NANO LETTERS
Multiexciton Generation by a Single Photon in Nanocrystals
2006 Vol. 6, No. 12 2856-2863
A. Shabaev,*,†,‡ Al. L. Efros,*,† and A. J. Nozik*,§ Center for Computational Material Science, NaVal Research Laboratory, Washington, D.C. 20375, School of Computational Sciences, George Mason UniVersity, Fairfax, Virginia 22030, and Center for Basic Sciences, National Renewable Energy Laboratory, Golden, Colorado 80401 Received August 31, 2006; Revised Manuscript Received October 17, 2006
ABSTRACT We have theoretically shown that efficient generation of multi-electron−hole pairs by a single photon observed recently in semiconductor nanocrystals1-4 is caused by breaking the single electron approximation for carriers with kinetic energy above the effective energy gap. Due to strong Coulomb interaction, these states form a coherent superposition with charged excitons of the same energy. This concept allows us to define the conditions for dominant two-exciton generations by a single photon: the thermalization rate of a single exciton, initiated by light, should be lower than both the two-exciton state thermalization rate and the rate of Coulomb coupling between single and two exciton states. Possible experimental manifestations of our model are discussed.
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 * Correspondingauthors.E-mail:
[email protected],
[email protected],
[email protected]. † Center for Computational Material Science, Naval Research Laboratory. ‡ School of Computational Sciences, George Mason University. § Center for Basic Sciences, National Renewable Energy Laboratory. 10.1021/nl062059v CCC: $33.50 Published on Web 11/01/2006
© 2006 American Chemical Society
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 theoretically11,12 and experimentally13,14 that direct Auger processes are greatly enhanced in NCs. These observations allowed Nozik to suggest that the inverse Auger processess impact ionizationsmust also be enhanced.6 Indeed soon after this prediction, an efficient generation of multi-electronhole 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 theory7-10 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
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.
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, γ1, must be slower than both the energy relaxation rate of the multiexciton, γ2, and the rate of Coulomb coupling between the two states, WC/p, where WC is the matrix element of Coulomb interaction between the single and multiexciton states. In the case of strong coupling, γ2 < WC/p, the MEG by a short light pulse is accompanied by quantum beats with the frequency WC/p, which are connected with the creation of the coherent superposition. These oscillations are damped in the opposite case of weak coupling, γ2 > WC/p. 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, Eg ) 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 Nano Lett., Vol. 6, No. 12, 2006
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), .... In order to understand how a single photon creates many excitons, let us first consider a single electron occupying the 2Pe 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, E2Pe - E1Se, between the 2Pe and the ground 1Se 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: Eg(a) ) E1Se - E1Sh. As a result, a single electron occupying the 2Pe level has the same energy as a trion consisting of two electrons and a hole occupying the ground quantum size 1Se,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 nLL+1/2 and nLL-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 2Pe 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 (2Pe) and right (1Se1Se1Sh) multielec2857
tron 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 2Pe state and the 1Se1Se1Sh 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 2Pe state. As an example, in Figure 1, we show the multielectron configuration and the total energy of the 1Pe1Se1Sh trion state which creates a coherent superposition with the 2De electron level. A single electron state with energy larger than (n + 1)Eg(a) where n g 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 2Eg(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 Φ(1) ) R j (1) ex φex +
(1) (j) (1) (j) φbi-ex +∑β φtr-ex + ... h tr-exj ∑j βh bi-exj j
Φ(2) ) R j (2) ex φex +
(2) (j) (2) (j) φbi-ex +∑β h tr-exj φtr-ex + ... ∑j βh bi-exj j
l Φ(n) ) R j (n) ex φex +
l
l
l
(n) (j) (n) (j) φbi-ex +∑β h tr-exj φtr-ex + ... ∑j βh bi-exj j
(1)
Here we expand the eigenfunctions in a basis of uncoupled multiexciton states: the single exciton, φex, biexciton, φbi-ex, triexciton, φtr-ex, etc. This implies that a direct optical excitation of the eigenstates would create a superposition of single and multiexcitons in NCs. The coupling between single and multiexciton states, however, does not change the absorption coefficient of a NC, R. Indeed, the absorption coefficient 2858
R∼
∫ dω ∑ |〈Φ(n)|pˆ Aω|0〉|2δ(pω - En) n
) |〈φex|pAω|0〉|2
∑n |R(n)ex |2 ) |〈φex|pˆ Aω|0〉|2
(2)
where En is the energy of a multiexciton state with the index n, pˆ is the momentum operator, and Aω is the vector potential of a photon with the frequency ω. The straightforward calculation shows that the matrix elements, 〈Φ(n)|pˆ Aω|0〉, are reduced to the matrix elements between the NC vacuum state, |0〉, and a single exciton component of the wave function, |R j (n) ex φex〉. Due to the completeness of the multiexciton basis, 2 ∑n|R j (n) ex | ) 1, the absorption coefficient in eq 2 is determined by unperturbed exciton states and it is not affected by the coupling with the multiexcitons. Equation 2 also shows that the interband optical selection rules are not affected by the multiexciton coupling to a firstorder approximation. The conservation of the selection rules explains the threshold values for very efficient MEG. This threshold was measured to be ∼3Eg(a) in PbSe, PbS, and PbTe NCs1-4 and ∼2.5Eg(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 2Pe electron levels are the first ones which have sufficient energy to generate the ground-state trion. The optical excitation of the 2Pe state is caused by the 2Pe-2Ph (PbSe) and 2Pe-1P3/2 (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, pω, to the range 3Eg(a) e pω < 4Eg(a) in PbSe NCs. According to the selection rules, a photon generates a nLhnLe electron-hole pairsthe excitonsin NCs with completely filled valence bands. From now on we will use the following notations: one exciton |1〉 ) |nLhnLe〉 and zero exciton |0〉 states. The single exciton created by the photon is coupled with the two-electron two-hole configurah tions, which are the excited biexciton states: |BnL 〉 ) |nLh, e + TnL〉 and |BnL〉 ) |nLe,TnL〉. In these biexciton states, one charge occupies the initial single electron/hole level |nLe/h〉, while the other three charges form the positively or nega( 〉. The electron-hole configuratively charged trion |TnL tions of the trions |T2P〉 ) |1Se1Se1Sh〉 and |T2D〉 ) |1Pe1Se1Sh〉, coupled to the 2Pe and 2De electron levels, respectively, Nano Lett., Vol. 6, No. 12, 2006
are shown in Figure 1a. The trion configurations |T+ 2L〉 coupled with holes can be obtained by the permutation of “e” and “h” indexes. h,e The coupling constants of both biexciton states |BnL 〉 to the initial single exciton are equal to each other. This symmetry allows us to introduce the symmetric and antih symmetric superpositions of biexciton states: |2〉 ≡ (|BnL 〉 e h e + |BnL〉)/21/2 and (|BnL〉 - |BnL〉)/21/2. 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, F is governed by the following equations: N˙ 1(t) ) F˘12(t) )
iWC V (F12 - F/12) + i (F10 - F/10) - γ1N1 p p
iWC ∆12 γ˘ 1 + γ2 V (N1 - N2) + i F12 - i F/20 F12 p p p 2 N˙ 2(t) ) -i
F˘10(t) ) i
WC (F - F/12) - γ2N2 p 12
pω - E1 WC γ1 V (N1 - N0) + i F10 - i F20 - F10 p p p 2
a complete time dependent description of MEG in PbSe NCs under arbitrary excitation conditions. 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, Nbi, and the single excitons, Nex, during time, t, which is much longer than the relaxation times of the excited states (1/γ1 and 1/γ2) but is much shorter than the relaxation time of the band edge excitons (1/Γex and 1/Γbi): 1/γ1 e1/γ2 , t , 1/Γbi , 1/Γex. For this time interval we can neglect the terms proportional to Γex and Γbi 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 Nbi γ2 ) P Nex γ1 1f2
V N˙ 0(t) ) -i (F10 - F/10) + ΓexNex + ΓbiNbi p pω -E1 + ∆12 WC γ2 V F20 + i F/12 - i F - F F˘20(t) ) i p p p 10 2 20
where P1f2 is the probability of the population of the excited biexciton state via its Coulomb coupling to the single exciton state excited by a photon
N˙ ex(t) ) γ1N1 + ΓbiNbi - ΓexNex N˙ bi(t) ) γ2N2 - ΓbiNbi
(3)
where N1, N2, Nex, Nbi, and N0, are the diagonal components of the density matrix, which describe the probabilities of finding a NC in one of the four excitonic states or with no excitons in it: N1 + N2 + Nex + Nbi + N0 ) 1; and F10 ) F/01, F20 ) F/02, and F12 ) F/21 are the off-diagonal components of the density matrix describing the phase coherence between the basis states. Here in eq 3, E1 is the energy of the allowed optical transition generating the nLenLh exciton, V is the matrix element of the optical transition, and ∆12 is the difference between the energies of a single electron (hole) and the trion coupled to this electron (hole). The decay rates of the excited exciton and biexciton, γ1 and γ2, are much higher than the decay rates of the ground exciton and biexciton: Γex and Γbi. The radiative decay time of excitons in PbSe NCs, 1/Γex is on the order of 100 ns.20 The biexciton decay is mainly determined by the nonradiative Auger recombination, which in PbSe NCs has typical times 1/Γbi on the order of 20-100 ps.1 The solution of eq 3 provides Nano Lett., Vol. 6, No. 12, 2006
(4)
P1f2(∆12) )
W2C(1 + γ1/γ2) ∆122 + W2C(1 + γ1/γ2) + (γ1 + γ2)2p2/4
(5)
The transition probability P1f2(∆12) has a Lorentzian shape. Its maximum value P1f2(0) at zero detuning, ∆12 ) 0, is always smaller than 1. The coupling constant, WC, appears in the denominator due to the dynamical broadening of the transition.21 For small coupling, WC/p , γ1,γ2, the width of the resonant transition is determined by the fastest of the two decay rates, γ1 or γ2. The transition probability P1f2(0) ≈ 4W2C/[p2γ2(γ1 + γ2)] is also small for the weak coupling. Equation 4 shows that efficient MEGsthe predominant generation of multiexcitonssis possible only if the relaxation rate of the biexciton is much larger than that of the exciton: γ2 . γ1, because P1f2 e 1. For strong coupling, the transition probability saturates at P1f2 ) 1 and the population ratio approaches its maximum γ2/γ1. This result has a transparent physical meaning: for a strongly coupled superposition state, the populations of Nbi and Nex are controlled by the state decay into two independent thermalization channels. How2859
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 δ ) 0, δ ) pγ2, and δ ) 2pγ2, respectively.
ever, MEG can be efficient even for a weak coupling regime, WC/p , γ2. In this case Nbi/Nex ) 4W2C/(p2γ2γ1) can be much larger than unity ifWC/p . γ1. The relative population Nbi/Nex is determined by the ratio of the biexciton creation rate, (2WC/p)2/γ2, to the direct exciton relaxation rate, γ1. In Figure 2, we show the MEG quantum yield, QY ) (2Nbi + Nex)/(Nbi + Nex) × 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 (2γ2 + γ1)/(γ2 + γ1) × 100% in the range of coupling 0.4 < WC/(pγ2) < 1 for comparable values of γ1 and γ2. However, for γ1/γ2 < 0.01 the QY approaches 200% even if the coupling is small. The above analysis of the QY was conducted for the resonant coupling where WC,γ2 > ∆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 P1f2(∆12) over detuning ∆12 energy distribution, which we assume has a Lorentzian form P
(δ + ∆)∆
(x) dx
∫-∞+∞ πδ π(x -1f2∆ )2 + δ2 ) P1f2(0) ∆ 2 + (δ + ∆)2 0
(6)
0
where ∆0 is the average detuning for the average size and δ is the detuning dispersion for the size dispersion in the NC ensemble; and ∆ ) (W2C(1 + γ1/γ2) + (γ1 + γ2)2p2/4)1/2. One can see that even in the resonance case: ∆0 ) 0, the size dispersion leads to an additional decrease of the MEG efficiency by the factor ∆/(δ + ∆). The effect of the size dispersion on the QY is shown in Figure 2 by the dashed and dotted lines. 2860
Figure 3. Time evolution of the absorption bleach after short pulse excitation, calculated for (a) γ2 ) 10γ1, (b) γ2 ) 3γ1, and (c) γ2 ) γ1. The other parameters are selected to demonstrate a gradual transition from a strong to weak coupling.
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 Se,h electron and hole levels can be written as -
∆R ) N1F1 + N2F2 + NexFex + NbiFbi R
(7)
where F1, F2, Fex, and Fbi are the band edge filling factors for the initial exciton, excited biexciton, ground exciton, and ground biexciton states that contribute to the transient bleach. For each of these electron-hole pair configurations the filling factors are determined as F ) (n1Se + n1Sh)/8, where n1Se e 8 and n1Sh e 8 are the occupation numbers of the 8-fold degenerate 1Se and 1Sh states in PbSe nanocrystals. Calculation shows that F1 ) 0, Fex ) 1/4, and Fbi ) 1/2, while F2 ) 3/8 and F2 ) 1/4 for the 2Pe-2Ph and 2De-2Dh excitations, respectively. Using eq 3, we find the time dependence of the diagonal components of the density matrix, N2, Nex, and Nbi after a short pulse excitation of the 2Pe - 2Ph transition and calculate the time dependence of the band edge absorption bleach, -∆R/R (see Figure 3). Figure 3 shows that the strong Nano Lett., Vol. 6, No. 12, 2006
coupling (WC/p > γ2)significantly shortens the bleach rise time ∼ p/WC 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 WC ∼ γ2, and the coupling still substantially mixes the exciton states. In the weak coupling regime, WC/p < γ2, the bleach rise time is slowed down by the biexciton decay rate, and it is proportional to γ2p2/W2C. 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, WC, 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 nLe level of the conduction band always interacts with another electron that occupies a mLmh 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, nLe f k Lke, and the electron from the valence band is transferred to the conduction band, mLmh f iLie, 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 WC(nLe,mLmh ;kLke,iLie) )
〈 〈
| |
| |
〉 〉
e2 nLe(r1) κ|r1 - r2| e2 k iLih(r2)mLm (r )kL (r ) nLh(r1) e 2 h 1 κ|r1 - r2|
iLie(r2)mLmh (r2)kLke(r1)
(8)
where κ is the effective dielectric constant and r1 and r2 are the coordinates of the first and second electrons, respectively. The Coulomb interaction mixes the trion wave functions with the single electron and hole wave functions. In NCs the eigenfunctions of the multielectron nanocrystal Hamiltonian can be always written as V,e V,e ) RnL |nLe〉 + ΨnL
∑ βnLV,e|kLkeiLiemLmh 〉
k,i,m V,h V,h ) RnL |nLh〉 + ΨnL
∑ βnLV,e|kLkhiLihmLme 〉
(9)
k,i,m
This admixture becomes important for the resonant electrontrion configurations, similar to the ones shown in Figure 1b. Due to the spherical symmetry of NCs there are selection Nano Lett., Vol. 6, No. 12, 2006
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 κ ) 1. The straight solid lines show the asymptotic 1/d behavior of these dependences, red and blue, respectively.
rules that determine the conditions for the trion state to be coupled with a single electron or hole: WC(nLe,mLmh ;kLke,iLie) * 0. For the trions which are created by Lke the excitation of 1Se1Sh excitons the initial electron cannot change its angular momentum by more than 1, and therefore WC(nLe,1Sh; kLke,1Se) ∼ δLke,Le(1, where δi,j is the Kronecker symbol. Figure 4 shows the size dependence of the matrix elements for resonant coupling of 2Pe,h and 2De,h states with 1Se,h1Se,h1Sh,e and 1Pe,h1Se,h1Sh,e trions, which were calculated for bare Coulomb interaction with the dielectric constant κ ) 1. As we have mentioned above, due to a weak spinorbit coupling, the P levels are split into the P1/2 and P3/2 levels and the D levels into D3/2 and D5/2. 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 WC for the 2P and 2D manifolds. 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 κ in the matrix element WC is a high-frequency dielectric constant controlled by the electron polarization. The highfrequency 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 κ∞. 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 γ1 to be much slower than the rate of 2861
biexciton relaxation γ2. 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 (nLenLh) 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 electronhole pairs created by light with phonons suppresses their relaxation and could result in the efficient MEG because condition γ1 , γ2 is fulfilled. Efficient MEG requires also a quite strong coupling between exciton and biexciton states relative to γ1: WC/p . γ1. 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 2862
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.
measured results which have been reported in several papers to be 1/γ1 ∼ 2-6 ps, depending on the NC size.20,22,24 Comparing the γ1 size dependence24 with the size dependence of WC in Figure 4 corrected relative to the bulk value of dielectric constant κ∞ ) 23, we find that condition WC/p . γ1 is fulfilled even for the lowest estimation of WC. In NCs, the dielectric constant, κ∞(a), is smaller than that in bulk, and consequently the coupling WC is stronger. We calculate size dependence of the carrier multiplication quantum yield, QY, using size dependence of γ1 measured in ref 24 and assuming that high-frequency dielectric constant of NCs, κ∞(a), is inversely proportional to the energy gap of NCs κ∞(a) ) 23Eg/Eg(a) like in narrow gap bulk semiconductors. Here Eg ) 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 γ1/γ2. 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 2Pe-2Ph or 2De-2Dh transition energy in each NC size. Surprisingly, QY does not show any size dependence if we assume that the ratio γ1/γ2 does not depend on size. At the same time, as we expect, the increase of γ2/γ1 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 pω/Eg(a), where pω is the exciting photon energy. In ref 24, the single exciton relaxation time, 1/γ1, 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, γ2, 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 γ2 and γ1. Nano Lett., Vol. 6, No. 12, 2006
Existing experimental data, unfortunately, do not allow us to extract the biexciton decay rate γ2, and the calculated WC size dependence does not give us a conclusive answer to the question if WC > pγ2 or WC < pγ2. 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 1Se,h-1Ph,e or 1Pe,h-1Ph,e would allow one to avoid some of these complications. These transitions are affected by the coherent coupling of the 2De2Dh exciton with the excited biexciton |B2Dh,e〉, which contains one carrier occupying the 1Pe,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 2Pe,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 2Pe,h-2Ph,e transitions with energy close to 3Eg(a) or by optically forbidden 1Se,h-2Ph,e transitions with energy close to 2Eg(a). The broken selection rules that allow observation of strong 1Se,h-1Ph,e transitions in PbSe NCs should allow also the 1Se,h-2Ph,e transitions, which have the same symmetry. These experiments may reveal resonancetype 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.
Nano Lett., Vol. 6, No. 12, 2006
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. References (1) Schaller, R. D.; Klimov, V. I. Phys. ReV. Lett. 2004, 92, 186601. (2) Ellingson, R. J.; Beard, M. C.; Johnson, J. C.; Yu, P.; Micic, O. I.; Nozik, A. J.; Shabaev, A.; Efros, Al. L. Nano Lett. 2005, 5, 865. (3) Schaller, R. D.; Sykora, M.; Pietryga, J. M.; Klimov, V. I. Nano Lett. 2006, 6, 424. (4) Murphy, J. E.; Beard, M. C.; Norman, A. G.; Ahrenkiev, S. P.; Johnson, J. C.; Yu, P.; Micic, O. I.; Ellingson, R. J.; Nozik, A. J. J. Am. Chem. Soc. 2006, 128, 3241 . (5) Shockley, W.; Queisser, H. J. J. Appl. Phys. 1961, 32, 510. (6) Nozik, A. J. Physica E 2002, 14, 115. (7) Werner, J. H.; Kolodinski, S.; Queisser, H. J. Phys. ReV. Lett. 1994, 72, 3851. (8) Kolodinski, S.; Werner, J. H.; Wittchen, T.; Queisser, H. J. Appl. Phys. Lett. 1993, 63, 2405. (9) Landsberg, P. T.; Nussbaumer, H.; Willeke, G. J. Appl. Phys. 1993, 74, 1451. (10) Gel’mont, B. L.; Zinov’ev, N. N.; Kovalev, D. I.; Kharchenko, V. A.; Yaroshetskii, I. D.; Yassievich, I. N. SoV. Phys. JETP 1988, 67, 613. (11) Chepic, D. I.; Efros, Al. L.; Ekimov, A. I.; Ivanov, M. G.; Kudriavtsev, I. A.; Kharchenko, V. A.; Yazeva, T. V. J. Lumin. 1990, 47, 113. (12) Efros, Al. L. In Semiconductor Nanocrystals: from Basic Principles to Applications; Efros, Al. L., Lockwood, D. J., Tsybeskov, L., Eds.; Kluwer Academic/Plenum Publishers: New York, 2003, Chapter 2, pp 52-72. (13) Roussignol, P.; Ricard, D.; Rustagi, K. C.; Flytzanis, C. Opt. Commun. 1985, 55, 143. (14) Klimov, V. I.; Mikhailovsky, A. A.; Xu, S.; Malko, A.; Hollingsworth, J. A.; Leatherdale, C. A.; Eisler, H.-J.; Bawendi, M. G. Science 2000, 290, 314. (15) Landolt-Bo¨rnstein, Numerical Data and Functional Relationships in Science and Technology Semiconductors-Group III: Condensed Matter 41; Madelung, O., Ro¨ssler, U., Schulz, M., Eds.; SpringerVerlag: Berlin, 1998; Chapter 9. (16) Mitchell, D. J.; Wallis, R. F. Phys. ReV. 1966, 151, 581. (17) Kang, I.; Wise, F. W. J. Opt. Soc. Am. B 1997, 14, 1632. (18) We describe our absorption spectra assuming a complete symmetry of the conduction and valence with remote band contribution to the electron and hole masses m0/me ) m0/mh ) 1.5 and Kane energy parameter 2P2/m0 ) 3.15 eV and bulk energy gap Eg ) 0.28 eV (see ref 15). m0 is the mass of free electron. (19) Schaller, R. D.; Petruska, M. A.; Klimov, V. I. Appl. Phys. Lett. 2005, 87, 253102. (20) Guyot-Sionnest, P.; Shim, M.; Matranga, C.; Hines, M. Phys. ReV. B 1999, 60, 2181(R). (21) Berestetskii, V. B.; Lifshitz, E. M.; Pitaevskii, L. P. Quantum electrodynamics; Pergamon Press: Oxford, New York, 1982. (22) Schaller, R. D.; Pietryga, J. M.; Goupalov, S. V.; Petruska, M. A.; Ivanov, S. A.; Klimov, V. I. Phys. ReV. Lett. 2005, 95, 196401. (23) Guyot-Sionnest, P.; Wehrenberg, B.; Yu, D. J. Chem. Phys. 2005, 123, 074709. (24) Harbold, J. M.; Du, H.; Krauss, T. D.; Cho, K.-S.; Murray, C. B.; Wise, F. W. Phys. ReV. B 2005, 72, 195312. (25) Schmitt-Rink, S.; Miller, D. A. B.; Chemla, D. C. Phys. ReV. B 1987, 35, 8113. (26) Klimov, V. I. J. Phys Chem. B 2000, 104, 6112.
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