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Metal halide perovskites are an important class of emerging semiconductors. Their charge carrier dynamics is poorly understood due to limited knowledge of defect physics and charge carrier recombination mechanisms. Nevertheless, classical ABC and Shockley-Read-Hall (SRH) models are ubiquitously applied to perovskites without considering their validity. Herein, an advanced technique mapping photoluminescence quantum yield (PLQY) as a function of both the excitation pulse energy and repetition frequency is developed and employed to examine the validity of these models. While ABC and SRH fail to explain the charge dynamics in a broad range of conditions, the addition of Auger recombination and trapping to the SRH model enables a quantitative fitting of PLQY maps and low-power PL decay kinetics, and extracting trap concentrations and efficacies. However, PL kinetics at high power are too fast and cannot be explained. The proposed PLQY mapping technique is ideal for a comprehensive testing of theories and applicable to any semiconductor.
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ARTICLE
Are Shockley-Read-Hall and ABC models valid for
lead halide perovskites?
Alexander Kiligaridis1, Pavel A. Frantsuzov 2, Aymen Yangui 1, Sudipta Seth 1, Jun Li 1, Qingzhi An 3,
Yana Vaynzof 3& Ivan G. Scheblykin 1
Metal halide perovskites are an important class of emerging semiconductors. Their charge
carrier dynamics is poorly understood due to limited knowledge of defect physics and charge
carrier recombination mechanisms. Nevertheless, classical ABC and Shockley-Read-Hall
(SRH) models are ubiquitously applied to perovskites without considering their validity.
Herein, an advanced technique mapping photoluminescence quantum yield (PLQY) as a
function of both the excitation pulse energy and repetition frequency is developed and
employed to examine the validity of these models. While ABC and SRH fail to explain the
charge dynamics in a broad range of conditions, the addition of Auger recombination and
trapping to the SRH model enables a quantitative tting of PLQY maps and low-power PL
decay kinetics, and extracting trap concentrations and efcacies. However, PL kinetics at high
power are too fast and cannot be explained. The proposed PLQY mapping technique is ideal
for a comprehensive testing of theories and applicable to any semiconductor.
https://doi.org/10.1038/s41467-021-23275-w OPEN
1Chemical Physics and NanoLund, Lund University, P.O. Box 118, Lund 22100, Sweden. 2Voevodsky Institute of Chemical Kinetics and Combustion, Siberian
Brunch of the Russian Academy of Science, Institutskaya 3, Novosibirsk 630090, Russia. 3Integrated Center for Applied Physics and Photonic Materials
(IAPP) and Centre for Advancing Electronics Dresden (CFAED), Technical University of Dresden, Dresden, Germany. email: pavel.frantsuzov@gmail.com;
yana.vaynzof@tu-dresden.de;ivan.scheblykin@chemphys.lu.se
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Semiconducting materials often exhibit complex charge
dynamics, which strongly depends on the concentration of
charge carriers due to the co-existence of both linear and
non-linear charge recombination mechanisms1,2. The emergence
of novel semiconductors like metal halide perovskites (MHPs),
exhibiting intriguing and often unexpected electronic
properties310, triggered a renewed interest in revisiting the
classical textbook theories of charge recombination and the
development of more complete, accurate models1117. Moreover,
modern technical advances in experimental and computational
capabilities4,1821 allow for a detailed quantitative comparison
between experiment and theory, far beyond what was once
possible.
MHP are a novel solution-processable material class with
enormous promise for application in a broad range of optoelec-
tronic devices2224. Driven in particular by their remarkable per-
formance in photovoltaics, with power conversion efciencies
surpassing 25% demonstrated to date25, signicant research efforts
have been devoted to study the fundamental electronic properties
of these materials4,5,7,13,15,18,2630. It was established that for many
MHP compositionswith the most notable example being the
methylammonium lead triiodide (MA =CH
3
NH
3
+, also referred
to as MAPbI
3
or MAPI)they can be considered as classical
crystalline semiconductors at room temperature, in which pho-
toexcitation leads to the formation of charge carriers that exist
independently from each other due to the low exciton binding
energy30. Consequently, conventional models that describe the
charge carrier dynamics are ubiquitously used to describe the
dynamics of charge carriers in MHPs11,1315,3139.
Historically, the rst model describing the kinetics of charge
carrier concentrations in a semiconductor was proposed by
Shockley and Read40 and independently by Hall41, and is known
as the ShockleyReadHall (SRH) model. This model considers
only the rst-order process (trapping of electrons or holes) and
the second-order kinetic processes (radiative electronhole
recombination and non-radiative (NR) recombination of the
trapped electrons and free holes). It is noteworthy that the SRH
model allows the concentrations of free charge carriers to differ
due to the presence of trapping. In an intrinsic semiconductor,
trapping of, for example, electrons generated by photoexcitation
creates an excess of free holes at the valence band. This effect is
often referred to as photodoping, in analogy with chemical
doping, with the important difference, however, that the material
becomes doped only under light irradiation and the degree of
doping depends on the light irradiation intensity.
Third-order processes, such as NR Auger recombination, via
which two charge carriers recombine in the presence of a third
charge that uptakes the released energy, have been recognized as
particularly important at a high charge carrier concentration
regime. To account for this process, Shen et al., instead of adding
the Auger recombination term into the SRH model, proposed a
simplied ABC model named after the coefcients A, B and C for
the rst-order (monomolecular), second-order (bi-molecular)
and third-order Auger recombination, respectively42. These
coefcients are also sometimes referred to as k
1
,k
2
and k
3
.
Importantly, the concentrations of free electrons and holes in the
ABC model are assumed to be equal, thus neglecting the possible
inuences of chemical and photodoping effects. The ABC model
is widely applied in a broad range of semiconductors and in
particular, is commonly used to rationalize properties and ef-
ciency limits of LEDs42,43. The simplicity of the ABC model led to
its extreme popularity also for MHPs (see e.g. ref. 15 and refer-
ences therein) with fewer reports employing SRH or its
modications13,14,16,17,31,36,39,44,45.
The ABC and SRH kinetic models are typically employed to
describe experimentally acquired data such as the excitation
power density dependence of photoluminescence (PL) quantum
yield (PLQY) measured upon continuous wave (CW) or pulsed
excitation, time-resolved PL decay kinetics and kinetics of the
transient absorption signal. These models are applied to semi-
quantitatively explain the experimental results and extract dif-
ferent rate constants1315,20,3134,46,47, often without necessarily
considering the modelslimitations. Despite the very large
number of published studies describing electronic processes in
MHPs using the terminology of classical semiconductor physics,
to the best of our knowledge, there have been only very few
attempts to t both PL decay and PLQY dependencies of exci-
tation power using ABC/SRH-based models or at least compare
the experimental data with theory14,16,17,31,39. These attempts,
however, were of limited success because large discrepancies
between the experimental results and the theoretical ts were
often permitted.
These observations raise fundamental questions concerning the
general validity of the SRH and ABC models to MHPs and the
existence of a straightforward experimental method to evaluate
this validity. To address these concerns, it is necessary to char-
acterize experimentally the PLQY and PL decay dynamics not
only across a large range of excitation power densities, but also
simultaneously over a large range of the repetition rates of the
laser pulses. We note that PL is sensitive not only to the con-
centrations of free charge carriers but also, indirectly, to the
concentration of trapped charge carriers, as the latter inuence
the former via charge neutrality. Such trapped carriers may also
lead to other non-linear processes, for example, between free and
trapped charge carriers (Auger trapping2), which should also be
considered, but are excluded from both the ABC and classical
SRH models. To expose and probe these processes, it is most
crucial to scan the laser repetition rate frequency in the PLQY
measurements, with such measurements, to the best of our
knowledge, have not been reported to date.
In this work, we developed a new experimental methodology
that maps the external PLQY in two-dimensional space as a
function of both the excitation pulse uence (P, in photons/cm2)
and excitation pulse frequency (f, in Hz). Due to scanning of the
excitation pulse frequency over a very broad range, this novel
technique allows to unambiguously determine the excitation
regime of the sample (single pulse vs. quasi-CW), which is cri-
tically important for data interpretation and modelling. Obtaining
a two-dimensional PLQY(f,P) map complemented with PL decays
provides a clear and unambiguous criterion to test kinetic models:
a model is valid if the entire multi-parameter data set can be tted
with xed model parameters.
By applying this method to a series of high-quality MAPbI
3
thin
lm samples, which when integrated in photovoltaic devices reach
power conversion efciencies of >20% (Supplementary Note 5),
we demonstrate that despite MAPbI
3
being extensively studied in
numerous publications before, neither ABC nor classical SRH
model can t the acquired PLQY maps across the entire excitation
parameter space. To tackle this issue, we develop an enhanced
SRH model (in the following, the SRH+model), which accounts
for Auger recombination and Auger trapping processes and
demonstrates that SRH+is able to describe and quantitatively t
the PLQY(f,P) maps over the entire range of excitation conditions
with excellent accuracy. PL decays can be also tted, albeit, with
more moderate accuracy. The application of the SRH+model
allowed us to extract the concentration of dominant traps in high
electronic quality MAPbI
3
lms to be of the order of 1015 cm3and
to demonstrate that surface treatments can create a different type
of trapping states of much higher concentration. Beyond the
quantitative success of the extended SRH+model, we reveal that
even this model is not capable to describe the PL decay at high
charge carrier concentrations. This means that there must be
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further non-linear mechanisms that inuence charge dynamics at
high charge carrier concentrations in MAPbI
3
. Therefore, further
theoretical work is necessary to identify the additional physical
process or processes which must be considered in order to com-
pletely elucidate the charge dynamics in MAPbI
3
.
Results
PLQY(f,P) mapping and elucidation of the excitation regime.
The acquisition of a PLQY(f,P) map occurs by measuring the
intensity of PL as a function of pulse repetition rate (f,s
1) for a
series of xed pulse uences (P, photons/cm2). PL intensity is
then plotted as a function of the time-averaged excitation power
density W¼fPhv (W/cm2), where hv is the excitation photon
energy, see further details in Supplementary Notes 13. Figure 1b
presents PLQY(f,P) map for a bare MAPbI
3
lm, while Fig. 1a
presents the same data in the traditional way as a series of PLQY
(W) dependencies for different frequencies. We use 19 fre-
quencies ranging from 100 Hz to 80 MHz, which corresponds to a
lag between pulses varying from 10 ms to 12.5 ns. In our
experiments after scanning the frequency for a certain value of P,
it is then changed to the next value and the scanning procedure is
repeated. The pulse uence ranges over four orders of magnitude
(P1=4.1 × 108,P2=4.9 × 109,P3=5.1 × 1010,P4=5.5 × 1011
and P5=4.9 × 1012 photons/cm2). Such uences, in the single-
pulse excitation regime (see below), correspond to charge carrier
densities of 1.04 × 1013, 1.24 × 1014, 1.3 × 1015, 1.37 × 1016 and
1.24 × 1017 cm3, respectively. For clarity, in Fig. 1and in all the
Fig. 1 PLQY(f,P) map and illustration of the difference between scanning the pulse repetition rate (f) and scanning of the pulse uence (P). a PLQY
(W) dependence plotted in the traditional way (P-scanning) for 19 different pulse repetition rates. The datapoints measured at the same frequency are
connected by lines, the sample is MAPbI
3
lm grown on glass (G/MAPI). The apparent slope of these dependencies (m, PLQY ~ Wm) depends on the range
of Wand the value of fand can be anything from 1 to 0.77 for this particular sample. bThe same data plotted in the form of a PLQY(f,P) map where data
points measured at the same pulse energy (P1, P2, ..., P5) are connected by lines (f-scanning). Data points measured at 50 kHz frequencies are connected
by a dashed-dotted line. cThe excitation scheme. Illustrations of PL decays in the single pulse (d) and quasi-CW (e,f) excitation regimes. Here etrapping
is assumed leading to h+photodoping.
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following gures in the manuscript, the data points measured at
the same pulse uence Pare shown by the same colour: P1
violet, P2blue, P3green, P4orange, P5red. The family of
lines connecting points with P=constant and fscanned make
together a pattern that resembles a horse neck with maneas
illustrated in Fig. 1b.
The acquisition of a PLQY(f,P) map is fully automated
(Supplementary Note 2) and includes precaution measures that
minimize the exposure of the sample to light, while controlling
for photo-brightening or darkening of the samples (Supplemen-
tary Note 4). Such measures ensure that PLQY maps are fully
reproducible when re-measured again on the same spot (see
Supplementary Fig. 4.2). These precautions were absolutely
necessary for obtaining a consistent data set, because light-
assisted transformation of defect states due to ion migration may
signicantly inuence the photophysics of perovskite
materials10,48,49 making any theoretical analysis unfeasible. We
also note that the high degree of uniformity of our samples leads
to very similar PLQY maps being measured on different areas of
the sample (see Supplementary Notes 2 and 4).
PLQY(f,P) map for bare MAPbI
3
samples (G/MAPI) is
presented in Fig. 1. A traditional representation of these data is
shown in Fig. 1a, which displays a series of PLQY(W) curves
each for one of the 19 different repetition rates used in our
experiment. Overall, such representation shows only minor
differences between the curves, in terms of their slope and
curvatures, apart from a noticeable horizontal shift at sufciently
low frequencies. By approximating the PLQY to vary as Wmover a
limited power interval, we observe that the slope mvaries between
0.77 at high repetition rates to approximately 1 at low repetition
rates for this sample. Traditional representations of PLQY(W)
plots for the other samples investigated in our study are shown in
Supplementary Note 8, where, for example, a PMMA coated
MAPbI
3
sample shows the slope mranging from 0.5 to 1 (see
Supplementary Fig. 8.1c). Such a traditional representation of the
PLQY(f,P) map does not offer a clear interpretation of the data,
making it difcult to elucidate the charge carrier dynamics.
An alternative representation of the PLQY(f,P) map is shown
in Fig. 1b, in which the data points for each laser uence Pare
presented as a single curve. Interestingly, the data points for each
value of Pfollow a characteristic line with a specic shape. At low
frequencies, and especially at high uences, the curves are rather
horizontal, yet once the frequency fexceeds a certain value, all
data points start to follow a certain common dependency, at
which the PLQY depends solely on the averaged power density
W¼fPhv. The frequency at which this happens depends on P,
such that, for example, the data obtained at pulse uence P5 joins
at ca. 500 kHz, while the data collected at P1 joins at below 50
kHz (see Fig. 1b).
Critically, such presentation of the PLQY(f,P) map allows us to
immediately distinguish between two principally different excita-
tion regimes for a semiconductor:
1. Single-pulse regime: In this regime the repetition rate of the
laser is so low, that PLQY values and PL decays do not
depend on the lag between consecutive laser pulses. In other
words, the excited state population created by one pulse had
enough time to decay to such a low level, that it does not
inuence the decay of the population generated by the next
pulse (Fig. 1d). In this case, PLQY does not depend on the
lag between pulses (i.e. the pulse frequency). This regime is
observed when PLQY follows the horizontal lines upon
frequency scanning (highlighted in green in Fig. 1b).
2. Quasi-continuous wave (quasi-CW) regime: In this regime,
the decay of the population generated by one pulse is
dependent on the history of the excitations by previous
pulses. This happens when some essential excited species
did not decay completely during the lag time between the
laser pulses (Fig. 1e, f). In this regime, the data points follow
the same trend and fall on the line highlighted in yellow in
Fig. 1b. The transition between the two regimes occurs
when the data points at xed values of Pstart to match with
each other upon increasing the f.
Examining the vast literature of MHPs reveals that, to the best of
our knowledge, no study has utilized such a broad range of pulse
repetition rates fwhen measuring PLQY(W). Without scanning of
fover a signicant range of values, a distinction between the single
pulse and quasi-CW regimes is not possible, and this reects the
current situation in the literature where the standard scanning
over Pis implemented with, at best, a few different repetition
rates of a pulsed laser, which is sometimes complemented by
excitation by a CW laser16,17,39. For example, Trimpl et al.39
studied FA
0.95
Cs
0.05
PbI
3
with the focus on temperature-dependent
PL decay kinetics measured at three repetition rates (61.5, 250, and
1000 kHz) and PLQY at one repetition rate and three pulse
uences approximately corresponding to P2, P3 and P4 in our
experiments. A qualitative similarity between PLQY predicted from
PL decay kinetics and experiments data was obtained and
temperature dependence of the model parameters was extracted.
The condition for charge accumulation (photodoping) in this work
was addressed solely using PL decays where an initial fast drop at
the ns time scale clearly visible at low temperature was assigned to
trapping39. In another example, Kudriashova et al. studied PL
decay over a quite broad pulse repetition rate range (10 kHz10
MHz) in order to distinguish between surface and bulk charge
recombination in MAPbI
3
lms with charge transport layers,
however, PLQY was not measured in this study34. In general, these
studies addressed the important question of the excitation regime
within the limits of their experimental approaches, however, the
only robust way to clarify the excitation condition for a given pulse
uence is to explicitly scan fwhile detecting PLQY. One may
assume that choosing a low repetition rate guarantees that the
excitation is in the single-pulse regime. However, this is not true.
As Fig. 1a and b show, if the system is at the single-pulse regime at
ahighpulseuence at a given repetition rate, there is always such
low pulse energy that the excitation regime becomes quasi-CW.
The cause for this effect is the presence of a non-exponential decay
of the excited state population as will be discussed later. Thus, at a
very low repetition rate, the excitation may still be in a quasi-CW
regime so long as the pulse uence is low enough. Without
scanning the pulse frequency, this cannot be disentangled. To
illustrate this, the data points measured at 50 Hz were connected by
a dash-dot line in the PLQY(f,P) map in Fig. 1b. For a pulse uence
P1 (i.e. the lowest uence), the excitation regime is quasi-CW.
Increasing the pulse uence by an order of magnitude (P2) brings
the system close to the single-pulse regime, with further increase of
the pulse uence (P3, P4andP5) making the excitation fall purely
in the single-pulse regime. We highlight the existence of a rather
extended intermediate region, at which the regime is neither a
single pulse nor quasi-CW. For example, for the pulse uence P2
(charge carrier concentration 1014 cm3), this intermediate
region starts at 50 kHz and continues down to at least 2 kHz.
We underscore that in order to identify the excitation regime
without any additional assumptions, one must scan the pulse
frequency and measure PLQY. As a result, the PLQY(f,P)mapping
technique described here allows for an unambiguous and very easy
discernment between the single pulse and quasi-CW excitation
regimes.
PLQY(f,P) maps and PL decays(f,P) of polycrystalline MAPbI
3
.
Figure 2compares the PLQY(f,P) maps measured for MAPbI
3
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lms prepared with four different combinations of the interfaces
(Fig. 2e and Supplementary Note 5): MAPbI
3
deposited on glass
(G/MAPI), MAPbI
3
deposited on PMMA-coated glass (G/P/
MAPI), MAPbI
3
deposited on glass and coated with PMMA (G/
MAPI/P) and MAPbI
3
deposited on PMMA/glass and then
coated by PMMA (G/P/MAPI/P). All samples exhibit the same
PL and absorption spectra (Supplementary Note 6). Scanning
electron microscopy (SEM) images show that all samples exhibit a
very similar microstructure, which is not affected by the presence
of PMMA layers (Supplementary Note 6). Despite all these
similarities, the PLQY(f,P) maps are clearly different. To
emphasize the differences, we added three horizontal lines that
mark the PLQY at the single pulse regimes for the pulse uences
P3, P4, and P5 for the G/MAPI sample in Fig. 1a. Black arrows
highlight the reduction in PLQY in the single pulse regimes when
compared with G/MAPI sample.
The decrease of PLQY upon the addition of PMMA differs for
different values of P. Moreover, when comparing the slope mof
the quasi-CW region in (a) and (b) with that of (c) and (d), it is
evident that it is strongly inuenced by the exact sample stack.
To visualize this difference, a line with the slope of m=0.5
(i.e. PLQY ~ W0.5) is shown in each plot. The PLQY(f,P) map is
most affected when MAPbI
3
lm is coated by PMMA, while its
presence at the interface with the glass substrate has only a minor
effect.
Similar to the PLQY maps, PL decay kinetics also depends on
the pulse uence and excitation regime (single pulse vs. quasi-
CW). Such kinetics should be considered together with PLQY(f,P)
map to complete the physical picture of charge recombination.
Figure 3shows PL decays measured at f=100 kHz and pulse
uences P2 (low) and P5 (high). MAPbI
3
lms deposited on glass
(G/MAPI) exhibit the slowest of all PL decay kinetics both at a
low and a high pulse uences. The addition of PMMA to the
sample stack accelerates the PL decay with the shortest decays
observed for G/P/MAPI/P samples.
The observation that modication of the sample interfaces by
PMMA results in a faster PL decay not only for the low, but also
for the high (P5) pulse uence is particularly interesting. While
the inuence of surface modication on NR recombination at low
charge carrier concentrations is expected due to the changes in
trapping, the same is not expected to occur at high pulse uences.
It is generally considered that at such uences, the decays will be
P2
P3
P4
P5
P1
Single pulse regime
0.4
0.2
0.02
PLQY
10
-6
10
-4
10
-2
10
0
10
2
(a)
0.4
0.2
0.02
PLQY
Power density, W/cm2 Power density, W/cm2
G/MAPI
PMMA, 30 nm
MAPbI3, 260 nm
PMMA, 30 nm
Glass
G/P/MAPI/P
G/MAPI/P
G/P/MAPI
Pulsed laser PL detecon
G/MAPI
e
10-6 10-4 10-2 100102
G/MAPI/P
(c)
10-6 10-4 10-2 100102
10
-3
10
-2
10
-1
10
0
G/P/MAPI
(b)
10-6 10-4 10-2 100102
10-3
10-2
10-1
100
G/P/MAPI/P
(d)
a b
c d
Fig. 2 PLQY maps of the samples under study plotted in the same scale for comparison. a Glass/MAPI, bglass/PMMA/MAPI, cglass/MAPI/PMMA,
and dglass/PMMA/MAPI/PMMA. The horizontal grey lines show the values of PLQY (0.4, 0.2 and 0.02) in the single-pulse regime for the glass/MAPI
sample (a) to set the benchmarks. Deviations from these values for other samples are shown by black arrows. The tilted grey line is the W0.5 dependence
as predicted by the SRH model. It is shown to see better the difference in the quasi-CW regime from sample to sample. The pulse uence (P1P5) is
indicated by the same colour code (shown in a) for all PLQY maps. Panel eshows the structure of the samples and geometry of the measurements.
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solely determined by non-linear processes such as Auger
recombination and are thus not inuenced by surface treatments.
However, the change in decay dynamics in PMMA interfaced
MAPbI
3
serves as the rst indication that additional non-linear
processes that involve trap states must be at play.
The second interesting observation is that according to the
PLQY(f,P) map, the repetition frequency 100 kHz used for the PL
decay measurements falls in the quasi-CW excitation regime for
the low pulse uence P2, but in the single-pulse excitation regime
for high pulse uence P5. It is remarkable, however, that the PL
intensity in the quasi-CW regime (Fig. 3a) decays until the next
laser pulse by almost two orders of magnitude for MAPbI
3
without PMMA and by four orders of magnitude for the sample
coated with PMMA. This is an excellent example for the inability
to correctly assign the excitation with P2uence to the quasi-CW
excitation regime without the knowledge gained from the PLQY
(f,P) map, considering the population observed in the PL kinetics
decays completely prior to the arrival of the next pulse. The cause
for the quasi-CW regime, in this case, is the presence of a
population of trapped carriers which lives much longer than
10 microseconds and that inuences the dynamics via
photodoping9,13,14. This example illustrates the hidden quasi-
CW regimeshown schematically in Fig. 1e (see also Supple-
mentary Note 7). These effects will be quantitatively explained by
the theory detailed in the next section.
Theory and modelling
Kinetic models: from ABC and SRH to SRH+. Figure 4a sche-
matically illustrates the key processes included in the ABC, SRH
and extended SRH (SRH+) kinetic models. The SRH+model
contains terms for radiative (second-order k
r
np) and NR (all
other terms) recombination of charge carriers. Note, that the
processes included in the SRH+model also naturally include
photon re-absorption and recycling as discussed in detail in
Supplementary Note 9.1 and Supplementary Note 10 leading to
effective renormalization of k
r
and k
E
rate constants. NR
recombination occurs via a trap state or due to Auger recombi-
nation. The trapping process can be linear and quadratic (Auger
trapping). We note that we consider only one type of band-gap
states. It is assumed that these states are placed above the Fermi
level (electron traps), but that they are deep enough to make
thermally activated de-trapping negligible. Similarly, one could
consider hole traps instead under the same conditionsthe
equations are symmetric in this regard. Auger trapping refers to
the process by which the trapping of a photoexcited electron
provides excess energy to an adjacent photoexcited hole2. The
possible importance of this process in perovskites has been sug-
gested in a few publications46,50. The complete set of equations
and additional description is provided in Supplementary Note 9.
We note that in the SRH and SRH+models, the complete set of
equations for free and trapped charged carriers is solved, contrary
to the studies where equations for only one of the charge carriers
(e.g. electrons) are used (see ref. 44). The latter simplication can
work only if the concentration of holes is very large and constant
(for example, in the case of chemical doping) which is not
applicable for intrinsic MAPbI
3
and other perovskites, see also
below. Due to the inclusion of Auger trapping in the SRH+
model, setting the parameter k
n
to innity reduces it to the ABC
model (see Supplementary Note 9.6), where the coefcient B
contains both radiative and NR contributions. Finally, the SRH+
model reduces to the SRH model by ignoring all Auger processes.
In the considered models the origin of the difference in the
concentration of free electrons and holes is the trapping of one of
the charge carriers, i.e. photodoping. We do not assume any
unintentional chemical doping45, and this assumption is
supported by solid experimental evidence. Indeed, in the case of
chemical doping and the presence of electron traps, the PLQY(W)
in the quasi-CW regime should change from its square root
dependence on Wto either linear (n-doping) or become
independent of W(p-doping) upon further decreasing of W
(see Supplementary Note 8). Note also that the situation is
symmetrical relative to the type of traps in the material. This
behaviour, however, was never observed in our samples where
PLQY /Wmat low excitation power with the slope mbeing
either 0.5 or 0.77, depending on the sample, without changing
upon decreasing of W(Figs. 1and 2). This means, that even if
there was unintentional doping in our samples, its level was so
low, that we do not observe any of its effects in the PLQY(f,P)
maps (Supplementary Note 9.7).
Photon reabsorption and recycling are considered to be
important processes inuencing the charge dynamics in MHPs11,46.
In our experimental study, we compare samples of very similar
geometries and microstructure ensuring that the effects of photon
reabsorption/recycling remain similar, such that they cannot serve
as the reasons for the differences between PLQY(f,P)mapsandPL
decay kinetics amongst the different samples. As we discuss in detail
012345678910
10-4
10-3
10-2
10-1
100
012345678910
10
-4
10
-3
10
-2
10
-1
10
0
G/MAPI
G/P/MAPI
G/MAPI/P
G/P/MAPI/P
Pulse fluence P2
Quasi-CW
excitaon regime
G/MAPI
G/P/MAPI
G/MAPI/P
G/P/MAPI/P
Pulse fluence P5
Single pulse
excitaon regime
Repeon rate f=100 kHz
b
a
Adding PMMA
Adding
PMMA
Time, μs
PL intensity, normalized PL intensity, normalized
Fig. 3 PL decays of all samples at 100 kHz repetition rate (10 µs distance
between the laser pulses). a Low pulse uence (P2). bHigh pulse uence
(P5). Note that all decays in aare in the quasi-CW excitation regime, while
all decays in bare in the single-pulse excitation regime. Adding PMMA
accelerates the PL decay.
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in Supplementary Note 10, effects on the charge dynamics related to
photon recycling in broad terms (both far-eld (photon reabsorp-
tion in the perovskite) and near eld (energy transfer) effects), are
included in our models via renormalized radiative rate k
r
and the
Auger trapping coefcient k
E
, respectively. We also do not explicitly
include charge diffusion in the model. The rationale here is that
charge carrier diffusion in MAPbI
3
occurs so fast that equilibrated
homogeneous distribution of charge carriers over the thickness of
the lm can be assumed at a time scale of 10 ns and longer
(Supplementary Note 9.1).
Applying the ABC, SRH and SRH+models to the quasi-CW
excitation regime.Werst consider the CW excitation regime at
low power densities. In this regime, the SRH and SRH+models
are identical since the contribution of Auger processes is largely
negligible. Figure 4b shows the experimental dependencies of
PLQY on the power density (W) for G/MAPI and G/P/MAPI/P
samples and Fig. 4c and d show the dependence calculated based
on the three different models.
At low power densities, the concentration nis small and
PLQY is low. In the ABC model, the main contribution to the
recombination rate comes from the rst-order term, which is
equal to the photogeneration rate. Thus, An /Wand conse-
quently n/W. Therefore, we can write:
PLQY ¼flux of emitted photons
flux of absorbed photons ¼krn2
Bn2þAn krn2
An ¼krn
A/W
In the SRH model, at a very low excitation power density the
fastest process is that of trapping of electrons. With most of the
electrons trapped and ntp, the trapping rate is equal to the
photogeneration rate ktnN /W, and the remaining electron
density n/W. The limiting step in the charge carrier kinetics is
the NR recombination of the trapped electrons and holes. The
rate of this process is equal to the generation rate, therefore
knntp¼knp2/W, and p/
ffiffiffiffiffi
W
p. Thus
PLQY ¼flux of emitted photons
flux of absorbed photons ¼krnp
krnp þknntp
krnp
knntp¼krn
knntkrn
knp/
ffiffiffiffiffi
W
p
We refer the reader to the Supplementary Note 9.6 for the
detailed derivation of these equations and their applicability
conditions. To summarize, at low power densities when PLQY 1,
Fig. 4 CW regimes of the ABC, SRH and SRH+models and their comparison with the experiment. a The energy level scheme, the processes and
parameters of all models (see the text and Supplementary Note 9 for details). bThe experimental dependence (G/MAPI and G/P/MAPI/P samples) of
PLQY on the excitation power density Win the quasi-CW excitation regime, mis the exponent in the dependence Wm.cPLQY(W) in the CW regime for
different models and trap feeling conditions. -A”— adding Auger recombination, -ATr”—adding Auger trapping (Supplementary Note 11). dEvolution of
the PLQY(W) upon the transformation of the SRH model with Auger recombination to the ABC model with increasing of the parameter k
n
(see
Supplementary Note 11 for the model parameters).
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PLQY(W) is a straight line in the double logarithmic scale
(PLQY /Wm) with the slope m=0.5 for the SRH and SRH+
models with no trap lling effect (see below) and m=1 for the
ABC model1.
Experimentally, we observe m0.45 for those perovskite
samples which are coated with PMMA (e.g. G/MAPI/P is shown
in Fig. 4b). This value is in a good agreement to the m=0.5
predicted by the SRH/SRH+models in the case of the absence of
trap lling. However, the other two samples, in which the
MAPbI
3
surface is not coated with PMMA, exhibit m0.77 (e.g.
G/MAPI sample is shown in Fig. 4b), which lies between the
values of 1 and 0.5 predicted by the ABC and SRH/SRH+models,
respectively. These slopes are observed over at least four orders of
magnitude in the excitation power density. Based on these results,
we must conclude that MAPbI
3
samples with and without
PMMA coating behave very differently in the quasi-CW regime.
In the framework of the SRH/SRH+models, there are two
possibilities that would lead to an increase in the coefcient m: (i)
transformation toward the ABC model and (ii) trap lling effect
in the SRH model. Figure 4d shows the transformation of the
SRH model, which includes Auger recombination to the ABC
model by increasing the parameter k
n
. At the condition k
n
k
r
,k
t
there is a limited range of excitation power where one can obtain
an intermediate slope mlaying between 0.5 and 1 for a limited
range of W(Supplementary Notes 9.6 and 11).
The second possibility is to allow for the trap lling effect to
occur at the excitation power densities which are below the
saturation of the PLQY due to the radiative recombination and
Auger processes. The effect of trap lling is caused when the
number of available traps starts to decrease with increasing W.
Consequently, the PLQY increases not only because the radiative
process becomes faster (quadratic term), but also because the NR
recombination (trapping and further recombination) becomes
smaller. As the result, PLQY grows faster than W0.5 over a certain
range of W. The effect is not trivial, because it is not the
concentration of traps N as one would think, but rather the
relation of k
t
to k
r
and k
n
(the necessary condition is k
t
k
r
,k
n
),
which determines if the trap-lling effect is observed in PLQY
maps or not (Supplementary Notes 9.6, 9.8 and 11).
The trap lling effect is illustrated in Fig. 4c, in which the
parameter k
t
is increased whilst maintaining all other parameters
xed. Obviously, the resulting dependence is too strong and
occurs over a too narrow range of excitation power densities (one
order of magnitude) to t the experimental data directly.
Nevertheless, as will be shown below, such processes are present
in MAPbI
3
samples which are not coated with PMMA, where
PLQY(W) in the quasi-CW regime deviates from the straight line
bending upwards before reaching saturation at high power.
At high excitation densities, non-linear recombination pro-
cesses begin to be particularly important. Since Auger processes
are NR, with further increase of Wthe PLQY cannot reach unity
and instead decreases after reaching a certain maximum. SRH
cannot account for this effect considering it does not include any
NR non-linear terms and leads to PLQY =1 at high W. The ABC
and SRH+models can potentially describe this regime since they
contain Auger recombination terms (Fig. 4c and d).
Fitting of the PLQY(f,P) maps and PL decays kinetics by ABC, SRH
and SRH+models. To examine the validity of the three theories,
we attempt to t the experimental PLQY(f,P) plots and PL decays
using all models and the results are shown in Fig. 5. Before we
discuss the tting results, it is important to stress that each
simulated value of PLQY(f,P) at the PLQY maps and each PL
decay curve shown in Fig. 5are obtained from a periodic solution
of the kinetic equations of the corresponding model under pulsed
excitation with the required pulse uence Pand repetition fre-
quency f. In practice it means that we excited the system again
and again until the solution PL(t) stabilizes and begins to repeat
SRH+ model
SRH model
ABC model
Power density, W/cm2
PLQY
PLQY
PL intensity, a.u.
PL intensity, a.u.
Power density, W/cm2
Power density, W/cm2Time, μs
P1
P2
P3
P4
P6
abcd
efgh
Trap concentraon
N=1.75× 1017 cm-3
Trap concentraon
N=1.15× 1015 cm-3
G/MAPI
G/P/MAPI/P
0 1 2 3 4 5 6 7 8
0 0.2 0.4 0.6 0.8 1.0
Fig. 5 Fitting of the PLQY(f,P) maps and PL decays by all models. a ABC, bSRH, cSRH+models applied to the MAPbI
3
lm and eABC, fSRH,
gSRH+models applied to the MAPbI
3
lm with PMMA interfaces. In PLQY maps the symbols are experimental points, the lines of the same colour are the
theoretical curves. dand hshow experimental and theoretical (black lines) PL decays according to the SRH+model for both samples, laser repetition rate
100 kHz. The pulse uences are indicated according to the colour scheme shown in ein the whole gure. Theoretical CW regime is shown by the yellow
lines in all PLQY maps. The model parameters can be found in Supplementary Note 13.
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itself after each pulse. Details of the simulations are provided in
Supplementary Note 12.
When tting experimental data, it is important to minimize the
number of tting parameters and maximize the number of
parameters explicitly calculated from the experimental data. We
exploit the experimental data to extract several parameters. First,
considering that in all three models, the decay of PL at low pulse
uences is determined exclusively by linear trapping and is thus
mono-exponential, we can extract the parameter ktNof the SRH
and SRH+models. Indeed, such behaviour is observed
experimentally for the studied samples (see Fig. 3a) allowing us
to use the decays at low pulse energies (P1P3) to directly
determine the trapping rates ktN. We note, however, to obtain the
best t using the ABC model, the PL decays were not used to x
the parameter A. Secondly, in a single pulse excitation regime (i.e.
the horizontal lines in the PLQY map), the magnitudes of PLQY
at pulse uence P3 and P4 allow to determine the ratio kr
ktNin the
SRH/SRH+models and the ratio kr
Afor the ABC model. Detailed
block schemes of the tting procedures are provided in
Supplementary Note 12.
As has been discussed above, MAPbI
3
samples coated with
PMMA cannot be described using the ABC model due to
mismatch of the slope within the quasi-CW regime (Fig. 5a),
while both SRH and SRH+models are well-suitable in this case
(Fig. 5b, c). However, at a high excitation regime (i.e. the
saturated region of the quasi-CW and the single pulse regime at
P5 pulse uence) SRH+works much better, highlighting the
limitations of the SRH model on its own. Consequently, the entire
PLQY(f,P) map of the PMMA-coated lms can be tted using the
SRH+model with excellent agreement between the theoretical
and experimental data (Fig. 5c).
The behaviour of MAPbI
3
samples whose surface is left bare
(where the PLQY(W) dependence in quasi-CW shows extra up-
bending before reaching saturation) can be approximated using the
ABC model (Fig. 5e) and well-tted by the SRH+(Fig. 5g) model.
ABC indeed works quite well with, however, an obvious
discrepancy in the tilt of the quasi-CW regime. Very good tcan
be obtained by the SRH/SRH+models by adjusting of the k
t
,k
n
and
Nto allow for the trap lling effect to occur in the medium
excitation power range and, at the same time, making the dynamics
closer to that in the ABC model by a relative increase of the
recombination coefcient k
n
(see the section above and Fig. 5g).
As was mentioned above, the PL decay rate at low power
densities (P1P3) was used to extract the product ktN. This is the
only occasion for which the PL decays are used in the tting
procedure of the SRH and SRH+models. In the tting procedure
for the ABC model the PL decays are not used at all. Upon
determining the t parameters for each of the models, it is
possible to calculate the PL decays at each condition and compare
them with those decays measured experimentally. Importantly,
PL decay rates calculated using the ABC model signicantly
underestimate the measured decay dynamics at all uencies
(Supplementary Note 13). On the other hand, as is shown in
Fig. 5d and h, the SRH+model (as well as SRH, Supplementary
Note 13) t well the low uence decay dynamics, but system-
atically underestimate the decay rate at high power uences. It is
noteworthy that the mismatch of the initial decay rate at the
highest pulse uence reaches a factor ranging from three to ve
depending on the sample, still signicantly outperforming the t
using the ABC model. Insights regarding the applicability of the
ABC, SRH and SRH+models to the PLQY maps and PL decays
are summarized in Table 1, see also Supplementary Fig. 13.3.
Discussion
Scanning the excitation pulse repetition rate as proposed herein
represents a novel experimental approach that transforms routine
power-dependent PLQY measurements to a universal metho-
dology for elucidating charge carrier dynamics in semi-
conductors. Adding the second dimension of pulse repetition rate
to the standard PLQY(W) experiment is not just an update, it is a
principle, qualitative change of the information content of the
experiment. The difference between PLQY(f,P) mapping and the
standard PLQY(W) experiment in the CW regime or at some
xed pulse repetition rate is analogous to the difference between
the standard NMR spectrum and 2D NMR spectrum. In our
method, we monitor not only the concentrations of free charge
carriers, but also the concentration of trapped charges due to the
total electro-neutrality of the system. Therefore, together with the
time-resolved PL decays, the PLQY map in the repetition fre-
quencypulse uence 2D parameter space comprise an experi-
mental series which contains all the information concerning the
charge dynamics in a given sample.
We stress the absolute necessity of the unambiguous deter-
mination of the excitation regime of the experiment, which would
Table 1 Comparison of the ability of the three models to describe the PLQY(f,P) maps and PL decays.
Observables/regimes Low and medium excitation pulse uence
power density (W< 0.1 Sun)
High excitation power density
(1300 Sun), high pulse uence
ABC model
PLQY(f,P) Quasi-CW regimepoor or very poor t strongly
depending on the sample. Good t in the single
pulse regime
Very good t in all regimes
PL decays for given PLQY(f,P)Cannot predict the PL decays Cannot predict the PL decays
SRH model
PLQY(W,f) Very good t in all regimes Discrepancy due to exclusion of
high order processes
PL decays for given PLQY(f,P) Very good match Underestimation of the initial decay
SRH+model
PLQY(f,P) Very good t in both the quasi-CW and
single pulse excitation regimes
Very good t in both the quasi-CW
and single pulse excitation regimes
PL decays for given PLQY(f,P) Very good match Underestimation of the initial decay
Bold font emphasizes serious deciencies of the tting.
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not be possible without scanning the pulse repetition rate. For
example, PL intensity decay kinetic showing the signal decay by
several orders of magnitude prior to the arrival of the next laser
pulse (Fig. 3a) can still be in the quasi-CW regime due to pre-
sence of long-lived trapped charges (darkcharges). Such trap-
ped charges cause the so-called photodoping effect, which lingers
until the millisecond timescale, and thus holds the memoryof
the previous laser pulse, leading to a stark inuence on the PLQY
map. While the importance of distinguishing between the single
pulse and quasi-CW regimes has been noted in several publica-
tions before9,31, it has never been accomplished for MHPs
experimentally. Indeed, in none of the published works present-
ing theoretical ts of experimental PLQY(W) dependencies was
this determination possible simply because either only CW
excitation14,44 or pulse excitation with only one17,31,39 or two (20
MHz and 250 kHz)16 repetition rates of the laser pulses were
employed.
Understanding the excitation conditions is also critically
important for interpretation of the classical experiments in which
the PL intensity (or PLQY) is measured as a function of excitation
power density (W) using a CW light source or a pulsed laser with
axed repetition rate. Traditionally the intensity of PL is
approximated using a Wmþ1dependence or in case PLQY is
measured, with Wm(because PLQY /PL=W), with both leading
to a straight line in the double logarithmic scale1,13,31,33,47.
According to the SRH and ABC models, approximations like
these can be valid for a large range of Wat low excitation power
density only, when there is no trap-lling effect, Auger processes
can be neglected and PLQY is far from saturation. In all other
cases, the dependence is not linear in the double logarithmic
scale. As discussed above, SRH predicts m=0.5 in the CW
excitation regime while ABC always predicts m=1. However, our
experiments reveal that when the excitation is pulsed, one can
obtain intermediate mvalues because upon increasing the power
density, the experimental excitation regime is almost certainly
switched from a quasi-CW to a single pulse. The change of the
slope can be seen in Fig. 1a and in Supplementary Note 8.
Consequently, the extracted mcannot be reliably used for inter-
pretation of the photophysics of the sample since any value of m
can be obtained depending on the conditions of the pulsed
excitation.
The main message of our work is that any model of charge
carrier dynamics which is considered to be correct should be able
to t not only standard one-dimensional PLQY(W) data, but also
the full PLQY(f,P) map and PL decays at different powers and
pulse repetition rates. This criterion is strict and universally
applied. With standard one-dimensional PLQY(W) dataeven
upon the inclusion of the PL decay dataone can nd several
principally different models that are capable of tting the data.
However, when the multi-dimensional data space consisted of the
PLQY(f,P) map and PL decays is available, this ambiguity
becomes highly unlikely.
As we have shown above, neither the standard ABC nor the
SRH model are capable of describing the complete PLQY maps
and predicting PL decays of the investigated MAPbI
3
samples. On
the other hand, the addition of Auger recombination and Auger
trapping processes to the SRH model (SRH+model) leads to an
excellent t of PLQY maps of all the studied samples. We
emphasize that the (f,P) space used in this work is very large with
fvarying from 100 Hz to 80 MHz (6 orders of magnitude) and
pulse uence Pchanging over 4 orders of magnitude corre-
sponding to charge carrier densities in the single-pulse excitation
regime from ca. 1013 to 1017 cm3. SRH+model also agrees
well with the PL decay kinetics for low and medium pulse
energies (charge carrier concentrations from 1013 to 1015 cm3).
However, for high pulse uences (10161017 cm3) the model
underestimates the initial decay rate by up to a factor of ve for
the higher pulse energies. The Auger rates obtained from the
ttings (2.8 × 1029 cm6s1for the PMMA-coated MAPbI
3
sample and 1.7 × 1029 cm6s1for the bare MAPbI
3
)
are in a reasonable agreement with theoretical estimation 7.3 ×
1029 cm6s1for MAPbI
3
from ref. 51 which is at the low limit
from 2 × 1029 to 1 × 1027 cm6s1range reported in
literature52. Note that increasing of the Auger rate constant
cannot help because a t of PL decay will result in lower PLQY
than experimentally observed. Therefore, we must conclude that
the SRH+model has limitations.
One possible explanation for the mismatch of decay rates at
high excitation powers might be provided by considering
experimental errors. It is well documented that the PL of per-
ovskite samples is sensitive to both illumination and environ-
mental conditions, which, may lead to both photodarkening or
photobrightening of the sample9,48,49,53. To account for these
effects, we paid a special attention to monitoring the evolution of
the sample under light irradiation throughout the entire mea-
surement sequence. As is shown in Supplementary Note 4, the
maximum change in PL intensity during the entire measurement
series is smaller than a factor of two. Taking this uncertainty
together with other errors inherent to absolute PLQY and exci-
tation power density measurements, missing the decay rates by
several times at the highest pulse uence is not impossible.
However, there is strong indication that the discrepancy reects a
problem of the model rather than in the experiment: the deviation
between the theoretical and experimental PL decays is systematic.
Experimental PL decay rates at high charge carrier concentrations
are faster than predicted for all samples despite of the excellent
matching of the PLQY(f,P) maps.
In our future work we are going to test several additional
concepts which might help to increase the PL decay rate without a
strong effect on PLQY. One of them is based on the idea that at
high charge carrier density, the time (few ns) required to reach an
equilibration of the charge carrier concentration over the thick-
ness of the sample (300 nm) becomes comparable with the initial
fast PL decay induced by Auger. In other words, the diffusion
length becomes smaller in the high excitation regime54. In this
case, diffusion cannot be ignored and an additional PL decay
should appear reecting the decreasing charge carrier con-
centration due to their diffusion from the initially excited layer
determined by the excitation light penetration depth (100 nm)
towards the opposite surface of the 300 nm-thick lm. This
process is often discussed in the context of charge carrier
dynamics in large single crystals regardless of the excitation
conditions12. Supporting this notion is the fact that in order to
model a MAPbI
3
solar cell under operation55, a much lower
charge carrier mobility (around 102cm2Vs
1) than that
obtained spectroscopically (130 cm2Vs
1)15 has to be assumed,
which suggests that the actual diffusion coefcient might be
smaller than expected.
Another possible contributing factor originates from a local
charge carrier distribution inside the sample, caused by, for
example, funnelling of charge carries due to the energy landscape
or/and variations of charge mobilities13. Presence of a small
fraction of charge carriers concentrated in local nano-scale
regions can lead to an apparent fast PL decay at early times,
accompanied by a relatively small effect on the total PLQY. In
addition, high charge concentrations may cause carrier degen-
eracy effects. This happens because charge carriers occupy all the
possible states with energies below kT (degenerated Fermi gas).
Considering that the effective density of states in perovskite
materials is relatively low56, such degeneracy effects should be
seriously examined. If present, all rate constants would depend on
the charge concentration, which may lead to unexpected effects.
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Further investigations will reveal which of theseor other
mechanisms can help in describing of the PLQY(f,P) and PL
decay data space.
Despite of the moderate success at high charge concentration
regime, the results of the SRH+tting still signicantly outper-
form all previous attempts to explain charge carrier dynamics in
MAPbI
3
samples and allow us to gain valuable insights concerning
the photophysics of the samples investigated herein and the
roles of traps within them. This is supported by the fact that
the effect of charge trapping is the most crucial in the low and
middle power ranges where the SRH+model works very well for
both the PLQY maps and PL decays. Note that with the current
experimental accuracy we have no reason to complicate the SRH+
model by adding another type of traps and/or thermal de-
trapping.
The analysis of PLQY maps reveals that the concentration of
dominant traps in high-quality MAPbI
3
lms (without PMMA
coating) is ~1.2 × 1015 cm3. Very recently, practically the same
value for trap concentration was obtained using impedance
spectroscopy and deep-level transient spectroscopy for MAPbI
3
samples prepared by exactly the same method57. This con-
centration is also in excellent agreement with the range of values
previously proposed by Stranks et al.14, where the trap con-
centration was estimated by assuming that PL decays become
non-exponential exclusively due to the trap lling. We note,
however, that trap lling is not a necessary condition to observe
non-exponentiality in a PL decay. For that to occur, the non-
linear recombination rate (radiative, Auger, etc.) should just be
faster than the trapping rate, which is determined not only by the
trap concentration, but also by the capture coefcient. All these
and related effects are considered when the data is modelled by
the SRH+model developed and employed here, thus allowing the
extraction of the trap concentrations without any special
assumptions. Note, however, that for the bare MAPbI
3
sample the
estimation of the trap concentration is reliable, because the
inuence of trap concentration alone is clearly decoupled from
that of the trapping constant in the regime of trap lling and
conversion from the SRH to ABC model as observed for the bare
MAPbI
3
sample at a moderate excitation power.
Several studies have established the important role that surface
defects play in determining the optoelectronic properties of per-
ovskite thin lms5860, yet traditional PLQY measurements do not
offer a reliable method to extract the defect density in perovskite
lms and investigate how surface modications inuence this
density of defects. Considering that a PLQY(f,P) mapping allowed
us to extract the density of defects in bare MAPbI
3
lms, we apply
the same analysis to the PMMA-coated samples. We nd that
coating the top surface of MAPbI
3
with PMMA changes the pic-
ture drastically in terms of both the concentration and the nature
of dominant traps. No indication of trap lling is observed in the
PLQY(f,P) maps, which allows us to provide only the lower esti-
mate for the trap density in these samples (2×10
17 cm3s1).
The only part of the PLQY(f,P) map where the trap concentration
and the trapping rate are decoupled is the region in which PLQY
saturates, so the estimation of the high-limit of the trap con-
centration is not reliable due to dependence of this region on
parameters related to the Auger processes. The strong increase in
the trap concentration is accompanied by a decrease of the trap-
ping rate constant ktand the nonradiative recombination rate
constant knby at least one order of magnitude. This can
be interpreted by considering the traps in the PMMA-coated
sample to be more prevalent, yet weakerthan those in the bare
MAPbI
3
sample in terms of the trapping and recombination rate
introduced by each of these traps. These results suggest that the
addition of PMMA at the top surface leads to the creation of weak
traps, which, however, due to their very large concentration
override the effect of the stronger, yet less common, traps present
in MAPbI
3
lms that did not undergo the surface treatment.
We note that coating with polymers (including PMMA) and
organic molecules in general is a common method employed in
literature to protect MAPbI
3
samples from environmental effects
when performing PL studies61,62, and also to reduce NR recom-
bination and improve PLQY of the material63. Yet our results,
using PMMA as an example, reveal that such a treatment fun-
damentally modies the photophysics in the perovskite layer.
More importantly, the supreme sensitivity of PLQY(f,P) mapping
method to the inuences of interfacial modications illustrates its
efcacy for studying charge carrier dynamics not only in lms,
but also in multilayers and complete photovoltaic devices.
Another question that remains under debate in the perovskite
community is the role of bulk defects on charge carrier dynamics
in perovskite lms. While some reports claim that such bulk
defects, found for example at the grain boundaries, do not
inuence charge recombination64,65 other reports suggest such
defects inuence the optoelectronic quality of the perovskite
layer66,67. Considering these contradicting reports, it is clear that
traditional PLQY measurements are not capable to idenitfy the
role of bulk defects. We believe that PLQY(f,P) map is the best
possible ngerprint of the sample in the context of its charge
recombination pathways and may aid at resolving this and other
open questions in the eld. We predict that this non-invasive,
simple and non-expensive method will nd practical applications
in controlling and optimizing semiconducting materials and the
devices that are based on them.
Conclusions
To summarize, we examined the validity of the commonly
employed ABC and SRH kinetic models in describing the charge
dynamics of metal halide perovskite MAPbI
3
semiconductor. For
this purpose, we developed a novel experimental methodology
based on PL measurements (PLQY and time resolved decays)
performed in the two-dimensional space of the excitation energy
and the repetition frequency of the laser pulses. The measured
PLQY maps allow for an unmistakable distinction between
samples, and more importantly, between the single-pulse and
quasi-continuous excitation regimes.
We found that neither the ABC nor the SRH model can explain
the complete PLQY maps for MAPbI
3
samples and predict the PL
decays at the same time. Each model is valid only in a limited
range of parameters, which may strongly vary between different
samples. On the other hand, we show that the extension of the
SRH model by the addition of Auger recombination and Auger
trapping (SRH+model) results in an excellent t of the complete
PLQY maps for all the studied samples. Nevertheless, even this
extended model systematically underestimate the PL decay rates
at high pulse uences pointing towards the existence of additional
processes in MAPbI
3
which must be considered to fully explain
the charge carrier dynamics.
Our study clearly shows that neither PL decay nor PLQY data
alone are sufcient to elucidate the photophysical processes in
perovskite semiconductors. Instead, a combined PLQY mapping
and time-resolved PL decays should be used to elucidate the
excitation dynamics and energy loss mechanisms in luminescent
semiconductors. Our experimental approach provides a strict
criteria for testing any theoretic model of charge dynamics which
is the requirement to be able to t PLQY(f,P) map and PL decays
at different powers and pulse repetition rates.
Methods
Thin lm preparation. All samples were prepared from same perovskite precursor
which was prepared with 1:3 molar ratio of lead acetate trihydrate and methy-
lammonium iodide dissolved in dimethylformamide (Supplementary Note 5). For the
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samples with PMMA between the glass and perovskite layer, PMMA was spin-coated
on the clean substrates at 3000 rpm for 30 s and annealed at 100 °C for 10 min. The
perovskite precursor was spin-coated at 2000 rpm for 60 s on glass or glass/PMMA
substrates, following by a 25s dry air blowing, a 5 min room temperature drying and
a 10 min 100 °C annealing. For the samples with PMMA on top of the perovskite
layer, no further annealing was applied after the PMMA deposition.
PL measurements. Photoluminescence microscopy measurements were carried
out using a home-built wide-eld microscope based on an inverted uorescence
microscope (Olympus IX-71) (Supplementary Note 1). We used 485 nm pulsed
laser (ca. 150 ps pulse duration) driven by Sepia PDL 808 controller (PicoQuant)
for excitation with repetition rate tuned from 100 Hz to 80 MHz. The laser irra-
diated the sample through an objective lens (Olympus ×40, NA =0.6) with ~30 µm
excitation spot size. The emission of the sample was collected by the same objective
and detected by an EMCCD camera (Princeton Instruments, ProEM 512B). Two
motorized neutral optical density (OD) lter wheels were used: one in the exci-
tation beam path in order to vary the excitation uence over 4 orders of magnitude
and one in the emission path to avoid saturation of the EMCCD camera. The
whole measurement of a PLQY(f,P) map was fully automated and took ~3 h (see
Supplementary Note 2 for details). Automation was crucial for avoiding human
errors in the measurements of so many data points (about 100 data points per
PLQY(f,P) map and to minimize light exposure of the sample.
Time-resolved photoluminescence (TRPL) measurements were carried out
using the same microscope, by adding a beam splitter in front of the EMCCD and
redirecting a part of the emission light to a single photon counting detector
(Picoquant PMA Hybrid-42) connected to a TCSPC module (Picoharp 300).
Absolute PLQY measurements were performed using a 150 mm Spectra lon
Integrating Sphere (Quanta-φ, Horiba) coupled through an optical bre to a
compact spectrometer (Thorlabs CCS200). Sample PL was excited by the same
laser with 80 MHz excitation repetition rate and 0.01 W/cm2excitation power
density. This reference point was then used to calculate the absolute PLQY for all
pulse uences and frequency combinations (Supplementary Notes 2 and 3).
It is important to stress that the whole acquisition of PLQY(f,P) was fully
automated and the sample was exposed to light only for the measurements. This
led to a rather small total irradiation dose of about 200 J/cm2(equivalent to 2000 s
of 1 Sun power) per one PLQY(f,P) map which accumulated over 85 acquisitions
during about 4 h for one PLQY map. Note, that 90% of this doze was accumulated
with the maximum power P5 which gives 1600 Sun when the highest frequency 80
MHz is used. This allowed us to have minimal effects of light induced PL
enhancement/bleaching on the measurements.
Data availability
The data that support the ndings of this study are available from the corresponding
authors upon reasonable request.
Code availability
The codes and algorithms used for data tting are available from the corresponding
authors upon reasonable request.
Received: 5 February 2021; Accepted: 1 April 2021;
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Acknowledgements
This work was supported by the Swedish Research Council (2016-04433) and Knut and
Alice Wallenberg foundation (2016.0059). J.L. thanks China Scholarship Council (CSC
No. 201608530162) for a Ph.D. scholarship. Theoretical work was supported by the
Russian Science foundation Project (20-12-00202). P.A.F. and S.S. thank the Wenner-
Gren foundation for the visiting (GFOh2018-0020) and postdoctoral (UPD2019-0230)
scholarships. This work was supported by the European Research Council (ERC) under
the European Unions Horizon 2020 research and innovation programme (ERC Grant
Agreement No. 714067, ENERGYMAPS). Y.V. and Q.A. also thank the Deutsche For-
schungsgemeinschaft (DFG) for funding the <PERFECT PVs> project (Grant No.
424216076) in the framework of SPP 2196. We thank Dr. Fabian Paulus for performing
and analysing the XRD measurements and Prof. Jana Zaumseil for providing access to
the XRD facilities.
Author contributions
I.G.S. conceived and planned the experiments with input from P.A.F. and A.K., A.K.
designed and built the automated PLQY mapping setup with contributions from I.G.S.,
A.K., and A.Y. performed PLQY mapping and PL decay measurements. Q.A., J.L., and
S.S. prepared the samples and carried out sample characterization. Q.A. performed the
UVvis and SEM measurements. P.A.F. developed the theory, carried out the modelling
and wrote the theoretical part of the manuscript. I.G.S., P.A.F. and Y.V. determined the
main ideas of the study and supervised the project. I.S. wrote the manuscript with great
contributions by Y.V. and P.A.F. All authors have discussed the results and commented
on the nal manuscript.
Funding
Open access funding provided by Lund University.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s41467-021-23275-w.
Correspondence and requests for materials should be addressed to P.A.F., Y.V. or I.G.S.
Peer review information Nature Communications thanks Michele Saba and the other,
anonymous, reviewer(s) for their contribution to the peer review of this work. Peer
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Point defects in metal halide perovskites play a critical role in determining their properties and optoelectronic performance; however, many open questions remain unanswered. In this work, we apply impedance spectroscopy and deep-level transient spectroscopy to characterize the ionic defect landscape in methylammonium lead triiodide (MAPbI3) perovskites in which defects were purposely introduced by fractionally changing the precursor stoichiometry. Our results highlight the profound influence of defects on the electronic landscape, exemplified by their impact on the device built-in potential, and consequently, the open-circuit voltage. Even low ion densities can have an impact on the electronic landscape when both cations and anions are considered as mobile. Moreover, we find that all measured ionic defects fulfil the Meyer–Neldel rule with a characteristic energy connected to the underlying ion hopping process. These findings support a general categorization of defects in halide perovskite compounds.
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Lead halide perovskites are a remarkable class of materials that have emerged over the past decade as being suitable for application in a broad range of devices, such as solar cells, light‐emitting diodes, lasers, transistors, and memory devices. While they are often solution‐processed semiconductors deposited at low temperatures, perovskites exhibit properties one would only expect from highly pure inorganic crystals that are grown at high temperatures. This unique phenomenon has resulted in fast‐paced progress toward record device performance. Unfortunately, the basic science behind the remarkable nature of these materials is still not well understood. This review assesses the current understanding of the photoluminescence properties of metal halide perovskite materials and highlights key areas that require further research. Furthermore, the need to standardize the methods for characterization of PL in order to improve comparability, reliability, and reproducibility of results is emphasized. The study of photoluminescence properties of metal halide perovskites is of great importance for both the fundamental understanding of their photophysics and further optimization of perovskite‐based optoelectronic devices. This review provides an overview of the current state of knowledge related to the photoluminescence of perovskites and highlights the dire need for standardization of characterization methodologies in order to improve the reproducibility, reliability, and comparability of results.
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Low-cost solution-based synthesis of metal halide perovskites (MHPs) invariably introduces defects in the system, which could form Shockley-Read-Hall (SRH) electron-hole recombination centers detrimental to solar conversion efficiency. Here, we investigate the nonradiative recombination processes due to native point defects in methylammonium lead halide (MAPbI 3 ) perovskites using ab initio nonadiabatic molecular dynamics within surface-hopping framework. Regardless of whether the defects introduce a shallow or deep band state, we find that charge recombination in MAPbI 3 is not enhanced, contrary to predictions from SRH theory. We demonstrate that this strong tolerance against defects, and hence the breakdown of SRH, arises because the photogenerated carriers are only coupled with low-frequency phonons and electron and hole states overlap weakly. Both factors appreciably decrease the nonadiabatic coupling. We argue that the soft nature of the inorganic lattice with small bulk modulus is key for defect tolerance, and hence, the findings are general to other MHPs.
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We compare three representative high performance PV materials: halide perovskite MAPbI3, CdTe, and GaAs, in terms of photoluminescence (PL) efficiency, PL lineshape, carrier diffusion, and surface recombination and passivation, over multiple orders of photo-excitation density or carrier density appropriate for different applications. An analytic model is used to describe the excitation density dependence of PL intensity and extract the internal PL efficiency and multiple pertinent recombination parameters. A PL imaging technique is used to obtain carrier diffusion length without using a PL quencher, thus, free of unintended influence beyond pure diffusion. Our results show that perovskite samples tend to exhibit lower Shockley–Read–Hall (SRH) recombination rate in both bulk and surface, thus higher PL efficiency than the inorganic counterparts, particularly under low excitation density, even with no or preliminary surface passivation. PL lineshape and diffusion analysis indicate that there is considerable structural disordering in the perovskite materials, and thus photo-generated carriers are not in global thermal equilibrium, which in turn suppresses the nonradiative recombination. This study suggests that relatively low point-defect density, less detrimental surface recombination, and moderate structural disordering contribute to the high PV efficiency in the perovskite. This comparative photovoltaics study provides more insights into the fundamental material science and the search for optimal device designs by learning from different technologies.
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Perovskite solar cells have attracted intense attention over the past decade due to their low cost, abundant raw materials and rapidly growing power conversion efficiency (PCE). However, nonradiative charge carrier losses still constitute a major factor limiting the PCE to well below the Shockley–Queisser limit. This perspective summarizes recent atomistic quantum dynamics studies on the photoinduced excited-state processes in metal halide perovskites (MHPs), including both hybrid organic-inorganic and all-inorganic MHPs, and three- and two-dimensional MHPs. The simulations, performed using a combination of time-domain ab initio density functional theory and nonadiabatic molecular dynamics, allow emphasis on various intrinsic and extrinsic features, such as component, structure, dimensionality and interface engineering, control and exposure to various environmental factors, defects, surfaces, and their passivation. The detailed atomistic simulations advance our understanding of electron-vibrational dynamics in MHPs, and provide valuable guidelines for enhancing the performance of perovskite solar cells.
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In this perspective, we explore the insights into the device physics of perovskite solar cells gained from modeling and simulation of these devices. We discuss a range of factors that influence the modeling of perovskite solar cells, including the role of ions, dielectric constant, density of states, and spatial distribution of recombination losses. By focusing on the effect of non-ideal energetic alignment in perovskite photovoltaic devices, we demonstrate a unique feature in low recombination perovskite materials – the formation of an interfacial, primarily electronic, self-induced dipole that results in a significant increase in the built-in potential and device open-circuit voltage. Finally, we discuss the future directions of device modeling in the field of perovskite photovoltaics, describing some of the outstanding open questions in which device simulations can serve as a particularly powerful tool for future advancements in the field.
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Halide perovskites have remarkable properties for relatively crudely processed semiconductors, including large optical absorption coefficients and long charge carrier lifetimes. Thanks to such properties, these materials are now competing with established technologies for use in cost‐effective and efficient light‐harvesting and light‐emitting devices. Nevertheless, the fundamental understanding of the behavior of charge carriers in these materials—particularly on the nano‐ to microscale—has, on the whole, lagged behind empirical device performance. Such understanding is essential to control charge carriers, exploit new device structures, and push devices to their performance limits. Among other tools, optical microscopy and spectroscopic techniques have revealed rich information about charge carrier recombination and transport on important length scales. In this progress report, the contribution of time‐resolved optical microscopy techniques to the collective understanding of the photophysics of these materials is detailed. The ongoing technical developments in the field that are overcoming traditional experimental limitations in order to visualize transport properties over multiple time and length scales are discussed. Finally, strategies are proposed to combine optical microscopy with complementary techniques in order to obtain a holistic picture of local carrier photophysics in state‐of‐the‐art perovskite devices. There has been rapid progress in halide perovskite device performance but further improvements require a firm understanding of charge carrier photophysics. This article details the recent uses of time‐resolved optical microscopy techniques to understand nanoscale charge carrier transport and recombination mechanisms. Ongoing technical developments and future strategies to fill gaps in understanding of carrier behavior in perovskites are discussed.
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The invention and development of the laser have revolutionized science, technology, and industry. Metal halide perovskites are an emerging class of semiconductors holding promising potential in further advancing the laser technology. In this Review, we provide a comprehensive overview of metal halide perovskite lasers from the viewpoint of materials chemistry and engineering. After an introduction to the materials chemistry and physics of metal halide perovskites, we present diverse optical cavities for perovskite lasers. We then comprehensively discuss various perovskite lasers with particular functionalities, including tunable lasers, multicolor lasers, continuous-wave lasers, single-mode lasers, subwavelength lasers, random lasers, polariton lasers, and laser arrays. Following this a description of the strategies for improving the stability and reducing the toxicity of metal halide perovskite lasers is provided. Finally, future research directions and challenges toward practical technology applications of perovskite lasers are provided to give an outlook on this emerging field.