ARTICLE pubs.acs.org/JPCC
Multiresonant Coherent Multidimensional Electronic Spectroscopy of Colloidal PbSe Quantum Dots Lena A. Yurs, Stephen B. Block, Andrei V. Pakoulev, Rachel S. Selinsky, Song Jin, and John Wright* Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706, United States
bS Supporting Information ABSTRACT: We demonstrate the use of multiresonant coherent multidimensional spectroscopy (CMDS) for obtaining 2D spectra of the diagonal and cross-peaks of the 1S and 1P excitons and biexcitons in PbSe quantum dots and their coherent and incoherent dynamics. We show that multiresonant CMDS line narrows the inhomogeneous broadening and resolves the excitonic peaks from the background that often obscures peaks. We develop theoretical methods that extract details of the homogeneous and inhomogeneous broadening, the Coulombic coupling within the biexciton, and the relative exciton and biexciton transition moments. The population dynamics are measured by scanning the excitation pulse time delays over all time orderings. Phase modulations of individual coherences are observed because of heterodyning between scattered light and the four-wave mixing signal. The experiments demonstrate that CMDS can be used to obtain quantum state resolved dynamics of the electronic states in complex nanostructures and other important materials.
’ INTRODUCTION Quantum confined nanostructures containing heterojunctions between donor and acceptor materials are interesting candidates for photovoltaic and photocatalytic applications. Typically, absorption of light in the donor material creates electron/ hole pairs that are separated by direct injection of the electron into the acceptor material. Ultrafast laser methods such as pump probe,13 transient absorption,4 transient grating, and photon echo5 are typical approaches to measuring this key process. The development of time domain coherent multidimensional spectroscopy (CMDS) extends these time domain methods to provide 2D and 3Dspectroscopy of the coherent and incoherent dynamics.619 Multiresonant CMDS2026 is an alternative mixed frequency/time domain approach to spectroscopy that can potentially form multiple quantum coherences involving the diverse quantum states involved in complex donoracceptor nanostructures and resolve their dynamics. This Article reports the first application of multiresonant CMDS to PbSe quantum dots. Physical parameters determined by this technique are state energies, biexciton Coulombic coupling energy, homogeneous and inhomogeneous spectral widths, population lifetime, and transition dipole moment strengths relative to the lowest optically active excitonic transition. Multiresonant CMDS uses multiple coherent excitations to create quantum mechanical superposition states on time scales that are faster than the dephasing rates.26 Pairs of the states then re-emit coherent beams at the frequency differences between the pairs. The emission is directional and occurs during the excitation process (the “driven process”) and continues after excitation r 2011 American Chemical Society
until dephasing occurs (free induction decay). The direction is defined by phase matching conditions that conserve momentum. An aperture and a monochromator spatially and spectrally isolate the output beam of interest. The output beam intensity is measured as a function of the excitation frequencies, the monochromator frequency, and the time delays between the different excitation pulses. Typical experiments obtain multidimensional spectra by scanning the excitation frequencies with constant time delays or obtain the coherent or incoherent dynamics of specific spectral features by scanning the excitation pulse time delays. Homodyne detection is used so the method is not constrained by the need for long-term phase stability between the excitation pulses. However, the excitation pulses do need to retain their short-term phase coherence during the creation of the quantum mechanical superposition states. The freedom from long-term phase coherence permits the use of excitation beams with very different frequencies, where it would be very difficult to maintain the necessary coherence between beams. The optimum excitation pulse duration for multiresonant CMDS corresponds to the dephasing time.24 That choice spectrally resolves individual quantum states and temporally resolves their dynamics. In this work, we report the first application of multiresonant CMDS to quantum confined nanostructures using PbSe quantum dots as a model system. Our excitation pulse durations are much longer than the dephasing Received: July 29, 2011 Revised: September 12, 2011 Published: October 06, 2011 22833
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The Journal of Physical Chemistry C times, so it is not possible to resolve the coherent dynamics of individual quantum states, nor is it possible to resolve the individual coherence pathways that are responsible for the output signal. Nevertheless, we demonstrate that it is possible to obtain line-narrowed 2D spectra for diagonal and cross-peaks involving the 1S and 1P excitons.27 We further show that the multiresonant CMDS spectra extract the quantum confined excitonic peaks from the background absorption that often obscures these features. This capability allows multiresonant CMDS to provide quantum state resolution in the multidimensional spectra. In addition to the diagonal and cross-peaks, there is a narrow resonance when the excitation pulses are temporally overlapped and the frequencies are identical. It arises from a resonance that is similar to dephasing induced resonances that depend on the relative size of pure dephasing and population relaxation.28 The line shapes and intensities of the line-narrowed 2D spectral features are controlled by the relative transition moments, Coulombic coupling within the biexciton state, dephasing rates, and inhomogeneous broadening of the exciton and biexciton quantum states. Each variable exerts a different influence on the features. We use an approach previous described29,30 to derive closed form equations for the multiple enhancements associated with all of the nonlinear processes assuming a steady-state approximation and Lorentzian inhomogeneous broadening. Although inhomogeneous broadening is not described by a Lorentzian distribution, its use does allow the formulation of an analytic solution to the differential equations that make it easier to obtain a clear understanding of how the different variables influence the spectral data. We performed least-squares fits of the data to the equations and extracted values for the relative transition moments, Coulombic coupling, dephasing rates, and inhomogeneous broadening that control the spectral features. We also show that changing the time delays between the excitation pulses while maintaining the same excitation frequencies measures the dynamics. The time delay data are fit with a model that is based on Domcke’s theoretical approach to directly integrate the Schrodinger equation.3134 We find that the coherence dephasing times in our experiments are much faster than the excitation pulses so the dephasing rates cannot be measured in these experiments, but we can measure the population relaxation rates. In addition, we show that it is possible for the multiresonant CMDS methods to resolve directly the rapid phase oscillations in the coherence created after the first interaction in the time delay experiments. These modulations appear in both 2D plots of the dependence on the two time delays and a Wigner representation on frequency and time delay.35 All of these effects can also be quantitatively modeled using Domcke’s approach.
’ THEORY Our experiments use the B k4 = B k1 B k2 + B k 20 phase matching k i are the wave condition, where B k 4 is the output wave vector and B k 20 beams are derived vectors of the excitation pulses. The B k 2 and B by splitting the original excitation pulse. The subscripts designate the frequencies of the ω1 and ω2excitation pulses. Figure 1 shows the 16 fully resonant coherence pathways that must be considered with this phase matching condition.36 Because our excitation pulse durations are long compared with the dephasing times, all 16 pathways contribute to the output signal. In Figure 1, the
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Figure 1. Liouville diagrams for all the pathways creating eg; e0 g; (e0 + e),e; and ev output coherences. The arrows represent transitions induced by ω1, ω2, or ω20 , the boxes designate the gg initial population or the output coherences. The green, blue, red, and black colors designate zero, single, and double quantum coherences and populations, respectively.
letters label the ket and bra states of density matrix element (i.e., k 1, B k 2, B k 20 ba f cbca*), the numbers and arrows label the B beams, and the boxes identify the initial ground-state population (gg) and the output coherences. Six of the pathways are partially coherent because they involve intermediate populations that can relax to other states and create new pathways and spectral features. These pathways are particularly valuable for measuring relaxation dynamics. Ten pathways are fully coherent because they do not involve intermediate populations. They are valuable for measuring the coherent dynamics including coherence transfer where environmental fluctuations induce states to evolve coherently to new states. Because the excitation pulse durations are long compared with the coherence dephasing times, we will use the steady-state limit for fully resonant four-wave mixing to describe the experiments.21 State designations e and e0 are arbitrary but are chosen in this Article to evoke two distinct excited electronic states. We first consider the fully coherent pathways. The fully coherent pathways involving a zero quantum intermediate coherence are 2 1 0 20 0 1 0 2 0 20 0 gg f e g f e e f e g or (e' + e); gg f ge f e e f e g or (e0 + e),e. The fully coherent pathways0 involving double quantum coher1 2 2 ence0 pathways are gg f e0 g fðe0 þ eÞ, g f e0 gor(e0 + e),e; and 2 1 2 gg f eg fðe0 þ eÞ, g f e0 gor(e0 + e),e. These pathways assume that ω1 is resonant with a transition involving state e0 and ω2 is resonant with a transition involving state e. These fully coherent pathways do not involve any intermediate populations and allow measurement of the coherent dynamics of the zero, single, and double quantum coherences involving e and e0 . The output coherence resulting from the third interaction is either e0 g or (e0 + e),e. The former returns the system to the ground state after the output emission. It is a parametric process that does not transfer energy to the system.37 The latter leaves the system in an excited state. It is a nonparametric process that does transfer energy. These pathways create coherences that interfere and change the nature of the observed resonant enhancements. In the steadystate limit, the enhancements from transitions where ca f ba are proportional to μba/(ωba ω iΓba) where μba, ωba, and Γba are the transition moment, frequency, and dephasing rate of 22834
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the ba coherence, respectively, while the ω is the excitation frequency creating the transition. In the following equations, there are three resonances and multiple pathways. The multiple pathways interfere so it is important to combine common resonances to understand the net effects.28 The resonances for the fully coherent pathways with zero quantum coherences are proportional to 0 10 1 μe0 g μeg μeg μe0 g μeg μe0 g μe0 þe, e0 μe0 þe, e @ A@ A þ ð1Þ ð1, 2Þ ð1, 2Þ ð1, 2, 20 Þ ð1, 2, 20 Þ Δe0 g Δe0 e Δð2Þ Δe0 g Δe0 þ e, e ge Δe0 e ¼
1
iðΓe0 e Γe0 g Γeg Þ
!
1 þ ð1Þ ð1, 2Þ Δe0 g Δð2Þ Δe0 e ge 0 1 μ2eg μ2e0 g μeg μe0 g μe0 þe, e0 μe0 þe, e A @ ð1, 2, 20 Þ ð1, 2, 20 Þ Δe0 g Δe0 þ e, e
ð1Þ
0
) where the resonance denominators are Δ(1,2,2 ωba ω1 + ba 0 ω2 ω2 iΓba and μba, ωba, and Γba are the transition moment, frequency, and dephasing rate of the ba coherence, respectively, whereas the ωi are the excitation frequencies creating the coherence or population. The fully coherent pathways with double quantum coherences are proportional to 0 10 1 μe0 g μe0 þe, e0 μeg μe0 þe, e μe0 þe, e0 μe0 g μeg μe0 þe, e @ A@ þ ð1, 2, 20 Þ A ð1Þ ð1, 20 Þ ð1, 2, 20 Þ ð20 Þ ð1, 20 Þ Δe0 g Δe0 þ e, g Δeg Δe0 þ e, g Δe0 g Δe0 þ e, e
ð2Þ 0
0
) (1,2,2 ) 0 0 In both cases, the factor involving Δe(1,2,2 and Δe+e g ,e shows that the output signal depends on the difference in two terms. The transitions associated with the two terms differ only in whether the e0 transition originates from the ground state or an excited state. If the excited state does not affect the e0 transition, then the transition moments and the resonance denominators will be equal and the output coherence exactly cancel. Coupling is therefore required for a nonzero output. Both sets of pathways have enhancements when ω1 and ω2 are resonant with the g fe0 or g fe transition. In addition, the zero quantum coherence pathways contain a (Γe0 e Γe0 g Γeg) factor 0 that determines the importance of the Δe(1,2) e resonance enhancement. The e0 e zero quantum coherence creates a diagonal feature when ω1 ω2 = ωba. However, eq 1 shows that it may not appear because of destructive interference between the different pathways. This enhancement is analogous to the dephasinginduced resonance predicted and observed previously.28 Because Γeg = (Γgg + Γee)/2 + Γ*eg, where Γ*eg is the pure dephasing rate of the eg coherence, then
Γe0 e Γe0 g Γeg ¼ Γe0 e Γgg Γe0 g Γeg
ð3Þ
This term vanishes if the pure dephasing and ground-state bleaching rates are small. Without pure dephasing or bleaching, the output coherence depends on the triple resonance enhancement from the three single quantum coherences. The partially coherent pathways involve intermediate populations. 20 2 1 Those pathways relevant to cross-peaks are gg f egf gg f e0 g; 0 0 2 2 1 2 2 1 gg f ge f gg f e0 g; gg f eg f ee fðe0 þ eÞ, e ; and
Figure 2. Simulations of the diagonal peak under the following conditions. (a) ωeg = 7000 cm1, Γeg = 100 cm1, Γgg = 0.25 cm1, σ = 0 cm1, ω2e,e 2ωeg = 0, μ2e,e/μeg = 1. (b) ωeg = 7000 cm1, Γeg = 50 cm1, Γgg = 0.25 cm1, σ = 400 cm1, ω2e,e 2ωeg = 0, μ2e,e/ μeg = 1. (c) ωeg = 7000 cm1, Γeg = 100 cm1, Γgg = 0.25 cm1, σ = 250 cm1, ω2e,e 2ωeg = 0, μ2e,e/μeg = 1. (d) ωeg = 7000 cm1, Γeg = 100 cm1, Γgg = 0.25 cm1, σ = 400 cm1, ω2e,e 2ωeg = 125 cm1, μ2e,e/μeg = 1. 20
2
1
gg f ge f ee fðe0 þ eÞ, e. Together, the four pathways result in an e0 g or (e0 + e),e output coherence that is resonantly enhanced by the equation 0 1 μeg 2 μe0 g 2 1 1 @ þ ð20 Þ ð2, 20 Þ A ð1, 2, 20 Þ ð2Þ ð2, 20 Þ Δ Δ Δ Δ Δe0 g ge gg eg gg
μeg 2 μe0 þe, e 2 ð1, 2, 20 Þ Δe0 þ e, e
0 @
1 1 ð2, 20 Þ Δð2Þ ge Δee
þ
1 ð20 Þ ð2, 20 Þ Δee Δeg
A
0 1 μeg 2 μe0 þe, e 2 2Γeg @ μeg 2 μe0 g 2 A ¼ ð20 Þ ð2Þ 0 0 Δeg Δge Γgg Δeð1,0 g 2, 2 Þ Γee Δeð1,0 þ2, e,2 eÞ
ð4Þ
The pathways have three resonances involving the initial and final coherences and the ground- and excited-state populations. The output coherences created by these pathways destructively interfere and cancel unless coupling between e and e0 changes their transitions. The pathways are partially coherent pathways because they involve an intermediate population. They allow measurement of the incoherent population dynamics. In the general case, there are also Raman and fluorescence pathways that are not usually important for describing cross peaks involving electronic resonances, but they become important when describing diagonal peaks or experiments involving vibra1 2 20 tional resonances. These pathways are gg f eg f vg f eg, 2
1
20
2
20
1
20
2
gg f ge f gv f ev, gg f ge f ee f ev, and gg f eg f 1 ee f ev, where v indicates a vibrational state (for diagonal peaks in CMDS spectra, v can be the ground state, g). We have assumed that ω1 and ω2 can be resonant with state e and that their difference is resonant with the e fv transition. These pathways 22835
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will also interfere, and a similar treatment shows that the eg output coherence is resonantly enhanced, as described by the equation μeg 2 μev 2 ð1, 2, 20 Þ
ð1, 2Þ Δð1Þ eg Δvg Δeg
þ
μeg 2 μev 2 ð1, 2, 20 Þ
ð1, 2Þ Δð2Þ ge Δgv Δev
μeg 2 μev 2 ¼ ð20 Þ ð2, 20 Þ ð1, 2, 20 Þ Δeg Δee Δev μeg 2 μev 2 2Γeg þ ð20 Þ ð1, 2, 20 Þ Γee Δð2Þ ge Δeg Δev
þ
þ
μeg 2 μev 2 0
0
ð2, 2 Þ ð1, 2, 2 Þ Δð2Þ Δev ge Δee
μeg 2 μev 2 ð1Þ ð1, 2Þ ð1, 2, 20 Þ Δeg Δvg Δeg
þ
μeg 2 μev 2 ð2Þ ð1, 2Þ ð1, 2, 20 Þ Δge Δgv Δev
ð5Þ
Together with the previous eight pathways, they describe the cross-peaks in a multidimensional spectrum. The resonance enhancements expected for the diagonal peak are identical to those in eqs 1, 2, 4, and 5 if e and e0 are the same states and v and g are the same states. The final expression for the enhancements of the diagonal peak becomes 20 1 μ2eg μ2 iðΓ 2Γ Þ gg eg 4 @1 þ A eg 0 ð2Þ 2Þ 2, 2 Þ Δð1, Δð1Þ Δð1, eg Δge gg eg 0 13 ! 2 2 μ μ iðΓee 2Γeg Þ @ eg 2e, e þ 1 þ ð1, 2, 20 Þ A5 ð1, 2Þ ð1, 2, 20 Þ Δee Δeg Δ2e, e 2 3 ! 2 2 2Γeg μ2eg μ μ 1 1 eg 2e, e 5 þ ð20 Þ ð2Þ 4 þ ð1, 2, 20 Þ ð1, 2, 20 Þ Γgg Γee Δeg Δeg Δge Γee Δ2e, e 0 10 1 μ2eg μ2eg μ22e, g 1 1 ð6Þ þ ð1, 2Þ @ ð1Þ þ ð20 Þ A@ ð1, 2, 20 Þ ð1, 2, 20 Þ A Δeg Δeg Δ2e, g Δeg Δ2e, e The resonances are very similar to those previously discussed. Note that they include Γgg 2Γeg and Γee 2Γeg terms that and Δ(1,2) dephasing-induced define the importance of the Δ(1,2) gg ee resonances previously described. Z
0 σ
μeg 2 μe0 g 2
When inhomogeneous broadening is present, it becomes important to consider the line-narrowing effects in the multidimensional spectra.21,30,3840 We assume that the inhomogeneous broadening is larger than the homogeneous broadening. Line-narrowing occurs because resonance with one excitation frequency will create an additional enhancement for the sample component within the inhomogeneously broadened distribution that is resonant with that frequency. Because successive resonant enhancements are multiplicative, the component will be selectively enhanced over other components that were not resonant with the excitation frequency. This effect is complicated by interference between the selected component and the other less resonant components. If the interference is destructive, then the less resonant components suppress the more resonant component, and no narrowing occurs.21 If it is constructive, then the less resonant components enhance the more resonant component, and narrowing is observed. It is possible to obtain analytical expressions for the enhancements in the presence of inhomogeneous broadening if the distribution is Lorentzian. We assume that the broadening perturbation affects all electronic states equally. In this case, the frequencies within the inhomogeneous distribution are ωag(ξ) = ωago + ξ, and the distribution is defined by σ ð7Þ 2 πðξ þ σ2 Þ where σ defines the width of the distribution and ωoag is the center frequency. Contour integration of eq 7 with the enhancements given by eqs 1, 2, 4, and 5 provides expressions for the enhancements of each pathway in the presence of inhomogeneous broadening. We first consider the effects of inhomogeneous broadening on the four fully coherent pathways with e0 e zero quantum coher(2) 0 Δ ences. These pathways create the 1/(Δe(1) g ge )resonance in eq 1. The contour integration over the inhomogeneous distribution gives
μeg μe0 g μe0 þe, e μe0 þe, e0
1
iðΓe0 e Γeg Γe0 g Þ
!
@ A 1 þ dξ 0 ð1Þ ð1, 2, 20 Þ ð1, 2Þ 2 2 Þ ðΔð1, 2, 2 Þ þ ξÞ πðΔe0 g þ ξÞðΔð2Þ ξÞðξ þ σ ðΔ þ ξÞ Δe0 e 0 0 ge eg ðe þ eÞ, e 9 8 2iσ > > > > > > ð1, 2Þ ð2Þ 2 2 > > 0 g þ Γeg Þ½ðΔ ½δ iðΓ Þ þ σ > > 0 e ge e e > > > > 0 1 > > > > > !> 2 2 > > μeg μe0 g μeg μe0 g μe0 þe, e μe0 þe, e0 = < @ A iðΓe0 e Γeg Γe0 g Þ x ð1, 2Þ ð1, 2Þ ¼ 1 þ ð1, 2Þ ½δe0 e iðΓe0 g þ Γeg Þ ½δðe0 þ eÞ, ðe þ eÞ iðΓðe0 þeÞ, e þ Γeg Þ > > Δe0 e > 0 1> > > > > > > 2 2 > > μ μ μ μ μ μ 0 0 0 0 0 > > 1 eg e g eg e g e þe, e e þe, e > > @ A > > > > 0Þ 0Þ ð1Þ ð1, 2, 2 ð1, 2, 2 > > ; : ðΔe0 g iσÞðΔð2Þ iσÞ ðΔ 0 iσÞ ge þ iσÞ ðΔe0 g
ð8Þ
ðe þ eÞ, e
0
where δ1,2,2 ωba ω1 + ω2 ω20 and δ(1,2) ba (e0 + e),e+e ωe0 +e,g 2ωeg ω1 + ω2. The factors in eq 8 that depend only on the dephasing rates of the e0 g, eg, and (e0 + e),g coherences describe the line-narrowing. The factors that depend on the ω1 ω2 frequency difference have a diagonal character in graphs that depend on ω1 or ω2. The diagonal character reflects the correlation established when resonance with ω1 or ω2 excites a subset of the quantum dot distribution and the other frequency establishes a resonance associated with the same subset.
The factors that describe the line-narrowing also involve a factor that contains both the dephasing rate and the inhomogeneous broadening. It creates a Lorentzian profile that controls the overall intensity of the line-narrowed peak. The factors in the second term contain both the dephasing rate and the inhomogeneous broadening so the second term creates a weaker, inhomogeneously broadened background. The relative size of each term scales as the square of the ratio of the dephasing rates to the inhomogeneous broadening, Γ2/σ2. The resonances for the in22836
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homogeneously broadened term depend only on ω1 or ω2. All enhancements depend on the difference between the two output resonances, again reflecting the importance of coupling between e and e0 . 0 Z
Integration over the inhomogeneous broadening for the four fully coherent pathways with (e0 + e),g double quantum coherences gives
μe0 g 2 μe0 þe, e0 2
μeg μe0 g μe0 þe, e μe0 þe, e0
1
þ C B ð1Þ 2, 20 Þ ð1, 2, 20 Þ B ðΔe0 g þ ξÞðΔð1, þ ξÞ ðΔð2Þ þ ξÞ C 0g eg þ ξÞðΔe0 g e C B C dξ B μeg μe0 g μe0 þe, e μe0 þe, e0 μeg 2 μe0 þe, e 2 C þ 2ξÞðξ2 þ σ2 ÞB A @ ð1Þ ð1, 2, 20 Þ ð1, 2, 20 Þ ð2Þ ðΔe0 g þ ξÞðΔðe0 þ eÞ, e þ ξÞ ðΔeg þ ξÞðΔðe0 þ eÞ, e þ ξÞ σ
ð1, 20 Þ
πðΔðe0 þ eÞ, g
0
μe0 g 2 μe0 þe, e0 2
1
μeg μe0 g μe0 þe, e μe0 þe, e0
þ C B ð1Þ 0 ð1, 2, 20 Þ B ðΔe0 g iσÞðΔeð1,0 g 2, 2 Þ iσÞ ðΔð2Þ iσÞ C eg iσÞðΔe0 g C B ¼ ð1, 20 Þ C B 2 2 μ μ μ μ 0 g μe0 þe, e μe0 þe, e0 0 þe, e C B eg e eg e ðΔðe0 þ eÞ, g 2iσÞ@ A 0 0 ð1Þ ð1, 2, 2 Þ ð1, 2, 2 Þ ð2Þ ðΔe0 g iσÞðΔðe0 þ eÞ, e iσÞ ðΔeg iσÞðΔðe0 þ eÞ, e iσÞ 1
This equation has only inhomogeneously broadened resonant enhancements and will not exhibit line-narrowing. Integration of Z
2Γeg σ
0
μeg 2 μe0 g 2
the four partially coherent pathways described by eq 4 over the inhomogeneous distribution gives μeg 2 μe0 þe, e 2
1
@ A dξ ð20 Þ ð1, 2, 20 Þ ð1, 2, 20 Þ 2 2 πðΔe0 g þ ξÞðΔð2Þ þ ξÞ Γee ðΔðe0 þ eÞ, e þ ξÞ ge ξÞðξ þ σ Þ Γgg ðΔe0 g 1 0 10 2 2 2 2 μ μ μ μ 0 0 2σ eg e g eg e þe, e A A@ ¼ @ 2 ð1, 2Þ ð1, 2Þ 2 Þ þ σ ½ðΔð2Þ 0 Γgg ½δe0 e iðΓe g þ Γeg Þ Γee ½δðe0 þ eÞ, e þ e iðΓðe0 þeÞ, e þ Γeg Þ ge 0 1 μeg 2 μe0 g 2 μeg 2 μe0 þe, e 2 2Γeg @ A ð20 Þ ð1, 2, 20 Þ ð1, 2, 20 Þ ðΔe0 g iσÞðΔð2Þ þ iσÞ Γ ðΔ iσÞ Γ ðΔ iσÞ gg ee ge e0 g ðe0 þ eÞ, e
The line-narrowing for these pathways is similar to that in eq 8, except it occurs only for the output coherence resonance. Simulations of the diagonal peaks in CMDS spectra also require inclusion of the Raman and fluorescence pathways. In this case, the v quantum state in eq 5 is taken as the ground state. The resonance enhancement from the Raman and fluorescence pathways for the diagonal peaks in the presence of inhomogeneous broadening is then given by the expression 2iσμeg 4 2 2 ð ω1 þ ω2 2iΓeg ÞððΔð2Þ ge Þ þ σ Þ ! μeg 4 1 i ð2Þ þ ω1 þ ω2 iΓgg Γee ðΔge þ iσÞðΔð2Þ eg iσÞ 0 1 ω þ ω 2iΓ 2Γ 1 2 eg eg A @ þ ð ω1 þ ω2 iΓgg ÞðΔð1Þ Γee ðΔð2Þ eg iσÞ eg iσÞ
ð11Þ These equations can simulate the effect of the different variables on the 2D spectra. We use parameters that are typical of quantum dot spectroscopy. Figure 2a shows the appearance of the diagonal peak for the 1S exciton when there is no inhomogeneous broadening and no Coulombic coupling or other factors to shift the energy of the biexciton from twice the single exciton
ð9Þ
ð10Þ
energy. The peak has the diamond shape expected for a resonance with a Lorentzian line profile. Figure 2b shows the effects of inhomogeneous broadening. The line shape takes on a diagonal appearance because different size quantum dots have different electronic energies. Figure 2c shows the line shape when the inhomogeneous and homogeneous broadening effects are comparable. The line shape still has a diagonal character, but it is rotated and is an intermediate between that in Figure 2a,b. The rotation serves as a measure of the relative amounts of inhomogeneous and homogeneous broadening. The addition of Coulombic coupling shifts the biexciton energy relative to the single exciton and causes an asymmetry to the line shape. If the biexciton energy is lowered, then the coherence pathways creating the biexciton output coherence result in a peak that is shifted to lower ω1 values. This peak also has diagonal character, and its destructive interference narrows the line shape more at lower ω1 values than at higher values. The Supporting Information shows additional examples that illustrate these effects. We will use detailed fitting of the spectra with these equations to understand the role of these different parameters that control the line shape. The coherent and incoherent dynamics are measured by fixing the excitation and detection frequencies while scanning the excitation pulse time delays. The dependence on the two time delays is simulated using the strategy developed by Gelin et al.,3134 where one integrates the Liouville equation in the rotating wave 22837
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Figure 3. Absorption spectrum of PbSe quantum dots in CCl4.
approximation using the superoperator approach to propagate the individual density matrix elements. The superoperator matrix contains the operators that define the transitions in the 16 pathways of Figure 1. It implements the Liouville equation in the rotating wave approximation
i μik E iωik t μjk E iωkj t ~_ij ¼ iðωRW ωij þ iΓij Þ~ ~kj e ~ Fij þ F F Fik e 2 p p
ð12Þ ~ijeiωRWt, ωRW is the frequency in the rotating wave where Fij = F approximation, and E is electric field created by the sum of three excitation fields, that is E¼
Eoi ðt σ ti Þ2 iωi ðt ti Þ e ðe þ eiωi ðt ti Þ Þ 0 2
∑ i ¼ 1, 2, 2
ð13Þ
The output intensity is then proportional to μba2|Fba|2, where Fba is the output coherence.
’ EXPERIMENTAL SECTION The experiments used one of two different spherical, monodisperse oleic acid (OA)-capped PbSe quantum dots with size distributions centered on 8 or 5 nm and dispersed in anhydrous carbon tetrachloride solvent in a 1 mm thick cuvette under an inert nitrogen atmosphere.41 The PbSe sample concentration was adjusted so the maximum absorbance of the first excitonic peak in the experiments did not exceed 0.5. Figure 3 shows the absorption spectrum of the 8 nm quantum dot sample. The 1S exciton peak at 6870 cm1 is the strongest feature while the 1P exciton peak at 8650 cm1 is also clearly visible. Using the estimated sample concentration, dot size, density, and sample path length, the absorption cross-section calculated for this sample is 3 1015 cm2.42 This value is comparable to that reported in the literature for CdSe quantum dots of a comparable size.19 The multiresonant CMDS experiments used two optical parametric amplifiers (frequencies of ω1 and ω2) that were excited by a 1 kHz regenerative amplifier/Ti:sapphire oscillator. The typical excitation pulses had pulse durations of ∼1 ps; spectral widths 20 cm1 (fwhm) and were tunable from ∼70009000 cm1. The lower end of the tuning range was limited by the need to scan only the OPA signal beam. It was not feasible to switch between the signal and idler beams during a continuous frequency scan with the current experimental system. The ω2 beam was split into two beams labeled 2 and 20 . Typical pulse energies were 10, 2, and 3 μJ for ω1, ω2, ω20 , respectively. The beams were directed without focusing into the sample at angles appropriate for phase matching. The typical beam diameter was 1.25 mm. These conditions were selected to avoid
damage to the sample. Occasional formation of quantum dot clusters was evidenced by the slow accumulation of PbSe at the bottom of the cuvette. These clusters caused scatter of the excitation beams and hence some background noise (for example, as seen in Figure 5a). The output signal intensity was measured with a monochromator at the frequency ωm and an InSb diode detector. The intensity was corrected for fluctuations in the ω1 excitation pulse intensity and the square of the ω2 excitation pulse intensity. Two-dimensional spectra were obtained by measuring the output intensity while scanning the ω1 and ωm frequencies (ω1 = ωm) for different fixed values of ω2. The spectra were normalized to the most intense feature in each 2D spectrum. The dynamics were measured by fixing the ω1, ω2, and ωm frequencies and varying the excitation pulse time delays, τ20 1 τ20 τ1and τ21 τ2 τ1.
’ RESULTS Figure 4 shows the 2D (ω1, ω2) spectra in the region of 70007500 cm1 for the 8 nm PbSe quantum dot sample. The excitation pulse time delays are temporally overlapped in Figure 4a, and the B k 1 beam is delayed by 1 and 2 ps from the other beams in Figure 4b,c, respectively. There is a very strong and sharp diagonal feature when the excitation pulses are temporally overlapped, but it disappears quickly when the B k 1 beam is delayed, revealing a broader feature that also has a diagonal character. The latter feature is reaching a maximum at the lower end of the ω1, ω2 tuning range corresponding to the 1S peak in the infrared spectrum. Figure 5 shows a similar spectrum for the region, where ω1 varies between 8000 and 8860 cm1 and ω2 varies between 6800 and 7400 cm1. The diagonally shaped feature is a cross-peak between the 1S exciton band centered near 7000 cm1 and the 1P exciton band near 8650 cm1. Figure 3 shows that the quantum-confined excitonic states appear on a rising background that obscures the quantumconfined exciton states. This background originates from a high density of delocalized excitonic states.43 However, this background is not important in the CMDS spectra of Figures 4 and 5. Figure 6 shows two cross sections of the CMDS spectra in the ω1 = ωm direction with ω2 = 7000 and 8600 cm1 for the 1S and 1P excitonic peaks, respectively, along with the square of the absorption spectrum both with and without the exponentially rising background numerically removed. The scan range for the 1P excitonic peak is large enough to see that the peak is nearly symmetrical and contains little or none of the rising background. The scan range for the 1S peak is not large enough to see the complete line shape. The comparison with the square of the absorption spectrum is justified because the CMDS intensity and the square of the absorption coefficient depend on similar factors. If we treat the inhomogeneously broadened absorption coefficient in the same manner as the CMDS treatment, then the absorption coefficient will scale as αba ≈
μ2ba ðΓba þ σÞ ðδba Þ2 þ ðΓba þ σÞ2
ð14Þ
Similarly, the nonlinear polarization created by typical pathways 2 2 1 in eqs 811 depend on [(Δ(2) ge ) + σ ] . Although the absorption and CMDS spectra have similar factors, the relationship is much more complex for CMDS because the features depend on interference effects between different pathways. Nevertheless, the comparison in Figure 6 shows that the 1S and 1P exciton 22838
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Figure 4. Two-dimensional multiresonant CMDS spectrum of PbSe quantum dots scanning the ω1 = ωm frequency for a series of different ω2 frequencies over the diagonal 1S exciton state. The top spectra shows the experimental data for delay times of (a) τ20 1 = τ21 = 0 ps, (b) τ20 1 = τ21 = 1 ps, and (c) τ20 1 = τ21 = 2 ps. The bottom spectra show the corresponding simulations. The color bar scale is logarithmic for panels a and d and linear for panels b, c, e, and f.
Figure 5. Two-dimensional multiresonant CMDS spectrum of PbSe quantum dots scanning the ω1 = ωm frequency for a series of different ω2 frequencies over the 1S, 1P exciton cross-peak. The top spectrum shows the experimental data for delay times of τ20 1 = τ21 = 2 ps. and the bottom spectrum shows the simulation. The color bar scale is linear.
CMDS peak shapes do not contain a significant contribution from the rapidly rising background. Figure 7 shows the dependence of the output intensity on the τ20 1(x axis) and τ21(y axis) temporal delay times. The graphs have six regions demarcated by the three solid lines. Each region
Figure 6. Comparison of the squared absorption spectrum (dotted line), the squared absorption spectrum with the background numerically removed (dashed line), and ω1 = ωm cross sections of the CMDS spectra (solid line) with (a) ω2 = 7000 cm1 and (b) ω2 = 8600 cm1.
corresponds to one of the six time orderings of the three excitation pulses. The temporal dependence along each of the positive and negative τ20 1 and τ21directions and the positive τ20 1 = τ21direction measures the dephasing rates of the different coherences. Similarly, the temporal dependence along the negative τ20 1 = τ21direction measures the population relaxation rate. For the case where ω1 = ω2 = ωm, the temporal dependence along the positive τ20 1direction is also a measure of the population 22839
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Figure 7. Dependence of the CMDS intensity on τ20 1 and τ21 with 200 fs temporal increments. (a) ω1 = ωm = ω2 = 7000 cm1 and (b) ω1 = ωm = 8650 cm1 and ω2 = 6950 cm1. The experimental data appear on the top and the simulations appear on the bottom. The color bar scales are linear.
Figure 9. Dependence of the CMDS intensity on τ21 and ω2 with τ20 1 = 1 ps and ω1 = ωm = 8650 cm1. The ω2 steps were 5 cm1 and the τ21 steps were 100 fs. The experimental data appear on the top and the simulations appear on the bottom. The color bar scales are linear.
Figure 8. Dependence of the CMDS intensity on τ20 1 and τ21 with 20 fs temporal increments for two different regions of Figure 7a. The experimental data appear on the top and the simulations appear on the bottom. The color bar scales are linear.
relaxation rate. Figure 7a shows the dependence of the output intensity when ω1 = ω2 = ωm = 7000 cm1. The intensity is maximized along the negative τ20 1 = τ21 and the τ21 = 0 directions where an excited 1S population is created by the first two interactions. The decay along these directions measures the population relaxation times. The intensity falls very quickly along the other directions, where coherences are created by the first and second interactions. This rapid decay corresponds to dephasing times that are much shorter than the excitation pulse durations. Figure 7b shows the dependence of the output intensity for the 1S1P exciton cross-peak where (ω1, ω2) = (8650, 6950) cm1. The only time ordering that provides significant signal cork 2 and responds to negative values of τ20 1 = τ21, where the B +k B20 beams create a 1S exciton population. All other time orderings are fully coherent and do not contribute when the dephasing rates are fast. A closer examination of Figure 7 shows that the intensity is modulated as a function of the delay times. Figure 8 shows example regions of the same data as Figure 7a, but the time delay
step size has been shortened from 200 to 20 fs. The modulations are clearly seen to be periodic and rapid. The modulations are also dependent on the wavelength of the excitation lasers and are responsible for intensity variations in the 2D spectra of Figures 4 and 5. Figure 9 shows how the modulations depend on both the τ21 time delay and ω2 frequency when ω1 = ωm = 8300 cm1. The quantum dot sample used for Figure 9 was different. It had a diameter of 5 nm, and the 1S exciton peak appeared at 8390 cm1.
’ DISCUSSION Equation 6 describes the resonance enhancements in the region of the diagonal peak in Figure 4 neglecting inhomogeneous broadening. All enhancements depend on the difference between the nonlinear polarizations of the eg and 2e,e output transition moment coherences. In vibrational CMDS, the μ2e,e √ 0 ) = involving the overtone is larger than μeg by 2, and Δ(1,2,2 eg 0 (1,2,2 ) Δ2e,e for a single harmonic vibrational mode. As a result, the interference from the eg and 2e,e output coherence pathways exactly cancel if there is no anharmonic coupling. For electronic states, there is no similar relationship between the transition 44 moment creating the biexciton and that creating the exciton. (1,2,20 ) The spectrum will contain two peaks, one from the Δeg factor 0 ) factor. If there is no Coulombic and one from the Δ(1,2,2 2e,e coupling, then these factors will be equal, and the peaks will overlap. Because the nonlinear polarizations for the peaks have opposite signs, the intensity of the peak will depend on the difference in the transition moments. If the output transition √ moments have the same 2 ratio as harmonic vibrational modes, 22840
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Table 1. Parameter Values Used in Simulations Shown in Figures 4, 5 and 79 Using eqs 813a ωeg
6990 cm1
ωe0 g
8697 cm1
ω2e,e 2ωeg
124 cm1
Γeg
114 cm1
Γe0 g
146 cm1
Γ2e,e
133 cm1
Γe+e0 ,e
228 cm1
μ2e,e/μeg μ(e+e0 ),e/μe0 g
1.4 0.64
σ
429 cm1 0.058 ps1
1/T1 a
0
0
States e, e , 2e, and e + e represent the 1S exciton, 1P exciton, 1S biexciton, and 1S, 1P biexciton, respectively. ωe0 +e,e 2ωe0 g is the Coulombic coupling in the biexciton state. The mean and standard deviation are listed for the 1S diagonal peak simulations with τ20 1 = τ21 delay times of 0, 1, and 2 ps. The values for the 1S, 1P cross-peak simulation were ωeg = 7016 cm1 and ωe0 +e,e0 2ωe0 g = 5 cm1, respectively.
then the diagonal peak will vanish, but, in general, such a ratio would be accidental. For the cross peak, the Raman and fluorescence pathways are not important. The pathways associated with the e0 g and (e0 +e),e output coherences still destructively interfere and will cancel if the frequencies, transition moments, and dephasing rates are the same. The appearance of cross peaks therefore depends on Coulombic coupling within the (e0 + e) biexciton changing these parameters. If there is no coupling and if the output coherence transition moments are equal, then the cross peak will vanish. The usefulness of CMDS methods for characterizing materials depends on their ability to obtain information about the processes that control the material characteristics. To extract this information from the CMDS spectra in Figures 4 and 5, we performed a simultaneous least-squares fitting of the data for the three time delays in Figure 4 using eqs 811 for the diagonal peak and eqs 810 for the cross peak in Figure 5. Because the sample absorption introduces a systematic change in the CMDS intensity, the intensity dependence of the simulation was corrected for the changes in sample absorbance using the equation 0 12 A1 þ A ð Þ 2 B C B10A1 =2 10 2 C ð11Þ C M ¼B @ A A2 ln 10 where Ai is the frequency-dependent sample absorbance.45 The simulations are compared with the experimental data in Figures 4 and 5. Table 1 summarizes the values for electronic energies, dephasing rates, transition moments, inhomogeneous broadening, and Coulombic coupling that provided the best fits to the data. The values for the 1S exciton state in Table 1 were used for the least-squares fitting of the cross-peak data. Figure 10 shows a second comparison between the experimental data and the simulation. Cross sections of the experimental data and simulations for constant ω2 values were fit to Lorentzian line shapes as a function of ω1 using the LevenbergMarquardt method.46 The Figure shows how the half width at half-maximum
Figure 10. Half width at half-maximum, central frequency, and normalized peak intensity in the fitted 1D ω1 cross sections of the experimental (solid) and simulated (dotted) CMDS spectrum in Figure 4c,f as a function ω2.
(hwhm), the peak frequency, and the amplitude of the Lorentzian fits vary for different values of ω2 when τ21 = τ21 = 2 ps. The Supporting Information contains similar figures for the other delay times. There is good agreement over most of the ω2 values until the peak becomes quite weak. The variation of the hwhm is caused by the interference between the pathways involving the biexciton and those involving only the exciton. The simulations assume the dephasing rate is a constant independent of ω2. It is expected that the dephasing rate depends on the quantum dot size.47,48 In fact, the experimental peak width in Figure 10a is somewhat larger than the width in the simulation at higher ω2 values. This difference may be caused by a size-dependent dephasing rate or ratio of the transition moments for the exciton and biexciton transitions. Size-dependent dephasing was observed by Scholes.47 However, we did not find that including size-dependent dephasing rates improved the fitting of the experimental data. The sharp diagonal feature in Figure 4a arises from the Δ(1,2) gg ω ω iΓ factors that become ω2 ω1 iΓgg and Δ(1,2) ee 2 1 ee resonant when the frequencies are equal. The feature corresponds to pathways where interactions with the ω1 and ω2 frequencies create a population. The relative peak intensity of the sharp feature depends on (1 (2Γeg)/(Γgg)) and (1 (2Γeg)/(Γee)). These factors become large when the dephasing rate is much larger than the population relaxation rate. The sharp diagonal feature vanishes in Figures 4b,c, where the ω1 beam follows the ω2 beam by 2 ps 22841
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The Journal of Physical Chemistry C and cannot create an excited-state population. It is replaced by the weaker and broader resonance from the 1S exciton. The 1S exciton peak has a diagonal character that arises from line-narrowing of the inhomogeneous broadening. The inhomogeneous broadening is dominated by the distribution of different size quantum dots in the sample so a resonance with a narrow size distribution within the entire distribution will result in multiple enhancements from the sizes that are most resonant. In eq 10, the (1,2) 0 narrowing arises from the δe(1,2) e i(Γe0 g + Γeg) and δ(e0 +e),e+e i(Γ(e0 +e),e + Γeg) resonance factors. Cross sections of the 2D spectra taken with either ω1 or ω2 constant will have a width dependent on the (Γe0 g + Γeg) and (Γ(e0 +e),e + Γeg) homogeneous 2 2 broadening parameters. The ((δ(2) ge ) + (Γeg + σ) ) factor in the line-narrowing terms of eq 11 controls the intensity variation of the line-narrowed feature. If a cross-section of the 2D spectrum is 0 taken for a constant value of ω1 ω2, then the δe(1,2) e in the linenarrowing term will remain constant but the intensity will change as a function of ω2 and will have a width defined by (Γeg + σ) as a result of our assumption of a Lorentzian inhomogeneous distribution. In addition, the slope of the diagonal feature depends on the relative amounts of homogeneous and inhomogeneous broadening. The simulations show that when the inhomogeneous broadening and the dephasing rates are both important, the spectral feature loses its diagonal character and provides a spectral signature of the relative amounts of homogeneous and inhomogeneous broadening (compare Figure 2b,c). If coupling within the biexciton causes small changes in the energy, transition moment, or dephasing rate, then the line profile will be symmetric along the diagonal. The data show that the feature is not symmetrical and the width increases as the ω2 frequency increases. Both effects result from the interference between the pathways involving the exciton and biexciton output coherences. The changes in the broadening and asymmetry are clearly seen in Figure 2d. The changes in symmetry and width of the feature are spectral signatures of the Coulombic coupling within the biexciton state. Dephasing in quantum dots is often ascribed to electron phonon coupling, but three-pulse echo peak shift (3PEPS) experiments show that dephasing has a dominant contribution from relaxation within the excitonic fine structure and that the dephasing rate depends on the size of the quantum dots.12,47 Our simulations show that the inclusion of a size-dependent dephasing rate does not improve the fitting, but we also point out that the simulations do not quantitatively reproduce the changes of the line width over the entire frequency range. (See Figure 10a.) The ability to extract the excitonic peaks from the rapidly rising background is potentially a very important capability for multiresonant CMDS. The comparison between the absorption spectra and the CMDS cross sections shown in Figure 6 and the agreement between the multidimensional spectra and the simulations shown in Figures 4 and 5 both show that the background does not play an important role in CMDS spectra. The CMDS intensity from the quantum-confined exciton states and the background depends on the eighth power of the transition moments and the sixth power of dephasing rates. There has been little work done to determine these parameters for the states associated with the background, but the absence of any contributions to the CMDS spectra is evidence that they are responsible for the small contribution from these states. It is also important to extract the population relaxation rates from the intensity dependence on the delay times. The data in Figure 7 were simulated by integrating the Liouville equation
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using the superoperator approach to propagate the individual density matrix elements. The dephasing rates that describe the 2D line-narrowed CMDS data were used for the simulation. The population relaxation times were adjusted to match the data. Example results of the simulations are shown in Figure 7, and the population relaxation rates (T11) are summarized in Table 1. There is excellent agreement between the experimental data and the simulations, and the T1 values from fitting Figure 7a,b are identical. The simulations are not sensitive to the values chosen for the dephasing rates, indicating that the dephasing rates are much faster than the excitation pulse length. The population relaxation rates vary for different samples and even vary when the same sample is measured at different times. The rates are much faster than those associated with radiative relaxation processes and are much slower than the dephasing rates. It is well known that the population relaxation rates depend on the sample synthesis and may be controlled by acoustical phonons, surface states, and interactions with the solvent.44 It is unusual to see the rapid modulations in the time delay data shown in Figures 8 and 9 because homodyne detection with multiresonant CMDS methods are insensitive to the phase oscillations of the coherences created in the nonlinear pathways. Measurement of phase oscillations forms the basis for time domain CMDS methods where a local oscillator heterodynes with the output field and Fourier transformation of the oscillations identifies the frequencies of the coherences.49,50 We attribute the oscillations to interference between the scattered light from the ω1 beam and the output field. To verify this hypothesis, we simulated the time delay data using the expression for the field in eq 13 and the superoperator approach previously described. The simulations are shown in Figures 8 and 9. The simulations match the observed modulations. For Figure 8, the coherence frequency is 2.1 1014 Hz and the sampling frequency is 5 1013 Hz. The observed signal is then aliased to a frequency of 1013 Hz that characterizes that in Figure 8. Similarly, the frequency of the modulations that appear in Figure 9 change as a function of ω2 because the sampling rate remains constant but the frequency of the coherence changes.
’ CONCLUSIONS This work has used PbSe quantum dots as a model for defining how multiresonant CMDS can be applied to provide quantumstate-resolved spectra of the electronic states of important materials. Multiresonant CMDS resolves the quantum confined excitonic states from the broad background that obscures most of the excitonic state, and it line-narrows the inhomogeneous broadening. The narrowing resolves the inhomogeneous and homogeneous broadening of the diagonal and cross peaks in the 2D spectra and allows measurement of the coherent and incoherent dynamics as well as the coupling and transition moments of the biexcitonic states. Furthermore, the dependence of the CMDS intensity on the excitation pulse time delays measures the population relaxation rates, and with faster excitation pulses, it can also measure the dephasing rates of zero, single, and double quantum coherences. Closed form equations were developed to simulate the multidimensional spectra using Lorentzian distributions as an approximation for the effects of inhomogeneous broadening. The simulations showed how the CMDS spectra reflected the different properties of the quantum dots. The relative amount of inhomogeneous and homogeneous broadening controlled both the relative widths of the 2D features along the 22842
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The Journal of Physical Chemistry C diagonal and antidiagonal directions as well as the symmetry of the feature relative to the diagonal. The Coulombic coupling in the biexcitonic state created an asymmetry in the line shape of the feature and also caused the line-width to depend on the excitation frequency. The relative transition moments of the exciton and biexciton transitions determined the relative intensity of the pathways responsible for the output coherences involving excitonic or biexcitonic transitions. Each of these effects created spectral signatures in the 2D spectra that allowed the simulations to extract the inhomogeneous broadening, dephasing rates, Coulombic coupling in the biexciton states, and the relative size of the transition moments. Simulation methods were also developed to fit quantitatively the temporal dependence of the intensity on the excitation pulse time delays and the modulations resulting from resolving the phase oscillations of the individual coherences. The excitation pulse durations used in these experiments are not optimal for multiresonant CMDS experiments in quantum dots. The durations were much longer than the dephasing times of the quantum dots. As a result, all of the coherence pathways in Figure 1 contribute to the signal, and the resonant enhancements expected from the different pathways are modified because of quantum mechanical interference between pathways. In addition, the samples were much more susceptible to laser-induced damage. Klimov has demonstrated that the excitation fluence controls the amount of damage.35 If F is the fluence required to avoid damage and τ is the excitation pulse duration, then the CMDS intensity will be proportional to the (F/τ)3 ratio. Ideally, the pulse durations should be comparable to the dephasing times so they are long enough to resolve spectrally individual quantum states but short enough to resolve the dynamics. For the experiments in this work, the picosecond pulse durations make the signal intensities more than four orders of magnitude weaker than the ideal case. It is clear that future multiresonant CMDS experiments need to use much faster excitation pulses. It will then be possible to resolve the individual coherence pathways in Figure 1 by using appropriate pulse time-orderings for the different fully coherent and partially coherent pathways. In addition to providing better resolution of the different excitonic and biexcitonic states and eliminating interfering pathways, it will also allow one to follow coherence transfer effects with the fully coherent pathways and population transfer dynamics with the partially coherent pathways. The most important capability of multiresonant CMDS is creating multiple quantum coherences involving states with very different energies.25,51 Multiresonant methods use independently tunable pulses to excite quantum states and require only short-term phase relationships. As the demands for characterizing more complex nanoscale heterostructures increase, the ability to create multiple quantum coherences involving different quantum states will provide the selectivity required to probe the dynamics of individual parts of the heterostructures. This selectivity is analogous to heteronuclear multiple quantum coherence nuclear magnetic resonance (HMQC-NMR) that is used to define protein structures containing tens of thousands spins.52 In the same way that NMR has revolutionized the measurement capabilities for biological materials, multiresonant CMDS potentially has the same importance for the electronic and vibrational states of technologically important materials.
’ ASSOCIATED CONTENT
bS
Supporting Information. Simulations of multiresonant CMDS spectra under different conditions and graphs that show a
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comparison of the half width at half-maximum, the peak frequency, and the amplitude of the Lorentzian fits vary for different values of ω2 when τ21 = τ21 has different values. This material is available free of charge via the Internet at http://pubs.acs.org.
’ ACKNOWLEDGMENT This work was supported by the National Science Foundation under grant DMR-0906525. Rachel Selinsky and Song Jin synthesized and characterized the quantum dot samples used for this work. They were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under SISGR award no. DE-FG0209ER46664. ’ REFERENCES (1) Kovalevskij, V.; Gulbinas, V.; Piskarskas, A.; Hines, M. A.; Scholes, G. D. Phys. Status Solidi B 2004, 241, 1986. (2) Kambhampati, P. Acc. Chem. Res. 2011, 44, 1. (3) Cooney, R. R.; Sewall, S. L.; Dias, E. A.; Sagar, D. M.; Anderson, K. E. H.; Kambhampati, P. Phys. Rev. B 2007, 75, 245311. (4) Asbury, J. B.; Ellingson, R. J.; Ghosh, H. N.; Ferrere, S.; Nozik, A. J.; Lian, T. Q. J. Phys. Chem. B 1999, 103, 3110. (5) Scholes, G. D. J. Chem. Phys. 2004, 121, 10104. (6) Carter, S. G.; Chen, Z.; Cundiff, S. T. Phys. Rev. B 2007, 76, 121303. (7) Doust, A. B.; Marai, C. N. J.; Harrop, S. J.; Wilk, K. E.; Curmi, P. M. G.; Scholes, G. D. J. Mol. Biol. 2004, 344, 135. (8) Colonna, A. E.; Yang, X. J.; Scholes, G. D. Phys. Status Solidi B 2005, 242, 990. (9) Kim, J.; Wong, C. Y.; Nair, P. S.; Fritz, K. P.; Kumar, S.; Scholes, G. D. J. Phys. Chem. B 2006, 110, 25371. (10) Kuznetsova, I.; Meier, T.; Cundiff, S. T.; Thomas, P. Phys. Rev. B 2007, 76, 153301. (11) Li, X. Q.; Zhang, T. H.; Borca, C. N.; Cundiff, S. T. Phys. Rev. Lett. 2006, 96, 057406. (12) Salvador, M. R.; Hines, M. A.; Scholes, G. D. J. Chem. Phys. 2003, 118, 9380. (13) Scholes, G. D.; Kim, J.; Wong, C. Y. Phys. Rev. B 2006, 73, 195325. (14) Spivey, A. G. V.; Borca, C. N.; Cundiff, S. T. Solid State Commun. 2008, 145, 303. (15) Spivey, A. G. V.; Cundiff, S. T. J. Opt. Soc. Am. B 2007, 24, 664. (16) Stone, K. W.; Gundogdu, K.; Turner, D. B.; Li, X. Q.; Cundiff, S. T.; Nelson, K. A. Science 2009, 324, 1169. (17) Stone, K. W.; Turner, D. B.; Gundogdu, K.; Cundiff, S. T.; Nelson, K. A. Acc. Chem. Res. 2009, 42, 1452. (18) Tokunaga, E.; Terasaki, A.; Kobayashi, T. J. Opt. Soc. Am. B 1995, 12, 753. (19) Cho, B.; Peters, W. K.; Hill, R. J.; Courtney, T. L.; Jonas, D. M. Nano Lett. 2010, 10, 2498 . (20) Nguyen, D. C.; Wright, J. C. Chem. Phys. Lett. 1985, 117, 224. (21) Wright, J. C.; Carlson, R. J.; Hurst, G. B.; Steehler, J. K.; Riebe, M. T.; Price, B. B.; Nguyen, D. C.; Lee, S. H. Int. Rev. Phys. Chem. 1991, 10, 349. (22) Zhao, W.; Murdoch, K. M.; Besemann, D. M.; Condon, N. J.; Meyer, K. A.; Wright, J. C. Appl. Spectrosc. 2000, 54, 1000. (23) Wright, J. C. Int. Rev. Phys. Chem. 2002, 21, 185. (24) Pakoulev, A. V.; Rickard, M. A.; Meyers, K. A.; Kornau, K.; Mathew, N. A.; Thompson, D. C.; Wright, J. C. J. Phys. Chem. A 2006, 110, 3352. (25) Mathew, N. A.; Yurs, L. A.; Block, S. B.; Pakoulev, A. V.; Kornau, K. M.; Wright, J. C. J. Phys. Chem. A 2009, 113, 9261. (26) Pakoulev, A. V.; Block, S. B.; Yurs, L. A.; Mathew, N. A.; Kornau, K. M.; Wright, J. C. J. Phys. Chem. Lett. 2010, 1, 822. 22843
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dx.doi.org/10.1021/jp207273x |J. Phys. Chem. C 2011, 115, 22833–22844
