Scaling Laws of Exciton Recombination Kinetics in Low Dimensional Halide Perovskite Nanostructures

Article abstract of DOI:10.1021/jacs.0c02000

Carrier recombination is a crucial process governing the optical properties of a semiconductor. Although various theoretical approaches have been utilized to describe carrier behaviors, a quantitative understanding of the impact of defects and interfaces in low dimensional semiconductor systems is still elusive. Here, we develop a model system consisting of chemically tunable, highly luminescent halide perovskite nanocrystals to illustrate the role of carrier diffusion and material dimensionality on the carrier recombination kinetics and luminescence efficiency. Our advanced synthetic methods provide a well-controlled colloidal system consisting of nanocrystals with different aspect ratios, halide compositions, and surface conditions. Using this system, we reveal the scaling laws of photoluminescence quantum yield and radiative lifetime with respect to the aspect ratio of nanocrystals. The scaling laws derived herein are not only a phenomenological observation but proved a powerful tool disentangling the carrier dynamics of microscopic systems in a quantitative and interpretable manner. The investigation of our model system and theoretical formulation bring to light the dimensionality, as a hidden constraint on carrier dynamics, and identify the diffusion length as an important parameter that distinguishes nanoscale and macroscale carrier behaviors. The conceptual distinction in carrier dynamics in different dimensionality regimes informs new design rules for optical devices where complex microstructures are involved.

Full text of DOI:10.1021/jacs.0c02000

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Article
Scaling Laws of Exciton Recombination Kinetics in
Low Dimensional Halide Perovskite Nanostructures
Mengyu Gao, Hao Liu, Sunmoon Yu, Sheena Louisia, Ye
Zhang, David P. Nenon, A. Paul Alivisatos, and Peidong Yang
J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.0c02000 • Publication Date (Web): 16 Apr 2020
Downloaded from pubs.acs.org on April 16, 2020
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TEM characterizations on growth kinetics and crystallography. a, Representative TEM images of CsPbBr3
nanocrystals with different aspect ratios. b, Length of NCs as a function of reaction time. c,d, HR TEM
images of CsPbBr3 and CsPbI3 nanowires.
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Representative optical characterizations and scaling laws of PLQY of CsPbX3 NCs. a, PL and absorption
spectra of CsPbBr3 nanowires (green curves) and CsPbI3 nanowires (red curves). b, Time-resolved PL decay
curves of CsPbBr3 nanowires (green curve) and CsPbI3 nanowires (red curve). c, Power-dependent PL
measurement for CsPbBr3 quantum dots, nanowires (diameter ~10 nm, length ~ 1000 nm), and microwires
(
diameter > 100 nm). Slopes of the log plots are indicated by k-values for NCs with different dimensions. d,
Aspect ratio dependent PLQY (scaling law of PLQY) of CsPbBr3 nanocrystals (green data points) and CsPbI3
nanocrystals (red data points). The error bar for x-axis showing the deviation of length and that for y-axis
showing the measurement deviation of quantum yield (inset shows the linear plotting with the same data
points, omitting the standard deviations).
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Comparison of simulation results with experimental results. a, Green solid data points are PLQY measured
from CsPbBr3 NCs, while red solid data points are that from CsPbI3 NCs. Blue/Orange hollow data points are
simulated for CsPbBr3/CsPbI3 NCs. b, Experimental and simulation results for averaged lifetime of CsPbBr3
NCs. Width of the shaded area is from the simulation of lifetime with different parameters (radiative time
and defect density).
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Simulated results where defect density and diffusivity are controlled. a, Aspect-ratio-dependent PLQY with
various defect density. b, Aspect-ratio-dependent PLQY with various carrier diffusivity. Green-shaded area is
the division of spatially-confined regime and blue-shaded area is that of diffusion-limited regime. c,
Demonstration of the linear relation between the initial slope of the QY-length plot and defect density
(dashed line is a linear plot with slope of unity). Data points with different colors are from different
diffusivities. This figure demonstrates that the initial slope is less dependent on diffusivity when defect
density is lower. Two experimental results are from CsPbBr3 and CsPbI3 samples, indicating the defect
density of our system is within the linear region where the slope of the scaling law of PLQY reflects the
defect density. d, Demonstration of the linear relation between critical length (the length upon which PLQY
reaches a plateau) and diffusion length. When diffusivity and defect density are smaller, this linear relation
is more prominent.
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Scaling Laws of Exciton Recombination Kinetics in Low
Dimensional Halide Perovskite Nanostructures
†
,§
‡
†,§
‡,§
‡,§
Mengyu Gao, Hao Liu, Sunmoon Yu, Sheena Louisia, Ye Zhang, David
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‡
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†,‡,§,
†,‡,§, ,*
∥
∥
P. Nenon, A. Paul Alivisatos,
and Peidong Yang
†
Department of Materials Science and Engineering, University of California, Berkeley,
Berkeley, California 94720, United States
Department of Chemistry, University of California, Berkeley, Berkeley, California 94720,
United States
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California
‡
§
94720, United States
∥ Kavli Energy NanoScience Institute, Berkeley, California 94720, United States
* Corresponding author. E-mail: p_yang@berkeley.edu (P.Y.)
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ABSTRACT
Carrier recombination is a crucial process governing the optical properties of a semiconductor.
Although various theoretical approaches have been utilized to describe carrier behaviors, a
quantitative understanding of the impact of defects and interfaces in low dimensional
semiconductor systems is still elusive. Here we develop a model system consisting of chemically
tunable, highly luminescent halide perovskite nanocrystals to illustrate the role of carrier
diffusion and material dimensionality on the carrier recombination kinetics and luminescence
efficiency. Our advanced synthetic methods provide a well-controlled colloidal system
consisting of nanocrystals with different aspect ratios, halide compositions, and surface
conditions. Using this system, we reveal the scaling laws of photoluminescence quantum yield
and radiative lifetime with respect to the aspect ratio of nanocrystals. The scaling laws derived
herein are not only a phenomenological observation but proved a powerful tool disentangling the
carrier dynamics of microscopic systems in a quantitative and interpretable manner. The
investigation of our model system and theoretical formulation bring to light the dimensionality as
a hidden constraint on carrier dynamics and identify the diffusion length as an important
parameter that distinguishes nanoscale and macroscale carrier behaviors. The conceptual
distinction in carrier dynamics in different dimensionality regimes informs new design rules for
optical devices where complex microstructures are involved.
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INTRODUCTION
The candidacy of halide perovskites in optoelectronic applications originates from their ability to
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realize highly-efficient energy conversion. Perovskite-based solar cells, light-emitting
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devices, and radiation detectors are all excellent examples. Such wide applications of halide
perovskites also stimulate fundamental studies in diverse fields, among which carrier dynamics,
because of its relevance to luminescence efficiency, is of great importance.6
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Early works on halide perovskites demonstrate their intriguing carrier transport properties,
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including their long charge carrier diffusion length, long lifetime, low deep-trap density, etc.
Recent findings regarding the microstructures of thin-film devices, however, have shown the
complexities related to the inhomogeneity of real systems.11–13 Due to the involvement of various
chemical species and complicated microenvironment, the mechanisms governing the specific
light-matter interactions that impair the device performance are under debate.6,14,15 A better
understanding of the impact of defects and dimensionality requires well-defined model systems
that permit precise structural determination and systematic study of optical properties.
Precise control on the synthesis of nanocrystals (NCs) is a promising approach to unravel the
complexities associated with heterogeneous interfaces. Due to their high uniformity and
excellent luminescence efficiency, together with molecular-level probes that afford accurate
structural characterizations, NCs have been utilized to understand the electronic structures,16
thermodynamics,17,18 and transport properties19,20 of semiconductors. On the other hand,
investigations of colloidal NCs carried within the field of halide perovskites can simultaneously
uncover other aspects looking into the “localized” carrier dynamics within single domain of each
NC.21–26 Systems consisting of well-controlled NCs, therefore, open new opportunities in the
investigation of optical properties of halide perovskites.
With extensive synthetic efforts and theoretical formulations in the halide perovskite research,
different models are employed to describe optical phenomena in (1) low-dimensional, confined
systems22,27 or (2) extended, macroscopic systems.28,29 However, little attention has been paid to
connect the two seemingly irreconcilable length scales. Integrating the local and global pictures
of carrier dynamics is thus essential to the better understanding of the electronic processes,
which will inform improved design of perovskite-based devices.
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In this work, we establish a model system for the systematic investigation of the carrier dynamics
in halide perovskites using highly luminescent and well-controlled NCs. Relying on the
advanced synthesis of halide perovskite colloidal NCs, we take advantage of the large flexibility
of tunable variables including aspect ratio, composition and surface condition to access physical
properties of halide perovskite nanostructures. Static photoluminescence quantum yield
measurement (PLQY) and time-resolved PL lifetime measurement reveal the intriguing aspect
ratio dependent carrier recombination kinetics of NCs, highlighting the distinguishable 0D and
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D carrier dynamics. The direct interpretation of size scaling behaviors in terms of carrier
diffusion and recombination is corroborated by time-resolved PL analysis and theoretical
modelling.
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RESULTS AND DISCUSSION
Building a model system of halide perovskite NCs. To avoid the interference of
microstructural heterogeneity, we established a platform for the systematic property studies
consisting of high-quality halide perovskite NCs with different aspect ratios and compositions.
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Modified based on our previous recipe,30,31 the time-controlled synthesis of colloidal CsPbBr
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NCs yields nanowires with high purity and size homogeneity. We observed that nanowire length
grew linearly with reaction time at the fixed reaction temperature, with the growth speed at
around 3.7 nm/second (Fig. 1b). Colloidal CsPbBr quantum dots with a similar diameter
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dispersion were also synthesized to construct a library of NCs with different dimensions. NCs
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(including nanowires and quantum dots) with aspect ratio ranging from 1 to 10 show decent size
homogeneity (with diameter of 9.5 ± 1.5 nm, and standard deviation of length of around 20 %),
and are morphologically pure (Fig. 1a). The crystallinity and growth directions of CsPbBr NCs
3
were confirmed by powder X-ray diffraction (Fig. S5), atomic resolution high-angle annular
dark-field imaging (Fig. S2), conventional high resolution transmission electron microscopy (HR
TEM) images (Fig. 1c).
Although this synthetic protocol yields single-crystalline, phase-pure nanowires with controllable
aspect ratios, the luminescence efficiencies are initially poor (PLQY ~1%), as the centrifugation
as a purification method is known to remove the majority of passivating ligands. To investigate
the carrier dynamics in systems with excellent optical properties, we turned to surface
passivation strategies to improve PLQY. Due to the soft-hard mismatch of oleate and lead
cations, the routinely used oleic acid-oleylamine (OA-OAm) pair could only improve the
efficiency up to 20% for short nanowires32,33. The use of softer anionic ligand (hexylphosphonic
acid, HPA) together with OAm for surface treatment on halide perovskite quantum dots proved
3
3
as the best way to improve the PLQY of perovskite NCs . We therefore adopted the similar
ligand exchange method, where the as-synthesized nanowires were purified thoroughly to
remove the native ligand shell, and then HPA-OAm was used as new ligand treatment to restore
the surface passivation. The resulting improvement of PLQY by ligand engineering is significant
(from ~1% to 20-60%). We also reproduced the surface treatment on quantum dots whose PLQY
could be increased up to unity. Though difficulties exist for probing the surface condition of
halide perovskite NCs, owing to the dynamic interaction of surface ligands with halide
perovskite NCs surface atoms, we were able to probe the strong interaction of the soft acid-base
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pair with halide perovskite NCs surfaces through 2D nuclear Overhauser effect spectroscopy (2D
NOESY) experiments. The negative cross peaks in the 2D NOESY spectrum indicated the
tumbling of ligands with slowed relaxation rate due to the interaction between ligands and NC
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surfaces (Fig. S8), which demonstrates the generality of this ligand exchange method on both
D quantum dots and 1D nanowires.
Anion exchange of perovskite NCs is well established31,35–39 and allows us to expand current
CsPbBr NCs system into other halide compositions, which helps to determine whether our
findings are generalizable beyond pure CsPbBr compositions. We fully converted CsPbBr
into CsPbI , without altering the morphology, size distribution, and crystallinity (Fig. 1d). The
same anion exchange reaction was done on the NCs of different aspect ratios (Fig. S1).
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NCs
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Exchanging the anionic sublattice from bromide to iodide produces the expected red shift in both
absorbance and PL spectra. The PL peak positions shift from 515 nm to 680 nm by fully
converting bromide NCs into iodide NCs (Fig. 2a).31
The advanced synthetic methods afford an ideal model system consisting of CsPbBr
exchanged CsPbI NCs with the same diameter and different aspect ratios. After surface
treatment, all those NCs were passivated with the same ligand pairs and exhibited high PLQY
100% for 0D NCs, and up to 85% for 1D NCs). The improved quality of NCs due to well-
3
and anion-
3
(
designed surface chemistry enables the reproducible and quantitative studies, while the purity
and controllability of halide perovskite NCs allow us to study their photophysics in a systematic
way.
Scaling laws of optical properties of halide perovskite NCs. With a model system in hand, we
used steady-state PLQY and time-resolved PL lifetime measurements to investigate carrier
dynamics across a wide range of aspect ratios. Together, these two techniques determine the
recombination kinetics, thereby informing the key carrier dynamics of the system of interest.
40
PLQY was determined by a home-built integrating sphere system, and PL lifetime was
calculated from time-resolved PL measurement (Fig. 2b). These measurements revealed an
intriguing dependence on the aspect ratio.
The aspect-ratio-dependent data are shown in Fig. 2c. From the experimental data, there is a
clear trend that NCs with larger aspect ratio have lower PLQY, and this trend also applies to
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different colloidal systems, such as those with different compositions, different surface
conditions and different synthetic methods (Fig. S13, Fig. S14). The linear plot of PLQY as a
function of aspect ratio shows that PLQY decreases with increased aspect ratio, and reaches a
plateau upon a certain critical length. The estimated critical length scale for CsPbBr
NCs is around 500-800 nm. The PLQY plateau for CsPbBr NCs is around 20% and that for
CsPbI NCs is around 50%. For the lowest aspect ratio, which is for quantum dots with cubic
shape, PLQY is almost unity for both CsPbBr and CsPbI NCs. The PL lifetime of those
nanocrystals also exhibits a length-dependent trend (Fig. S9). For CsPbBr NCs with increased
aspect ratio, the averaged PL lifetime increased from 5 ns to tens of nanoseconds, and for CsPbI
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and CsPbI
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NCs, the averaged PL lifetime increased from 20 ns to hundreds of nanoseconds, both consistent
with reported PL lifetime for quantum dots and nanowires.31,32 We summarized the size-
dependent optical properties of halide perovskite NCs as “scaling laws”. Explicitly, the scaling
law describes the relation between a property (f) and the size (V) of a material, f(V). In our
scenario, the size V is linearly proportional to the length. The scaling laws defined in this way are
not only an important attribute of NCs, but also an essential characteristic component that
defines low-dimensional semiconductors.41,42
Validity of the scaling laws of halide perovskite NCs. Before the theoretical analysis of the
observed scaling laws, the validity and generality of the experimental observations need to be
demonstrated. Our later analysis and modelling are based on the assumption that all NCs have
similar surface condition and defect density. As mentioned in previous sections, the diameters of
all NCs are uniform at around 9.5 nm (Fig. S4), so both the surface area and volume linearly
scale with the aspect ratio or length of NCs. X-ray photoemission spectroscopy (XPS) was
conducted to probe the surface condition of NCs with different aspect ratios, which revealed the
halide-lean surface for halide perovskite NCs, and NCs of different dimensions all shared similar
surface stoichiometry (Fig. S7, Tab. S1). By tuning only NC length and composition, with all
other synthesis parameters being well-controlled, the measurements from such samples will
highlight length- and composition-specific information only. Other evidence of the aspect ratio
related scaling law was done by sonicating as-synthesized long nanowires. Sonication broke
down the nanowires into shorter ones and this treatment resulted in increased PLQY (Fig. S13).
Considering the deleterious effect usually imposed by sonication, the improvement of PLQY
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indicates the generality of our experimental observation. Similar scaling law of PLQY has also
been observed for NCs with different surface environments (Fig. S14). Power dependent PL
measurements (Fig. 2c) together with power dependent time-resolved PL measurements
demonstrate that the linear kinetics governs the recombination pathway in halide perovskite NCs
in our study. This linearity rules out the possible higher order recombination pathways, such as
bimolecular free-carrier recombination, Auger recombination, exciton-exciton interaction, etc.
This linear kinetics also suggests that excitons are main carriers in the diffusion and
recombination processes. With the assumption of commensurability among all dimensions and
the generality of the scaling law, we now turn to theoretical modelling to identify the key factors
that underlie our experimental observations.
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Modelling of PLQY and carrier dynamics. To understand the size scaling of optical
properties, we resorted to the theoretical frameworks of carrier dynamics. The continuity
equation with spatial and temporal evolution of carrier density is one of the formal ways to
describe carrier behaviors in bulk semiconductors. Although the continuity equation generally
remains valid for the confined system, it requires reformulation to account for the large
heterogeneity of nonradiative recombination kinetics and explicit spatial boundary conditions
(see Supporting Information). As will be shown later, the dimensionality will play a signification
role modulating the carrier dynamics in confined systems when defect density is relatively low.
The incompatibility of models in extended and confined systems poses a difficulty43–45 in using a
single continuity equation to describe our system, where the aspect ratio spans from 0D to 1D.
As a result, instead of solving the set of continuity equations, we conducted a Monte Carlo
simulation with a exciton-diffusion-defect-encounter random walk model.
As observed in the scaling law of PL lifetime, the NCs with large aspect ratio exhibits prolonged
lifetime, while the power dependent time-resolved PL measurements preclude the existence of
free-carrier-dominant recombination channels. So in addition to the typical radiative and
nonradiative recombination processes, we introduced a detrapping process, through which the
trapped excitons can be released with some probability. Trapping-detrapping mechanisms are
classically adopted in the semiconductor community46,47 and have been recently brought up as an
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alternative explanation to the PL blinking phenomena and prolonged radiative lifetimes. In the
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context of halide perovskite, trapping/detrapping process is highly relevant to its defect
tolerance.
In each random walk, we tracked the trajectory of each exciton and collected all the dynamic
information. From this, PLQY and carrier recombination decays as ensemble observables can be
easily retrieved. We performed both a “local” fitting and a “global” fitting, where the former
technique was intended to extract carrier dynamical parameters (defect density, exciton
diffusivity, exciton radiative recombination rate, etc.) from the time-resolved PL curves, while
the later one was aimed at reproducing the scaling laws by assuming the constant aforementioned
carries dynamical parameters.
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From the local fitting of time-resolved PL experiments (Fig. S11, Fig. S12), we extract the main
carrier dynamical parameters from six representative batches of CsPbBr NCs (Tab. 1). We
3
found that although the apparent averaged lifetime is increased from shorter (~8) to longer aspect
ratios (~180), the radiative recombination rate is almost constant in the NWs regimes.
Considering that the Bohr radius of CsPbBr is around 7 nm, the reduced dielectric screening and
3
quantum confinement due to elongation of NWs plays little role altering the exciton radiative
recombination kinetics. The local fitting results also proved the homogeneity of our model
system, that is, the consistency of carrier dynamical parameters. With the assumption that NCs
with different aspect ratios share similar carrier dynamical parameters, we performed the global
fitting by varying the aspect ratio, therefore, reproducing the scaling laws observed
experimentally. The global fitting is shown in Fig. 3a, whereby we determined the linear defect
density as 0.005 effective defects per nanometer (abbreviated as 1/nm thereafter) and 0.0016
1/nm for CsPbBr
3
and CsPbI
3
respectively. We transformed the linear defect density into bulk
defect density taking into account the diameter of NCs. The bulk defect density is determined as
16
3
16
3
5
.0×10 1/cm for CsPbBr
3
and 1.6×10 1/cm for CsPbI , which respectively share a similar
3
3
1
order of magnitude with our previous results. One usually overlooked fact about the defect
density, as reported in the literature, lies where the defect density only counts the time average of
fully-functioning trap states. Dominant defect species in halide perovskite (especially in CsPbBr
3
and CsPbI ), such as halide or cesium vacancies, only introduce shallow trapping levels or barely
3
contribute to luminescence efficiency.10,32,33 So in general electrical or optically defined and
determined defect density will be significantly lower than the density of physically present
3
3
vacancies, and the apparent discrepancy between the two types of values is a signature of the
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well-known defect tolerance of halide perovskites. The exciton diffusivity were determined as
2
7,8,10,49,50
0.4 cm /s, which is comparable with the literature.
As mentioned above, in order to
explain prolonged PL lifetime, we need to include detrapping process. With delayed time of
around 100 ns, we were able to fit the lifetime results with our simulation (Fig. 3b). Without a
detrapping process, however, simulated results showed a decreased lifetime with increasing
aspect ratio (Fig. 3b), which is contradictory to the experimental observations. The physical
meaning of this number is still under investigation but comparable with other NCs systems.48,51
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Size scaling in the NCs literature. The aspect ratio dependent optical properties of 1D
semiconductor NCs have been reported in diverse systems, notably in traditional covalent II-VI
or III-V semiconductor nanorods/nanowires and carbon nanotubes. Because of the synthetic
limitations and phase complexity, the aspect ratio of single crystalline nanorods is usually limited
below 100 or even 20.52–58 There are some recent results also concerning the low-aspect-ratio
perovskite nanorods, but those NCs also suffer from both limited length scale and unknown
surface condition.58,59 In such small aspect ratio regime, wave function extension will be
modified with changing aspect ratio, thus giving rise to the renormalized electronic energy
levels,60,61 and imposing more complexities on the analysis of PLQY60,62,63 and recombination
dynamics.58,60,64–66 In our system, due to advanced synthetic control, the aspect ratio of halide
perovskite NCs ranges from 1 to 1000, far exceeding the quantum mechanical regime. We
therefore exclude complexities due to the band gap renormalization, and the carrier dynamics can
be treated semi-classically.
The PLQY scaling in large aspect ratio system is well documented in the optical studies of
carbon nanotubes.67–70 However, the size scaling trend of carbon nanotubes is the opposite to our
case, where a smaller aspect ratio results in a lower PLQY. This is due to the quenching sites
found on the ends of the carbon nanotube.69,71 In other words, the defect density decreases as the
nanotube grows longer. In our system, however, due to the well-controlled surface chemistry, we
could modify the surface and maintain a constant defect density among different aspect ratios.
This was corroborated by the carrier dynamics analysis (Tab.1) and XPS measurement (Tab. S1).
Due to a better synthetic control over dimensionality, composition and surface condition of
halide perovskite NCs, high crystal quality and size uniformity help to exclude potential
complications from microstructures. The realization of the continuity from quasi-0D to 1D NCs,
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simply put, that of the chemical tunability, reveals the hidden constraint of dimensionality on
carrier dynamics. We are therefore able to tackle the scaling laws from a new perspective, while
constructing a revised and unified view on carrier dynamics in low-dimensional semiconductors.
The scaling laws of optical properties in halide perovskite and theoretical analysis suggest that
the description of carrier dynamics in nanoscale compared with that in bulk semiconductors
requires careful reformulation, notably when the non-radiative recombination rate knr turns out to
be size-dependent as the excitons are confined by microscopic boundaries (see Supporting
Information). That is why we resorted to a Monte Carlo random walk model to simulate the
global fitting of the scaling laws. With this model, we bridged the gap between quasi-0D and 1D
exciton behaviors directly while circumventing the complexity of continuity equation and the
difficulty in solving the first passage process analytically.43–45 In the preceding sections, we not
only reproduced the experimental results, but identified a critical length scale that distinguishes
the carrier dynamical behaviors in 0D and 1D. Given the advantages of our NCs system and the
flexibility of our simulation method, in the following section, we will examine the physical
origin of the scaling laws and demonstrate the generics of this size-scaling methodology.
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Interpreting the scaling laws. To interpret the scaling laws in an intuitive way, we first need to
investigate the very length scale on which physical processes are happening. In our system, three
types of length scale are of interest, (1) the length of the nanowire l, (2) the average distance
between defects l
d
(inverse of the linear defect density ρ), and (3) the exciton diffusion length l
r
2
(which is here defined as l
r
= Dτ
r
, where D is the exciton diffusivity, and τ is the radiative
r
recombination lifetime). Assuming ρ is small, we derived an asymptotic expression based on two
extreme limits of l (see Supporting Information): When l approaches infinity, PLQY = 1− αl /l
where α is a proportional constant); when l is infinitesimally small, PLQY = 1− l/l
When l goes to the bulk limit, in both experimental and simulation results, we observed the
r
d
(
d
.
length-independent plateau where NCs with large aspect ratio behave more like bulk materials
than nanomaterials, which is corresponding to a length-independent and diffusion-limited PLQY.
The diffusivity plays an important role because the mobility of excitons determines how efficient
the radiative recombination compete with the interruption of diffusion and radiative
recombination by defects. We call this regime the diffusion-limited regime.
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In the length scale where l is on a similar order of l
r
, we observed a drastically different behavior,
that is, the PLQY starts to be less dependent on the exciton diffusivity (Fig. 4b, Fig. S18). When
l approaches zero, the influence of diffusion disappears. Due to the confined nature in this length
scale, we call this regime the spatially-confined regime. In this regime, the free diffusion of
excitons is impaired by the impermeable boundaries imposed by the length of certain NC, thus
preventing excitons being influenced by defects in other NCs. The interplay between defect
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density and the length of NCs is manifested in the expression, PLQY = 1− l/l
Fig. 4c show that the initial slope of PLQY-length curve is proportional to the defect density
ρ=1/l ), while almost independent of the exciton diffusivity.
d
, and Fic. 4a and
(
d
The demarcation between diffusion-limited regime and spatially-confined regime is graphically
determined as the intersection between the initial slope and final plateau and is denoted as lcritical
.
Based on the asymptotic analysis, this critical length is proportional to diffusion length, lcritical
αl . Fig. 4d demonstrates the near linearity of diffusion length and critical length, and this
=
r
linearity can be derived through the classical semiconductor statistics as well (see Supporting
Information).
The direct interpretability of the observed scaling laws lies in the simplicity of the asymptotic
expression in either regime and the small number of parameters involved. The important carrier
dynamical parameters can be extracted from those geometrical characteristics of the scaling laws
in a straightforward manner. Due to the generality of our theoretical derivation, we propose that
this methodology could be applied to other materials systems and other dimensionalities.
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CONCLUSIONS AND NEW PERSPECTIVES
The capacity of realizing the morphologically- and compositionally-controlled system in a
flexible and systematic way enables us to investigate the defect-efficiency relation of halide
perovskite from a new perspective. We take this advantage to demonstrate the size scaling as a
powerful tool to elucidate the carrier dynamics in low-dimensional semiconductors.
Distinguished from other techniques, the scaling laws derived here afford both simplicity and
interpretability to the analysis of carrier dynamics. Experimentally, our study bridges two
dimensions (0D and 1D or, in a general sense, nanoscale and macroscale), and identifies a
critical length scale that distinguishes the signatures of carrier dynamics exhibited in different
dimensionality regimes. As revealed in our theoretical and numerical analysis, the distinct
exciton behaviors and formulations in different dimensionality regimes are the manifestation of
how defects locally or globally influence the carrier dynamics and luminescence efficiency. We
explicitly denote this distinction of length scales by using the diffusion length, which is
highlighted due to its relevance to unifying nanoscale phenomena with the macroscopic systems.
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The microstructure of semiconductor devices is one of the key factors that influence the device
efficiency. Our system is a novel prototype to mimic the micro-to-macro transition, while
eliminating interfacial interferences. The new understanding of carrier dynamics provides a
promising approach for the design of high-efficiency semiconductor devices. Informed by the
fact that both material dimensionality and carrier diffusion are key factors contributing to the
carrier dynamics, engineering the diffusion pathways such as implementing impermeable
boundaries or immobile excitons might serve as another mechanism for increased utilization of
charge carriers or excitons.
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ACKNOWLEDGEMENTS
The authors thank Prof. E. Rabani, Dr. D. Zhang, Dr. Q. Kong, Dr. A. D. Balan, and D. An for
fruitful discussions on theoretical formulation, experimental design and computational strategies.
This work was primarily supported by the U.S. Department of Energy, Office of Science, Office
of Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE-
AC02-05-CH11231 within Physical Chemistry of Inorganic Nanostructures Program (KC3103).
Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy
Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05-CH11231. M.G.
acknowledges the Ning Fellowship granted by University of California, Berkeley. Y.Z.
acknowledges the fellowship from Suzhou Industrial Park. S.Y. acknowledges support from
Samsung Scholarship.
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NOTES
The authors declare no competing financial interest.
ASSOCIATED CONTENT
Supporting Information. Detailed synthetic procedures, characterization, theoretical and
simulation methods, and additional figures and tables (PDF).
.
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FIGURES
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Fig. 1 | TEM characterizations on growth kinetics and crystallography. a, Representative
TEM images of CsPbBr nanocrystals with different aspect ratios. b, Length of NCs as a
function of reaction time. c,d, HR TEM images of CsPbBr and CsPbI nanowires.
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Fig. 2 | Representative optical characterizations and scaling laws of PLQY of CsPbX NCs.
3
a, PL and absorption spectra of CsPbBr
curves). b, Time-resolved PL decay curves of CsPbBr
nanowires (red curve). c, Power-dependent PL measurement for CsPbBr
3
nanowires (green curves) and CsPbI
nanowires (green curve) and CsPbI
quantum dots,
3
nanowires (red
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nanowires (diameter ~10 nm, length ~ 1000 nm), and microwires (diameter > 100 nm). Slopes of
the log plots are indicated by k-values for NCs with different dimensions. d, Aspect ratio
dependent PLQY (scaling law of PLQY) of CsPbBr
3
nanocrystals (green data points) and CsPbI
3
nanocrystals (red data points). The error bar for x-axis showing the deviation of length and that
for y-axis showing the measurement deviation of quantum yield (inset shows the linear plotting
with the same data points, omitting the standard deviations).
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Tab. 1 | Main carrier dynamical parameters of CsPbBr NCs with different aspect ratios.
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Averaged Radiative
Carrier
defect density Detrapping Detrapping
1/cm3)
probability time (ns)
Length (nm)
lifetime
ns)
lifetime
(ns)
diffusivity
(
2
(
(cm /s)
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5.00×1016
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.5 (QDs)
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120
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14.6
18.5
16.3
17.6
122 (NWs)
229 (NWs)
120
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150
160
537 (NWs)
1800 (NWs)
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Fig. 3 | Comparison of simulation results with experimental results. a, Green solid data
points are PLQY measured from CsPbBr
NCs. Blue/Orange hollow data points are simulated for CsPbBr
and simulation results for averaged lifetime of CsPbBr NCs. Width of the shaded area is from
the simulation of lifetime with different parameters (radiative time and defect density).
3
NCs, while red solid data points are that from CsPbI
3
3
/CsPbI NCs. b, Experimental
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Fig. 4 | Simulated results where defect density and diffusivity are controlled. a, Aspect-ratio-
dependent PLQY with various defect density. b, Aspect-ratio-dependent PLQY with various
carrier diffusivity. Green-shaded area is the division of spatially-confined regime and blue-
shaded area is that of diffusion-limited regime. c, Demonstration of the linear relation between
the initial slope of the QY-length plot and defect density (dashed line is a linear plot with slope
of unity). Data points with different colors are from different diffusivities. This figure
demonstrates that the initial slope is less dependent on diffusivity when defect density is lower.
Two experimental results are from CsPbBr
3
and CsPbI samples, indicating the defect density of
3
our system is within the linear region where the slope of the scaling law of PLQY reflects the
defect density. d, Demonstration of the linear relation between critical length (the length upon
which PLQY reaches a plateau) and diffusion length. When diffusivity and defect density are
smaller, this linear relation is more prominent.
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