair electron-hole as one particle interacting with phonons. Using the excitonic polaron Fro¨ hlich Hamiltonian given by: iA Hex ph ¼ He ph þ Hh ph ¼

q

X ðei~q~re a~q e i~q~rei a~þ qÞ ~ q

X

ðei~q~rhe a~q e i~q~rhi a~þ qÞ

(8)

~ q

we are able to deduce, by the same technique, the excitonic polaron spectrum.

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217

Fig. 4. Energies of the excitonic polarons (solid lines) as the function of the radius of the cone. The zero of the energy is chosen for the state Sh ; 0qi. The long dashed lines represent the exciton states and the short dashed lines represent the excitonic states plus one phonon. jSe ^

Similar results are obtained for excitonic polaron levels. In fact, two large anticrossings (11 and 13 meV) are obtained. In fact, Fig. 4 shows, for a self assembled CdSe/ZnSe quantum dot, the r dependence of the polaron energies measured from the ground exciton state (solid lines). We have also plotted, in this figure, the excitonic energy states (long dashed lines) and the excitonic energy states plus one phonon (short dashed states). We find two crossing. The first is for ˚ and it is between the ground state plus one r ¼ 95 A phonon jSe ^ Sh ; 1qi and the first excited state. The ˚ . It results from the second second is for r ¼ 155 A excited state plus one phonon jPe ^ Sh ; 1qi and the ^ h ; 0qi. After diagonalization third excited state jPe P the polaron Hamiltonian in the two subspaces ^ h ; 0qig and fjPe ^ ^ h ; 0qig, fjSe ^Sh ; 1qi; jSe P Sh ; 1qi; jPe P these two crossings are replaced by two large anticrossings: 11 and 13 meV, respectively. In conclusion, we have plotted the polaron energy spectrum in a self assembled II–VI quantum

dot. We find that the magnitude of the splitting between the coupled modes in these semiconductor quantum dots are enhanced in comparison with the III–V ones.

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