Probing drift velocity dispersion in MAPbI₃ photovoltaic cells with nonlinear photocurrent spectroscopy

Knowledge of transport mechanisms facilitates the development of materials for solar energy applications. Conventional transient absorption spectroscopies are commonly employed for determining the rates of short-range energy and charge transfer dynamics,^1–6 whereas long-range carrier motions can be imaged with time-resolved optical microscopies.^7–11 Although valuable insights are derived from optical spectroscopies, the understanding of a device’s operation mechanisms can be limited in ex-situ measurements. For example, the signals measured in a transient absorption experiment reflect the concentrations and extinction coefficients of all photoexcited species regardless of functional relevance. Elimination of such ambiguities has motivated the development of “action spectroscopy” techniques in which the response of a photovoltaic device to a sequence of laser pulses is directly detected.^12–22 Early versions of photocurrent action spectroscopies resembled (two-pulse) pump–probe experiments,^12–16 whereas (four-pulse) two-dimensional Fourier transform spectroscopies have been applied to a variety of systems in recent years.^17–20,22 With inspiration from this work, we have developed a related technique capable of determining charge carrier velocities in photovoltaic cells with time resolution that is several orders of magnitude shorter than that of a conventional time-of-flight (TOF) method (i.e., 0.1 ns vs 0.1 µs timescales).^23–25 While our initial applications involved a complex optical setup and femtosecond laser system, a simpler technical approach that will be accessible to non-specialists is motivated by the practical knowledge of carrier mobilities in photovoltaic devices.

Conventional TOF methods were initially developed for applications to low-mobility solids.^26–31 In this class of techniques, carrier drift is initiated when a single laser pulse is absorbed by the active layer of a photovoltaic cell. Charge carriers then traverse the active layer in response to the bias applied to the device while undergoing numerous cycles of capture and release at trap sites.^29,32–34 Trap-induced dispersion of carrier drift velocities is reflected by the temporal profile of the photocurrent detected at an acceptor electrode. Because the time resolution of a conventional TOF method is limited by the RC time constant of a device, which is on the order of 0.1–1 µs for the MAPbI₃ systems targeted in the present work,^35–37 applications to photovoltaic cells with active layer thicknesses of 100’s of nm are limited by time resolution. For example, MAPbI₃ devices with 100 nm thick active layers have carrier transit times on the order of 10 ns when the electric field is ∼10 V/μm (i.e., magnitude of potential equal to 1.0 V).^24,25 Thus, the time resolution of a conventional TOF technique must be improved by roughly 1000 times to resolve carrier transit in such thin-film devices.

Our two-pulse TOF method, termed nonlinear photocurrent spectroscopy (NLPC), achieves time resolution on the order of picoseconds by applying a pair of laser pulses to a photovoltaic cell.^23–25 In this approach, the first laser pulse initiates carrier drift in the active layer of the device, whereas a second laser pulse is applied with an experimentally controlled delay time to resolve drift velocities. When the delay time is shorter than the carrier transit time, the total amount of charge collected from the device saturates due to recombination processes involving carriers photoexcited by separate laser pulses. The technique yields TOF information because the nonlinearity in the photocurrent vanishes when carriers photoexcited by the first laser pulse transfer into the electrodes. The full capabilities of the method were leveraged with applications to mixtures of layered perovskite quantum wells that exhibit exciton resonances spanning the 500–700 nm spectral range.^23–25 Because the wavelengths of the two laser pulses are independently tunable, transient occupation of the various quantum wells was resolved during carrier transit by acquiring two-dimensional NLPC spectra of layered perovskite systems. Although the nonlinearity originates in spontaneous recombination processes, the spectroscopic signatures resemble those of two-dimensional Fourier transform experiments applied to photosynthetic light-harvesting complexes.^38–43 

In the present work, we describe an experimental approach for conducting NLPC spectroscopy with synchronized 40 ps diode lasers. Development of this instrument is motivated by overall ease of use and applicability to devices with arbitrary active layer thicknesses. Our earlier NLPC measurements were limited to photovoltaic cells with transit times of 15 ns because the delay time between laser pulses was controlled with a motorized translation stage.^23–25 In contrast, delay times up to 100 µs are readily introduced with electronics at the 5 kHz repetition rates employed in the experiments presented below. The capabilities of this instrument are leveraged to understand how velocity dispersion influences the length scale on which carrier drift is probed in MAPbI₃ films. Model calculations demonstrate that NLPC spectroscopy is sensitive to such dispersive effects because the nonlinear response of the photocurrent weakens as carrier densities broaden during transit.²⁵ This aspect of the signal generation mechanism is explored with measurements conducted on MAPbI₃ photovoltaic devices possessing a range of thicknesses in both co-propagating and counter-propagating laser beam geometries to probe carrier drift near opposing electrodes. Together, our calculations and experimental measurements are used to establish the length scale of drift velocity dispersion in the MAPbI₃ photovoltaic cells.

CH₃(CH₂)₃NH₃I (BAI) was prepared by combining n-butyl-amine in ethanol (1:1 by volume) with hydriodic acid (HI) (57 wt. % in water without stabilizer) at 0 °C in an ice water bath for 1 h. To obtain the crude product, the solvent was slowly evaporated under reduced pressure at 60 °C for 1 h. The resulting white powder was recrystallized in ethanol and washed with diethyl ether three times before drying it in a vacuum oven at 60 °C overnight. The powder was then transferred into a glove box filled with nitrogen gas for future use. CH₃NH₃I (MAI) was synthesized by combining a methylamine solution (40 wt. % in H₂O) with hydriodic acid and 57 wt. % in water without stabilizer).

Glass substrates patterned with indium doped tin oxide (ITO) were purchased from Thin Film Devices, Inc., with a sheet resistance of 20 Ω/square. Prior to use, the substrates were cleaned with an ultrasonic bath using deionized water, acetone, and 2-propanol (15 min for each solvent in sequence). The substrates were treated with UV–ozone for 15 min and dried under a stream of nitrogen gas before being transferred into a glove box filled with a nitrogen atmosphere. We then spin-coated a solution of PTAA [poly(triaryl amine) from Sigma-Aldrich] in toluene (2 mg/ml) onto cleaned ITO substrates at 4000 rpm for 30 s and baked at 100 °C for 10 min. The perovskite precursor solution was spun-cast on the PTAA-coated substrate after cooling down to room temperature.

To fabricate MAPbI₃ perovskite solar cells, perovskite precursor solutions were prepared by dissolving PbI₂ and MAI in a 9:1 mixture of N, N-Dimethylformamide (DMF) and Dimethyl Sulfoxide. The concentrations of Pb²⁺ required to produce 90, 240, and 460 nm thick films are 0.4, 0.8, and 1.3M, respectively. The MAPbI₃ precursor solution was spin-coated onto a PTAA-coated substrate (pre-wet by spin coating pure DMF on top at 2000 rpm for 3 s twice) at 2000 rpm for 2 s and 4000 rpm for 20 s. The sample was then drop-cast with 0.3 ml toluene at 8 s for a second step of spin-coating in a glove box filled with nitrogen gas. The sample was annealed at 65 °C for 10 min and 100 °C for 10 min. The device fabrication process was completed by thermally evaporating 40 nm of C₆₀, 4 nm of BCP (Bathocuproine), and 23 nm of copper at a base pressure of 3 × 10⁻⁷ Torr with evaporation rates of 0.1–0.3, 0.1, and 0.1–0.2 Å/s, respectively. The same C₆₀, BCP and copper layers were also thermally evaporated onto precleaned glass slides to determine a transmittance of ∼10% at 400 nm for the semi-transparent electrode (see the supplementary material). The active area of 0.13 cm² was controlled by a shadow mask.

The photovoltaic cells considered in this work are grouped according to the targeted active layer thicknesses of 90, 240, and 460 nm. SEM measurements and linear absorbance data were used to confirm the active layer thicknesses with an estimated uncertainty of ∼15%. Because accurate absorbance coefficients for MAPbI₃ films are known,⁴⁴ we find that linear absorbance is generally more reliable than SEM characterization for determining the film thicknesses because it does not require destruction of the photovoltaic cells. Moreover, variation in the NLPC mobilities determined for the devices primarily originates from qualities of the electrode contacts (i.e., not from the ∼15% uncertainties in the active layer thicknesses).^23–25 For this reason, the averaged mobilities reported in this work include measurements conducted on various devices and with multiple electrodes on individual devices.

Device characterization was carried out under AM 1.5G irradiation with an intensity of 100 mW/cm² (Oriel 91160, 300 W) calibrated by a NREL certified standard silicon cell. Current density vs voltage (J-V) curves were recorded with a Keithley 2400 digital source meter. The scan rates were 0.05 V/s.

The NLPC instrument employed in the present work is based on three PicoQuant diode lasers producing 400 nm, 40 ps laser pulses with energies of 100 pJ (LDH-P-C-405). The repetition rates of the diode modules range from single shot to 80 MHz; however, the current amplifier used for signal detection (Stanford Research Systems 570) does not fully recover between laser shots at repetition rates greater than 10 kHz. For this reason, all experiments are conducted at 5 kHz repetition rates to establish an accurate baseline. In principle, charge carriers can accumulate in the active layers of photovoltaic cells if the 200 µs time interval between laser shots is shorter than the amount of time required to recover the initial conditions. We find that all devices investigated in this work are fully refreshed within ∼100–200 ns after photoexcitation. Therefore, the bandwidth of the current amplifier limits the repetition rate of the laser system. The three 400 nm diode lasers are operated with a single Picoquant laser driver module (SEPIA II SLM 828) housed in a mainframe (PDL 828-L SEPIA II). The delay times between laser pulses produced by separate diode modules are varied with a minimum step size of 24 ps using a PicoQuant oscillator module (SEPIA II SOM 828-D).

The layout of the NLPC instrument is summarized in Fig. 1. Laser beams generated by the three pulsed diode lasers, which are mounted on a common optical table, are directed over ∼2.2 m distances to the sample with silver-coated mirrors. Fluences of 1 µJ/cm² are produced at the sample position by focusing 65 pJ pulses to FWHM spot sizes of 90 µm using singlet lenses with 30 cm focal lengths. Here, the circular area of the laser spot is set equal to 45 µm to compute the laser fluence (i.e., half of the FWHM). Because the relative phases of the pulse trains are controlled electronically, it is not necessary to precisely match the distances between the diode lasers and the photovoltaic cell.

As shown in Fig. 1, the nonlinear response of the photocurrent is obtained by acquiring signals with a sequence of four conditions: pulses 1 and 2 [$S1+2τ$], pulse 1 only (S₁), pulse 2 only (S₂), and both pulses blocked (S₀).^23–25,45 Carrier drift is initiated at t′ = 0 when the first laser pulse arrives at the photovoltaic device, whereas the amount of time elapsed after arrival of the first pulse is decomposed into an experimentally controlled delay time, τ, and current integration time, t.^24,25 The individual signal components are acquired sequentially; S₁₊₂ is acquired before S₁, etc. To cycle through the four conditions, the individual laser beams are turned on and off using LabVIEW software interfaced with the PicoQuant control system. This cycle of four conditions is repeated 15 times, and the signals are averaged to optimize the data quality. The NLPC signal is defined as $SNLPCτ=S1+2τ−S1−S2+S0$ to reflect the component of saturation induced by recombination processes involving carriers photoexcited by separate laser pulses. The S₁ and S₂ conditions are independent of the delay time because a single laser pulse photoexcites carriers in the active layer, whereas a transient saturation effect is produced when the photocurrent is detected with both pulses present, $S1+2τ$⁠.

The total amount of charge collected from a device is determined by time-integrating signals acquired with a Stanford Research Systems 570 current amplifier. The voltage output of the current amplifier scales linearly with the photocurrent input; however, the signal pulses are broadened to ∼20 µs with the 2 µA/V sensitivity and gain mode (low noise) settings employed in the present measurements. Signals are processed with a National Instruments data acquisition board (NI USB-6341), which yields data points with 4 µs intervals when operated at 250 kHz. The total amount of charge collected from a device can be computed by multiplying the time-integrated voltage output by the current-to-voltage amplification factor. As in earlier work,^24,25 the external bias is cycled over a 0.4 V range with step sizes of 0.1 V for devices with active layer thicknesses of 90 and 240 nm. The bias is cycled over a 0.2 V range with 0.05 V steps in experiments conducted with 460 nm thick active layers. The total bias applied across the active layer is given by the sum of the variable external and constant internal biases.

NLPC signal generation mechanisms were explored with a numerical model developed for a co-propagating laser beam geometry in previous work.^24,25 In this section, we further characterize the effects of drift velocity dispersion on NLPC measurements with an approach that is based on empirical parameters. In addition, the model is extended to compare signals acquired with co-propagating and counter-propagating laser beams.

Models that incorporate multiple trapping events have successfully accounted for drift velocity dispersion in conventional TOF measurements.^{28,29,32,34,46–48} In this context, “multiple trapping” suggests that carriers must be thermally excited from localized states to reinitiate transport after trapping, whereas alternate “hopping” descriptions involve carrier tunneling between trap sites.^49,50 When the active layer of a photovoltaic device absorbs light at photon energies above the bandgap, the drift velocity is initially given by the product of the electric field and “trap-free” carrier mobility; however, the effective mobility subsequently decreases as carriers are temporarily immobilized by traps during transit across the active layer. The carrier mobility determined by conventional TOF measurements, $μTOFt$⁠, can be expressed in terms of the intrinsic carrier mobility, μ₀, as^28,29,32

where $Nfreet$ and $Ntrapt$ are time-dependent populations of free and trapped carriers, respectively. Free electrons (holes) undergo long-range drift while occupying spatially extended states with energies above (below) the mobility edge of the conduction (valence) band. In contrast, numerous cycles of capture and release at localized trap states reduce the average drift velocity in addition to broadening the spatial distributions of charge density.

NLPC experiments previously conducted on solution-processed perovskite films are consistent with the predictions of conventional TOF models;^23–25 however, NLPC measurements are specially equipped to reveal “instantaneous” carrier mobilities with picosecond time resolution by cycling the external bias applied to a photovoltaic cell.²⁵ To understand the dynamics observed by NLPC spectroscopy, it is instructive to consider that the electric field applied to a 100 nm thick active layer is ∼10 V/μm based on the ratio between the magnitude of the 1 V internal bias and active layer thickness.²⁵ Therefore, the potential energy varies by roughly 10 meV on the length scale of a defect with a size of 1 nm, which is much less than the 150–500 meV depths of traps found in perovskite films.^1,51,52 For illustration, consider that cycling the external bias in the range of −0.2 to 0.2 V with a 100 nm thick active layer translates to 8–12 V/µm electric field magnitudes. Because the magnitudes of the local potential energy differences are small compared to k_BT at ambient temperatures, the drift velocity determined for the full ensemble of carriers decreases as the population of trapped carriers increases [see Eq. (1) and Fig. 2].

As in conventional TOF measurements,³⁰ the effective transit time, T_eff, determined by NLPC spectroscopy can be divided into intervals dominated by “free” and “trapped” charge carriers,^24,53

The initial and terminal phases of carrier drift are described by T_free and T_trap because the populations of trap states increase with the amount of time elapsed after photoexcitation. Under the assumption that carrier drift exhibits these two phases, we decompose the effective transit time into free (initial) and trap-limited (terminal) mobilities, μ_free and μ_trap, using

where l_trap is the length scale of trap-induced velocity dispersion, d is the active layer thickness, and E is the electric field applied to the active layer. The drift velocity is then given by

To relate this formula to our experimental procedure, we next consider the difference in drift velocities, Δv_drift, measured if the electric field applied to a device is cycled between values of E₋ = E_int − E_ext and E₊ = E_int + E_ext,^23–25 

In a sufficiently thin device, d = l_trap and the measurement directly reflects the intrinsic free-carrier mobility via Δv_drift = 2μ_freeE_ext; however, if d ≫ l_trap, the difference in drift velocities is given by Δv_drift = 2μ_trapE_ext. Using Eq. (5), we estimate that the ∼1 m/s detection threshold of our NLPC instrument corresponds to a trap-limited mobility of μ_trap = 2.5 × 10⁻³ cm²/V/s for an external bias with a magnitude of E_ext = 2 V/μm (i.e., external bias of 0.2 V applied to a 100 nm thick active layer). For a complementary perspective, we note that Eq. (2) was combined with the same device parameters in earlier work to show that our Δv_drift ≈ 1 m/s detection threshold suggests trapping of ∼75% of the carriers in the active layer.²⁵ 

With the goal of establishing a model based on experimentally accessible parameters, the distance traversed by carriers in the active layer can be partitioned into short- and long-range components by integrating the drift velocity up to the effective transit time,

The total path length is decomposed into phases associated with free- and trap-limited motions using

under the assumption of exponential relaxation in the carrier mobility,

where t_trap is a phenomenological timescale for carrier accumulation in traps (see Fig. 2). Multiple trapping models typically treat the time-dependent decay of the mobility using a power law;^28,29,32,48 however, exponential temporal profiles are reasonable approximations for the line shapes of the “instantaneous” drift velocities determined with NLPC experiments.²⁵ Finally, the free and trap-limited drift velocities are given by

and

on length scales short and long compared to l_trap. In the limit t_trap ≪ T_eff, the trapping time can be written as

We next outline a numerical model for simulating NLPC signals in co-propagating and counter-propagating geometries. Unlike our earlier approach,²⁵ the present method incorporates drift velocity dispersion and decomposes the total signal into carrier densities associated with separate laser pulses. Given that the first laser pulse, which is incident on the transparent ITO electrode, travels in the direction of positive x′, the carrier density profile generated at t′ = 0 can be written as

where f is the laser fluence (energy/area), ω₁ is the frequency of the first laser pulse, $αω1$ is the absorption coefficient, d is the active layer thickness, and t′ is the total amount of time elapsed after the first laser pulse arrives at the device, t′ = τ + t. The position variable, x′, is equal to 0 and d at the boundaries of the active layer near the ITO and copper electrodes, respectively. The experimentally controlled delay, τ, and detection, t, times are defined in Fig. 1. The carrier densities produced by the second laser pulse are given by

and

for co-propagating and counter-propagating laser beam geometries, respectively.

We have previously distinguished the recombination-induced nonlinearity targeted in NLPC spectroscopy and the fourth-order perturbative responses probed in phase-sensitive multidimensional photocurrent spectroscopies.²⁵ In the fourth-order perturbative response of the photocurrent, the second laser pulse must interact with the subset of carriers photoexcited by the first laser pulse. We estimate that the fourth-order component of the total photocurrent is on the order of the 10⁻⁵–10⁻⁴ transient absorbances of similar perovskite films induced by laser pulses with fluences near 1 µJ/cm²;^54,55 however, the signal generation mechanism associated with the fourth-order response additionally requires that the doubly excited carriers either enhance or suppress the amount of charge extracted from the device, so it is not clear that such a response will be efficiently produced with all materials.^17–20,22 In contrast, the recombination-induced changes in the time-integrated photocurrent measured in NLPC spectroscopy are on the order of 10⁻² with a laser fluence 1 µJ/cm².²⁵ Therefore, we neglect contributions of fourth-order perturbative laser–matter interactions in the model outlined in this section. Our estimations of the relative magnitudes of fourth-order perturbative and recombination-induced nonlinearities agree with earlier experimental work.⁵⁶ Because of its greater magnitude, the recombination-induced nonlinearity leveraged in NLPC spectroscopy is generally seen as a nuisance in multidimensional action spectroscopies targeting fourth-order perturbative responses.^56,57

Dynamics in the carrier densities photoexcited by the first laser pulse, p₁ and n₁, are written as

and

where $vdt′=E⋅μt′$ is the drift velocity computed using the form of $μt′$ given in Eq. (8), t_trap is the timescale of carrier trapping defined by Eq. (11), β is the two-body recombination coefficient, and Δt′ is the temporal step size. We assume that photoexcitation produces only free charge carriers due to the ∼16 meV exciton binding energy determined for MAPbI₃.⁵⁸ Two-body recombination is assumed in this model because transient absorption experiments conducted on our solution-processed perovskite films revealed dominant quadratic dependence of recombination rates on the carrier densities.^54,55,59 Quadratic scaling is consistent with radiative and trap-assisted Auger recombination, which are both accounted for with the β parameter.⁶⁰ Trap-assisted band-to-band recombination scales linearly in the carrier density;^60,61 however, it is fundamentally a two-body process and may represent a significant contribution to β. Similarly, the dynamics in the carrier densities photoexcited by the second laser pulse are described with

and

where the Heaviside step function, $θt′−τ$⁠, is equal to zero before the second pulse arrives at the device. It is not necessary to convolute these expressions for the carrier densities with laser pulse envelopes when the pulse durations are much shorter than the carrier transit times. For example, the 40 ps laser pulses employed in this work are 50–500 times shorter than the carrier transit times measured for the various devices. The S₁₊₂ chopper condition (see Fig. 1) is simulated using Eqs. (12)–(18), which account for interactions between carriers associated with separate laser pulses through the βp₁n₂ and βn₁p₂ terms.

To define the saturation effect targeted in an NLPC experiment, it is also necessary to compute the carrier densities generated when individual laser pulses interact with the device under the S₁ and S₂ conditions. The electron and hole densities photoexcited by pulse k (k = 1, 2) are written as

and

where the tildes denote that single laser pulses are incident on the device. Heterogeneity in the carrier transit times is incorporated using

where k is the index of the laser pulse (1 or 2) and ξ_k is a general carrier density (p_k, $p̃k$⁠, n_k, or $ñk$⁠). The phenomenological broadening parameter, w, is defined as w/Δt′ = $25⋅vdt′$ to prevent unrealistic flat edges from appearing in the temporal profiles of NLPC signals.^23,24,27 Notably, similar sharp features (i.e., flat edges in temporal profiles) also appear in conventional TOF profiles of systems with minimal drift velocity dispersion.²⁷ 

The NLPC signal is obtained by summing over differences in the amounts of charge collected from the photovoltaic cell using

where signal components associated with holes and electrons photoexcited by laser pulse k are given by

and

respectively. The accumulated charges, ΔQ_pk and ΔQ_nk, have units of Coulombs and A is the area of the laser pulse on the device. Finally, it is also convenient to represent the signal magnitude as the ratio in the amount of charge collected from the photovoltaic cell with and without recombination processes involving separate laser pulses,

where $Φω1,τ,ω2$ is proportional to the ratio of experimental signal components, $SNLPC/S1+S2$⁠. Consideration of $Φω1,τ,ω2$ is practical during data acquisition because it can be computed without knowledge of the laser spot sizes and line shape of the amplified photocurrent.

Components of the carrier densities responsible for signal generation are next considered from various perspectives to establish a context for interpreting the experimental measurements presented below. For this reason, we employ parameters that closely approximate our experimental conditions. Calculations are conducted with active layer thicknesses of 90, 240, and 460 nm, whereas the penetration depth, $1/αω$⁠, is set equal to 34 nm for 400 nm light.⁴⁴ The parameters of the mobility function in Eqs. (8) and (11) are set equal to μ_free = 0.025 cm²/V/s, μ_trap = 0.015 cm²/V/s, and l_trap = 60 nm for consistency with the present measurements and earlier work on similar systems.²⁵ The drift velocities and trapping times are evaluated under the assumption of a uniform electric field E = V/d, where V is the total bias and d is the active layer thickness. Parameters of the model are summarized in Table I.

As shown in Eqs. (23) and (24), contributions to NLPC signals associated with holes and electrons are computed by integrating $pk−p̃k$ and $nk−ñk$ at the interfaces with the electrodes. These quantities are plotted at delay times of τ = 3 ns (t′ = τ + t) in Figs. 3 and 4, which correspond to co-propagating and counter-propagating laser beam geometries, respectively. In all figure panels, the initial distributions of the carrier densities on the position axis, x, are determined solely by the light penetration depth of α⁻¹ = 34 nm, where a laser fluence of 1 µJ/cm² has been assumed [see Eqs. (12)–(14)]. The spatial profile of the laser beam, which is uniform in this model, is not represented in Figs. 3 and 4.⁴⁴ The potential is taken to linearly increase from x = 0 (interface near the ITO electrode) to x = d (interface near the copper electrode), and the electric field is computed using the ratio of the potential difference and active layer thickness. Thus, in response to the electric field, the holes and electrons drift downward and upward along the x axis until injecting into their respective electrodes. All calculations display two dynamic effects as the time variable, t′, increases. First, the carrier densities drift to the respective electrodes according to the mobility given in Eq. (8) and the parameters summarized in Table I. Second, the differential carrier densities decrease in magnitude and broaden along the x axis as t′ increases due to carrier trapping and drift velocity dispersion [see Sec. III A and Eqs. (15)–(21)]. These “differential carrier densities,” $pk−p̃k$ and $nk−ñk$⁠, reflect the saturation effect at the origin of the NLPC signal generation mechanism, wherein carriers photoexcited by the first pulse are “probed” by carriers associated with the second pulse. In other words, the differential carrier densities couple the recombination rates, βn₁p₂ and βp₁n₂, and drift velocities of the individual carrier densities.

In Fig. 3, differential carrier densities calculated with a 90 nm thick active layer and co-propagating beam geometry show that recombination processes are localized near the ITO (⁠$p1−p̃1$⁠) and copper (⁠$n1−ñ1$⁠) electrodes because the n₁ carrier density drifts over more than half of the active layer thickness during the 3 ns delay time. As the active layer thickness increases, the carrier distributions broaden, and the drift velocities decrease due to reductions in the electric fields. In addition, as d increases, the peak value of $p1−p̃1$ grows relative to that of $n1−ñ1$ because the amount of overlap between n₁ and p₂ decreases during the experimentally controlled delay time, thereby reducing the yield of the βn₁p₂ recombination process. For example, the maxima of $n1−ñ1$ and $p1−p̃1$ differ by factors of 0.8, 0.6, and 0.4 at τ = 3 ns for active layer thicknesses of 90, 240, and 460 nm, respectively. Because the probability of trapping increases with the distance traversed by a carrier density, the efficiency with which electrons reach the copper electrode decreases as the active layer thickness increases. This aspect of carrier trapping is evidenced by the differences in the temporal widths of $pk−p̃k$ and $nk−ñk$ at x = 0 and x = d, respectively. In effect, the abilities of holes and electrons to reach their respective electrodes are governed by the ratios, $ltrapαω$ (holes) and l_trap/d (electrons), under co-propagating conditions. Holes always traverse a distance equal to the optical penetration depth [$ltrapαω>1$ for the present systems], whereas electrons must drift over the full thickness of the active layer (l_trap/d < 1 for the present systems).

In the counter-propagating laser beam geometry simulated in Fig. 4, the first and second laser pulses enter the active layer through the ITO and copper electrodes at x = 0 and x = d, respectively. Similar behaviors are computed for the differential carrier densities in both laser beam geometries when the active layer thickness is 90 nm; however, the recombination processes have lower rates under counter-propagating conditions because only 10% of the second laser pulse is transmitted through the copper electrode. Unlike the co-propagating beam geometry, contributions from the βn₁p₂ terms outweigh those of the βp₁n₂ terms. For example, the peak magnitudes of $n1−ñ1$ and $p1−p̃1$⁠, differ by factors of 2.8, 3.4, and 3.0 for active layer thicknesses of 90, 240, and 460 nm, respectively. The $n2−ñ2$ and $p2−p̃2$ densities exhibit similar discrepancies because of the two-body nature of the recombination mechanism. The relative efficiencies of the βn₁p₂ and βp₁n₂ recombination processes reflect two competing effects. First, the magnitudes of the βp₁n₂ terms decrease with the active layer thickness because the laser pulses penetrate only $1/αω$ = 34 nm into active layer near the ITO electrode, which results in poor overlap between p₁ and n₂ at all times. Second, the probabilities that the n₁ and p₂ carrier densities trap in the active layer during transit increase with the active layer thickness, which reduces the yield of the βn₁p₂ recombination process. Because of these competing factors, the ratio in peak densities computed for $nk−ñk$ and $pk−p̃k$ exhibits a maximum value of 3.4 at the intermediate active layer thickness of 240 nm.

The total amount of charge collected from a photovoltaic device is decomposed into contributions from the four classes of charge carriers in Fig. 5. The integrated charges, ΔQ_j, are related to the differential carrier densities, $pk−p̃k$ and $nk−ñk$⁠, by Eqs. (23) and (24). Calculations conducted with the co-propagating beam geometry show that contributions from the four classes of carriers to the total NLPC signals are relatively similar for all active layer thicknesses. In contrast, the signal component corresponding to the βn₁p₂ recombination mechanism outweighs that associated with βp₁n₂ for all systems under counter-propagating conditions, which is consistent with the peak values of differential carrier densities displayed in Fig. 4. Due to poor spatial overlap between the p₁ and n₂ carrier densities, the ΔQ_n1 and ΔQ_p2 signal components dominate the nonlinear response if the active layer thickness exceeds the light penetration depth with counter-propagating beams. Relative to ΔQ_p2, contributions from ΔQ_n1 increase with the active layer thickness because the n₁ carrier density “flattens” during the experimentally controlled delay time as a result of trapping and drift velocity dispersion. In Fig. 4, trapping dynamics cause the differential carrier densities, $n1−ñ1$ and $p2−p̃2$⁠, to localize near the copper electrodes in devices with 240 and 460 nm thick active layers.

The delay-dependent overlaps in carrier densities, which contribute to the βp₁n₂ and βn₁p₂ recombination processes, are illustrated with schematics in Figs. 6(a) and 6(b), where the drift velocity is taken to be constant to maintain a simple physical picture. The directions of carrier drift are indicated with arrows, and the shades reflect the concentration profiles associated with light absorption in the active layer (i.e., darker shades represent greater carrier densities). In the co-propagating geometry, the βp₁n₂ and βn₁p₂ terms contribute with nearly equal weights to NLPC signals regardless of the active layer thickness because the pairs of carrier densities involved in the recombination processes exhibit similar spatial overlaps at all delay times. The schematics show that the p₂ and n₂ carrier densities penetrate a distance of $1/αω$ into the active layer near the ITO electrode when the second laser pulse arrives at the device. Therefore, although the p₁ and n₁ densities drift in opposite directions as τ increases, these two carrier densities possess similar overlaps in the βp₁n₂ and βn₁p₂ terms. In the counter-propagating geometry, the βp₁n₂ term produces a smaller contribution when $αωd>1$ because the p₁ and n₂ carrier densities are never well-overlapped. In contrast, the n₁ and p₂ carrier densities drift towards each other and recombine throughout the full thickness of the active layer under counter-propagating conditions in Fig. 5.

Figure 6(c) illustrates an important aspect of the NLPC signal generation mechanism that is not represented in Figs. 6(a) and 6(b). In counter-propagating beam geometry, the n₁ carrier density is “probed” by the p₂ carrier density while traversing the full thickness of the active layer, whereas the length scale of the NLPC probing process is limited by the light penetration depth in the co-propagating beam geometry. Therefore, the distance over which trapped carriers and drift velocity dispersion accumulate is greater when counter-propagating beams are employed. In measurements conducted with the goal of obtaining TOF information, it must be recognized that NLPC signals decay with τ for two reasons under counter-propagating conditions: (i) the n₁ carrier density arrives at the copper electrode and (ii) the nonlinear response associated with βn₁p₂ recombination terms weakens due to flattening of the n₁ carrier density.

In summary, the calculations presented in this section suggest that the p₁ and n₁ carrier densities contribute in similar proportions to NLPC signals with the co-propagating laser beam geometry. Because the four classes of carriers have the same initial conditions, the light penetration depth into the active layer is the relevant length scale for TOF measurements. In contrast, the subset of carriers responsible for the nonlinear response under counter-propagating conditions, n₁ and p₂, travel in opposite directions after photoexcitation near separate electrodes. For systems in which $αωd>1$⁠, trapped carriers and drift velocity dispersion accumulate while n₁ and p₂ traverse the full thickness of the active layer, which reduces the magnitudes of the βn₁p₂ terms as a function of τ [see Fig. 6(c)]. As a practical matter, broadening of the n₁ carrier density shortens the decay times of NLPC signals and must be accounted for in the analysis of experimental data. For this reason, we suggest an empirical approach for estimating the relevant length scale of carrier drift for the counter-propagating beam geometry in Sec. IV.

In this section, we present NLPC data acquired for photovoltaic cells with 90, 240, and 460 nm active layer thicknesses in both co-propagating and counter-propagating laser beam geometries. Although mobilities have been determined for MAPbI₃ films by alternate methods,^7,37,62–66 the present experiments complement this earlier work by probing drift velocity dispersion from a unique perspective. Moreover, establishing general physical insights into NLPC signal generation mechanisms will facilitate applications of the technique. Averaged data are summarized below because experiments must be conducted on multiple devices (and with multiple electrodes on a given device) to meaningfully characterize behaviors for each active layer thickness and beam geometry. All individual datasets (i.e., a total of 100 NLPC decay curves) are summarized in the supplementary material.

In Fig. 7, co-propagating NLPC signals are represented with respect to saturation percentages and the total amounts of charge collected from the devices. Signals acquired using the smallest, intermediate, and largest values of the total biases demonstrate the effects of the applied electric fields on the NLPC decay profiles. For each system, the electric fields are calculated using the ratio of the total bias and active layer thickness, E = V/d. As in earlier works,^23,24 TOF information is obtained by fitting the decay curves to sums of exponential functions,

Weighted averages of time constants are then computed using

The general behaviors exhibited by the co-propagating signals in Fig. 7 resemble those observed in our earlier experiments.^23–25 For a given active layer thickness, the NLPC decay times shorten as the magnitudes of the electric fields and drift velocities increase. The NLPC signals decay more slowly as the active layer thickness, d, increases; however, this trend does not reflect transit across the full thickness of the active layer under co-propagating conditions, where the 34 nm penetration depth of the incident light [i.e., $α−1ω$] is always the relevant length scale. Rather, the decay times increase with the active layer thicknesses because the drift velocities scale linearly with the applied electric field, E = V/d. For example, Fig. 6(a) shows that the p₁ and n₁ carrier densities drift away from the illuminated region of the active layer where they are subsequently “probed” by p₂ and n₂.

NLPC signals acquired under counter-propagating conditions are presented in Fig. 8. For each active layer thickness, the NLPC decay times shorten as the magnitudes of the electric fields and carrier drift velocities increase. Compared to the co-propagating data, the timescales on which the NLPC signals decay are more sensitive to the active layer thickness under counter-propagating conditions. For example, the decay times measured in the second column of Fig. 8 are 2.6, 10.8, and 17.8 ns for active layer thicknesses of 90, 240, and 460 nm, respectively. In contrast, the co-propagating decay times derived from the second column of Fig. 7 are 2.5, 6.5, and 10.1 ns for the 90, 240, and 460 nm active layer thicknesses, respectively. The carrier drift velocities must be the same for the two laser beam geometries because the applied electric fields have the same magnitudes; however, the relevant length scales for carrier drift differ when the active layer thickness is greater than the light penetration depth. As illustrated in Fig. 6, the counter-propagating path lengths are sensitive to the total thickness of the active layer and the extent to which the carrier density flattens during the experimentally controlled delay time, whereas the light penetration depth is always the relevant length scale with co-propagating beams.

Like our earlier works,^23–25 TOF information is extracted from the experimental data by computing carrier drift velocities. Under co-propagating conditions, the drift velocities summarized in Fig. 9 are calculated using $vdrift=α−1ω⋅Tav−1$⁠, where $αω$ is the absorbance coefficient and T_av is the weighted average of time constants determined by fitting the NLPC decay curves. The averaged velocities and uncertainty ranges in Fig. 9 correspond to measurements conducted on different devices and with different electrodes on individual devices (see the supplementary material). The carrier mobilities obtained by fitting these data are summarized in Table II. The mobility computed for the 90 nm thick active layer is equal to 0.013 cm²/V/s, whereas the mobilities associated with the 240 and 460 nm devices are 0.023 and 0.025 cm²/V/s. The increase in the carrier mobility with the active layer thickness is consistent with behaviors established for MAPbI₃ devices, wherein the mobility maximizes near an active layer thickness of 300 nm due to a combination of crystallinity and structural orientation.⁶⁵ For example, using a technique based on a field-effect transistor, mobilities of ∼2 cm²/V/s were determined for film thicknesses near 240 and 460 nm, whereas a mobility near 1 cm²/V/s was found with a film thickness near 90 nm.⁶⁵ While these values are about a factor of 100 larger than those obtained with NLPC spectroscopy, we note that mobilities measured using different samples and techniques can vary by orders of magnitude.⁶⁴ For example, a mobility of 0.012 cm²/V/s was determined for MAPbI₃ using the steady-state space charge-limited current produced by a device.⁶⁶ In addition, the present measurements are in good agreement with the mobilities of 0.016–0.026 cm²/V/s measured for similar MAPbI₃ photovoltaic cells using an NLPC instrument employing femtosecond laser pulses.^23,24

Determining carrier mobilities with the counter-propagating beam geometry is complicated by contributions of both the transit times and drift velocity dispersion to T_av [see Fig. 6(c)]. As expected for carrier drift, our data show that $Tav−1$ scales linearly with the applied electric field for all active layer thicknesses under counter-propagating conditions. Therefore, the challenge is to determine the effective length scale over which the carrier density has traveled when the signal decays to 1/e of its initial value. Because the drift velocity must be independent of the laser beam geometry, we calculate the effective length scale of carrier drift under counter-propagating conditions empirically using

where T_ctr and T_co are the values of T_av obtained under the intermediate bias conditions displayed in the middle columns of Figs. 7 and 8. The two key assumptions made in Eq. (28) are (i) the length scale of transit is well-defined with co-propagation [see Fig. 6(a)] and (ii) the drift velocities are independent of the beam geometry due to equivalent electric field magnitudes. Using this approach, the mobilities obtained for the three active layer thickness are essentially identical for the two beam geometries.

The values of l_ctr, which are summarized in Table III, suggest a characteristic length scale for the localized states that induce drift velocity dispersion. Although the specific nature of the dispersion mechanism is not clear, the trap densities determined for band-to-band recombination processes, for which 10¹⁵–10¹⁶ cm⁻³ is a reasonable estimate,^67,68 can be considered as a potential source of dispersion. Because trap densities of 10¹⁵–10¹⁶ cm⁻³ correspond to an average distance of ∼100 nm between defects, the frequency of encounters cannot be high on the 60 nm length scale of l_ctr. Therefore, our data suggest that drift velocity dispersion may originate from different aspects of the structure, such as short-range thermal fluctuations.^69–71 Density functional theory calculations conducted on MAPbI₃ predict that thermal fluctuations generate partially localized states on the length scale of 5–10 nm, which suggests that numerous encounters are plausible over a path length of 60 nm.⁷² Moreover, the local potential energy differences associated with our applied biases, which are less than k_BT on the 1 nm length scale (see Fig. 2), are also consistent with contributions of thermal disorder. While thermal fluctuations may play an important role, it is likely that drift velocity dispersion originates from a combination of factors, including thermal disorder, trap sites, grain boundaries, and structural defects.

In the present work, we have leveraged a new experimental approach for conducting NLPC experiments to determine carrier mobilities and characteristic length scales for drift velocity dispersion in photovoltaic cells with MAPbI₃ active layers. Operation of our former NLPC setup, which incorporated a femtosecond laser system, required specialized knowledge of optics and ultrafast spectroscopy.^23–25 In addition, studies of photovoltaic devices were limited to a time window of 15 ns because the delay time between laser pulses was controlled with a motorized translation stage. We have simplified the experimental design in this work using synchronized picosecond diode lasers (see Fig. 1). The present setup involves only a few mirrors and lenses, possesses superior laser stability, and enables facile control of delay times up to 100 µs at a 5 kHz repetition rate. Although the present setup involves fixed 400 nm wavelengths, this is not a fundamental limitation for an instrument that employs picosecond diode lasers because continuum sources are commercially available. With color tunability, NLPC spectra can additionally reveal transient occupation of separate phases of an active layer when the two spectral dimensions are scanned (e.g., mixtures of layered perovskite quantum wells).^24,45

Our measurements demonstrate that the sensitivity of an NLPC experiment to drift velocity dispersion increases with the distance between the regions of the active layer illuminated by the two laser pulses. Therefore, in addition to determinations of carrier mobilities, this aspect of the nonlinear response can be exploited to establish the length scale on which the carrier densities broaden during transit. Our NLPC measurements demonstrate several aspects of carrier transport in MAPbI₃ photovoltaic devices. First, we find the smallest mobilities in devices with 90 nm thick active layers (see Fig. 9). This observation is consistent with an established effect in which the MAPbI₃ crystal grain sizes and orientations are poorly optimized with active layer thicknesses less than 200 nm.⁶⁵ Second, we find mobilities on the order of 0.025 cm²/V/s with closer-to-optimal active layer thicknesses of 240 and 460 nm. Third, we estimate that the length scale of drift velocity dispersion, l_ctr, is ∼60 nm based on measurements conducted with different laser beam geometries. The magnitude of l_ctr is consistent with contributions of thermal disorder in the perovskite structure that has been shown to generate partially localized states on the 5–10 nm length scale.⁷² 

See the supplementary material for transmission spectrum for a semitransparent electrode and the 100 NLPC decay profiles accounted for in the averaged mobilities presented in Fig. 9.

This work was supported by the National Science Foundation under Grant No. CHE-1763207 (Z.O. 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). This work was performed, in part, at the Chapel Hill Analytical and Nanofabrication Laboratory, CHANL, a member of the North Carolina Research Triangle Nanotechnology Network, RTNN, which is supported by the National Science Foundation, Grant No. ECCS-2025064, as part of the National Nanotechnology Coordinated Infrastructure, NNCI.

The authors have no conflicts to disclose.

Zhenyu Ouyang: Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Liang Yan: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (supporting). Wei You: Data curation (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (supporting). Andrew M. Moran: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – original draft (equal).

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