Solar irradiance can potentially become an unlimited source of clean and renewable energy if a way could be found to greatly increase the efficiency of solar energy conversion. Today, the power conversion efficiency of the standard photovoltaic solar cells based on a single p-n junction, in which one photon generates one electron-hole pair, does not exceed the Shockley-Queisser limit⁵ of 32%. The major limiting factor is connected with a transfer of the largest part of the energy of solar photons into heat, because such a cell utilizes only part of the absorbed photon energy equal to the free energy available from the semiconductor energy gap. One of the possibilities to increase the efficiency of photovoltaic conversion is to produce a second electron-hole pair through impact ionization of the valence band.⁶

Impact ionization is a process during which an electron with kinetic energy larger than the semiconductor energy gap excites a second electron from the valence band creating an electron-hole pair.^7-9 This process is inverse to a direct Auger process during which an electron-hole pair annihilates nonradiatively, transferring its energy to another electron. Both the direct and inverse Auger processes are controlled by the same transition matrix element where the "initial" and "final" states are coupled via a multielectron Coulomb interaction. In bulk semiconductors, the coupling is suppressed due to a substantial mismatch of the initial and final quasi momenta of carriers.¹⁰

Strong confinement of carriers, however, significantly enhances the Coulomb coupling because quasi momentum is a bad quantum number in nanocrystals (NCs) and all carriers are localized in the same volume. It was shown both theoretically^11,12 and experimentally^13,14 that direct Auger processes are greatly enhanced in NCs. These observations allowed Nozik to suggest that the inverse Auger processes-impact ionization-must also be enhanced.⁶ Indeed soon after this prediction, an efficient generation of multi-electron-hole pairs by a single photon was observed in several semiconductor NCs.^1-4

In bulk, both the direct and inverse Auger processes can be considered in the framework of perturbation theory^7-10 because final states undergo fast decay into a continuous spectrum of the three-dimensional space. The strong confinement in NCs not only increases the strength of the multielectron Coulomb interaction but also changes the physics of various transition processes. In any transitional process the "initial" and "final" states always interact via a Coulomb potential. As a result, in small NCs, the large energy of confined carriers is an insufficient condition for pertubative consideration of the transitions caused by the Coulomb interaction. The perturbative approach can be used only if the decay of the "final" state due to dephasing or energy relaxation is much faster than the transition time. Otherwise, the Coulomb coupling returns the excitation back to the "initial" state mixing the two states into a superposition.

In this Letter, we show that efficient multiple exciton generation (MEG) by a single photon in NCs is connected with the formation of new excitations, which are described as a coherent superposition of single and multiexciton states. The developed theory uses a time-dependent density matrix approach, which allows simultaneous consideration of an arbitrary strength coupling between single and multiexciton states, different dephasing rates for these states, and short pulse excitation of NCs. The steady-state solution of the density matrix equations gives us restrictive conditions for efficient MEG: the energy relaxation rate of a single exciton initiated by light, ₁, must be slower than both the energy relaxation rate of the multiexciton, ₂, and the rate of Coulomb coupling between the two states, W_C/, where W_C is the matrix element of Coulomb interaction between the single and multiexciton states. In the case of strong coupling, ₂ < W_C/, the MEG by a short light pulse is accompanied by quantum beats with the frequency W_C/, which are connected with the creation of the coherent superposition. These oscillations are damped in the opposite case of weak coupling, ₂ > W_C/. The MEG, however, can still be very efficient.

The density matrix approach is the only self-consistent method that takes into account the diverse processes responsible for the MEG in NCs. Although this approach can be applied to NCs with arbitrary symmetry of conduction and valence bands, below we discuss the MEG only in PbSe NCs. Bulk PbSe is a narrow gap semiconductor, in which the energy gap, E_g = 0.28 eV at room temperatures, is determined by the four equivalent L points of the Brillouin zone and where conduction and valence bands have almost mirror symmetry.¹⁵ The energy spectrum in each L valley is described very well by the Mitchell and Wallis 4-band theory.¹⁶ The energy level structure in PbSe NCs can be calculated using the multiband effective mass theory of Kang and Wise.¹⁷ The size dependence of the several lowest electron and hole levels is shown in the inset of Figure 1.¹⁸ The energy levels are labeled by a level number, n = 1, 2, 3, ..., and an angular momentum, L = 0(S), 1(P), 2(D) 3(F), 4(G), ....

Figure 1 Coherent superpositions of single electron and trion states with total energy larger than two effective energy gaps of the NCs. (a) Electron-hole configurations of the coherent superpositions created by 2Pe (red) and 2De (blue) electrons in PbSe NCs with diameter d. (b) Energy size depenedence of the 2Pe> electron and 1Se1Se1Sh> trion coupled states (red); and of the 2De> electron and 1Pe1Se1Sh> trion coupled states (blue). Inset: Size dependence of the lowest electron and hole levels calculated in PbSe NCs. The arrows show the resonant transitions caused by Coulomb interactions between conduction 2Pe (red) and 2De (blue) electrons with valence band electrons occupying the 1Sh level.

In order to understand how a single photon creates many excitons, let us first consider a single electron occupying the 2P_e electron level. The reason for this level selection will be clarified later. The calculation shows that in a certain range of NC radii, the energy difference, E_2Pe - E_1Se, between the 2P_e and the ground 1S_e electron levels, is almost equal to the lowest excitation energy of a single electron-hole pair, or the effective energy gap of the NC with radius a: E_g(a) = E_1Se - E_1Sh. As a result, a single electron occupying the 2P_e level has the same energy as a trion consisting of two electrons and a hole occupying the ground quantum size 1S_e,h levels: these resonant electron-hole configurations are shown in Figure 1a. For these two configurations, the calculated size dependence of the total energy is shown in Figure 1b. The size dependencies shown in Figure 1b are simplified. Although the Kang and Wise model¹⁷ gives different energies for the nL^L+1/2 and nL^L-1/2 levels with different total angular momenta J = L ± 1/2, we do not show separately the fine structure of the levels connected with spin-orbit splitting. In addition, the intervalley interaction splits the 24 times degenerate P and 40 times degenerate D states forming "P" and "D" bands. An additional splitting can be caused by the effects of nonspherical NC shape, differences in the surface facets, and surface passivation. Not withstanding, one can see that the energy of the 2P_e electron is very close to the energy of the trion over the whole range of NC sizes, which are important for applications.

On the other hand, the calculations of the Coulomb matrix element between left (2P_e) and right (1S_e1S_e1S_h) multielectron configurations in Figure 1a show that the two states are coupled because the matrix element is not zero. Generally, an average energy of interparticle Coulomb interaction is smaller than a typical energy of a particle confined in a small NC because the energy of confinement increases as 1/a² while the Coulomb energy only as 1/a with a decrease of the NC radius a. In the lowest order perturbation theory, the interaction can be considered in terms of the direct or inverse (impact ionization) Auger process. However, for the states with almost the same energy, the coupling is not small and cannot be considered perturbatively. The superposition of the single electron 2P_e state and the 1S_e1S_e1S_h trion state is an eigenstate of the total multielectron Hamiltonian of a NC and these states cannot be described within a single particle approximation.

Similar consideration can be applied to all other single electron levels with an energy larger than the energy of the 2P_e state. As an example, in Figure 1, we show the multielectron configuration and the total energy of the 1P_e1S_e1S_h trion state which creates a coherent superposition with the 2D_e electron level. A single electron state with energy larger than (n + 1)E_g(a) where n 2 can not only be coupled with trions but also can create a superposition with multielectron excitations consisting of n holes and n + 1 electrons. In PbSe NCs, similar consideration can be applied to holes due to the mirror symmetry of the valence and conduction bands. This means that all hole states with energy larger than 2E_g(a) also form superpositions with multi-electron-hole states.

Let us consider now an optical excitation of these mixed states in NCs. A single photon can only generate a single electron hole pair which is not an eigenstate of the multielectron Hamiltonian. From the above analysis of single carrier states, it follows that a single electron-hole pair state (an exciton) is coupled with multiexciton states. The corresponding eigenfunctions of the multielectron Hamiltonian can be generally written

The coupling between single and multiexciton states, however, does not change the absorption coefficient of a NC, . Indeed, the absorption coefficient

A

Equation 2 also shows that the interband optical selection rules are not affected by the multiexciton coupling to a first-order approximation. The conservation of the selection rules explains the threshold values for very efficient MEG. This threshold was measured to be ~3E_g(a) in PbSe, PbS, and PbTe NCs^1-4 and ~2.5E_g(a) in CdSe NCs for some range of NC sizes;¹⁹ for the ensuing discussion we focus on PbSe and CdSe NCs. The analysis of the coupling of the single electron state with the band edge trions (see Figure 1a) and the energy spectrum shows that the 2P_e electron levels are the first ones which have sufficient energy to generate the ground-state trion. The optical excitation of the 2P_e state is caused by the 2P_e-2P_h (PbSe) and 2P_e-1P_3/2 (CdSe) electron-hole transitions, whose energies are very close to the observed efficient MEG threshold in PbSe and CdSe NCs.

For strongly coupled superposition of single and multiexciton states generated by light, the question of MEG efficiency is reduced to the question of the state relaxation. The single and multiexciton components of the superposition have different decay rates, and the ratio of these rates determines what a single photon creates: a single exciton or a multiexciton.

The initial excitation of strongly coupled multi-exciton states by a short light pulse and the following complex relaxation of this state can be consistently described by the density matrix method. Here, we consider only the strongly coupled excitons and biexcitons which limits the photon excitation energy, , to the range 3E_g(a) < 4E_g(a) in PbSe NCs. According to the selection rules, a photon generates a nL_hnL_e electron-hole pair-the exciton-in NCs with completely filled valence bands. From now on we will use the following notations: one exciton 1> = nL_hnL_e> and zero exciton 0> states. The single exciton created by the photon is coupled with the two-electron two-hole configurations, which are the excited biexciton states: > = nL_h,> and > = nL_e,>. In these biexciton states, one charge occupies the initial single electron/hole level nL_e/h>, while the other three charges form the positively or negatively charged trion >. The electron-hole configurations of the trions > = 1S_e1S_e1S_h> and > = 1P_e1S_e1S_h>, coupled to the 2P_e and 2D_e electron levels, respectively, are shown in Figure 1a. The trion configurations > coupled with holes can be obtained by the permutation of "e" and "h" indexes.

The coupling constants of both biexciton states > to the initial single exciton are equal to each other. This symmetry allows us to introduce the symmetric and antisymmetric superpositions of biexciton states: 2> (> + >)/2^1/2 and (> - >)/2^1/2. The new basis significantly simplifies the density matrix equation, because the antisymmetric superposition is uncoupled from the single exciton state. Both the single and two-exciton excited states relax to the ground states, ex> and bi>, of a one exciton and a biexciton, respectively. The coupling between the excited states and the ground states is negligibly weak. In the basis of the five states, which control the major physical processes leading to MEG, the 5 × 5 time dependent density matrix, is governed by the following equations:

The effect of the MEG by a single photon could be utilized in solar cell elements only if the photoexcited carriers are extracted from the nanocrystals prior to undergoing the nonradiative Auger recombination. The nonradiative Auger recombination also limits opportunities to measure the carrier multiplication. Experimentally, the MEG was determined from the time-dependent change of the band edge absorption bleach and the photoinduced intraband infrared absorption after short pulse excitation.^1-4 The efficiency of the MEG can be determined by the difference in the transmission or the photoinduced intraband absorption caused by one exciton and multiexciton populations.

To calculate the efficiency of the MEG, we should compare populations of the ground biexcitons, N_bi, and the single excitons, N_ex, during time, t, which is much longer than the relaxation times of the excited states (1/₁ and 1/₂) but is much shorter than the relaxation time of the band edge excitons (1/_ex and 1/_bi): 1/₁ 1/₂ t 1/_bi 1/_ex. For this time interval we can neglect the terms proportional to _ex and _bi and find a "steady state" solution of eq 3 by setting the left-hand side of this equation to zero. The description of efficiency of the MEG by a single photon requires the solution in the lowest order of optical excitation, V. This approach gives the ratio of the band edge exciton and biexciton populations, which does not depend on the light intensity

Equation 4 shows that efficient MEG-the predominant generation of multiexcitons-is possible only if the relaxation rate of the biexciton is much larger than that of the exciton: _{2 1}, because P₁₂ 1. For strong coupling, the transition probability saturates at P₁₂ = 1 and the population ratio approaches its maximum ₂/₁. This result has a transparent physical meaning: for a strongly coupled superposition state, the populations of N_bi and N_ex are controlled by the state decay into two independent thermalization channels. However, MEG can be efficient even for a weak coupling regime, W_C/ ₂. In this case N_bi/N_ex = 4/(²₂₁) can be much larger than unity ifW_C/ ₁. The relative population N_bi/N_ex is determined by the ratio of the biexciton creation rate, (2W_C/)²/₂, to the direct exciton relaxation rate, ₁. In Figure 2, we show the MEG quantum yield, QY = (2N_bi + N_ex)/(N_bi + N_ex) × 100%, defined in refs 1 and 2 as a function of the coupling strength calculated in the photon energy interval where no more than two electron-hole pairs can be generated. One can see that the QY approaches its maximum value of (2₂ + ₁)/(₂ + ₁) × 100% in the range of coupling 0.4 < W_C/(₂) < 1 for comparable values of ₁ and ₂. However, for ₁/₂ < 0.01 the QY approaches 200% even if the coupling is small.

Figure 2 Dependence of the MEG quantum yield (QY) on the coupling between excited exciton and biexciton states for the excitation frequency range where only one or two excitons can be generated. Calculations are conducted for the three parameter sets. The solid, dashed, and dotted lines show the QY for dispertion = 0, = 2, and = 22, respectively.

The above analysis of the QY was conducted for the resonant coupling where W_C,₂ > ₁₂. Equation 5 shows clearly that detuning from the resonance reduces the efficiency of MEG. As a result, the NC size distribution always decreases the average efficiency of MEG. The exciton and biexciton relaxation rates should be size independent for small size dispersion of NCs. This allows us to estimate the magnitude of this decrease in the efficiency by averaging P₁₂(₁₂) over detuning ₁₂ energy distribution, which we assume has a Lorentzian form

As we have mentioned above in the experiments,^1-4 MEG has been observed by measuring the bleach of the optical absorption at the band gap frequency. Although the initially excited exciton, whose energy is above the MEG threshold, does not directly populate the states at the band gap, the bleach increases rapidly after the excitation. The time scale of the bleach evolution should provide the information about the strength of the coupling between the initially excited state and the multiexcitons which populate the low energy states at the band gap. The effective MEG leads to the bleach that exceeds significantly the bleach of the single band edge exciton. In the excitation frequency range where only two electron hole pairs can be generated, the relative change of the band edge absorption coefficient connected with filling of the lowest 1 S_e,h electron and hole levels can be written as

Using eq 3, we find the time dependence of the diagonal components of the density matrix, N₂, N_ex, and N_bi after a short pulse excitation of the 2P_e - 2P_h transition and calculate the time dependence of the band edge absorption bleach, -/ (see Figure 3). Figure 3 shows that the strong coupling (W_C/ > ₂)significantly shortens the bleach rise time ~ /W_C and leads to quite pronounced oscillations (quantum beats) connected with the creation of the coherent superposition of the single and biexciton excited states. The coherent oscillations become overdamped even for a relatively strong coupling if W_C ~ ₂, and the coupling still substantially mixes the exciton states. In the weak coupling regime, W_C/ < ₂, the bleach rise time is slowed down by the biexciton decay rate, and it is proportional to ₂²/.

Figure 3 Time evolution of the absorption bleach after short pulse excitation, calculated for (a) 2 = 101, (b) 2 = 31, and (c) 2 = 1. The other parameters are selected to demonstrate a gradual transition from a strong to weak coupling.

The detuning also affects the bleach rise time behavior. Generally, detuning increases the frequency of the beats and slows down the formation of coherent superposition. As a result the NC size dispersion washes out the pronounced beats and slows down the average rise of the bleach signal.

Now let us discuss the parameters of our coherent superposition model. We begin with the most important parameter: the matrix element, W_C, which describes the Coulomb coupling between an electron and a negatively charged trion (see Figure 1a). To describe this coupling, we implement the approach used for the description of Auger processes in exciton complexes¹⁰ and nanocrystals.¹¹ In this approach, an electron occupying a nL_e level of the conduction band always interacts with another electron that occupies a m level of the hole in the valence band via a direct Coulomb potential. As a result of this interaction, both particles undergo the following virtual transition: the first electron goes to another level of conduction band, nL_e k, and the electron from the valence band is transferred to the conduction band, m i, producing an exciton. The exciton and the electron form a negatively charged trion. A similar effect occurs with the hole, which becomes coupled with the positively charged trion. The interaction strength between the negative (positive) trion and the initial electron (hole) is determined by the Coulomb matrix element

Figure 4 shows the size dependence of the matrix elements for resonant coupling of 2P_e,h and 2D_e,h states with 1S_e,h1S_e,h1S_h,e and 1P_e,h1S_e,h1S_h,e trions, which were calculated for bare Coulomb interaction with the dielectric constant = 1. As we have mentioned above, due to a weak spin-orbit coupling, the P levels are split into the P^1/2 and P^3/2 levels and the D levels into D^3/2 and D^5/2. This splitting increases the number of possible superposition states to 2 and 3 for the 2P and 2D levels, respectively. The splitting of these states is not significant, and in Figure 4 we show the average value of W_C for the 2P and 2D manifolds.

Figure 4 Size dependence of the Coulomb matrix element, WC defined by eq 8, that decribes coupling between two pairs of states: 2Pe> and 1Se1Se1Sh> (red points), and 2De> and 1Pe1Se1Sh> (blue points). The matrix element, WC, is calculated for the unscreened Coulomb potential with = 1. The straight solid lines show the asymptotic 1/d behavior of these dependences, red and blue, respectively.

How large, however, is the screening of the Coulomb potential in NCs? It is clear, that phonons do not contribute to the charge screening because the typical frequency of carrier motion is much larger than the phonon frequencies, and in the matrix element W_C is a high-frequency dielectric constant controlled by the electron polarization. The high-frequency dielectric constant in bulk narrow gap semiconductors, like PbSe, is inversely proportional to the semiconductor energy gap. In PbSe nanocrystals the effective energy gap increases up to three times from its bulk value with a decreasing radius, and this effect should decrease . Due to the discrete character of electron-hole energy spectrum in nanocrystals, the self-consistent consideration of this effect is complicated because different multielectron configurations play a role in the screening for the various effects and processes.

As we determined above, efficient MEG requires the rate of exciton relaxation ₁ to be much slower than the rate of biexciton relaxation ₂. These characteristics are not yet defined, however, because despite almost 20 years of research, we still do not have a proper description of the major mechanisms of carrier relaxation in NCs. The energy separation between electron levels and hole levels in PbSe NCs is significantly larger than typical phonon energies in semiconductors. Theoretically, carrier relaxation in NCs should be strongly suppressed because the relaxation should be accomplished by a simultaneous emission of many phonons, a process which has a very low probability. This, however, contradicts experiments that measured single exciton relaxation time to be on the order of several picoseconds. Currently, there is no conventional explanation how carriers overcome the phonon bottleneck during their relaxation.

Only recently the Klimov,²² Guyot-Sionnest,²³ and Wise²⁴ groups reported several breakthroughs in their studies of carrier thermalization in CdSe and PbSe NCs. First, all these studies show that carrier relaxation is faster in small NCs, despite the rise in level spacing. This suggests a polar character of carrier interaction with phonons, which increases with decreasing NC size. Studies of electron relaxation from the 1P to 1S levels of PbSe NCs show a strong effect of the surface ligands that helps overcome the phonon bottleneck.²³ Finally, the efficient multiphonon relaxation between these levels has been reported in ref 22 to be triggered at high temperature by a large, size-dependent intraband Huang Rhys parameter. The origin of this effect, however, is not yet known.

Although the major mechanism of carrier thermalization in NCs is not known, we find that the biexciton relaxation rate is much faster than the exciton relaxation rate assuming that the relaxation rate for various electron-hole pair configurations is proportional to their coupling with phonons. Our calculations show that polar interactions of intrinsic semiconductor phonons in CdSe and PbSe NCs with asymmetric e-h pair configuration are 10 to 40 times stronger than that of their coupling with symmetric electron-hole pair configurations (nL_enL_h) created by light. This is because charge distributions of the optically created electron and hole compensate each other almost exactly at each point of the NC, and the NC thus retains its local neutrality even after exciton creation. As a result, exciton interactions with the polar optical phonons that are sensing the total charge are very weak.²⁵ It also is quite obvious that the coupling of asymmetric e-h configurations with phonons of organic molecules at the NC surface²³ should be significantly stronger as well. In both cases, weak coupling of symmetric electron-hole pairs created by light with phonons suppresses their relaxation and could result in the efficient MEG because condition _{1 2} is fulfilled.

Efficient MEG requires also a quite strong coupling between exciton and biexciton states relative to ₁: W_C/ ₁. The fulfillment of this condition satisfies the obvious requirement that the exciton thermalization time should be longer than the coupling time between the optically created exciton and the asymmetric biexciton state.⁶ Instead of calculating the exciton relaxation time, we use experimentally measured results which have been reported in several papers to be 1/₁ ~ 2-6 ps, depending on the NC size.^20,22,24 Comparing the ₁ size dependence²⁴ with the size dependence of W_C in Figure 4 corrected relative to the bulk value of dielectric constant = 23, we find that condition W_C/ ₁ is fulfilled even for the lowest estimation of W_C. In NCs, the dielectric constant, (a), is smaller than that in bulk, and consequently the coupling W_C is stronger.

We calculate size dependence of the carrier multiplication quantum yield, QY, using size dependence of ₁ measured in ref 24 and assuming that high-frequency dielectric constant of NCs, (a), is inversely proportional to the energy gap of NCs (a) = 23E_g/E_g(a) like in narrow gap bulk semiconductors. Here E_g = 0.28 eV is the energy gap of bulk PbSe. The resulting QY for the size interval from 3.7 to 6.2 nm is shown in Figure 5 for the four ratios of ₁/₂. Note, that calculations are conducted only for the optical excitation interval, where no more than two electron-hole pairs can be excited, and excitation frequency follows the 2P_e-2P_h or 2D_e-2D_h transition energy in each NC size. Surprisingly, QY does not show any size dependence if we assume that the ratio ₁/₂ does not depend on size. At the same time, as we expect, the increase of ₂/₁ always leads to the increase of QY. The surprising size independence of the average number of electron-hole pairs generated via excitation of the same optical transition is consistent with the results of experimental measurements, which show the universal dependence of the QY on the one parameter only:³ /E_g(a), where is the exciting photon energy.

Figure 5 Size dependence of the quantum yield of MEG for the excitation frequency range where only one or two excitons can be generated. The calculation uses the size dependence of the single exciton relaxation time measured at room temperatures in ref 24 and the size dependence of the matrix element WC from Figure 4.

In ref 24, the single exciton relaxation time, 1/₁, was measured at room tempeartures. The relaxation rate in PbSe NCs, however, is strongly suppressed at tempertaures below 130-170 K.²² The suppression should lead to the larger value of QY at low temperatures. Our calculations show also that the supression of the biexciton relaxation rate, ₂, increases the rise time of the bleach. This suggests that a temperature decrease may slow down the bleach rise time if there is a correlation between temperature dependence of ₂ and ₁.

Existing experimental data, unfortunately, do not allow us to extract the biexciton decay rate ₂, and the calculated W_C size dependence does not give us a conclusive answer to the question if W_C > ₂ or W_C < ₂. As a result, we cannot tell if the formation of coherent superposition would lead to the quantum beats in the band edge transient bleach. The quantum beats in the band edge transient absorption connected with formation of the coherent superposition are generally masked by the bleach from the ground excitons and biexcitons which are filled in a process of incoherent thermalization (see eq 7). In addition to the filling factor bleach there is an instant Stark shift of band edge energy levels due to their interactions with photoexcited excitons,²⁶ which also obscures the detection of any quantum beats. Studying the transient bleach of excited interband optical transitions such as 1S_e,h-1P_h,e or 1P_e,h-1P_h,e would allow one to avoid some of these complications. These transitions are affected by the coherent coupling of the 2D_e2D_h exciton with the excited biexciton B_2D^h,e>, which contains one carrier occupying the 1P_e,h levels (see Figure 1a). Although the interband transitions are also affected by the thermalization, the coherent contribution can be more appreciable because the relaxation is expected to be fast, and most of the thermalized excitons should accumulate at the band edge states.

Due to large energy of photoexcited transitions required for MEG, the discrete character of electron-hole energy spectra in PbSe NCs has never been revealed in the MEG experiments. It still makes sense to study the energy dependence of the MEG near the transitions to the 2P_e,h levels. These are the lowest levels that are strongly coupled with trions, and the level spacing in this range of energies is quite significant. These levels can be excited by optically allowed 2P_e,h-2P_h,e transitions with energy close to 3E_g(a) or by optically forbidden 1S_e,h-2P_h,e transitions with energy close to 2E_g(a). The broken selection rules that allow observation of strong 1S_e,h-1P_h,e transitions in PbSe NCs should allow also the 1S_e,h-2P_h,e transitions, which have the same symmetry. These experiments may reveal resonance-type energy dependence of the MEG efficiency.

To summarize, we have developed a model that explains the MEG of excitons in NCs by a single photon as a result of the formation of a coherent superposition of single and multiexciton states. Within this model the efficient MEG is a consequence of a suppressed thermalization rate of an exciton created by light. Calculations conducted for the set of model parameters estimated for PbSe NCs demonstrate efficient MEG, which is consistent with experimental observations.

Acknowledgment

In acknowledgment, the authors thank M. C. Beard, R. J. Ellingson, J. C. Johnson, and V. I. Klimov for the multiple stimulating discussions. Al.L.E. was supported by the Office of Naval Research, and A.S. and A.J.N. were supported by the U.S. Department of Energy, Office of Basic Energy Science.

Note Added after ASAP Publication. There was an incorrect affiliation symbol with a name in the author byline, an error in an equation in the text following eq 9, and a modification (from exciton to biexciton) in the third paragraph from the end of the text in the version published ASAP November 1, 2006; the corrected version was published ASAP November 9, 2006.