## INTRODUCTION

On one hand, the theoretical efficiency limit of 29% (*1*) ultimately restricts further development of Si photovoltaic (PV) technology. On the other hand, there is a need to reduce the cost of PV devices to compete with fossil fuels. Therefore, the current research aims at searching for alternative materials with high efficiencies and low costs simultaneously. Recently, the inexpensive organometallic halide perovskite (OMHP) semiconductors have emerged as a new class of PV materials with highly efficient light absorption and charge transport properties. The efficiency of the OMHP solar cells increased significantly from 3.8% in 2009 to 25.2% (*2*) in 2020. Another great advantage of OMHP semiconductors is the fabrication capability on flexible substrates, which offers an additional opportunity for the development of portable power sources (*3*) and new PV architectures (*4*). Among the wide compositional range of OMHPs, methylammonium lead tribromide perovskite (MAPbBr_{3}) has attracted great interest for its potential applications in tandem solar cells (*5*), as well as in other optoelectronic devices such as high-energy radiation sensors (*6*), photodetectors (*7*), and light-emitting diodes (*8*).

One of the most critical factors in the performance of multifunctional OMHP devices is the presence of trapping centers resulting in the loss of charge collection efficiency in a solar cell or a detector. Trapping centers in a semiconductor lattice form energy states in the bandgap. These energy states affect the relaxation dynamics of free carriers by trapping and, therefore, detrimentally influence the free charge transport properties such as lifetime and drift mobility. The detection and characterization of these traps and their associated relaxation dynamics are highly challenging. The dominant nonradiative nature of these energy transitions does not allow the measurement of key parameters related to trapping/detrapping by optical spectroscopies including photoluminescence (*9*) because the optical and thermal transition energies of traps in semiconductors are different (*10*). In addition, optical spectroscopies cannot detect shallow traps with energies *E*_{t} < 0.3 eV due to the strong Urbach tail absorption (*11*). Other techniques, such as thermal emission, can be limited by the low activation energy of shallow traps. The presence of several phase transitions in OMHPs also prevents adequate cooling of the sample to reveal the properties of traps via thermal relaxation of traps.

Recently, by combining the time-of-flight (ToF) current waveform (CWF) and the photo-Hall effect spectroscopy, we revealed deep levels and their relative positions in the bandgap of MAPbBr_{3} single-crystal devices (*12*). Several studies observed similar deep energy transitions by optical excitation methods (*13*, *14*). In addition to deep levels, the presence of multiple shallow levels in OMHPs has been estimated theoretically (*15*, *16*). However, the trapping parameters—trapping and detrapping time constants—of these levels and the interplay of free charge carriers with these energy levels have not been shown experimentally. The primary aim of this study is to uncover the effect of traps on charge transport dynamics in MAPbBr_{3} single-crystal devices using the ToF current spectroscopy.

It is known that the predicted multiple trapping states in the bandgap of OMHP complicate the dynamics of charge transport beyond the classical model of trap-controlled mobility (*17*). Such a model, also known as the effective mobility model, considers a semiconductor with a single trap delaying free charge carriers. Therefore, a new approach is necessary to unambiguously describe the dynamics of free charge carrier transport in OMHP semiconductors. To do this, we use Monte Carlo (MC) simulations to investigate the delay of charge carriers and identify the effective transit time by tracking the center of the charge cloud affected by traps. We then reassess the definition of the effective mobility (

) according to the effective transit time. Such an approach allows us to study charge transport properties in any semiconductor with any number of active traps. Last, we analyze the effect of traps on the effective mobility of carriers (holes) and its relationship with the electric field, thickness, and temperature in OMHP devices.

In addition, to demonstrate the effect of traps on the performance of MAPbBr_{3} single-crystal devices, we explore the influence of traps on the mobility-lifetime product. Understanding the impact of traps on charge transport properties is necessary to control the traps and to further improve the performance of OMHP devices.

## RESULTS

### Characterization of charge transport in a semiconductor by ToF spectroscopy and MC simulation

The ToF current spectroscopy is based on analyzing a transient photocurrent generated by charge carriers drifting through the sample under an applied bias, as demonstrated in Fig. 1A. In general, traps in a material affect the charge cloud and, as a result, the measured CWFs. Thus, CWF provides valuable information about free carrier relaxation including trapping and detrapping times from the trap states in the bandgap. The working principle of this method is shown schematically in Fig. 1B. Here, a laser pulse of 1 μs generates electron-hole pairs near the illuminated electrode. Then, an applied bias pulse of 0.5 ms separates these electron-hole pairs. As a result of free holes drifting toward negative cathode, a current is generated. During the drift process, free holes interact with defects in the bulk of the material, which affects the dynamics of the measured current. More information describing the ToF technique can be found in Methods, the Supplementary Materials, and in our previous studies (*12*).

The analytical calculation of ToF CWFs based on the current continuity, simplified Shockley-Reed-Hall (SRH) model, and drift-diffusion equation is limited to only simple examples (e.g., a solution of the drift-diffusion equation with a single trap) (*18*). Thus, numerical simulations are necessary in the case of multiple trapping states.

MC simulation is known as a powerful and convenient method for studying charge dynamics. To model the experimental results of ToF spectroscopy and to decipher the effect of trapping and detrapping of carriers from defect states, we first solve charge transport equations by one-dimensional (1D) MC simulation (*19*). In this simulation, each MC particle represents either a free hole drifting with a velocity of μ_{h}*E*(*x*) in the valence band (where μ_{h} is the drift mobility and *E*(*x*) is the applied electric field) or a trapped hole in one of the states in the bandgap. Here, using the MC simulation, we develop a charge transport model, including nonradiative energy transitions associated with traps. We simulate how the trapping and detrapping of photogenerated charge carriers limit the drift of free carriers in a MAPbBr_{3} single-crystal device at different electric biases relevant to the device operation. In this approach, the simulated trapping and detrapping of charge carriers provide insight into the free carrier dynamics and charge transport across the bulk of a MAPbBr_{3} single-crystal device. The proposed model is further validated by the ToF measurement.

The temporal dynamics of free carriers is considered as the most crucial characteristic of a material, defining the efficiency of a semiconductor device. According to SRH model (*20*, *21*), the temporal dynamics, given by Eq. 1, is mainly affected by energy states in the bandgap

(1)

The transitions between traps in single crystals are neglected, considering the low probability of such a process with respect to the band-to-trap transition (*20*). The details of SRH model are shown in the Supplementary Materials (eqs. S9 to S11). The effect of traps on the free hole concentration (*p*) is described by the specific trapping (τ_{Ti}) and detrapping (τ_{Di}) times of the *i*th trap, and *p*_{ti} is the concentration of holes trapped at the *i*th trap. Depending on the trapping and detrapping times, defects can induce short-term and long-term trapping of free carriers. The presence of shallow traps, causing fast trapping and detrapping, delays the free carrier drift and consequently reduces the drift mobility. Virtually, the long-term trapping is commonly associated with carrier lifetime (τ_{life}). However, in reality, a trapping center releases the trapped carriers after **τ**_{Di}, and the detrapped free carriers continue moving through a semiconductor. This effect, which is missing in the conventional carrier lifetime and effective mobility models, leads to errors in the study of transport features and charge collection properties in semiconductor devices.

Here, we show the results of MC simulation for a semiconductor with one deep trap and semiconductor with one shallow trap: (i) Figure 1 (C and D) demonstrates the MC simulation for charge transport in a semiconductor with one deep trap (long-term trapping). The deep trap supports a trapping and long-term detrapping process, where the detrapping time is longer than the trapping time and typical transit time of free carriers (τ_{D1} > τ_{T1} > *T*_{R}). The Gaussian profile of the free charge cloud does not evolve in the time since no detrapping events take place. Only the thermal diffusion process produces charge cloud broadening. The green crossed circles represent long-term trapped holes that are not detrapped in the valence band during charge cloud drift to the opposite electrode. The CWFs induced by drifting free holes have characteristic exponential decay in *t* < *T*_{R} time region. Electric current continues the exponential decay at lower biases, e.g., *U*/2, as the free carriers need additional time to reach the collecting electrode. (ii) Figure 1 (E and F) demonstrates the MC simulation depicting the effect of a shallow trap (short-term trapping-detrapping events) on free holes drifting from anode to cathode. At time *t* = 0, the free hole cloud, generated by the absorbed light at the anode, has the Gaussian profile localized at spatial point *x* = 0 in the device. The free charge cloud further drifts toward the cathode and evolves due to the presence of a shallow trap with trapping and detrapping times similar to the transit time of the charge cloud. At the time *t*_{k}, when free holes nearly reach the collecting electrode, the holes’ cloud accumulates in the vicinity of the electrode, revealing a large tail of delayed holes shown by cyan circles. These delayed holes were previously detrapped from the shallow trap back to the valence band. The shape of the charge cloud deviates from the Gaussian distribution due to the presence of delayed carriers. CWF induced by drifting free holes in the material with a shallow trap has a specific CWF relaxation profile, as shown in Fig. 1F. The sharp decrease at the beginning and the long tail at the end of CWF is a typical signature of the fast trapping-detrapping process induced by the shallow level defect.

### Charge transport dynamics in single crystals of MAPbBr_{3} perovskite

By probing ToF signals in p-type MAPbBr_{3} single-crystal devices, we found reliable hole signals in the CWFs (Fig. 2A), but the electron signals could not be revealed. Therefore, here, we only study the free hole transport. By studying the transit time (*T*_{R}) of the ToF CWFs, i.e., the time required for the holes to transit through the semiconductor and the relaxation dynamics before and after *T*_{R}, we can reveal the information of traps that interact with holes.

Figure 2A shows the ToF transient current from holes collected by a 100-V electric bias. The 20-nm semitransparent Cr electrode (anode) immediately collects the electron cloud, and only free holes drifting through the bulk of MAPbBr_{3} toward the cathode induce ToF signals. The profile of the hole drift in MAPbBr_{3} reveals complex relaxation dynamics: two exponentially decaying regions before the transit time (*T*_{R} = 32 μs) and a long current tail after *T*_{R}. The distinct transit time region (at *t* = *T*_{R}) indicates that a dominant fraction of the charge cloud reached the cathode and produced a transit time bending (*22*) [change of the *I*(*t*) curvature] of CWF. The long *T*_{R} indicates a long lifetime of holes (τ_{life} > *T*_{R}) and a relatively low free hole mobility in the MAPbBr_{3} single-crystal device.

We applied the MC simulation in combination with the least square regression analysis to explore the complex charge dynamics and to assess transport parameters of MAPbBr_{3} single crystals. The details of MC simulation can be found in the Supplementary Materials (fig. S2 and eq. S1 to S6). The fitting results are shown in Fig. 2G, Table 1, and table S1. To fit the CWF, we initially consider four models based on the number of traps. As can be seen in Fig. 2G, the fitting with one and two traps shows a substantial deviation from ToF CWF. By comparing the shape of CWF with the shape of MC fit, the presence of a third trap is evident. This additional trap is responsible for the current tail broadening in the *T*_{R} region. The fitting with three- and four-trap models in Fig. 2G shows the least deviation from the CWF. However, the four-trap model does not improve the fit. By analyzing the three- and four-trap models, we found that trap E_{1} (in three-trap model) splits into two traps (E_{1} and E_{4} in a four-trap model) with nearly the same parameters, τ_{T} = 50 μs and τ_{D} = 3 μs (table S1). This analysis confirms that an additional trap (four-trap model) is not necessary to describe the relaxation dynamics in MAPbBr_{3}.

Here, we discuss the three-trap model describing the ToF CWF results. The red line in Fig. 2 (A, C, and E) represent the MC transport model (three-trap model) with minimum deviations from the ToF spectra. This MC model with parameters summarized in Table 1 reveals that three-trap states affect the hole transport. According to the simulated MC transport model (Fig. 2H), the two traps, E_{1} and E_{2}, cause the fast trapping of free holes with nearly the same trapping time of 23 and 24 μs, respectively. These traps shortly release the trapped holes to the valence band with detrapping times of 3 and 14 μs, respectively. The trap level, E_{3}, gives rise to the long-lasting carrier trapping with a trapping time of 90 μs. This trap shows a relatively slow (τ_{D3} > *T*_{R}) detrapping time of 120 μs.

To study the effect of traps on charge transport, it is convenient to separate the free carrier profiles from carrier trapping and detrapping phenomena. Using the MC simulation, we divide the free holes into three groups: never trapped holes (holes that did not interact with any traps), delayed holes (holes detrapped at least once by any traps), and long-term trapped holes (holes trapped by the trap E_{3}). Figure 2 (B, D, and F) represent the simulated evolution of a free hole cloud and its interaction with traps during the drift through a MAPbBr_{3} single-crystal device under the electric bias of 100 V at different times (*t*_{1} = 6 μs, *t*_{2} = 14 μs, and *t*_{3} = 23 μs). The E_{1} and E_{2} traps decelerate free holes by relatively fast trapping-detrapping processes, and as a result, the fraction of delayed holes increases with time. As can be seen in the bottom panel of Fig. 2F, when the charge cloud reaches the collecting electrode, the concentration of delayed holes is comparable with the concentration of never trapped holes. Thus, the detrapped carriers create an extended profile of delayed holes, which deviate from the total charge cloud in the Gaussian distribution (see blue and red lines in Fig. 2F). In contrary to the fast trapping-detrapping dynamics, the trap E_{3} causes the long-term trapping of holes, which remain trapped in this defect (*p*_{t}) during the whole drifting process (see Fig. 2F).

We estimated the effective transit time of 48 μs (see Fig. 2E) describing the average delaying of the hole cloud by traps from the MC simulation, while *T*_{R} = 32 μs reflects the transit time of the never trapped holes. Here, we revealed the presence of traps, their parameters, and their roles in delaying free hole transport in a MAPbBr_{3} single-crystal device under the bias of 100 V. It is expected that the traps E_{1}, E_{2}, and E_{3} are located near the valence band as they have a direct influence on free hole transport in MAPbBr_{3}.

### Validation of the charge transport model and uniform electric field profile

It is well established that the drift velocity (*v*_{dr}) of charge carriers is directly proportional to the electric field and the free carrier mobility. Therefore, free carriers driven by a lower bias need longer time to be collected by the electrode. To validate the simulated MC transport model and to study the effect of electric field on the transit time and hole trapping dynamics, we performed ToF CWF measurements at different biases (Fig. 3A). The results of MC simulations (based on parameters in Table 1) agree very well with the experimentally measured, bias-dependent CWFs. MC results follow the main trends of ToF results, including the sharp current decrease at the beginning of CWF and the long tail after *T*_{R}. The effect of the trap E_{3} is even more evident at lower biases between 20 and 80 V. As can be seen in Fig. 3B, both experimental CWFs and MC simulations follow the same trend of a single trap with slow detrapping (trap E_{3}) in the charge transit region, *t* < *T*_{R}. The good agreement between the simulated and measured results at all biases confirms the reliability of the MC simulation and supports the proposed explanation of charge transport dynamics.

In addition to short- and long-term charge trapping, defects can induce an electric field distortion by creating a depletion region near the metallic electrode (*22*). Several studies demonstrated the presence of mobile defects in OMHPs and discussed that the collection of these low-mobility species at the interface could lead to the deformation of the electric field profile (*23*). To suppress the possible formation of the space charge in the sample during a ToF experiment, we use a short voltage pulse of 0.5 ms, synchronized with a light pulse of 1 μs. Note that the drift of photoinduced carriers across the material can also result in an electric field distortion. Here, by integrating the CWF in Fig. 3A, a low carrier concentration of ~10^{6} cm^{−3} and an electric field distortion of 0.6 V cm^{−1} are obtained (see details in the Supplementary Materials and eq. S8). Therefore, the effect of free charge carriers on the electric field distortion is negligible. In general, mobile ions have much lower mobility than free carriers. Recent studies found the mobility of halide ions to be 4 × 10^{−7} cm^{2} s^{−1} V^{−1} (*24*). Using ion mobility in eq. S7, we found the transit time of halide defects for a thick crystal with 2-mm distance between the electrodes to be around 10^{3} s. Thus, the contribution of ions in ToF CWFs is negligible, considering that the maximum transit time for free holes is 250 μs and the pulse bias width is 500 μs.

CWFs at different biases can be used to verify the electric field profile and the presence of the space charges. The presence of non-negligible space charges deforms the electric field. The deformed electric field systematically prolongs the transit time of the never trapped charge cloud at different biases (*22*). Thus, the transit times in ToF CWFs follow Eq. 2 in a semiconductor with space charges

(2)

According to ToF CWFs and MC simulations in Fig. 3C, all CWFs have the same *T*_{R}(*U*) ∙ *U* product, which confirms the uniform distribution of the electric field in MAPbBr_{3} single-crystal devices.

### Charge distribution in MAPbBr_{3} induced by trap states at different biases

In the previous section, we showed that the fast and slow trapping-detrapping processes, induced by three energy states in the bandgap, affect the charge dynamics in MAPbBr_{3} devices. To further understand the impact of each trap on the charge transport at different electric biases (100, 20, and 1 V), we studied the time evolution of the charge cloud using both the MC simulation and ToF CWF (Fig. 4). We found that the relaxation dynamics of free holes changes with bias (Fig. 4, A, C, and E) and the number of trapping-detrapping events increases at lower electric bias. We attribute the changes in relaxation dynamics and number of trapping-detrapping events to the interplay between traps and free holes, which varies at different biases. As can be seen in Fig. 4 (B, D, and F), at lower bias, the traps generate higher concentration of delayed holes in MAPbBr_{3}. This is because the charge cloud needs more time to drift across the material, and as a result, there is a higher possibility of their interaction with traps.

The interplay between free holes and traps follows four regions as illustrated in Fig. 4 (A, C, and D). After a light pulse generates the free holes, they start to drift through the bulk. All traps actively capture the free holes, leading to the occupation of all traps and, consequently, the reduced concentration of free holes in the region (i). In this region, the charge trapping induced by the traps E_{1} and E_{2} dominates with faster trapping times. The trap E_{1} first reaches a saturation point (a steady-state condition, in which the trapping and detrapping rates are equal) due to the faster detrapping time from this trap (τ_{D1} > τ_{D2} > τ_{D3}). Next, the holes detrapped from E_{1} are retrapped by the traps E_{2} and E_{3}. Usually, in a material with a single trap, the occupation of a trap does not change after a steady-state condition is reached. Because of the presence of three traps, the occupation of the trap E_{1} decreases after the saturation due to the trapping by other traps (E_{2} and E_{3}). We note that the cross-retrapping process of the delayed holes can play an important role in the further delaying of the charge cloud.

Similar to the trap E_{1}, the trap E_{2} reaches a steady-state condition at region (ii). When the traps E_{1} and E_{2} both attain their steady-state conditions, they do not capture additional free holes. Therefore, the trap E_{3} further dominates the free carrier trapping, which leads to an exponential decay of the free hole concentration in the region (ii) with the time constant τ_{T3}. At the time *T*_{R}, the never trapped holes reach the cathode in the region (iii). Since a fraction of holes were collected at the electrode, the occupation of all traps decreases after *T*_{R}. A substantial fraction of delayed holes reaches the cathode after *T*_{R}, as shown in the bottom panels of Fig. 4 (B, D, and F), due to the presence of several traps participating in trapping-detrapping events and cross-retrapping in MAPbBr_{3}. Here, the delaying of the charge cloud results in the prolongation of the effective transit time,

at lower biases.

Last, at biases lower than 1 V (*E* < 5 V cm^{−1}) in region (iv), all traps reach a steady-state condition after processes in the regions (i) and (ii); therefore, the occupation of traps and the free hole concentration do not further change with time. Because of the long *T*_{R} (*>*τ_{D3}), the defect E_{3} also participates in the hole detrapping in the region (iv). The steady-state regime in the time region (iv) is qualitatively similar to the steady-state photoconductivity or solar cell operation regime. The continuous illumination used in PV devices leads to the redistribution of free charges, and a substantial fraction of the photogenerated carriers remains trapped in defects. Note that at low electric field (Fig. 4F), the charge cloud does not reach the halfway of the path in the bulk at the transit time *T*_{R} (corresponding to the never trapped holes) due to the delaying effect of traps. This example explicitly demonstrates the highly detrimental influence of traps on the charge drift in OMHP devices. Besides decreasing the charge collection efficiency, the trapped carriers also contribute to the memory effect and the variation of device transport parameters based on the rate of scanning (up to a few milliseconds) and the illumination intensity.

### Delaying effect of traps on the charge transport

Using the ToF combined with the MC simulation, we demonstrated that the activity of traps leads to a noticeable delay of the charge cloud. The effective mobility,

${\mathbf{\mu}}_{\mathbf{h}}^{\mathbf{\text{eff}}}$, can qualitatively describe this delaying process. Therefore, the conventional model (*17*) considering a single trap in the material can be modified for the case of multiple traps (see eqs. S9 to S16). If the drift of holes is limited by shallow traps, then the effective hole mobility

is given by

$${\mathrm{\mu}}_{\mathrm{h}}^{\text{eff}}={\mathrm{\mu}}_{\mathrm{h}}\frac{1}{1+{\Sigma}_{i}\frac{{\mathrm{\tau}}_{\mathrm{D}i}}{{\mathrm{\tau}}_{\mathrm{T}i}}}$$(3)

However, this definition is only valid when all traps reach a saturation point, the condition attainable only at low biases of <1 V (*E* < 5 V cm^{−1}). Therefore, a new definition is necessary to describe

at higher biases as well.

To modify the effective mobility,

${\mathbf{\mu}}_{\mathbf{h}}^{\mathbf{\text{eff}}}$, we track the center of the charge cloud, which drifts in a semiconductor with multiple traps. The MC simulation allows us to determine the effective transit time,

${T}_{\mathrm{R}}^{\text{eff}}$, of the total charge cloud and to calculate the corresponding effective mobility. The effective mobility can be defined by the following equation

$${\mathrm{\mu}}_{\mathrm{h}}^{\text{eff}}={\mathrm{\mu}}_{\mathrm{h}}\frac{{T}_{\mathrm{R}}}{{T}_{\mathrm{R}}^{\text{eff}}}=\frac{{L}^{2}}{{T}_{\mathrm{R}}^{\text{eff}}U}$$(4)where the effective transit time,

${T}_{\mathrm{R}}^{\text{eff}},$ describes how the traps delay the drift of free charge cloud. Note that the trapping/detrapping distorts the charge cloud substantially as demonstrated in Fig. 4 (B, D, and E; bottom). Thus, the drift mobility (μ_{h}) cannot be determined by only ToF measurements without considering the delaying effect of traps. The value of μ_{h}, unaffected by shallow traps, can be determined by considering the trapped and detrapped carriers in the MC simulation.

We performed the MC simulation to analyze the influence of each trap on hole transport and the effective hole mobility

${\mathbf{\mu}}_{\mathbf{h}}^{\mathbf{\text{eff}}}$and to obtain the dependence of

${\mathbf{\mu}}_{\mathbf{h}}^{\mathbf{\text{eff}}}$ on the electric field at different MAPbBr_{3} thicknesses (Fig. 5A). At a high electric field, the electrode collects the holes so rapidly that the effect of traps is negligible. Figure 5A shows that

for all material thicknesses converges to the drift mobility, unaffected by traps, i.e., 12.4 cm^{2} V^{−1} s^{−1}. The drift mobility obtained by ToF with MC simulations agrees well with experimentally measured drift mobility of 10 to 20 cm^{2} V^{−1} s^{−1} (*25*), which is unaffected by traps.

Reducing the electric field leads to stronger interactions between holes and traps, thereby reducing

${\mathbf{\mu}}_{\mathbf{h}}^{\mathbf{\text{eff}}}$ as seen in Fig. 5A. The traps E_{1} and E_{2} cause the initial decrease in

to

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{1}}$and further down to

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{2}}$as the electric field is reduced. Here,

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{1}}$and

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{2}}$ are the effective hole mobilities limited by traps E_{1} and E_{2} under the steady-state condition. The values of

and

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{2}}$ are 11.0 and 7.8 cm^{2} V^{−1} s^{−1}, respectively, based on Eq. 3 and trapping/detrapping ratios in Table 1. After

drops below

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{2}}$ (7.8 cm^{2} V^{−1} s^{−1}), the cross-retrapping of detrapped holes by both traps E_{1} and E_{2} primarily drives further decrease in

down to

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{1}+\mathbf{E}\mathbf{2}}$ (7.2 cm^{2} V^{−1} s^{−1}). As demonstrated in Fig. 4 (A, C, and D), trap E_{1} is the first trap that reaches the steady-state condition. After *t* = τ_{D1} = 3 μs, this trap (E_{1}) actively detraps holes. Trap E_{2} further captures and releases detrapped free holes. Thus, both traps actively delay the drift of free hole cloud in the bulk of MAPbBr_{3}.

At low electric fields, e.g., 300 V cm^{−1}, the *T*_{R} of the charge cloud is higher than τ_{D3}; therefore, trap E_{3} can effectively participate in the delay of hole cloud and further reduce the effective hole mobility. The cross-retrapping processes involving all three traps interacting with delayed holes reduces

below the effective hole mobility limited by a single trap E_{3},

= 5.3 cm^{2} V^{−1} s^{−1}. At sufficiently low electric fields,

saturates at

${\mathbf{\mu}}_{\mathbf{\text{eff}}}^{\mathbf{E}\mathbf{1}+\mathbf{E}\mathbf{2}+\mathbf{E}\mathbf{3}}$ (4.0 cm^{2} V^{−1} s^{−1}). The value of

agrees well with the effective mobility found from Eq. 3 (i.e., 4.0 cm^{2} V^{−1} s^{−1}), which is valid when all three traps reach the steady-state condition. These results demonstrate that the proposed model of effective mobility correctly describes the complicated carrier trapping, detrapping, and retrapping, involving multiple traps during the charge transport process in MAPbBr_{3} single crystals.

Next, we discuss the thickness dependence of the effective mobility. At the constant electric field, a thicker MAPbBr_{3} device prolongs transit time of the drifting charge cloud and, consequently, results in a lower

(*E*) according to eq. S7 (see Fig. 5A). In thin-film devices, the

(*E*) rapidly converges to the drift mobility as the free holes reach the collecting electrode with little trap interaction. In contrast, a large number of trapping-detrapping events take place in thick samples where the charge cloud interacts substantially with traps along its long drift path. Therefore,

slowly converges to the drift mobility with increasing electric field in thick samples. In addition,

${\mathbf{\mu}}_{\mathbf{h}}^{\mathbf{\text{eff}}}$has a lower absolute value compared to those in thin devices under the same electric field.

In general, phonon scattering dominates the temperature dependence of the drift mobility (following the power law of μ ~ *T*^{−1.5}), which, in turn, influences the temperature dependence of the effective mobility. In addition, the trap activities also affect the temperature dependence of the effective mobility,

(*T*). We modeled

(*T*) under the steady-state condition following Eq. 3 and eqs. S10 and S11. The charge-trapping rate varies slightly with temperature, whereas the detrapping rate decreases more rapidly with decreasing temperature. Therefore, although lowering temperature reduces phonon scattering, which tends to increase the effective mobility, it also suppresses detrapping of charge carriers from traps thereby lowering the effective mobility. These two effects combine to give the temperature dependence of the effective mobility shown in Fig. 5B. We observe a strong deviation of the effective mobility from the *T*^{−1.5} (*26*) dependence as temperature decreases, especially for *T* < 250 K, below which the detrapping activity is frozen. Previous experiments based on time-resolved methods (e.g., ToF, transient space charge–limited currents, etc.) (*27*) showed deviations of the measured drift mobility from the *T*^{−1.5} dependence. These experiments, however, did not address the effect of traps on the charge delaying. The present work based on MC simulations provides an explanation to this deviation. Here, we demonstrate a theoretical prediction that agrees with the temperature effect on the effective mobility reported in organic semiconductors (*28*).

### Analysis of drift mobility, lifetime, and mobility-lifetime product (μτ) by ToF and MC simulation in MAPbBr_{3} single-crystal device

Several ToF studies used the inflection (trap-free approach) or intersection (dispersive photocurrent) points between transit and tail regions of CWF (see details in fig. S3) to evaluate the drift mobility in both organic and inorganic semiconductors. We evaluated the possible errors in the determination of the drift mobility using the above approach. Figure 5C compares the drift mobilities calculated by a MC simulation and by the standard methods. The hole mobilities found from inflection (μ_{inf}) and intersection (μ_{inter}) transit times show notable deviations (up to 7.8 cm^{2} V^{−1} s^{−1} at 100 V cm^{−1}) from the drift mobility in the MC simulation. The deviation increases at low biases, which agrees with the MC simulation. Note that the MC simulation predicts a larger number of trapping-detrapping events induced by traps and more notable deformation of the hole cloud at lower biases demonstrated in Fig. 4 (B, D, and F). Thus, the deformed charge cloud leads to an incorrect treatment of ToF results by simplified approaches, which do not include the effect of traps on the deformation of the charge cloud.

The trapping time of 90 μs from the trap E_{3} obtained in this work is in agreement with our previous calculation of the lifetime of free holes limited by the long-term trapping in MAPbBr_{3} (*12*). Such a low trapping time of free holes highlights the advanced transport properties of MAPbBr_{3} devices. However, the interplay of free charge carriers with traps has a detrimental effect on the performance of OMHP detectors due to the decreased effective mobility and long-term trapping. It is demonstrated that the free carrier dynamics follow a complex nonexponential decay with fast (23 and 24 μs) and slow (90 μs) trapping characters accompanied by the fast (3 and 14 μs) and slow (120 μs) detrapping of carriers, respectively. The release of trapped charge carriers from defects in MAPbBr_{3} single crystals leads to a notable divergence of lifetime values measured under dynamical [0.3 (*29*) and 1 (*30*) μs] and steady-state [>1 ms (*31*, *32*)] conditions. The fast trapping is typically interpreted as the charge carrier lifetime in time-resolved measurements, while the long-term (slow) trapping influences a steady-state lifetime. Thus, the detrapping and the resulting complex carrier decay must be considered for the correct characterization of OMHPs. The important role of detrapping in the lifetime measurements was also recently discussed by Lang *et al.* (*33*). A trapping/detrapping ratio of 3.7 estimated by averaging the values in Table 1 is in agreement with 3.4 in mixed hybrid perovskites reported previously (*33*). In addition, Lukosi *et al.* (*34*) estimated a trapping time of 26 μs and a detrapping time of 15 μs using a single-trap model to describe the carrier relaxation in MAPbBr_{3} single-crystal detectors. Despite the limitation of the single-trap model, these results agree with parameters of the trap E_{2} in Table 1.

In general, the trapping and detrapping phenomena are scarcely studied in OMHPs. Thus, it is useful to compare the trap parameters (τ_{T}/τ_{D}) in MAPbBr_{3} with those in inorganic semiconductors. Using ToF measurements in combination with MC simulation, we found the ratio of trapping/detrapping (τ_{T}/τ_{D}) from shallow traps to be ~250 ns/40 ns and from deep traps to be ~150 ns/3 μs in GaAs devices (*35*) and 13 ns/11 ns and 2 μs/20 μs in CdZnTe devices (*35*), respectively. Noticeably, the trapping-detrapping times in inorganic semiconductors are much faster; thus, the exceptional performance of halide perovskites optoelectronics can be attributed to the low capture cross-section of defects. The low capture cross section typically corresponds to the strong screening of the charged defects in OMHPs, leading to a low nonradiative recombination rate, as previously proposed (*36*). This effect can be also attributed to the polaronic nature of charge carriers in OMHPs (*37*) or unstable defects configuration after trapping (*38*). In addition to the suppressed capture cross section, we can explain the low trapping time in OMHPs by a low concentration of deep defects with long detrapping time.

The interplay between free charge carriers and traps can affect the charge collection properties and mobility-lifetime product, μτ. Figure 5D demonstrates the results of ToF and MC simulations fitted by the Hecht equation (*39*) to obtain μτ-product in a MAPbBr_{3} single-crystal device. Because of the delay of the hole transport by traps, μτ shows a strong dependence on the collection time (10^{−1} to 10^{−4} cm^{2} V^{−1}), which is the time needed to detect high-energy absorption events. Engineering a lower concentration of traps, particularly trap E_{3}, can cancel the detrimental influence of traps as shown by MC simulations (Fig. 5D, blue dash-dotted line). The longer detrapping time of 120 μs from the defect E_{3} can compete with the transit time of the free charge cloud and the typical charge collection time of classical inorganic semiconductors [typically <200 μs (*40*)]. By considering the collection of free holes in 200 μs, the MC simulation in combination with ToF result gives μτ = 10^{−3} cm^{2} V^{−1} in MAPbBr_{3} devices. This is competitive with the best mobility-lifetime product found with similar methods in inorganic detectors such as CdZnTe (10^{−3} cm^{2} V^{−1}) (*22*) and GaAs (6 × 10^{−4} cm^{2} V^{−1}) (*35*).

We calculated a diffusion coefficient of 0.32 cm^{2} s^{−1} and a diffusion length of 54 μm in MAPbBr_{3} single-crystal devices by considering the mobility and long-term trapping found from our ToF measurements and MC simulations. The results agree well with the previously reported diffusion coefficient of 0.27 cm^{2} s^{−1} (*27*) and a diffusion length of 2.6 to 650 μm (*31*). A solar cell is usually fabricated using a very thin active layer of ~300 to 500 nm. Thus, a rather large internal electric field induced by the work function difference between electrodes (~1 eV) separates the charge carriers. Consequently, the transit time in a solar cell is on the order of nanoseconds, which is much shorter than the trapping times found in this study. Therefore, the demonstrated traps are not as critical in a PV device as in ionizing radiation detectors made from bulky single crystals.

However, in solar cells, the high intensity of illumination yields a steady-state hole density of more than 10^{13} cm^{−3} (*p =* Photon Flux ·*T*_{R} per thickness). Such a high density of photogenerated carriers could completely fill the traps (holes) up to the position of the quasi–Fermi energy *E*_{F} = *E*_{v} + 0.35 eV. This result has three crucial implications. First, the strong carrier trapping induces substantial space charge that could completely screen the photovoltage at a density above 10^{14} cm^{−3}. Second, traps populated by holes could subsequently trap electrons, resulting in nonradiative recombination and reduce *V*_{OC} in OMHP solar cells. The trapping-detrapping activity decreases the effective mobility, which can consequently lead to instability of photovoltage (after light is switched on/off), and causes a memory effect (hysteresis). Third, the typically higher concentration of defects in thin films, roughly by two orders of magnitude (*41*), can lead to a much faster trapping times of defects E_{1} and E_{2}. The decrease in the trapping time can transform the traps with shallow character into fast trapping centers, since the detrapping time does not depends on the trap concentration. For example, the trap E_{2} can increase its trapping time up to ~240 ns in thin films.

### Nature and chemistry of traps

Cation vacancies (V_{Pb} and V_{MA}) and Br interstitial (Br_{i}) are the main acceptors in MAPbBr_{3} identified by first-principles calculations (*15*). Similar results have been also demonstrated in MAPbI_{3} (*16*, *42*). We tentatively assign E_{1}, E_{2}, and E_{3} to V_{MA}, V_{Pb}, and Br_{i} considering the results of previous density functional theory calculations. The formation of V_{MA} does not involve Pb─Br bond breaking, and thus, V_{MA} is likely the shallowest acceptor with relatively short trapping and detrapping times as found for the shallow hole trap E_{1} (Table 1). On the other hand, the formation of V_{Pb} requires Pb─Br bond breaking, which should lead to a slightly deeper acceptor level than that of V_{MA} as found by calculations (*15*). In addition, the −2 charged V_{Pb} has a stronger Coulomb attraction to holes than the −1 charged V_{MA}. A deeper level tends to make both the trapping and detrapping times longer, while a stronger Coulomb attraction tends to increase the detrapping time but decrease the trapping time. The combination of the above two effects should lead to a longer detrapping time for V_{Pb} compared to that of V_{MA} but partially cancel each for hole trapping, resulting in similar trapping times for the two cation vacancies. These characteristics are consistent with the trapping/detrapping times of E_{1} and E_{2}. In contrast to V_{MA} and V_{Pb}, which introduce shallow hydrogenic levels, the hole trapped by Br_{i} is strongly localized (*16*) as also found for I_{i} in MAPbI_{3} (*42*). Strong localization of Br_{i} defect leads to deeper hole trapping and long trapping/detrapping times, consistent with the behaviors of trap E_{3}.

The calculated hole trapping level [the (0/−) transition level] by Br_{i} in MAPbBr_{3} is about *E*_{v} + 0.13 eV (*15*), comparable to the calculated (0/−) level (*E*_{v} + 0.15 eV) of I_{i} (*42*). However, the calculated shallow defect levels (*15*, *16*) are not expected to be converged with respect to the supercell size because delocalized wave functions associated with shallow levels cannot be modeled accurately using relatively small supercells. The direct assignment of the trap parameters (*E*_{t}, σ, and *N*_{t}) found from the experiment to the specific defect nature is a longstanding challenge in the field of OMHP materials. Previous studies detected several deep traps in MAPbBr_{3} with activation energies of 1.05, 1.5, and 0.7 eV (*12*–*14*), a concentration of 10^{11} cm^{−3} (*12*), and capture cross sections in the range of 10^{−17} to 10^{−15} cm^{2} (*12*, *43*). Less studied shallow traps demonstrate activation energies in the range of 0.15 to 0.3 eV (*12*, *33*, *43*). Estimation of the activation energy and other parameter of defects such as concentration is beyond the scope of this study and will be considered in our future studies. Materials processing such as optimization of growth parameters, doping, or purification can reduce the trap density. Recently several studies showed that combining Cs and Rb in quadruple cation (Rb-Cs-FA-MA) perovskite mixtures increases the effective mobility and decreases the trap density, resulting in solar cells with the highest stabilized power efficiency (*33*).

**Acknowledgments: **A.M., J.P., P.P., M.B., E.B., and R.G., thank the Institute of Physics of Charles University for providing necessary facilities to conduct this research. **Funding:** A.M., J.P., P.P., M.B., E.B., and R.G acknowledge financial support from the Grant Agency of the Czech Republic, grant no. P102/19/11920S, and the Grant Agency of Charles University, project no. 1234119. M.A. and B.D. acknowledge financial support from U.S. Department of Homeland Security (grant no. 16DNARI00018-04-0). M.-H.D. is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences, and Engineering Division. **Author contributions:** A.M. and M.A. wrote the paper. A.M. conceived the idea and conducted the experiments. M.A. and B.D. grew the perovskite single crystals and performed the device fabrication. P.P., E.B., and A.M. contributed to the design of the experimental setup. J.P. and A.M. performed the theoretical calculations. A.M. and R.G. supervised the theoretical simulation. M.-H.D. contributed to the defect nature assignment. M.B., J.P., and A.M. contributed to the figures and preparation of video materials. All authors contributed to results’ discussion and approved the final version. **Competing interests:** The authors declare that they have no competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in this paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the corresponding author, A.M.