Resonant Third-Order Susceptibility of PbSe Quantum Dots

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C: Physical Processes in Nanomaterials and Nanostructures

Resonant Third-Order Susceptibility of PbSe Quantum Dots Determined by Standard Dilution and Transient Grating Spectroscopy Daniel D. Kohler, Blaise J. Thompson, and John C. Wright J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04462 • Publication Date (Web): 10 Jul 2018 Downloaded from http://pubs.acs.org on July 11, 2018

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The Journal of Physical Chemistry

Resonant Third-Order Susceptibility of PbSe Quantum Dots Determined by Standard Dilution and Transient Grating Spectroscopy Daniel D. Kohler, Blaise J. Thompson, and John C. Wright∗ Department of Chemistry, University of Wisconsin-Madison Madison, WI 53706, USA E-mail: [email protected] Phone: (608) 262-0351

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ACS Paragon Plus Environment

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Abstract We present a methodology for obtaining quantitative ultrafast multiresonant coherent multidimensional spectra. Using the solvent and cuvette as an internal standard, we extract the quantitative third-order susceptibility of PbSe quantum dots through a method of standard dilution. The resonant susceptibility is extracted using a few parameter fit. After accounting for differences in pulse width and experimental geometry, we find agreement between our measured susceptibility and prior measurements. We demonstrate how the measured susceptibility can be directly compared with theoretical models of non-linearity (state-filling). Our internal standard methodology should be generally applicable to nonlinear spectroscopy.

Introduction Coherent multidimensional spectroscopy (CMDS) provides a wealth of information on the structure, energetics, and dynamics of solution phase systems. By using multiple excitations to probe resonances, CMDS elucidates correlations and couplings between electronic, vibrational, and vibronic states. It is now commonplace to interrogate such structures with femtosecond time resolution. The time-resolved nature of the measurement allows characterization of transient states that are unresolvable in steady-state methods. Multiresonant CMDS (MR-CMDS) is a frequency-domain technique whereby tunable lasers are scanned to obtain multidimensional spectra. It has been demonstrated on semiconductors using femtosecond pulses. 1,2 Traditionally, CMDS analyzes the line shape (peak center, width, sign, homogeneous and inhomogeneous line width) and dynamics (time constants, amplitudes) to characterize the fundamental properties of a material. The magnitudes of optical non-linearities, though commonly ignored, are intrinsic properties that also inform on microscopic properties. The microscopic mechanisms for optical nonlinearities are determined by a sequence of fieldmatter interactions (Liouville pathways) that depend on linear properties (cross-sections, 2


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etc.) with each interaction. Some methods, like Coherent Anti-Stokes Raman Spectroscopy and Triple Sum Frequency Spectroscopy, isolate single Liouville pathways so that the magnitude of the non-linearity is a simple product of cross-sections. Some of these cross-sections are easily measured with conventional experiments (Raman, absorption), but others depend on transitions that are not normally accessible. Techniques like CMDS must be employed to measure the cross-sections of these inaccessible transitions. Other methods, such as 2DElectronic Spectroscopy, Transient Grating, and Transient Absorption (2DES, TG, TA), have multiple Liouville pathways that constructively and destructively interfere. This interference is directly related to the microscopic properties. For example, saturation effects, such as state-filling, depend directly on the degeneracy of a transition. This paper describes the measurement of state-filling. The third-order susceptibility is directly measurable only in a subset of non-linear techniques, such as the z-scan 3,4 and transient absorption. Internal standards are a convenient alternative to direct measurement 5 which can be implemented in any spectroscopic measurement. For solution phase samples, the implicit solvent and window provide a natural internal standard. Internal standard approaches typically require a full characterization (phase and amplitude) of the spectral properties of a sample in order to relate the analyte and internal standard signals. For CMDS methods that possess a multitude of Liouville pathways, this characterization typically requires a large number of parameters that complicate fitting. This approach was previously used for MR-CMDS of quantum dots with picosecond excitation pulses. 6 This work details a few-parameter extraction of the third-order susceptibility of the 1S exciton band of PbSe quantum dots (QDs) using femtosecond pulses. We utilize standard dilutions for characterizing the third-order susceptibility of resonant signals, using window and solvent contributions as a reference. Applying simple approximations, we extract the susceptibility without explicitly modeling Liouville pathways of an excitonic manifold. We connect the common phenomenologies of optical non-linearities to the theory of state-filling.

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Finally, we compare our measurements with previous literature values. Once pulse-duration and absorption effects are accounted for, we find that the measured non-linear susceptibility of these femtosecond experiments is in good agreement with previously published values.

Theory These experiments consider the MR-CMDS signal resulting from a chromophore resonance in a transparent solvent. We consider three input pulses, E1 , E2 , and E20 , where E2 and E20 have the same carrier frequency ω2 , and isolate the phase matched output at ~kout = ~k1 − ~k2 + ~k20 , as in transient grating spectroscopy. We first formulate the CMDS intensity in terms of the separate contributions from the solvent and solute using the steady state expressions that define the third-order susceptibility, χ(3) , and the hyperpolizability, γ. We then connect the well-known theory of optical bleaching of the 1S band to our measurements of γ.

Extraction of susceptibility In the Maker-Terhune convention, the relevant third-order polarization, P (3) , is related to the non-linear susceptibility, χ(3) , by 7 P (3) (z, ω) =Dχ(3) (ω; ω1 , −ω2 , ω20 )

(1)

× E1 (z, ω1 )E2 (z, −ω2 )E20 (z, ω20 ), where z is the optical axis coordinate (the experiment is approximately collinear) and Ei and ωi are the real-valued electric field and frequency of pulse i, respectively. The degeneracy factor D = 3!/(3 − n)! accounts for the permutation symmetry that arises from the interference of n distinguishable excitation fields. Permutation symmetry depends on the experimental configuration. 8 Including D in our convention allows comparison of χ(3) with different experiments. The non-linear polarization launches an output field. The intensity of this output depends 4


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on the accumulation of polarization throughout the sample. For a homogeneous material, the output intensity, I, is proportional to 9 Z 2 (3) I ∝ P (z, ω)dz 2 ∝ M P (3) (0, ω)`

(2)

= |M Dχ(3) E1 (0, ω1 )E2 (0, −ω2 )E3 (0, ω20 )|2 . Here ` is the sample length and M is a frequency-dependent factor that accounts for phase mismatch and absorption effects. Phase mismatch is negligible in these experiments (see Supporting Information). For the remaining absorptive effects, M may be written as 9,10 e−α1 `/2 1 − e−α2 ` M (ω1 , ω2 ) = , α2 `

(3)

where αi = σi NQD is the absorptivity of the sample at frequency ωi . To account for pulse bandwidths, we compute M using the weighted average of absorptivities within our pulse bandwidths (see Supporting Information for details). Absorption effects disrupt the proportional relationship between I and χ(3) . Equation 2 shows that we can derive spectra free of pulse propagation effects by normalizing the output intensity by M 2 . The distortions incurred by optically thick samples are well-known and have been treated in similar CMDS experiments. 11–14 For cuvettes, the sample solution is sandwiched between two transparent windows. Rather than Eqn. 2, the total polarization has three distinct homogeneous regions: the front window, the solution, and the back window. The windows each have the same thickness, `w , (3)

and susceptibility, χw . The absorption-corrected output intensity is proportional to: 2 −α2 `s I 1 + e (3) (3) (3) α2 `w ∝ χ + χ + χ w QD sol I1 I2 I20 M 2 1 − e−α2 `s (3)

(3)

where χQD is the QD susceptibility and the χsol is the solvent susceptibility. 5


(4)


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Each susceptibility depends on the chromophore number density and local field enhancements for each wave: (3)

χi = f (ω1 )2 f (ω2 )2 Ni γi ,

(5)

where γi is the intrinsic (per-QD/per-molecule, in vacuo) hyperpolarizability, Ni is the number density of species i, and f (ω) is the local field enhancement factor. Since QDs constitute a negligible number/volume fraction of the solution, the field enhancement is derived entirely from the solvent: f (ω) = (nsol (ω) + 2) /3, where nsol is the solvent refractive index. Both n and f are frequency dependent, but both vary slowly (∼ 0.1%) over the frequency ranges considered here. We approximate both as constants, and remove the frequency argument from further equations. Equation 4 can be expressed as the classic signal-local oscillator interference,

2 ∗ I ∝ |ELO |2 + NQD f 8 |γQD |2 + 2NQD f 4 Re ELO γQD

(6)

(3) (3) where we have used the substitutions ELO = α2 `w 1 + e−α2 `s / 1 − e−α2 `s χw + χsol . The character of the interference depends both on the amplitude of the QD field and on the phase relationship between the two fields. The QD field amplitude depends on NQD . At low concentrations there is a linear dependence on NQD , but this changes at high optical densities due to an α2 dependence on the window contribution. The phase relationship cannot be controlled externally and is frequency dependent: it is defined by the resonant character of each material. The phase is defined by electronic resonances in QDs and by Raman resonances in the solvent and the windows. In general, the local oscillator and signal fields are non-additive.

Optical bleaching and dependencies on experimental conditions Most non-linear experiments on QDs extract pulse propagation parameters such as the nonlinear absorptivity, β, or the non-linear index of refraction, n2 . These parameters are con6


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nected to the third-order susceptibility (in cgs units) by 32π 2 Dω (3) Im χ n20 c2 16π 2 D (3) . Re χ n2 = n20 c β=

(7) (8)

These relations are derived in the Supporting Information. At the band edge, the non-linear absorptivity of semiconductor QDs is dominated by state-filling, 15 where absorption is reduced in proportion to the average number of excitons on a quantum dot, hni. Due to an 8-fold degeneracy, lead chalcogenide (PbX) QDs are bleached fractionally by the presence of an exciton. An 8-fold degeneracy predicts a bleach fraction of φ = 0.25 for singly-excited quantum dots. 16–20 For a Gaussian pump pulse of peak intensity I, p frequency ω, and full-width at half-maximum (FWHM) of ∆t , hni = ∆t Iσ/ (2¯ hω) π/ ln 2 where σ is the QD absorptive cross-section at frequency ω. This equation assumes the fluence, ∼ ∆t I2 , is small enough to avoid population saturation effects. For larger fluence, multiexcitons are created and more advanced treatments are needed to evaluate hni. 21,22 We can now write the non-linear change in absorptivity as I2 β √ = −φhniα 3 r π σ1 σ2 = −φNQD ∆t I2 ln 2 2¯ hω

(9)

where the indices 1 and 2 denote properties of the probe and pump fields, respectively. The √ factor of 3 on the left-hand side arises because β corresponds to the average intensity incurred over the pulse duration and not the peak intensity (see the Supporting Information for the derivation of this factor). In some techniques (e.g. z-scan), both probe and pump fields are the same, in which case the subscripts not necessary. By combining Eqns. 5, 7, and 9, we can relate the bleach factor directly to the hyperpo-

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larizability: r Im [γQD ] = −φ

3π n 2 c2 σ1 σ2 ∆t . ln 2 64π 2 Df 4 h ¯ ω1 ω 2

(10)

Equation 10 shows that when the Im [γQD ] depends on state-filling, it becomes dependent on ∆t because of its dependence on the excitonic population. This linear dependence is true only when the population is far from saturation. Equation 10 also shows that the ratio Im [γQD ] /∆t is a constant and is directly related to the bleach factor, φ.

Experimental Sample preparation and characterization Quantum dots were created using a standard solution-phase technique. 23 QDs were washed in ethanol-toluene before being immersed in carbon tetrachloride (CCl4 ) and stored in a nitrogen-pumped glovebox. Successive dilutions created aliquots of different concentration. Aliquots were stored in 1 mm path length fused silica cuvettes with 1.25 mm thick windows. Each aliquot was characterized by absorption spectroscopy. The spectra are consistent between all dilutions (no agglomeration, see Supporting Information). The 1S exciton peak at 0.937 eV has a FWHM of 92 meV. Concentrations were extracted using Beer’s law and published cross-sections. 24,25 The peak optical density ranges from 0.06 to 0.86 (QD densities of ∼ 1016 − 1017 cm−3 ). MR-CMDS An ultrafast oscillator (Tsunami, Spectra-Physics) produced an 80 MHz train of 35 fs pulses, which were amplified (Spitifire Pro XP, Spectra-Physics, 1kHz) and split to pump two independently tunable OPAs (TOPAS-C, Light Conversion): OPA1 and OPA2. The OPA output pulses were 50 fs wide. The frequency-dependent OPA power output was measured (407-A Thermopile, Spectra-Physics) and used to normalize the non-linear spectra. The OPA2 pulse was split to create a total of three excitation pulses: E1 , E2 , and E20 . Motorized (Newport 8


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MFA-CC) retroreflectors controlled the relative delay times, defined as τ21 = τ2 − τ1 and τ220 = τ2 − τ20 . The three excitation pulses were focused (1m FL spherical mirror) into the sample using a BOXCARS geometry 26 (∼ 1 deg angle of incidence for all beams). All input fields were identically polarized. The coherent output at ~k1 − ~k2 + ~k20 was isolated using apertures and passed into a monochromator (HORIBA Jobin Yvon MicroHR, 140 mm focal length, 150 grooves per mm grating, 150 µm slits) that followed the driven output frequency, ωm = ω1 . The intensity was measured with an InSb photodiode detector (Teledyne-Judson J10D-M204R01M-3C-SP28).

Results & Discussion In this section we describe the measurement of the QD hyperpolarizability, γQD , through standard dilution and comparison with the known γsol of the solvent. First, we examine the window and solvent response, which acts as a local oscillator that heterodynes with the sample response. Second, we isolate the pure QD response, using temporal discrimination, to validate the correction factors used to account for the concentration dependence. Third, we use the interference between the solvent and QDs at pulse overlap in order to determine the ratio of the hyperpolarizabilities, γQD /γsol . As Eqn. 10 shows, this ratio is dependent on ∆t . Knowing γsol then defines γQD for our specific pulse width. Finally, a comparison is made between our result and literature.

Solvent response Carbon tetrachloride is an ideal solvent because of the high QD solubility, transparency in the near infra-red, and its well-studied non-linear properties. The four-wave mixing (FWM) response of transparent solvents like carbon tetrachloride has components from nuclear and electronic non-linearities. 27,28 The nonresonant electronic perturbations dephase rapidly and

9



Figure 1: CMDS amplitude of neat CCl4 . In all plots, E1 and E2 are coincident (τ21 = 0 fs). Spectra are not normalized by the frequency-dependent OPA input powers. (a) The 2D frequency response at pulse overlap (τ21 = τ220 = 0). (b) Same as (a), but E20 is latent by 200 fs. (c) The τ220 dependence on CMDS amplitude (thin blue line) is tracked at (¯hω1 , h ¯ ω2 ) = (0.905, 0.955) eV, so that the ν1 Raman mode is resonantly excited. The fit to the measured transient (thick blue line) is described further in the text. The ω1 , ω2 frequency combination is represented in (a) and (b) as a blue dot.

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are only present during pulse overlap. The nuclear response depends on the vibrational dephasing times (ps and longer). 29,30 Vibrational features appear in the 2D spectra when stimulated Raman pathways resonantly enhance the FWM at constant (ω1 − ω2 ) frequencies. Fig. 1 summarizes the nonlinear measurements performed on neat CCl4 . In general, our results corroborate with impulsive stimulated Raman experiments. 30,31 When all pulses are temporally overlapped (Fig. 1a), the electronic response creates a featureless 2D spectrum. 32 The weak diagonal enhancement observed may result from overdamped nuclear libration. The broad spectrum is present during temporal pulse overlap and quickly disappears at finite delays. Temporally overlapping pulses E1 and E2 while delaying pulse E20 (Fig. 1b) resolves the contributions from the Raman resonances. These “TRIVE-Raman” 33 resonances have been observed in CCl4 previously. 1 The bright feature seen at ω1 − ω2 = ±50 meV is the ν1 symmetric stretch mode (459 cm−1 ). 34 If Raman resonances are important, their spectral phase needs to be characterized and included in modeling. 6 Characterization of the solvent response at pulse overlap can be simplified if Raman resonances are negligible. To estimate the relative magnitude of Raman components at pulse overlap, we consider a delay trace. Figure 1c shows the signal dependence on τ220 with pulse frequencies resonant with the large ν1 resonance. The transient was fit to two components: a fast Gaussian (electronic) component and an exponential decay (Raman) component. 35 We determined the fast (non-resonant) component amplitude to be 4.0 ± 0.7 times larger than the long (Raman) contributions at zero delay (see Supporting Information). At most colors, the ratio is much smaller (Fig. 1b). Since the Raman features are weak and spectrally sparse, we assume the CCl4 spectrum near pulse overlap is well-approximated by its non-resonant response (γsol is constant and real-valued). This approximation simplifies Eqn. 4 because the dispersion of the interference term is completely ∗ determined by the real component of quantum dot response: Re ELO γQD = ELO Re [γQD ].

11



amplitude

a

NQD × 1016 (cm−3 ); OD1S 0.8; 0.06 1.3; 0.10 2.5; 0.20 5.9; 0.47 10.7; 0.86

0

amplitude

b

0

c

amplitude

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0

0.80

0.85

0.90

0.95

~ω1 (eV)

1.00

1.05

Figure 2: The three panels show the changes in the FWM spectra of the five QD concentrations when corrected for concentration and absorption effects. The legend at the top identifies each QD loading level by the number density (units of 1016 cm−3 ). In all plots a representative QD absorption spectrum is overlaid (gray). Top: I/I1 I2 I20 spectra (intensity q level). Middle: FWM amplitude spectra after normalizing by the carrier concentration 2 ( I/ I1 I2 I20 NQD ). Bottom: same as middle, but with the additional normalization by the absorptive correction factor (M ).

12


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Concentration-dependent corrections The CMDS spectra are strongly dependent on the QD concentration. Fig. 2 shows ω1 spectra gathered at the QD concentrations explored in this work. All spectra were gathered at delay values τ21 = −200 fs, τ220 = 0 fs, and ω2 is tuned to the exciton resonance. We choose these pulse delays to remove all solvent and window contributions, i.e. the signal is due entirely to QDs (χw , χsolvent = 0 in Eqn. 4). The output amplitudes in Fig. 2a are normalized to changes in the excitation intensities and are positively correlated with QD concentration. The output amplitudes in Fig. 2b are further normalized to changes in the QD concentration, NQD , and are negatively correlated with concentration because of changes in the absorption at both the ω1 and ω2 frequencies of the excitation pulses. The absorption of the ω1 and ω2 frequency dependence is corrected using the M factor (Fig. 2c). After these corrections, the output amplitudes of all QD concentrations are consistent with each other. The robustness of these corrections (derived from accurate absorption spectra) shows that CMDS data can be taken at large concentrations and corrected to provide reliable spectra over a large dynamic range. Note that the asymmetry of the CMDS line shape about the peak frequency differs from that of the absorption line shape; the difference arises from excited state broadband refraction and will be addressed in a future publication. 36

Quantum dot response We now consider the behavior at pulse overlap, where solvent and window contributions are also important. Figure 3a shows the (absorption-corrected) spectra for all samples at zero delay. The spectrum changes qualitatively with concentration, from a symmetric line shape at high concentration (purple), to a dispersed and asymmetric line shape at low concentration (yellow). This behavior contrasts with the signals at finite τ21 delays, where the sample spectra are independent of concentration (Fig. 2c). Pulse overlap is complicated by the interference of multiple time-orderings and pulse effects. 37–39 These line shapes are not easily related to material properties, such as inhomogeneous broadening and pure dephasing. 13



I/I1 I22 M 2 (a. u. )

a

Z

I/I1 I22 M 2 dω1 (a. u. )

b

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100

10-1

0.8

0.9

ω1 (eV)

1.0

100

NQD (10 −16 cm 3 )

101

Figure 3: FWM with temporally overlapped pulses (τ21 = τ220 = 0 fs), with ω2 = ω1S . (a) Absorption-corrected ω1 spectra for each of the concentrations, offset for clarity. Yellow is most dilute, purple is most concentrated. Each spectrum is individually normalized. (b) The integrals of the FWM line shapes in part (a) are plotted against the QD concentration. The dashed black line is a fit according to Eqn. 11. The concentration dependence in Fig. 3 can be understood with our knowledge of the solvent/window character and Eqn. 6. We approximate the solvent and window susceptibilities as real and constant, such that the frequency dependence of the interference is solely from the real projection of the QD nonlinearity. As a consequence, the interference term will be the Kramers-Kronig counterpart of the peaked transient absorption spectrum. The real contribution from the solvent and windows is responsible for the observed asymmetric and dispersed line shape at low concentrations. We analyze these spectra through two different methods: spectral integration and global line shape fitting.

Spectral integration If we integrate Eqn. 4, the integral of the solvent-QD interference term disappears and the contributions are additive:

14


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Z a

a+∆

!2 (3) I χw dω1 =A∆ 1 + (3) g(NQD ) I1 I2 I20 M 2 χsol Z a+∆ 2 NQD |γQD |2 dω1 , +A 2 2 Nsol γsol a

(11)

−σ2 NQD `s

where A is a proportionality factor and g(NQD ) = σ2 NQD `w 1+e −σ N ` . Care must be taken 1−e 2 QD s when choosing integral bounds a and a + ∆ so that the odd character of the interference is adequately destroyed. In our case, we suppress the interference term by keeping the integration window (0.825 - 1.065 eV) centered about the peak resonance. Figure 3b shows the integral values for all five concentrations considered in this work (colored circles). At high concentrations the QD intensity dominates and we see quadratic scaling with NQD . The lower intensities converge to a fixed offset due to the solvent and window contributions. Eqn. 11 (black dashed line) fits the data well. Notably, our fit fails to distinguish between window and solvent contributions. The solvent integral does not depend on NQD , while the window contribution changes moderately over the five concentrations (g(NQD ) varies by ∼ 0.3x). In contrast, the QD integral will change by ∼ 100x over these concentration ranges, overwhelming the changes in window behavior. The approximation of a constant g(NQD ) fits the data equally well. In order to distinguish between window and solvent contributions, we take literature values from Kerr lensing z-scan measurements of χw /χsol ≈ 0.13. 40 The peak QD susceptibility can now be determined by assuming a Lorentzian line shape,

γQD = γ1S,peak

so that |γ1S,peak | =

q

Γ−1 π −1

R

Γ , ω1S − ω1 − iΓ

(12)

|γQD |2 dω1 . The line width parameter Γ can be inferred from

the high QD concentration spectrum with ∼ 25 meV HWHM. This method then gives a ratio of |γ1S, peak |/γsol = 6.5 × 105 . 15



6 NQD × 1016 (cm−3 )

5

0.78 1.27 2.48 5.91 10.79

4

|γtot| (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3 2 1 0

0.85

0.95

~ω1 (eV)

1.05

Figure 4: FWM with temporally overlapped pulses (τ21 = τ220 = 0 fs), with ω2 = ω1S . The 2 spectra are normalized by NQD . The thick, lighter lines are the result of a global fit using Eqn. 13. Global line shape fitting The integration approach provides a simple means to separate the resonant and non-resonant contributions to the non-linearity, but it relies on QDs having a Lorentzian line shape. This approximation may not be appropriate for PbX QDs. Many studies have reported a broadband contribution, attributed to excited state absorption of excitons, in addition to the narrow 1S bleach feature. 6,41,42 To account for a broad feature, we perform a global fit of Eqn. 4 with the QD line shape definition

γQD = γ1S,peak

Γ + B, ω1S − ω1 − iΓ

(13)

where Γ is a line width parameter and B is the broadband QD contribution. The results of 2 the fit are overlaid with our data in Fig. 4. The data is normalized by NQD (as in Fig. 2c) so

that least-squares fitting weighs all samples on similar scales. The fit parameters are listed in Table 1. Again, we use the same literature value for χw /χsol . This method gives a ratio of |γ1S, peak | γsol

= −7.7×105 , ∼ 20% larger than the value from the integral analysis. This difference 16


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is expected since the integral analysis does not distinguish between the resonant and nonresonant contributions in Eqn. 4. Unlike the integral method, the line shape fit extracts the sign of γQD . The negative sign is consistent for a nonlinearity created by a photobleach. The broadband contribution has a positive imaginary component which is consistent with excited state absorption. Table 1: Parameters and extracted values from the integral and global line shape analyses. Γ and B were extracted by least squares minimization. All other values were fixed parameters. 6.23 × 1021 Nsol /cm−3 h ¯ ω1S /eV 0.945 σ2 /cm2 1.83 × 10−16 24,25 χw /χsol 0.13 40 Γ (meV) 25 |γ1S,peak |/γsol (integral) 6.5 × 105 γ1S,peak /γsol (global) −7.7 × 105 B/γQD,peak 0.10 − 0.13i

Comparison with Literature Table 2: Comparison of these measurements with PbX measurements in literature. γsol,ν1 refers to the 465 cm−1 (symmetric stretch) mode of CCl4 . The third column, thisYurs , shows the ratios between the first and second columns work (unitless). QD ∆t [fs] σpeak [10−16 cm2 ] |γsol,ν1 | / γsol |γ1S,peak | / γsol |B| / γsol h i 6 γ1S,peak 10−30 cm h erg i cm6 γ1S,peak /∆t 10−18 erg s φ 6φ/D

this work PbSe ∼ 50 1.83 0.25 ± 0.04 7.7 · 105 1.3 · 105

Yurs et. al. PbSe ∼ 1250 1.74 5.1 1.3 · 107 1.6 · 105

-6 0.8 0.8

∓4 ± 0.6 ± 0.6

-0.3

∓5

Yurs this work

∼ 25 0.95 21 18 1.2 18

Omari et. al. PbS ∼ 2500 9.2 43

-130 -50 0.15 0.3

Table 2 compares the results of this work with Yurs et. al. 6 and Omari et. al. 44 Since 17



these studies were performed with different excitation pulse widths, experimental configurations, and/or samples, their values for the hyperpolarizability must be corrected to obtain a parameter common to each. Yurs et. al. performed the picosecond pulse analogue of this work. They too used CCl4 solvent as an internal standard. 45 The picosecond pulses are roughly 25x longer in duration than those of this work. This factor directly manifests in the non-resonant:resonant ratio (see Eqn. 10), as can be seen in γsol,ν1 /γsol (Table 2, third row) and γQD,peak /γsol (fourth row). B/γsol is similar for both works, suggesting that the B feature is short-lived (> 50 fs). This feature may arise from zero or double quantum coherences, ultrafast population relaxation, or non-resonant transitions. This driven feature is different from the broadband absorption observed in transient absorption experiments because the broadband absorption has a long lifetime. Obtaining γ1S, peak from the ratio requires a value for |γsol |. The hyperpolarizability of the solvent, |γsol |, has been previously measured. 5,28,29,40,46–50 The mean for the γsol literature values is 4 × 10−37 cm6 /erg but the values have a large spread of ±50% (see Supporting Information). This uncertainty transfers to the calculations of all absolute nonlinear quantities (Table 2, rows 6-9) for both this work and Yurs et. al. With that uncertainty noted, the peak QD hyperpolarizability for this work is 3 × 10−31 cm6 /erg. Table 2 reports the resonant γ1S,peak /γsol ratio that depends on ∆t and the γ1S,peak /γsol ∆t ratio that removes the dependence on ∆t . The agreement with Yurs et. al. γ1S,peak /γsol ∆t values shows that each approach produces consistent results, even though the excitation pulse widths differ by 25x. Omari et. al. 44 performed z-scan measurements of PbS QDs to quantify the non-linear parameters (see right-hand column of Table 2). While PbS QDs have a larger cross-section (and a correspondingly large γQD /∆t , row 7) than PbSe QDs, the 1S feature has the same 8-fold degeneracy, so direct comparison with φ should be possible. We can approximate 51 φ for all experiments using Eqn. 10. Omari et. al. predict a smaller bleach fraction 52 than the theoretical value of φ = 0.25, while this work and Yurs et. al. both measure larger values

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(Table 2, row 8). Part of this discrepancy is due to the smaller D of the z-scan measurement (a transient absorption measurement of Omari’s sample would give φ = 0.3). A factor of φ/D (Table 2, row 9) provides a direct comparison between all experiments. The 6φ/D ratio for this work and Yurs are higher than the theoretical ratio and the ratio of the z-scan measurement, but both have a large (∼ 50%) uncertainty due to the uncertainty of γsol . A more precise determination of agreement between this methods, theory, and the z-scan technique motivates a careful examination of the non-linear susceptibility of CCl4 .

Conclusion Ultrafast, multiresonant CMDS is a powerful technique that can interrogate the energetics and dynamics of electronic and vibrational states in molecular and semiconductor systems. We demonstrate that MR-CMDS can also extract absolute, quantitative susceptibilities of PbSe quantum dots using internal standards and fitting measured spectra to simple, fewparameter models. The analyte susceptibility is measured using tabulated literature susceptibilities for the solvent and windows. Quantification enables direct comparisons between spectroscopic methods that characterize different aspects of a sample and allows more detailed comparisons with theoretical treatments.

Supporting Information Available Analysis of absorptive correction factors and phase mismatch; absorption spectra for all aliquots used; details of the CCl4 analysis shown in Fig. 1, and a tabulation of γsol values gathered from literature. The full data is available online at http://dx.doi.org/10.17605/OSF.IO/3VPRB.

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Acknowledgement This work was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under award DE-FG02-09ER46664.

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(45) Yurs et. al. did not account for the effects of the windows. The results for Yurs et. al. have been corrected to account for the contribution of the windows to the non-resonant background and the different value of γsol used in their work. (46) Levine, B. F.; Bethea, C. G. Second and Third Order Hyperpolarizabilities of Organic Molecules. J. Chem. Phys. 1975, 63, 2666–2682. (47) Cherlow, J.; Yang, T.; Hellwarth, R. Nonlinear Optical Susceptibilities of Solvents. IEEE J. Quantum Electron. 1976, 12, 644–646. (48) Thalhammer, M.; Penzkofer, A. Measurement of Third-Order Nonlinear Susceptibilities by Non-Phase Matched Third-Harmonic Generation. Appl. Phys. B 1983, 32, 137–143. (49) Etchepare, J.; Grillon, G.; Thomazeau, I.; Migus, A.; Antonetti, A. Third-Order Electronic Susceptibilities of Liquids Measured by Femtosecond Kinetics of Optical Kerr Effect. J. Opt. Soc. Am. B 1985, 2, 649. (50) Nibbering, E.; Franco, M.; Prade, B.; Grillon, G.; Le Blanc, C.; Mysyrowicz, A. Measurement of the Nonlinear Refractive Index of Transparent Materials by Spectral Analysis after Nonlinear Propagation. Opt. Commun. 1995, 119, 479–484. (51) We assume the maximum bleach occurs at ω1 = ω2 = ω1S . (52) Omari et. al.’s analysis of “Sample D” using their Eqn. 11 is used to discern their value of φ and all other non-linear parameters.

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four wave mixing concentrated intensity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ω1

dilute

TOC Graphic

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Resonant Third-Order Susceptibility of PbSe Quantum Dots

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