6154
2007, 111, 6154-6157 Published on Web 04/05/2007
Radiative Recombination of Triexcitons in CdSe Colloidal Quantum Dots A. Franceschetti*,† and M.C. Troparevsky†,‡ National Renewable Energy Laboratory, Golden, Colorado 80401, and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 ReceiVed: January 10, 2007; In Final Form: March 1, 2007
Recent experimental observations of multiexciton recombination in colloidal CdSe quantum dots have revealed a high-energy emission band that has been attributed to the radiative recombination of a p-like electron with a p-like hole in a triexciton complex (three holes + three electrons). The reason for the occupation of p-like hole states in the triexciton, however, has remained elusive. Using atomistic pseudopotential calculations, we show here that p-like hole states are populated even at low temperature because of the relatively small Coulomb repulsion between s-like and p-like hole states, which leads to a non-Aufbau occupation sequence of the hole levels. We also show that the observed temperature dependence of the p-p emission band originates from the dark-bright splitting of the triexciton ground state. Our results provide a consistent explanation of the physical origin of the triexciton emission lines in CdSe quantum dots.
Radiative recombination of multiparticle excited states (e.g., multiexcitons and charged excitons) in semiconductor quantum dots leads to peculiar spectroscopic signatures, such as the appearance of multiple emission lines and the shift of the emission energies. These phenomena are driven by the enhancement of interparticle Coulomb and exchange interactions upon quantum confinement. Radiative recombination of multiexcitons and charged excitons has been widely reported in self-assembled quantum dots made of III-V1-3 and II-VI4,5 semiconductors. In colloidal semiconductor quantum dots, however, the primary decay channel for multiparticle excitations consists of nonradiative Auger recombination of electron-hole pairs.6 For example, a biexciton decays nonradiatively into a single exciton by promoting one of the two electrons to a higher-energy conduction-band level or one of the two holes to a lower-energy valence-band level. This energy-conserving process is enabled by the electronic coupling of exciton and biexciton states via screened Coulomb interactions.7 Typical nonradiative Auger recombination lifetimes of multiexcitons in CdSe quantum dots range between ∼10 and ∼100 ps.6,7 As a result, even at high excitation intensity the time-integrated photoluminescence signal comes almost entirely from the recombination of single excitons, created either by direct optical excitation, or by ∼ps nonradiative recombination of multiexcitons. The presence of this ultrafast, nonradiative recombination channel has been utilized to detect carrier multiplication in quantum dots.8,9 Recently, optical emission from short-lived, multiparticle excited states has been observed in CdSe colloidal quantum dots10-15 using either time-resolved, femtosecond photoluminescence spectroscopy10-14 or quasi-continuous-wave optical pumping.15 In those experiments, the appearance of additional emission peaks, both to the red and to the blue of the single* Corresponding author. E-mail:
[email protected]. † National Renewable Energy Laboratory. ‡ Oak Ridge National Laboratory.
10.1021/jp070223c CCC: $37.00
exciton emission peak, at high excitation intensity was attributed to multiexciton radiative recombination. In particular, a wellresolved, high-energy emission bandsabout 100-250 meV to the blue of the single-exciton emission peakswas assigned11,12,14,15 to the radiative recombination of a p-like electron with a p-like hole (p-p recombination) in a triexciton (TX) complex (three holes and three electrons). This interpretation, however, is problematic because in CdSe quantum dots up to several nanometers in diameter the two energy levels at the top of the valence band are both s-like16,17 and can accommodate up to four holes. As a result, one expects that in the TX ground state the p-like valence-band states are not occupied by holes, thus making p-p radiative recombination impossible. Several mechanisms have been proposed in the literature11,14 to explain the observed high-energy TX emission band. Caruge et al.11 suggested that p-p recombination could be enabled by thermal occupation of p-like hole states. Fisher et al.14 pointed out that in sufficiently large CdSe quantum dots the second highest hole state is p-like because in large dots the confinement energy of p-like hole states becomes smaller than the crystalfield splitting that separates the s-like hole states. This s-p hole crossing would allow direct p-to-p recombination from the TX ground state in large CdSe quantum dots. However, the highenergy TX emission band has been observed in CdSe dots as small as 2.7 nm in diameter,12 for which the second hole state is s-like.16,17 Another possibility, discussed by Caruge et al.,11 is that the high-energy TX emission band may originate from a formally forbidden p-to-s transition, which could be allowed by configuration mixing. It is also possible that p-like hole states can be populated as a result of a non-Aufbau occupation sequence of the hole single-particle levels. Indeed, non-Aufbau occupation of hole levels (in the absence of electrons) was inferred from hole-charging experiments for InAs colloidal nanocrystals18 and InAs/GaAs self-assembled quantum dots19 and was predicted theoretically for InAs and InP colloidal © 2007 American Chemical Society
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J. Phys. Chem. C, Vol. 111, No. 17, 2007 6155
Figure 2. Schematic diagram of the radiative recombination channels for a ground-state triexciton (a) in CdSe quantum dots. Note the nonAufbau configuration of the holes in the triexciton ground state. The triexciton can decay radiatively by s-s electron-hole recombination, leading to an excited biexciton (b), or by p-p recombination, leading to a ground-state biexciton (c).
Figure 1. Calculated triexciton (TX) emission spectra at room temperature (T ) 300 K, red solid line) and low temperature (T < 1 K, blue solid line) of a series of spherical CdSe quantum dots. Also shown is the calculated single-exciton (SX) emission spectrum at room temperature (dashed line). The emission lines have been broadened by a 40 meV Gaussian. The two main TX emission peaks are denoted as A and B. For comparison purposes, the room-temperature emission spectra have been shifted rigidly on the energy axis so that the band gap at room temperature coincides with the band gap at low temperature.
nanocrystals20 and for InAs/GaAs self-assembled quantum dots.21 Notwithstanding these suggestions, the origin of the highenergy emission band and the mechanisms of p-p radiative recombination in CdSe quantum dots remain largely unexplained. A fundamental understanding of multiexciton radiative recombination processes also has practical implications because time-resolved photoluminescence spectroscopy of multiexcitons is currently being utilized22 to detect the occurrence and measure the efficiency of carrier multiplication in quantum dots. In the present work, we use atomistic, many-body pseudopotential calculations to reveal the physical origin of the TX emission peaks in CdSe quantum dots. Figure 1 shows the calculated TX emission spectrum at low temperature and at room temperature for a series of nearly spherical CdSe quantum dots ranging in size from 3.0 to 5.6 nm. We find that (i) the emission spectrum shows two main peaks (denoted A and B in Figure 1), which shift to higher energy as the size of the dot decreases. (ii) The low-energy peak (A) is due to s-s recombination of the TX, while the high-energy peak (B) originates from p-p
recombination of the TX, as shown schematically in Figure 2. (iii) In the TX ground state, one of the holes always occupies a p-like state (Figure 2), even in the smallest quantum dots considered here. This is a result of reduced hole-hole Coulomb repulsion when one of the holes is in a p state, compared to the case where all three holes occupy s-like states. (iv) For R < 2.5 nm the two highest hole levels are both s-like, while for R > 2.5 nm the second highest hole level is p-like. (v) The p-p emission peak is thermally activated (Figure 1). This temperature dependence is not due to thermal occupation of p-like hole levels but rather to the dark-bright splitting of the TX ground state induced by electron-hole exchange interactions. The calculations were performed using the atomistic manybody pseudopotential approach described in ref 17. We consider nearly spherical CdSe quantum dots having the wurtzite lattice structure. The single-particle energy levels are obtained using the semiempirical pseudopotential method.23 In this approach, the potential experienced by the electrons in the quantum dot is given by the superposition of screened atomic pseudopotentials, which are fitted to reproduce transition energies, effective masses, and deformation potentials of bulk CdSe. The Cd and Se atoms at the surface of the quantum dot are passivated by ligand-like potentials (centered along the direction of the surface dangling bonds), which act to remove surface-state energy levels from the dot band gap. The single-particle energies and wave functions are obtained by solving the Schro¨dinger equation in a plane-wave basis set and are then used as input for a manybody configuration-interaction (CI) calculation of the excited states.17 Coulomb and exchange interactions between carriers (holes and electrons) are screened by a distance-dependent and size-dependent dielectric function.17 The CI basis set is constructed using the 24 highest-energy single-particle hole states and the 8 lowest-energy electron states. The diagonalization of the CI Hamiltonian yields the excited-state energies {En} and wave functions {Ψn} of the quantum dot. The emission spectrum is then calculated as
I(pω) ∝
∑i,f ni|〈Ψi| rˆ|Ψf〉|2 δ(pω - Ef + Ei)
(1)
where pω is the photon energy, ni is the occupation of the initial state i (which we assume to occur thermally and to follow the
6156 J. Phys. Chem. C, Vol. 111, No. 17, 2007
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Figure 4. Energy shift ∆TX between the TX p-p recombination peak and the single-exciton emission peak as a function of the energy of the first absorption peak (black line and symbols). ∆ABS is the calculated difference between the first p-p linear-absorption peak and the singleexciton emission peak (red line and symbols).
Figure 3. The four highest-energy hole levels of CdSe quantum dots are shown in part a as a function of the dot radius. Note the crossing between the hs2 and hp1 energy levels for R ∼ 2.5 nm. Part b shows the direct Coulomb energies between two holes in the states hs1 and hs2 (red solid line) and between two holes in the states hs1 and hp1 (blue solid line).
Boltzmann distribution), and the sums run over the initial (e.g., triexciton) and final (e.g., biexciton) states. Our results are discussed in the following. Origin of p-p Radiative Recombination. The four singleparticle energy levels at the top of the valence band are shown in Figure 3a as a function of the dot radius. Each energy level is doubly degenerate because of spin multiplicity. The energy levels are labeled hs1, hs2, hp1, and hp2 according to the dominant angular-momentum character (s or p) of their envelope function. The two s-like energy levels hs1 and hs2 are separated by the crystal-field splitting induced by the CdSe wurtzite lattice structure. The crystal-field splitting depends weakly on the dot size. The splitting between s and p levels, however, is due to quantum confinement effects and decreases rapidly as the size of the dot increases. As a result, the levels hs2 and hp1 cross around R ) 2.5 nm. This crossing was qualitatively predicted by Fisher et al.14 A similar crossing between s and p hole levels was reported in pseudopotential calculations of CdS quantum dots.24 On the basis of the single-particle energy levels of Figure 3a, one would expect that, for R < 2.5 nm, the three holes in the TX complex would assume the Aufbau configuration (hs1,hs1,hs2). An analysis of the CI many-particle wave functions shows instead that in the TX ground state the holes have the non-Aufbau configuration (hs1,hs1,hp1), while the electrons are in the (es,es,ep) configuration, as shown schematically in Figure 2a. The contribution of this configuration to the TX groundstate correlated wave function ranges from ∼71% for the 2.8nm-radius quantum dot to ∼80% for the 1.5-nm-radius dot. The reason that the (hs1,hs1,hp1) configuration is energetically more favorable than the (hs1,hs1,hs2) configuration is that the Coulomb
energy Js1,s2 (interaction between a hole in the hs1 state and a hole in the hs2 state) is significantly larger than Js1,p1 (interaction between a hole in hs1 and a hole in hp1), as shown in Figure 3b. In fact, the difference Js1,p1 - Js1,s2 (Figure 3b) is larger than the difference between the hs2 and hp1 single-particle energies (Figure 3a) for all of the dot sizes considered here. As a result, one of the holes always occupies a p-like leVel in the TX ground state. The occupation of a p level in the TX ground state enables two different channels for TX radiative recombination, as shown schematically in Figure 2. In the s-s recombination channel, the final state is a biexciton (BX) in the excited configuration (hs1,hp1;es,ep). This transition corresponds to the low-energy emission peak (peak A) in Figure 1. The s-s transition splits into multiple emission lines because of the threefold degenerate character of the electron p levels and the exchange splitting of the final BX states. The fine structure of the s-s recombination peak is visible, even after broadening, in the emission spectrum of the smallest dots (see Figure 1). In the p-p recombination channel, the final state is a ground-state BX in the (hs1,hs1;es,es) configuration. This channel corresponds to the high-energy, thermally activated emission peak (peak B) in Figure 1. Because the ground state of the BX is nondegenerate, the p-p transition does not split into multiple lines. Our results suggest that the p-p emission peak does not originate from thermal occupation of excited p-like hole states or from radiative recombination of a p-like electron with an s-like hole, as proposed previously.11 Instead, the p-p emission peak originates from the occupation of a p-like hole state in the TX ground state. Comparison with Experimental Data. We compare next our results with available experimental data. Figure 4 shows the calculated energy difference ∆TX between the triexciton p-p emission peak and the single-exciton s-s emission peak (see Figure 1) as a function of the energy of the first absorption peak. Our results are in very good agreement with the experimental results of Caruge et al.11,14 and Bonati et al.12 Note that the experimental error bars for ∆TX are larger than 20 meV for quantum dots in this size range.11 Oron et al.15 pointed out that their quasi-continuous-wave optical pumping method tends to emphasize larger dots because they have a larger absorption cross section. That might explain the smaller values of ∆TX reported in ref 15 compared to the experimental results of ref 12 and to our calculated results (Figure 4). Also shown in Figure
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J. Phys. Chem. C, Vol. 111, No. 17, 2007 6157 In conclusion, using atomistic many-body pseudopotential calculations we have shown that the observed high-energy emission band in highly excited CdSe colloidal quantum dots originates from p-p recombination of triexcitons. This transition is made possible by the non-Aufbau occupation of p-like hole states in the triexciton ground state. We have also determined that the temperature dependence of the p-p recombination peak is due to the dark-bright splitting of the TX ground state, induced by electron-hole exchange interactions. Our results provide a consistent explanation of the different peaks observed in the triexciton emission spectrum.
Figure 5. Calculated exchange splitting ∆ex of the TX ground state as a function of the dot radius.
4 is the energy difference ∆ABS between the first p-to-p linearabsorption peak and the single-exciton emission peak. Again, we find very good agreement with the experimental data of refs 11 and 12. The blue shift of ∆ABS with respect to ∆TX originates from the large interparticle interactions in the TX ground state. Temperature Dependence of TX Radiative Recombination. Finally, we address the origin of the temperature dependence of the TX emission peaks. We see from Figure 1 that, although the s-s emission peak (peak A) is rather insensitive to temperature, the p-p emission peak (peak B) is observed only at room temperature. Interestingly, Caruge et al.11 reported quenching of the p-p emission peak in 2.3-nm-radius CdSe quantum dots at low temperature. Our calculations show that the temperature dependence of the p-p transition is not due to thermal occupation of p-like hole excited states because p-like hole levels are occupied even in the TX ground state. Instead, we find that the temperature dependence originates from the dark-bright splitting of the TX ground state. Indeed, electronhole exchange interactions split the fourfold degenerate TX ground state into two doublets, separated by the exchange splitting ∆ex. This splitting is similar in origin to the singleexciton exchange splitting25,26 and is due to the presence of an unpaired electron and an unpaired hole in the TX ground state (see Figure 2). Figure 5 shows the exchange splitting ∆ex as a function of the dot radius. We find that ∆ex decreases from ∼7 meV for R ) 1.5 nm to ∼2 meV for R ) 2.8 nm. In the p-p recombination channel, the final state is the BX ground state (Figure 2c), which is a spin singlet state. Optical recombination from the low-energy TX doublet is spin-forbidden (“dark”), while recombination from the high-energy TX doublet is allowed (“bright”). Thus, a temperature kBT > ∆ex is necessary to populate the bright state, which explains why the p-p emission peak is not observed at low temperature (Figure 1). In the s-s recombination channel, however, transitions from the TX ground-state doublet to BX excited states (Figure 2b) are optically allowed. Therefore, the s-s emission peak is observed even at low temperature (Figure 1).
Acknowledgment. This work was funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Contract No. DE-AC36-99GO10337 to NREL. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. References and Notes (1) Dekel, E.; Gershoni, D.; Ehrenfreund, E.; Spektor, D.; Garcia, J. M.; Petroff, P. M. Phys. ReV. Lett. 1998, 80, 4991. (2) Warburton, R. J.; Scha¨flein, C.; Haft, D.; Bickel, F.; Lorke, A.; Karrai, K.; Garcia, J. M.; Schoenfeld, W.; Petroff, P. M. Nature 2000, 405, 6789. (3) Ware, M. E.; Stinaff, E. A.; Gammon, D.; Doty, M. F.; Bracker, A. S.; Gershoni, D.; Korenev, V. L.; Baˇdescu, S. C.; Lyanda-Geller, Y.; Reinecke, T. L. Phys. ReV. Lett. 2005, 95, 177403. (4) Besombes, L.; Kheng, K.; Marsal, L.; Mariette, H. Europhys. Lett. 2004, 65, 144. (5) Akimov, I. A.; Flissikowski, T.; Hundt, A.; Henneberger, F. Phys. Status Solidi A 2004, 201, 412. (6) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011. (7) Wang, L. W.; Califano, M.; Zunger, A.; Franceschetti, A. Phys. ReV. Lett. 2003, 91, 056404. (8) Schaller R. D.; Klimov, V. I. Phys. ReV. Lett. 2004, 92, 186601. (9) Ellingson, R. J.; Beard, M. C.; Johnson, J. C.; Yu, P. R.; Micic, O. I.; Nozik, A. J.; Shabaev, A.; Efros, Al. L. Nano Lett. 2005, 5, 865. (10) Achermann, M.; Hollingsworth, J. A.; Klimov, V. I. Phys. ReV. B 2003, 68, 245302. (11) Caruge, J. M.; Chan, Y.; Sundar, V.; Esler, H. J.; Bawendi, M. G. Phys. ReV. B 2004, 70, 085316. (12) Bonati, C.; Mohamed, M. B.; Tonti, D.; Zgrablic, G.; Haacke, S.; van Mourik, F.; Chergui, M. Phys. ReV. B 2005, 71, 205317. (13) Fisher, B.; Caruge, J. M.; Zehnder, D.; Bawendi, M. G. Phys. ReV. Lett. 2005, 94, 087403. (14) Fisher, B.; Caruge, J. M.; Chan, Y. T.; Halpert, J.; Bawendi, M. G. Chem. Phys. 2005, 318, 71. (15) Oron, D.; Kazes, M.; Shweky, I.; Banin, U. Phys. ReV. B 2006, 74, 115333. (16) Efros, Al. L. Phys. ReV. B 1992, 46, 7448. (17) Franceschetti, A.; Fu, H.; Wang, L. W.; Zunger, A. Phys. ReV. B 1998, 60, 1819. (18) Banin, U.; Cao, Y.; Katz, D.; Millo, O. Nature 1999, 400, 542. (19) Reuter, D.; Kailuweit, P.; Wieck, A. D.; Zeitler, U.; Wibbelhoff, O.; Meier, C.; Lorke, A.; Maan, J. C. Phys. ReV. Lett. 2002, 94, 026808. (20) Franceschetti, A.; Zunger, A. Europhys. Lett. 2000, 50, 243. (21) He, L. X.; Bester, G.; Zunger, A. Phys. ReV. Lett. 2005, 95, 246804. (22) Schaller, R. D.; Sykora, M.; Jeong, S.; Klimov, V. I. J. Phys. Chem. B 2006, 110, 25332. (23) Wang, L. W.; Zunger, A. Phys. ReV. B 1995, 51, 17398. (24) Demchenko, D. O.; Wang, L. W. Phys. ReV. B 2006, 73, 155326. (25) Nirmal, M.; Norris, D. J.; Kuno, M.; Bawendi, M. G.; Efros, Al. L.; Rosen, M. Phys. ReV. Lett. 1995, 75, 3728. (26) Franceschetti, A.; Wang, L. W.; Fu, H.; Zunger, A. Phys. ReV. B 1998, 58, R13367.
