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Spatiotemporal coupling of excited state dynamics in time-resolved microscopies

Erik M. Grumstrup

Open Access Open Access

Abstract

In the high-density excitation limit, as is often probed with ultrafast spectroscopies, spatial and temporal evolution of photogenerated excited states are strongly coupled, giving rise to artifacts that influence experimentally-determined material parameters. The interplay between spatial and temporal degrees of freedom is especially pronounced in pump-probe microscopy, where small laser spot sizes amplify the effects of spatiotemporal coupling on spectroscopic observables. To quantitatively model these effects, a continuum model is developed that accounts for laser spot size as well as nonlinear excited state decay and diffusion. It is shown that effective excitation densities cannot be used to determine quantitatively correct rate constants. Significant error is introduced unless experimental data is fit with a numerical model that accounts for spatial anisotropy in the excitation density. Furthermore, the quantitative determination of material diffusion coefficients is shown to be highly sensitive to experimental parameters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Recent efforts aimed at characterizing individual nanostructures and nanoscale regions of materials samples have driven significant interest in non-linear, time-resolved microscopies [1–4]. One appeal of the technique is that by reducing the excitation and probe volume to the µm length scale, spectroscopists can interrogate a sub-ensemble whose observable would otherwise be averaged together with that of neighboring (bulk-like) sample regions. Alternatively, single-crystal bulk-like regions can be selectively probed while eliminating contributions from structural defects, edge states, and grain boundaries in disordered or polycrystalline materials. By alternatively isolating or eliminating structural and chemical heterogeneities, researchers have leveraged time-resolved microscopies to correlate local structure to excited state spatial and temporal evolution in individual nanoparticles [5,6], nanoscale domains and edges [7–9], at grain boundaries and interfaces [10,11], and in thin films [12–14].

The use of time-resolved microscopy as a nano and microscale spectroscopic tool requires careful consideration of the intrinsic coupling between spatial and temporal degrees of freedom when interpreting observables. Relative to their traditional ensemble analogues, time-resolved microscopies amplify the effects of spatiotemporal coupling because the spot sizes of the laser pulses at the sample position are near the diffraction limit. For a far-field microscope, more than an order of magnitude difference in excitation density occurs between the center of the excitation pulse and at one full width at half maximum (FWHM) – a distance that can be as small as ∼ 200 nm for visible light excitation. If the excited states interact with one another in such a way that the overall population is influenced by that interaction (for example, via Auger recombination or exciton-exciton annihilation), the temporal evolution of the photogenerated ensemble will vary on a similarly short length scale. Thus, experimentally measured decay kinetics will reflect not a single kinetic process, but rather a (spatial) superposition of many decay processes. Spatiotemporal coupling is also manifest in time-resolved microscopies due to the diffusive motion of excited states. If the decay kinetics of a photogenerated population are of interest, it is necessary to account for diffusion of excited state density out of the probe volume to avoid significant underestimation of the excited state lifetime.

These examples of spatiotemporal coupling are intrinsic to time-resolved microscopies, even in the case of an ideal instrument with no aberrations. While the effects have been qualitatively discussed in the literature [14–16], a systematic and quantitative analysis has not been reported as it pertains to time-resolved microscopy. Here a numerical model describing the coupled temporal and spatial evolution of excited states is developed and several experimentally common examples of spatiotemporal coupling are examined. This manuscript focusses on pump-probe microscopy, however the findings are applicable to other linear and nonlinear time-resolved microscopies, provided the light-matter interaction is suitably modified to account for differences in spatial resolution. The goal of this manuscript is not to exhaustively describe the complicated many-body effects that give rise to excited state dynamics in micro- and nanoscale materials systems, but rather to provide a straightforward, practical framework for experimentalists to gauge the extent to which excited state spatial anisotropy and evolution affect time-domain spectroscopic measurements. For that reason, common units for temporal and spatial dimensions are adopted throughout the manuscript, and the use of dimensionless parameters is avoided except where they add particular insight.

2. Methods

To model the optical response expected from a typical pump-probe microscopy measurement, several simplifying assumptions are adopted. First, it is assumed that the spatial profile of both pump and probe pulses are azimuthally symmetric Gaussians. Second, that the photogenerated population density is accurately approximated by a continuum model and that it spatially evolves according to simple (constant, isotropic) diffusion. Finally, it is assumed that relaxation of the excited state is governed by a simple sum of linear and nonlinear (2nd and 3rd order) rate processes. Under such conditions the evolution of the excited state population density, $\; n({r,z,\Delta t} )$, can be described by Eq. (1):

$$\frac{{\partial n({r,z,\Delta t} )}}{{\partial t}} = \frac{1}{r}{\nabla _{\textrm{r},\textrm{z}}}\cdot {D_c}\; r\; {\nabla _{r,z}}n - {k_1}n - {k_2}{n^2} - {k_3}{n^3}$$
In Eq. (1), Dc is the diffusion coefficient (a scalar, given assumption 2 above [17]) and ${\nabla _{\textrm{r},\textrm{z}}}$ is the del operator with respect to r and z, which describe the radial and axial coordinates of the system. The temporal coordinate, Δt, describes the pump-probe delay time and k1, k2, and k3 are rate constants parameterizing linear, 2nd, and 3rd order excited state decay. Unless otherwise noted, model parameters for the results presented in this manuscript are k1 = 0.001 ps−1, k2 = 1.0 × 10−10 µm3/ps, and k3 = 1.0 × 10−17 µm6/ps, values selected to be generally consistent with inorganic semiconductor thin films and nanostructures [18]. For clarity, some figures are shown with cartesian axes, where $r = {({{x^2} + {y^2}} )^{1/2}}$.

Equation (1) is numerically solved in Mathematica to obtain the time- and spatially- dependent population density, subject to the following boundary conditions: 1) the plane defined by z = 0 is reflective (no surface recombination), 2) The initial excited state population profile, $n({r,z,\Delta t = 0\; ps} )$, matches the pump radial profile and has an axial dependence determined by Beer-Lambert absorption ($n(z )\propto \alpha Exp[{ - \alpha \; z} ]$, $\alpha = 1$ µ ${m^{ - 1}}$ throughout), and 3) The population density at the spatial limits of the simulation domain, $n({{R_{max}},{z_{min}},\Delta t} )= 0$ µm−3. The signal intensity, $I({\Delta t} )$, is calculated according to Eq. (2), which assumes the probe is not attenuated by the sample (non-resonant),

$$I({\Delta t} )\propto \; \mathop \smallint \nolimits_0^{{R_{max}}} \mathop \smallint \nolimits_0^{{\textrm{z}_{max}}} {G_{probe}}({r;\gamma } )n({r,z,\Delta t} )\; r\; \partial z\; \partial r$$
Here, ${G_{probe}}({r;\gamma } )$ is the Gaussian probe profile, which, like the pump, is parameterized by its FWHM, $\gamma $. The quantity determined by Eq. (2) is referred to throughout the manuscript as the signal, as it is the observable measured experimentally. The excitation wavelength is fixed at 425 nm throughout.

3. Results and discussion

3.1 Kinetics

One challenge to describing spatiotemporal coupling in pump-probe microscopy is rigorously defining the excitation density. Experimentally, it is common practice to report an excitation density by averaging over (for example) the 1/e2 width and depth of the excitation profile. In the limit the pump is spatially much larger than the probe, this approach is appropriate because the probe samples an excitation density that is approximately uniform along the radial coordinate. However, when the probe spot size is comparable to that of the pump, the measured signal averages over the kinetics occurring throughout the anisotropic excitation profile.

Figure 1 outlines the effects of an anisotropic excitation profile for a pump size of γpump = 0.4 µm and a probe size of γprobe = 0.5 µm (Dc = 0 µm2/s). At a low pump pulse energy (25 femtojoules/pulse, fJ, $5.4\; \times {10^4}$ excitations), the nonlinear decay terms (k2 and k3) have negligible contribution to the overall kinetics, so the entire population profile decays with an identical 1/e lifetime, ${\tau _{1/e}} = 1/{k_1} = 1000\; ps$ (black solid line). On the other hand, when the pulse energy is increased (2.5 picojoules/pulse, pJ), the center of the profile decays with a significantly shorter lifetime than the edges (green, solid). The consequences of this anisotropic decay can be seen in Fig. 1(b). The black solid line shows the low-fluence (25 fJ/pulse) kinetics, which decay with a first order rate of 0.001 ps−1 at both the edge and center of the profile. When the pump fluence is 2.5 pJ/pulse, the excitation profile center (green dashed) decays more quickly than the edge (green dotted) due to the nonlinear terms, k­2 and k3. However, as shown by the green solid line, the signal (calculated with Eq. (2)) exhibits decay kinetics that are intermediate between the two spatial limits, reflecting a linear combination of decay processes occurring throughout the excitation profile.

 figure: Fig. 1.

Fig. 1. Kinetic effects of anisotropic excitation density. (a) Solid lines show spatially dependent 1/e lifetimes (plotted on left axis) for 25 fJ (dark blue), 250 fJ (light blue), and 2.5 pJ (green), along the y = 0 profile. The black dashed line shows the excitation profile and is plotted on the right axis. (b) At low fluence, the excitation density, $n({r,\Delta t} )$., and the signal (Eq. (2)) decay with identical kinetics (black solid line). At 2.5 pJ, the center of $n({r,\Delta t} )$ (green, dashed) decays more quickly than the edge (r = .91γpump, where the excitation density is an order of magnitude lower, green dotted). The signal (solid line) decays at a rate between these two limits.

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By integrating over the anisotropic population profile, the experimental measurement process (Eq. (2)) yields decay kinetics that are qualitatively intermediate between the decay dynamics occurring at the two spatial limits described above. It is therefore reasonable to ask whether an effective average excitation density, $\langle{n}\rangle_{\textrm{eff}}$, defined by some optimal radius reff and absorption depth, zeff, might be sufficient to reproduce the experimentally observed kinetics (and accurately determine the excited state rate constants).

$$\langle{n}\rangle_{\textrm{eff}} = \mathop \smallint \nolimits_0^{2\pi } \mathop \smallint \nolimits_0^{{r_{eff}}} \mathop \smallint \nolimits_0^{{z_{eff}}} \frac{{m\; \alpha \; Ln[{16} ]}}{{\pi \; \gamma _{pump}^2}}Exp\left[ {\frac{{ - Ln[{16} ]\; {r^2}}}{{\gamma_{pump}^2}}} \right]Exp[{ - \alpha \; z} ]r\; \partial \theta \; \partial r\; \partial z/\left( {\mathop \smallint \nolimits_0^{2\pi } \mathop \smallint \nolimits_0^{{r_{eff}}} \mathop \smallint \nolimits_0^{{z_{eff}}} r\; \partial \theta \; \partial r\; \partial z} \right)$$
To address this question, a series of kinetics were numerically modeled with pump energies ranging between 125 fJ and 25 pJ. The kinetics, shown as dotted lines in Fig. 2(a), were calculated using model parameters of γpump = 0.4 µm, γprobe = 0.5 µm. The set of power-dependent kinetics was globally fit to Eq. (1), under the assumption that the excitation density could be represented as a scalar (Eq. (3)), rather than the position-dependent function solved for in the full numerical model. The best fit models, assuming a simple effective density, are shown as solid lines in Fig. 2(a). Not surprisingly, the fits are excellent at low fluences when k1 determines the (density-independent) decay kinetics. However, as the power is increased, deviations of the effective density fit with respect to the full numerical model are evident. Although only on the order of 3-4% and thus below the signal-to-noise level of many experiments, the deviations from the full numerical model suggest that a simple effective density model will not yield accurate values for k2 and k3.

 figure: Fig. 2.

Fig. 2. Effective density fitting (a) Dotted lines show numerically modeled kinetics for γpump = 0.4 µm, γprobe = 0.5 µm with pulse energies of 0.125 pJ (blue), 0.25 pJ (brown), 1.25 pJ (red), 2.5 pJ (orange), 12.5 pJ (yellow) and 25 pJ (green). Global best fits to the power dependent kinetics are shown as correspondingly colored solid lines. (b) Normalized rate constants plotted as a function of the normalized probe width, assuming an effective excitation density defined by the 1/e2 width and $1/\alpha $ absorption depth of the pump. (c) and (d) show normalized kinetics assuming a reduced pump-probe width (see text) and differ by zeff. Panel c assumes zeff = 1/α, whereas panel d assumes zeff = 2/α.

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To understand its limitations, the effective density model was examined for different values of reff and zeff, as well as for different probe/pump width ratios. Figures 2(b)-2(d) show fit-determined rate constants normalized by the nominal value input as a parameter in the full numerical simulation (k1 = 0.001 ps−1, k2 = 1.0 × 10−10 µm3/ps, and k3 = 1.0 × 10−17 µm6/ps). For all definitions of the effective density explored, the fit-determined first order rate constant was found to be within 1% of the modeled value, however the accuracy of the fit-determined k2 and k3 varied with reff and zeff.

Figure 2(b) takes the simplest definition for the effective excitation density, averaging over the 1/e2 width and $1/\alpha $ absorption depth of the pump pulse. While the fits to the individual kinetics traces are of similar quality to those shown in Fig. 2(a), the determined rate constants are generally incorrect, with k3 deviating from the true value by more than a factor of five for some probe widths. The general cause of the deviation derives from two main effects. For a large probe spot size, the signal (Eq. (2)) is an average over kinetics arising from both high and low excitation densities, similar to the case outlined in Fig. 1(b). The contribution from the fast, high-density kinetics are mitigated somewhat by the slower kinetics arising from the edges of the excitation volume and therefore, the fit-determined k2 and k3 are lower than the true values. On the other hand, at very small (relative) probe sizes, the signal reports on dynamics occurring only at the very center of the pump spot, where the excitation density is highest. The fit-determined rate constants are therefore significantly larger than in the large probe limit. Note that even at the smallest probe widths simulated, the fit with a simple effective density model does not converge to the rates of the full numerical model. This deviation is not due to anisotropy along the radial coordinate, but rather due to axial density anisotropy, which is not well described by the effective density model. Large magnitude extinction coefficients ($\alpha > 1$ µ ${m^{ - 1}}$) are not uncommon in materials samples, suggesting that excitation anisotropy must also be treated carefully in ensemble measurements where radial excitation gradients are small.

Figures 2(c) and 2(d) show fit determined rate constants, using an effective density determined by the reduced 1/e2 width of the pump and probe spots, ${\gamma _{pp}} = {({2\; Ln[2 ]} )^{ - 0.5}}{({1/{\gamma_{pump}} + 1/{\gamma_{probe}}} )^{ - 1}}$. The reduced pump-probe width properly accounts for the fact that the signal magnitude is dependent on the product of pump and probe spatial profiles (Eq. (2)). Figure 2(c) differs from Fig. 2(d) only by the axial depth that is used to calculate the effective density. In Fig. 2(c), $1/\alpha $ is used, whereas in Fig. 2(d), $2/\alpha $ is used. In either case, the determined rate constants are systematically biased from the true values, reflecting the inadequacy of an effective density model. Nevertheless, while the fit-determined rate constants are not quantitatively correct, they are, especially in the case of Fig. 2(d), within a factor of approximately two of the correct value. Furthermore, there is little dependence on the relative probe size, provided the reduced pump-probe width is used to calculate the density. These results suggest that if the pump and the probe spot sizes are accurately characterized, nonlinear decay processes can be quantified, to within a factor of two, using a simple effective density model. The results further highlight that a comparison between rate constants determined using different instruments requires knowledge of pulse energies, pump and probe spot sizes, and ideally the spatial profile of both pump and probe (i.e. deviations from gaussian due to aberrations).

The scenarios discussed in Fig. 2 assumed a diffusion coefficient, Dc = 0 cm2/s. However, as outlined in Fig. 3, diffusion can strongly impact measured decay kinetics and the determination of rate constants. Here, only first order kinetics were modeled (k2 = 0, k3 = 0), but the diffusion coefficient was varied between Dc = 0.0 and 50 cm2/s. Figure 3(a) shows kinetics for ${\gamma _{probe}} = 1.0$ µm with ${\gamma _{pump}}$ = 0.4 µm (dashed) and ${\gamma _{pump}}$ = 0.8 µm (solid). As Dc is increased, the kinetics lose their single exponential character as diffusive motion of the excited state density out of the probe region begins to contribute to the signal decay. The effect is slightly more pronounced for a narrower pump width because of the probe’s radial weighting effect – the relative contribution of the excited state density decreases at greater r. Figure 3(b) shows the kinetic dependence on probe diameter for the same series of diffusion coefficients. The pump width is ${\gamma _{pump}}$ = 0.4 µm, and the dashed and solid lines show the simulated kinetic decay traces for probe widths of 0.5 µm and 1.0 µm, respectively. Again, the effective decay lifetime is significantly shorter due to diffusion.

 figure: Fig. 3.

Fig. 3. Effects of excited state diffusion on measured decay kinetics with a finite probe size. (a) Kinetics traces with ${\gamma _{probe}} = 1.0$ µm and ${\gamma _{pump}}$ = 0.4 µm (dashed) or ${\gamma _{pump}}$ = 0.8 µm (solid) for Dc = 0.0 (dark blue), 0.1(light blue), 1.0 (brown), 10 (orange), and 50 cm2/s (yellow). (b) Kinetics traces with ${\gamma _{pump}} = 0.4$ µm and ${\gamma _{probe}}$ = 0.5 µm (dashed) or ${\gamma _{probe}}$ = 1.0 µm (solid) for Dc = 0 to 50.0 cm2/s (same values and colors as in panel a).

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Together, these results highlight the importance of both the pump and the probe spot sizes on the measured kinetics when the material has an appreciable diffusion coefficient. Generally, the spatial motion of the excited state density out of the probe volume will significantly shorten the observed lifetime when the (1-dimensional) diffusion length is comparable to the reduced width of the pump-probe volume, $\sqrt {{D_c}\Delta t} \gtrsim {\gamma _{pp}}$. Thus, it is possible to mitigate these effects by enlarging the spot sizes at the sample position. However, provided the pump and probe spot sizes and the diffusion coefficient are well-characterized, the effects of diffusive motion on the kinetics can be well described by Eq. (4) (in the low fluence limit) [15].

$$I{({\Delta t} )_{\Delta r = 0}} \propto {({\gamma_{pump}^2 + \gamma_{probe}^2 + 16Ln(2 ){D_c}\Delta t} )^{ - 1/2}}Exp({ - {k_1}\Delta t} )$$

3.2 Transport

Pump probe microscopy provides the ability to directly measure the transport of excitons, plasmons, or charge carriers by direct imaging of the excited state spatial evolution. As illustrated in Fig. 4(a), a series of excited state distribution “snapshots” is obtained by fixing the pump pulse at a particular sample location while the probe is spatially scanned over the field of view of the microscope at a fixed delay time. If both pump and probe pulses are Gaussian shaped, the double convolution of the pump spatial profile, diffusion equation, and the probe spatial profile yields a time and spatially dependent response, which is given (for the one-dimensional case) by Eq. (5).

$$\textrm{I}({\Delta{x},\Delta{t}})= \frac{{{a_0}}}{{{\beta}({\Delta \textrm{t}})}}\exp \left( {\frac{{ - 4\ln (2 )\Delta{x^2}}}{{\; \beta{{({\Delta{t}})}^2}}}} \right)$$
$$\beta ({\Delta{t}})= {({\gamma_1^2 + \gamma_2^2 + 16{D_c}\Delta{t}\; ln(2)} )^{\frac{1}{2}}}$$

 figure: Fig. 4.

Fig. 4. Spatiotemporal coupling in diffusion imaging. (a) Scanning the probe over the photoexcited population density produces a spatially-separated profile at a given delay time. (b) Spatially separated profiles at low fluence (25 fJ, solid) and high fluence (12.5 pJ, dashed) at Δt = 0 ps (dark blue) Δt = 200 ps (light blue) and Δt = 600 ps (maroon). Note the broadening of the high-fluence measurements relative to the low-fluence. (c) A plot of FWHM2 vs. Δt should produce a linear trend in the limit of purely diffusive broadening. However at higher fluences, the trend is nonlinear due to nonlinear recombination. Dark blue, light blue, maroon, and red traces show results from 25 fJ, 2.5 pJ, 12.5 pJ, and 25 pJ, respectively. (d) Such behavior could be interpreted (erroneously) as a time-dependent diffusion process due to non-equilibrated excited states. Trace colors and powers correspond to those in panel c.

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Based on Eqs. (5a) and (5b), it is common practice to plot the squared width of these spatially-separated pump-probe profiles as a function of increasing pump-probe delay time. In the limit of diffusive motion, the squared FWHMs increase linearly with time, and by finding the slope of this trend, the diffusion coefficient can be quantitatively determined (Eq. (5b)).

Spatiotemporal coupling influences such measurements through nonlinear excited state decay, which causes the center of the excitation profile to decay more quickly than the edges [14,16]. Figure 4(b) shows this effect for two modeled cases: the solid lines show spatially separated pump-probe profiles at a series of delay times for low excitation density. The dashed lines show the same profiles under conditions in which nonlinear decay (k2 and k3 processes) contributes to the evolution of the excited state density. At Δt = 0, the dashed and solid profiles are identical, however as the delay time is increased, the widths of the profiles undergoing nonlinear decay are significantly broader than those in the low-density limit. The anomalous broadening arises in the high excitation density limit because the center of the profile decays more quickly than the edges. Note that although the profile will no longer be gaussian in shape, differences in shape can be challenging to discern under experimental conditions.

If unaccounted for, nonlinear recombination leads to apparently higher diffusion coefficients because a gaussian fit of the high excitation density profile will yield speciously larger widths. This effect is illustrated in Figs. 4(c) and 4(d). Figure 4(c) shows plots of the FWHM2 vs. Δt as determined by Gaussian fits to the spatially-separated profiles, for a series of increasing excitation densities. At low fluences the trend is linear, accurately reflecting the excited state diffusive transport. However at high fluences, the squared widths increase nonlinearly, particularly at early Δt, when nonlinear decay processes contribute most strongly to the kinetics. Interpreted incorrectly, the decay-induced broadening illustrated in Fig. 4(c) appears to show diffusion coefficients which are significantly faster at early times than at later delay times (Fig. 4(d)). This effect is purely an artifact of spatiotemporal coupling and will inflate measured diffusion coefficients when excitation densities are high enough that k2 and k3 decay processes are relevant. As a result, nonlinear power-dependence must be carefully ruled out when non-equilibrium or ballistic transport is suspected, as the experimental observable of larger early-time diffusion coefficients would be indistinguishable in the two cases.

4. Conclusions

Coupling of spatial and temporal excited state evolution was modeled for a number of scenarios commonly encountered with pump-probe microscopy. It was shown that quantitative determination of nonlinear rate constants cannot be achieved through an effective density approximation, however fit-determined values converge to within a factor of ∼2, provided the reduced pump-probe width and absorption coefficients are well-characterized. The effects of nonlinear recombination were examined under conditions mimicking experimental determination of transport properties. It was shown that nonlinear recombination causes an apparently larger diffusion coefficient, particularly at early delay times.

The described simulations show that even in the simplest scenarios, two factors must be considered to eliminate the possibility of spatiotemporal artifacts influencing spectroscopic observables: the diffusion length and nonlinear decay of the excited state. Although reducing excitation fluences to a density-independent regime will eliminate the effects of nonlinear decay, excited state transport can still impact spectroscopic measurements if the diffusion length is comparable to the spot size at the sample.

Some scenarios likely to be encountered when characterizing materials samples were not directly investigated here. For example, determination of nonlinear rate constants with a finite diffusion coefficient, a density-dependent diffusion coefficient, and anisotropic diffusion are all relevant to many semiconducting materials systems. For these more complicated situations, a full numerical propagation of Eq. (1), with well-characterized experimental parameters and boundary conditions (including the effect of surface recombination), is likely necessary to accurately reproduce the excited state dynamics. Given the sensitive nature of coupling between temporal and spatial degrees of freedom, exceptional care will be needed to ensure that variations in material parameters are not conflated in such scenarios.

Funding

Arnold and Mabel Beckman Foundation; Beckman Young Investigator Program.

Disclosures

The authors declare no conflicts of interest.

References

1. E. M. Grumstrup, M. M. Gabriel, E. E. M. Cating, E. M. Van Goethem, and J. M. Papanikolas, “Pump–probe microscopy: Visualization and spectroscopy of ultrafast dynamics at the nanoscale,” Chem. Phys. 458, 30–40 (2015). [CrossRef]  .

2. M. C. Fischer, J. W. Wilson, F. E. Robles, and W. S. Warren, “Invited Review Article: Pump-probe microscopy,” Rev. Sci. Instrum. 87(3), 031101 (2016). [CrossRef]  

3. D. y. Davydova, A. de la Cadena, D. Akimov, and B. Dietzek, “Transient absorption microscopy: advances in chemical imaging of photoinduced dynamics,” Laser Photonics Rev. 10(1), 62–81 (2016). [CrossRef]  

4. T. Zhu, J. M. Snaider, L. Yuan, and L. Huang, “Ultrafast Dynamic Microscopy of Carrier and Exciton Transport,” Annu. Rev. Phys. Chem. 70(1), 219–244 (2019). [CrossRef]  

5. H. Staleva and G. V. Hartland, “Transient Absorption Studies of Single Silver Nanocubes,” J. Phys. Chem. C 112(20), 7535–7539 (2008). [CrossRef]  

6. E. E. M. Cating, C. W. Pinion, J. D. Christesen, C. A. Christie, E. M. Grumstrup, J. F. Cahoon, and J. M. Papanikolas, “Probing Intrawire, Interwire, and Diameter-Dependent Variations in Silicon Nanowire Surface Trap Density with Pump-Probe Microscopy,” Nano Lett. 17(10), 5956–5961 (2017). [CrossRef]  

7. G. Grancini, D. Polli, D. Fazzi, J. Cabanillas-Gonzalez, G. Cerullo, and G. Lanzani, “Transient Absorption Imaging of P3HT:PCBM Photovoltaic Blend: Evidence For Interfacial Charge Transfer State,” J. Phys. Chem. Lett. 2(9), 1099–1105 (2011). [CrossRef]  

8. V. Tiwari, Y. A. Matutes, A. T. Gardiner, T. L. C. Jansen, R. J. Cogdell, and J. P. Ogilvie, “Spatially-resolved fluorescence-detected two-dimensional electronic spectroscopy probes varying excitonic structure in photosynthetic bacteria,” Nat. Commun. 9(1), 4219 (2018). [CrossRef]  

9. H. Patel, L. Huang, C. J. Kim, J. Park, and M. W. Graham, “Stacking angle-tunable photoluminescence from interlayer exciton states in twisted bilayer graphene,” Nat. Commun. 10(1), 1445 (2019). [CrossRef]  

10. C. Y. Wong, B. L. Cotts, H. Wu, and N. S. Ginsberg, “Exciton dynamics reveal aggregates with intermolecular order at hidden interfaces in solution-cast organic semiconducting films,” Nat. Commun. 6(1), 5946 (2015). [CrossRef]  

11. J. M. Snaider, Z. Guo, T. Wang, M. Yang, L. Yuan, K. Zhu, and L. Huang, “Ultrafast Imaging of Carrier Transport across Grain Boundaries in Hybrid Perovskite Thin Films,” ACS Energy Lett. 3(6), 1402–1408 (2018). [CrossRef]  

12. M. J. Simpson, B. Doughty, B. Yang, K. Xiao, and Y. Z. Ma, “Spatial Localization of Excitons and Charge Carriers in Hybrid Perovskite Thin Films,” J. Phys. Chem. Lett. 6(15), 3041–3047 (2015). [CrossRef]  

13. S. Nah, B. Spokoyny, C. Stoumpos, C. M. M. Soe, M. Kanatzidis, and E. Harel, “Spatially segregated free-carrier and exciton populations in individual lead halide perovskite grains,” Nat. Photonics 11(5), 285–288 (2017). [CrossRef]  

14. A. H. Hill, K. E. Smyser, C. L. Kennedy, E. S. Massaro, and E. M. Grumstrup, “Screened Charge Carrier Transport in Methylammonium Lead Iodide Perovskite Thin Films,” J. Phys. Chem. Lett. 8(5), 948–953 (2017). [CrossRef]  

15. E. M. Grumstrup, M. M. Gabriel, E. M. Cating, C. W. Pinion, J. D. Christesen, J. R. Kirschbrown, E. L. Vallorz, J. F. Cahoon, and J. M. Papanikolas, “Ultrafast Carrier Dynamics in Individual Silicon Nanowires: Characterization of Diameter-Dependent Carrier Lifetime and Surface Recombination with Pump–Probe Microscopy,” J. Phys. Chem. C 118(16), 8634–8640 (2014). [CrossRef]  

16. Z. Guo, J. S. Manser, Y. Wan, P. V. Kamat, and L. Huang, “Spatial and temporal imaging of long-range charge transport in perovskite thin films by ultrafast microscopy,” Nat. Commun. 6(1), 7471 (2015). [CrossRef]  

17. J. Crank, The Mathematics of Diffusion (Oxford University, 1956).

18. P. T. Landsberg, Recombination in Semiconductors (Cambridge University, 1991).

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Figures (4)
Fig. 1.
Fig. 1. Kinetic effects of anisotropic excitation density. (a) Solid lines show spatially dependent 1/e lifetimes (plotted on left axis) for 25 fJ (dark blue), 250 fJ (light blue), and 2.5 pJ (green), along the y = 0 profile. The black dashed line shows the excitation profile and is plotted on the right axis. (b) At low fluence, the excitation density, $n({r,\Delta t} )$., and the signal (Eq. (2)) decay with identical kinetics (black solid line). At 2.5 pJ, the center of $n({r,\Delta t} )$ (green, dashed) decays more quickly than the edge (r = .91γpump, where the excitation density is an order of magnitude lower, green dotted). The signal (solid line) decays at a rate between these two limits.

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Fig. 2.
Fig. 2. Effective density fitting (a) Dotted lines show numerically modeled kinetics for γpump = 0.4 µm, γprobe = 0.5 µm with pulse energies of 0.125 pJ (blue), 0.25 pJ (brown), 1.25 pJ (red), 2.5 pJ (orange), 12.5 pJ (yellow) and 25 pJ (green). Global best fits to the power dependent kinetics are shown as correspondingly colored solid lines. (b) Normalized rate constants plotted as a function of the normalized probe width, assuming an effective excitation density defined by the 1/e2 width and $1/\alpha $ absorption depth of the pump. (c) and (d) show normalized kinetics assuming a reduced pump-probe width (see text) and differ by zeff. Panel c assumes zeff = 1/α, whereas panel d assumes zeff = 2/α.

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Fig. 3.
Fig. 3. Effects of excited state diffusion on measured decay kinetics with a finite probe size. (a) Kinetics traces with ${\gamma _{probe}} = 1.0$ µm and ${\gamma _{pump}}$ = 0.4 µm (dashed) or ${\gamma _{pump}}$ = 0.8 µm (solid) for Dc = 0.0 (dark blue), 0.1(light blue), 1.0 (brown), 10 (orange), and 50 cm2/s (yellow). (b) Kinetics traces with ${\gamma _{pump}} = 0.4$ µm and ${\gamma _{probe}}$ = 0.5 µm (dashed) or ${\gamma _{probe}}$ = 1.0 µm (solid) for Dc = 0 to 50.0 cm2/s (same values and colors as in panel a).

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Fig. 4.
Fig. 4. Spatiotemporal coupling in diffusion imaging. (a) Scanning the probe over the photoexcited population density produces a spatially-separated profile at a given delay time. (b) Spatially separated profiles at low fluence (25 fJ, solid) and high fluence (12.5 pJ, dashed) at Δt = 0 ps (dark blue) Δt = 200 ps (light blue) and Δt = 600 ps (maroon). Note the broadening of the high-fluence measurements relative to the low-fluence. (c) A plot of FWHM2 vs. Δt should produce a linear trend in the limit of purely diffusive broadening. However at higher fluences, the trend is nonlinear due to nonlinear recombination. Dark blue, light blue, maroon, and red traces show results from 25 fJ, 2.5 pJ, 12.5 pJ, and 25 pJ, respectively. (d) Such behavior could be interpreted (erroneously) as a time-dependent diffusion process due to non-equilibrated excited states. Trace colors and powers correspond to those in panel c.

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Equations (6)

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∂ n ( r , z , Δ t ) ∂ t = 1 r ∇ r , z ⋅ D c r ∇ r , z n − k 1 n − k 2 n 2 − k 3 n 3
I ( Δ t ) ∝ ∫ 0 R m a x ⁡ ∫ 0 z m a x ⁡ G p r o b e ( r ; γ ) n ( r , z , Δ t ) r ∂ z ∂ r
⟨ n ⟩ eff = ∫ 0 2 π ⁡ ∫ 0 r e f f ⁡ ∫ 0 z e f f ⁡ m α L n [ 16 ] π γ p u m p 2 E x p [ − L n [ 16 ] r 2 γ p u m p 2 ] E x p [ − α z ] r ∂ θ ∂ r ∂ z / ( ∫ 0 2 π ⁡ ∫ 0 r e f f ⁡ ∫ 0 z e f f ⁡ r ∂ θ ∂ r ∂ z )
I ( Δ t ) Δ r = 0 ∝ ( γ p u m p 2 + γ p r o b e 2 + 16 L n ( 2 ) D c Δ t ) − 1 / 2 E x p ( − k 1 Δ t )
I ( Δ x , Δ t ) = a 0 β ( Δ t ) exp ⁡ ( − 4 ln ⁡ ( 2 ) Δ x 2 β ( Δ t ) 2 )
β ( Δ t ) = ( γ 1 2 + γ 2 2 + 16 D c Δ t l n ( 2 ) ) 1 2
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