Origin of the Size-Dependent Stokes Shift in CsPbBr3 Perovskite Nanocrystals

Article abstract of DOI:10.1021/jacs.7b05683

The origin of the size-dependent Stokes shift in CsPbBr₃ nanocrystals (NCs) is explained for the first time. Stokes shifts range from 82 to 20 meV for NCs with effective edge lengths varying from ～4 to 13 nm. We show that the Stokes shift is intrinsic to the NC electronic structure and does not arise from extrinsic effects such as residual ensemble size distributions, impurities, or solvent-related effects. The origin of the Stokes shift is elucidated via first-principles calculations. Corresponding theoretical modeling of the CsPbBr₃ NC density of states and band structure reveals the existence of an intrinsic confined hole state 260 to 70 meV above the valence band edge state for NCs with edge lengths from ～2 to 5 nm. A size-dependent Stokes shift is therefore predicted and is in quantitative agreement with the experimental data. Comparison between bulk and NC calculations shows that the confined hole state is exclusive to NCs. At a broader level, the distinction between absorbing and emitting states in CsPbBr₃ is likely a general feature of other halide perovskite NCs and can be tuned via NC size to enhance applications involving these materials.

Full text of DOI:10.1021/jacs.7b05683

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Article
Origin of the Size-Dependent Stokes Shift in CsPbBr3 Perovskite Nanocrystals
Michael Christopher Brennan, John Herr, Triet S. Nguyen-Beck, Jessica
Zinna, Sergiu Draguta, Sergei Rouvimov, John Parkhill, and Masaru Kuno
J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.7b05683 • Publication Date (Web): 03 Aug 2017
Downloaded from http://pubs.acs.org on August 3, 2017
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Origin of the Size-Dependent Stokes Shift in
CsPbBr3 Perovskite Nanocrystals
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Michael C. Brennan,† John E. Herr,† Triet S. Nguyen-Beck,† Jessica Zinna,†
Sergiu Draguta,† Sergei Rouvimov,‡ John Parkhill∗,† and Masaru Kuno∗,†
†Department of Chemistry and Biochemistry, University of Notre Dame
251 Nieuwland Science Hall, Notre Dame, Indiana 46556, USA
‡Notre Dame Integrated Imaging Facility, University of Notre Dame, Notre Dame, Indiana
46556, USA
E-mail: mkuno@nd.edu,john.parkhill@nd.edu
Abstract
The origin of the size-dependent Stokes shift in CsPbBr3 nanocrystals (NCs) is
explained for the first time. Stokes shifts range from 82 to 20 meV for NCs with
effective edge lengths varying from ∼4 to 13 nm. We show that the Stokes shift is
intrinsic to the NC electronic structure and does not arise from extrinsic effects such as
residual ensemble size distributions, impurities or solvent-related effects. The origin of
the Stokes shift is elucidated via first-principles calculations. Corresponding theoretical
modeling of the CsPbBr3 NC density of states and band structure reveal the existence
of an intrinsic confined hole state 260 to 70 meV above the valence band edge state
for NCs with edge lengths from ∼2 to 5 nm. A size-dependent Stokes shift is therefore
predicted and is in quantitative agreement with the experimental data. Comparison
between bulk and NC calculations show that the confined hole state is exclusive to
NCs. At a broader level, the distinction between absorbing and emitting states in
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CsPbBr3 is likely a general feature of other halide perovskite NCs and can be tuned
via NC size to enhance applications involving these materials.
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Introduction
Hybrid lead halide perovskites have been successfully implemented as effective, low-cost light
absorbers in photovoltaic devices.1 Within the relatively short period of time since their
rediscovery,2,3 light-to-electricity conversion efficiencies have exceeded 20%.4 This success
has stimulated tremendous interest in using hybrid perovskites, such as methylammonium
lead iodide (MAPbI3, MA=CH3NH+3 ), for solar cells and for a host of other applications.5
The remarkable properties of bulk lead halide perovskites have motivated research into har-
nessing the intrinsic size and compositional control over the optical/electrical response of
corresponding nanostructured hybrid perovskites.6
Recent breakthroughs in colloidal chemistry now permit MAPbX3, FAPbX3 [FA = CH(NH2)2
+; X = I−, Br−, Cl−] and their all-inorganic counterparts (CsPbX3) to be made as high qual-
ity colloidal nanocrystals (NCs),7–9 nanowires,10–12 and nanosheets.13–16 These nanomateri-
als are especially attractive materials for next generation lighting and display technologies
due to their high photoluminescence (PL) quantum yields (QY ∼40–90%), narrow emis-
sion linewidths (70–140 meV), and size-/composition-tunable band gaps.7–9 Despite reports
that now exist on their electronic structure,17–19 much less is known about their underlying
photophysics and the nature of the emitting state.
In particular, ubiquitous Stokes shifts (∆Es) exist in hybrid8,13,20 and all-inorganic lead
halide perovskite nanostructures7,11,13,16,21–24 as well as in their thin-film counterparts.15,25
For CsPbBr3 NCs, the Stokes shift is size-dependent and ranges from 82 to 20 meV for
particles with effective edge lengths (l) between ∼4 and 13 nm, respectively. This suggests
that a distinction exists between absorbing and emitting states. Although recent studies
have discussed the spectral properties and excited state dynamics of CsPbBr3 NCs,20,23,26–29
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the origin of their size-dependent Stokes shift has yet to be explained.
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As a historical note, analogous size-dependent Stokes shifts were observed two decades
ago in colloidal CdSe quantum dots (QDs). Stokes shifts there likewise ranged from ∼100 to
10 meV for particle diameters between ∼1 and 5 nm.30 Corresponding research31 ultimately
explained this shift, yielding a more in-depth understanding about the electronic structure
of CdSe QDs, which are today an important model system.32,33 A similar elucidation of
the origin of the Stokes shift in CsPbBr3 NCs will improve our understanding about the
photophysics of this new system and will ultimately be crucial in promoting their eventual
use within applications.
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In a recent study, we have demonstrated that a size-dependent Stokes shift exists in
CsPbBr3 NCs.34 Here we establish that it is not caused by extrinsic factors and rationalize
its origin using detailed theoretical calculations of the size-dependent density of states (DOS)
and electronic structure of CsPbBr3 NCs. These computational results reveal the existence
of an inherent, size-dependent, confined hole state 260 to 70 meV above the valence band
edge state for particles with sizes between ∼2 and 5 nm. We also show that the confined
hole state is robust across NC size and crystal phase. We propose that this state is relatively
dark in absorption due to its low DOS, but bright in emission. This, in turn, explains the
experimentally-observed size-dependent Stokes shift. We further illustrate that the emit-
ting state possesses low cesium character, likely providing insight into the PL properties of
analogous nanostructured lead halide perovskites (e.g. MAPbX3).
Results and Discussion
Nineteen CsPbBr3 NC ensembles were synthesized using the technique described by Prote-
sescu et al.7 Resulting NCs adopt cuboidal morphologies and have surfaces passivated
with oleylammonium and oleate ligands.35 Figure 1 shows representative low- and high-
magnification TEM images of four ensembles, which span the size series. The low-magnification
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Figure 1: Representative low- [(a) l = 12.8 nm, (b) l = 8.5 nm, (c) l = 6.8 nm, (d) l = 4.2
nm] and high- [(e) l = 12.8 nm, (f) l = 8.5 nm, (g) l = 6.8 nm, (h) l = 4.1 nm] magnification
TEM images of CsPbBr3 NCs. Scale bars on low- and high-magnification images are 20 and
5 nm, respectively.
micrographs show that the NCs form close packed areas with residual size distributions in the
range of 5–20%. Figure S1 illustrates sizing histograms from where effective edge lengths
and size distributions have been obtained. Figure S2 illustrates an experimental sizing
curve extracted from the data. Powder X-ray diffraction (PXRD) patterns shown in Figure
S3 confirm the crystallinity of the NCs. Both TEM and PXRD data, in turn, suggest that
the NCs predominately adopt a cubic crystal structure, although it should be noted that
there is considerable debate36–38 in the literature as to the actual lattice adopted by these
NCs.
Size-dependent Stokes Shift
Figure 2a now shows ensemble absorption and emission spectra from seven representative
samples (corresponding edge lengths from ∼4 to 13 nm). NCs are dispersed in toluene and
emission is induced using an excitation energy far above the band edge (Eexc = 3.543 eV;
λexc = 350 nm). Excitation intensities are low (Iexc ∼1.4 µW/cm2) and are within the linear
(single exciton) regime. Additional details regarding this can be found in Figure S4. The
data reveal that, as l decreases, both band edge absorption and emission energies blueshift.
This has been explained as stemming from quantum confinement effects given a CsPbBr3
bulk exciton Bohr radius of a0 = 3.5 nm.7 Apart from the structured absorption, the high
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Figure 2: (a) Ensemble absorption (solid blue lines) and emission (dashed red lines) spectra
from a small size series of CsPbBr3 NCs dispersed in toluene. Above gap excitation (Eexc
=
3.543 eV, λexc = 350 nm) used to acquire PL spectra. All absorption/emission spectral pairs
offset for clarity. (b) Corresponding size-dependent Stokes shifts and those extracted from
existing literature.7
quality of the as-produced NCs is evident from measured PL quantum yields, which range
from 30 to 60% along with narrow emission line widths between 90 and 170 meV (Tables
S1 and S2 respectively).
The most notable observation in Figure 2a is the apparent Stokes shift between the band
edge absorption and emission in each case. This Stokes shift decreases with increasing edge
length as quantified by fitting acquired ensemble absorption/emission data. Representative
absorption fits and extracted data are shown in Figure S5 and Table S2 of the SI. What
emerges is a size-dependent Stokes shift (∆Es) that ranges from 115 to 17 meV as the
effective edge length decreases from ∼3 to 13 nm. Figure 2b summarizes these shifts and
contains additional data points extracted from the literature.7 Note that the bulk Stokes
shift is roughly 20 meV39 as is indicated by the dashed horizontal line in Figure 2b.
Room-temperature PL excitation (PLE) measurements confirm the size dependence of
∆Es. Namely, Figures 3a,b show representative spectra for l = 8.5 nm (Figure 3a)
and l = 4.2 nm (Figure 3b) ensembles. In each case, PLE spectra have been acquired by
monitoring the red edge of each sample’s band edge emission. Monitored energies are denoted
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Figure 3: PLE spectra acquired on (a) l = 8.5 nm and (b) l = 4.2 nm NC ensembles at
three energies on the red edge of their above gap excitation-derived emission spectra. (c) PL
spectra of a l = 4.1 nm sample recorded using band edge (λexc = 460 nm, dashed green line)
and above gap (λexc = 350 nm, dashed-dotted red line) excitation energies. Corresponding
absorption is given in solid blue. Extracted Stokes shifts (∆Es) provided at the top right. (d)
∆Es vs. edge length of l = 3.8, 4.1, 4.5, 5.5, 5.8, 6.2, 7.9, 9.2, 11.7, and 12.8 nm ensembles
using both band edge (closed blue circles) and above gap (Eexc = 3.543 eV, open red circles)
excitation. The dashed line is a fit to band edge excitation-acquired ∆Es values. (e) Results
of the convolution analysis (dashed-dotted line) for the l = 4.1 nm ensemble. Associated
absorption and emission spectra are shown as solid blue and dashed red lines, respectively.
Inset: Modeled single particle spectrum with a Huang-Rhys parameter of S = 0.11.
with colored symbols. Resulting PLE spectra match corresponding linear absorption spectra.
The correlation is especially evident in the 4.2 nm ensemble where the room temperature
band edge excitonic feature is clearly resolved. Corresponding average PLE-derived Stokes
shifts are ∼77 and 34 meV for the l = 4.2 and 8.5 nm ensembles respectively. Of note is that
these PLE-derived shifts are slightly smaller than corresponding values seen in Figure 2b.
This likely stems from the residual size distribution of each ensemble given that the PLE
measurement selectively monitors the largest NCs present.
To further verify the existence of a Stokes shift, we eliminate other possible extrinsic ori-
gins. First, we examine the time evolution of the emission maximum through time-correlated
single photon counting time-resolved emission spectroscopy (TCSPC-TRES). Measurements
were performed on l = 4.1 and 7.9 nm NC ensembles in toluene. These TCSPC-TRES results,
shown in Figure S6, display negligible (≤ 5 meV) redshifting of the emission maximum over
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the course of 4 ns following excitation. This suggests that solvatochromism or other effects
with slow degrees of freedom do not contribute to the apparent absorption/emission shift.
To establish that the Stokes shift is independent of dielectric media, we have addition-
ally recorded absorption/emission spectra in hexane and chloroform. Resulting Stokes shifts
quantitatively agree with values in Figure 2b and deviate by no more than 10 meV across
solvents. This is illustrated in Figure S7a, which plots Stokes shift data from CsPbBr3
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(ꢀ = 4.96)7 NC ensembles dispersed in hexane, toluene, and chloroform (associated static
∞
dielectric constants: ꢀhex = 1.88, ꢀtol = 2.38, and ꢀchlo = 4.81).40 The only observed solvent-
dependent effect is a blueshift of absorption/emission spectra in toluene and chloroform rela-
tive to that in hexane. This is summarized in Figure S7b, which shows absorption/emission
spectra of a l = 4.2 nm ensemble in the same three solvents. The effect likely stems from
dielectric contrast wherein increasing (decreasing) the permittivity difference between NCs
and their surrounding medium increases (decreases) exciton binding energies and results in
corresponding spectral redshifts (blueshifts).41
Next, we quantify contributions of the residual size distribution to ∆Es, something first
suggested by earlier PLE measurements (Figures 3a,b). This is done by exciting samples
at their respective band edge and comparing observed Stokes shifts to those obtained when
exciting further to the blue. Motivating this is the fact that larger NCs in an ensemble absorb
more strongly at large energies because of their sizable density of states.42 Consequently, the
emission of an above gap excited NC ensemble possesses an enhanced contribution from
larger NCs in the distribution.43
Figure 3c shows results of this measurement for a l = 4.1 nm ensemble excited at its
band edge (Eexc = 2.696 eV, λexc = 460 nm) and further to the blue at Eexc = 3.543 eV
(λexc = 350 nm). Evident is a decrease of the measured Stokes shift from ∆Es = 93 meV to
76 meV when exciting at the band edge. This clearly shows that the residual ensemble size
distribution influences observed Stokes shifts. Of added note, the extracted ∆Es = 76 meV
shift is in excellent agreement with the earlier PLE-derived ∆Es = 77 meV Stokes shift for
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a l = 4.2 nm ensemble.
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As a consequence, Figure 3d plots band edge excitation-derived ∆Es values against l
for 10 NC ensembles. Observed Stokes shifts (filled blue circles) range from 82 to 20 meV
for l-values between 3.8 and 12.8 nm. Associated shifts, obtained when exciting further to
the blue at Eexc = 3.543 eV are also provided using open red circles. In all cases, differences
between band edge and above gap excitation Stokes shifts are apparent. These differences
decrease with increasing l from ∼15 meV at l ≤ 7.0 nm to ≤ 5 meV at l ≥ 7.0 nm. As
added note, exciting samples progressively to the red, beyond their respective band edges,
causes monotonic decreases in observed ∆Es, as expected from Figure 3d. Representative
absorption/emission spectra from samples excited at the band edge are provided in Figure
S8. The data in Figure 3d thus indicates that actual size-dependent ∆Es values in CsPbBr3
NCs range from 82 to 20 meV for l = 3.8–12.8 nm. A corresponding emission-based sizing
curve is provided in the SI.
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As added verification of size distribution effects on ∆Es, we model the Stokes shift by
convoluting a theoretical single particle emission spectrum with a Gaussian function repre-
sentative of the residual size distribution.31 The Gaussian takes a ∼90–140 meV linewidth
and is established through fits to the band edge linear absorption. The single particle spec-
trum consists of a thermally broadened phonon progression and uses a literature longitudinal
optical (LO) phonon energy of ωLO = 16 meV44–46 as well as a Huang-Rhys parameter (S)
that takes values between S = 0.1145 and S = 0.45.46 The inset in Figure 3e illustrates the
employed single particle spectrum for the case where S = 0.11.
Figure 3e plots the resulting theoretical emission spectrum (dashed-dotted line, S =
0.11, 140 meV linewidth) for a l = 4.1 nm sample relative to corresponding experimental
absorption and emission spectra. An apparent Stokes shift of 7 meV is evident. For S = 0.45,
∆Es increases to 9 meV. These shifts, in turn, agree with ∆Es differences seen in Figures
3c,d for samples excited at the band edge and above gap. More importantly, the modeled
Stokes shift differs significantly from the experimentally-determined ∆Es value of 76 meV
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(Eexc = 2.696 eV). Analogous results have been obtained for l = 5.3 nm and l = 11.7 nm
ensembles (Figure S9). In all cases, modeled Stokes shifts are less than 10 meV enabling us
to conclude that while residual NC size distributions influence observed Stokes shifts they
do not account for either their origin or size dependence.
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Theoretical Models
Having evidenced that the size-dependent Stokes shift in CsPbBr3 NCs is not explained
by extrinsic factors, we now focus on rationalizing an intrinsic origin by employing first-
principles calculations. This requires a description of the electronic structure of CsPbBr3.
Figures 4a,b show the calculated band structure and projected DOS of bulk, cubic CsPbBr3.
The band gap occurs at the R point [k = (1/2, 1/2, 1/2)] and has an estimated gap of
Eg = 2.16 eV. This agrees well with prior work suggesting Eg = 2.39 eV.47,48 The conduction
band (CB) edge is largely Pb p-character whereas the valence band (VB) edge predominately
arises from the overlap of Br p-orbitals. Dangling Br (Pb) bonds lead to defect states below
(above) the VB maximum (CB minimum). Bulk CsPbBr3 therefore possesses a surface
defect-tolerant electronic structure similar to that seen in other lead-based semiconductors
(e.g. PbSe).49 Furthermore, a bulk Stokes shift of 20 meV39 largely results from lattice-
induced carrier stabilization and remains distinct from the size-dependent shifts observed in
NCs. Details of these bulk CsPbBr3 calculations can be found in the SI and in Table S3.
Given that the size-dependent electronic structure of CsPbBr3 NCs remains relatively
unexplored,19 our modeling efforts are premised on the idea that many geometrical models
must be examined to consider the results robust. NC models with edge lengths of l = 2.05,
2.64, 3.23, 3.82, and 4.40 nm were therefore sampled across different morphologies and de-
fects as summarized in Table S4. These calculations employ hybrid exchange functionals
to suppress well known over-delocalization artifacts of generalized gradient approximations
(GGA). Furthermore, they include spin-orbit50,51 and electron-hole exchange interactions
to account for the existence of excitonic fine structure.24,52–54 In all cases, an adiabatic as-
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(c)
(a)
(b)
VBES
CHS
CBES
|2i
3
ΔES
ESOC
CBES
5
0
T
|1i
l = 2.64 nm
l = 3.82 nm
S
)
2
1
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e
(
1
Bulk
y
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CHS
e
n
CHS
E
3
ΔES
VBES
|0i
-5
With
Without
spin-orbit interaction spin-orbit interaction
Bulk
R
ꢀ
X
M
R
-1
0
1
2
Energy (eV)
Energy (eV)
Figure 4: (a) Electronic band structure of bulk, cubic CsPbBr3 calculated with the spin-
orbit coupling interaction. (b) DOS for two NC sizes and the bulk material. The CHS shifts
towards the VB edge as the NC size increases, eventually becoming indistinguishable from
the bulk VB edge. (c) Model Jablonski diagrams show the fine structure resulting from SOC.
sumption —namely, that the electron-hole polaronic lattice relaxation is size-independent—is
invoked.14,55
Figure S10 shows resulting molecular orbitals (MOs) for the two lowest occupied VB
states of l = 2.64, 3.23, 3.82, and 4.40 nm NC models. In all cases, NC VB frontier states pos-
sess nodeless, cuboidal spatial distributions delocalized over the entire particle. This state,
hereafter referred to as the confined hole state (CHS), exists in both cubic and orthorhombic
models. Existence of the CHS is thus insensitive to NC crystal phase (Figure S11). Similar
electronic structure has been observed in prior theoretical work conducted on CsPbBr3 NCs,
further supporting its existence.19 The most important result of these calculations is the fact
that the CHS is delocalized over the entire NC. Consequently, it is spatially confined by the
NC size and hence possesses size-dependent energies. This provides a basis for rationalizing
observed size-dependent Stokes shifts, as discussed below.
These calculations simultaneously reveal that the next occupied NC level (260–70 meV
above the CHS) closely resembles the bulk CsPbBr3 VB maximum at the R point, hereafter
referred to as the VB edge state (VBES). In contrast with the nodeless and diffuse nature
of the CHS, the singly-noded VBES localizes near NC corners with a distinctive nodal plane
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Figure 5: (a) Size-dependent NC CBES and VBES energies plotted relative to CHS energies,
which are set to zero. State energies calculated at the Γ-point. Dashed lines are correspond-
ing interpolation fits. (b) Experimental band edge excitation-derived ∆Es values (closed
blue circles) plotted against theoretical Stokes shifts (open red triangles).
cutting diagonally through the NC core. This is evidenced by an apparent phase change in
calculated MOs (Figure S10). The calculations also show that the CB edge state (CBES)
mirrors the bulk CB minimum at the R point. Figure 4b summarizes size-dependent CHS
energies relative to complementary VBES and CBES levels for l = 2.64 nm and l = 3.82
nm NCs. Calculated bulk levels are shown for comparison purposes. A projected DOS for a
l = 2.64 nm NC is provided in Figure S12.
CsPbBr3 NC fine structure has simultaneously been investigated to assess its impact
on observed Stokes shifts. This has been done by building a Breit-Pauli spin-orbit cou-
pling (SOC) Hamiltonian50,51,56 with electron-hole exchange interactions between the four
spin-adapted excitations linking the CHS and CBES. The four resulting fine-structure states
are close in character to the three triplet and singlet states they are built from and are
separated by 10 meV. This is consistent previously described fine structure for MAPbBr3
and MAPbI3.57 Figure 4c (left) summarizes the energetic ordering of resulting fine struc-
ture states and depicts relevant absorption and emission transitions within a fine structure
descriptor of CsPbBr3 NCs. The calculations reveal that the magnitude of any spin-orbit
induced fine structure splitting (ESOC) ranges from 10.3 meV for a l = 2.64 nm NC to
0.13 meV for a l = 3.82 nm NC. Triplet/singlet fine structure splitting is therefore neg-
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ligible for experimentally-relevant sizes (i.e. ∼4–13 nm). We conclude that fine structure
spin-orbit coupling/electron-hole exchange cannot explain observed size-dependent ∆Es val-
ues. In what follows, we therefore do not invoke fine structure when rationalizing observed
Stokes shifts. Note that this conclusion differs from the case of CdSe QDs where band edge
fine structure was found to be responsible for intrinsic size-dependent “resonant” and “non-
resonant” Stokes shifts.31 Details of the CsPbBr3 fine structure modeling can be found in
the SI.
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At this point, NC band edge absorption in the model primarily occurs as a VBES→CBES
transition. This is followed by non-radiative relaxation of the photoexcited hole to the CHS
whereupon emission occurs as a CBES→CHS transition. A Stokes shift is therefore predicted
with a magnitude that reflects the size-dependent energy difference between the CHS and
VBES. Absorption/emission processes are summarized in Figure 4c (right). In support
of this origin for the Stokes shift, the ratio of calculated VBES→CBES and CHS→CBES
absorption strengths are substantial and differ 10- to 50-fold. This originates from differences
in their underlying DOS as suggested by Figure 4b. As a result, the CHS→CBES transition
is relatively dark in absorption whereas the VBES→CBES transition is bright. The generally
small energetic separation of the CHS from the VBES along with significant VBES→CBES
line broadening, stemming from residual size distributions, is likely the reason why this
transition is not seen in absorption. Further discussion and details of the absorption strength
estimates can be found in the SI.
Figure 5a now illustrates calculated CHS/VBES energy differences for NCs with edge
lengths between l = 2.64 nm and l = 3.82 nm. Apparent is an increase from 260 to 70 meV
for models with l ∼2 to 5 nm. Theoretical ∆Es values are therefore size-dependent and of
similar magnitude to experimental Stokes shifts. Figure 5b shows an explicit comparison
of experimental and theoretical Stokes shifts and reveals that they are in quantitative agree-
ment. The origin of a size-dependent Stokes shift in CsPbBr3 NCs is therefore rationalized.
Remarkably, existence of the Stokes shift is robust with respect to crystal phase, NC
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size, and defect type throughout the calculations (see Figures S10, S11 and Table S4).
Size-dependent CHS→VBES energy differences exist in both cubic and orthorhombic models
and across different degrees of surface vacancies. An example is provided in the SI. Both
experimental and computational results therefore indicate that absorbing and emitting states
are distinct in CsPbBr3 NCs. Of note is that the CHS is exclusive to NCs and does not
appear in the bulk. Furthermore, the calculations reveal that CsPbBr3 NC CHS, VBES,
and CBES levels possess minimal cesium orbital character. Consequently, conclusions from
the current CsPbBr3 NC model are likely applicable to other hybrid lead-halide perovskites.
Note that lattice relaxation in NCs does not explain observed size-dependent Stokes shifts
since calculated (upper limit) exciton binding energies are generally small and range from
Ebind = 30 to 60 meV. This corroborates a picture wherein diffuse electron-hole pairs weakly
interact with the nuclear lattice. Details of these calculations and a further discussion of
CsPbBr3 NC exciton binding energies can be found in the SI.
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Conclusion
In summary, we establish for the first time the origin of a size-dependent Stokes shift in
CsPbBr3 NCs. These shifts range from 82 to 20 meV for particles with effective edge lengths
ranging from ∼4 to 13 nm. We provide experimental proof to show that the shift is intrinsic
to the NC electronic structure. Using first-principles modeling, we identify an inherent,
size-dependent CHS level above the VBES, with the CHS dark in absorption and bright in
emission. Subsequently computed Stokes shifts from various geometric models quantitatively
agree with experimentally-measured values. The study therefore indicates that absorbing
and emitting states are distinct in CsPbBr3 NCs and simultaneously rationalizes observed
Stokes shifts. At a broader level, since cesium orbitals do not significantly contribute to the
CHS, this state is likely a general feature of other perovskite NCs and can be tuned via NC
size to influence their response within photovoltaic or light-emitting applications.
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Experimental
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Synthesis and Purification of CsPbBr3 NCs
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Ensemble NC samples were synthesized with average edge lengths ranging from l = 3.0–12.8
nm using a hot injection method previously reported by Protesescu et al.7 Briefly, 0.18 mmol
of PbBr2 (99.999% trace metals basis, Sigma-Aldrich) and 5 mL of octadecene (ODE, 90%
technical grade, Acros) were placed in a 10-mL three-neck flask and degassed at 120◦C for
1 hour. The reaction vessel was back-filled with N2. Then 1 mL of oleylamine (80-90%
C-18 content, Acros) and 1 mL of oleic acid (90% technical grade, Sigma-Aldrich) were
injected into it. The temperature was raised to 130–200◦C in order to tune the size with
higher (lower) temperatures favoring larger (smaller) NCs. Once the temperature stabilized,
1 mL of a Cs-oleate precursor solution (125 mM in ODE, prepared as described below) was
injected. The reaction was then immediately quenched with an ice bath. The precipitate was
separated by centrifuging the suspension at 7500 rpm for 10 minutes and was subsequently
washed with a 4:1 mixture of toluene:acetone. Note that the addition too much acetone
strips the surface ligands, causing them to aggregate. NCs were then dispersed in toluene
and isolated using further centrifugation at 7500 rpm for 10 minutes. Recovered NCs were
stored in toluene for further analysis. For smaller NC ensembles (in the range of l ∼3–6 nm),
reactions were scaled up 10-fold and the Cs-oleate precursor was injected at temperatures
between 130◦ and 140◦C.
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Cs-oleate was synthesized according to a previous report.7 Briefly, 2.5 mmol of Cs2CO3
(99.9%, Sigma-Aldrich) was placed in a 100 mL three neck flask with 40 mL of ODE and
2.5 mL of oleic acid. The reaction mixture was degassed at 120◦C for 1 hour and was then
back-filled with N2 and heated at 150◦C for 6–8 hours. Cs-oleate exists as a precipitate at
room temperature and must be heated to 100◦C in order to dissolve it prior to injection.
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Optical Measurements
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Absorption Spectroscopy
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Ensemble absorption spectra were obtained on a Cary 50 Bio UV-Visible spectrophotometer.
Spectra were recorded under ambient condition in a fused silica cuvette using dilute solutions
of CsPbBr3 NCs in toluene. Optical densities were less than 0.3 in all cases.
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Photoluminescence and Photoluminescence Excitation Spectroscopy
PL and PLE spectra were recorded under identical conditions as absorption spectra using a
Jobin Yvon Fluorolog-3. In both PL and PLE measurements, excitation and emission slit
widths were set to 2 nm and 1 nm, respectively. The integration time was 0.25 s with 1
nm increments. For PL measurements, the excitation wavelength was λexc = 350 nm. For
band edge excitation measurements, λexc was set to the peak energy of each sample’s band
edge absorption. In PLE measurements, emission positions were taken on the red edge of
each sample’s band edge emission. PLE spectra were corrected for wavelength-dependent
variations of the lamp intensity.
Photoluminescence Quantum Yield Measurements
PL QY measurements were performed according to a standardized procedure58 with appro-
priate organic dyes [coumarin 6 (QY = 78%)59 and coumarin 102 (QY = 74%)60 for the green
and blue regions of the spectrum respectively] dissolved in ethanol (100%, Sigma-Aldrich).
The NC solvent was toluene. Optical densities were below 0.2 for both dye and NC samples.
Time-Correlated Single Photon Counting Time-Resolved Emission Spectroscopy
TCSPC-TRES measurements were obtained with a Horiba Jobin Yvon TCSPC Spectrometer
with a NanoLED 371 nm excitation source. Time-resolved PL decays were measured over
120 nm in 2 nm increments. TCSPC-TRES spectra were constructed by taking slices of PL
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decay series to obtain the spectrum at a given time after photoexcitation. Note that the
instrument response time is approximately 300 ps. Optical densities were below 0.2 in all
cases.
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Solvatochromism Measurements
Ensemble samples for solvatochromism experiments were dried via vacuum and were re-
dispersed in hexanes and chloroform. Absorption and emission spectra were measured as
described above.
Excitation Intensity-Dependent Emission Measurements
NC ensembles for excitation intensity (Iexc)-dependent emission measurements were were
spin coated onto glass substrates (VWR micro cover glass, 25 mm). Emission intensities
were collected as a function of Iexc using a 405 nm continuous wave (CW) laser [Coherent,
Obis] focused onto samples using a high numerical aperture objective (0.65 NA, Nikon).
Excitation intensities ranged from 0.01-1.5 W/cm2. The resulting emission was collected
with the same objective using a barrier filter (425LP, Chroma) to reject the laser. The
emission was subsequently detected with a single photon counting avalanche photodiode
(PerkinElmer, SPCM AQR-14).
Powder X-Ray Diffraction Measurements
Powder XRD measurements were acquired with a Bruker D8 Discover diffractometer with a
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Cu Kα source (λ = 1.5418 A) in the Bragg-Brentano geometry. Measurements were recorded
from 10–60◦ 2θ with a step size of 0.01◦ and an integration time of 4 s per step.
Transmission Electron Microscopy Measurements
TEM images were acquired using both a FEI Titan 80-300 kV microscope with an acceler-
ating voltage of 300 kV and a JEOL 2011 microscope with an accelerating voltage of 200
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kV. Samples were prepared by drop casting dilute NC solutions in toluene onto ultrathin
amorphous carbon substrates with copper supports (Ladd).
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Theoretical Modeling
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NC models were created starting with a n × n × n supercell, where n is an integer repre-
senting the number of cubic unit cells. Due to the cubic nature of experimentally-observed
particles, as evidenced in this work and others,7 theoretical models are restricted to reflect
this morphology. For the cubic lattice, the (100), (010), and (001) directions are identical.
Each has two unique layers; one with only Cs and Br atoms and another with only Pb and
Br atoms. These are referred to as Cs-Br and Pb-Br layers, respectively. Orthorhombic
models were cut to supercells identical in stoichiometry to the cubic models, with the only
difference being the tilting of lead-bromide octahedra.
Only models terminated on all faces by the same layer, either a Cs-Br or a Pb-Br layer
were considered. Pure supercells have three faces terminated by a Cs-Br layer and three faces
terminated by a Pb-Br layer. To rectify this with our restriction that surface-terminating
layers be identical on all faces, each supercell model was extended by adding a Cs-Br (Pb-Br)
layer over the Pb-Br (Cs-Br) surface-terminating faces of the supercell.
To correctly simulate these NCs, geometrical models-of-interest must possess the fol-
lowing features: (1) charge neutrality, with near zero dipole moment and a stoichiometry
which closely follows the bulk molecular formula, (2) a crystalline structure observed in
experiments, and (3) surface defects known to this material. The first requirement puts
limitations on the surface-terminating layers for different-sized models unless defects deeper
than the surface layer are considered. For n × n × n supercell models, an even integer n
requires that the surface be terminated with a Pb-Br layer. An odd integer n requires sur-
faces to be terminated with a Cs-Br layer. Minimal surface vacancy defects were made to
charge neutralize the models while maintaining a near zero dipole. Additional surface va-
cancy defects were considered to examine their effect on electronic structure. CsPbBr3 NCs
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have high surface defect tolerance, thus defects that break this high tolerance have large
formation energies.18 To validate that this defect tolerance fits with the Stokes shift model
posited in the study, a range of surface vacancy defects was explored. Models therefore
considered varied placements as well as amounts of Cs, Pb, and Br surface vacancies.
Given the debate as to whether CsPbBr3 NCs adopt cubic or orthorhombic phases, models
of both phases were prepared. We explore models with two types of optimization: (1) only
allowing surface atoms to relax, leaving the core atoms frozen, and (2) allowing the whole
structure to undergo full geometry optimization. Doing so enabled us to rationalize whether
there could be characteristics of either morphology or unique NC aspects which explained
the Stokes shift.
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NC calculations were done in CP2K 4.161 with the Heyd-Scuseria-Ernzerhof (HSE) func-
tional,62 a double-ζ quality basis set63 and core potentials including relativistic effects.64
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Each model was calculated in a periodic cell with at least 15 A of vacuum in each direction.
Periodic calculations for the bulk cubic and orthorhombic structures were performed in the
Quantum Espresso package65 with the Perdew-Burke-Ernzerhof hybrid (PBE0) functional
and a 500 eV plane-wave cutoff; scalar relativistic and spin-orbit interactions were included.
Acknowledgments
This work was supported by the Division of Materials Sciences and Engineering, Office
of Basic Energy Sciences, U.S. Department of Energy under award DE-SC0014334. J.Z.
acknowledges support from the Arthur J. Schmitt Fellowship Foundation. T. S. N.-B. ac-
knowledges the support of the NSF Graduate Research Fellowship under grant DGE-131358
and the Patrick and Jana Eilers Graduate Student Fellowship for Energy Related Research.
J. P. thanks the University of Notre Dame’s College of Science and Department of Chemistry
and Biochemistry, ND Energy and the Honeywell Corporation for generous start-up funding.
S. R. thanks the University of Notre Dame Integrated Imaging Facility for financial support
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of the work conducted on this project.
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Author Information Notes
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M.C.B and J.E.H. contributed equally to this work.
Associated Content
Supporting Information Available. NC sizing histogram, absorption- and emission-derived
NC sizing curves, NC PXRD data, excitation intensity-dependent emission data and dis-
cussion, table of PL quantum yields, table of relevant experimental absorption and emission
data, linear absorption fits, TCSPC-TRES data, solvatochromism data, band-edge excitation
emission and absorption spectra, convolution analysis of other ensemble NC sizes, discus-
sion of bulk electronic structure calculations, discussion of geometries and band structure
calculations, table of calculated bulk band gaps/lattice parameters, MOs for VBES/CHS of
different NC sizes, MOs for CHS of NC models with various crystal phases, projected DOS
for l = 2.64 nm NC model, discussion of spin-orbit coupling calculations, absorption strength
calculations discussion, table of NC model data, discussion of exciton binding energies. This
material is available free of charge via the Internet at http://pubs.acs.org.
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