Quantitative size-dependent structure and strain determination of CdSe
  nanoparticles using atomic pair distribution function analysis
APS/123-QED
Quantitative size-dependent structure and strain determination of CdSe nanoparticles
using atomic pair distribution function analysis
A. S. Masadeh, E. S. Božin, C. L. Farrow, G. Paglia, P. Juhas and S. J. L. Billinge∗
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-1116, USA
A. Karkamkar and M. G. Kanatzidis
Department of Chemistry, Michigan State University, East Lansing, Michigan 48824-1116, USA
The size-dependent structure of CdSe nanoparticles, with diameters ranging from 2 to 4 nm, has
been studied using the atomic pair distribution function (PDF) method. The core structure of
the measured CdSe nanoparticles can be described in terms of the wurtzite atomic structure with
extensive stacking faults. The density of faults in the nanoparticles ∼ 50% . The diameter of the
core region was extracted directly from the PDF data and is in good agreement with the diameter
obtained from standard characterization methods suggesting that there is little surface amorphous
region. A compressive strain was measured in the Cd-Se bond length that increases with decreasing
particle size being 0.5% with respect to bulk CdSe for the 2 nm diameter particles. This study
demonstrates the size-dependent quantitative structural information that can be obtained even
from very small nanoparticles using the PDF approach.
PACS numbers: 61.46.Df, 61.10.-i, 78.66.Hf, 61.46.-w
I. INTRODUCTION
Semiconductor nanoparticles are of increasing interest
for both applied and fundamental research. Wurtzite-
structured cadmium selenide is an important II-VI semi-
conducting compound for optoelectronics.1 CdSe quan-
tum dots are the most extensively studied quantum
nanostructure because of their size-tunable properties,
and they have been used as a model system for inves-
tigating a wide range of nanoscale electronic, optical,
optoelectronic, and chemical processes.2 CdSe also pro-
vided the first example of self-assembled semiconductor
nanocrystal superlattices.3 With a direct band gap of
1.8 eV, CdSe quantum dots have been used for laser
diodes 4, nanosensing 5 , and biomedical imaging.6 In
fundamental research, particles with a diameter in the 1-
5 nm range are of particular importance since they cover
the transition regime between the bulk and molecular
domains where quantum size effects play an important
role. Significant deviation from bulk properties are ex-
pected for particles with diameter below 5 nm, and were
observed in many cases 6,7 as well as in this study.
Accurate determination of atomic scale structure, ho-
mogeneous and inhomogeneous strain, structural defects
and geometrical particle parameters such as diameter and
shape, are important for understanding the fundamen-
tal mechanisms and processes in nanostructured materi-
als. However, difficulties are experienced when standard
methods are applied to small nanoparticles. In this do-
main the presumption of a periodic solid, which is the ba-
sis of a crystallographic analysis, breaks down. Quantita-
tive determinations of the nanoparticle structure require
methods that go beyond crystallography. This was noted
early on in a seminal study by Bawendi et al.8 where they
used the Debye equation, which is not based on a crys-
tallographic assumption, to simulate semi-quantitatively
the scattering from some CdSe nanoparticles. How-
ever, despite the importance of knowing the nanoparticle
structure quantitatively with high accuracy, this work
has not been followed up with application of modern lo-
cal structural methods9,10 until recently.11,12,13,14,15,16 In
this study we return to the archetypal CdSe nanoparti-
cles to investigate the extent of information about size-
dependent structure of nanoparticles from the atomic
pair distribution function (PDF) method. This is a lo-
cal structural technique that yields quantitative struc-
tural information on the nanoscale from x-ray and neu-
tron powder diffraction data.10 Recent developments in
both data collection17,18 and modeling19,20 make this a
potentially powerful tool in the study of nanoparticles.
Additional extensions to the modelling are necessary for
nanoparticles, and some of these have been successfully
demonstrated.11,12,21
In this paper, we present a detailed analysis of the
structural information available from PDF data on (2-
4 nm) CdSe nanoparticles. The PDF method is demon-
strated here as a key tool that can yield precise structural
information about the nanoparticles such as the atomic
structure size of the core, the degree of crystallinity, lo-
cal bonding, the degree of the internal disorder and the
atomic structure of the core region, as a function of the
nanoparticle diameter. Three CdSe nanoparticle sam-
ples with different diameters that exhibit different opti-
cal spectra have been studied. The purpose of this paper
is not only to explain the PDF data of CdSe nanoparti-
cles through a modeling process, but also to systemati-
cally investigate the sensitivity of the PDF data to subtle
structural modifications in nanoparticles relative to bulk
material.
The measurement of the nanoparticle size can lead to
significantly different results when performed by differ-
ent methods, and there is no consensus as to which is the
most reliable.22,23 It is also not clear that a single diam-
http://arxiv.org/abs/0704.1288v2
eter is sufficient to fully specify even a spherical particle
since the presence of distinct crystalline core and disor-
dered surface regions have been postulated.8
Powder diffraction is a well established method for
structural and analytical studies of crystalline materi-
als, but the applicability to such small particles of stan-
dard powder diffraction based on crystallographic meth-
ods is questionable and likely to be semi-quantitative at
best. Palosz et al.24 have shown that the conventional
tools developed for elaboration of powder diffraction data
are not directly applicable to nanocrystals.24 There have
been some reports8,23,25 in the past few years extract-
ing nanoparticle diameter from x-ray diffraction (XRD)
using the Scherrer formula, which is a phenomenologi-
cal approach that considers the finite size broadening of
Bragg-peaks.26 This approach will decrease in accuracy
with decreasing particle size, and for particle sizes in the
range of a few nanometers the notion of a Bragg peak
becomes moot.24 At this point the Debye formula27 be-
comes the more appropriate way to calculate the scatter-
ing.8 The inconsistency between the nanoparticle diame-
ter determined from the standard characterization meth-
ods and the diameter obtained by applying the Scherrer
formula have been observed by several authors.8,23,28
Previous studies of CdSe nanoparticle structure have
demonstrated the sensitivity of the XRD pattern to the
presence of planar disorder and thermal effects due to
nano-size effects.8,29 The diffraction patterns of CdSe
nanoparticles smaller than 2.0 nm have been observed
to appear markedly different from those of the larger
diameters (see Ref. 29 Fig. 11), the large attenuation
and broadening in the Bragg reflections in these small
nanoparticles, making the distinction between wurtzite
and zinc-blende hard using conventional XRD methods.
Murray et al.29 reported that the combination of X-ray
studies and TEM imaging yields a description of the av-
erage CdSe nanoparticle structure. Strict classification
of the CdSe nanoparticles structure as purely wurtzite or
zinc-blend is potentially misleading.29 Bawendi et al.8 re-
ported that CdSe nanoparticles are best fit by a mixture
of crystalline structures intermediate between zinc-blend
and wurtzite. Here we apply the PDF method to CdSe
nanoparticles and refine quantitative structural parame-
ters to a series of CdSe nanoparticles of different sizes.
Strain in nano systems has been observed before in dif-
ferent studies, as well as in this study. Using combined
PDF and extended X-ray-absorbtion fine structure (EX-
AFS) methods, Gilbertet al.11 observed a compressive
strain compared to the bulk in ZnS nanocrystals. Us-
ing an electric field-induced resonance method, Chen et
30 detected the enhancement of Young’s modulus of
ZnO nanowires along the axial direction when the diam-
eters are decreased. Very recently, Quyang et al.31 de-
veloped an analytical model for the size-induced strain
and stiffness of a nanocrystal from the perspective of
thermodynamics and a continuum medium approach. It
was found theoretically that the elastic modulus increases
with the inverse of crystal size and vibration frequency
is higher than that of the bulk.31 Experimentally, the
CdQ (Q=S, Se, T e) first-neighbor distances have been
studied using both XRD and EXAFS methods.32 The
distances were found smaller than those in the bulk com-
pounds by less than 1.0%. Herron et al.33 studied CdS
nanocrystals and showed a bond contraction of ∼ 0.5%
compared to the bulk. Carter et al.34 studied a series
of CdSe nanoparticles using the EXAFS method. In
the first shell around both the Se and Cd atoms, they
found essentially no change in the first-neighbor distance.
Chaure et al.35 studied the strain in nanocrystalline CdSe
thin films, using Raman scattering and observed a peak
shift with decrease in particle size, which was attributed
to the increase in stress with decreasing particle size.35
Local structural deviations or disorder mainly affect
the diffuse scattering background. The XRD experiments
probe for the presence of periodic structure which are
reflected in the Bragg peaks. In order to have information
about both long-range order and local structure disorder,
a technique that takes both Bragg and diffuse scattering
need to be used, such as the PDF technique. Here we
apply the PDF method to study the structure, size and
strain in CdSe nanoparticles as a function of nanoparticle
diameter. The core structure of the CdSe nanoparticles
can be described by a mixture of crystalline structures
intermediate between zinc-blend and wurtzite, which is
wurtzite containing a stacking fault density (SFD) of up
to ∼ 50%, with no clear evidence of a disordered surface
region, certainly down to 3 nm diameter. The structural
parameters are reported quantitatively. We measure a
size-dependent strain on the Cd-Se bond which reaches
0.5% at the smallest particle size. The size of the well-
ordered core extracted directly from the data agrees with
the size determined from other methods.
II. EXPERIMENTAL DETAILS
A. Sample preparation
CdSe nanoparticles were synthesized from cadmium
acetate, selenium, trioctyl phosphine and trioctyl phos-
phine oxide. Sixty four grams of trioctylphosphine ox-
ide (TOPO) containing cadmiumacetate was heated to
360◦C under flowing argon. Cold stock solution (38.4 ml)
of (Se:trioctylphosphine = 2:100 by mass) was quickly
injected into the rapidly stirred, hot TOPO solution.
The temperature was lowered to 300◦C by the injection.
At various time intervals, 5-10 ml aliquots of the reac-
tion mixture were removed and precipitated in 10 ml of
methanol. The color of the sample changed from bright
yellow to orange to red to brown with time interval vari-
ation from 20 seconds to 1200 seconds. Three nanopar-
ticle sizes, CdSeI (small), CdSeII (medium) and CdSeIII
(large), were used for this study, as well as a bulk CdSe
sample for reference.
The samples were further purified by dissolving and
centrifuging in methanol to remove excess TOPO. This
FIG. 1: TEM image of CdSe nanocrystal prepared using the
method described in the text. CdSe obtained by 1200 seconds
(left) and 15 seconds (right) nucleation. The line-bar is 10 nm
in size in both images.
FIG. 2: (a) Room temperature UV-vis absorption and
(b) photoluminescence spectra from the sample of CdSe
nanocrystals. (©) CdSeI, (△) CdSeII, (�) CdSeIII.
process also resulted in a narrower particle size distri-
bution. The transmission electron micrograph (TEM)
images (Fig. 1) show uniformly sized nanoparticles with
no signs of aggregation. The ultraviolet visible (UV-vis)
absorption and photoluminescence (PL) spectra of the
aliquots were recorded by redissolving the nanocrystals
in toluene. The spectra are shown in Fig. 2.
The band-gap values obtained for the measured sam-
ples can be correlated with the diameter of the nanopar-
ticles based on the table provided in supplementary infor-
mation of Peng et al.36 using the data on exciton peaks
TABLE I: CdSe nanoparticle diameter as determined using
various methods.
CdSeIII CdSeII CdSeI
Nucleation time (s) 1200 630 15
Diameter (nm)
TEM 3.5(2) 2.7(2) 2.0(2)
UV-vis 3.5(4) 2.9(3) ≤ 1.90
PL 3.6(4) 2.9(3) ≤ 2.1
PDF 3.7(1) 3.1(1) 2.2(2)
measured with UV-visible light absorption, and photolu-
minescence peaks. The particle sizes were measured by
TEM as well. The measured values of particle diameter
using these various methods are summarized in Table I.
B. The atomic PDF method
The atomic PDF analysis of x-ray and neutron pow-
der diffraction data is a powerful method for studying
the structure of nanostructured materials.9,10,37,38,39,40
Recently, it has been explicitly applied to study the
structure of discrete nanoparticles.11,12,40,41,42 The PDF
method can yield precise structural and size information,
provided that special care is applied to the measurement
and to the method used for analyzing the data. The
atomic PDF, G(r), is defined as
G (r) = 4πr [ρ (r)− ρ0] , (1)
where ρ(r) is the atomic pair-density, ρ0 is the average
atomic number density and r is the radial distance.43
The PDF yields the probability of finding pairs of atoms
separated by a distance r. It is obtained by a sine Fourier
transformation of the reciprocal space total scattering
structure function S(Q), according to
G (r) =
Q[S(Q)− 1] sinQr dQ, (2)
where S(Q) is obtained from a diffraction experiment.
This approach is widely used for studying liquids, amor-
phous and crystalline materials, but has recently also
been successfully applied to nanocrystalline materials.10
C. High-energy x-ray diffraction experiments
X-ray powder diffraction experiments to obtain the
PDF were performed at the 6IDD beamline at the Ad-
vanced Photon Source at Argonne National Labora-
tory. Diffraction data were collected using the recently
developed rapid acquisition pair distribution function
(RAPDF) technique17 that benefits from 2D data col-
lection. Unlike TEM, XRD probes a large number of
crystallites that are randomly oriented. The powder
samples were packed in a flat plate with thickness of
FIG. 3: Two dimensional XRD raw data collected using im-
age plate detector from (a) CdSe bulk and (b) nanoparticle
CdSeIII samples.
1.0 mm sealed between kapton tapes. Data were col-
lected at room temperature with an x-ray energy of
87.005 keV (λ = 0.14248 Å). An image plate camera
(Mar345) with a diameter of 345 mm was mounted or-
thogonally to the beam path with a sample to detec-
tor distance of 208.857 mm, as calibrated by using sil-
icon standard sample.17 The image plate was exposed
for 10 seconds and this was repeated 5 times for a to-
tal data collection time of 50 seconds. The RAPDF ap-
proach avoids detector saturation whilst allowing suffi-
cient statistics to be obtained. This approach also avoids
sample degradation in the beam that was observed for the
TOPO coated nanoparticles during longer exposures, on
the scale of hours, that were required using conventional
point-detector approaches. To reduce the background
scattering, lead shielding was placed before the sample
with a small opening for the incident beam.
Examples of the raw 2D data are shown in Fig. 3.
These data were integrated and converted to intensity
versus 2θ using the software Fit2D,44 where 2θ is the an-
gle between the incident and scattered x-ray beam. The
integrated data were normalized by the average monitor
counts. The data were corrected and normalized9 using
the program PDFgetX245 to obtain the total scattering
structure function, S(Q), and the PDF, G(r), which are
shown in Figs. 4 (a) and (b) respectively. The scattering
signal from the surfactant (TOPO) was measured inde-
pendently and subtracted as a background in the data
reduction.
In the Fourier transform step to get from S(Q) to the
PDF G(r), the data are truncated at a finite maximum
value of the momentum transfer, Q = Qmax. Different
values of Qmax may be chosen. Here a Qmax = 25.0 Å
was found to be optimal. Qmax is optimized such as to
avoid large termination effects and to reasonably mini-
mize the introduced noise level as signal to noise ratio
decreases with Q value.
Structural information was extracted from the PDFs
using a full-profile real-space local-structure refinement
method46 analogous to Rietveld refinement.47 We used
an updated version48 of the program PDFfit19 to fit
the experimental PDFs. Starting from a given structure
model and given a set of parameters to be refined, PDF-
FIG. 4: (a) The experimental reduced structure function
F (Q) of CdSe nanoparticle with different diameters and (b)
the corresponding PDF, G(r), obtained by Fourier transfor-
mation of the data in (a) with Qmax = 25.0 Å
−1, from top to
bottom: bulk, CdSeIII, CdSeII and CdSeI.
fit searches for the best structure that is consistent with
the experimental PDF data. The residual function (Rw)
is used to quantify the agreement of the calculated PDF
from model to experimental data:
ω(ri)[Gobs(ri)−Gcalc(ri)]2
ω(ri)G
. (3)
Here the weight ω(ri) is set to unity which is justified
because in G(r) the statistical uncertainty on each point
is approximately equal.49,50
The structural parameters of the model were unit cell
parameters, anisotropic atomic displacement parameters
(ADPs) and the fractional coordinate z of Se/Cd atom.
Non structural parameters that were refined were a cor-
rection for the finite instrumental resolution, (σQ), low-r
correlated motion peak sharpening factor (δ),51,52 and
scale factor. When estimating the particle size, a new
version of the fitting program with particle size effects in-
cluded as a refinable parameter53 was used. The sample
resolution broadening was determined from a refinement
to the crystalline CdSe and the silicon standard sample
and fixed and the particle diameter refined, as described
below. Good agreement between these results was ob-
tained.
III. RESULTS AND DISCUSSION
The reduced structure functions for the bulk and
nanocrystalline samples are shown plotted over a wide
range of Q in Fig 4(a). All of the patterns show signifi-
cant intensity up to the highest values of Q, highlighting
the value of measured data over such a wide Q-range. All
of the diffraction patterns have peaks in similar positions
reflecting the similarity of the basic structures, but as the
nanoparticles get smaller the diffraction features become
broadened out due to finite size effects.26
The PDFs are shown in Fig. 4(b). What is apparent is
that, in real-space, the PDF features at low-r are compa-
rably sharp in all the samples. The finite size effects do
not broaden features in real-space. The finite particle size
is evident in a fall-off in the intensity of structural fea-
tures with increasing-r. Later we will use this to extract
the average particle size in the material. The structure
apparent in the G(r) function comes from the atomic or-
der within the nanoparticle. The value of r where these
ripples disappear indicates the particle core region diam-
eter; or at least the diameter of any coherent structural
core of the nanoparticle. By direct observation (Fig. 9)
we can put a lower limit on the particle diameters to be
3.6, 2.8 and 1.6 nm for CdSeIII, II and I, respectively,
where the ripples can be seen to die out by visual inspec-
tion. These numbers will be quantified more accurately
later.
A. Nanoparticle structure
Features in the PDF at low-r reflect the internal struc-
ture of the nanoparticles. The nanoparticle PDFs have
almost the same features as in the bulk in the region
below 8.0 Å, reflecting the fact that they share a simi-
lar atomic structure on average. In the finite nano-size
regime, local structural deviations from the average bulk
structure are expected.
A large number of semiconductor alloys, especially
some sulfides and selenides, do not crystallize in the cu-
bic zinc-blende structure but in the hexagonal wurtzite
structure54. Both wurtzite and zinc-blende structures are
based on the stacking of identical two-dimensional pla-
nar units translated with respect to each other, in which
each atom is tetrahedrally coordinated with four nearest
neighbors. The layer stacking is described as ABABAB...
along the [001] axis for wurtzite and asABCABC... along
FIG. 5: Fragments from the (a) wurtzite structure, space
group (P63mc) and (b) zinc-blende structure, space group
(F 4̄3m).
the [111] axis for zinc-blende. As can be seen in the Fig. 5,
each cadmium and selenium is tetrahedrally coordinated
in both structures. However, the next nearest and more
distant coordination sequences are different in the two
structures.
The largest changes in structure are expected in the
smallest nanoparticles. In these small nanoparticles, the
proportion of atoms on the surface is large making the
notion of a well-ordered crystalline core moot. The frac-
tion of atoms involved in the surface atoms was estimated
as 0.6, 0.45 and 0.35 for 2 nm, 3 nm and 4 nm nanoparti-
cle diameters, respectively. This was estimated by taking
different spherical cuts from bulk structure, then count-
ing the atom with coordination number 4 as core atom
and the one with less than 4 as surface atom. For the
smallest particles the small number of atoms in the core
makes it difficult to define a core crystal structure, mak-
ing the distinction between wurtzite and zinc-blende dif-
ficult using the conventional XRD methods as nanopar-
ticle size decreases.29 The principle difference between
these structures is the topology of the CdSe4 connec-
tions, which may also be becoming defective in the small
nanoparticles.
Two structure models wurtzite (space group P63mc)
and zinc-blende (space group F 4̄3m), were fit to the PDF
data. The results of the full-profile fitting to the PDF
data are shown Fig. 6. In this figure we compare fits
to the (a) wurtzite and (b) zinc-blende structure mod-
els using isotropic atomic displacement factors (Uiso) in
both models. The wurtzite structure gives superior fits
for the bulk structure. However, for all the nanoparticle
sizes, the fits of wurtzite and zinc-blende are comparable
as evident from the difference curves in Fig. 6 and the
Rw-values reported in Table II. This indicates that clas-
sification of the CdSe nanoparticles structure as purely
wurtzite or zinc-blend is misleading29 and it is better de-
scribed as being intermediate between the two structures,
as has been reported earlier8.
Introducing anisotropic ADPs (U11 = U22 6= U33) into
the wurtzite model, resulted in better fits to the data.
The refined parameters are reproduced in Table III and
the fits are shown in Fig. 7(a). The values for the
nanoparticles are rather close to the values in the bulk
wurtzite structure. The model with anisotropic ADPs
FIG. 6: (Color online) The experimental PDF, G(r), with
Qmax = 19.0 Å
−1(blue solid dots) and the calculated PDF
from refined structural model (red solid line), with the dif-
ference curve offset below (black solid line). PDF data are
fitted using (a) wurtzite structure model, space group P63mc
and (b) zinc-blende model with space group F 4̄3m. In both
models isotropic atomic displacement factors (Uiso) are used.
TABLE II: The refined residual (Rw) values obtained from
PDF analysis assuming the wurtzite and zinc-blend structure
models with space group P63mc and F 4̄3m, respectively. In
both models isotropic atomic displacement factors (Uiso) are
used.
CdSe-bulk CdSeIII CdSeII CdSeI
Wurtzite (Rw) 0.16 0.31 0.28 0.31
Zinc-blende (Rw) 0.52 0.32 0.30 0.35
resulted in lower Rw. There is a general increase in the
ADPs with decreasing particle size. This reflects inho-
mogeneous strain accommodation in the nanoparticles
as we discuss below. However, the values of the ADPs
along the z-direction for Se atoms (U33) are four times
larger in the nanoparticles compared with the bulk where
FIG. 7: (Color online) The experimental PDF, G(r), with
Qmax = 19.0 Å
−1(blue solid dots) and the calculated PDF
from refined structural model (red solid line), with the differ-
ence curve offset below (black solid line). PDF data are fitted
using wurtzite structure model (a) with no stacking fault and
(b) with 33% stacking fault density for bulk and 50% for all
nanoparticle sizes. In both cases anisotropic atomic displace-
ment factors (Uaniso) are used
they are already unphysically large. The fact that this
parameter is large on the Se site and small on the Cd
site is not significant, since we can change the origin of
the unit cell to place a Cd ion at the (1/3,2/3,z) position
and the enlarged U33 shifts to the Cd site in this case.
The unphysically large U33 value on the Se site is likely
to be due to the presence of faults in the basal plane
stacking. For example, similar unphysical enlargements
of perpendicular thermal factors in PDF measurements
are explained by the presence of turbostratic disorder in
layered carbons55, which is a similar effect to faults in
the ABABAB wurtzite stacking. Also, the presence of
stacking faults in the nanoparticles has been noted pre-
viously.8 It is noteworthy that this parameter is enlarged
in EXAFS analyses of bulk wurtzite structures, probably
FIG. 8: The the enlargement in the the ADPs along the z-
direction for Se site U33, as a function of the stacking fault
density.
for the same reason.32,56,57 We suspect that the enlarge-
ment in this parameter (U33) is related to the stacking
fault density present in bulk and that is increasing in the
nanoparticles.
To test this idea we simulated PDF data using the
wurtzite structure containing different stacking fault den-
sities. The stacking faults were simulated for different
densities (0.167, 0.25, 0.333, and 0.5) by creating wurtzite
superlattices with different stacking sequences along the
C-axis. The program DISCUS58 was used to create the
stacking fault models and PDFgui48 was used to gener-
ate the corresponding PDFs. The PDFs were simulated
with all the ADPs fixed at Uii = 0.0133 Å
2, the value ob-
served in the experimental bulk data collected at room
temperature (see Table III).
To see if this results in enlarged U33 values we refined
the simulated data containing stacking faults using the
wurtzite model without any stacking faults. Indeed, the
refined Se site U33 increased monotonically with increas-
ing stacking fault density. The results are plotted in
Fig. 8.
Fig. 8 can be considered as calibration curve of stacking
fault density in the wurtzite structure, based on the en-
largement in the ADPs along the z-direction U33. From
this we can estimate a stacking fault density of ∼ 35%
for our bulk CdSe sample, and ∼ 50% for each of the
nanoparticles.
It is then possible to carry out a refinement using a
structural model that contains an appropriate stacking
fault density. The PDF data of bulk CdSe was therefore
fit with a wurtzite model with a 33% density, and the
nanoparticle PDF fit with a model with 50% of stacking
faults. The refinements give excellent fits, as is evident in
Fig. 7(b). The results are presented in Table III. The en-
larged U33 parameter on the Se site is no longer present
and it is now possible to refine physically reasonable val-
ues for that parameter. As well as resulting in physically
reasonable ADPs, the quality of the fits to the data are
excellent, though the Rw value is slightly larger in the
nanoparticles.
Attempts to characterize the structure changes using
direct measurements such as TEM technique for such
small CdSe nanoparticles59 were unsuccessful due to the
poor contrast. However, in the present study we were
successful in exploring the local atomic structure for
CdSe nanoparticles, in real space, at different length
scales. The PDF fits clearly indicate that the structure
can be described in terms of locally distorted wurtzite
structure containing ∼ 50% stacking fault density (i.e.,
intermediate between wurtzite and zinc-blende) even for
the 2 nm diameter particles, Fig. 7.
Interestingly, there is little evidence in our data for a
significant surface modified region. This surface region
is sometimes thought of as being an amorphous-like re-
gion. Amorphous structures appear in the PDF with
sharp first neighbor peaks but rapidly diminishing and
broadening higher neighbor peaks. Thus, in the presence
of a surface amorphous region, we might expect to see ex-
tra intensity at the first-peak position when the wurtzite
model is scaled to fit the higher-r features coming just
from the crystalline core. As evident in Fig. 7, this is
not observed. Furthermore, as we describe below, the
diameter of the crystalline core that we refine from the
PDF agrees well with other estimates of nanoparticle size,
suggesting that there is no surface amorphous region in
these nanoparticles. The good agreement in the intensity
of the first PDF peak also presents a puzzle in the op-
posite direction since we might expect surface atoms to
be under-coordinated, which would result in a decrease
in the intensity of this peak. It is possible that the com-
peting effects of surface amorphous behavior and surface
under coordination perfectly balance each other out, and
this cannot be ruled out, though it seems unlikely that it
would work perfectly at all nanoparticle diameters. This
is also not supported by the nanoparticle size determina-
tions described below.
B. Nanoparticle size
We describe here how we extracted more accurate
nanoparticle diameters. This determination is impor-
tant since the physical proprieties are size dependent.
It is also important to use complementary techniques to
determine particle size as different techniques are more
dependent on different aspects of the nanoparticle struc-
ture, for example, whether or not the technique is sensi-
tive to any amorphous surface layer on the nanoparticle.
More challenges are expected in accurate size determi-
nation as nanoparticle diameter decreases, due to poor
contrast near the surface of the nanoparticle.
In the literature, CdSe nanoparticles with a diame-
ter of 2.0 nm have been considered to be an especially
stable size with an associated band edge absorption cen-
tered at 414 nm60, that size was observed earlier29,61 with
TABLE III: The refined parameters values obtained from PDF analysis assuming the wurtzite structure , space group P63mc,
with different stacking fault densities (SFDs).
CdSe-bulk CdSeIII CdSeII CdSeI
Stacking fault density (%) 0.0 33.0 0.0 50.0 0.0 50.0 0.0 50.0
a (Å) 4.3014(4) 4.3012(4) 4.2997(9) 4.2987(9) 4.3028(9) 4.3015(9) 4.2930(9) 4.2930(8)
c (Å) 7.0146(9) 7.0123(9) 7.0145(4) 7.0123(4) 6.9987(9) 6.9975(9) 6.9405(9) 6.9405(7)
Se Z-frac. 0.3774(3) 0.3771(3) 0.3761(9) 0.3759(9) 0.3751(6) 0.3747(6) 0.3685(9) 0.3694(9)
Cd U11 = U22 (Å
2) 0.0108(2) 0.0102(2) 0.0146(7) 0.0149(7) 0.0149(6) 0.0112(5) 0.0237(9) 0.0213(8)
U33 (Å
2) 0.0113(3) 0.0112(3) 0.0262(9) 0.0241(9) 0.0274(9) 0.0271(9) 0.0261(9) 0.0281(9)
Se U11 = U22 (Å
2) 0.0109(9) 0.0102(9) 0.0077(7) 0.0138(7) 0.0083(7) 0.0121(7) 0.0110(9) 0.0191(9)
U33 (Å
2) 0.0462(9) 0.0115(9) 0.1501(9) 0.02301(9) 0.1628(9) 0.0265(9) 0.1765(9) 0.0311(9)
NPa diameter (nm) ∞ ∞ 3.7(1) 3.7(1) 3.1(1) 3.1(1) 2.4(2) 2.2(2)
Rw 0.12 0.09 0.20 0.14 0.18 0.15 0.27 0.21
aNP refers to nanoparticle.
an estimated diameter of ≤2.0 nm. There are some re-
ported difficulties in determining the diameter of such
small CdSe nanoparticles. Attempts to characterize the
structure changes by TEM and X-ray diffraction tech-
niques59 were unsuccessful due to the small diameter of
the particles relative to the capping material.
If we assume the nanoparticle to have spherical shape
(a reasonable approximation based on the TEM in Fig. 1)
cut from the bulk, then the measured PDF will look like
the PDF of the bulk material that has been attenuated by
an envelope function given by the PDF of a homogeneous
sphere, as follows62
G (r, d)
= G (r) f (r, d) , (4)
where G(r) is given in Eq. 1, and f(r, d) is a sphere en-
velope function given by
f (r, d) =
Θ(d− r), (5)
where d is the diameter of the homogeneous sphere, and
Θ(x) is the Heaviside step function, which is equal to 0
for negative x and 1 for positive.
The approach is as follows. First we refine the bulk
CdSe data using PDFfit. This gives us a measure of
the PDF intensity fall-off due to the finite resolution of
the measurement.9 Then the measured value of the finite
resolution was kept as an unrefined parameter after that,
while all the other structural and non structural param-
eters were refined. To measure the PDF intensity fall-off
due to the finite particle size, the refined PDF is atten-
uated, during the refinement, by the envelope function
(Eq. 5) which has one refined parameter, the particle di-
ameter. The fit results are shown in Fig. 9 and the result-
ing values of particle diameter from the PDF refinement
are recorded in Table I. The insets show the calculated
and measured PDFs on an expanded scale. The accuracy
of determining the nanoparticle size can be evaluated di-
rectly from this figure. Features in the measured PDFs
that correspond to the wurtzite structure are clearly seen
disappearing smoothly attenuated by the spherical PDF
envelope function. The procedure is least successful in
the smallest nanoparticles, where the spherical particle
approximation on the model results in features that ex-
tend beyond those in the data. In this case, the spherical
approximation may not be working so well.
The particle diameters determined from the PDF are
consistent with those obtained from TEM, UV-vis and
photoluminescence measurements. In particular, an ac-
curate determination of the average diameter of the
smallest particles is possible in the region where UV-vis
and photoluminescence measurements lose their sensitiv-
ity.23 In this analysis we have not considered particle size
distributions, which are small in these materials. The
good agreement between the data and the fits justify this,
though some of the differences at high-r may result from
this and could contribute an error to the particle size.
Several additional fits to the data were performed to test
the sphericity of the nanoparticles. Attempts were made
to fit the PDF with oblate and prolate spheroid nanopar-
ticle form factors. These fits resulted in ellipticities very
close to one, and large uncertainties in the refined elliptic-
ity and particle diameters, which suggests that the fits are
over-parameterized. Another series of fits attempted to
profile the PDF with a lognormal distribution of spherical
nanoparticles. Allowing the mean nanoparticle diameter
and lognormal width to vary resulted in nonconvergent
fits, which implies that the particle sizes are not lognor-
mal distributed. Therefore, there appears to be little ev-
idence for significant ellipticity, nor a significant particle
size distribution, as fits assuming undistributed spherical
particles give the best results.
The simple fitting of a wurtzite structure with ∼ 50%
SFD to the data will result in an estimate of the coherent
structural core of the nanoparticle that has a structure
can be described by a mixture of crystalline structures
intermediate between zinc-blend and wurtzite. Compar-
ing the nanoparticle core diameter extracted from PDF
analysis with the diameter determined from the standard
characterization methods yields information about the
existence of a surface amorphous region. The agreement
between the core diameter extracted from PDF and that
determined from the standard methods (Table I), indi-
cates that within our measurement uncertainties, there is
FIG. 9: (Color online) The experimental PDF, G(r), shown
as solid dots. Sphere envelope function (Eq. 5) is used to
transform the calculated PDF of bulk CdSe, using wurtzite
structure containing 50% stacking fault density, to give a best
fit replication of the PDF of CdSe nanoparticles (red solid
line). The inset shows on an expanded scale for the high-
r region of experimental G(r) on the top of simulated PDF
data for different diameters of CdSe nanoparticles (solid line).
(a) CdSeIII, (b) CdSeII, (c) CdSeI. Dashed lines are guides
for the eye.
FIG. 10: (a) The first PDF peak, (•) bulk, (◦) CdSeIII,
(�) CdSeII and (△) CdSeI fitted with one Gaussian (—).
Dashed line represents the position of first PDF peak in the
bulk data. (b)(N) The first PDF peak width vs nanoparticle
size, obtained from one Gaussian fit. Dashed line represents
the width of first PDF peak in the bulk data. (c) Strain in
Cd-Se bond (∆r/r)(%) vs nanoparticle size. (�) Bond values
obtained from the local structure fitting and (•) obtained from
one Gaussian fit to the first PDF peak. Dotted curves are
guides for the eye.
no significant heavily disordered surface region in these
nanoparticles, even at the smallest diameter of 2 nm
(Fig. 9). In contrast with ZnS nanoparticles11 where
the heavily disordered surface region is about 40% of the
nanoparticle diameter for a diameter of 3.4 nm, the sur-
face region thickness being around 1.4 nm.11
C. Internal strain
The local bonding of the tetrahedral Cd-Se building
unit was investigated vs nanoparticle diameter. The
nearest neighbor peaks at r = 2.6353(3) Å come from
covalently bonded Cd-Se pairs. The positions and the
width of these peaks have been determined by fitting a
Gaussian (Fig. 10(a)) and the results presented in Ta-
ble IV. The results indicate that there is a significant
compressive strain on this near-neighbor bond length,
and it is possible to measure it with the PDF with high
accuracy. The bond length of Cd-Se pairs shorten as
nanoparticle diameter decreases, suggesting the presence
of an internal stress in the nanoparticles. The Cd-Se
bond lengths extracted from the PDF structural refine-
ment are also in good agreement with those obtained
from the first peak Gaussian fit, as shown in Fig. 10(c).
Thus we have a model independent and a model depen-
TABLE IV: The first PDF peak position (FPP) and width
(FPW) for different CdSe nanoparticle sizes and the bulk.
CdSe-bulk CdSeIII CdSeII CdSeI
PDF FPP (Å) 2.6353(3) 2.6281(3) 2.6262(3) 2.6233(3)
PDF FPW (Å) 0.1985(09) 0.1990(19) 0.2021(25) 0.2032(25)
dent estimate of the strain that are in quantitative agree-
ment. The widths of the first PDF peaks have also been
extracted vs nanoparticle diameter from the Gaussian
fits (Table IV). They remain comparably sharp as the
nanoparticles get smaller, as shown in Fig. 10(b). Ap-
parently there is no size-dependent inhomogeneous strain
measurable on the first peak. However, peaks at higher-r
do indicate significant broadening (Fig. 4(b)) suggesting
that there is some relaxation taking place through bond-
bending. This is reflected in enlarged thermal factors
that are refined in the nanoparticle samples. This is sim-
ilar to what is observed in semiconductor alloys where
most of the structural relaxation takes place in relatively
lower energy bond-bending distortions.63,64
IV. CONCLUSION
The PDF is used to address the size and structural
characterization of a series of CdSe nanoparticles pre-
pared by the method mentioned in the text. The core
structure of the measured CdSe nanoparticles was found
to possess a well-defined atomic arrangement that can
be described in terms of locally disordered wurtzite struc-
ture that contains∼ 50% stacking fault densit, and quan-
titative structural parameters are presented.
The diameter of the CdSe nanoparticles was extracted
from the PDF data and is in good agreement with the
diameter obtained from standard characterization meth-
ods, indicating that within our measurement uncertain-
ties, there is no significant heavily disordered surface re-
gion in these nanoparticles, even at the smallest diame-
ter of 2 nm . In contrast with ZnS nanoparticles11 where
the heavily disordered surface region is about 40% of the
nanoparticle diameter for a diameter of 3.4 nm, the sur-
face region thickness being around 1.4 nm.11
Compared with the bulk PDF, the nanoparticle PDF
peaks are broader in the high-r region due to strain
and structural defects in the nanoparticles. The near-
est neighbor peaks at r = 2.6353(3) Å which come from
covalently bonded Cd-Se pairs, shorten as nanoparticle
diameter decreases resulting in a size-dependent strain
on the Cd-Se bond that reaches 0.5% at the smallest
particle size.
Acknowledgments
We would like to acknowledge help from Didier Wer-
meille, Doug Robinson, Mouath Shatnawi, Moneeb Shat-
nawi and He Lin for help in collecting data. We are
grateful to Christos Malliakas for the valuable assistance
with the transmission electron microscopy. May thanks
to HyunJeong Kim for useful discussion. We are grateful
to Prof. Reinhard Neder for the valuable help with the
stacking fault simulation. This work was supported in
part by National Science Foundation (NSF) grant DMR-
0304391. Data were collected at the 6IDD beamline
of the MUCAT sector at the Advanced Photon Source
(APS). Use of the APS is supported by the U.S. DOE,
Office of Science, Office of Basic Energy Sciences, under
Contract No. W-31-109-Eng-38. The MUCAT sector at
the APS is supported by the U.S. DOE, Office of Sci-
ence, Office of Basic Energy Sciences, through the Ames
Laboratory under Contract No. W-7405-Eng-82.
∗ Electronic address: billinge@pa.msu.edu
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