Carrier Diffusion in Thin-Film CH3NH3PbI3 Perovskite Measured using Four-Wave Mixing

TL;DR

This study uses femtosecond four-wave mixing spectroscopy to measure carrier diffusion in perovskite thin films, revealing that carriers diffuse across multiple grains before recombination, aiding rapid transport property characterization.

Contribution

It demonstrates a novel application of FWM spectroscopy for direct measurement of carrier diffusion and transport properties in perovskite thin films.

Findings

Carrier diffusion coefficient measured via FWM.

Carriers traverse multiple grain boundaries.

FWM enables rapid transport characterization.

Abstract

We report the application of femtosecond four-wave mixing (FWM) to the study of carrier transport in solution-processed CH3NH3PbI3. The diffusion coefficient was extracted through direct detection of the lateral diffusion of carriers utilizing the transient grating technique, coupled with simultaneous measurement of decay kinetics exploiting the versatility of the boxcar excitation beam geometry. The observation of exponential decay of the transient grating versus interpulse delay indicates diffusive transport with negligible trapping within the first nanosecond following excitation. The in-plane transport geometry in our experiments enabled the diffusion length to be compared directly with the grain size, indicating that carriers move across multiple grain boundaries prior to recombination. Our experiments illustrate the broad utility of FWM spectroscopy for rapid characterization of macroscopic film transport properties.

Equations

$$\text{I(T)}\propto \exp \left(-\text{T}/{\text{T}}_{\text{s}}\right).$$

$$\frac{1}{{\text{T}}_{\text{s}}}=\frac{2}{{\text{T}}_{\text{L}}}+\frac{8{\pi }^2\text{D}}{{\text{d}}^2},$$

Full text

All-optical techniques provide an effective approach to studying carrier transport within a wide range of photovoltaic materials as no carrier extraction layers or ohmic contacts are required. Time-domain terahertz spectroscopy (TDTHz) and time-resolved microwave conductivity (TRMC) have provided valuable insight into carrier scattering processes in the organometal halide perovskites in recent years.Ponseca:2014 ; Valverde:2015 ; Hutter:2015 For such techniques, the quantitative determination of mobility requires modeling of the Drude response of the multi-component carrier system. Transient absorption microscopyGuo:2015 provides a more direct, all-optical probe of carrier transport. In this technique, carrier motion away from the laser excitation spot is imaged as a function of time after excitation using a diffraction-limited focusing geometry, enabling the influence of variations in sample morphology on the local transport characteristics to be assessed. A complementary all-optical technique that probes carrier transport on macroscopic length scales, while also relying on the direct detection of the spatial motion of carriers, would provide a valuable characterization tool for the rapid survey of photovoltaic materials.

Here we apply femtosecond four-wave mixing (FWM) to investigate carrier transport in CH3NH3PbI3, highlighting the utility of this technique for application to a wide range of organic-inorganic perovskites. Our experiments illustrate several advantages of this approach: (i) it involves a direct measurement of the ambipolar diffusion coefficient without any need for modeling; (ii) it is an all-optical (contactless) technique, applicable to the survey of a wide range of photovoltaic materials; (iii) the carrier lifetime may be accessed through fitting the grating period dependence or directly through differential transmission with the same apparatus, exploiting the versatility of the boxcar geometry; (iv) diffusive and trap-limited transport are easily distinguished through the interpulse delay dependence of the four-wave mixing signal; and (v) the in-plane transport configuration enables the motion of carriers to be directly compared with grain sizes and allows anisotropy of transport characteristicsLiu:2016 to be evaluated through simple rotation of the sample. As an additional feature, transport may be studied using this technique with high time resolution at low carrier densities ($\sim$1015 to 1016 cm*-3*), reflective of solar cell operating conditions. Our experimental results on solution-processed CH3NH3PbI3 indicate diffusive transport within the first nanosecond following excitation, with a diffusion coefficient (D) of 1.7 $\pm$ 0.1 cm2 s*-1* and diffusion length (L${}_{\textrm{D}}$) of 0.95 $\pm$ 0.07 $\mu$m. As this technique probes the ambipolar diffusion length, which is dominated by the transport of the slower carrier type (electrons or holes), the observation of L${}_{\textrm{D}}\sim$four times larger than the measured average grain size of 250 nm indicates that defects tied to grain boundaries have a limited impact on transport, in agreement with recent experimentsChen:2015 ; Liu:2016 and theoretical predictions of predominantly shallow traps.YinAdvMater:2014 ; YinAPL:2014 The application of nonlinear optical FWM techniques to the study of transport in CH3NH3PbI3 presented here builds upon recent studies of carrier-carrier interactions and exciton localization in this material using a two-pulse degenerate FWM configuration,March:2016 ; MarchTwo:2016 as well as measurements of the third-order nonlinear susceptibility in perovskite thin films.Johnson:2016

In the four-wave mixing experiments, three noncollinear optical pulses with electric fields $\vec{\textrm{E}}_{1}$(t), $\vec{\textrm{E}}_{2}$(t-$\uptau$), and $\vec{\textrm{E}}_{3}$(t-($\uptau$+T)) with corresponding wavevectors $\vec{\textrm{k}_{1}}$, $\vec{\textrm{k}_{2}}$, and $\vec{\textrm{k}_{3}}$ were focused onto the sample in a noncollinear boxcar geometry,Bristow:2009 resulting in a fourth field emitted in the phase-matched direction $\vec{\textrm{k}_{3}}+\vec{\textrm{k}_{2}}-\vec{\textrm{k}_{1}}$ (See Fig. 1(c)). Interference between pulses $\vec{\textrm{E}}_{1}$(t) and $\vec{\textrm{E}}_{2}$(t-$\uptau$) creates a periodic spatial modulation of the carrier density in the material, forming a transient grating. The grating undergoes decay due to recombination and lateral carrier transport in the material. A third delayed pulse $\vec{\textrm{E}}_{3}$(t-($\uptau$+T)), was used to probe the grating modulation depth by monitoring the diffracted signal as a function of time delay T between pulses $\vec{\textrm{E}}_{2}$ and $\vec{\textrm{E}}_{3}$. Provided carrier transport is diffusive,Salcedo:1978 ; Ruzicka:2010 the FWM signal decays exponentially versus T,Salcedo:1978 ; Schwab:1992 with the signal intensity proportional to:

[TABLE]

and the decay constant T${}_{\textrm{s}}$ determined by:

[TABLE]

In Eqn. 2, T${}_{\textrm{L}}$ is the carrier lifetime, D is the ambipolar diffusion coefficient, $\lambda$ is the center wavelength of the laser pulse, and the grating constant $\textrm{d}=\frac{\lambda}{2\sin{\left(\uptheta/2\right)}}$, where $\uptheta$ is the angle between the two incident beams. The slope of a linear fit of the inverse grating decay time versus $\frac{8\pi^{2}}{\textrm{d}^{2}}$ yields the ambipolar diffusion coefficient D, while the offset yields the carrier lifetime. The experimental boxcar geometry also permits differential transmission (DT) measurements using the same apparatus for a separate verification of the carrier lifetime. In this case, $\vec{\textrm{E}}_{1}$(t) was blocked, $\vec{\textrm{E}}_{2}$(t-$\uptau$) served as the pump pulse (setting $\uptau$ = 0), and $\vec{\textrm{E}}_{3}$(t-T) attenuated by approximately 100$\times$ served as the probe pulse.

The laser source used in these experiments was a Ti:sapphire oscillator producing 100 fs pulses with a 10 nm bandwidth. The grating period d was varied by controlling the angle $\uptheta$ between the beams $\vec{\textrm{E}}_{1}$(t) and $\vec{\textrm{E}}_{2}$(t-$\uptau$) by changing the effective focal length of a two-lens system. The excitation density in the sample was held constant as a function of d by measuring the beam diameter (150 - 200 $\upmu$m) for each lens configuration and adjusting the laser power using neutral density filters to maintain a constant pulse fluence. All three optical pulses were s-polarized. At each delay T, the FWM signal as a function of the delay $\uptau$ between $\vec{\textrm{E}}_{1}$(t) and $\vec{\textrm{E}}_{2}$(t-$\uptau$) was detected with a fast amplified photodiode using a rapid scan technique mediated by a retroreflector attached to a vibrating speaker cone. This allowed the FWM signal at $\uptau=0$ fs to be easily separated from scattered light sources propagating along the same direction. The delay T between $\vec{\textrm{E}}_{2}$(t-$\uptau$) and $\vec{\textrm{E}}_{3}$(t-($\uptau$+T)) was varied using a computer-controlled delay stage.

The CH3NH3PbI3 sample under investigation in this work was prepared using a modified sequential deposition procedure described in detail in Ref. March:2016, . Scanning electron microscopy (SEM) measurements showed good uniformity, and x-ray diffraction (XRD) indicated full conversion to perovskite with no residual lead iodide. For the XRD and SEM studies, the exposure of the sample to air was limited to 1-2 hours. The linear absorption spectrum of a companion sample prepared at the same time using an identical procedure was measured with a Cary UV-Vis spectrometer, and is shown alongside the laser spectrum in Fig. 1(b). For all FWM experiments, exposure to air was avoided by mounting the sample in a compact optical cryostat within an argon glove box and sealing prior to transport to the FWM apparatus.

Fig. 1(d) shows the FWM signal at $\uptau$ = 0 fs versus pulse delay T at 300 K for a grating period of d = 4.14 $\upmu$m. For these experiments, the laser was tuned to 760 nm, exciting carriers $\sim$10-20 meV above the band gapEven:2014 and the excitation density was 1.6$\times$1016 cm*-3*. With the exception of the region of overlap of the excitation laser pulses, the four-wave mixing signal exhibits exponential decay over the full measurement range of 1 ns. The red curve in Fig. 1(d) shows a fit to Eqn. 1, yielding a grating decay time of T${}_{\textrm{s}}$ = 840 ps. Fig. 2 shows a logarithmic plot of the FWM signal as a function of delay for grating constants d = 4.14, 4.65, 5.50, and 6.45 $\upmu$m. As shown by the quality of the linear fits, the FWM signal decays exponentially in all cases. The inverse decay time T${}_{\textrm{s}}^{-1}$ is plotted versus $8\pi^{2}/\textrm{d}^{2}$ in Fig. 3. The error bars represent the standard deviation of ten delay scans taken for each grating constant. The solid line in Fig. 3 is a linear fit using Eqn. 2. The slope (intercept) yields the ambipolar diffusion coefficient (carrier lifetime), corresponding to D = 1.7 $\pm$ 0.1 cm2 s*-1* and T${}_{\textrm{L}}$ = 5.3 $\pm$ 0.5 ns. As a self-consistency check, differential transmission measurements were also carried out using the same apparatus for the same total excitation carrier density (Fig. 4). These measurements yield a carrier lifetime of $\textrm{T}_{\textrm{L}}$ = 5.4 $\pm$ 0.2 ns, in good agreement with the value of 5.3 ns obtained from fitting the transient grating results. Using a one dimensional diffusion model, for which L${}_{\textrm{D}}$ = $\sqrt{\textrm{D}\textrm{T}_{\textrm{L}}}$, these fit results yield a diffusion length of 0.95 $\pm$ 0.07 $\mu$m. This value is in line with measurements of the ambipolar diffusion length on similar films using transient absorption microscopy techniques.Guo:2015

The good agreement between the experimental results and Eqn. 1 and  2 over the full range of accessible delays indicates that carrier transport is within the diffusive regime. Carrier trapping, which would lead to subdiffusive behavior,Akselrod:2014 therefore has a negligible influence on the carrier kinetics within the first nanosecond after excitation. This is in agreement with studies of CH3NH3PbI3 thin films using transient absorption microscopy,Guo:2015 in which a diffusive model accounted for the spatiotemporal carrier occupation over a 3 ns time interval after excitation. Using broadband THz spectroscopy, Valverde-Chavez et al. observed a rapid initial decay of mobility on a subpicosecond time scale in single crystal CH3NH3PbI3 attributed to the charging of trap states and/or a charge-mediated phonon scattering mechanism.Valverde:2015 The experiments in Ref. Valverde:2015, were carried out at a much larger excited carrier density ($\sim$1018 cm*-3*), suggesting that these charge-mediated effects turn on under stronger excitation conditions than our experiments.

Scanning electron microscopy measurements on the same CH3NH3PbI3 film used for FWM experiments are shown in Fig. 3(b). A spatial Fourier analysis indicates the presence of grains up to 500 nm in diameter, with a mean grain size of 250 nm. The observation of LD $\sim$4 $\times$ larger than the mean grain diameter, together with the fact that this ambipolar diffusion length is limited by the slower carrier species,Ruzicka:2010 indicates that grain boundaries do not constitute a significant barrier to carrier motion. This result is in agreement with recent measurements on solution processed CH3NH3PbI3 using AC Photo HallChen:2015 and photocurrent imaging techniquesLiu:2016 as well as theoretical calculations suggesting that the most prevalent point defects represent shallow traps and that grain boundaries have a relatively benign effect.YinAdvMater:2014 ; YinAPL:2014 In addition, screening by the methylammonium ions may play a role in reducing the rate of trapping by charged defects,Chen:2015 in line with the recently proposed large polaron model of carrier transport in perovskites.XYZ

Reported values of the diffusion length in solution processed methylammonium lead iodide films have varied over the range 0.1 to 23 $\mu$m.Stranks:2013 ; Xing:2013 ; Wehrenfennig:2014 ; GonzalezPedro:2014 ; Li:2015 ; Guo:2015 ; Hutter:2015 ; Chen:2015 ; Liu:2016 ; Edri:2014 Differences in both the microstructure of the film and the optically-excited carrier density contribute to this broad range of values as these together dictate the carrier lifetime.Semonin:2016 The application of a variety of experimental techniques to extract the transport characteristics also plays an important role in the observed variability. For instance, photoluminescence-based techniques only probe the transport of radiative species, TRMC and TRTHz techniques measure the sum of the electron and hole mobilities ($\mu_{e}$+$\mu_{h}$), AC Photo Hall measurements detect $\mu_{h}$-$\mu_{e}$, and TA microscopy and the FWM technique applied in this work both probe the ambipolar diffusion coefficient ($\mu_{a}=\frac{2\mu_{e}\mu_{h}}{\mu_{e}+\mu_{h}}$). In addition, while techniques that rely on electrical contacts can offer the advantage of probing transport in a working device,Edri:2014 uncertainties tied to the energetics of the contact interfaces can significantly influence the extracted results for L${}_{\textrm{D}}$.Hodes:2015 This highlights the need for a survey tool for probing transport in a wide range of photovoltaic materials such as the FWM transient grating technique presented here. We note that a steady-state photocarrier grating method relying on a change of photocurrent induced by interference of continuous wave laser beamsRitter:1986 ; Balberg:1988 was recently applied to organic-inorganic perovskite films.Adhyaksa:2016 In contrast to the approach used in Ref. Adhyaksa:2016, , the transient grating technique applied in this work does not require electrical contacts to the sample and provides access to dynamic information about the photocarrier response.

In conclusion, we have applied the all-optical four-wave mixing transient grating technique using a boxcar geometry to the study of carrier transport in a solution processed film of CH3NH3PbI3, illustrating the power and flexibilty of this approach for application to emerging photovoltaic materials based on the organic-inorganic family of perovskites. Our experiments yield values for the carrier lifetime of 5.3 $\pm$ 0.5 ns and the ambipolar diffusion coefficient of 1.7 $\pm$ 0.1 cm2 s*-1*, corresponding to a diffusion length of 0.95 $\pm$ 0.07 $\mu$m. These fit values are obtained from the measured transient grating signal without the need for modeling, a consequence of the direct optical detection of the spatial motion of carriers. Our experiments show that this method provides a valuable rapid survey tool for probing macroscopic transport properties in the hybrid perovskites.

This research is supported by the Natural Sciences and Engineering Research Council of Canada, the Canada Foundation for Innovation, and the Canada Research Chairs program.