Surface Science 361/362 (1996) 818-821
Effects of collective modes on single particle electronic excitations of semiconductor quantum wells and quantum dots P.A. Knipp a,,, Stephen W. Pierson b, T.L. Reinecke b • Department of Physics and Computer Science, Christopher Newpo~ University, Newport News, VA 23606-2998, USA b Naval Research Laboratory, Washington, DC 20375-5347, USA Received 21 June 1995; accepted for publication 20 September 1995
Almraet We have studied the interaction of collective excitations with single particle electronic excitations in semiconductor quantum wells and quantum dots. It is found that "hybrids" of these excitations are formed, and we calculate their energies. In the case of the quantum wells, one of them is a coupled intersubband plasmon-LO phonun mode which is localized at a neutral donor, and the other is a donor electronic excitation which is ~dressed" by coupled intersubband pl~qmon-LO phonon mode~ For quantum dots, these excitations are hybrids of LO phonon modes and a transition between different states of an electron confined to the dot. For dots we include both confined and interface phonons. We comment on the comparison of these results with experimental data.
Keywords: Ousters; Copper chloride; Gallium arsenide; Many body and quasi-particle theories; Photoluminescence; Quantum effects; Quantum wells; Semiconductor-semiconductor heterostructures
The possibility of an LO phonon hybridizing with the ls to 2p excitation of a neutral donor in bulk semiconductors has been studied extensively theoretically  and experimentally . The resulting excitations have been interpreted as phohens bound in the vicinity of the donor and as donor electronic excitations which are "dressed" by phonons. As the methods for growing samples and for producing semiconducting nanostructures has improved, the interest in phonons (or other collective excitations such as plasmons) bound to neutral donors has extended to lower dimensions. For example, in Raman studies of Si-doped * Corresponding author. Fax: + 1 804 5947919, e-mall: p k n i ~ c n u . e d u
GaAs/AI~GaI_~As quantum wells, excitations which depend on well width and carrier density have been observed 13] and have been attributed to coupled plasma-LO phonon modes bound at the neutral donors. Rew.ently we have made a quantum mechanical treatment of the binding of these collective excitations at neutral donors in quantum wells 14] and have shown that the results account for the well-width and carrier density dependence of these data. In quantum dots, a related phenomena occurs in the absence of neutral donors: the LO phonons hybridize with the transition between the groundand first-excited electronic state of the dot. In this case, the quantum dot acts as an artificial atom in which the confinement of the electrons is due to geometry. Recent photolumlnescence (PL) experimeats I-5] on CuC1 nanocrystals suggest that the
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P.A. Kntpp et al./Surface Science 361/362 (1996) 818--821
optical transitions in these structures involve such hybrid excitations. Here we briefly describe how the hybridization of LO phonons (or other collective excitation) with electronic transitions is modified as one goes to the quantum well and to the quantum dot geometries. Note that in the case of the quantum well, the single-particle electronic excitation involves low-lying states of a donor, and in the case of a quantum dot it involves electronic states confined to the dot. The mechanism for the hybridization is similar in both cases.
2. Hybrid energies In this section we give results of calculations of (A) the energies of hybrid excitations involving intersubband plasmon-LO phonon modes coupled with electrons at neutral donors in quantum wells, and (B) confined and interface LO phonons coupled with electrons in quantum dots. We use almost-degenerate BriUouln-Wigner perturbation theory  to obtain the coupling of the collective excitations and the single-particle excitations. In the present work we go beyond our previous work for wells  by finding exactly all of the hybrid excitations of these systems without approximations for the denominators in the perturbation theory.
2.1. Quantumwell Here we consider a neutral hydrogenic donor located in the center of a GaAs/Al=Gal_xAs quantum well, which is the system studied experimentally in Ref. . We consider a two-dimensional carrier density n~, in the quantum well, which in the experiment originates from other donors. In order to obtain the coupled excitations we extend the formulation of Ref. , which was used to study LO phonons bound to donors in bulk. In this way we derive the following equation for the hybrid energies, E: 2
E--E~--I(EIAE+E_AE_2E=,~) =0, (1)
where E,~o is the energy of the collective excitation [=ho~±, the upper (+) and lower (--) branches of the coupled intersubband plasmon-LO phonon modes of the quantum well], AE is the splitting of 'the ground- and first-excited state of the neutral donor in the well and I is given in Ref. :
\~l, + %1,./
(o~. - ~ o ) ~
x 2x,'n~-~-~ita+(ca2+- ta2__)(co~--o~)'
where ~ol is the bare transition energy between the ground- and first-excited subbands of the quantum welL torero) is the LO (TO) phonon frequency, and oht=8OL3n2Deo/27~aBc~,. ~u (~2p.) is the radial extent of the ls (2p.) donor wave function, aB is the donor Bohr radius, and L is the quantum well width. The frequencies of the coupled plasmon-LO phonon modes are given
by: o~: = ( o ~ + o~,)/2 + ~/(C~o + ~,)2 _ 4 a ~ , ( ~
+ ,.,,, O~o)/e/2, (3)
where c%l=~ol(1 + ~11)1/2/~ is the intersubband plasmon frequency, which is assumed to be dispersioniess here, as are the phonon frequencies. Eq. (1) is essentially a cubic equation having three real solutions for E. Because I is su~ciently small, one of these solutions is near E,~o, one is near AE, and one is near AE + 2E~.~. The first is a collective excitation (here, a plasmon-LO phonon) which is bound to the donor, and the second is an electronic excitation of the donor which is dressed by the collective excitation. Our results for the binding energies (Eb± = E-~u~±) of the collective,-excitation-like solutions are shown in Fig. 1 along with the data from Ref. . The general trend of the experimental data and of our results agree reasonably well. We find that the well-width dependence of the binding energies is richer than suggested by the experimental data, which was obtained for a limited range of well widths.
P.A. Knipp et at~Surface Science 361/362 (1996) 818-821
o % o
2 s~ ~',
ss ~ s S s~
I-JaB Fig. 1. The binding energy of the hybridized mode which is associated with the upper (lower) plm~mon-LO phonon is plotted as a clashed (solid) line as a function of the quantum well width. The experimental data of Ref.  are also included. The diamonds ($qUaX~) represent the binding energy of the upper (lower) mode. In the absence of coupling to the donor transition these binding energie, would vanish_
state [(l,m) = (0,0)] to one of its threefold-degenerate first-excited states [(l,m)=(1,0) or (1,+1)]. Conservation of angular momentum requires that the phonons participating in these transitions are those for which (l,m) = (1,0) or (1,+ 1). This condition is satisfied by an infinite number of confined phonon modes (having different radial "confinement numbers") and by three IF phonon modes. The energies of the three (degenerate) IF phonons are smaller than that of the confined phonons, which we take to be the dispersionless fU~o of the bulk. We generalize the approach of Ref.  to obtain the following extension of Eq. (1) above to find that the determinant of the following matrix must equal zero:
"E - En~ --
+ F - 2E----~+ F - 2ELo/
1 )I (4)
Now we consider the coupling o f LO phonons with an electronic transition of a spherical quantum dot of CmAs of diameter d embedded in a nonpolar material having e~o the same as that of GaAs. The carrier states are confined by the dot shape, not by an impurity. We are interested in the hybridization of this electronic transition with both "interface" (IF) phonon modes as well as "confined" phonon modes of the dot. This model of a spherical dot illustrates well the concepts involved in the case of quantum dots. Other dot shapes and materials can be treated by a genera!iTation of the present work. Previous workers  have considered such hybridiTation with the confined modes alone, but electron-phonon interactions are in fact dominated by interface phonons for nanostructure sizes small enough [typically d < O (10 nm)] that the energetic splitting between the electronic states exceeds the phonon energy . For the spherical dots studied here, the optical phonons and the electronic states can be classified by angular quantum numbers (l,m). We consider virtual transitions of an electron from its ground
E - - ELO -- k--F--
F - - 2ELo
~ ] l
where In, and Ioo~r are the quantum dot analogs of Eq. ( 2 ) , F = E - AE, AE = 20.6~/(mofrd2) is the transition energy between the ground- and firstexcited states, and m ~ is the effective mass of the electron. Eq. (4) is essentially a ninth degree polynomial in E, having nine real roots. These energies correspond to hybrids of the electronic excitations, the IF phonon modes, and the confined phonon modes of the quantum dot system. Because Iw and I~onr are small, one of the solutions for E lies near Ew, one near ELO, and one near AE, for most values of AE. The first (second) solution is an IF (confined) phonon mode whose energy is shifted owing to its hybridization with the electronic excitation. The third solution is an electronic excitation of the dot which is dressed by a combination of IF phonons and confined phonons. The remaining six solutions form three nearly-degenerate doublets which are energetically near the sum of AE and respectively
P.A. Kn#Tp et aL/~urface Science 361/362 (1996) 818--821
action) to charge carriers, and whose energy lies only slightly above the bulk TO energy. Detailed calculations of the optical intensities of these transitions should clarify this identification.
24 d (nm)
Fig. 2. Energies of a spherical GaAs dot as a function of the dot cli~tmctcrd. Solid lilies represent the energies of the hybrid excitations, short-,~qhod lines represent the bare IF- and confined-phonon energies, and the long-dashed line represent the bare electronic excitation of the dot.
either 2Ew, 2ELo, or Ew + ELO, and they are not the main interest of the present work. We have calculated these energies E for varying values of the dot diameter d, and the results are plotted in Fig. 2. For dot diameters d such that AE lies either considerably above or considerably below ELO and EnF, the hybrid energies lie very close to either Ew, EI~, AE, or a simple sum of these. However for AE sufficiently near ELO or Em we see avoided crossings, indicating that for quantum dot systems of this size the collective excitations can hybridize strongly with the singleparticle electronic excitations. In their recent PL studies of CuC1 dots, Itch et al.  have seen Stokes shifts whose energies lie close to the bulk LO phonon energy of CuC1. In addition they see features which they attribute to coupling to TO phonons of CuC1. However, from our results here we suggest that these features may not be TO phonons, which couple only weakly to charge carriers, but instead may be IF phonons, which couple more strongly (via the Fr6hlich inter-
The effects of confinement on the energies of hybrid excitations formed from collective modes and localized electronic transitions in quantum wells and quantum dots are studied here. In quantum wells, the collective excitations are cou= pled intersubband plasmon-LO phonon modes, and the electrons are in hydrogenic states of neutral donors. In quantum dots, the collective excitations are confined and interface LO phonons, and the electrons are confined to the quantum dot, an artificial atom.
Acknowledgements This work was supported in part by contract of the U.S. Office of Naval Research and by a National Research Council Research Associate-
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