Attorney Docket 4210.0526WO TITLE METHODS, DEVICES AND SYSTEMS FOR NONLINEAR OPTICAL DETERMINATION OF ELECTRON MOBILITIES IN SOLAR CELLS CROSS-REFERENCE TO RELATED APPLICATION [0001] This application claims priority to U.S. Provisional Patent Application Serial No.63/405,024, filed on September 9, 2022, the disclosure of which is incorporated herein by reference in its entirety. STATEMENT OF GOVERNMENT INTEREST [0002] This invention was made with government support under Grant No. CHE-1763207 awarded by the National Science Foundation and Grant No. DE-AC36-08GO28308 awarded by the Department of Energy. The government has certain rights in the invention. BACKGROUND [0003] 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, whereas long-range carrier motions can be imaged with time-resolved optical microscopies. 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 without any regard to the functional relevance thereof. 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. Early versions of photocurrent action spectroscopies resembled pump-probe (i.e., two-pulse) experiments, whereas two-dimensional Fourier transform spectroscopies (i.e., four-pulse) have been applied to a variety of systems in recent years. [0004] Conventional time-of-flight (TOF) methods were initially developed for applications to low-mobility solids. In this class of techniques, carrier drift Attorney Docket 4210.0526WO 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. 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 microseconds (μs) for the MAPbI ₃ systems that are used in the present disclosure, applications to photovoltaic cells with active layer thicknesses of 100’s of nm have, until now, been limited by time resolution. For example, MAPbI3 devices with 100 nanometer (nm) thick active layers have carrier transit times on the order of 10 nanoseconds (ns) when the electric field is ~10 Volts per micrometer (V/μm) (e.g., magnitude of potential equal to about 1.0 V). 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. [0005] With these deficiencies of such conventional TOF methods in mind, a related technique is disclosed herein, in which charge carrier velocities in photovoltaic cells can be determined with a time resolution that is multiple orders of magnitude shorter than can possibly be achieved in a conventional TOF method (i.e., picosecond vs. microsecond timescales). A system is disclosed herein as well for use in performing such novel TOF methods. SUMMARY [0006] A system for performing non-linear photocurrent (NLPC) spectroscopy on a photovoltaic cell comprises the photovoltaic cell comprising a transparent indium doped tin oxide (ITO) electrode and a copper electrode, first and second picosecond lasers, wherein the first picosecond laser is configured to emit a first laser beam therefrom and the second picosecond laser is configured to emit a second laser beam therefrom, a first lens configured to focus the first laser beam to have a specified spot size on the ITO electrode of the photovoltaic cell, a second lens configured to focus the second laser beam to have a specified spot size on the ITO electrode of the Attorney Docket 4210.0526WO photovoltaic cell or on the copper electrode of the photovoltaic cell, and a controller configured to control a time delay between emission of the first laser beam from the first picosecond laser and emission of the second laser beam from the second picosecond laser. [0007] A method for performing non-linear photocurrent (NLPC) spectroscopy on a photovoltaic cell comprises providing first and second picosecond lasers, arranging a first lens at a first position to focus a first laser beam emitted from the first picosecond laser to have a specified spot size on a transparent indium doped tin oxide (ITO) electrode of the photovoltaic cell, arranging a second lens at a second position to focus a second laser beam emitted from the second picosecond laser to have a specified spot size on the ITO electrode of the photovoltaic cell or on a copper electrode of the photovoltaic cell, emitting the first laser beam from the first picosecond laser, applying, using a controller, a time delay after emission of the first laser beam; and emitting, immediately after a duration of the time delay has elapsed, the second laser beam from the second picosecond laser. BRIEF DESCRIPTION OF THE DRAWINGS [0008] FIG. 1 is a schematic illustration of an example embodiment of a system for producing nonlinear photocurrent spectroscopy (NLPC) measurements. [0009] FIGS. 2A-C are graphical representations of examples of laser pulses used for producing NLPC measurements. [0010] FIGS. 3A-B are graphical plots showing the influence of carrier trapping on mobilities determined by NLPC spectroscopy. [0011] FIGS. 4A-L are graphical plots of carrier densities computed for different thickness values of the active layer in a co-propagating laser beam geometry at a delay time of 3 nanoseconds (ns). [0012] FIGS. 5A-L are graphical plots of carrier densities computed for different thickness values of the active layer in a counter-propagating laser beam geometry at a delay time of 3 ns. Attorney Docket 4210.0526WO [0013] FIGS. 6A-C are graphical plots of NLPC signal profiles computed with a co-propagating laser beam geometry for the active layer thickness value noted in each of FIGS.6A-C. [0014] FIGS. 6D-F are graphical plots of NLPC signal profiles computed with a counter-propagating laser beam geometry for the active layer thickness value noted in each of FIGS.6D-F. [0015] FIG. 7A shows schematic illustrations showing key aspects of NLPC signal generation mechanisms, in which carrier densities are represented with the delay time set to be equal to the fractions of the transit time indicated in the respective figure for co-propagating laser beam geometries. [0016] FIG. 7B shows schematic illustrations showing key aspects of NLPC signal generation mechanisms, in which carrier densities are represented with the delay time set to be equal to the fractions of the transit time indicated in the respective figure for counter-propagating laser beam geometries. [0017] FIG.7C shows graphical plots of the relationship between n1 and p2 carrier density vs propagation distance (x) across the thickness of the active layer. [0018] FIGS. 8A-I are graphical plots of NLPC signals measured for the active layer thicknesses indicated in the respective graphical plots in a co- propagating laser beam geometry, as well as the electric fields associated therewith. [0019] FIGS. 9A-I are graphical plots of NLPC signals measured for the active layer thicknesses indicated in the respective graphical plots in a counter-propagating laser beam geometry, as well as the electric fields associated therewith. [0020] FIGS.10A-C are graphical plots showing carrier drift velocities with co-propagating laser beam geometries with the parameters indicated in the respective graphical plots. [0021] FIGS.10D-F are graphical plots showing carrier drift velocities with counter-propagating laser beam geometries with the parameters indicated in the respective graphical plots. Attorney Docket 4210.0526WO [0022] FIG.11 is a table of parameters of effective path length for counter- propagating laser beam geometry conditions. [0023] FIGS. 12A and 12B schematically show aspects of a three- dimensional example embodiment of the system shown in FIG. 1, for performing the novel method of NLPC spectroscopy disclosed herein. DETAILED DESCRIPTION [0024] The presently disclosed subject matter now will be described more fully hereinafter, in which some, but not all embodiments of the presently disclosed subject matter are described. Indeed, the presently disclosed subject matter can be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will satisfy applicable legal requirements. [0025] To address the aforementioned deficiencies of conventional TOF methods, novel analytical TOF methods and systems are disclosed herein. The method disclosed herein is generally referred to as a two-pulse TOF method and is referred to herein generally using the term nonlinear photocurrent spectroscopy (NLPC). Using such methods, it is now possible to achieve time resolutions on the order of picoseconds by applying a pair of laser pulses to a photovoltaic cell. Using this novel approach, the first laser pulse initiates carrier drift in the active layer of the photovoltaic cell, while a second laser pulse is applied at a later time (i.e., delayed by an experimentally controlled and predetermined delay time) to resolve carrier 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 the carriers that are photoexcited by separate laser pulses. This novel analytical technique yields TOF information because the signal vanishes when the carriers that are photoexcited by the first laser pulse transfer into the electrodes. [0026] Further capabilities of the novel TOF method have been applied to mixtures of layered perovskite quantum wells, which exhibit exciton resonances spanning the 500-700 nm spectral range. Because the Attorney Docket 4210.0526WO wavelengths of the two laser pulses are independently selectable or tunable (i.e., can each have different wavelengths), transient occupation of the various quantum wells is disclosed herein as being 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 obtained or generated resemble those of two- dimensional Fourier transform experiments applied to photosynthetic light- harvesting complexes. [0027] Thus, an experimental approach for conducting NLPC spectroscopy with synchronized 40 ps diode lasers is disclosed herein. Development of the system for carrying out the novel TOF methods disclosed herein advantageously provide greater ease of use and demonstrate applicability for photovoltaic cells with arbitrary active layer thicknesses. Earlier work regarding NLPC measurements was required to be limited to photovoltaic cells with transit times of 15 ns because the delay time between laser pulses was controlled with a motorized translation stage. In contrast with such earlier systems and methods, according to the presently disclosed subject matter, delay times of up to about 100 μs are readily introduced with electronics at repetition rates of about 5 kHz employed in the example embodiments disclosed herein. The capabilities of the presently disclosed systems can also be used 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 weakens as carrier densities broaden during transit. 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, the calculations and example embodiments disclosed herein are shown to be useful in establishing the length scale of drift velocity dispersion in such example MAPbI ₃ photovoltaic cells. While the following terms are believed to be well understood by one Attorney Docket 4210.0526WO having ordinary skill in the art, the following definitions are set forth to facilitate explanation of the presently disclosed subject matter. [0028] While the following terms are believed to be well understood by one having ordinary skill in the art, the following definitions are set forth to facilitate explanation of the presently disclosed subject matter. [0029] Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood to one having ordinary skill in the art to which the presently disclosed subject matter belongs. Although, any methods, devices, and materials similar or equivalent to those described herein can be used in the practice or testing of the presently disclosed subject matter, representative methods, devices, and materials are now described. [0030] Following long-standing patent law convention, the terms “a”, “an”, and “the” refer to “one or more” when used in this application, including the claims. Thus, for example, reference to "a vial" can include a plurality of such vials, and so forth. [0031] Unless otherwise indicated, all numbers expressing quantities of length, diameter, width, and so forth used in the specification and claims are to be understood as being modified in all instances by the terms “about” or “approximately”. Accordingly, unless indicated to the contrary, the numerical parameters set forth in this specification and attached claims are approximations that can vary depending upon the desired properties sought to be obtained by the presently disclosed subject matter. [0032] As used herein, the terms “about” and “approximately,” when referring to a value or to a length, width, diameter, temperature, time, volume, concentration, percentage, etc., is meant to encompass variations of in some embodiments ±20%, in some embodiments ±10%, in some embodiments ±5%, in some embodiments ±1%, in some embodiments ±0.5%, and in some embodiments ±0.1% from the specified amount, as such variations are appropriate for the disclosed apparatuses and devices. [0033] The term “comprising”, which is synonymous with “including” “containing” or “characterized by” is inclusive or open-ended and does not exclude additional, unrecited elements or method steps. “Comprising” is a Attorney Docket 4210.0526WO term of art used in claim language which means that the named elements are essential, but other elements can be added and still form a construct within the scope of the claim. [0034] As used herein, the phrase “consisting of” excludes any element, step, or ingredient not specified in the claim. When the phrase “consists of” appears in a clause of the body of a claim, rather than immediately following the preamble, it limits only the element set forth in that clause; other elements are not excluded from the claim as a whole. [0035] As used herein, the phrase “consisting essentially of” limits the scope of a claim to the specified materials or steps, plus those that do not materially affect the basic and novel characteristic(s) of the claimed subject matter. [0036] With respect to the terms “comprising”, “consisting of”, and “consisting essentially of”, where one of these three terms is used herein, the presently disclosed and claimed subject matter can include the use of either of the other two terms. [0037] As used herein, the term “and/or” when used in the context of a listing of entities, refers to the entities being present singly or in combination. Thus, for example, the phrase “A, B, C, and/or D” includes A, B, C, and D individually, but also includes any and all combinations and sub-combinations of A, B, C, and D. [0038] Materials and Device Fabrication [0039] CH ₃ (CH ₂ ) ₃ NH ₃ I (BAI) was prepared by combining n-butylamine 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 hour. To obtain the crude product, the solvent was slowly evaporated under reduced pressure at 60°C for 1 hour. The resulting white powder was then 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 H ₂ O) with hydriodic acid and 57 wt% in water without stabilizer). Attorney Docket 4210.0526WO [0040] Glass substrates patterned 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-proponal (15 minutes 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. [0041] To fabricate MAPbI ₃ perovskite solar cells, perovskite precursor solutions were prepared by dissolving PbI2 and MAI in DMF:DMSO (9:1). The concentrations of Pb ₂ + required to produce films having respective thicknesses of 90, 240, and 460 nm are approximately 0.4, 0.8, and 1.3 M, respectively. The MAPbI ₃ 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 the 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 Å/s, 0.1 Å/s 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 Supplementary Material). The active area of 0.13 cm ² was controlled by a shadow mask. [0042] Characterization of the resultant photovoltaic cells was carried out under AM 1.5G irradiation with an intensity of 100 mW/cm ² (Oriel 91160, 300 W) calibrated by a NREL certified standard silicon cell. Current density versus voltage (J-V) curves were recorded with a Keithley 2400 digital source meter. The scan rates were 0.05 V/s. Attorney Docket 4210.0526WO [0043] Picosecond Diode Lasers [0044] The example embodiment of the system, generally designated 10, for performing NLPC that is disclosed herein is shown in FIGS.1, 12A, and 12B. The system 10 comprises three (3) PicoQuant diode lasers 100A, 100B, 100C producing 400 nm, 40 ps laser pulses with energies of 100 pJ (LDH-P- C-405). The lasers are generally designated, as a unit, as 100. The repetition rates of the diode lasers 100A, 100B, 100C are within the range of a single emission to recurrent emissions at about 80 MHz. It was noted, however, that the current amplifier 150 that was used for signal detection herein (Stanford Research Systems 570) is not fully refreshed between laser shots at repetition rates greater than about 10 kHz. For this reason, all experiments are conducted at 5 kHz repetition rates to provide comparable data. The three (3) 400 nm diode lasers lasers 100A, 100B, 100C are operated with a single (e.g., only one) PicoQuant laser driver module (SEPIA II SLM 828) housed in a mainframe (PDL 828-L SEPIA II), also referred to herein as controller 110. The delay times between laser pulses produced by the individual diode lasers 100A, 100B, 100C was controlled by the controller 110 to have a minimum step size of 24 ps using a PicoQuant oscillator module (SEPIA II SOM 828- D). [0045] System Configuration [0046] The configuration of the system 10 for performing the novel NLPC methods disclosed herein is schematically illustrated in FIG. 1. As shown therein, the system 100 comprises three diode lasers 100A, 100B, 100C that are all controlled by a controller 110. The diode lasers 100A, 100B, 100C are all mounted on the same surface, such as an optical table, and generate, in the configuration shown in FIG. 1, respective laser beams that are directed over distances of about 2.2 meters to the photovoltaic cell 140 being evaluated. Before striking the photovoltaic device 140, each of the laser beams generated from a respective one of the diode lasers 100A, 100B, 100C is reflected along a prescribed beam path using mirrors 120, which are silver- coated mirrors in the example embodiment disclosed herein. Other types of mirrors can be used. The mirrors 120 used for reflecting the laser beams are positioned to provide a prescribed distance between one of the diode lasers Attorney Docket 4210.0526WO 100A, 100B, 100C from which the laser beam to be reflected by the mirror 120 is emitted, respectively, and the target, which is the photovoltaic cell 140. Thus, by changing a position of the mirrors 120 of one or all of the beam paths, the distance (and, thus necessarily, time) travelled by an associated laser beam traversing such beam path can be changed and controlled. The system 10 also comprises, within each of the beam paths and, optionally, between a last one of the mirror 120 that defines the beam path and the photovoltaic cell 140, a lens 130 for focusing the laser beam onto the photovoltaic cell 140. In the example embodiment disclosed herein, the lens 130 of each beam path is a singlet lens with a 30 cm focal length. Using such a lens 130, fluences of 1 μJ/cm ² can be produced at the photovoltaic cell 140 by focusing 65 picoJoule (pJ) pulses from a respective one of the diode lasers 100A, 100B, 100C to FWHM spot sizes of 90 μm. Because the relative phases of the pulse trains are controlled electronically, it is not necessary to precisely match the distances of the beam paths between each of the diode lasers 100A, 100B, 100C and the photovoltaic cell 140. [0047] FIGS. 2A-C show the laser pulse sequences that are used to generate the nonlinear response of the photocurrent. Thus, NLPS signals can be acquired with a laser pulse sequence of laser pulses 1 and 2 ( ^S _{1 ^ 2 ^} ^{^} _^ ) (FIG. 2A), laser pulse 1 only (S ₁ ) (FIG. 2B), laser pulse 2 only (S ₂ ) (FIG.2C), and both laser pulses blocked ( _{S 0} ). Carrier drift is initiated at _t ^{^} _{^ 0} when the first laser pulse arrives at the photovoltaic device and the amount of time elapsed after arrival of the first pulse is decomposed into an experimentally controlled delay time ( ^ ) and current integration time, (t). The individual signal components are acquired sequentially; thus, S _{1 ^ 2} is acquired before S ₁ , S ₁ is acquired before S ₂ , and S ₂ is acquired before S ₀ . To cycle through the four laser pulse 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 NLPC signals were averaged to optimize the data quality. The NLPC signal is defined according to the equation ^S _{NLPC ^} ^{^} _^ ^{^ S} _{1 ^ 2 ^} ^{^} _^ ^{^ S} ₁ ^{^ S} ₂ ^{^ S} ₀ to reflect the component of saturation induced by recombination processes involving carriers photoexcited by Attorney Docket 4210.0526WO separate laser pulses. The S ₁ and S ₂ 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 integrated with both pulses present, ^S _{1 ^ 2 ^} ^{^} _^ . [0048] As shown in the example embodiment of FIG.1, Pulse 1 is emitted from diode laser 100A and enters the photovoltaic cell 140 through the transparent ITO electrode of the photovoltaic cell 140. The second laser pulse (“Pulse 2”) is emitted from one of the other diode lasers 100B, 100C and is incident on, in the case of the laser diode 100B, the transparent ITO electrode of the photovoltaic cell 140 (the beam path designated “Pulse 2 (co)” in FIGS. 1 and 12B for co-propagation) or, in the case of the laser diode 100C, a copper electrode (e.g., with 10% transmission) of the photovoltaic cell 140 (designated “Pulse 2 (ctr)” in FIGS.1 and 12B for counter-propagation). [0049] The total amount of charge collected from a photovoltaic cell 140 is determined by time-integrating signals acquired therefrom with a Stanford Research Systems 570 current amplifier 150. The voltage output of the current amplifier 150 scales linearly with the photocurrent input, however, the signal pulses are broadened to approximately 20 µs with the 2 μA/V sensitivity and gain mode (low noise) settings that were employed in obtaining the presently disclosed measurements. The signals from the current amplifier 150 are then processed with a National Instruments data acquisition board 160 (e.g., NI USB-6341), which yields data points with 4-μs intervals when operated at 250 kHz. The total amount of charge collected from the of the photovoltaic cell 140 can be computed by multiplying the time-integrated voltage output by the current-to-voltage amplification factor. 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 was cycled over a 0.2 V range with 0.05 V steps in experiments that were 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. [0050] NLPC Signal Generation Mechanisms [0051] The effects of drift velocity dispersion on NLPC measurements are evaluated herein using an approach that is based on empirical parameters. Attorney Docket 4210.0526WO Furthermore, signals acquired with co-propagating laser beams (“Pulse 2 (co)”, from diode laser 100B) are compared to signals acquired using counter- propagating laser beams (“Pulse 2 (ctr)”, from diode laser 100C). [0052] Drift Velocity Dispersion in NLPC Spectroscopy [0053] Models that incorporate multiple trapping events have successfully accounted for drift velocity dispersion in conventional TOF measurements. As used in this context, the term “multiple trapping” means or suggests that carriers must be thermally excited from localized states to reinitiate transport after the occurrence of such trapping, whereas alternate “hopping” descriptions involve carrier tunneling between trap sites. When the active layer of a photovoltaic device absorbs light at photon energies above the band gap, the drift velocity is initially determined by the product of the electric field and the “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, ^{^} _{TOF ^} ^t _^ , can be expressed in terms of the intrinsic carrier mobility, ^ ₀ , as N ^ TOF ^{^} t ^{^ ^ ^} free ^{^} t ^{^ 0} Equation (1), ^ _{^ ^ ^ ^} free and 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. [0054] NLPC measurements are specially equipped to reveal “instantaneous” carrier mobilities with picosecond time resolution by cycling the external bias applied to a photovoltaic cell. In FIGS. 3A and 3B, the influence of carrier trapping on mobilities determined by NLPC spectroscopy is shown. 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 approximately 10 V/μm based on the ratio between the magnitude of Attorney Docket 4210.0526WO the 1 V internal bias and active layer thickness. Thus, as shown in FIG.3A, the potential energy varies by roughly E ^ ^ x _trap = 10 meV on the 1 nm length scale of a defect, ^x _trap , in a device with an active layer that has a thickness of about 100 nm, the energy of 10 meV being much less than the 150- 500 meV found in perovskite films. Therefore, most trapped carriers are unaffected by cycling of the external bias applied to the device in the range of -0.2 to 0.2 V. Merely by way of an illustrative example, it can be considered that cycling the external bias in the range of -0.2 to 0.2 V with a 100 nm thick active layer translates to electric field magnitudes in the range of about 8-12 V/μm. Because the magnitudes of the local potential energy differences are small compared to kBT at ambient temperatures, the drift velocity determined for the full ensemble of carriers decreases as the population of trapped carriers increases (see Equation (1) and FIGS.3A and 3B). As shown in FIG. 3B, the carrier mobility relaxes from initial ( ^ _free ) to terminal ( ^ _trap ) values as charge carriers accumulate in traps on the t _trap . As in conventional TOF measurements, the effective transit time,T _eff , determined by NLPC spectroscopy can be divided into intervals dominated by “free” and “trapped” charge carriers, according to the equation below. T _eff ^ T _free ^ T _trap Equation (2) and terminal phases of the carrier drift are described byT _free and T _trap because the populations of trap states increase with the amount (e.g., or length) of time that has elapsed after photoexcitation. Assuming that carrier drift exhibits these two phases, the effective transit time can be decomposed into free (initial) and trap-limited (terminal) mobilities, ^ _free and ^ _trap , respectively, using the equation below. (3) Attorney Docket 4210.0526WO [0057] In Equation (3), l _trap is the length scale of trap-induced velocity dispersion, d is the active thickness, and E is the electric field applied to the active layer. The is then defined according to the equation below. d ^ ^trap ^ ^fr E v ^{ee drift ^} Equation (4) ^ _trap l _{trap ^ ^ free} ^ d _^ l _trap ^ to the experimental procedure disclosed herein, the difference in drift velocities, ^v _drift , is measured when the electric field applied to a device is cycled between values of E - = E _int - E _ext andE _^ = E _int ^ E _ext , thereby producing the following equation. d ^ ^ ^ ^ E v ^{^} E d ^{rift ^ trap free ^ ^} ^ Equation (5) ^ ^ _^ ^ ^ _^ ^ device,d ^ l _trap and the measurement directly reflects the intrinsic free-carrier mobility via ^v _drift ^ 2 ^ _free E _ext . However, if d ^ ^ l _trap , the difference in drift velocities is by the relationship ^v _drift ^ 2 ^ _trap E _ext . Using Equation (5), it is possible to estimate that the ~1 m/s detection threshold of the NLPC instrument used in the example embodiment disclosed herein corresponds to a trap-limited mobility in which ^ _trap =2.5×10- ³ cm ² /V/s for an external bias with a magnitude of E _ext =2 V/μm (e.g., an external bias of 0.2 V applied to a 100 nm thick active layer). The ^v _drift of ~1 m/s detection threshold suggests trapping of approximately 75% of the carriers in the active layer. [0060] 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-range and long-range components by integrating the drift velocity up to the effective transit time, shown in the below equation. d ^{^ E} ^ ^T 0 ^{^} ^ ^t ^ ^dt Equation (6) [0061] The total path length can be decomposed into phases associated with free and trap-limited motions using the equation Attorney Docket 4210.0526WO d ^{^ E ^} _free ^t _v ^{^} _^ ^{1 ^ exp} _^ ^{^ T} _eff ^{/ t} _{trap ^} ^{^} _^ ^{^ E ^} _{trap ^} ^T _eff ^{^ t} _v ^{^} _^ ^exp _^ ^{^ T} _eff ^{/ t} _{trap ^} ^{^ 1 ^} _{^ ^} in the carrier mobility, ^ _^ ^t _^ ^{^ ^} _free ^exp _^ ^{^ t / t} _{trap ^} ^{^ ^} _^ ^{1 ^ exp} _^ ^{^ t / t} _{trap ^} ^{^} _^ ^{^} _trap Equation (8) timescale for carrier accumulation in (see FIGS. 3A and 3B). Multiple trapping models typically treat the dependent decay of the mobility using a power law; however, exponential temporal profiles are reasonable approximations for the line shapes of the “instantaneous” drift velocities determined with NLPC experiments. Finally, the free and trap-limited drift velocities are given by the equations l t ^ 1 E ^ trap tr free ^{^} ap Equation (9) ^ _^ ^ _^ ^ ^{^} t _trap ^ ^ T _eff , the trapping time can be written as l t trap ^{^} trap Equation (11) ^ [0065] A numerical model for simulating NLPC signals in co-propagating and counter-propagating laser beam geometries. Unlike in known methods, the presently disclosed method incorporates drift velocity dispersion and decomposes the total signal into carrier densities associated with separate laser pulses. Defining the first laser pulse, which is incident on the transparent ITO electrode, as traveling in the direction of positive _x ^{^} , the carrier density profile generated at _t ^{^} _{^ 0} can be written as follows: Attorney Docket 4210.0526WO n ₁ ( ^{^} ₁ , x ^{^} f , t ^{^ ^} 0) ^{^} p ₁ ( ^{^} ₁ , x ^{^} , t ^{^ ^} 0) ^{^ ^ ^ ^1 ^} exp ^ _^ ^{^ ^ ^} _^ ^{^} _^ x ^{^ ^} Equation 1 _{^ 1 ^} (12) , ₁ frequency of the first laser pulse, ^{^} _^ ^{^} _{1 ^} is the absorption coefficient, ^d is the active layer thickness, and t ^ total amount of time elapsed after the first laser pulse arrives at the _t ^{^} _{^ ^ ^ t} . The experimentally controlled time delay, _^ , and detection, t , times are defined in FIG.2. The carrier densities produced by the second laser pulse are defined by the following equations: f n ₂ ( _{^ 2} , x ^{^} , t ^{^} _^ 0) _^ p ₂ ( _{^ 2} , x ^{^} , t ^{^} _^ 0) _^ ^{^ ^ ^2 ^} exp _{^ ^ ^ ^ ^ ^ ^ ^ 2 ^} x ^{^} _{^ ^} Equation (13) and n ₂ ( _{^ 2} , x ^{^} , t ^{^} _^ 0) _^ p ₂ ( _{^ 2} , x ^{^} , t ^{^} _^ 0) _^ ^{f ^ ^ ^2 ^} exp ^ _{^ ^ ^ ^ ^ ^ ^ 2 ^ ^} d _^ x ^{^} _{^ ^ ^} for co-propagating and counter-propagating laser beam geometries, respectively. [0066] Dynamics in the carrier densities, p ₁ and n ₁ , photoexcited by the first laser pulse are expressed, respectively, as follows: p 1 ^ ^ 1 , ^ 2 , x ^{^} , t ^{^} ^ _^ p 1 ^ ^ 1 , ^ 2 , x ^{^} _^ v d ^ t ^{^} ^ _^ t ^{^} , t ^{^} _{^ ^} t ^{^} ^ _^ p 1 ^ ^ 1 , ^ 2 , x ^{^} _^ v d ^ t ^{^} ^ _^ t ^{^} , t ^{^} _{^ ^} t ^{^} ^ t ^{^ ^} _^ t ^{^} where ^v _{d ^} ^{t ^} _^ ^{^ E ^ ^} _^ ^{t ^} _^ is the drift velocity computed using the form of ^{^} _^ ^{t ^} _^ in , _{t trap} is the timescale of carrier trapping defined by Equation (11), ^ is body recombination coefficient, and ^ t ^ is the temporal step size. Two-body recombination is assumed in this model Attorney Docket 4210.0526WO because transient absorption experiments conducted on the solution- processed perovskite films revealed dominant quadratic dependence of recombination rates on the carrier densities. Quadratic scaling is consistent with radiative and trap-assisted Auger recombination, which are both accounted for with the ^ parameter. Trap-assisted band-to-band recombination scales linearly in the carrier density; however, it is fundamentally a two-body process and such contributions to ^ cannot be ruled out entirely. [0067] Similarly, the dynamics in the carrier densities, p ₂ and n ₂ , photoexcited by the second laser pulse are expressed, respectively, as follows: p 2 ^{^} ^ 1 , ^ 2 , x ^ , t ^ ^{^} ^ ^ ^{^} t ^ ^ ^ ^{^} _^ p 2 ^{^} ^ 1 , ^ 2 , x ^ ^ v d ^{^} t ^ ^ ^ ^{^} ^ t ^ , t ^ ^ ^ t ^ ^{^} ^ t ^{^} _^ where the Heaviside step function, ^{^} _^ ^{t ^ ^ ^} _^ , is equal to zero before the second pulse arrives at the device. condition (see FIG.2A) is simulated using Equations (12)-(18), which account for interactions between carriers associated with separate laser pulses through the ^p ₁ n ₂ and ^n ₁ p ₂ terms. [0068] 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 S ₁ and S ₂ conditions. The Attorney Docket 4210.0526WO electron and hole densities photoexcited by pulse k ( k =1,2) are expressed, respectively, as follows: p ^ k ^ ^ k , x ^{^} , t ^{^} ^ _^ p ^ k ^ ^ k , x ^{^} _^ v d ^ t ^{^} ^ _^ t ^{^} , t ^{^} _{^ ^} t ^{^} ^ _^ p ^ k ^ ^ k , x ^{^} _^ v d ^ t ^{^} ^ _^ t ^{^} , t ^{^} _{^ ^} t ^{^} ^ ^ ^{^ ^1} ^ ^ ^ _^ ^ ^{^} ^ ^{^ ^ ^} ^ ^{^ ^} ^ ^ ^{^} _^ ^{^} ^ ^ ^{^} t ^{^} ^ are on the target device (i.e., photovoltaic cell). [0069] Heterogeneity in the carrier transit times is incorporated according to the following equation: ^ 2 ^ ^ ^{^} x ^{^} x ^{^ ^ ^ k} _^ x _, t ^{^} _^ ^{^ 1} ^ ^ ^ _exp ^{^ ^} 2 ^{^ ^ k} _^ x ^{^} _, t ^{^} _^ dx ^{^} Equation (21) (e.g., from the first laser (1) or from one of the second lasers (2)) and _{^ k} is a general carrier density ( _{p k} , _p ^{^} _k ,n _k , or n ^ _k ). The phenomenological broadening parameter, w , is defined asw / ^ t ^{^} =25 ^v _{d ^} t ^{^} _^ to prevent unrealistic flat edges from appearing in the temporal profiles of NLPC signals. Notably, similar sharp features (i.e., flat edges in temporal profiles) also appear in conventional TOF profiles of systems with minimal drift velocity dispersion. [0070] The NLPC signal is obtained by summing over differences in the amounts of charge collected from the photovoltaic cell, using the following equation: 2 ^{^} ^ ^ ₁ , ^ , ^ ₂ ^ ^{^} ^ ^{^} Q ^ ^ ₁ , ^ , ^ ₂ ^ ^{^ ^} Q ^ ^ ₁ , ^ , ^ ₂ ^ Equation (22) holes and electrons photoexcited by a laser pulse k are expressed according to the following equations: Attorney Docket 4210.0526WO 2 ^{^} Q _NLPC ^ ^ ₁ , ^ , ^ ₂ ^ ^{^} ^ ^{^} Q _pk ^ ^ ₁ , ^ , ^ ₂ ^ ^{^ ^} Q _nk ^ ^ ₁ , ^ , ^ ₂ ^ Equation (23) k ^ 1 p _{k nk} , and A is the area of the laser pulse on the device. The signal magnitude can also be represented as the ratio in the amount of charge collected from the photovoltaic cell with and without recombination processes involving separate laser pulses, as follows: ^ _{^ ^} ¹ , _^ , _^ ^{^} Q NLPC ^{^ ^} 1 , ^{^} , ^{^} 2 ^{^ 2} _{^ ^ 2} ^{^} Equation (25) v _^ t _^ components, ^S _NLPC ^/ _^ ^S ₁ ^{^ S} _{2 ^} . Consideration of ^{^} _^ ^{^} ₁ ^{, ^ , ^} _{2 ^} is practical during data acquisition because it can be computed without knowledge of the laser spot sizes and line shape of the amplified photocurrent. [0072] 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 made with active layer thicknesses of 90 nm, 240 nm, and 460 nm, while the penetration depth, ^{1/ ^} _^ ^{^} _^ , is set equal to 34 nm for 400 nm light. The parameters of the mobility function in Equations (8) and (11) are set equal to ^ _free =0.025 cm ² /V/s, ^ _trap =0.015 cm ² /V/s, and l _trap =60 nm for consistency with the present measurements and earlier work on similar systems. 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 1 below. Attorney Docket 4210.0526WO (a) ^(b) _{^ ^ ^ ^} ^{A ^ (c) ^} _free ^{(c) ^} _trap ^l _trap f ^ ^ ^ ^ / h ^ ent on the ITO electrode. Under counter-propagating conditions, the initial carrier density associated with the second laser pulse is ^{f ^} _^ ^{^} _^ ^{/ h ^} =5.9×1016 cm- 3; ^(b) is the absorbance coefficient at 400 nm; ^(c) is parameterized using Equation (8). [0074] In FIGS.4A-4L, carrier densities are graphically plotted, the carrier densities being computed for active layers of different thicknesses in a co- propagating laser beam geometry at a delay time of 3 ns. In FIGS. 4A, 4D, 4G, and 4J, the active layer thickness of 90 nm. In FIGS.4B, 4E, 4H, and 4K, the active layer thickness of 240 nm. In FIGS.4C, 4F, 4I, and 4L, the active layer thickness of 460 nm. Differential hole and electron densities are expressed as _{n k ^ n} ^{^} _k and _{p k ^ p} ^{^} _k , where ^k is the index for the laser pulse ( ^k =1, 2). These differential carrier densities represent the recombination processes at the origin of the NLPC signal generation mechanism. The greyscale bars at the top of each of FIGS. 4A-4L are scaled by factors of 10 ¹⁴ cm ^-3 . Model parameters are summarized in Table 1. [0075] In FIGS.5A-5L, carrier densities are graphically plotted, the carrier densities being computed for active layers of different thicknesses in a co- propagating laser beam geometry at a delay time of 3 ns. In FIGS. 5A, 5D, 5G, and 5J, the active layer thickness of 90 nm. In FIGS.5B, 5E, 5H, and 5K, the active layer thickness of 240 nm. In FIGS.5C, 5F, 5I, and 5L, the active layer thickness of 460 nm. Differential hole and electron densities are expressed as _{n k ^ n ^ k} and _{p k ^ p ^ k} , where ^k is the index for the laser pulse ( ^k =1, 2). These differential carrier densities represent the recombination processes at the origin of the NLPC signal generation mechanism. The greyscale bars at the top of each of FIGS. 4A-4L are scaled by factors of 10 ¹⁴ cm ^-3 . Model parameters are summarized in Table 1. Attorney Docket 4210.0526WO [0076] As shown in Equations (23) and (24), contributions to NLPC signals associated with holes and electrons are computed by integrating p _k ^ p ^ _k and _{n k ^ n} ^{^} _k at the interfaces with the electrodes. These quantities are plotted at delay times of 3 ns ( _t ^{^} _{^ ^ ^ t} ) in FIGS.4A-4L and 5A-5L, which correspond to co-propagating and counter-propagating laser beam geometries, respectively. The ITO and copper electrodes are located at the bottoms and tops of the photovoltaic cells ( x =0 and x = d ). Thus, in response to the electric field, the holes and electrons drift downward and upward until injecting into their respective electrodes. The “differential carrier densities,” _{p k ^ p} ^{^} _k and _{n k ^ n} ^{^} _k , 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. [0077] In FIGS.4A-4L, 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 (p ₁ ^ p ^ ₁ ) and copper (n ₁ ^ n ^ ₁ ) electrodes, because the n ₁ carrier density drifts over more than half of the active layer thickness 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 ofp ₁ ^ p ^ ₁ grows relative to that of n ₁ ^ n ^ ₁ because the amount of overlap between n ₁ and p ₂ decreases during the experimentally controlled delay time, thereby reducing the yield of the ^n ₁ p ₂ recombination process. For example, the maxima of _{n 1 ^ n} ^{^} ₁ and _{p 1 ^ p} ^{^} ₁ differ by factors of 0.8, 0.6, and 0.4 at a 3 ns time delay of 90 nm, 240 nm, 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 _{p k ^ p} ^{^} _k and _{n k ^ n} ^{^} _k 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, _{ltrap ^ ^ ^ ^} (holes) and ^l _trap ^{/ d} (electrons), under co- Attorney Docket 4210.0526WO propagating conditions. Holes always traverse a distance equal to the optical penetration depth ( ^l _trap ^{^} _^ ^{^} _^ ^{^ 1} for the example embodiment of the system disclosed herein), whereas electrons must drift over the full thickness of the active layer (l _trap / d ^ 1 for the example embodiment of the system disclosed herein). [0078] In the counter-propagating laser beam geometry simulated in FIGS. 5A-5L, the first and second laser pulses ( k =1, 2) 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 ^n ₁ p ₂ terms outweigh those of the ^p ₁ n ₂ terms. For example, the peak magnitudes of _{n 1 ^ n} ^{^} ₁ and _{p 1 ^ p} ^{^} ₁ differ by factors of 2.8, 3.4, and 3.0 for active layers having thicknesses of 90 nm, 240 nm, and 460 nm, respectively. The _{n 2 ^ n} ^{^} ₂ and _{p 2 ^ p} ^{^} ₂ densities exhibit similar discrepancies because of the two-body nature of the recombination mechanism. The relative efficiencies of the ^n ₁ p ₂ and ^p ₁ n ₂ recombination processes reflect two competing effects. First, the magnitudes of the ^p ₁ n ₂ terms decrease as a function of active layer thickness because the laser pulses penetrate only ^{1/ ^} _^ ^{^} _^ =34 nm into the active layer near the ITO electrode, which results between p ₁ and _{n 2} at all times. Second, the probabilities that the ⁿ ₁ and ^p ₂ carrier densities trap in the active layer during transit increase with the active layer thickness, which reduces the efficiency of the ^n ₁ p ₂ recombination process. Because of these competing factors, the ratio in peak densities computed forn _k ^ n ^ _k and p _k ^ p ^ _k exhibits a maximum value of 3.4 at the intermediate active layer thickness of 240 nm. [0079] FIGS. 6A-6F show NLPC signal profiles that are computed with a co-propagating (FIGS.6A-6C) and counter-propagating (FIGS.6D-6F) laser Attorney Docket 4210.0526WO beam geometry for the active layer thicknesses, d , shown in each of FIGS. 6A-6F. In the co-propagating laser beam geometry configuration, the four classes of carrier densities exhibit similar contributions to the total NLPC signals for all active layer thicknesses. In contrast, the ^n ₁ p ₂ recombination mechanism dominates the nonlinear response when ^{^} _^ ^{^} _^ ^{d ^ 1} under counter-propagating conditions. Model parameters are in Table 1. [0080] The total amount of charge collected from a photovoltaic device is decomposed into contributions from the four classes of charge carriers in FIGS. 6A-6F. The integrated charges, ^Q _j , are related to the differential carrier densities, p _k ^ p ^ _k and n _k ^ n ^ _k , by Equations (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, contributions from the ^n ₁ p ₂ recombination mechanism outweigh those associated with ^p ₁ n ₂ for all systems under counter-propagating conditions, which is consistent with the peak values of differential carrier densities displayed in FIGS.5A-5L. Due to poor spatial overlap between the p ₁ and n ₂ carrier densities, the ^Q _{n 1} and ^Q _{p 2} signal components dominate the nonlinear response if the active layer thickness exceeds the light penetration depth with counter-propagating beams. Relative to ^Q _{p 2} , contributions from ^Q _{n 1} increase with the active layer thickness because the n ₁ carrier density “flattens” during the experimentally controlled delay time as a result of trapping and drift velocity dispersion. In FIGS. 5A-5L, trapping dynamics cause the differential carrier densities, _{n 1 ^ n} ^{^} ₁ and _{p 2 ^ p} ^{^} ₂ , to localize near the copper electrodes in devices with 240 nm and 460 nm thick active layers. [0081] The delay-dependent overlaps in carrier densities, which contribute to the ^p ₁ n ₂ and ^n ₁ p ₂ recombination processes, are illustrated with schematics in FIGS.7A and 7B, where the drift velocity is assumed to be a constant value to maintain a simple physical picture. The directions of carrier Attorney Docket 4210.0526WO 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 ^p ₁ n ₂ and ^n ₁ p ₂ 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 p ₂ and n ₂ carrier densities penetrate a distance of ^{1/ ^} _^ ^{^} _^ into the active layer the ITO electrode when the second laser at the device. Therefore, although the p ₁ and _{n 1} densities drift in opposite directions as ^{^} increases, these two carrier possess similar overlaps in the ^p ₁ n ₂ and ^n ₁ p terms. In the ₂ counter-propagating geometry, the ^p ₁ n ₂ term produces a smaller contribution when ^{^} _^ ^{^} _^ ^{d ^ 1} because the _{p 1} and _{n 2} carrier densities are never well-overlapped. In contrast, the n ₁ and p ₂ carrier densities drift towards each other and recombine throughout the full thickness of the active layer under counter-propagating conditions in FIGS.6A-6F. [0082] In FIGS. 7A-7C, key aspects of NLPC signal generation mechanisms. Carrier densities are represented with the delay time ( ^ ) set equal to various fractions of the transit time, T _transit , for co-propagating (FIG. 7A) and counter-propagating (FIG. 7B) geometries ( t =0 in all devices). Calculations show that recombination processes originating in the ^p ₁ n ₂ and ^n ₁ p ₂ terms possess comparable weights in the co-propagating geometry, whereas the ^n ₁ p ₂ recombination mechanism dominates under counter-propagating conditions when ^{^} _^ ^{^} _^ ^{d ^ 1} . FIG.7C shows how trapping and drift velocity dispersion flattens the n ₁ carrier density as a function of propagation across the full thickness of the active layer in the counter- propagating laser beam geometry (FIG.7B) which decreases the magnitude of the nonlinear response. These densities are taken from the calculation presented in FIGS.5A-5L. Attorney Docket 4210.0526WO [0083] FIG. 7C illustrates an important aspect of the NLPC signal generation mechanism that is not represented in FIGS.7A and 7B. In counter- propagating beam geometry, the ⁿ ₁ carrier density is “probed” by the _{p 2} 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 laser beam geometries are used. In measurements conducted with the goal of obtaining TOF information, NLPC signals decay with ^ for two reasons under counter-propagating conditions, the first reason being that the n ₁ carrier density arrives at the copper electrode and the second reason being that the nonlinear response associated with ^n ₁ p ₂ recombination terms weakens due to flattening of the n ₁ carrier density. [0084] In summary, the calculations presented in this section suggest that the _{p 1} and ⁿ ₁ 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, n ₁ and p ₂ , travel in opposite directions and are initiated near separate electrodes. For systems in which ^{^} _^ ^{^} _^ ^{d ^ 1} , trapped carriers and drift velocity dispersion accumulate while n ₁ and p ₂ traverse the full thickness of the active layer which reduces the magnitudes of the ^{^n} ₁ ^p ₂ terms as a function of _^ (see FIG. 7C). As a practical matter, broadening of the n ₁ carrier density shortens the decay times of NLPC signals and must be accounted for in the analysis of experimental data. For this reason, an empirical approach for estimating the relevant length scale of carrier drift for the counter-propagating beam geometry is disclosed herein. [0085] Time-of-Flight Measurements [0086] Empirically-derived NLPC data acquired for photovoltaic cells with 90 nm, 240 nm, and 460 nm active layer thicknesses in both co-propagating Attorney Docket 4210.0526WO and counter-propagating laser beam geometries is disclosed herein. Although mobilities have been determined for MAPbI ₃ films by alternate methods, the presently disclosed subject matter is a novel approach for probing drift velocity dispersion. Moreover, establishing general physical insights into NLPC signal generation mechanisms facilitates applications of the technique. Averaged data are summarized herein because experiments must be conducted on multiple photovoltaic cells, each of which has multiple electrodes, to meaningfully characterize behaviors for each active layer thickness and beam geometry. [0087] FIGS.8A-8I are graphical plots of NLPC signals are measured for active layer thicknesses of 90 nm (FIGS.8A-8C), 240 nm (FIGS.8D-8F), and 460 nm (FIGS.8G-8I) in a co-propagating laser beam geometry. The smallest, intermediate, and largest values of the electric fields are displayed in separate columns; however, the external biases applied to the devices are cycled over five (5) points for each measurement. [0088] In FIGS.8A-8I, 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 . TOF information is processed by fitting the decay curves to sums of exponential functions, according to the equation below. S _^ ^{^} _^ ^{^ A} ₀ ^{^ A} ₁ ^exp _^ ^{^ ^ / T} _{1 ^} ^{^ A} ₂ ^exp _^ ^{^ ^ / T} _{2 ^} Equation (26) can then be computed using the equation below. T ^{^ A} 1 ^T 1 ^{^ A} 2 ^T 2 av Equation (27) 2 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 Attorney Docket 4210.0526WO depth of the incident light (i.e., ^{^ ^1} _^ ^{^} _^ ) is the relevant length scale. Rather, the decay times increase with the active layer thicknesses because the drift velocities scales linearly with the applied electric field, E ^ V / d . For example, FIG. 7A shows that the p ₁ and n ₁ carrier densities drift away from the illuminated region of the active layer where they are subsequently “probed” by _{p 2} and ⁿ ₂ . [0091] FIGS.9A-9I show NLPC signals that were measured for active layer thicknesses of 90 nm (FIGS. 9A-9C), 240 nm (FIGS. 9D-9F), and 460 nm (FIGS. 9G-9I) in a counter-propagating laser beam geometry. The smallest, intermediate, and largest values of the electric fields are displayed in separate columns. However, the external biases applied to the devices are cycled over 5 points for each measurement. [0092] NLPC signals acquired under counter-propagating conditions are presented in FIGS. 9A-9I. 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 experimental data shown in FIGS. 8A-8I, the timescales on which the NLPC signals decay are more sensitive to the active layer thickness under counter-propagating laser beam geometry configurations. For example, the decay times measured in FIGS.9B, 9E, and 9H are 2.6 ns, 10.8 ns, and 17.8 ns for active layer thicknesses of 90 nm, 240 nm, and 460 nm, respectively. In contrast, the co-propagating decay times measured in FIGS. 8B, 8E, and 8H are 2.5 ns, 6.5 ns, and 10.1 ns for the same 90 nm, 240 nm, 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 FIGS. 7A-7C, 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. [0093] TOF information is extracted from the experimental data by computing carrier drift velocities. Under co-propagating conditions, the drift Attorney Docket 4210.0526WO velocities in FIGS.9A-9I are calculated using ^{v ^1 ^ 1} d _rift ^{^ ^} _^ ^{^} _^ ^{^ T} _av , where ^{^} _^ ^{^} _^ is the absorbance coefficient and T _av is the constants determined by fitting the NLPC decay curves. The averaged velocities and uncertainty ranges in FIGS. 10A-10F correspond to measurements conducted on different devices and with different electrodes on individual photovoltaic cells. The mobility computed for the 90 nm thick active layer is equal to 0.013 cm ² /V/s, whereas the mobilities associated with the 240 nm and 460 nm devices are 0.023 and 0.025 cm ² /V/s, respectively. The increase in carrier mobility as a function of the active layer thickness is consistent with behaviors established for MAPbI3 devices, in which carrier mobility is maximized near an active layer thickness of about 300 nm due to a combination of crystallinity and structural orientation. For example, using techniques based a field-effect transistor, mobilities of approximately 2 cm ² /V/s were determined for film thicknesses of about 240 and 460 nm, whereas a mobility near 1 cm ² /V/s was found with a film thickness of about 90 nm. While these values are about a factor of ten larger than those obtained with NLPC spectroscopy, it is noted that mobilities measured using different samples and techniques can vary by more than an order of magnitude. For example, a mobility of 0.012 cm ² /V/s was determined for MAPbI3 using the steady-state space charge-limited current produced by a photovoltaic cell. In addition, the presently disclosed empirical measurements are in good agreement with the mobilities of 0.016-0.026 cm ² /V/s measured for similar MAPbI3 photovoltaic cells using an NLPC instrument employing femtosecond laser pulses. [0094] Determining carrier mobilities with the counter-propagating laser beam geometry configuration is complicated by contributions of both the transit times and drift velocity dispersion to T _av (see FIG.7C). As expected for carrier drift, the data disclosed herein shows that T^ 1 a _v scales linearly with the applied electric field for all active layer thicknesses under counter-propagating laser beam geometry conditions. Therefore, the effective length scale over which the carrier density has traveled when the signal decays to 1/e of its initial value was determined. Because the drift velocity must be independent of the Attorney Docket 4210.0526WO laser beam geometry, the effective length scale of carrier drift under counter- propagating laser beam geometry conditions empirically were calculated using the following equation: l ^{ctr ^ Tctr} ^ Equation (28) ^ _^ ^ _{^ T co} 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.8A-8I and 9A-9I. Two assumptions are made in Equation (28), namely, that the length scale of transit is well-defined with co-propagation (see FIG. 7A) and that the drift velocities are independent of the laser beam geometry (e.g., co- propagating or counter-propagating) due to equivalent electric field magnitudes. Using this approach, the mobilities obtained for the three active layer thickness are essentially identical for the two laser beam geometries discussed herein. [0095] FIGS. 10A-10F are graphical plots of carrier drift velocities measured with a co-propagating laser beam geometry (FIGS.10A-10C) and a counter-propagating laser beam geometry (FIGS.10D-10F). Data acquired with active layer thicknesses of 90 nm, 240 nm, and 460 nm thick are displayed in FIGS. 10A and 10D, 10B and 10E, and 10C and 10F, respectively. Carrier mobilities, ^ , obtained by the linear fits are included in the corresponding graphical plots for each of FIGS.10A-10F. The uncertainty ranges shown therein represent standard deviations for measurements conducted on separate photovoltaic cells, in addition to separate electrodes on individual devices. [0096] The values of l _ctr , which are summarized in the table shown in FIG. 11, 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 10 ¹⁵ -10 ¹⁶ cm ^-3 is a reasonable estimate, can be considered as a potential source of dispersion. Because trap densities of 10 ¹⁵ -10 ¹⁶ 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 Attorney Docket 4210.0526WO of l _ctr . Therefore, the data presented herein from the empirical testing suggests that drift velocity dispersion may originate from different aspects of the structure, such as short-range thermal fluctuations. Density functional theory calculations conducted on MAPbI3 suggest that thermal fluctuations generate partially localized states on the length scale of 5-10 nm, suggesting that numerous encounters are plausible over a path length of 60 nm. Moreover, the local potential energy differences associated with the applied biases in the experimental data disclosed herein, which are less than kBT on the 1 nm length scale (see FIGS. 3A and 3B), are also consistent with contributions of thermal disorder. While thermal fluctuations may play a contributing role, it is thought that drift velocity dispersion originates from a combination of factors including thermal disorder, electronic traps, grain boundaries, and structural defects. [0097] In FIG.11, columns designated ^(a) indicate measurements that were conducted with intermediate biases shown in Figures 8A-8I and 9A-9I. Uncertainty ranges are standard deviation for the three separate measurements. Columns designated ^(b) are for effective path lengths for counter-propagating measurements computed using Equation (28) with ^{^} _^ ^{^} _^ equal to 29.2 μm ^-1 . [0098] FIGS. 12A and 12B show aspects of an example embodiment of the system 10 for performing NLPC spectroscopy for a photovoltaic cell 140, in which the paths of the laser beams emitted from the picosecond diode lasers 100A, 100B, 100C are shown reflecting off of 2 sets of mirrors 120, each laser beam from one of the diode lasers 100A, 100B, 100C passing through a designated lens 130 to focus the laser beam onto the desired location (e.g., electrode) on the photovoltaic cell 140. The example embodiment of the system 10 is configurable on an optical table measuring about 5 ft by 5 ft, in comparison to known NLPS systems that require construction on a much larger optical table of about 5 ft by 20 ft due to the requirement of multiple optical manipulations being required in such known systems. In the example embodiment shown in FIGS. 12A and 12B, the system 10 is configured as a two-pulse, time-of-flight instrument based on 40- picosecond diode lasers 100A, 100B, 100C. Experiments using such a Attorney Docket 4210.0526WO system were conducted by generating two laser beams (e.g., only lasers 100A, 100B or 100A, 100C). Thus, in some embodiments, the system 10 may comprise as few as two (2) diode lasers (e.g., 100A, 100B or 100A, 100C); however, it is advantageous to use the three (3) diode lasers 100A, 100B, 100C shown in FIGS. 12A and 12B to aid in reducing experimental time by not having to refocus the second laser beam between the co-propagating and counter-propagating geometries. Delay times between laser activations are controlled electronically by the controller 110. It is convenient to leave empty space between the optics for instrument development; however, it is possible for all components of the system to fit into an area as small as about 1 ft by about 2 ft. [0099] In the presently disclosed subject matter, systems 10 are disclosed that enable a new experimental approach for conducting NLPC experiments to determine carrier mobilities and characteristic length scales for drift velocity dispersion in photovoltaic cells 140 with MAPbI3 active layers. Operation of other known NLPC systems and methods, such as those that incorporate a femtosecond laser system, require specialized knowledge of optics and ultrafast spectroscopy. Additionally, studies of photovoltaic devices in previously known NLPC systems and methods are known to be limited to a time window of 15 ns because the delay time between laser pulses is controlled with a motorized translation stage. In the improved systems and methods disclosed herein, synchronized picosecond diode lasers are used to achieve different time windows than was previously possible (see FIG.1). The presently disclosed system 10 comprises only a few (e.g., as low as 2) sets of mirrors 120 and sets (e.g., as low as 1) of lenses 130, possesses superior laser stability, and enables easy control of delay times up to 100 μs at a 5 kHz repetition rate. Although the example embodiment disclosed herein is stated as using fixed 400 nm wavelength laser beams, the operation of the system 10 and the associated methods is not limited to this or any specific range of wavelengths for the lasers 100A, 100B, 100C, especially because, for an instrument that employs picosecond diode lasers 100A, 100B, 100C, continuum sources are commercially available. With color tunability, NLPC spectra can additionally reveal transient occupation of different phases of an Attorney Docket 4210.0526WO active layer when the two spectral dimensions are scanned (e.g., mixtures of layered perovskite quantum wells). [0100] The empirical data disclosed herein demonstrates that the sensitivity of an NLPC experiment to drift velocity dispersion increases as a function of the distance between the regions of the active layer of the electrode of the photovoltaic cell 140 illuminated by the two laser pulses from whichever of the diode lasers 100A, 100B, 100C. Therefore, in addition to determinations of carrier mobilities, this aspect of the nonlinear response has been shown herein to be capable of being exploited to establish the length scale on which the carrier densities broaden during transit. The example NLPC measurements disclosed herein also demonstrate several aspects of carrier transport in MAPbI ₃ photovoltaic cells 140, or devices. First, the smallest mobilities in devices with 90 nm thick active layers were determined (see FIGS.10A, 10D). 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. Second, mobilities on the order of about 0.025 cm ² /V/s were detected with closer-to-optimal active layer thicknesses of 240 nm and 460 nm. Third, the length scale of drift velocity dispersion, l _ctr , was estimated to be approximately 60 nm based on conducted with different laser beam geometries (e.g., co- propagating and counter-propagating). The magnitude of l _ctr was also found to be consistent with contributions of thermal disorder in the perovskite structure. which has been shown to generate partially localized states on the 5-10 nm length scale. [0101] The description herein describes embodiments of the presently disclosed subject matter, and in some cases notes variations and permutations of such embodiments. This description is merely an example of the numerous and varied embodiments. The description or mentioning of one or more representative features of a given embodiment is likewise example. Such an embodiment can typically exist with or without the feature(s) mentioned; likewise, those features can be applied to other embodiments of the presently disclosed subject matter, whether listed in this summary or not. Attorney Docket 4210.0526WO [0102] The subject matter disclosed herein can be implemented in or with software in combination with hardware and/or firmware. For example, the subject matter described herein can be implemented in or with software executed by a processor or processing unit. In one example implementation, the subject matter described herein can be implemented using a computer readable medium having stored thereon computer executable instructions that when executed by a processor of a computer control the computer to perform steps. Example computer readable mediums suitable for implementing the subject matter described herein include non-transitory devices, such as disk memory devices, chip memory devices, programmable logic devices, and application specific integrated circuits. In addition, a computer readable medium that implements the subject matter described herein can be located on a single device or computing platform or can be distributed across multiple devices or computing platforms. [0103] While at least one example embodiment of the invention(s) is disclosed herein, it should be understood that modifications, substitutions and alternatives may be apparent to one of ordinary skill in the art and can be made without departing from the scope of this disclosure. This disclosure is intended to cover any adaptations or variations of the example embodiment(s). In addition, in this disclosure, the terms “comprise” or "comprising" do not exclude other elements or steps, the terms "a", “an” or "one" do not exclude a plural number, and the term “or” means either or both. Furthermore, characteristics or steps which have been described may also be used in combination with other characteristics or steps and in any order unless the disclosure or context suggests otherwise. This disclosure hereby incorporates by reference the complete disclosure of any patent or application from which it claims benefit or priority.