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Excitons, Biexcitons, and Trions in CdSe Nanorods F. Rajadell,† J. I. Climente,† J. Planelles,*,† and A. Bertoni‡ Departament de Quı´mica Fı´sica i Analı´tica, UniVersitat Jaume I, Box 224, E-12080, Castello´, Spain, and CNR-INFM S3, Via Campi 213/A, 41100 Modena, Italy ReceiVed: March 24, 2009; ReVised Manuscript ReceiVed: May 6, 2009
We calculate the recombination energies and probabilities of neutral excitons (X), singly charged excitons (X(), and biexcitons (XX) in CdSe quantum dots of variable length. For spherical dots the relative position of the emission lines is determined by the confinement. As the dot is elongated, however, Coulomb correlation overcomes single-particle effects and the emission line of X becomes more energetic than that of any other excitonic complex. Likewise, the recombination probability (τ-1) of spherical dots is characteristic of the strong confinement regime: τ-1(XX) ∼ 2τ-1(X() ∼ 4τ-1(X). However, these ratios are reduced with increasing length, as correlations enhance the emission of each excitonic complex at a different rate. Our results are compared with available experimental data. Introduction Colloidal semiconductor nanocrystals (NCs) are wet-chemically synthesized quantum dots with remarkable light emission properties in a wide spectral range, which renders them appealing for applications in photonic structures, quantum optics, photovoltaic technologies, lasing, or biolabeling.1-3 The optical emission of these nanostructures arises from the recombination of conduction band electrons with valence band holes, which prior to recombination exist as excitons (Coulomb interacting electron-hole pairs) or excitonic complexes, such as charged excitons or multiexcitons (multiple electron-hole pairs). For this reason, the physics of excitons and excitonic complexes in NCs has been a subject of intense study over the last few years.4 The synthesis of semiconductor nanorods (NRs), an elongated variant of colloidal dots,5,6 brought further interest to these nanoemitters. Compared to traditional spherical NCs, the anisotropic shape of NRs provides several advantages, which include the possibility of emitting linearly polarized light on a macroscopic scale,7,8 improved photostability and optical gain,9 reduced Auger recombination rates,10 and higher photovoltaic conversion efficiency.11 To date, however, the vast majority of works dealing with NRs refer to single excitons. This is in spite of the fact that more complex excitonic species are also relevant in many processes. For instance, biexcitons are usually involved in lasing from NCs12 and trions (singly charged excitons) are spontaneously generated when the system is subject to continuous excitation.13 Only recently the progress in single NR spectroscopy has enabled experimental studies focusing on such species.14,15 For these studies, little theoretical assessment is currently available, as only energetic aspects of InAs structures have been investigated.16,17 In this work, we use an effective mass-configuration interaction scheme to calculate the length dependence of the recombination energies and probabilities of various excitonic complexes in NRs, namely neutral excitons, positive and negative trions, and biexcitons. We investigate CdSe structures, which is regarded as the most prominent system for light emission1 * E-mail:
[email protected]. † Universitat Jaume I. ‡ CNR-INFM S3.
and is the material employed in the experiments of ref 14. The effective Bohr radius of this material is smaller than that of InAs. This brings CdSe NRs into a regime of strong Coulomb correlation,18 which is responsible for a different behavior as compared to that of InAs NRs.8,19 The effect of the rod radius and that of a passivating ZnS shell are also addressed. The general trend we find is that, contrary to the case in spherical NCs, in NRs the emission peak of the exciton is always higher in energy than that of the excitonic complexes. This is a result of the strong Coulomb correlation, which overrides singleparticle effects such as electron-hole confinement asymmetry. The recombination probabilities are also significantly affected by the correlation, so that important deviations from the strong confinement limit are observed. Theory The electron and hole single-particle states are described with three-dimensional effective mass Hamiltonians which, in cylindrical coordinates and atomic units, read as follows:20
Hi ) -
( (
)
1 F ∂ ∂ 1 1 ∂ ∂ + 2 F ∂F m* (F, z) ∂F ∂z m* (F, z) ∂z i,F i,z li2 * F2mi,F (F, z)
)
+ Vi(F, z)
(1)
where i ) e, h is a subscript denoting electron or hole, respectively, li is the azimuthal angular momentum, and Vi(F,z) is the spatial confinement potential, which is zero inside the outside. m*i,F(F,z) and m*i,z(F,z) are the nanostructure and Vout i position-dependent transversal and longitudinal effective masses, respectively. For electrons we use isotropic masses, m*e,F(F,z) ) For holes, however, the mass anisotropy is m*e,z(F,z) ) m*(F,z). e important, as it is responsible for the heavy hole-to-light hole ground state transition that occurs as the aspect ratio of the nanostructure increases.21,22 Thus, for heavy holes we use m*h,F ) 1/(γ1 + γ2) and m*h,z ) 1/(γ1 - 2γ2), while for light holes we use m*h,F ) 1/(γ1 - γ2) and m*h,z ) 1/(γ1 + 2γ2). Here, γ1 and γ2
10.1021/jp902652z CCC: $40.75 2009 American Chemical Society Published on Web 06/05/2009
Excitons, Biexcitons, and Trions in CdSe Nanorods
J. Phys. Chem. C, Vol. 113, No. 26, 2009 11269
are the Luttinger parameters.22 The Hamiltonian in eq 1 is integrated numerically using finite differences. The Hamiltonian for excitons is as follows (in atomic units):
HX ) He + Hh -
1 ε|re - rh |
(2)
where ε is the semiconductor dielectric constant. The Hamiltonian for positive trions (one electron, two holes) is as follows:
HX+ ) He +
∑ Hh(j) - ∑ ε|re -1 rh(j)| +
j)1,2
j)1,2
1 ε|rh(1) - rh(2)|
where erσ is the annihilation operator for an electron with spin σ (v or V) in the single-particle orbital r, hsσ′ is the annihilation operator for a hole with spin σ′ in the single-particle orbital s, x is a factor and 〈r|s〉 is the overlap between the two orbitals. Kσσ′ coming from the dipole coupling of electron (|Be〉) and hole (|Bh〉) Bloch functions. These functions are |Sv〉 and |SV〉 for electrons, |J+v〉 and |J-V〉 for heavy holes, and (-2/3)|Zv〉 + (1/3)|J+V〉 and (-2/3)|ZV〉 - (1/3)|J-v〉 for light holes, with J( ) (X ( iY)/2. For longitudinal light K| ) 〈Be|Z|Bh〉, and for the two transversal light components K⊥ ) 〈Be|J(|Bh〉. The resulting factors for heavy holes are
Kvv⊥ ) KVV⊥ ) K
(7)
KvV⊥ ) KVv⊥ ) 0
(8)
| Kσσ′ )0
(9)
(3)
The Hamiltonian for negative trions (two electrons, one hole) is identical but exchanging electron and hole indexes. Finally, the Hamiltonian for biexcitons is:
HXX )
∑
He(i) +
i)1,2
∑
Hh(j) -
j)1,2
∑ ε|re(i) 1- rh(j)| + i,j
1 1 + ε|rh(1) - rh(2)| ε|re(1) - re(2)|
(4)
The Hamiltonians in eqs 2-4 are solved using a configuration interaction (CI) procedure on the basis of the single-particle states. That is, we choose a basis of single-particle electron and hole eigenstates. If there is more than one electron or hole, we form all possible Slater determinants, including spin degrees of freedom. Next, we form Hartree products between the electron and hole Slater determinants (or spin-orbitals). The few-body Hamiltonian is then spanned on this basis and diagonalized. Coulomb integrals are evaluated on a fivedimensional real space using Monte Carlo routines.23 Note that the CI approximation is convenient to account for the strong Coulomb correlation of long CdSe NRs. One may expect Hartree-Fock-like approximations like that of refs 16 and 17 to fall short in these structures. For simplicity, the electron-hole exchange interaction24 is neglected in this study. The inclusion of this term splits the 4-fold degenerate ground state of the exciton, inducing changes in recombination probabilities. Yet, this will only produce a small departure from the general trends we report. The emission spectra are calculated within the dipole approximation and Fermi’s golden rule.25 The recombination probability from an initial state |i〉 to a final state |f〉 with one less electron-hole pair (e.g., the |i〉 state of X- and the |f〉 state of a single electron), at an emission frequency ω, is then given by the following: -1 τfri (ω)
(5) ˆ x is the polarization operator for transversal (x ) ⊥) or Here P longitudinal (x ) |) polarized light, x ∑ ∑ erσhsσ′〈r|s〉Kσσ′ rσ
sσ′
KvV⊥ ) -KVv⊥ )
(6)
-1 K √3
(10)
Kvv⊥ ) KVV⊥ ) 0
(11)
KvV| ) KVv| ) 0
(12)
Kvv| ) KVV| ) -
23 K
(13)
so that light holes emit both transversally and longitudinally polarized light. In the expressions above K is the Kane matrix element.26 Note that we have identified the crystolagraphic direction Z with the elongated axis of the NR. This is because CdSe NRs are grown along the c axis of the wurtzite crystal.5 We assume zero temperature and study the fundamental transition, which involves the ground states of the initial and final species.27 The energy of these transitions is calculated as follows:
EPL(X) ) EX,
EPL(X+) ) EX+ - Esh
EPL(X-) ) EX- - Ese,
ˆ ⊥ |i〉|2 + |〈f|P ˆ | |i〉|2) δ(Ei - Ef - pω) ) (|〈f|P
ˆ x) P
so that heavy holes only emit transversally polarized light. For light holes instead
EPL(XX) ) EXX - EX
where EX, EX(, and EXX are the ground state energies of the s neutral exciton, trion, and biexciton, respectively, while Ee and s Eh are the single-electron and single-hole ground state energies. The binding energies, which give the relative positions of the emission lines with respect to that of X, are as follows:28
Eb(X-) ) (E es + EX) - EX-
(14)
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Rajadell et al.
Eb(X+) ) (E hs + EX) - EX+
(15)
Eb(XX) ) 2EX - EXX
(16)
Note that the binding energies are defined with respect to the ground state energy of the dissociated excitonic complexes. Thus, positive (negative) values imply bound (unbound) complexes. Results and Discussion We study CdSe NRs composed of a cylinder of radius R nm and variable length L nm, attached to two hemispherical caps of the same radius R, as illustrated in the upper part of Figure 1. The material parameters are m*(F,z) ) 0.13 inside the e structure and 1 outside and (γ1, γ2) ) (1.66, 0.41) inside and (1.0, 0.0) outside.29 The confinement potential of the outer out medium is set to Vout e ) Vh ) 4 eV. For the CI expansion we build a basis set using the lowest-energy spin-orbital with angular momenta le(lh) ) ( 1 and the eight lowest with le(lh) ) 0. In all cases we calculate heavy and light hole excitonic complexes and then choose the lowest energy one as the ground state. Typically the ground state is formed by heavy holes for the nearly spherical dots limit, but it switches to light holes as the structure is elongated.21,22 Recombination Energies. Figure 2 shows the emission energies of a NR with R ) 2 nm, as a function of the rod length. The inset represents the corresponding binding energies (relative positions of the emission lines with respect to the neutral exciton). In the spherical NC limit (L ) 0), the neutral exciton resonance is in between the positive trion and the biexciton ones. This is in agreement with the experimental measurements of Oron et al.30 The sign of the binding energy reflects whether Coulomb attractions or Coulomb repulsions dominate the system (negative or positive energy, respectively).31,32 Thus, the higher energy of the X+ resonance as compared to that of X suggests that hole-hole repulsion is larger than electron-hole attraction.
Figure 1. Top: Schematic cross section of the structure under study. Bottom: Radial profile of the electron and hole ground state wave functions for R ) 2 nm (z ) 0).
Figure 2. Recombination energies in a NR of R ) 2 nm as a function of the rod length. The ground state is formed by heavy holes up to L ) 2.5 nm and by light holes from L ) 5 nm onward. The inset shows the binding energies. Lines are a guide to the eye.
Since the spherical NC is in the strong confinement regime, this can be readily understood from the single-particle orbitals involved: as shown in the bottom part of Figure 1, the hole ground state is more confined than the electron one. This asymmetry reduces the electron-hole overlap, as opposed to the hole-hole case, where the overlap remains large. This results in stronger Coulomb repulsion than attraction. In our calculations, for the dominant electronic configuration (two holes in the lowest lh ) 0 orbital and one electron in the lowest le ) 0 orbital), the hole-hole and electron-hole Coulomb integrals are Vhh ) 118.5 meV and Veh ) -106.8 meV, respectively. The question arises of why the X- resonance is lower in energy than that of X in spite of the electron-hole asymmetry. This is because the weaker confinement of electrons reduces their repulsion. In our calculations, at L ) 0, Vee ) 98.3 meV. With increasing L, as the dot changes from NC to NR, all the resonances red shift. This is due to the decrease of kinetic energy that follows from weakening the longitudinal confinement. After a certain length, the emission lines of X-, XX, and X+ are close to each other and show different behaviors (see Figure 2 inset).34 The fine details of this region are likely to differ in real samples with atomistic effects, confinement fluctuations, etc. Indeed, in quantum wires, the relative positions of the X+ and X- lines are known to be very sensitive to the confinement details.35,36 A more robust result is the fact that X+, which is initially bound (Eb(X+) > 0), becomes rapidly unbound as the dot is elongated. This a consequence of the increasing Coulomb correlation, which allows the system to minimize Vhh efficiently by localizing the holes on opposite sides of the rod.18 An important implication of the latter result is that the highenergy peak observed in recent single-NR photoluminescence spectra by Le Thomas et al.14 can no longer be assigned to a positive trion, as it shows up several millielectronvolts above the X peak and their rod is far from the spherical limit (R ∼ 2 nm, L ∼ 20 nm). We have further tested that the inclusion of a thin ZnS shell around the CdSe NR, as used in the experiment, barely changes the energies presented in Figure 2 (not shown).33 On the other hand, the presence of biexciton and negative trion resonances below that of a neutral exciton is in agreement with the experiment. A biexciton resonance a few millielectronvolts below the exciton line also agrees with measurements for long InGaAs NRs,15 as opposed to the theoretical estimates in ref 17, which, using a Hartree-Fock-like method, predicted a splitting over 100 meV. Another aspect of Figure 2 worth mentioning is that the binding energies reach a relative minimum and then increase
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J. Phys. Chem. C, Vol. 113, No. 26, 2009 11271
Figure 3. Recombination energies in a NR of R ) 2 nm with a typeII shell as a function of the rod length. The ground state is formed by heavy holes at L ) 0 nm and by light holes from L ) 2.5 nm onward. The inset shows the binding energies.
Figure 5. Recombination probabilities in a NR of R ) 2 nm as a function of the rod length. Top panel: absolute values. Bottom panel: relative values with respect to the neutral exciton. The ground state is formed by heavy holes up to L ) 2.5 nm and by light holes from L ) 5 nm onward. Lines are guides to the eye.
Figure 4. Recombination energies in a NR of R ) 4 nm as a function of the rod length. The ground state is formed by heavy holes for all values of L. The inset shows the binding energies.
again. This is due to the interplay between Coulomb and confinement energies, as explained in ref 16 for InAs rods. Here the maximum takes place for shorter rod lengths than in InAs (e.g., L ∼ 15 nm vs L ∼ 20 nm for X+), owing to the smaller effective Bohr radius of CdSe. Recent atomistic calculations of CdSe/ZnS NRs suggested that a thin ZnS shell may originate a type-II band confinement, such that the valence band minimum would be in the core and the conduction band one in the shell.37 To explore the effect a type-II shell would have on the previous results, in Figure 3 we illustrate the corresponding recombination and binding (inset) energies.33 The main effect of the shell is to increase the binding energies of X+ and XX. Indeed, both species are now bound near the L ) 0 limit. This result is consistent with the trends reported in experiments with CdTe/CdSe NCs,38 and it occurs because the type-II confinement potential enhances the electron-hole single-particle asymmetry, which translates into unbalanced Coulomb attractions and repulsions (see Figure 2 discussion). Still, with increasing L, Coulomb correlation overcomes this asymmetry again and the two species become unbound. It then follows that a type-II shell does not suffice to support the assignment of X+ as the high-energy peak in ref 14. Next, we investigate the effect of the NR radius by comparing the previous results with those of a structure with larger radius, namely R ) 4 nm. The recombination and binding energies for the latter structure are shown in Figure 4. Three main differences arise with respect to the results of R ) 2 nm: (i) there is an overall red shift due to the weaker radial confinement; (ii) X+
is no longer bound in the L ) 0 limit, because the weaker confinement makes the electron-hole asymmetry less relevant; (iii) for long L, X( resonances are lower in energy than XX ones, indicating that Coulomb repulsions are smaller. Recombination Probabilities. In this section we compare the recombination probabilities of X, X(, and XX as a function of the NR length. We consider a NR with R ) 2 nm. Thus, the optical transitions correspond to the resonances shown in Figure 2. The upper panel in Figure 5 illustrates the total recombination probability, including longitudinal and transversal light components. Except for a small damping between L ) 2.5 and L ) 5 nm, which is due to the heavy hole-to-light hole ground state crossover, τ-1 increases with L in all cases. This is due to Coulomb correlations, which are known to enhance excitonic emission,39 and it is one of the reasons why NRs exhibit stronger luminescence than spherical NCs.40 The figure reveals a different behavior of CdSe NRs as compared to that reported for selfassembled InGaAs QDs. In self-assembled QDs the recombination probability of neutral excitons increases with the dot size, but that of trions decreses because Coulomb repulsions may dominate over attractions.32 In NRs, however, the increasing length reduces repulsions so efficiently that not only the recombination probability of excitons but also that of trions increases. The lower panel in Figure 5 shows the ratio of the recombination probability of XX and X( as compared to that of X. For the spherical NC (L ) 0), τ-1(XX) ∼ 2τ-1(X() ∼ 4τ-1(X). These ratios are characteristic of the strong confinement regime and can be understood because (i) the biexciton ground state has twice as many decay channels as trions or excitons, and (ii) the exciton ground state is 4-fold degenerate, with two dark configurations.28,41 As the longitudinal confinement is weakened, however, the ratios decrease until an asymptotic limit
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Figure 6. Same as top panel of Figure 5 but for longitudinally polarized light only.
is reached. This is due to the fact that correlations favor exciton emission more than trion or biexciton emission.32 In the asymptotic limit, τ-1(XX) ∼ 2.6τ-1(X) and τ-1(X() ∼ 1.3τ-1(X). NRs are interesting for their capability to emit longitudinally polarized light. In Figure 6 we show the probabilities of recombining to emit longitudinally polarized light. As can be seen in eqs 6-9, only light holes contribute to this emission. Therefore, before L ) 5 nm, when the ground state is built from heavy holes, the recombination probability is zero. From L ) 5 nm on, the probability increases gradually for all complexes. Conclusions We have calculated the recombination energies and probabilities of excitons, trions, and biexcitons in CdSe nanocrystals with different aspect ratios. In the spherical dot limit, the system is in the strong confinement regime and the relative position of the optical resonances depends on the single-particle confinement details, such as the dot radius or the core-shell band offset. As the dot is elongated and correlation effects start dominating, Coulomb repulsions are minimized and attractions are maximized, with the former process being more efficient than the latter. Consequently, the neutral exciton resonance becomes the highest in energy regardless of the confinement details. This result rules out the tentative assignment of the high energy peak in spectra in ref 14 as coming from the positive trion. As for the recombination probability, in spherical nanocrystals the strong confinement imposes the following ratios: τ-1(XX) ∼ 2τ-1(X() ∼ 4τ-1(X). With increasing dot length, Coulomb correlation enhances the emission of all excitonic complexes. This effect is stronger for neutral excitons, so that the above ratios are reduced. Our results agree with different features observed in photoluminescence experiments of spherical and rod-shaped dots. Acknowledgment. Support from MCINN project CTQ200803344, UJI-Bancaixa project P1-1B2006-03, the Ramon y Cajal program (JIC), and Cineca Calcolo Parallelo is acknowledged. A.B. acknowledges UJI-Bancaixa for the INV-2008-01 grant. References and Notes (1) Woggon, U. J. Appl. Phys. 2007, 101, 081727. (2) Wang, Y. J. Nanosci. Nanotechnol. 2008, 8, 1068.
Rajadell et al. (3) Gomez, D. E.; Califano, M.; Mulvaney, P. Phys. Chem. Chem. Phys. 2006, 8, 4989. (4) Klimov, V. I. Annu. ReV. Phys. Chem. 2007, 58, 635. (5) Peng, X.; Manna, L.; Yang, W.; Wickham, J.; Scher, E.; Kadavanich, A.; Alivisatos, A. P. Nature 2000, 404, 59. (6) Kan, S.; Mokari, T.; Rothenberg, E.; Banin, U. Nat. Mater. 2003, 2, 155. (7) Hu, J.; Li, L.; Yang, W.; Manna, L.; Wang, L.; Alivisatos, A. P. Science 2001, 292, 2060. (8) Li, L.; Hu, J.; Yang, W.; Alivisatos, A. P. Nano Lett. 2001, 1, 349. (9) Htoon, H.; Hollingworth, J. A.; Malko, A. V.; Dickerson, R.; Klimov, V. I. Appl. Phys. Lett. 2003, 82, 4776. (10) Htoon, H.; Hollingworth, J. A.; Dickerson, R.; Klimov, V. I. Phys. ReV. Lett. 2003, 91, 227401. (11) Huynh, W. U.; Dittmer, J. J.; Alivisatos, A. P. Science 2002, 295, 2425. (12) Klimov, V. I.; Ivanov, S. A.; Nanda, J.; Achermann, M.; Bezel, I.; McGuire, J. A.; Piryatinski, A. Nature 2007, 447, 441. (13) Berciaud, S.; Cognet, L.; Lounis, B. Nano Lett. 2005, 5, 2160. (14) Le Thomas, N.; Allione, M.; Fedutik, Y.; Woggon, U.; Artemyev, M. V.; Ustinovich, E. A. Appl. Phys. Lett. 2006, 89, 263115. (15) Sek, G.; Podemski, P.; Misiewicz, J.; Li, L. H.; Fiore, A.; Patriarche, G. Appl. Phys. Lett. 2008, 92, 021901. (16) Baskoutas, S. Chem. Phys. Lett. 2005, 404, 107. (17) Baskoutas, S.; Terzis, A. F. J. Appl. Phys. 2005, 98, 044309. (18) Climente, J. I.; Royo, M.; Movilla, J. L.; Planelles, J. Phys. ReV. B 2009, 79, 161301(R). (19) Steiner, D.; Katz, D.; Millo, O.; Aharoni, A.; Kan, S.; Mokari, T.; Banin, U. Nano Lett. 2004, 4, 1073. (20) In eq 1 we have considered that the mixing of heavy hole and light hole subbands is weak. For the low-lying states of the structures we study, this can be shown to be true. (21) Wang, X.; Zhang, J.; Nazzal, A.; Darragh, M.; Xiao, M. Appl. Phys. Lett. 2002, 81, 4829. (22) Sercel, P. C.; Vahala, K. J. Phys. ReV. B 1991, 44, 5681. (23) Climente, J. I.; Planelles, J.; Rajadell, F. J. Phys.: Condens. Matter 2005, 17, 1573. (24) Zhao, Q.; Graf, P. A.; Jones, W. B.; Franceschetti, A.; Li, J.; Wang, L. W.; Kim, K. Nano Lett. 2007, 7, 3274. (25) Jacak L.; Hawrylak P.; Wo´js, A. Quantum Dots; Springer-Verlag: Berlin, 1998. (26) Kane, E. O. J. Phys. Chem. Solids 1957, 1, 249. (27) In quasi-one-dimensional NRs several states converge to the lowest energy subband. At finite temperature, the thermal population of these states modifies the emission probability in a complex way, see ref 17. (28) Narvaez, G. A.; Bester, G.; Zunger, A. Phys. ReV. B 2005, 72, 245318. (29) Laheld, U. E. H.; Einevoll, G. T. Phys. ReV. B 1997, 55, 5184. (30) Oron, D.; Kazes, M.; Shweky, I.; Banin, U. Phys. ReV. B 2006, 74, 115333. (31) Dalgarno, P. A.; Smith, J. M.; McFarlane, J.; Gerardot, B. D.; Karrai, K.; Badolato, A.; Petroff, P. M.; Warburton, R. J. Phys. ReV. B 2008, 77, 245311. (32) Climente, J. I.; Bertoni, A.; Goldoni, G. Phys. ReV. B 2008, 78, 155316. (33) The core NR was the same as that for Figure 2, and a 5-Å-thick ZnS shell was added. The ZnS parameters used are m*(F,z) ) 0.28 and e (γ1, γ2) ) (1.95, 0.78). For a type-I (type-II) shell, we use VeZnS ) 1.4 (VeZnS ZnS )-0.5) eV and Vh ) 0.6 eV. (34) The seemingly discontinuous behavior of the binding energies in Figure 2 inset at L ) 5 nm is due to the heavy hole-to-light hole ground state transition. (35) Szafran, B.; Chwiej, T.; Peeters, F. M.; Bednarek, S.; Adamowski, J. Phys. ReV. B 2005, 71, 235305. (36) Tsuchiya, T. Int. J. Mod. Phys. B 2001, 15, 3985. (37) Mohr, M.; Thomsen, C. Phys. Status Solidi B 2008, 245, 2111. (38) Oron, D.; Kazes, M.; Banin, U. Phys. ReV. B 2007, 75, 035330. (39) Corni, S.; Brasken, M.; Lindberg, M.; Olsen, J.; Sundholm, D. Phys. ReV. B 2003, 67, 045313. (40) Shabaev, A.; Efros, Al. L. Nano Lett. 2004, 4, 1821. (41) Wimmer, M.; Nair, S. V.; Shumway, J. Phys. ReV. B 2006, 73, 165305.
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