Multidimensional time-of-flight spectroscopy

Investigations of energy and charge transport mechanisms in novel materials inform the development of optoelectronic devices, such as photovoltaic cells and light-emitting diodes.^1–10 Conventional time-resolved laser spectroscopies may be employed for this purpose; however, there is no guarantee that the dynamics observed in these experiments hold relevance for the operation of a device. The signal components detected with transient absorption spectroscopies applied to isolated materials (e.g., solutions, films, and crystals) depend on the concentrations and extinction coefficients of all photoexcited species regardless of their practical significances. For this reason, several experimental methods have been developed in which a variable number of laser pulses induce a nonlinear response of the photocurrent produced by a device.^11–20 These techniques, which are applied to working photovoltaic cells, can be designed to elucidate processes ranging from electronic dephasing to long-range transport of charge carriers. Tremendous opportunities exist for interdisciplinary discoveries because interpretations of photocurrent spectroscopies borrow concepts from different areas of specialization (e.g., materials, device measurements, and nonlinear spectroscopy).

In this Perspective, we present a nonlinear photocurrent (NLPC) spectroscopy in which charge carrier motions are resolved in the delay time between a pair of laser pulses applied to a photovoltaic cell.^21–23 Our approach takes inspiration from conventional time-of-flight (TOF) methods in that we determine the timescale of long-range carrier transport across the active layer of a device;^24–30 however, the experiment yields additional insights into mechanisms by probing nonequilibrium charge distributions. This aspect of the technique is shared with several multi-pulse photocurrent spectroscopies demonstrated over the past 25 years.^11–22 Whereas these earlier NLPC spectroscopies were employed in studies of short-range dynamics, our newly developed method is sensitive to carrier drift over length scales of 100s of nm.²³ In this approach, charge carrier velocities and mobilities are extracted from NLPC signals by cycling the external biases applied to photovoltaic cells.²³ The capability of NLPC measurements to yield TOF information, which is distinct from our earlier work,^21,22 is the focus of this Perspective.

The development of TOF approaches for measuring drift velocities in photovoltaic devices was initially motivated by challenges associated with the characterization of highly resistive, low-mobility solids.^24–30 In these techniques, a short pulse of light initiates charge carrier drift within the active layer of a device, as indicated in Fig. 1. The distribution of arrival times at an electrode is then imprinted on the transient profile of the photocurrent. Drift velocities may be determined with knowledge of the sample thickness and transit time; however, the temporal profile of the photocurrent also encodes information regarding trap densities, trap depths, carrier diffusion, and excited-state deactivation processes (e.g., Auger recombination and spontaneous emission).³¹ Because the drift velocity varies linearly with the applied electric field, the carrier mobility may be specifically targeted by examining the dependence of the transit time on the external bias applied to the device.^32,33 Although TOF techniques are quite versatile, potential drawbacks include the requirement that the active layer of a device is thick enough to (i) resolve the transit time with an oscilloscope and (ii) localize the initial distribution of photoexcited carriers due to large optical density.

Earlier experimental work has demonstrated that charge carrier dynamics can be probed within a photovoltaic device using a pair of femtosecond laser pulses.^11–15 As in a conventional transient absorption experiment, the first pulse photoexcites the material within the active layer of a device, whereas the color of the second pulse may be tuned to probe the concentrations of photoexcited species. Information specific to the operation of a photovoltaic device is obtained by measuring the component of the current that depends on the presence of both laser pulses (i.e., the nonlinear response). This approach has been applied to photovoltaic cells based on organic materials to reveal the timescale of exciton dissociation into free carriers;^12,15 however, these experiments did not yield drift velocities because the delay times were scanned over hundreds of ps. Nonetheless, these earlier works were critically important for demonstrating that the nonlinear optical response of the active layer of a device is reflected by the photocurrent.

Four-pulse Fourier transform 2D photocurrent spectroscopies (2DFT-PC) were developed in recent years to determine the phases of individual signal components and to enhance the time-frequency resolution for this class of experiments.^16–20 The differences between two- and four-pulse photocurrent spectroscopies resemble the distinctions between conventional transient absorption and photon echo methods.^34,35 2DFT-PC experiments employ a sequence of four broadband laser pulses, which arrive at the device in pairs separated by a population time. The delays between members of the two pairs of laser pulses correspond to electronic coherence times.³⁶ Two-dimensional Fourier transformation of these coherence times yields 2DFT-PC spectra as a function of population time (i.e., the delay between the second and third laser pulses). Similar to Fourier transform electronic spectroscopies,^37–45 2DFT-PC spectroscopy avoids a trade-off between time and frequency resolution associated with the spectral widths of the laser pulses (e.g., the frequency resolution is limited by the inverse of the pulse duration in a two-pulse photocurrent experiment). Access to shorter timescales has facilitated investigations of ultrafast processes, such as multiple exciton generation in PbS photocells¹⁷ and electronic decoherence in a polymer-based system.¹⁸ 

With inspiration from these related methods, we have recently developed a two-pulse NLPC spectroscopy capable of determining drift velocities and carrier mobilities.²³ Because Fourier transformation is not well-motivated on the ns timescale of carrier drift, the experiment employs narrowband color-tunable laser pulses to record 2D NLPC spectra at variable delay times. As in previous two-pulse photocurrent spectroscopies,^11–15 the first pulse photoexcites charge carriers and initiates drift, whereas the second pulse probes the concentrations of carriers that remain in the active layer at later times [see Fig. 1(b)]. Transit times through the active layer may then be determined by scanning the delay time, τ, over a sufficient range. Our data show that NLPC spectroscopy can yield transport mechanism-specific mobilities when applied to photovoltaic devices with heterogeneous active layers. For example, in a recent study of layered perovskite systems,²³ the first and second laser pulses were tuned into the resonances of separate quantum wells to extract the mobilities associated with quantum well-specific pathways through the devices. The signal generation mechanism will be further clarified in this work.

Signals presented in this Perspective article were acquired with those reported in earlier publications.^22,23 We refer readers to these original studies for the standard procedures used to prepare materials and fabricate devices. In this section, the NLPC experimental setup is described to define the method.

Our experiments are conducted with a 45-fs, 4-mJ Coherent Libra with a 1-kHz repetition rate. ∼1.5 mJ of the 800-nm fundamental is focused into a 2-m long tube filled with argon gas to generate a visible continuum. The continuum is split into two beams and passed through replica all-reflective 4F spectral filters, which are based on 1200-g/mm gratings and 20-cm focal length mirrors. The desired portions of the spectra are filtered with motorized slits at the Fourier planes. The spectrally filtered pulses have 5-nm widths and 300-fs durations.

The 15-ns delay range is achieved using a motorized translation stage (Zaber X-LDQ0600C-AE53D12). Four retroreflectors (Thorlabs) are mounted on the stage to maintain a small footprint (one of the four retroreflections is shown in Fig. 2 for clarity). The setup is housed in an enclosure to minimize air currents. Collimation of the laser beam over this distance is achieved using a telescope with +100 and −50 mm singlet lenses. We have confirmed that transmission of the laser beam through a 100-µm diameter pinhole at the sample position varies by less than 5% over the full range of parameters covered in our experiments. The laser pulses have 55-pJ energies and 82-µm spot sizes. The laser fluence of 1.0 µJ/cm² is on the same order of magnitude as those employed in earlier NLPC-like experiments.^16,19,20

The NLPC instrument and signal detection schemes are shown in Fig. 2. To begin, the two laser beams are chopped at 500 and 250 Hz to establish a cycle of four conditions: pulses 1 and 2 (S₁₊₂), pulse 1 only (S₁), pulse 2 only (S₂), and both pulses blocked (S₀). The NLPC signal is defined as S_NLPC = S₁₊₂ − S₁ − S₂ + S₀. Each photocurrent difference is averaged over a total of 800 laser shots (0.8 s). The photocurrent produced by the device is amplified at either 2 µA/V (layered perovskite) or 20 µA/V (bulk perovskite) using a Stanford Research 570 current preamplifier. Signals are then processed with a National Instruments data acquisition board (NI USB-6221), which is synchronized to the laser system at 250 Hz. An adequate density of points is obtained by setting the sampling rate of the data acquisition board to 500 kHz (time intervals of 2 µs) because the widths of the signal pulses are ∼80 µs. The temporal widths of the signal pulses processed at the data acquisition board primarily reflect the 3 dB bandwidth of the bandpass filter within the preamplifier.

Carrier mobilities are determined by measuring NLPC decay profiles at five external biases: −0.2, −0.1, 0, 0.1, and 0.2 V. The delay time, τ, and wavelengths, λ₁ and λ₂, are also scanned in these experiments. We cycle through the full set of conditions 30 times to produce a single dataset with a total data acquisition time of 24 h. To establish reproducibility, the experiments are repeated multiple times with separate devices.

The time resolution of an NLPC experiment is determined by the laser pulse durations,¹² whereas the time resolution of a conventional TOF measurement is limited by the RC time constant of the photovoltaic cell.^25,28 This distinction is important because RC time constants for perovskite-based photovoltaic devices are on the order of microseconds.^46,47 As shown in Fig. 1, the nonlinear response of the photocurrent is processed as a function of the experimentally controlled delay time, τ, in NLPC spectroscopy. The width of the instrument response can be estimated using a configuration in which the two pulses are centered at different wavelengths. Two-color pulse sequences produce an asymmetry in the response, wherein the amount of resonance enhancement differs on opposite sides of time-zero. This asymmetry is maximized when one pulse is resonant with the material at equilibrium, and the second pulse is resonant only with the excited system (i.e., off-resonant with the equilibrium system).¹² 

In Fig. 2(c), we present an NLPC signal profile measured at time-zero for the layered perovskite system discussed in Sec. IV. The wavelengths of the two laser pulses are 610 and 680 nm. The signal magnitude is greater at positive delay times (i.e., when the 610 nm pulse arrives at the device first) because the interaction with the 680 nm pulse is resonance-enhanced by a relaxation-induced red shift in the signal spectrum. This measurement is sufficient to show that the time resolution of NLPC spectroscopy is not limited by the RC time constant of the photovoltaic cell. For our measurements, the ns timescale of interest is three orders of magnitude larger than the width of the instrument response. Notably, absorptive media, such as that employed for the measurement shown in Fig. 2(c), can broaden the instrument response by way of sub-ps photoinduced relaxation processes. Therefore, the 1.7 ps width of the response should be understood as an upper limit for the time resolution.

In this section, we describe the NLPC signal generation mechanism by combining a perturbative description of the nonlinear optical response with numerical simulations of long-range carrier drift. The model takes inspiration from earlier applications of TOF measurements to organic crystals, which established experimental signatures of carrier drift, diffusion, trapping, and many-body recombination.^25,27–29 Photoexcitation induces the same relaxation processes in both NLPC and conventional TOF measurements; however, only NLPC spectroscopy is able to resolve dynamics ranging from the sub-ps to ns timescales.

To begin, it is useful to consider the physical insights gathered by applying conventional TOF techniques to low-mobility solids. In Fig. 3, we sketch temporal profiles of TOF signals measured with and without trap-induced dispersion in the drift velocities.²⁵ In general, TOF measurements are conducted on devices in which the penetration depth of the incident light is small compared to the thickness of the active layer. Large optical densities localize the initial photoexcited carrier distributions near one of the electrodes, thereby enabling separate determinations of the electron and hole mobilities. Without carrier trapping, the drift velocities of the photoexcited carriers exhibit a narrow distribution, which is evidenced by a step function-like signal profile [see Fig. 3(a)].²⁵ In contrast, trap-induced drift velocity dispersion inflates the average transit time and gives rise to quasi-exponential decay profiles, as indicated in Fig. 3(b). The effective transit time observed by TOF measurements can be written as²⁹ 

where T_transit is the trap-free transit time, B is the number of trapping incidents, and T_trap is the average residence time at shallow trap sites.

Transient absorption microscopies conducted on solution-processed bulk and layered perovskites films show that traps and disorder suppress carrier diffusion and enhance trap-assisted recombination processes.^49,50 Because of the fluctuation–dissipation relation between carrier diffusion and mobility,⁵¹ it is predictable that trap-induced velocity dispersion will be a prominent effect in NLPC experiments. For this reason, it is useful to consider the model summarized in Fig. 4, which has been used to describe drift velocity dispersion in photovoltaic devices based on thin films.^{23,27,48,52,53} Electrons are treated in Fig. 4, but the concepts can be generalized to holes. Below the conduction band minimum (i.e., the mobility edge), the density of trap states is exponentially distributed. On a timescale shorter than thermal release from a shallow trap, the populations of trapped electrons mirror the density of states under the assumption that the capture cross sections are independent of the depths of the traps. Transport occurs when the electrons are thermally excited into spatially extended states above the mobility edge. At later times, the shallow traps exhibit a Boltzmann distribution of populations due to numerous cycles of capture and release; however, the populations of the deep traps do not thermalize because of low probabilities for escape.

The dependence of the thermal release time, T_trap, on the depth of a trap, E, is given by⁴⁸ 

where T₀ is the time interval between attempts at thermal release. This parameter was taken to be 1 ps in earlier studies,^27,28 and we make the same assumption in Fig. 4(c). The thin film devices investigated in our recent work have transit times on the order of 10 ns,²³ which suggests a practical demarcation between shallow and deep traps. According to the model, traps with depths on the order of 150 meV may undergo 25 cycles of capture and release during the transit time. In contrast, traps with depths greater than 230 meV are unlikely to release electrons on the timescale of our experiments. Traps with depths of 150–500 meV have been reported for bulk and layered perovskite systems.^54–56 The demarcation energy between shallow and deep traps is on the lower end of this range.

In previous work, we have simulated the NLPC signal generation mechanism numerically on a one-dimensional grid by finite differences.²³ Carrier drift and two-body recombination are treated explicitly, whereas a phenomenological approach is used to incorporate trap-induced velocity dispersion. Dynamics in the carriers photoexcited by the first laser pulse, p₁ and n₁, are given by

and

where v_d is the magnitude of the average drift velocity and β is the two-body recombination coefficient. Here, we introduce the time variable, t′ = τ + t, for convenience to describe processes that occur continuously after the system interacts with the first laser pulse (e.g., recombination, drift, etc.). The experimentally controlled delay, τ, and time elapsed after the interaction with the second laser pulses, t, are illustrated in Fig. 1. As depicted in Fig. 5, the hole- and electron-selective layers are located at lesser and greater values of the variable, x′. Similarly, carriers generated by the second laser pulse are described with

and

where the Heaviside step function, $θt′−τ$⁠, sets p₂ and n₂ equal to zero before the second pulse arrives at the device. Broadening of the carrier distributions induced by processes such as trap-induced velocity dispersion and diffusion is incorporated phenomenologically using

and

where k is the index of the laser pulse (1 or 2) and w is set equal to 0.85 nm to approximate the experimental NLPC decay profiles. The nonlinear response of the photocurrent is computed using

where ϕ_ex is the probability of light absorption, l is the thickness of the active layer, Δt′ is the temporal step size, and A is the laser spot area on the device. The photocurrent is then integrated over the detection time,

where χ is the amplification factor of the preamplifier (units of current/voltage). The capacitance, C, corresponds to the photovoltaic cell, preamplifier, and stray contributions. The integral on the right-hand side of Eq. (10) possesses units of charge; however, multiplication by the ratio, χ/C, yields units of current for $Jτ$⁠.

The initial electron and hole densities are written as

where f is the laser fluence, ν is the laser frequency, α is the absorption coefficient, l is the active layer thickness, and OD is the optical density. The subscript for the electron and hole densities, k, represents the index of the laser pulse (1 or 2), and the factor of 4/5 in the second term accounts for a reflection from the copper electrode near the electron-selective layer.⁵⁷ This is an appropriate description for the systems investigated in our recent studies, which have active layer thicknesses of ∼110 nm. The carrier densities are propagated forward in time using Eqs. (3)–(8) with a temporal step size, Δt, of 10 ps.

As shown in Fig. 5(a), the carriers photoexcited by the first laser pulse drift, diffuse, and undergo a variety of excited-state deactivation processes during the experimentally controlled delay time. Based on the absorbance cross section (10⁻²¹ to 10⁻²⁰ m²) and laser fluence (∼10¹⁶ photons/m²), the probability that an electron photoexcited by the first laser pulse will also interact with the second pulse is on the order of ϕ_ex = 10⁻⁵ to 10⁻⁴.²³ The second laser pulse also produces a new distribution of photoexcited carriers, as depicted in Fig. 5(b); however, these carriers do not contribute to the signal because linear responses are removed by chopping the two laser beams. The small fraction of carriers that interact with both laser pulses are responsible for the nonlinear response of the photocurrent.

Signal components of NLPC-like experiments are usually decomposed into terms analogous to those found in conventional transient absorption spectroscopies.^17,18 In recent work, we have proposed a perturbative model for simulating NLPC signals at delay times long compared to carrier cooling.²³ The fourth-order response functions factor into products of steady-state absorption and emission line shapes on this timescale. The magnitude of the ground state bleach (GSB)-like signal component for a bulk semiconductor can be written as

where $σabsω$ is the rate of light absorbance and τ is the experimentally controlled delay time. Similarly, the excited state emission (ESE)-like signal component is obtained by substituting the stimulated emission rate, $σemω2$⁠, for the linear absorbance rate, $σabsω$⁠,

Excited state absorption-like terms can be introduced by replacing $σemω2$ with the rate of light absorption between singly and doubly excited states. These formulas are derived under the assumption that extrinsic aspects of the material govern carrier transport, which makes the nonlinear photocurrent, $Jτ$⁠, independent of the quantum states. Transient absorption microscopies conducted on films processed from solution are consistent with this approximation.^49,50

In Fig. 5(c), dynamics in the electron and hole densities are simulated using our numerical model with the parameters summarized in Table I. The magnitude of the electric field, E₀, is given by the ratio of internal bias (−1.0 V) and active layer thickness (110 nm). A drift velocity of 14 m/s is calculated using the product of the electric field and carrier mobility, which is set equal to 0.015 cm²/V/s based on the timescale of the signal decay observed experimentally. The two-body recombination parameters determined with transient absorption microscopy experiments (β = 3.3 × 10⁻⁹ cm³/s for bulk perovskite)^49,50 account for all processes that scale as the square of the carrier density (e.g., radiative and trap-assisted Auger recombination).³¹ Dispersion in the drift velocity broadens the carrier distributions, thereby inducing quasi-exponential decay profiles. In contrast, the carrier distributions possess flat edges at all delay times in calculations conducted without drift velocity dispersion and two-body recombination. The resulting NLPC signal exhibits a step function-like profile consistent with that presented in Fig. 3.

As shown in Fig. 5(b), carriers photoexcited by separate laser pulses can interact and recombine during signal detection. The effects that these processes have on NLPC decay profiles depend on the laser fluence and the magnitudes of the two-body recombination coefficients. Bulk perovskites have two-body recombination coefficients, β, on the order of 10⁻⁹ to 10⁻⁸ cm³/s,^49,58–62 whereas the values of β determined for layered perovskites are sensitive to the sizes of the quantum wells (β = 10⁻⁹ to 10⁻⁷ cm³/s).⁵⁰ If we assume β = 10⁻⁹ cm³/s, the carrier density will initially decay at a rate of 9 × 10²⁵ cm⁻³ s⁻¹ for a laser fluence of 1 µJ/cm². At this rate, the carrier density decreases by roughly 3% in the first 100 ps after photoexcitation. The two-body recombination rate increases linearly with β, so a decrease in the carrier density of ∼30% is anticipated within 100 ps of photoexcitation if β = 10⁻⁸ cm³/s. The influence of two-body recombination on the signal profiles may be reduced by decreasing the laser fluence; however, we find that the signal detection threshold is approached near 1 µJ/cm² at a 1 kHz repetition rate. Related 2DFT-PC experiments suggest that lower fluences may be employed at higher laser repetition rates.¹⁷ 

The nonlinear responses of the photocurrent shown in Fig. 6 are computed over the full range of β parameters previously determined for bulk and layered perovskite systems.^49,58–62 These calculations show that recombination between carriers photoexcited by separate pulses inflates the overall decay time by flattening the profiles at short delay times (i.e., when the carrier concentrations are greatest). The discrepancies between decay curves are negligible with β = 10⁻⁹ cm³/s, whereas clear differences are predicted with β = 10⁻⁷ cm³/s. Primarily, the differences in the decay profiles presented in Fig. 6 reflect recombination processes involving carriers photoexcited by the first laser pulse during the experimentally controlled delay time [see Fig. 5(a)]. The decay profiles are less sensitive to recombination mechanisms associated with carriers photoexcited by separate laser pulses [see Fig. 5(b)]. Because recombination processes can have significant effects on NLPC decay profiles, we cycle the external bias during our measurements to extract information specific to the electric field-induced drift of charge carriers. Similar approaches have been employed to determine carrier mobilities with conventional TOF measurements.^32,33

We have recently applied NLPC spectroscopies to photovoltaic cells based on mixtures of layered organohalide perovskite quantum wells with different sizes.^21,22 The systems investigated in these earlier studies are described by the general chemical formula BA_nMA_n−1Pb_nI_3n+1, where BA is butylammonium, MA is methylammonium, and the subscript n represents the number of stacked lead-iodide octahedra within the quantum wells. The linear absorbance spectrum of the film presented in Fig. 7 reveals well-resolved exciton resonance wavelengths at 570, 600, 640, and 680 nm for quantum wells with n = 2, 3, 4, and 5, respectively. Because the average thicknesses of the quantum wells vary monotonically between the electrodes in photovoltaic cells,^50,63,64 long-range energy and charge transport processes are driven by both the electronic structures and concentration distributions of quantum wells with different sizes.^1,2,65–68 Energy and electron transfer cascades proceed from thinner to thicker quantum wells because the bandgaps and conduction band minima decrease as the value of n increases. A consensus has not been reached regarding the preferential direction of hole transfer; however, recent theoretical work suggests that a mixture of type I and II band alignments may be found in the same films due to structural disorder.⁶⁸ 

Knowledge of photoinduced relaxation mechanisms in layered perovskites has largely been gathered from transient absorption experiments.^{63,64,69–74} It is now understood that the energy transfer concentrates electronic excitations in the thickest quantum wells on the sub-ns timescale, whereas the charge transfer between quantum wells occurs at later times.^{22,64,70,73,74} Here, we suggest a sub-ns timescale for the energy funneling processes in systems with heterogeneous concentrations of quantum wells [see Fig. 7(c)]; energy transfer transitions between individual donor–acceptor pairs are thought to occur within tens of ps.⁷⁴ Transient absorption spectroscopies conducted with visible light primarily yield information related to energy transfer processes for multiple reasons. First, the nonlinear optical responses of the quantum wells are dominated by excitons because of their large transition dipoles, narrow linewidths, and large binding energies.⁷⁵ Second, continuum states, which give rise to the broad feature in Fig. 7(b), have small contributions in transient absorption spectra due to their relatively small extinction coefficients and broad linewidths. For these reasons, it is difficult to extract information regarding charge transport from the responses of continuum states in transient absorption spectra. By contrast, the hole and electron funneling dynamics illustrated in Fig. 7(d) can be detected without interference from undesired signal components using NLPC spectroscopy.^21,22 Species that relax by way of lossy recombination processes do not contribute to NLPC signals because they do not generate the photocurrent.

Two-dimensional NLPC spectra acquired for both layered and bulk perovskite-based devices are presented in Fig. 8. At sub-ns delay times, the layered system exhibits exciton resonances; however, the linewidths are much broader than those detected with transient absorption and nonlinear fluorescence spectroscopies.^{63,64,69,70,72,73,76} We attribute these differences to enhanced contributions from continuum states in photocurrent spectroscopies.²¹ In comparison to transient absorption data, the responses of excitons in the smallest quantum wells are suppressed by spontaneous emission due to their large binding energies and transition dipoles.⁷⁰ As the delay time increases, the signal red-shifts along the λ₂ axis with λ₁ = 550–600 nm, which is consistent with electron funneling mechanisms initiated in the n = 2 and 3 quantum wells [see Fig. 7(d)]. We consider charge carrier funneling to be the most likely assignment for these dynamics because the 3-ns timescale of this process is 2–3 orders of magnitude slower than that determined for energy transfer dynamics in closely related systems.^{64,70,72–74} Of course, this difference in timescales is predictable because the energy transfer is driven by longer-range donor–acceptor couplings than is charge transfer.^22,64,70 In addition, we find that the diagonal peak associated with the n = 4 quantum well (λ₁ = λ₂ = 640 nm) disappears on the ns timescale due to exciton dissociation.^22,50

In contrast to the layered system, the line shapes of NLPC spectra associated with the bulk perovskite (i.e., three-dimensional perovskite) are essentially independent of the delay time. The magnitude of the nonlinear response of the photocurrent is greatest at longer wavelengths because of contributions from the (red-shifted) excited state emission-like nonlinearity [see Eq. (13)]. We interpret these delay-independent spectral line shapes as reflecting the quasi-homogeneous nature of the active layer. That is, the bulk perovskite does not possess material components with distinct electronic resonances, which can give rise to dynamic changes in the spectral line shapes. Therefore, we consider NLPC measurements conducted with the bulk perovskite system as a control to confirm that quantum well-specific mobilities are obtained in photovoltaic devices based on the layered perovskite material.

The numerical model presented in Sec. III suggests that the NLPC temporal profiles are influenced by carrier drift, two-body recombination, and trap-induced velocity dispersion. The drift velocity can be distinguished from other processes by varying the external bias applied to a photovoltaic cell;^32,33 however, the NLPC decay profiles must be moderately sensitive to carrier drift for this approach to be successful. To experimentally establish the influence of carrier drift on the signal profiles, the measurements presented in Fig. 9 are conducted with active layers possessing different thicknesses. In effect, the transit times are varied while holding the excited state deactivation rates constant in these control experiments. For both layered and bulk perovskite systems, the decay times become shorter as the thicknesses of the active layers decrease. The differences in decay times reflect the shorter path lengths traversed by carriers in the perovskite materials before injection into the transport layers. Importantly, the discrepancies in the decay profiles are significant, which suggests that carrier drift is a significant component of the NLPC decay profiles.

Signals measured over a range of external biases are displayed for devices based on layered and bulk perovskites in Fig. 10. For both systems, the overall timescale of the decay decreases as the magnitude of the potential increases. As shown in Figs. 10(a) and 10(d), the decay times are determined by fitting the NLPC decay profiles at five values of the external bias with the following equation:

The average relaxation times, T_av, are then computed by taking a weighted average of the time constants,

The velocity is given by the ratio of the active layer thickness and T_av. The mobilities are then obtained by fitting the velocities with respect to the electric field using v_d = μE [see Figs. 10(b) and 10(e)]. Processes such as two-body recombination and diffusion contribute to the intercepts of these fits but not the slopes. The magnitudes of the electric fields, E, are computed by summing the internal (−1.0 V) and external (−0.2 to 0.2 V) biases then dividing by the thicknesses of the active layers (110–140 nm). The shortest decay times are measured with an external bias of −0.2 V because the internal bias has a negative sign. This procedure is repeated for the 16 grid points overlaid on the 2D NLPC spectrum in Fig. 8(a).

In our earlier work, we found that the NLPC temporal profiles exhibit signatures of carrier funneling when measured above the diagonal of the 2D NLPC spectrum for the layered perovskite system.^22,23 For example, when the laser beams are tuned into the cross peak between the n = 2 and 5 quantum wells (λ₁ = 570 nm and λ₂ = 680 nm), the signal magnitude rises over 3 ns before gradually decaying from 3 to 15 ns.²³ The measurements presented in Fig. 10(c) show that the NLPC profiles measured in this region of the spectrum for the layered material are independent of the external bias, which corresponds to a mobility of zero. We interpret this behavior as a signature of carrier trapping at the interfaces between quantum wells. Notably, the laser beams are incident on the indium tin oxide (ITO) substrate and must transmit through the smaller quantum wells concentrated near the hole-selective layer.⁵⁰ Therefore, the trapping effects are most prominent when the first pulse is tuned to shorter wavelengths and absorbed by the smallest quantum wells. In contrast, the signals acquired with the same set of wavelengths for a bulk perovskite-based device [see Fig. 10(f)] reflect electric field-induced drift rather than carrier trapping (i.e., the timescale of the decay decreases when the magnitude of the electric field increases).

Mobilities determined at the 16 grid points on the 2D spectrum in Fig. 8(a) are compared for the layered and bulk perovskite systems in Fig. 11. The mobilities of the layered system exhibit a significant asymmetry above and below the diagonal of the spectrum. We attribute the small mobilities measured above the diagonal to carrier trapping at the organic/inorganic interfaces between quantum wells with different sizes. The interstitial organic layers constitute >2.0-eV potential energy barriers⁷⁷ with ∼0.8-nm thicknesses,¹ thereby suggesting slow barrier crossing processes with butylammonium spacer cations.^78,79 The potential energy changes by ∼9 meV over a 1-nm distance with an electric field of 9.1 × 10⁶ V/m. Increasing or decreasing the external bias by 0.2 V translates to a 2-meV potential energy difference on the 1-nm length scale. Therefore, we suggest the barrier crossing dynamics are insensitive to the external bias. In effect, the mobility becomes small when the second term in Eq. (1), BT_trap, is large compared to T_transit. In support of this interpretation, the carrier mobilities measured in the planes of n = 3 quantum wells are on the order of 1.2 cm²/V/s, whereas the out-of-plane transport through interstitial organic layers corresponds to a mobility of 1.5 × 10⁻⁴ cm²/V/s.⁶⁷ 

The larger mobilities measured below the diagonal for the layered perovskite system represent processes in which the first laser pulse photoexcites the thickest quantum wells. We suggest that the behaviors observed below the diagonal do not signify extraordinary hole transfer mechanisms [see Fig. 7(d)]. Rather, because the thickest quantum wells are concentrated near the electron-selective layer, the carrier concentrations in this phase of the material are maximized when the first laser pulse is tuned to longer wavelengths (i.e., light may then transmit through the smaller quantum wells located near the ITO substrate on which the laser beams are incident). In addition, the larger quantum wells respond below the diagonal because light absorption induces a bleach of continuum states throughout the visible spectral range. Thus, we conclude that the large mobilities found below the diagonal represent photoexcitation and transport within the phases of the thickest quantum wells (i.e., without inter-phase charge transfer). For reference, a mobility 0.2 cm²/V/s was determined using a conventional TOF method for a system in which the n = 4 quantum wells have the greatest concentration.⁶⁷ This mobility is a factor of 5–10 greater than that measured for the present n = 3 system with NLPC spectroscopy; however, this difference is reasonable considering that films become more conductive as the thicknesses of the quantum wells increase.

By contrast, the carrier mobility surface of the bulk perovskite is relatively flat and does not approach zero above the diagonal. Insensitivity of the mobilities to the excitation wavelengths is expected because of the quasi-homogeneous nature of the active layer. Carriers photoexcited at shorter wavelengths relax to the band edges on a timescale much shorter than the transit time through the active layer (i.e., ps carrier cooling vs ns carrier drift). The carrier mobilities determined for the bulk perovskite, which range from 0.016 to 0.026 cm²/V/s,²³ are in good agreement with the electron mobility derived from the steady-state space charge-limited current (0.012 cm²/V/s).⁸⁰ Photoluminescence quenching yields mobilities that are 50–100 times greater than those obtained within working devices.^32,81 This discrepancy likely reflects contributions from contacts with the electrodes, interfaces with transport layers, and space-charge effects. Therefore, while the shapes of the 2D mobility surfaces carry clear-cut physical insights, we find that the absolute magnitudes of the mobilities are not readily compared to those obtained by ex situ techniques applied to isolated films.

Recent experimental work shows that nonlinear photocurrent spectroscopies can be configured as TOF methods for determining drift velocities by cycling the external biases applied to photovoltaic cells.²³ Two-pulse photocurrent spectroscopies have been applied by our group and others;^{12–15,21,22} however, the TOF application is a new development first reported in Ref. 23. In this Perspective, we have explored the signal generation mechanism and presented additional data acquired for layered and bulk perovskites. While further development of the technique is of interest, we suggest that the core multidimensional TOF method summarized above may be widely applicable to solar cells with active layers possessing multiple material components (e.g., polymer-based heterojunction systems and organic semiconductors with molecular charge-transfer complexes).

The carrier mobility surfaces presented in Fig. 11 suggest that NLPC spectroscopy can be used to decompose the photocurrent into transport mechanism-specific contributions for devices with multiple materials in the active layer. The shapes of the surfaces offer physical insights that cannot be gathered from conventional TOF methods in which the initial conditions are established with a single laser pulse. The delay time is limited to 15 ns in our NLPC instrument, so the active layers of photovoltaic cells must be thin compared to the penetration depth of the incident light. Contributions from processes such as two-body recombination, which are most prominent in the sub-3-ns delay range in the present experiments, are significant if the transit time is short. Undesired contributions to the signal can be suppressed by varying the external bias to target carrier drift; however, it is preferable to apply this same procedure to solar cells possessing thicker active layers. As in conventional TOF studies, electron and hole mobilities can be determined separately if the penetration depth is small compared to the thickness of the active layer. Alternate laser systems can be employed to increase the accessible delay range by orders of magnitude. Light sources with higher repetition rates may also allow for a decrease in the laser fluence, which will facilitate studies of trap-related processes.⁸² Separate femtosecond laser systems can be synchronized;⁸³ however, pulsed diode lasers may be preferred if time resolution on the order of tens of ps is sufficient for the application.

We envision the development of a new family of experiments that combine multidimensional spectroscopies with device measurements (see Fig. 12). As discussed above, the present two-pulse approach for conducting NLPC spectroscopy is analogous to a conventional transient absorption experiment,^12,15,21,22 whereas four-pulse NLPC-like techniques resemble photon echo methods.^16–18 Additional parallels between nonlinear optical and photocurrent techniques can be explored.³⁵ For example, transient absorption spectroscopies can be conducted with wavelength-integrated or dispersed signal detection. Dispersing the probe pulse onto an array detector facilitates signal interpretations by distinguishing contributions from separate resonances. Similarly, NLPC experiments can be conducted in a dispersed detection mode using an oscilloscope with a sufficiently broad bandwidth (i.e., the signal is processed with a preamplifier in the present approach).²⁵ Obtaining the temporal profile of the photocurrent will provide additional information regarding carrier drift by adding a dimension to the experiment (e.g., correlations between drift velocities in separate time intervals with a minimal increase in data acquisition times). Hybrid resonance Raman/NLPC spectroscopies may also be useful for identifying vibrational promoting modes in photoinduced electron transfer reactions.^84–86 We hypothesize that vibrational resonances imprinted on the photocurrent will be understood to have practical significances for electron transport.

Z.O. and N.Z. contributed equally to this work.

This work was supported by the National Science Foundation under Grant No. CHE-1763207 (N.Z., Z.O., M.G.M., and A.M.M.). L.Y. and W.Y. were supported by the Center for Hybrid Organic Inorganic Semiconductors for Energy (CHOISE), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES). The authors thank Dr. Yi Yao for helpful discussions.

The data that support the findings of this study are available from the corresponding author upon reasonable request.