ARTICLE pubs.acs.org/JPCB
Probing Interband Coulomb Interactions in Semiconductor Nanostructures with 2D Double-Quantum Coherence Spectroscopy Kirill A. Velizhanin and Andrei Piryatinski* Center for Nonlinear Studies (CNLS), Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States ABSTRACT: Employing the interband exciton scattering model, we have derived a closed set of equations determining the 2D doublequantum coherence signal sensitive to the interband Coulomb interactions (i.e., many-body Coulomb interactions leading to the couplings between exciton and biexciton bands) in semiconductor nanostructures such as nanocrystals, quantum wires, wells, and carbon nanotubes. Our general analysis of 2D double-quantum coherence resonances has demonstrated that the interband Coulomb interactions lead to new cross-peaks whose appearance can be interpreted as a result of exciton and biexciton state mixing. The presence of the strongly coupled resonant states and weakly coupled background of off-resonant states can significantly simplify cross-peak analysis by eliminating the congested background spectrum. Our simulations of the 2D double-quantum coherence signal in PbSe NCs have validated this approach.
I. INTRODUCTION Many-body Coulomb interactions between carriers in a variety of semiconductor nanostructures including nanocrystals (NCs), quantum wires, wells, and carbon nanotubes play a central role in their carrier transport and optical properties.1-3 Introducing conduction and valence bands within mean field (e.g., effective mass) approximation and using the exciton representation, these interactions can be partitioned into two groups.4 The first group combines those interactions that conserve the number of electrons and holes. In other words, these interactions are acting within exciton bands of different multiplicity and therefore will be referred to as the intraband Coulomb interactions. The intraband interactions determine the binding energies of excitons, trions, and multiexcitons.5-9 Spectroscopically, these quantities are typically obtained from spectral energy shifts using a transientabsorption (pump-probe) technique. An important area where such interactions are widely exploited is wave function engineering in semiconductor heterostructures,10,11 with applications to the problem of the lasing in the single-exciton regime.12 The other group of many-body Coulomb interactions includes those which correspond to valence-conduction band transitions. These transitions change the number of electrons and holes but conserve the total charge. In other words, they are responsible for the photoexcited dynamics which involves transitions between exciton bands of different multiplicity. Hence, in the exciton representation, we refer to such interactions as the interband Coulomb interactions. Important examples involving the interband transitions are nonradiative Auger recombination and impact ionization processes.13 Recently, the problem of carrier multiplication (CM) in semiconductor NCs has received significant attention.14-18 r 2011 American Chemical Society
CM has also been reported in carbon nanotubes.19 Also referred to as multiexciton generation, CM is a process of more than one electron-hole pair production per single absorbed photon, and has great potential for applications in photovoltaic, photochemical, and energy storage devices.20-26 A number of models for CM dynamics based on the interband Coulomb interactions which go beyond the simple impact ionization model have been proposed recently.27-31 The development of experimental techniques capable of probing the interband Coulomb interactions in semiconductor nanostructures should greatly advance our understanding of CM by validating proposed models and shedding new light on the multiexciton photogeneration dynamics. Besides the CM problem, the direct probe of the interband interactions in various semiconductor heterostructures should advance our knowledge of the nonradiative relaxation processes, e.g., Auger recombination (specifically, the effect of carrier confinement and surface/ interface polarization on the nonradiative relaxation rates). The interband Coulomb interactions appear at excitation energies exceeding two band gaps, i.e., J2Eg. Above this value, the exciton and biexciton densities of states (DOS) are rapidly growing, and associated spectroscopic features could be difficult to resolve due to their strong overlap. However, if among the variety of coupled exciton and biexciton states there are a few states that are strongly coupled, their spectroscopic signatures could still be detectable. Special Issue: Shaul Mukamel Festschrift Received: October 1, 2010 Revised: January 27, 2011 Published: March 10, 2011 5372
dx.doi.org/10.1021/jp109453y | J. Phys. Chem. B 2011, 115, 5372–5382
The Journal of Physical Chemistry B
ARTICLE
Recently, Efros, Shabaev, and Nozik pointed out that, in semiconductor NCs among the large number of exciton and biexciton states in the energy range above 2Eg, there are Coulomb coupled quasi-degenerate states, i.e., resonant exciton-biexciton state pairs.29 Initially, these resonant states were expected to provide a major contribution to the CM dynamics in NCs, but subsequent studies showed that the effect is instead associated with the exciton and biexciton DOS and averaged interband Coulomb interactions.27,28,30 However, if these resonant states are spectroscopically resolved, they could be used to probe the upper limit of the interband Coulomb interactions. Now, the question is what spectroscopic technique would reveal the most complete information on the many-body Coulomb interactions by probing these states? The ultrafast techniques such as transient absorption and fluorescence decay carry highly averaged information on the many-body interactions. The resolution can be tremendously enhanced by using coherent multiple-pulse ultrafast nonlinear techniques. These techniques allow one to distinguish between various interaction pathways through the spatial phase-matching conditions for the induced nonlinear polarization.32 Further representing the coherent signal as a 2D Fourier transform of the nonlinear polarization with respect to various pulse delay times, one can gain direct insight into the hidden interactions by looking at the position, line-shape, intensity, and phase of the associated cross-peaks.33-37 These techniques are referred to as 2D correlation spectroscopies and represent an ultrafast optical counterpart of 2D NMR.38 Among 2D correlation spectroscopies, the double-quantum coherence technique proposed by Mukamel and co-workers is considered to be most sensitive to the many-body correlations.39 It has been applied to study the intraband Coulomb interactions leading to biexciton formation in organic molecular systems40,41 and semiconductor quantum wells.42-46 The application of this methodology to probe the interband Coulomb interactions requires additional theoretical study which is the focus of this paper. To model the nonlinear optical response of coupled excitonbiexciton states, we adopt the interband exciton scattering model previously developed by us to provide a unified treatment of CM dynamics in semiconductor NCs.31 This methodology is general and can be used to model the interband interactions and related carrier dynamics in various semiconductor nanostructures. Therefore, the general spectroscopic features that we discuss below should be common for many different nanostructures. A general theory of double-quantum coherence response based on the interband exciton scattering model is given in section II. To clarify the 2D spectroscopic features associated with the interband Coulomb interactions, we first consider the signal associated with the effective five-level system as discussed in section III. In section IV, we model the 2D signal from an ensemble of PbSe NCs in which the double-quantum coherences are prepared using the predicted resonant exciton-biexciton state pairs. The concluding remarks are presented in section V.
II. NONLINEAR RESPONSE FROM COUPLED EXCITON AND BIEXCITON STATES Let us consider an ensemble of nanostructures, in which the carrier dynamics is restricted to the coupled exciton and biexciton bands described by the following material Hamiltonian:31 ^ 0 þ V^ C ^M ¼ H H
ð1Þ
Here, the first term ^0 ¼ H
∑a jxa æpωxa Æxa j þ ∑k jxxk æpωxxk Æxxkj
ð2Þ
represents the noninteracting exciton, |xaæ, and biexciton, |xxkæ, states characterized by the energies pωxa and pωxx k , respectively. These quantities already include the corresponding binding energies associated with the intraband Coulomb interactions. The second term x, xx jxa æVa, k Æxxk j þ h:c: ð3Þ V^ C ¼
∑a ∑k
represents the interband Coulomb interactions between the states.47 An explicit representation for the interband interaction x,xx , in terms of the single-particle couplings, matrix elements, Va,k and related matrix equations defining many-body exciton and biexciton states are given in ref 31. In our interband exciton scattering model, the time evolution of coupled exciton and biexciton states is described by the Hilbert space Green function31 ! ^ x, xx ðtÞ ^ x ðtÞ G G ^ ð4Þ GðtÞ ¼ ^ xx, x ^ xx ðtÞ G ðtÞ G whose upper (lower) diagonal block describes the exciton (biexciton) intraband propagation renormalized by the even-order interband scattering events that do not change the exciton multiplicity. The offdiagonal components describe the odd-order interband scattering events changing the exciton multiplicity, i.e., leading to the interband transitions. A closed set of equations for this Green function based on the scattering matrix formalism is provided in Appendix A. The interaction of coupled exciton and biexciton states with the optical field is described by the following Hamiltonian: ^ ¼H ^M - μ ^ EðtÞ H
ð5Þ
^ M (eqs 1-3), and the optical containing the material part, H interaction term. This term depends on the electric field: Eðr, tÞ ¼
4
∑ E jðt - τj Þeik 3 r - iω t þ c:c: j¼1 i
0
ð6Þ
which is a sequence of four linearly polarized nonoverlapping ultrafast pulses. The first three pulses excite the third-order nonlinear polarization, and the fourth one is used to heterodyne the signal.32 Each pulse is characterized by the wave vector, ki, and the central frequency, ω0. The phase matching condition for the double-quantum coherence response, ks = k1 þ k2 - k3, is assumed, and the pulse central frequency is fixed for the whole train. Typically, in double-quantum coherence spectroscopy, the first and second pulses coincide corresponding to the additional constraint τ2 = τ1 used in eq 6. The dipole moment operator projected on the optical field direction entering eq 5 has the following block-matrix form: ! ^ x, xx ^x μ μ ^ ¼ ð7Þ μ ^ xx, x μ ^ xx μ where the upper (lower) diagonal block describes the interband ground-state-exciton transitions and the intraband exciton (biexciton) transitions. The off-diagonal blocks describe the interband exciton-biexciton transitions. In nanostructures, especially those in the strong confinement regime (e.g., NCs of 5373
dx.doi.org/10.1021/jp109453y |J. Phys. Chem. B 2011, 115, 5372–5382
The Journal of Physical Chemistry B
ARTICLE
Here, the operator Z ^ ¼ D
¥
0
dt2
-¥
Z
¥
0
^ ð - t20 Þ^ ^ ðt1 Þ^ dt1 G μþG μþF 0
0
Eðt2 - t1 ÞEðt1 Þe-iω0 ð2t2 - t1 Þ
Figure 1. Double-sided Feynman diagrams associated with (a) the first and (b) the second term under the integrals in eq 8. They represent the thirdorder nonlinear optical polarization induced by the pulse train (eq 6) and propagating in the wave vector, ks = k1 þ k2 - k3, direction. The vertices denoted by the components of the transition dipole operator μ^ = μ^þ þ μ^(eq 7) indicate the interactions between the associated optical pulses and the coupled exciton-biexciton states. The time evolution of the photoexcited ^ (eq 4). states is described by the Hilbert space Green function, G
small size), the intraband optical transitions could become allowed depending on the selection rules. Therefore, for the sake of generality, we will account for the intraband dipole transitions on equal footing and further identify their 2D spectroscopic signatures along with the signatures of the interband Coulomb interactions. Using the sum-over-states approach and the rotating-wave approximation, the following general expression for the nonlinear polarization propagating in the ks = k1 þ k2 - k3 direction can be derived:36 Z ¥ Z ¥ Z ¥ PðtÞ ¼ dt3 dt2 dt1 0
0 -^
0
-^
þ^
þ
ðTr½^ μ Gðt3 Þ^ μ Gðt2 Þ^ μ Gðt1 Þ^ μ F - ^† -^ þ^ μ Gðt3 þ t2 Þ^ μ Gðt1 Þ^ μ þ FÞ - Tr½^ μ G ðt3 Þ^ E ðt - t3 - τ3 ÞEðt - t3 - t2 - τ2 Þ Eðt - t3 - t2 - t1 - τ2 Þeiω0 t3 þ 2iω0 t2 þ iω0 t1
ð8Þ
Here, the dipole operator (eq 7) is partitioned into the sum of the two terms, μ^ = μ^þ þ μ^-, describing the optical transitions up and down, respectively.36 F = |x0æÆx0| is the equilibrium density operator for the ensemble of nanostructures in the ground state. Equation 8 is a convolution of the optical pulses with the nonlinear response function components given by two terms under the trace sign. Each response function is associated with the double-sided Feynman diagram shown in Figure 1, describing the Liouville space pathways contributing to the double-quantum coherence signal. These pathways have been studied before for exciton and biexciton manifolds, in which the intraband dipole transitions and the interband Coulomb interactions are forbidden.36,40,42,43,46 For our analysis of the effects of the interband Coulomb interactions and the intraband optical transitions, this expression is a starting point. For the analysis of the nonlinear response in the region of dense manifolds, the pulse finite spectral widths (i.e., finite duration time) should be accounted for. Since eq 8 contains a three-fold pulse convolution, it is not convenient for further analysis. Thus, taking into account the fact that the third pulse does not overlap with the first pair of pulses, we introduce the associated delay time, T2 = τ3 τ2. Further introducing the delay time, T3 = t - τ3, between the third pulse and the polarization measurement event (i.e., heterodyning), eq 8 can be represented in a form more convenient for analysis: ^ eiω0 ð2T2 þ T3 Þ ^ ðT3 ÞI ^1 G ^ ðT2 ÞD PðT3 , T2 Þ ¼ Tr½^ μ-G ^ eiω0 ð2T2 þ T3 Þ ð9Þ ^ † ðT3 ÞI ^2 G ^ ðT3 þ T2 ÞD - Tr½^ μ-G
ð10Þ
describes the preparation of the double-quantum coherence by the first pair of pulses, and the operators Z ¥ 0 0 ^ ð - t30 Þ^ ^ ðt30 ÞE ðt30 Þe-iω0 t3 dt3 G μ-G ð11Þ I ^1 ¼ -¥
I ^2 ¼
Z
¥
-¥
0
^ † ð - t3 Þ^ dt3 G μ - E ðt3 Þe-iω0 t3 0
0
0
ð12Þ
describe further transitions from the double-quantum coherence to the single-quantum coherences induced by the third pulse. The time evolution of the optically manipulated coherences during the pulse delay times is governed by the Hilbert space Green function (eq 4). Using the expressions for the nonlinear polarization derived above, we define the 2D double-quantum coherence signal in a conventional way: Z ¥ Z ¥ SðΩ3 , Ω2 Þ ¼ dt3 dt2 PðT3 , T2 ÞeiðΩ3 - ω0 ÞT3 þ iðΩ2 - 2ω0 ÞT2 0
0
ð13Þ For further convenience, the origin of the 2D signal represented in the (Ω3,Ω2)-plane is shifted to (ω0,2ω0), where ω0 is the pulse central frequency. Equations 9-13 along with eqs A1-A12 from Appendix A constitute a closed set for the calculations of the 2D signal of interest. However, this general approach does not give any insight into the cross-peak structure. Therefore, we perform further modifications by using the following representation for the Hilbert space Green function:31 ^ ðtÞ ¼ G
^ ξ Þe-iω~ t ∑ξ Λðω ξ
ð14Þ
Here, the complex frequencies, ω~ξ = ωξ - iγξ, are the poles of ^ (ω), giving the Coulomb coupled exciton and biexciton G eigenfrequencies, ωξ. Hereafter, we will refer to the associated eigenstates, |ωξæ, as quasiparticle states. ^ (ωξ) = Res(G ^ ,ω~ξ), entering eq 14 is the Green The quantity, Λ function residue which can be represented in the block-matrix form: 0 1 ^ x, xx ðωξ Þ ^ x ðωξ Þ Λ Λ ^ ξÞ ¼ @ A ð15Þ Λðω ^ xx ðωξ Þ ^ xx, x ðωξ Þ Λ Λ This quantity determines the probability amplitudes for various transitions within and between the exciton and biexciton manifolds associated with the time evolution of the quasiparticle state, |ωξæ. This representation provides a connection between the spectroscopic resonances determined by the quasiparticle frequencies and the bare exciton/biexciton transitions. Substitution of eqs 14 and 15 into eqs 9-13 leads to the following compact form for the 2D double-quantum 5374
dx.doi.org/10.1021/jp109453y |J. Phys. Chem. B 2011, 115, 5372–5382
The Journal of Physical Chemistry B coherence signal: SðΩ3 , Ω2 Þ ¼
∑ ∑ Rβ t
Cβ, t ~β Ω2 - ω
ARβ, t BR, t ~ R Ω3 - ω ~ βR Ω3 - ω
ARTICLE
!
where, without lack of generality, we set ωxr j ωxx r (an opposite case when ωxr > ωxx r is considered, in particular, in section IV). The resonant states are split due to the interband Coulomb interaction: V^ C ¼
ð16Þ where ω~βR = ω~β - ω~*R , and the coefficients are ~ βR Þ ARβ, t ¼ E ðω ~ RÞ BR, t ¼ E ðω Cβ , t ¼
∑pq μ0pΛpq ðω~ R Þμqt
∑pq μtq Λqp ðω~ R Þμp0
∑γ F βγ spq ∑ Λts ðω~ β ÞμspΛpq ðω~ γÞμq0
ð17Þ ð18Þ ð19Þ
Note that these coefficients depend on all the possible matrix elements of the transition dipole moment and the transition amplitudes. Thus, we dropped the superscripts describing the manifold each index characterizes. The superscripts can be easily restored in a particular case. Equations 17-19 also contain the convolution function between the overlapping first and second pulses Z ¥ Z ¥ dt dt1 EðtÞEðt - t1 Þeiðω~ β - 2ω0 Þt - iðω~ γ - ω0 Þt1 F β, γ ¼ -¥
0
ð20Þ and the Fourier transform of the third pulse envelope function: Z ¥ EðtÞeiðω - ω0 Þt dt ð21Þ EðωÞ ¼
2
∑ jxa æpωxa Æxa j þ k ¼∑1, 2 jxxkæpωxxk Æxxk j a¼0
ð22Þ
ð23Þ
^ ¼ jx0 æμx01 Æx1 j þ jx1 æμx12 Æx2 j þ jx1 æμx1,, xx μ 2 Æxx2 j xx, x þ jxx1 æμ1, 2 Æx2 j þ jxx1 æμxx 12 Æxx2 j þ h:c:
ð24Þ
which accounts for all the possible inter- and intraband transitions (Figure 2). For our five-level model, the 2D double-quantum coherence signal (eqs 16-21) becomes a sum of eight components: SFLS ðΩ3 , Ω2 Þ ¼
8
∑ Sj ðΩ3 , Ω2 Þ
ð25Þ
j¼1
with Sj ðΩ3 , Ω2 Þ ¼ Mj
∑
R, β ¼ (
Bj ~ β2 Ω2 - ω
Aj Aj ~ R1 Ω3 - ω ~ β2,, 1R Ω3 - ω
!
ð26Þ ω ~β,R 2,1
ω~β2
ω~R* 1 ,
= and the coupled exciton and biexciton Here, eigenfrequencies (determining the quasiparticle energies) are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2ffi x xx xx 2 ω þ ω ω ω Vr r r r r ( þ ð27Þ ω( r ¼ 2 2 p where r = 1, 2. The expressions for the coefficients Aj and Bj are listed in Table 1, and depend on the transition amplitude matrix elements explicitly represented as31
III. DOUBLE-QUANTUM COHERENCE SIGNAL IN A FIVELEVEL SYSTEM In this section, we analyze the 2D double-quantum coherence signal for the effective five-level system depicted in Figure 2. This model contains two pairs of resonant exciton, |xræ, and biexciton, |xxræ, states (r = 1,2) and the ground state, |x0æ. As a result, the noninteracting component of the material Hamiltonian (eq 1-3) becomes ^0 ¼ H
jxr æVr Æxxr j þ h:c:
where the shorthand notation, Vr Vx,xx r,r , is used. The well-separated in energy nonresonant states are naturally assumed to be uncoupled. In eq 22, the exciton frequencies are set to be in resonance with the optical pulses, i.e., ωx1 ∼ ω0 and ωx2 ∼ 2ω0. The spectral width of the pulse is assumed to be much less than the transition energies but broader than the Coulomb interactions, Vr, and the level widths, γxr and γxx r . Accordingly, the impulsive excitation regime can be adopted, allowing us to approximate the pulse envelope functions as E j(t - τi) = E 0δ(t - τi). Here, δ(t - τi) is the Dirac delta-function and E 0 is the effective amplitude which we set to unity. Finally, the transition dipole operator entering the optical Hamiltonian (eq 5) has the following form:
-¥
Equations 16-21 are our main result, providing the 2D double-quantum coherence signal for the general case of coupled exciton and biexciton states. Although these expressions explicitly contain the cross-peak resonances, their analysis and identification of the interband Coulomb interaction signatures is complicated. At this point, we use the notion of the resonant exciton-biexciton state pairs introduced in section I. By looking at the Feynman diagrams in Figure 1, one can see that both pathways involve two optical transitions up and two down. This suggests that, for the well-separated in energy pairs of the resonant states, the maximum number that signal calculations require is two pairs. Including the ground state, this leads to the effective five-level system whose response is considered in detail below. The presence of a large number of off-resonant states can be accounted for as perturbative corrections to the signal. This procedure is outlined in Appendix B. Finally, the analysis of the signal in the general case when the off-resonant levels also have strong couplings (e.g., for the carbon nanotubes) can be described using the same strategy that we discuss in the next section.
∑
r ¼ 1, 2
Λxrr ðω( r Þ ¼ (
xx ðω( r - ωr Þ þ ðωr - ωr Þ
ð28Þ
( Λxx rr ðωr Þ ¼ (
x ðω( r - ωr Þ ðωþ r - ωr Þ
ð29Þ
Vr pðωþ r - ωr Þ
ð30Þ
Λxr,, rxx ðω( r Þ ¼ (
The level diagrams and the associated double-sided Feynman diagrams for each signal component are shown in Figure 2. 5375
dx.doi.org/10.1021/jp109453y |J. Phys. Chem. B 2011, 115, 5372–5382
The Journal of Physical Chemistry B
ARTICLE
Figure 2. Level diagrams showing the optical transition and the double-sided Feynman diagrams representing associated Liouville space pathways for the corresponding components of the 2D double-quantum coherence signal given by eqs 25 and 26 and Table 1. The Feynman diagrams are obtained from those presented in Figure 1 by using the Hilbert space Green function representation given by eq 14 and including all allowed matrix elements. Note that, due to the adopted impulsive excitation regime, the delay time, t1, is explicitly set to zero.
According to these diagrams, turning off the intraband optical transitions, i.e., μx12 = μxx 12 = 0, and the interband Coulomb interactions, i.e., V1 = V2 = 0, leads to a nonvanishing contribution of the first component only. This component acquires the following simple form: ! x, xx jμx12 j2 jμ1, 2 j2 1 1 ð0Þ S1 ðΩ3 , Ω2 Þ ¼ ~ xx ~ x1 Ω3 - ω ~ 2xx, ,1x Ω3 - ω Ω2 - ω 2 ð31Þ and corresponds to the (Ω3, Ω2) cross-peaks such as ωxx 2 ) xx,x xx and (ω2,1 , ω2 ) shown in Figure 3a. This cross-peak structure (ωx1,
has been studied extensively.39-46 It has been demonstrated that the cross-peaks carry information about the biexciton binding energies (intraband Coulomb interactions) through their Ω3splitting. First, we describe the effect of the interband Coulomb interactions on the cross-peak structure in the absence of the intraband optical transitions, i.e., μx12 = μxx 12 = 0. Setting V1 6¼ 0 (V2 6¼ 0) and V2 = 0 (V1 = 0) leads to the mixing of |x1æ and |xx1æ (|x2æ and |xx2æ) states. As a result, mixed, i.e., quasi-particle, states þ |ωþ 1 æ and |ω1 æ (|ω2 æ and |ω2 æ) appear. The signal in this case is still due to the S1 component only (Figure 2). However, the level mixing and the associated oscillator strength transfer lead to the 5376
dx.doi.org/10.1021/jp109453y |J. Phys. Chem. B 2011, 115, 5372–5382
The Journal of Physical Chemistry B
ARTICLE
xx þþ xx appearance of two new cross-peaks: (ωþ 1 , ω2 ) and (ω2,1 , ω2 ) x xx,x ((ω1, ω2 ) and (ω2,1 , ω2 )) as shown in panel b (panel c) of xx,x xx Figure 3. Note that the cross-peaks (ωx1, ωxx 2 ) and (ω2,1 , ω2 ) in panel a further split in frequency due to the interband interaction, þ,xx xx and as shown in panels b and c become (ω1 , ω2 ) and (ω2,1 , ω2 ) cross-peaks, respectively. If both V1 and V2 interactions are turned on, the response becomes a superposition of the following two components, S1 þ S8 (Figure 3), and the spectrum shown in Figure 3d contains eight cross-peaks, which is their maximum amount possible for the five-level system. In this spectrum, six cross-peaks can be treated as the superposition of the state mixing effects illustrated in panels b and c. Additionally, two new resonances, (ω-þ 2,1 , ω2 ) , ω ), appear as a contribution from the S compoand (ω-2,1 2 8 nent. If V1 = V2 = 0, this component vanishes identically, since it
Table 1. List of Coefficients Entering eqs 25 and 26 Describing the 2D Double-Quantum Coherence Signal for the Five-Level System Sj
Mj
Aj
Bj
S1
2 |μx01|2|μx,xx 1,2 |
Λx11(ωR1 )
β Λxx 22(ω2 )
S2
|μx01|2|μx12|2
Λx11(ωR1 )
Λx22(ωβ2 )
S3
x |μx01|2μx,xx 1,2 μ21
Λx11(ωR1 )
β Λxx,x 2,2 (ω2 )
S4
|μx01|2μx21μx,xx 2,1
Λx11(ωR1 )
β Λx,xx 2,2 (ω2 )
S5 S6
x |μx01|2μxx,x 1,2 μ21 x 2 xx xx,x |μ01| μ12μ2,1
R Λx,xx 1,1 (ω1 ) x,xx Λ1,1 (ωR1 )
Λx22(ωβ2 ) β Λxx 22(ω2 )
S7
x |μx01|2μxx 21μ21
R Λx,xx 1,1 (ω1 )
β Λxx,x 2,2 (ω2 )
S8
xx,x |μx01|2μxx,x 1,2 μ2,1
R Λx,xx 1,1 (ω1 )
β Λx,xx 2,2 (ω2 )
R depends on the interband transition amplitudes Λx,xx 1,1 (ω1 ) and x,xx β Λ2,2 (ω2 ) (Table 1). Now, we turn our attention to the effects of the intraband optical transitions on the cross-peak structure. Assuming that V1 = V2 = 0 but all the dipole transitions in eq 24 are allowed, one can conclude from Figure 2 (and Table 1) that the total signal is determined by S1 þ S2 components. The other contributions vanish due to their dependence on the interband transition amplitudes. In this case, S1 is given by eq 31 and S2 simplifies to
ð0Þ S2 ðΩ3 , Ω2 Þ
jμx12 j2 jμx1, 2 j2 1 1 ¼ ð32Þ ~ x2 ~ x1 Ω3 - ω ~ x21 Ω3 - ω Ω2 - ω
In addition to the resonances shown in panel a of Figure 3, this component gives rise to two new cross-peaks, (ωx1, ωx2) and (ωx21, ωx2), as demonstrated in panel e. The effect of the Coulomb interactions on the cross-peak structure in panels f-h can be interpreted in the same way as in panels b-d. However, one has to keep in mind that in this case the signal in panels f, g, and h is the result of the contributions from S1 þ S2 þ S5 þ S6, S1 þ S2 þ S3 þ S4, and S1 þ S2 þ ... þ S8 pathways, respectively (Figure 2). This leads to different crosspeak patterns and also to significant variation of the cross-peak intensities associated with pathway interferences. In Figure 3, we present the most general cross-peak patterns associated with each considered case. However, the interference between the pathways can lead to the complete vanishing of certain crosspeaks present in the plot. Therefore, the identification of the resonances in more complex situations should be based on the modeling of the 2D double-quantum coherence spectra. The
Figure 3. The 2D double-quantum coherence spectrum for the five-level system. Black dashed lines show the frequencies of uncoupled exciton and biexciton states, while red dashed lines indicate frequency splittings due to the interband Coulomb interactions. In all panels, the interband dipole xx,x x xx transitions, μx,xx 1,2 and μ1,2 , are allowed. No intraband transitions are allowed, i.e., μ12 = μ12 = 0, in the following panels where the Coulomb interactions are (a) V1 = V2 = 0, (b) V1 6¼ 0 and V2 = 0, (c) V1 = 0 and V2 6¼ 0, and (d) V1 6¼ 0 and V2 6¼ 0. The intraband transitions are allowed, i.e., μx12 6¼ 0 and μxx 12 6¼ 0, in the following panels, and the interband Coulomb interactions are (e) V1 = V2 = 0, (f) V1 6¼ 0 and V2 = 0, (g) V1 = 0 and V2 6¼ 0, and (h) V1 6¼ 0 and V2 6¼ 0. 5377
dx.doi.org/10.1021/jp109453y |J. Phys. Chem. B 2011, 115, 5372–5382
The Journal of Physical Chemistry B
ARTICLE
Figure 4. Energies of the coupled exciton, |2Pe; 1Shæ, and biexciton, |1Se, 1Se; 1Sh, 1Shæ, states in PbSe NC as a function of the NC diameter. The biexciton binding energy of 30 meV is included.
modeling is also important, since the contributions of the interband Coulomb interactions and the intraband optical transitions cannot be distinguished based only on the cross-peak pattern. One can easily notice that panels c, e, and g in Figure 3 have similar crosspeak patterns. The same applies to panels d, f, and h. The interpretation of 2D double-quantum coherence spectra in the general case of the multilevel system introduced in section II can be performed in the same way as for the five-level system. Specifically, the identification of the cross-peaks in the case of no interband Coulomb interaction should be done first and their further splitting due to the interband interactions should provide the required information on the new expected cross-peaks. In the next section, we apply the methodology developed above to investigate the 2D double-quantum coherence spectra associated with an ensemble of PbSe NCs. As we demonstrate below, the partitioning into resonant and off-resonant states in the NCs can be performed, allowing us to map the problem onto the considered effective five-level system.
IV. SPECTRAL SIGNATURES OF THE INTERBAND COULOMB INTERACTIONS IN PbSe NANOCRYSTALS To calculate the electronic structure of PbSe NCs, we have implemented the k 3 p effective-mass model developed by Kang and Wise.48 Using this model, we evaluate the electron and hole wave functions, and then construct the exciton and biexciton states as their uncorrelated configurations. Among these states, the lowest in energy exciton and biexciton pair that can be tuned in resonance is identified as |2Pe; 1Shæ and |1Se, 1Se; 1Sh, 1Shæ, respectively. Here, the exciton state |2Pe; 1Shæ is ∼10 meV split by the spin-orbit interaction into two components characterized by the electron and hole angular momenta (Je = 1/2; Jh = 1/ 2) and (Je = 3/2; Jh = 1/2). Only the former component couples to the resonant biexciton state. In the strong confinement regime, the exciton binding energy is a small correction to the bare exciton energy. Thus, we safely drop it in the calculations. The calculation of the biexciton binding energy requires evaluation of the second- and higher-order four-particle Coulomb correlations along with the surface polarization contributions.11,49 To avoid this tedious task and the associated value and sign controversy,11,49 we set the value of attractive biexciton binding energy to 30 meV.50 The energy dependence of the selected exciton and biexciton states as a function of NC diameter is plotted in Figure 4. According to the plot, the resonance condition is satisfied for a NC diameter ranging between 6 and 7 nm. Thus, for our ensemble of NCs, we fix the mean diameter at 6.2 nm, and assume a Gaussian size distribution with 5% standard deviation.
Figure 5. Level diagram indicating the resonant |2Pe; 1Shæ-exciton and |1Se, 1Se; 1Sh, 1Shæ-biexciton states for PbSe NC of 6.2 nm diameter. Since the band-edge exciton state |1Se; 1Shæ is participating in the double-quantum coherence preparation, it is also shown in the plot. Allowed interband dipole transitions and the interband Coulomb coupling are marked as μx, μx,xx, and V, respectively. (b) Exciton and biexciton DOS containing the resonant levels (vertical arrow) calculated for the same NC. All the selection rule allowed off-resonant exciton and biexciton states with the energies
