MATERIALS SCIENCE Copyright © 2020 Deciphering the effect ... · beyond the classical model of trap-controlled mobility (17). Such a model, also known as the effective mobility model,

Musiienko et al., Sci. Adv. 2020; 6 : eabb6393 11 September 2020

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M A T E R I A L S S C I E N C E

Deciphering the effect of traps on electronic charge transport properties of methylammonium lead tribromide perovskiteArtem Musiienko1*, Jindřich Pipek1, Petr Praus1, Mykola Brynza1, Eduard Belas1, Bogdan Dryzhakov2, Mao-Hua Du3, Mahshid Ahmadi2*, Roman Grill1

Halide perovskites have undergone remarkable developments as highly efficient optoelectronic materials for a variety of applications. Several studies indicated the critical role of defects on the performance of perovskite devices. However, the parameters of defects and their interplay with free charge carriers remain unclear. In this study, we explored the dynamics of free holes in methylammonium lead tribromide (MAPbBr3) single crystals using the time-of-flight (ToF) current spectroscopy. By combining ToF spectroscopy and Monte Carlo simulation, three energy states were detected in the bandgap of MAPbBr3. In addition, we found the trapping and detrapping rates of free holes ranging from a few microseconds to hundreds of microseconds. Contrary to previous studies, we revealed a strong detrapping activity of traps. We showed that these traps substantially affect the transport properties of MAPbBr3, including mobility and mobility-lifetime product. Our results provide an insight on charge transport properties of perovskite semiconductors.

INTRODUCTIONOn 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. An-other 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 (MAPbBr3) 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 multifunc-tional 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 non-

radiative nature of these energy transitions does not allow the mea-surement 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 dif-ferent (10). In addition, optical spectroscopies cannot detect shallow traps with energies Et < 0.3 eV due to the strong Urbach tail absorp-tion (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 MAPbBr3 single-crystal devices (12). Several studies observed similar deep energy transitions by optical excitation methods (13, 14). In addi-tion 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 en-ergy levels have not been shown experimentally. The primary aim of this study is to uncover the effect of traps on charge transport dynamics in MAPbBr3 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 semicon-ductors. To do this, we use Monte Carlo (MC) simulations to inves-tigate 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 ( 𝛍 h eff ) according to the effective transit time. Such an approach allows us to study charge transport properties in any semiconductor with any number

1Charles University, Faculty of Mathematics and Physics, Institute of Physics, Ke Karlovu 5, CZ-121 16 Prague 2, Czech Republic. 2Joint Institute for Advanced Materials, Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA. 3Oak Ridge National Laboratory, Materials Science and Technology Division, Oak Ridge, TN 37831, USA.*Corresponding author. Email: [email protected] (A.M.); [email protected] (M.A.)

Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).

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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 MAPbBr3 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.

RESULTSCharacterization of charge transport in a semiconductor by ToF spectroscopy and MC simulationThe ToF current spectroscopy is based on analyzing a transient photocurrent generated by charge carriers drifting through the sam-ple under an applied bias, as demonstrated in Fig. 1A. In general,

traps in a material affect the charge cloud and, as a result, the mea-sured 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 s 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 solu-tion of the drift-diffusion equation with a single trap) (18). Thus, numerical simulations are necessary in the case of multiple trap-ping 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 detrap-ping of carriers from defect states, we first solve charge transport equations by one-dimensional (1D) MC simulation (19). In this sim-ulation, each MC particle represents either a free hole drifting with a velocity of hE(x) in the valence band (where 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 MAPbBr3 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 MAPbBr3 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 semi-conductor device. According to SRH model (20, 21), the temporal dynamics, given by Eq. 1, is mainly affected by energy states in the bandgap

dp

─ dt = i ( −

p ─ Ti +

p ti ─ Di ) (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 trap-ping (Ti) and detrapping (Di) times of the ith trap, and pti is the concentration of holes trapped at the ith 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 (life). How-ever, in reality, a trapping center releases the trapped carriers af-ter Di, and the detrapped free carriers continue moving through a

A B

DC

FE

Fig. 1. ToF spectroscopy and MC simulation. (A) A schematic of physical principle of ToF method. The light pulse generates free carriers (holes) near the electrode (anode). The photogenerated free holes further drift toward the cathode by an electric pulse bias. The drifting holes interact with traps, and their interaction affects the current transient. (B) ToF experimental setup. (C) Top: Simulation of the holes drifting from anode to cathode in the bulk of a semiconductor characterized by a deep trap. a.u., arbitrary units. The time t = 0 represents the initial hole distribution immediately after illumination. At t = tk, the evolving carrier cloud reaches in the vicinity of the collecting electrode at the right side. Trapped holes (green crossed circles) remain localized in the traps. Bottom: This graph quantitatively shows the spatial distribution of trapped holes and evolution of free holes cloud-drifting be-tween the two electrodes. (D) Normalized CWFs induced by drifting holes. The single exponential decay of CWF assigns the presence of a deep trap in the material. The current was normalized by the corresponding applied bias to show the influence of deep traps on the CWF shape at different biases. (E) Top: The visualization of free holes drifting in the semiconductor with one shallow trap and a high number of trapping-detrapping events. Free carriers drift toward the collecting electrode. The shallow traps delay the fraction of free carriers demonstrated by cyan circles. Bottom: Spatial distribution of never trapped and the delayed free holes drifting between the two electrodes. (F) CWF induced by drifting holes. The tails at the beginning and the end of CWF are affected by the shallow trap.


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semiconductor. This effect, which is missing in the conventional car-rier lifetime and effective mobility models, leads to errors in the study of transport features and charge collection properties in semi-conductor 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 trans-port in a semiconductor with one deep trap (long-term trapping). The deep trap supports a trapping and long-term detrapping pro-cess, where the detrapping time is longer than the trapping time and typical transit time of free carriers (D1 > T1 > TR). 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 repre-sent 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 < TR 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) demon-strates 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 tk, 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 de-layed 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 MAPbBr3 perovskiteBy probing ToF signals in p-type MAPbBr3 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 (TR) of the ToF CWFs, i.e., the time required for the holes to transit through the semicon-ductor and the relaxation dynamics before and after TR, we can re-veal 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 MAPbBr3 toward the cathode induce ToF signals. The profile of the hole drift in MAPbBr3 reveals complex relaxation dynamics: two exponentially decaying regions before the transit time (TR = 32 s) and a long current tail after TR. The distinct transit time region (at t = TR) indicates that a dom-inant fraction of the charge cloud reached the cathode and pro-duced a transit time bending (22) [change of the I(t) curvature] of CWF. The long TR indicates a long lifetime of holes (life > TR)

and a relatively low free hole mobility in the MAPbBr3 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 MAPbBr3 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

G H

E F

C D

A B

Fig. 2. Effect of traps on the dynamics of free holes depicted by ToF and MC simulation. (A, C, and E) ToF CWF measured at 100 V. The red curve shows the best MC simulation fit with three traps at different times t1 = 6 s, t2 = 14 s, and t3 = 23 s, respectively. (B, D, and F) Top: The time evolution of the free charge cloud (subdi-vided according to their trapping-detrapping history) during the drift process at t1, t2, and t3, respectively. Bottom: The respective normalized concentrations of never trapped, trapped, and delayed holes in the material. The attached video (movie S1) shows the evolution of the charge cloud computed by the MC simulation with a 0.1-s time resolution, which allows the direct visualization of the free charge move-ment in a MAPbBr3 device. (G) The ToF CWF and MC simulation at 80-V bias. MC simulation represents four models fitted according to the least square regression analysis. The inset demonstrates the residuals of fitting curves shown in (G). (H) Charge transport model in MAPbBr3. Upward and downward arrows illustrate energy transitions (trapping and detrapping). Table 1 summarizes the parameters of these transitions estimated by ToF measurements and MC simulations.


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responsible for the current tail broadening in the TR region. The fit-ting with three- and four-trap models in Fig. 2G shows the least devi-ation 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 E1 (in three-trap model) splits into two traps (E1 and E4 in a four-trap model) with nearly the same parameters, T = 50 s and D = 3 s (table S1). This analysis confirms that an additional trap (four-trap model) is not necessary to describe the relaxation dynamics in MAPbBr3.

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 trans-port 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, E1 and E2, cause the fast trapping of free holes with nearly the same trap-ping time of 23 and 24 s, respectively. These traps shortly release the trapped holes to the valence band with detrapping times of 3 and 14 s, respectively. The trap level, E3, gives rise to the long-lasting carrier trapping with a trapping time of 90 s. This trap shows a relatively slow (D3 > TR) detrapping time of 120 s.

To study the effect of traps on charge transport, it is convenient to separate the free carrier profiles from carrier trapping and de-trapping phenomena. Using the MC simulation, we divide the free holes into three groups: never trapped holes (holes that did not in-teract with any traps), delayed holes (holes detrapped at least once by any traps), and long-term trapped holes (holes trapped by the trap E3). 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 MAPbBr3 single-crystal device under the electric bias of 100 V at different times (t1 = 6 s, t2 = 14 s, and t3 = 23 s). The E1 and E2 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 E3 causes the long-term trapping of holes, which remain trapped in this defect (pt) during the whole drifting process (see Fig. 2F).

We estimated the effective transit time of 48 s (see Fig. 2E) de-scribing the average delaying of the hole cloud by traps from the MC simulation, while TR = 32 s reflects the transit time of the never trapped holes. Here, we revealed the presence of traps, their param-eters, and their roles in delaying free hole transport in a MAPbBr3 single-crystal device under the bias of 100 V. It is expected that the

traps E1, E2, and E3 are located near the valence band as they have a direct influence on free hole transport in MAPbBr3.

Validation of the charge transport model and uniform electric field profileIt is well established that the drift velocity (vdr) 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 mea-sured, 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 TR. The effect of the trap E3 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 E3) in the charge transit region, t < TR. The good agreement between the sim-ulated 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 pres-ence of mobile defects in OMHPs and discussed that the collection of these low-mobility species at the interface could lead to the defor-mation of the electric field profile (23). To suppress the possible for-mation 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 s. Note that the drift of photoinduced carriers across the material can also result in an electric field distortion. Here, by inte-grating the CWF in Fig. 3A, a low carrier concentration of ~106 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 cm2 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 103 s. Thus, the contribution of ions in ToF CWFs is negligible, considering that the maximum transit time for free holes is 250 s and the pulse bias width is 500 s.

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

Table 1. Charge transport parameters of single-crystal MAPbBr3 found from the combination of ToF and MC simulation.

Trap Trapping time (s) Detrapping time (s) Trapping/detrapping ratio h × nt product*10−3 cm−1

E1 23 3 7.7 1.31

E2 24 14 1.7 1.25

E3 90 120 0.75 0.34

*Hole capture cross section and trap density product.


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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

T R (U ) ⋅ U ≠ const (2)

According to ToF CWFs and MC simulations in Fig. 3C, all CWFs have the same TR(U) ∙ U product, which confirms the uniform dis-tribution of the electric field in MAPbBr3 single-crystal devices.

Charge distribution in MAPbBr3 induced by trap states at different biasesIn 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 MAPbBr3 devices. To further under-stand 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 MAPbBr3. 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 cap-ture the free holes, leading to the occupation of all traps and, conse-quently, the reduced concentration of free holes in the region (i). In this region, the charge trapping induced by the traps E1 and E2 domi-nates with faster trapping times. The trap E1 first reaches a saturation point (a steady-state condition, in which the trapping and detrap-ping rates are equal) due to the faster detrapping time from this trap (D1 > D2 > D3). Next, the holes detrapped from E1 are retrapped by the traps E2 and E3. 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 E1 decreases after the saturation due to the trapping by other traps (E2 and E3). 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 E1, the trap E2 reaches a steady-state condi-tion at region (ii). When the traps E1 and E2 both attain their steady-state conditions, they do not capture additional free holes. Therefore, the trap E3 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 T3. At the time TR, 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 TR. A substantial fraction of delayed holes reaches the cathode after TR, 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 MAPbBr3. Here, the delaying of the charge cloud results in the prolongation of the effective transit time, T R eff 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)

(..)

(..)

>

= =

>

= = .

(..)

(..)

= =

>

(..)

(..)

A B

C D

E F

T T

T T

T

(a.u.)

(a.u.)

(iv)

(iv)(ii)(i)

T

R

R

R

tt

t

(a.u.)

(a.u.)

μs

ms

Fig. 4. Spatial and temporal distribution of free and trapped charges in MAPbBr3. The evolution of free holes and the occupation of trap states in a MAPbBr3 device at electric biases of 100 V (A and B), 20 V (C and D), and 1 V (E and F). (A), (C), and (E) represent the temporal evolution of free carriers and traps occupation. Flag indicators separate the regions (i) to (iv) with different relaxation dynamics. (B), (D), and (F) show the simulated visualization of the spatial distribution of traps (top), the distribution of occupied trap density (middle), and the profile of the charge cloud in MAPbBr3 at different biases (bottom). The attached videos (movies S2 to S4) show the simulated evolution of the charge cloud and trap occupation with 0.1-s time resolution, allow-ing direct visualization of the interplay between free holes and a particular trap.

A B

C

Fig. 3. ToF and MC validation of the three-trap model. (A) Bias dependence of CWFs. The green curves represent the simulated MC fit. The inset shows electric field profiles between the two electrodes. (B) Normalized CWFs at different biases. (C) CWFs dependence on normalized time according to Eq. 2.


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and (ii); therefore, the occupation of traps and the free hole concentra-tion do not further change with time. Because of the long TR (>D3), the defect E3 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 photo-generated carriers remains trapped in defects. Note that at low elec-tric field (Fig. 4F), the charge cloud does not reach the halfway of the path in the bulk at the transit time TR (corresponding to the never trapped holes) due to the delaying effect of traps. This example ex-plicitly 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 mem-ory effect and the variation of device transport parameters based on the rate of scanning (up to a few milliseconds) and the illumina-tion intensity.

Delaying effect of traps on the charge transportUsing 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, 𝛍 h eff , can qualitatively describe this de-laying 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 𝛍 h eff is given by

h eff = h 1 ─ 1 + i

Di _ Ti (3)

However, this definition is only valid when all traps reach a sat-uration point, the condition attainable only at low biases of <1 V (E < 5 V cm−1). Therefore, a new definition is necessary to describe 𝛍 h eff at higher biases as well.

To modify the effective mobility, 𝛍 h 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 R eff , of the total charge cloud and to calculate the corresponding ef-fective mobility. The effective mobility can be defined by the follow-ing equation

h eff = h T R ─ T R eff

= L 2 ─ T R eff U

(4)

where the effective transit time, T R 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 (h) cannot be determined by only ToF measurements without considering the delaying effect of traps. The value of 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 𝛍 h eff and to ob-tain the dependence of 𝛍 h eff on the electric field at different MAPbBr3 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 𝛍 h eff for all material thicknesses converges to the drift mobility, unaffected by traps, i.e., 12.4 cm2 V−1 s−1. The drift mobility obtained by ToF with MC simulations agrees well with experimen-

tally measured drift mobility of 10 to 20 cm2 V−1 s−1 (25), which is unaffected by traps.

Reducing the electric field leads to stronger interactions between holes and traps, thereby reducing 𝛍 h eff as seen in Fig. 5A. The traps E1 and E2 cause the initial decrease in 𝛍 h eff to 𝛍 eff

E1 and further down to 𝛍 eff

E2 as the electric field is reduced. Here, 𝛍 eff E1 and 𝛍 eff E2 are the ef-

fective hole mobilities limited by traps E1 and E2 under the steady-state condition. The values of 𝛍 eff

E1 and 𝛍 eff E2 are 11.0 and 7.8 cm2 V−1 s−1,

respectively, based on Eq. 3 and trapping/detrapping ratios in Table 1. After 𝛍 h eff drops below 𝛍 eff

E2 (7.8 cm2 V−1 s−1), the cross- retrapping of detrapped holes by both traps E1 and E2 primarily drives further decrease in 𝛍 h eff down to 𝛍 eff

E1+E2 (7.2 cm2 V−1 s−1). As demon-strated in Fig. 4 (A, C, and D), trap E1 is the first trap that reaches the steady-state condition. After t = D1 = 3 s, this trap (E1) active-ly detraps holes. Trap E2 further captures and releases detrapped free holes. Thus, both traps actively delay the drift of free hole cloud in the bulk of MAPbBr3.

At low electric fields, e.g., 300 V cm−1, the TR of the charge cloud is higher than D3; therefore, trap E3 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 𝛍 h eff below the effective hole mobility limited by a single trap E3, 𝛍 eff

E3 = 5.3 cm2 V−1 s−1. At sufficiently low electric fields, 𝛍 h eff saturates at 𝛍 eff

E1+E2+E3 (4.0 cm2 V−1 s−1). The value of 𝛍 eff

E1+E2+E3 agrees well with the effective mobility found from Eq. 3

C D

A B

Fig. 5. Effect of traps on the effective mobility, drift mobility evaluation, and mobility-lifetime product. (A) The effective hole mobility as a function of the electric field, as determined by the MC simulation for various thicknesses (L) of the material. The effective hole mobilities measured by the ToF for L = 0.2 cm are shown by the stars. (B) The temperature dependence of the effective mobility obtained by theo-retical modeling. The observed behavior suggests that traps affect eff(T) dependence in MAPbBr3 single crystals especially at T < 250 K. Two magenta dots show the ex-perimental drift and effective mobilities [measured at room temperature (RT) under steady-state conditions]. (C) Hole drift mobility as a function of applied electric field obtained by different approaches using data from Fig. 3A and MC simulations. The inset demonstrates the linear fit of the transit time versus electric field. (D) Mobility- lifetime product as a function of collection time. The inset in (D) shows the Hecht fit according to data from Fig. 3A.


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(i.e., 4.0 cm2 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 MAPbBr3 single crystals.

Next, we discuss the thickness dependence of the effective mobility. At the constant electric field, a thicker MAPbBr3 device prolongs transit time of the drifting charge cloud and, consequently, results in a lower 𝛍 h eff (E) according to eq. S7 (see Fig. 5A). In thin-film de-vices, the 𝛍 h eff (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, 𝛍 h eff slowly converges to the drift mobility with increasing electric field in thick samples. In addition, 𝛍 h eff has a lower absolute value compared to those in thin devices under the same electric field.

In general, phonon scattering dominates the temperature depen-dence of the drift mobility (following the power law of ~ 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, 𝛍 h eff (T). We modeled 𝛍 h eff (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 scatter-ing, 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 de-viation 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 demon-strate 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 () by ToF and MC simulation in MAPbBr3 single-crystal deviceSeveral ToF studies used the inflection (trap-free approach) or in-tersection (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 mobil-ities found from inflection (inf) and intersection (inter) transit times show notable deviations (up to 7.8 cm2 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 de-formed 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 s from the trap E3 obtained in this work is in agreement with our previous calculation of the lifetime of free holes limited by the long-term trapping in MAPbBr3 (12). Such a low trapping time of free holes highlights the advanced transport prop-erties of MAPbBr3 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 fol-low a complex nonexponential decay with fast (23 and 24 s) and slow (90 s) trapping characters accompanied by the fast (3 and 14 s) and slow (120 s) detrapping of carriers, respectively. The release of trapped charge carriers from defects in MAPbBr3 single crystals leads to a notable divergence of lifetime values measured under dynamical [0.3 (29) and 1 (30) s] 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 detrap-ping and the resulting complex carrier decay must be considered for the correct characterization of OMHPs. The important role of de-trapping 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 s and a detrapping time of 15 s using a single-trap model to describe the carrier relaxation in MAPbBr3 single-crystal detectors. Despite the limitation of the single-trap model, these results agree with parameters of the trap E2 in Table 1.

In general, the trapping and detrapping phenomena are scarcely studied in OMHPs. Thus, it is useful to compare the trap parameters (T/D) in MAPbBr3 with those in inorganic semiconductors. Using ToF measurements in combination with MC simulation, we found the ratio of trapping/detrapping (T/D) from shallow traps to be ~250 ns/40 ns and from deep traps to be ~150 ns/3 s in GaAs devices (35) and 13 ns/11 ns and 2 s/20 s 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 typi-cally corresponds to the strong screening of the charged defects in OMHPs, leading to a low nonradiative recombination rate, as pre-viously proposed (36). This effect can be also attributed to the polaronic nature of charge carriers in OMHPs (37) or unstable de-fects 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, . Figure 5D demonstrates the results of ToF and MC simulations fitted by the Hecht equation (39) to obtain -product in a MAPbBr3 single- crystal device. Because of the delay of the hole transport by traps, shows a strong dependence on the collection time (10−1 to 10−4 cm2 V−1), which is the time needed to detect high-energy absorption events. Engineering a lower concentration of traps, particularly trap E3, can cancel the detrimental influence of traps as shown by MC simula-tions (Fig. 5D, blue dash-dotted line). The longer detrapping time of 120 s from the defect E3 can compete with the transit time of the free charge cloud and the typical charge collection time of classical


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inorganic semiconductors [typically <200 s (40)]. By considering the collection of free holes in 200 s, the MC simulation in combination with ToF result gives = 10−3 cm2 V−1 in MAPbBr3 devices. This is competitive with the best mobility-lifetime product found with sim-ilar methods in inorganic detectors such as CdZnTe (10−3 cm2 V−1) (22) and GaAs (6 × 10−4 cm2 V−1) (35).

We calculated a diffusion coefficient of 0.32 cm2 s−1 and a diffu-sion length of 54 m in MAPbBr3 single-crystal devices by consider-ing 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 cm2 s−1 (27) and a diffusion length of 2.6 to 650 m (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 be-tween 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. There-fore, 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 1013 cm−3 (p = Photon Flux ·TR per thickness). Such a high density of photogenerated carriers could completely fill the traps (holes) up to the position of the quasi–Fermi energy EF = Ev + 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 1014 cm−3. Sec-ond, traps populated by holes could subsequently trap electrons, re-sulting in nonradiative recombination and reduce VOC 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 E1 and E2. The decrease in the trapping time can transform the traps with shallow character into fast trapping centers, since the detrap-ping time does not depends on the trap concentration. For example, the trap E2 can increase its trapping time up to ~240 ns in thin films.

Nature and chemistry of trapsCation vacancies (VPb and VMA) and Br interstitial (Bri) are the main acceptors in MAPbBr3 identified by first-principles calculations (15). Similar results have been also demonstrated in MAPbI3 (16, 42). We tentatively assign E1, E2, and E3 to VMA, VPb, and Bri considering the results of previous density functional theory calculations. The for-mation of VMA does not involve Pb─Br bond breaking, and thus, VMA is likely the shallowest acceptor with relatively short trapping and detrapping times as found for the shallow hole trap E1 (Table 1). On the other hand, the formation of VPb requires Pb─Br bond break-ing, which should lead to a slightly deeper acceptor level than that of VMA as found by calculations (15). In addition, the −2 charged VPb has a stronger Coulomb attraction to holes than the −1 charged VMA. 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 combina-tion of the above two effects should lead to a longer detrapping time for VPb compared to that of VMA 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 E1 and E2. In contrast to VMA and VPb, which

introduce shallow hydrogenic levels, the hole trapped by Bri is strongly localized (16) as also found for Ii in MAPbI3 (42). Strong localiza-tion of Bri defect leads to deeper hole trapping and long trapping/detrapping times, consistent with the behaviors of trap E3.

The calculated hole trapping level [the (0/−) transition level] by Bri in MAPbBr3 is about Ev + 0.13 eV (15), comparable to the calcu-lated (0/−) level (Ev + 0.15 eV) of Ii (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 param-eters (Et, , and Nt) 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 MAPbBr3 with acti-vation energies of 1.05, 1.5, and 0.7 eV (12–14), a concentration of 1011 cm−3 (12), and capture cross sections in the range of 10−17 to 10−15 cm2 (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 concentra-tion 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. Re-cently 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).

DISCUSSIONIn conclusion, by combining the ToF current spectroscopy and the MC simulation, we have provided insights and detailed information about the dynamics of free charge carriers in MAPbBr3 single- crystal devices. We have demonstrated strong trapping activities in MAPbBr3 single crystals of three defects with trapping times of T1 (23 s), T2 (24 s), and T3 (90 s). The three found traps showed fast detrapping times of 3, 14, and 120 s. Our results reveal that traps have a notable impact on the free carrier transport and the collection efficiency in MAPbBr3 perovskite devices; the traps E1 and E2 act as shallow levels with short-term trapping-detrapping characters, while the trap E3 acts as a deep level causing long-term trapping of free holes.

We propose a new model for the effective mobility, 𝛍 h eff , to describe the delaying effect of traps on the charge transport properties of MAPbBr3 single crystals. By decreasing the electric field, 𝛍 h eff decreases from the drift hole mobility (12.4 cm2 V−1 s−1) to the effective mo-bility calculated from the classical theory (4 cm2 V−1 s−1). Detailed MC simulations show that, besides the trapping-detrapping events from defects, the retrapping of the delayed holes plays an important role in the delaying of the total hole cloud. In addition, the material thickness and temperature have a substantial impact on 𝛍 h eff and conse-quently on the free charge collection efficiency in MAPbBr3 devices. Our results demonstrate that the role of traps should be carefully consid-ered in the study of transport properties of OMHP devices, such as the assessment of drift mobility, lifetime, and mobility- lifetime product.

METHODSToF spectroscopyToF method is based on measurement of the current response in a planar semiconductor device with an external stimulation (such as


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an alpha particle, electron, short light pulse, etc.). In this experimental setup, the external stimulus is an above bandgap (450 nm) laser pulse (1 s) with an intensity power of 0.03 W. The attenuation depth of the optical photons is estimated to be 1 m. The optical pulse gen-erates electron-hole pairs near the illuminated electrode. By apply-ing an electric field, the electron-hole pairs are separated. Free holes drift toward the cathode and generate a current according to the Shockley-Ramo theorem. The anode immediately collects free elec-trons; therefore, only free holes drifting in the material generates CWFs signal. In these measurements, the CWFs are recorded by a synchronous triggering derived from the laser pulse. This setup results a much better signal-to-noise ratio as compare to untriggered sources such as alpha particles. Often, the enhanced continuous DC biasing (tens of voltage) in an OMHP device results in dynamic degradation of the sample, which makes the reliable record of CWF impossible. To overcome this detrimental effect, we apply a synchronized pulsed biasing. A light source with a photon energy of 2.8 eV (450 nm) is used to generate free carriers at the anode. The above-bandgap light pulse is preferably absorbed in less than 1-m-thick layer below the contact electrode. A positive bias, U, is applied between the two electrodes to collect free charge carriers. The current signal is detected using an oscilloscope synchronized with laser and voltage pulses. Using this pulse photoexcitation allows us an accumulation and averaging of multiple CWFs, resulting in a high signal-to-noise ratio. The additional description can be found in the Supplementary Materials (fig. S1).

MC simulationWe use MC calculation to simulate charge dynamics in MAPbBr3 single-crystal device. We developed 1D MC with the total number of particles, n = 105. The initial position x of the MC particle is generated according to Lambert-Beed law for light absorption. Our MC simula-tion also includes the diffusion of the carriers in addition to simulated drift process between two metal contacts. Each MC simulation step changes the state of the MC particle using random numbers according to probability given by trapping and detrapping time of particular trap level. We found parameters (trapping/detrapping time) of each trap in the bandgap by fitting experimental ToF results with MC simula-tion and least square regression analysis. CWFs are calculated using Shockley-Ramo theorem. The detailed description of the method is given in the Supplementary Materials (fig. S2 and eqs. S1 to S6).

The effective mobility is a useful approximation of free charge carrier cloud movement. When free carriers drift in the material and interact with traps, the center of the charge cloud moves with an effective mobility rather than with microscopic one. This approxi-mation relies on the central limit theorem, which states that when carriers undergo many trapping and detrapping events, a new equi-librium between traps and conduction band is established. The accuracy of the effective mobility depends on the number of trapping- detrapping events. If there is more than one trap level, then the thermalization of traps starts with the smallest detrapping time; after that, the trap with larger detrapping time is thermalized and so on until all traps are thermalized. To reliably apply MC simulation, the number of trapping-detrapping events has to be several hundred to use the effective mobility approach. The typical error of effective mobility in this study is about 0.3%.

Sample preparation: MAPbBr3 single-crystal growthA single crystal of MAPbBr3 is grown from ultrapurity precursors. The sample geometry is 5 mm by 5 mm by 2 mm. According to

six-probe Hall measurements, the free hole concentration and resis-tivity found under dark condition are 6 × 108 cm−3 and 109 ohm·cm−1, respectively. Several samples showed the same ToF and MC simula-tion results; thus, the results of one sample are presented in this work. Detail of single-crystal growth can be found in our previous work (44).

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/37/eabb6393/DC1

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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.

Submitted 8 March 2020Accepted 29 July 2020Published 11 September 202010.1126/sciadv.abb6393

Citation: A. Musiienko, J. Pipek, P. Praus, M. Brynza, E. Belas, B. Dryzhakov, M.-H. Du, M. Ahmadi, R. Grill, Deciphering the effect of traps on electronic charge transport properties of methylammonium lead tribromide perovskite. Sci. Adv. 6, eabb6393 (2020).


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lead tribromide perovskiteDeciphering the effect of traps on electronic charge transport properties of methylammonium

Ahmadi and Roman GrillArtem Musiienko, Jindrich Pipek, Petr Praus, Mykola Brynza, Eduard Belas, Bogdan Dryzhakov, Mao-Hua Du, Mahshid

DOI: 10.1126/sciadv.abb6393 (37), eabb6393.6Sci Adv

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MATERIALS SCIENCE Copyright © 2020 Deciphering the effect ... · beyond the classical model of trap-controlled mobility (17). Such a model, also known as the effective mobility model,