Interest in layered perovskite quantum wells is motivated by fundamental knowledge of their electronic structures and the potential for use in optoelectronic applications. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] In these systems, quantum wells composed of metal-halide octahedra are separated by layers of organic spacer cations. The general chemical formula for the class of systems targeted in the present work is given by BA 2 MA n-1 PbnI 3n+1 , where BA is butylammonium, MA is methylammonium, and the subscript n represents the number of stacked lead-iodide octahedra within the quantum wells. Out-of-plane confinement causes the properties of electronic excitations to vary with the thicknesses of the quantum wells. For example, quantum wells with n = 2, 3, 4, and 5 have exciton resonances near 570, 600, 640, and 680 nm. 2,3,[16][17][18] In addition, the exciton binding energies, which range from 250 to 125 meV for n = 2-5, 4 decrease as the thicknesses of the quantum wells increase. Incorporation of insulating organic spacer cations into the structures (e.g., butylammonium) also has significant implications for carrier transport in these materials. Because of the large potential energy barriers imposed by the interstitial organic phases, 19 the in-plane carrier mobilities are roughly four orders of magnitude greater than the out-of-plane mobilities. 9 The solution-processed films investigated in this work consist of mixtures of quantum wells with different sizes as depicted in Fig. 1. The smallest and largest quantum wells are concentrated near opposing interfaces of the films (i.e., near opposing electrodes in photovoltaic devices), 16,20 which produces gradients in the average values of the bandgaps and energy levels. Energy and charge funneling behaviors are promoted by the prototypical light-harvesting antenna structure; [16][17][18][21][22][23][24][25][26][27] however, our recent studies suggest that long-range charge transport primarily occurs within the phases
ARTICLE scitation.org/journal/jcp The in-plane carrier mobility is roughly four orders of magnitude greater than the out-of-plane mobility because the interstitial butylammonium molecules impose potential energy barriers between quantum wells on the order of 2 eV. 19 (c) Photoexcitation induces energy and charge carrier funneling in solution-processed mixtures of quantum wells. Type II band alignments are depicted in this diagram; however, it is also possible that type I band alignments contribute due to thermal fluctuations and static heterogeneity.
of the thickest quantum wells rather than exploiting a cascade of charge transfer transitions between quantum wells with different sizes. 12,28,29 We have hypothesized that the large potential energy barriers imposed by interstitial organic spacer cations suppress carrier funneling. 29 This interpretation is based on the analysis of a wide range of nonlinear optical and photocurrent spectroscopies applied to systems composed of quantum wells with similar size distributions. 17,20,23,28 While these measurements are self-consistent, the understanding of transport mechanisms in solution-processed films is challenged by heterogeneity. In-plane and out-of-plane transport cannot be distinguished because of quasi-random quantum well orientations. 20 Photovoltaic device efficiencies generally increase with the sizes of the quantum wells; however, it is not clear whether this trend reflects the in-plane mobilities of the quantum wells and/or the probabilities of trapping at interfaces with interstitial spacer cations.
In this paper, we use a newly developed nonlinear photocurrent (NLPC) spectroscopy to explore the effects that organic spacer cation concentrations have on carrier drift velocities in solutionprocessed mixtures of quantum wells. 23,28,30 The NLPC technique involves the application of two color-tunable laser pulses with an experimentally controlled delay time. The first laser pulse initiates long-range charge transport by photoexciting carriers in the active layer of a photovoltaic cell. The second laser pulse probes this set of carriers by way of two-body recombination mechanisms as they drift toward the electrodes, thereby yielding time-of-flight (TOF) information. 28 Whereas the time resolution of a conventional TOF measurement is limited by the RC time constant of a device, 31,32 the time resolution of an NLPC experiment is determined by a combination of the laser pulse durations and two-body recombination rate. 28,29,[33][34][35] Here, we leverage the superior time resolution of the NLPC technique to obtain instantaneous drift velocity profiles by cycling the external biases applied to photovoltaic devices. Our experimental data show that carriers are harvested less effectively from the active layer as the concentrations of organic spacer molecules increase (i.e., as the thicknesses of the quantum wells decrease). Measurements conducted on a variety of layered perovskite systems suggest that the differences in device efficiencies primarily reflect the timescales of carrier immobilization at the interfaces between quantum wells and interstitial organic molecules.
The multidimensional TOF technique applied in this work is an extension of a more general class of NLPC-like spectroscopies. [33][34][35][36][37][38][39][40][41][42] Although these approaches differ in appearance, these NLPClike experiments similarly induce a nonlinear response in the photocurrent with either two or four laser pulses. Two-pulse experiments essentially "pump" and "probe" the system like transient absorption spectroscopy, [33][34][35][36] whereas four-pulse experiments enhance the time-frequency resolution of the method by Fourier transformations. [37][38][39][40][41][42] The two-pulse technique applied here is distinct in that the external bias is cycled to probe long-range carrier drift through the active layer of a device on the nanosecond timescale. In contrast, NLPC-like studies of short-range processes such as exciton dissociation and charge transfer are motivated by the ability to distinguish useful and lossy mechanisms in photovoltaic cells. [33][34][35][36][37][38][39][40][41][42] Unlike transient absorption spectroscopy, photocurrent detection reduces ambiguities in signal interpretation by directly targeting the processes that are relevant to the operation of the device. In other words, NLPC-like experiments provide information specific to the fate of an excitation, whereas transient absorption signals reflect the concentrations of photoexcited species regardless of their functional significances.
scitation.org/journal/jcp
A. Synthesis and device fabrication CH 3 (CH 2 ) 3 NH 3 I (BAI) was synthesized 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 h. The solvent was slowly evaporated under reduced pressure at 60 ○ C for 1 h to obtain the crude product. The white powder was then recrystallized in ethanol and further 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. CH 3 NH 3 I (MAI) was synthesized by combining a methylamine solution (40 wt. % in H 2 O) with hydriodic acid and 57 wt. % in water without stabilizer).
Glass substrates coated with 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-propanol (15 min for each solvent in sequence). The substrates were then dried under a stream of nitrogen gas and treated with UV-ozone for 15 min before being transferred into a glove box filled with nitrogen gas. PTAA [poly(triaryl amine) from Sigma-Aldrich] solution in toluene (2 mg/ml) was then spin-coated onto cleaned ITO substrates at 4000 rpm for 30 s and baked at 100 ○ C for 10 min. After cooling down to room temperature, the perovskite precursor solution was spun-cast on the PTAA-coated substrate.
For BAI-based 2D perovskite solar cells with maximal concentrations of n = 1, 2, 3, and 4 quantum wells, the precursor solutions were made by dissolving BAI, MAI, and PbI 2 in dimethylformamide (DMF) with molar ratios of BAI:MAI:PbI 2 of 2:0:1, 2:1:2, 2:2:3, and 2:3:4, respectively. These solutions were stirred at 70 ○ C for 30 min. The concentration of Pb 2+ required to produce a 110-nm thick film is ∼0.5M. The 2D perovskite film was obtained by spin-coating the precursor solution at 70 ○ C onto a substrate (pre-wet by spin coating pure DMF on top at 2000 rpm for 3 s twice) at room temperature with 5000 rpm for 20 s in air. The resulting film was then quickly transferred to a hot plate at 80 ○ C for 1 min.
To fabricate 3D MAPbI 3 perovskite solar cells, the perovskite precursor solution was prepared by dissolving PbI 2 and MAI in DMF:dimethyl sulfoxide (DMSO) = 9:1. The concentration of Pb 2+ required to produce a 140-nm thick film is ∼0.65M. 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 dropcasted with 0.3 ml toluene at 8 s for the second-step spin-coating. Subsequently, the sample was annealed at 65 ○ C for 10 min and 100 ○ C for 10 min. The spin coating process for the 3D perovskite was conducted in a glove box filled with nitrogen gas. After cooling to room temperature, substrates with 2D and 3D perovskite film were also transferred to a glove box filled with nitrogen gas. To finish the device fabrication process, 40 nm C 60 , 3 nm BCP (Bathocuproine), and 80 nm copper are thermally evaporated at a base pressure of 3 × 10 -7 Torr. The active area of 0.13 cm 2 was controlled by a shadow mask.
Device characterization was carried out under AM 1.5 G irradiation with the intensity of 100 mW/cm 2 (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.
Our experiments are conducted using a 45-fs, 4-mJ Coherent Libra with a 1-kHz repetition rate. Approximately 1.5 mJ of the 800-nm fundamental is focused into a 2-m long tube filled with argon gas to generate a visible continuum. The continuum is then split into two beams and passed through replica all-reflective 4F spectral filters, which are based on 1200-g/mm gratings and 20-cm focal length mirrors. The desired portions of the spectra are filtered with motorized slits at the Fourier planes. The filtered pulses have 5-nm widths and 300-fs durations. This light source has been employed for transient absorption, 17,18,20,43 nonlinear fluorescence, 23,24 and NLPC experiments. [28][29][30] The delay time between laser pulses is controlled with a motorized translation stage (Zaber X-LDQ0600C-AE53D12). Four retroreflectors (Newport) are mounted on the translation stage to maintain a small footprint. Collimation of the laser beam over this distance is achieved using a telescope with +100 and -50 mm singlet lenses. Transmission of the laser beam through a 100-μm diameter pinhole at the sample position varies by less than 5% over the full range of wavelengths and delay times covered in the present experiments. The laser pulses have energies ranging from 55 to 110 pJ with 82-μm spot sizes (i.e., fluences of 1.0-2.0 μJ/cm 2 ). This fluence produces initial carrier densities of the order of 10 17 cm -3 . 29 As illustrated in Fig. 2, the two laser beams applied to the photovoltaic device are chopped at 500 and 250 Hz to establish a cycle of four conditions: pulses 1 and 2 [S 1+2 (τ)], pulse 1 only (S 1 ), pulse 2 only (S 2 ), and both pulses blocked (S 0 ). The NLPC signal is defined as SNLPC(τ) = S 1+2 (τ) -S 1 -S 2 + S 0 . The S 1 and S 2 conditions are independent of the delay time because a single laser pulse photoexcites carriers in the active layer; all carriers that reach the electrodes contribute to S 1 and S 2 regardless of their transit times. In contrast, signals generated with both pulses present, S 1+2 (τ), depend on the delay time because interactions between carriers photoexcited by separate laser pulses produce a transient saturation effect. Because S 0 ≈ 0, the sign of the signal is negative if S 1+2 (τ) < S 1 + S 2 . Like a pump-probe experiment, the signal, SNLPC(τ), must be zero if the two laser beams do not spatially overlap in the active layer. 30 Diffusion coefficients measured for our solution processed films with transient absorption microscopy are ∼0.01 cm 2 /s. 43 Therefore, at the maximum delay time of 15 ns, the lateral diffusion length of ∼0.1 μm will be negligible compared to the 82-μm diameter of the laser spot on the device.
The signal may be expressed with respect to the total charge and/or the magnitude of the photocurrent measured with a current amplifier (Stanford Research Systems 570). The voltage output of the preamplifier scales linearly with the photocurrent input; however, the signal pulses are broadened to ∼20 μs with our sensitivity (varied from 2 to 20 μA/V for the present measurements) and gain mode (low noise) settings. Signals acquired for a particular system can be expressed using the peak voltage measured at a data acquisition board because the widths of the signal pulses are constant. Alternatively, the voltage can be multiplied by the amplification factor (2-20 μA/V) to express the signal with respect to the peak photocurrent. The bandwidth of the preamplifier, which determines
The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp FIG. 2. Nonlinear photocurrent spectroscopy setup and detection scheme. (a) Two laser pulses are chopped at 250 and 500 Hz to isolate the nonlinear response. The external bias applied to the photovoltaic device is varied with a current amplifier. (b) The pulse sequence is cycled through four conditions with a pair of chopper wheels (both laser pulses are blocked in the S 0 condition).
the widths of the signal pulses, must be accounted for to compare signal magnitudes acquired with different sensitivity and gain mode settings. Therefore, we integrate the signal pulses over time in this work to obtain the total amounts of charge collected from the devices. Each photocurrent difference is averaged over a total of 800 laser shots (0.8 s). The photocurrents produced by the devices are amplified in the range of 2-20 μA/V using a Stanford Research Systems 570 current preamplifier. Signals are then processed with a National Instruments data acquisition board [NI code USB-6221], which is synchronized to the laser system at 250 Hz. An adequate density of points is obtained by setting the sampling rate of the data acquisition board to 500 kHz (time intervals of 2 μs) because the signal pulses are broadened to 15-30 μs, depending on the sensitivity and gain mode settings for the current amplifier.
The external bias is cycled through five settings for each of the four pulse sequences: -0.2, -0.1, 0, 0.1, and 0.2 V. The total bias within the active layer is given by the sum of the external bias and the internal bias (-1.0 V) associated with the transport layers. Thus, a total of four variables may be scanned in this NLPC setup: the excitation wavelengths, λ 1 and λ 2 , the delay time, τ; and the external bias. We cycle through the full set of conditions 30 times to produce a single dataset with a total data acquisition time of 24 h. To establish reproducibility, the experiments are repeated multiple times with separate devices.
In this section, we define the NLPC signal generation mechanism and present model calculations to demonstrate how TOF information can be extracted from the signal profiles. Signatures of transient carrier trapping are discussed in the context of models developed for conventional TOF measurements.
We begin by summarizing the processes that occur in the active layer of a photovoltaic cell during an NLPC experiment. As shown in Fig. 3, the first laser pulse initiates drift and two-body recombination processes (e.g., radiative and trap-assisted Auger recombination) 44 in the experimentally controlled delay time, τ. The second laser pulse photoexcites an additional set of carriers that drift and recombine during the signal detection time, t. Two-body recombination processes involving carriers photoexcited by separate laser pulses induce a transient saturation effect, which relaxes as carriers associated with the first laser pulse drift to the electrodes. Under our experimental conditions, two-body recombination is most prominent on the sub-ns timescale because the rate depends on the carrier density. For example, the 3D perovskite material considered here has a two-body recombination coefficient of the order of 3 × 10 -9 cm 3 /s (i.e., a range of 0.8-20 × 10 -9 cm 3 /s), 20,[45][46][47][48][49][50][51] whereas the initial carrier density is ∼1.5 × 10 17 cm -3 . With these parameters, the carrier concentration is reduced by 32% at 1 ns after photoexcitation; however, the recombination rate slows significantly with time. For example, compared to the initial value, the carrier densities are reduced by 70% and 82% at 5 and 10 ns after photoexcitation, respectively (i.e., a change of only 12% in 5 ns).
The The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp FIG. the direction of positive x ′ , the initial profile of the carrier density can be written as
where f is the laser fluence (energy/area), ω k is the laser frequency for pulse k, α(ω k ) is the absorption coefficient, d is the active layer thickness, OD is the optical density at frequency ω k , and t ′ is the total amount of time elapsed after the first laser pulse arrives at the device, t ′ = τ + t. The second term in Eq. ( 1) represents a reflection from the copper electrode. 52 The experimentally controlled delay time, τ, and detection time, t, are defined in Fig. 3. Dynamics in the carrier densities photoexcited by the first laser pulse, p 1 and n 1 , are calculated using
and
where β is the two-body recombination coefficient, v d is the magnitude of the average drift velocity, and Δt ′ is the temporal step size employed in numerical calculations (Δt ′ = 25 ps). The twobody recombination coefficient, β, accounts for both radiative and trap-assisted Auger recombination. 44 Notably, we have detected dominant two-body recombination dynamics for solution processed films using transient absorption methods; 17,20,43 however, contributions from radiative recombination and trap-assisted Auger were not distinguished in these measurements. Similarly, carriers generated by the second laser pulse are described with
and
where the Heaviside step function, θ(t ′τ), sets p 2 and n 2 equal to zero before the second pulse arrives at the device. Equations ( 2)-( 5) are required to simulate the photocurrent when both laser pulses are incident on the device (i.e., the S 1+2 chopper condition). We next consider the carrier densities generated when individual laser pulses interact with the device (i.e., the S 1 and S 2 chopper conditions). The electron and hole densities photoexcited by pulse k (k = 1, 2) can be written as
The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp
and
where the tildes denote that these carrier densities correspond to measurements conducted with single laser pulses. Carrier densities propagate with flat edges when trap-induced velocity dispersion is not accounted for. 29,31,32 Therefore, dispersion in the transit times is incorporated phenomenologically using
where k is the index of the laser pulse (1 or 2), q k is a general carrier density (p k , pk , n k , or ñk ), and w is parameterized to remove flat edges from the experimental NLPC decay profiles (w/v d ≈ 0.5 ns). 29,31 We remark that p k and n k possess two laser frequency parameters, whereas pk and ñk depend on only one of the laser frequencies.
The NLPC signal is obtained by computing differences in the amounts of charge collected from the device during separate chopper conditions,
where
and
The accumulated charges, Q j , have units of coulombs, and A is the area photoexcited by the laser pulse (A ≈ 5280 μm 2 in the present experiments). Equations ( 10)-( 12) correspond to the S 1 , S 2 , and S 1+2 chopper conditions defined in Sec. II B, respectively. In Fig. 4, we present carrier densities calculated using parameters consistent with the experimental data discussed in Sec. IV (see Table I). The first two columns in Fig. 4 show that the first laser pulse photoexcites the active layer at t ′ = 0. Differences in the calculated hole (p 1 and p1 ) and electron (n 1 and ñ1 ) densities do FIG. 4. Carrier densities are computed using Eqs. ( 2)-( 7) with the experimentally controlled delay time, τ, set equal to 3 ns. Light enters the active layer propagating in the direction of positive x. The initial carrier densities are 1.5 × 10 17 cm -3 (i.e., the colorbar scales represent factors of 10 17 cm -3 ). The electric field and mobility are 10 V/μm and 0.02 cm -2 V -1 s -1 , respectively. Other parameters are given in Table I. TABLE I. Parameters used for model calculations.
1.5 × 10 17 cm -3 4 μm -1 5280 μm 2 3 × 10 -9 cm 3 /s 100 nm 0.01 or 0.02 cm -2 V -1 s -1 8-12 V/μm a Absorbance coefficient at 640 nm. b The calculations in Fig. 4 are conducted with a mobility of 0.02 cm -2 V -1 s -1 . Mobilities of 0.01 and 0.02 cm -2 V -1 s -1 are used for the calculations presented in the top and bottom rows of Fig. 5, respectively. c The calculations in Fig. 4 are conducted with E = 10 V/μm, whereas E = 8, 10, and 12 V/μm in Fig. 5.
not develop until the second pulse arrives at t ′ = 3 ns. Recombination processes involving carriers photoexcited by separate laser pulses increase the rates at which p 1 and n 1 decay as evidenced by the steep losses of the carrier density computed at t ′ = 3 ns. In contrast, the carrier densities generated by the second laser pulse are less sensitive to interactions with carriers photoexcited by the first pulse. For example, the hole densities, p 2 and p2 , are almost indistinguishable in these contour plots. As discussed previously, the carrier densities, p 1 and n 1 , primarily contain the targeted TOF information. 29 The signals associated with separate chopper conditions, which are defined in Eqs. ( 10)-( 12), are plotted in the first column of Fig. 5. The Q 1 and Q 2 signal components are independent of the delay time, whereas Q 1+2 rises until it reaches an asymptotic value of
Drift velocities are then computed using d ⋅ T -1 transit /2, where d/2 represents the average path length of 50 nm (i.e., half of the active layer thickness). It is shown in the third column of Fig. 5 that the drift velocities scale linearly with the electric field. Linear fits to these points yield the "empirical" mobilities (i.e., obtained by processing simulated data) in fair agreement with the model's mobility parameters. This approach for determining carrier mobilities was employed in Ref. 29 with inspiration from methods of analyses for conventional TOF data. 53,54 FIG. 5. Summary of NLPC signal components and signatures of carrier transport. The total amounts of charge collected for individual chopper conditions are plotted in the first column. The second column presents the saturation effect underlying the NLPC signal [see Eq. ( 9)]. Increasing the magnitude of the electric field reduces the carrier transit times, which causes the signals to decay on shorter timescales. In the third column, drift velocities are calculated using the ratio of the path length and first moments of the decay profiles presented in the second column. The fitted slopes are in good agreement with the carrier mobility parameters.
Table I summarizes parameters not specified in the figure panels. The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp B. Perturbative description of NLPC signal generation mechanism for layered perovskite materials
The nonlinearities targeted in NLPC-like experiments can be represented in two spectral dimensions much like transient absorption and 2D electronic spectroscopies. 23,29,30,[37][38][39][40][41][42][55][56][57] Although NLPC and transient absorption spectroscopies correspond to different orders of time-dependent perturbation theory, their response functions both possess four field-matter interactions. For example, transient absorption is a third-order process because the fourth field-matter interaction corresponds to signal emission. 58 Fourthorder perturbative descriptions of NLPC are required if the species produced after four field-matter interactions either enhances or reduces the amount of charge collected from a photovoltaic device. For example, a fourth-order nonlinear optical effect is required to produce a doubly excited state, which may saturate the photocurrent and produce a response if it has a short lifetime. In contrast, the signal generation mechanism summarized in Sec. III A and Fig. 6 suggests that resonances between excited states cannot be observed because both laser pulses interact under quasi-equilibrium conditions.
For the present materials, we suggest that the nonlinearity in the photocurrent reflects sequential second-order processes because carriers photoexcited by separate laser pulses recombine after the second laser pulse arrives at the device. In other words, light is absorbed under quasi-equilibrium conditions at two locations in the active layer. Although the second laser pulse does not directly interact with a previously excited species in this interpretation, the model calculations presented in Figs. 4 and 5 show that NLPC signals still reflect excited-state dynamics in the spirit of a pump-probe experiment. This aspect of the signal generation mechanism distinguishes it from the cascades of lower-order optical responses observed in 2D Raman spectroscopies; [59][60][61][62][63][64][65] cascades are not interesting because they do not contain information beyond that associated with the lower-order response. In contrast, the signal generation mechanisms described in Sec. III A and Fig. 6 provide information regarding both short-range charge transfer and long-range carrier drift.
Models developed to describe conventional TOF measurements suggest that mobilities must decrease with the amount of time elapsed after photoexcitation because carriers progressively accumulate in the deepest traps. 32,66,67 The empirical mobilities are predicted to scale linearly with the fraction of "free" carriers, which is greatest at time zero under the assumption that shallow and deep traps have similar capture cross sections. 32,66 To generalize this conventional TOF model to NLPC experiments, we have suggested a practical demarcation in which "shallow" traps broaden the detected drift velocity distribution but have residence times less than our 15-ns detection window. 29 For example, as illustrated in Fig. 7, traps with depths of 150 meV allow ∼37 cycles of capture and release during a 15-ns time window, whereas a trap with a depth of 240 meV fully immobilizes carriers for 15 ns. 29 For layered perovskite systems, carriers may also accumulate at interfaces between quantum wells and organic layers. Because the 0.8-nm-thick organic layers impose potential energy barriers of ∼2 eV, 19 we suggest that the organic-inorganic interfaces must reduce the carrier mobilities in a manner similar to deep traps.
NLPC experiments differ from conventional TOF measurements in that the photocurrent is integrated over the detection time, t. 29 Therefore, carriers that are temporarily immobilized in the active layer during the 15-ns delay window may reach the electrodes at later times and contribute to the signal. For this reason, the relative The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp FIG. 7. Deep traps and interstitial organic layers increase carrier transit times and reduce average drift velocities. (a) Carriers undergo numerous cycles of capture and release from traps as they traverse the active layer. We categorize "deep" traps as having release times greater than or equal to the 15-ns delay window. Carriers are not released by deep traps when the electric field is cycled within the 8-12 V/μm range. Potential energy barriers between quantum wells and interstitial organic layers similarly immobilize carriers during the 15-ns delay window. (b) The average drift velocity is reduced when carriers are temporarily immobilized in the active layer. Carriers immobilized during our full 15-ns detection window may still reach the electrodes at later times and contribute to the NLPC signals.
concentrations of trapped and mobile carriers are reflected by the sensitivity of the signal's temporal profile to cycling of the applied bias. In the present experiments, the external bias is cycled from -0.2 to 0.2 V, whereas the built-in potentials for the devices are ∼-1.0 V. Therefore, the magnitude of the electric field is cycled in the 8-12 V/μm range under the assumptions of a uniform electric field and an active layer thickness of 100 nm. The average drift velocities of mobile carriers, which are defined as having transit times less than 15 ns, increase with the magnitude of the potential. In contrast, carriers that are immobilized at deep traps and/or organic-inorganic interfaces do not respond to cycling of the biases during the 15-ns delay window. For example, it is useful to consider that the potential energy changes by only 8 and 12 meV on the 1-nm length scale of an interstitial organic layer when the electric field is set equal to 8 and 12 V/μm, respectively. Because this 4-meV range of potential energy differences is negligible compared to 2-eV potential energy barrier heights 19 and/or trap depths of 100's of meV, [68][69][70] we suggest that cycling the electric field between 8 and 12 V/μm has a negligible effect on the probability of releasing carriers immobilized in deep traps and/or at inorganic-organic interfaces.
We next explore the influences that shallow and deep traps have on our measured drift velocities. First, with inspiration from conventional TOF experiments, 71 we partition the transit time into mobile and immobile time intervals to demonstrate the effects of shallow traps on drift velocity dispersion. This analysis will relate the range of drift velocities measured experimentally (i.e., at maximum and minimum applied potentials) to the fraction of the transit time for which carriers are immobilized at shallow traps. Second, we will express the range of measured drift velocities with respect to the fraction of carriers immobilized during the full 15-ns detection window of an NLPC measurement (e.g., immobilization at organic-inorganic interfaces and/or deep traps). This approach will relate the fraction of fully immobilized carriers to the range of drift velocities detected when the external bias is cycled in an experiment.
Contributions to the effective transit time, T eff , can be decomposed using 71 T eff = T drift + BTtrap, (
where T drift is the "trap-free" transit time, B is the number of trapping events, and Ttrap is the average amount of time required to traverse a trap site (see Fig. 7). For illustration, we suggest that BTtrap represents shallow trap sites with residence times less than the 15-ns delay window. Under this assumption, T drift scales as the inverse of the applied potential, whereas BTtrap is independent of the bias. The effective drift velocity is then given by the ratio of the active layer thickness, d, and transit time,
where d is the thickness of the active layer, μ is the carrier mobility, and Eint is the built-in electric field (∼10 V/μm for the present systems).
We next estimate the fraction of the effective transit time that must be spent at trap sites if the NLPC decay profiles are insensitive to the cycling of the applied biases. This behavior is anticipated when d/μE int ≪ BTtrap according to Eq. ( 15). For insights into the present experiments, we assume a 10-ns effective transit time, an active layer thickness of 100 nm, and biases spanning 0.8-1.2 V. The range of velocities detected experimentally is given by
where the magnitude of the electric field induced by the external bias, Eext, is 2 V/μm. Equation ( 16) can be rewritten as
The difference in drift velocities can now be expressed in terms of the fraction of the transit time for which carriers are immobilized at shallow traps, f ,
where
Finally, Eqs. ( 18) and ( 19) can be combined to show that the difference in measured velocities scales linearly with f ,
The Journal of Chemical Physics
If the NLPC signal profiles are insensitive to the applied bias, the fraction of the transit time that is spent traversing potential energy barriers is given by
The range of drift velocities, Δv eff , can now be set equal to the 1-m/s detection threshold of our NLPC experiments. 28,29 The fraction, f , is equal to 0.75 if Eint is 10 V/μm, Eext is 2 V/μm, d is 100 nm, and T eff is 10 ns. In other words, the present analysis suggests that our measurements will be insensitive to bias-induced changes in NLPC decay profiles if carriers spend greater than 75% of the effective transit times immobilized at shallow trap sites. The result obtained in Eq. ( 21), which is based on descriptions of conventional TOF measurements, suggests that shallow traps induce drift velocity dispersion; however, this approach does not account for carriers immobilized at organic-inorganic interfaces and/or deep traps during the full 15-ns delay window of our NLPC experiments. Therefore, we next consider how the instantaneous velocity, which will be measured with NLPC spectroscopy in Sec. IV, changes when the applied electric field is cycled between Eint -Eext and Eint + Eext. Assuming the light penetration depth is greater than the thickness of the active layer (i.e., a good approximation with 510-640-nm light and 100-nm active layer thicknesses), the total amount of charge collected from the device at delay time, τ, is given by
where y(τ) is the fraction of immobilized carriers, Δ Q±(τ) is the accumulated charge normalized to 1.0 at τ = 0, l± is the distance traversed in Δτ with positive and negative values of the external bias, and d is the active layer thickness. The amount of accumulated charge depends on the laser frequencies as shown in Sec. III A; however, the frequency arguments of Δ Q±(τ) are not written explicitly here to simplify the notation. With these definitions, Eq. ( 22) can be rewritten as
using the path lengths
The instantaneous drift velocities are equal to
The delay-dependent derivative of the drift velocity with respect to the applied electric field is then given by
which suggests that the empirical mobilities must decrease as carriers accumulate at organic-inorganic interfaces and/or deep traps. This result is consistent with the time-dependent mobilities predicted with conventional TOF models. 32,66,67 It is also practical because the left-hand side of Eq. ( 26) will be measured in the NLPC experiments presented in Sec. IV. For convenience, the fraction of trapped carriers can be expressed in terms of the empirical drift velocities if
is rewritten as
where we have substituted v drift (τ) = μ(τ)Eint.
In this section, we extract TOF information from NLPC experiments conducted on layered perovskite systems in which the quantum wells with the greatest concentrations range from ⟨n⟩ = 1-4. In addition, measurements performed with the corresponding 3D perovskite system, which is denoted as ⟨n⟩ = ∞, serve as a control in which interstitial organic layers cannot suppress transport. The background presented in Sec. III C suggests that carriers will more rapidly immobilize at organic-inorganic interfaces as the value of ⟨n⟩ decreases due to a concomitant increase in the concentration of organic-inorganic interfaces. Moreover, the timescale of carrier immobilization can be determined by measuring instantaneous drift velocities over a range of applied biases using Eq. ( 28). This approach differs from our earlier work in which average drift velocities were determined for the ⟨n⟩ = 3 and ∞ systems. 29 The NLPC signals presented in Fig. 8 are conducted by tuning the two laser pulses into the exciton resonances associated with the most concentrated quantum wells. The delay times, τ, are scanned, and the magnitudes of the electric fields are varied by cycling the external biases applied to the photovoltaic cells. The electric fields reported in these plots are given by the ratio of the sum of internal and external biases and the thicknesses of the active layers (i.e., assumption of uniform electric fields). Signals acquired for each system are normalized to -1 at τ = 0 to facilitate comparisons. As demonstrated in Figs. 4 and 5, the greatest signal magnitudes are found at τ = 0 when the carrier densities photoexcited by the first laser pulse, p 1 and n 1 , are ∼1.5 × 10 17 cm -3 . The NLPC temporal profiles decay to zero as p 1 and n 1 drift to the electrodes, thereby providing TOF information. Electric field-dependent differences in the signals develop as the delay times increase because the drift velocities are proportional to the potential. For this reason, the contour lines for all systems in Fig. 8 transition from vertical (at τ = 0) to slanted (at τ > 0) orientations as the delay time increases.
The data presented in Fig. 8 show that the NLPC decay times become more sensitive to the magnitude of the electric field as the value of ⟨n⟩ increases. For example, the exponential decay times associated with the ⟨n⟩ = 1 system decrease from ∼13.0 to 11.2 ns when the electric field increases from 5.7 to 8.6 V/μm. In contrast, the decay times measured for the ⟨n⟩ = ∞ system decrease from 13.7 to 5.2 ns over the same range of electric field magnitudes. Therefore, the average drift velocities measured for the ⟨n⟩ = ∞ system, which are estimated using the ratios of the 100-nm active layer thickness and decay times, are ∼7 and 19 m/s with electric field magnitudes The signals are normalized to -1.0 at τ = 0 to facilitate comparisons. Quantum wells with the greatest concentrations are denoted as ⟨n⟩. The index, ⟨n⟩ = ∞, represents the "three-dimensional" perovskite, MAPbI 3 . The wavelengths used to conduct these measurements are specified in the respective panels (λ = λ 1 = λ 2 ). These data show that the sensitivities of the decay profiles to the applied electric fields increase with the sizes of the quantum wells.
of 5.7 and 8.6 V/μm, respectively. An average mobility of roughly 0.04 cm -2 V -1 s -1 is calculated by dividing the 12-m/s difference in velocities by the 2.9-V/μm difference in electric field strengths. In contrast, an average mobility of roughly 0.006 cm -2 V -1 s -1 is computed for the ⟨n⟩ = 1 system using the 1.8-m/s difference in drift velocities (i.e., velocities are again estimated using a 100-nm active layer thickness). This trend in the average drift velocities is consistent with our prediction that increasing the concentrations of organic spacer cations suppresses carrier transport; however, deeper insights into the dynamics of carrier immobilization can be derived by processing the instantaneous velocities.
Our NLPC data are fit using the maximum entropy method in Fig. 9 to enable further signal processing. In the maximum entropy method, [72][73][74][75] the decay profiles are described with distributions of exponential functions rather than assuming a small number of terms (e.g., single exponential or biexponential). The Laplace integral is expressed in logarithmic space in the form of a discrete summation using 72,73
where hj is the weight of the component corresponding to the rate constant, kj. In our code, the rate constants range from 32 ps -1 to 32 ns -1 and the spacings between points, Δ log(kj ⋅ 1 ns), are equal to 0.03. For example, with a 1-ns unit of reference, the logarithmic rate axis possesses 100 points ranging from Δ log(0.032 ns -1 ⋅ 1 ns) to Δ log(3.2 ns -1 ⋅ 1 ns) (i.e., the axis ranges from -1.5 to 1.5 with 100 steps). The set of coefficients, hj, is determined by minimizing residuals between the fit and measurement with the constraint that the peaks in the rate spectra possess the broadest widths that can be justified by the data (i.e., noise broadens peaks in the rate spectra). For this reason, the maximum entropy method is said to be "maximally noncommittal with respect to unavailable information." 72 Example fits are shown in the supplementary material.
Carrier mobilities determined with conventional TOF techniques reflect average drift velocities within the active layers of photovoltaic devices. 31,32 The thicknesses of the active layers are well-defined, but the transit times exhibit distributions. Physical insights based on average transit times miss important effects such as trap-induced velocity dispersion in solution-processed films. Therefore, we next outline an approach in which instantaneous drift velocities are extracted from the fitted NLPC data presented in Fig. 9. For this purpose, the signal is written as a sum of three components,
where S Drift (τ, E) accounts for the decrease in the signal magnitude induced by carrier drift, S Space-Charge (τ, E) represents the accumulation of space charge in the active layer, and S Recombination (τ) represents recombination processes. 44 Importantly, S Drift (τ, E) and S Space-Charge (τ, E) are sensitive to the applied electric field. Control experiments in which transit times measured with NLPC spectroscopy are shown to reflect the active layer thicknesses suggest that assuming minimal electric field dependence for S Recombination (τ) is a good approximation. 29 To obtain drift velocities, all signals are normalized to one at time zero, SNLPC(τ = 0, E) = 1.0, and differentiated with respect to the delay time, The quantum wells with the greatest concentration are denoted as ⟨n⟩. The index, ⟨n⟩ = ∞, represents the threedimensional perovskite, MAPbI 3 . The wavelengths used to conduct these measurements are specified in the respective panels (λ = λ 1 = λ 2 ). These data are fit to enable processing of instantaneous drift velocities.
It is useful to consider that the instantaneous drift velocity can be obtained by multiplying Eq. ( 31) with the thickness of the active layer, d,
if SRecombination (τ) and SSpace-Charge (τ, E) are equal to zero. This is generally a bad assumption for laser fluences near 1 μJ/cm 2 because the many-body recombination processes are fast compared to the transit times (see p 1 and n 1 in Fig. 4).
To eliminate recombination processes from consideration, Eq. ( 31) is differentiated with respect to the applied electric field,
The instantaneous mobility, μ(τ), is obtained by multiplying the partial second derivatives by the thickness of the active layer,
We multiply μ(τ) by the electric field associated with the internal bias, Eint, to define the instantaneous drift velocity,
This expression for the instantaneous drift velocity, v drift (τ, E), suggests that the application of bias gives rise to two competing effects. The carrier drift velocities increase with the applied bias, whereas space-charge effects are induced by the transient immobilization of charge carriers in the active layer of a device. 76 When carriers are not efficiently removed from the active layer, increasing the electric field pushes the regions of positive and negative charge apart, thereby generating a secondary electric field that opposes carrier drift. In addition, spatial separation of the electron and hole densities may also enhance the carrier lifetimes by reducing the probability of electron-hole recombination; however, this effect is minimized when the light penetration depths are larger than the active layer thicknesses (i.e., a good approximation with 100-nm-thick active layers and 515-640 nm light).
In Fig. 10, we multiply the partial second derivatives of the signals presented in Fig. 9 by d and E
The first term on the right-hand side of Eq. ( 36) is the instantaneous drift velocity defined in Eq. ( 35), whereas the second term represents the accumulation of space charge in the active layer. Because these two mechanisms have competing effects on the drift velocities, they can be distinguished based on the signs of the terms in an expansion 36), is fit with the maximum entropy method to distinguish carrier drift (red curves) from space charge effects (blue curves). These two signal components have opposite signs because increasing the magnitude of the electric field increases the drift velocity in addition to enhancing space-charge effects. The electric field generated by the space-charge region counteracts the built-in electric field, thereby reducing the drift velocities.
of exponential functions (i.e., decay vs rise). To maximize flexibility, the fits presented in Fig. 10 are conducted using the maximum entropy method [see Eq. ( 29)]. [72][73][74][75] The decaying and rising components of the fits are attributed to trap-induced relaxation of the drift velocities and space-charge effects, respectively. The fits suggest that contributions from the space charge become more prominent as ⟨n⟩ decreases, which is consistent with the lower efficiencies of devices based on smaller quantum wells (i.e., less charge reaches the electrodes as ⟨n⟩ decreases). Systems in which ⟨n⟩ is greater than 2 yield profiles that are dominated by the drift velocities as evidenced by the relatively small magnitudes of the rising signal components. In Fig. 11(b), the fitting parameters associated with the drift velocities in Fig. 11(a) are used to compute average "trapping times," tv. Overall, the range of timescales displayed in Fig. 11(b) suggests that carrier transport is sensitive to the applied electric field for a small fraction of the 15-ns detection window for all layered perovskite systems. The drift velocities decrease with increasing delay times as predicted by models used to describe conventional TOF measurements. 32,66,67,77 These data indicate that the timescales of trap-induced relaxation in the drift velocities increase with the value of ⟨n⟩. In other words, mobile carriers can "freely" drift under the influence of the electric field for greater lengths of time in systems with smaller concentrations of organic-inorganic interfaces.
To put the measured drift velocities into the context, we next compare photocurrent generation efficiencies associated with these same devices and laser fluences (i.e., these data are acquired simultaneously with NLPC signals). The photovoltaic cell efficiencies are proportional to the ratios of the amount of charge collected with individual laser pulses, 12. Photocurrent generation efficiencies are characterized using the ratio of the amount of charge collected using a single laser pulse and number of absorbed photons [see Eq. ( 37)]. These data show that the device efficiencies increase as the concentrations of interstitial organic spacer cations decrease.
Here, the wavelength of the incident laser pulse is given by λ, F is the laser fluence (1 μJ/cm 2 ), OD is the absorbance of the film, and r is the beam radius (41 μm). The collected charge, Q, corresponds to the S 1 and S 2 conditions of the NLPC experiments [see Eqs. ( 10) and (11)]. We consider this approach for computing efficiencies most appropriate for interpreting the significance of the drift velocities because all quantities are obtained under the same operating conditions. Notably, the instantaneous velocities depend on the shapes of the NLPC decay profiles, whereas the efficiencies, ϕ, reflect the absolute signal magnitudes. The photocurrent generation efficiencies displayed in Fig. 12 suggest statistically indistinguishable performances for the ⟨n⟩ = 1-3 systems; however, the efficiencies increase significantly with ⟨n⟩ = 4 and ∞, which is consistent with the sensitivity of the decay times to the electric fields displayed in Fig. 8. Here, the uncertainty ranges reflect measurements averaged using multiple devices and separate electrodes on individual devices. The trend in device efficiencies displayed in Fig. 12 appears correlated with the carrier trapping times plotted in Fig. 11(b). This is an intuitive result because carriers drift for the shortest distances in systems with the smallest trapping times. Because the systems with the smallest values of tv possess the greatest concentrations of organic material, we suggest that the device efficiencies increase with ⟨n⟩ because carriers are less likely to encounter organic-inorganic interfaces as they traverse the active layer.
In summary, we have used NLPC spectroscopy to investigate carrier transport mechanisms in a family of layered perovskite systems exhibiting different size distributions of quantum wells. Because the interstitial organic spacer cations impose large potential energy barriers, 19 we hypothesized that carriers are likely to be trapped at organic-inorganic interfaces in these systems, thereby reducing the effective drift velocities. 28,29 The present results support this interpretation by correlating the timescales for carrier trapping to the concentrations of organic spacer cations in the various systems. We find that large concentrations of organic spacer cations rapidly immobilize charge carriers in systems for which ⟨n⟩ is less than 4. In addition, our comparison of trapping times and device efficiencies suggests that trap-induced drift velocity deceleration is the primary mechanism limiting the harvest of carriers from the devices. Overall, these results suggest that films composed of small quantum wells are poor candidates for photovoltaic cells because higher densities of organic-inorganic interfaces translate to lower efficiencies.
Several representations of NLPC responses have been explored by plotting signals with respect to the excitation wavelengths, delay times, and/or applied electric fields in our recent work. 23,28,29 Twodimensional spectra exhibit features such as those found in photosynthetic complexes and molecular aggregates; [78][79][80] however, interpretations of 2D NLPC spectra acquired for layered perovskites are challenged by the spectral overlap between excitons and the continuum states. For this reason, we find that the NLPC method is most powerful when configured as a TOF experiment. 28 Because the time resolution of NLPC spectroscopy is orders of magnitude better than a conventional TOF experiment, it is possible to extract instantaneous drift velocities from TOF profiles by cycling the external bias at each delay point. Access to this information motivates a variety of future applications. For example, multi-color pulse sequences can be used to reveal carrier transport mechanisms in systems with distinct material components (e.g., polymer heterojunctions and/or charge transfer crystals).
See the supplementary material for maximum entropy method fits for the data shown in Figs. 9 and 10.
Published under an exclusive license by AIP Publishing