crystal structure of defect-containing semiconductor nanocrystals – an x-ray diffraction study

$: Crystal structure of defect-containing semiconductor nanocrystals – an X-ray diffraction study$
research papers

660 doi:10.1107/S0021889809017476 J. Appl. Cryst. (2009). 42, 660–672

Journal of

AppliedCrystallography

ISSN 0021-8898

Received 26 January 2009

Accepted 9 May 2009

# 2009 International Union of Crystallography

Printed in Singapore – all rights reserved

Crystal structure of defect-containingsemiconductor nanocrystals – an X-raydiffraction study

Maja Buljan,a,b* Uros V. Desnica,b Nikola Radic,b Goran Drazic,c Zdenek Matej,a

Vaclav Valesa and Vaclav Holya

aFaculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague, Czech

Republic, bRuer Boskovic Institute, Bijenicka cesta 54, 10000 Zagreb, Croatia, and cJozef Stefan

Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia. Correspondence e-mail: [email protected]

Defects of crystal structure in semiconductor nanocrystals embedded in an

amorphous matrix are studied by X-ray diffraction and a full-profile analysis of

the diffraction curves based on the Debye formula. A new theoretical model is

proposed, describing the diffraction from randomly distributed intrinsic and

extrinsic stacking faults and twin blocks in the nanocrystals. The application of

the model to full-profile analysis of experimental diffraction curves enables the

determination of the concentrations of individual defect types in the

nanocrystals. The method has been applied for the investigation of self-

organized Ge nanocrystals in an SiO2 matrix, and the dependence of the

structure quality of the nanocrystals on their deposition and annealing

parameters was obtained.

1. Introduction

Semiconductor nanocrystals (NCs) have great relevance for

application in semiconductor technology and optoelectronics

(lasers compatible with CMOS technology based on quantum

dots, quantum memories, solar cells etc.) (Alivisatos, 1996;

Siegel et al., 1999). Technological applications of quantum dots

are based on their special physical properties, which are to a

large extent determined by their crystal structure, especially

by the presence of structure defects. Therefore, it is very

important to have a suitable method for the growth of nano-

structured materials with controlled structural properties, as

well as reliable methods for the characterization of their real

structure.

Diffraction-based methods are suitable for the determina-

tion of mean structure and crystal quality, since they provide

averaged information with excellent statistics (typically an

order of 1012 nano-objects in the irradiated sample volume).

For the investigation of NCs two approaches can be used, the

convolution approach using the well known Rietveld method

(Guinier, 1994) and the Debye-function method (Cervellino et

al., 2003, 2005). The former method is based on an expression

of the electron density by means of a Fourier series over

reciprocal lattice points. This method can be effectively

applied for NCs larger than approximately 5 nm; for smaller

objects the method cannot properly describe the overlap of

diffraction maxima belonging to different reciprocal lattice

points. The latter method calculates the Debye function of the

crystal without using the reciprocal lattice at all. Therefore, the

Debye-function method can be used for very small NCs and

even for an atomic or molecular aggregate without a

pronounced crystal structure. On the other hand, the Debye-

function method involves the summation of the contributions

of individual atoms to the diffracted intensity, so it is practi-

cally useless for large objects, unless special calculation tech-

niques are used (Cervellino et al., 2003).

The original Debye-function method has recently been

improved to include the size distribution of the NCs and an

inhomogeneous strain in the NC volume, or assuming a core–

shell type structure of the NCs (Cervellino et al., 2003).

Nevertheless, a little progress has been achieved so far in the

incorporation of structure defects, such as stacking faults or

twin boundaries in zincblende and diamond-type lattices, in

spite of the fact that these defects occur rather frequently in

NCs with the diamond structure (Wang, Poudel et al., 2005;

Wang, Smirani et al., 2005).

X-ray diffraction from face-centred cubic (fcc) poly-

crystalline metals containing stacking faults and twins has

been extensively studied since the 1950s, both theoretically

and experimentally (Warren, 1959; Leoni et al., 2004; Balogh et

al., 2006; Estevez-Rams et al., 2007; Ungar, 2007). All these

investigations are based on the first approach mentioned

above (the convolution method), so they cannot be directly

applied for very small NCs. Kumpf et al. (2005) included

stacking faults in the Debye function, generating the positions

of individual atoms in an NC with a given sequence of the

faulted {111} planes, i.e. without averaging over a statistical

ensemble of all possible positions of the faulted planes in the

NC volume. A similar procedure was used by Oddershede et

al. (2008), where a numerical ‘brute-force’ averaging was

performed by calculating the Debye formula for different

randomly chosen defect spatial configurations.

In the past decade, nanostructures such as nanoclusters or

nanowires have also been studied, using the pair-distribution

function of atoms obtained by X-ray diffraction at extremely

short wavelengths, and subsequent Fourier transformation of

the diffraction curve (see, for example, Billinge, 2007, 2008;

Petkov et al., 2008; Petkov, 2008). This method has proved to

be very useful for the investigation of the crystal structure of

various types of nano-objects and, in connection with the

reverse Monte Carlo method (Tucker et al., 2001; McGreevy &

Pusztai, 1988), it is possible to study atomic disorder in nano-

objects. On the other hand, in order to achieve the necessary

resolution in real space, this method requires very hard X-ray

radiation, which is not usually accessible in the laboratory.

In this paper we develop a new approach for the calculation

of the X-ray intensity diffracted from NCs with a diamond-like

structure containing randomly distributed stacking faults and

twin planes. We use the Debye-function approach and a direct

(analytic) ensemble averaging over all defect positions and

over the orientations of the NC lattice. In comparison with

other simulation methods mentioned above, the numerical

procedure based on our approach is much faster so it can be

easily incorporated into fitting software.

The paper is organized as follows. In x2 we present the

experimental data and demonstrate that a standard Debye

formula (without structural defects) does not properly

describe the diffracted intensity. A theoretical description of

the scattering from NCs with defects is given in x3. In x4 we

apply the method to the analysis of the diffraction curves of

Ge NCs embedded in Ge+SiO2 multilayers, obtained by

magnetron sputtering deposition and subsequent annealing at

different temperatures. Conclusions are given in x5.

2. Diffraction measurements on Ge nanocrystals in(Ge+SiO2)/SiO2 multilayers

In this section we present a description of the investigated

(Ge+SiO2)/SiO2 multilayers, and we show experimental

diffraction curves measured from them. We then apply the

standard Debye formula for the analysis of the X-ray

diffraction measured from Ge dots formed in (Ge+SiO2)/SiO2

multilayers after annealing at 1073 K.

The system investigated consists of Ge NCs buried in an

amorphous SiO2 matrix. From our previous studies based on

grazing-incidence small-angle X-ray scattering (GISAXS)

(Buljan et al., 2009a), it follows that the NCs are spatially

correlated in all three dimensions, forming a slightly disor-

dered three-dimensional superlattice. The as-deposited Ge

nanoclusters are amorphous and their regular ordering in an

amorphous SiO2 matrix is achieved during the deposition

process. In contrast with nano-objects in a crystalline matrix,

the ordering mechanism is explained by the influence of

surface morphology on the surface diffusion of deposited ad-

atoms and on the nucleation of the nanoclusters in the troughs

of the growing surface (Buljan et al., 2009a). The crystal-

lization of the nanoclusters is then achieved by subsequent

annealing of the deposited films at several temperatures up to

1173 K.

Transmission electron microscopy (TEM) images of the

cross sections of the (Ge+SiO2)/SiO2 multilayers and a high-

resolution transmission electron microscopy (HRTEM) image

of an individual NC, after annealing at 1073 K, are shown in

Figs. 1(a) and 1(b). The NCs formed by post-growth annealing

are spherical and fully crystalline, and their ordering exhibits a

regularity which is confirmed by GISAXS measurements.

A typical reciprocal-space distribution of scattered X-ray

intensity obtained in a GISAXS experiment is exhibited in

Fig. 1(c). Since the primary X-ray beam irradiates a large

number of NCs in a GISAXS experiment (typically about

1012), it contains averaged information from a large sample

volume. A correlation of the positions of the NCs gives rise to

lateral intensity maxima, and from their distance and shape we

determined the average distance of the NCs, their average size

and its r.m.s. deviation (Buljan et al., 2009a). The resulting size

distribution of the NCs is presented in Fig. 1(d), together with

the size distributions determined from TEM and low-

frequency Raman measurements (Buljan et al., 2009a). All

results are in a very good agreement [mean NC radius R0 =

2.7 (1) nm and its dispersion �R = 0.40 (5) nm] and the size

distribution is narrower than in similar systems, where self-

ordering was not found (Buljan et al., 2009b).

However, these techniques are not sensitive enough to

resolve the details of the crystalline structure of the NCs

formed, especially structure defects. GISAXS is not sensitive

to the crystalline state at all, Raman analysis can provide basic

research papers

J. Appl. Cryst. (2009). 42, 660–672 Maja Buljan et al. � Defect-containing nanocrystals 661

Figure 1Structural properties of the Ge dots formed in (Ge+SiO2)/SiO2 multi-layers after annealing at Ta = 1073 K (800�C). (a), (b) TEM cross sectionsand HRTEM images of typical NCs (insets). (c) Two-dimensionalGISAXS map of the same sample. (d) Size distributions of Ge NCsobtained by TEM, GISAXS and low-frequency Raman analysis.

parameters of crystalline structure such as lattice constant and

NC size, while HRTEM provides data on defects in a crys-

talline structure but usually with very bad statistics. Therefore,

another analytical method is needed for defect investigation,

and diffraction analysis is a very suitable choice for this

purpose. On the other hand, NC sizes and size distributions

are precisely determined by GISAXS, TEM and Raman

techniques, so analysis of the same parameters by diffraction

using the existing Debye formalism can be discussed.

X-ray diffraction from the as-grown Ge+SiO2 samples and

from samples after various post-growth annealing steps was

carried out on a laboratory diffractometer using a standard

Cu K� X-ray tube, parabolic multilayer optics, a parallel-plate

collimator and a secondary graphite crystal monochromator.

Fig. 2 shows the resulting curves. The diffraction maxima of

the NCs are clearly visible on a broad background stemming

from the scattering from the amorphous SiO2 matrix. In order

to remove the background, we have subtracted the curve of a

matrix layer annealed under the same conditions from the

measured diffraction curve. In particular, we first fitted the

curve of the SiO2 layer without any buried Ge atoms, and then

we put the resulting function into the fit of the diffraction

curve of samples with Ge NCs (Ge+SiO2) as a starting esti-

mate for the final fitting. We used a combination of three

Gaussian functions to carry out this fit. The contribution of the

amorphous Ge component was treated by two different

methods. First, we added two additional Gaussians to include

the contribution of amorphous Ge. The positions and widths

of these Gaussian functions were initially estimated from the

diffraction curve measured on the as-deposited (AS) sample

shown in Fig. 2(a), where the Ge is mostly amorphous

[according to the Raman results published by Buljan et al.

(2009a)]. In the second method we included the contribution

of the amorphous Ge phase via a size distribution of Ge NCs

used in the applied Debye formula. In this approach no

additional parameters were needed. The latter method is

found to be much better than the former, because it gives a

good description of the amorphous Ge contribution and the

number of parameters used for the background description is

considerably reduced. We have used this method for the

description of the data presented here.

An example of the amorphous background treatment is

shown in Fig. 2(b). The sample used for this example contains

both amorphous and crystalline Ge contributions, which

follows from the Raman data shown in the inset of Fig. 2(b), as

well as an amorphous SiO2 contribution. The amorphous SiO2

contribution is initially estimated, as described above, from

the data measured on the pure SiO2 layer and used for the

final fitting of the total Ge+SiO2 experimental curve. The SiO2

background contribution determined from the final fitting of

the total curve agrees well with the experimental data of the

pure SiO2 curve, showing that the applied procedure describes

our system well. The amorphous Ge contribution is included

in the fit via a size distribution of the Ge NCs. The mean NC

radius was found to be approximately 1.5 nm, while the

standard deviation of the size distribution was about 0.6 nm,

showing the presence of NCs with very small sizes comparable

to the Ge lattice parameter. Thus, both amorphous contribu-

tions (amorphous SiO2 and amorphous Ge) were taken into

account in the fitting process.

In Fig. 3 we present an example of the fit of the measured

data to the simple Debye formula of spherical NCs without

defects, after removal of the amorphous SiO2 background. In

addition to the background parameters, the fitting parameters

were the lattice constant, the mean radius of the NCs and its

r.m.s. dispersion. The lattice constant is found to be very close

to that of a bulk Ge crystal, the NC mean radius R0 =

2.18 (6) nm and its r.m.s. dispersion �R = 0.31 (4) nm.

However, the NC radius and r.m.s. dispersion values do not

match the same parameters determined by other techniques

(GISAXS, TEM, Raman scattering), as can be seen in the

inset of Fig. 3. GISAXS cannot resolve crystalline and amor-

phous states, so differences in the size-distribution parameters

could originate from that discrepancy if only the GISAXS

method were applied. However, several additional applied

techniques showed that the NCs formed are predominantly

crystalline (Buljan et. al., 2009a), so the size distribution

obtained by GISAXS is applicable for comparison with the

data obtained by diffraction. The size distribution of the NCs

obtained from low-frequency Raman measurements is even

more applicable, since the technique is predominantly sensi-

tive to the crystalline state and the measured data have

excellent statistics. Therefore, the size distributions obtained

by diffraction and Raman approaches should not be very

different. In addition to the discrepancy of the size-distribu-

tion parameters, the ratio of the heights of the measured

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662 Maja Buljan et al. � Defect-containing nanocrystals J. Appl. Cryst. (2009). 42, 660–672

Figure 2(a) Measured diffraction curves of the (Ge+SiO2)/SiO2 multilayersobtained after deposition (AS) and after post-growth annealing atdifferent temperatures indicated above each curve. The curve labelledSiO2 was measured from a pure SiO2 matrix without Ge nanoclusters. (b)An example of the background treatment during the fitting process forthe curve measured on the sample annealed at 873 K (denoted byGe+SiO2 Exp.), where both amorphous and crystalline Ge phases arepresent. The inset shows the Raman spectrum of the same sample, wherethe vibrational bands at 80 and 275 cm�1 show the presence ofamorphous Ge (a-Ge) in the sample, while the line close to 300 cm�1

shows the presence of crystalline Ge (c-Ge). SiO2 Exp. denotes theexperimentally measured curve from the pure SiO2 layer. The red curve isobtained by a fit using the Debye formula with the size distribution of theNCs included. The curve shown by a blue line corresponds to thebackground obtained by the fitting process. A detailed description isgiven in the text.

diffraction peaks (111 and 311) does not correspond to the

simulated curve (arrows in Fig. 3).

To summarize this section, the simulated curve does not fit

well with the experimental data, and the best-fit values do not

agree with the results obtained with other techniques. The

obvious explanation for this discrepancy is structure defects,

which are indeed found in NCs. This is demonstrated in Fig. 4,

where dots with twins and intrinsic and extrinsic stacking

faults are depicted by HRTEM images. The defects shown are

found for the multilayers annealed at 1073 K. The concen-

trations of the different defects (stacking faults and twins) are

also found to be very similar. For the lower annealing

temperatures, no defects were observed in the HRTEM

images. Of course, the statistics of the HRTEM data are not

sufficient to draw any general conclusions from these

measurements. This simple example demonstrates the neces-

sity of improving the standard Debye formula to include

structure defects. In the next section, we present a modified

Debye formula describing the X-ray intensity diffracted from

NCs with randomly distributed intrinsic and extrinsic stacking

faults, as well as twin boundaries.

3. Debye equation for nanocrystals with defects

We now derive expressions for the diffraction intensity from

an NC with fcc and diamond lattices containing stacking faults

(intrinsic and extrinsic) and twins. We also present the influ-

ence of each kind of defect on the diffraction curves.

We start from the definition of the positions of the atoms in

an fcc crystal with defects. In an ideal crystal with an fcc

structure, the (111) layers are stacked according to the well

known ABCABC scheme. Each (111) layer consists of a two-

dimensional hexagonal array of atoms and each subsequent

layer is shifted by a vector D with respect to the previous one.

In the following, we denote by d = a( 16,

16, �

13 ) the component

of D parallel to the (111) fault plane (a is the lattice para-

meter). It is worth noting that the vector 3d is a lattice vector

(i.e. it connects two points in the fcc lattice), so that all

expressions of the faulted atomic positions contain only

mod(n, 3)d instead of an integer multiple nd. In the following,

all sums containing the vector d will be considered in this

sense.

Fig. 5 presents a schematic sketch of the layer stacking in an

ideal crystal, and in crystals with intrinsic or extrinsic stacking

faults (SFs) or with twins. In an intrinsic SF (iSF), one layer is

missing at the SF position, while for an extrinsic SF (eSF) one

layer is inserted. In the case of a twin, the stacking sequence

ABCABC is inverted (CBACBA) at the twin boundary (TB).

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Figure 5Schematic sketch of the layer stacking in an ideal fcc lattice and in alattice with various defect types. In an intrinsic SF one layer is missing atthe defect position and in an extrinsic SF one layer is inserted, while for atwin the stacking sequence is inverted. l denotes the layer index and Cm

are the random defect coefficients; Cm = 1 if the layer is defective,otherwise Cm = 0.

Figure 4HRTEM images of individual NCs containing various defect types: (a)one twin, (b) two twins, (c) an intrinsic stacking fault and (d) an extrinsicstacking fault.

Figure 3Measured (circles) and fitted curves (lines) of the sample annealed at1073 K. The fitted curve was obtained without including structure defectsin the dot structure. The contribution of the amorphous matrix wassubtracted from the measured data. The differences between themeasured and fitted curves are most pronounced in the regions denotedby arrows. The inset shows the size distributions obtained by the fit of thediffraction data and from the GISAXS, Raman and TEM measurements.

In order to describe the positions of the atoms in an ideal fcc

lattice and in a lattice with defects, we use the basis vectors

a1 ¼ a1

2;�

1

2; 0

� �; a2 ¼ a 0;�

1

2;

1

2

� �; a3 ¼ a

1

2;

1

2; 0

� �;

where a is the lattice parameter. Using this choice, the atomic

positions in an ideal fcc lattice are integer linear combinations

of the vectors

Ridðh; k; lÞ ¼ ha1 þ ka2 þ la3; ð1Þ

where h, k, l are integers and vectors a1,2 lie in the (111) plane.

The presence of an SF in the mth (111) layer is described by

the coefficient Cm, which is 1 if the layer m is defective and 0

otherwise. All coefficients Cm for a given set of layers with

defects form a random vector built from 0s and 1s (see Fig. 5).

The positions of the TBs are described by the sequence of

their random integer coordinates along the [111] direction

perpendicular to the TB planes.

Following the above notation, the positions of the atoms in

a lattice with particular defect positions are given as

RiSFðh; k; lÞ ¼ ha1 þ ka2 þ la3 þPl

m¼0

Cmd; ð2Þ

ReSFðh; k; lÞ ¼ ha1 þ ka2 þ la3 þ

�2Pl

m¼0

Cm � Cl

�d; ð3Þ

RTBðh; k; lÞ ¼ ha1 þ ka2 þ la3 þ

�Pl�1

p¼0

ð�1ÞPpm¼0

Cm � l

�d: ð4Þ

The number of SFs or TBs (n) in a sequence of l layers is a

simple sum of all Cm for a given layer sequence:

n ¼Pl

m¼1

Cm: ð5Þ

We assume that the X-ray irradiated sample volume contains a

large number of NCs with randomly distributed defect layers.

Therefore, for a given concentration of defects c, the prob-

ability P(n, l, c) that n layers in a sequence of l layers are

defective is given by the binomial distribution function

Pðn; l; cÞ ¼l

n

� �cnð1� cÞðl�nÞ; ð6Þ

where c is the concentration of layers with defects in the whole

sample normalized to unity, i.e. the probability of finding a

defect in a given layer. We assume that the positions of the

defect layers in the NCs are random, i.e. the vectors Cm are

random and hCmi ¼ c. From the particular stacking of the

(111) planes in an fcc lattice that corresponds to a given

sequence of faults, we obtain directly the corresponding

positions of atoms in a diamond lattice by replacing each layer

by a bilayer with identical positions of atoms, the second

bilayer being shifted with respect to the first one by a vector

b ¼ a½1313

13� ¼ �a1=3þ 2a2=3þ a3.

The intensity of the diffraction from a system consisting of a

very large number of randomly oriented NCs is averaged over

all possible positions of defective layers, and for all orienta-

tions of the lattices in the NCs,

IðQÞ ¼ Aj f ðQÞj2�� P

i;j2�

exp ð�iQ �DijÞ

�def

�O

; ð7Þ

where Dij = R(h, k, l) � R(h0, k0, l0), and hiO and hidef denote

averaging over the lattice orientations and defect positions,

respectively. f(Q) = fat(Q)[1 + exp(�iQ�b)] is the scattering

factor of a pair of atoms displaced along [111] by a vector b,

and fat(Q) denotes the scattering factor of a single atom that

depends only on the length Q of the scattering vector Q [Q =

(4�/�)sin(�/2), where � is the scattering angle and � is the

wavelength of the incident radiation]; we assume that all

atoms in the NC are identical, i.e. they have the same scat-

tering factor. The factor A includes all other terms, such as the

intensity of the primary beam, polarization factor, classical

electron radius etc. The sumP

i;j2� runs over all pairs of atoms

denoted by indexes i � ðh; k; lÞ and j � ðh0; k0; l0Þ belonging to

one NC, the shape of which is described by the set �.

In deriving the final expressions for diffracted intensity,

averaging over different positions of SFs is performed first.

The details of the derivation procedure are given in

Appendix A, and we present here only the final expressions

for each type of defect.

3.1. Ideal lattice

For an ideal lattice, the averaging is performed only over all

possible lattice orientations, which gives rise to the usual

Debye formula

IðQÞid ¼ Aj f ðQÞj2P

i;j2�

sinc QDidij

� ; ð8Þ

where Didij is the distance between atoms i and j in an ideal

lattice, and sinc(x) � sin(x)/x. The diffracted intensity for

spherical Ge dots of different sizes, calculated from the Debye

formula, is shown in Fig. 6. A decrease in the NC size causes a

broadening of the diffraction peaks and a decrease in their

intensity, but the ratio of particular peak heights remains

constant, as can be seen from Figs. 6(a) and 6(b).

3.2. Intrinsic stacking faults

For the derivation of the equation for iSFs we use the fact

that the sumPl

m¼l0 Cmd equals any integer value n between 0

and l. A particular random value of n has the probability P(n,

l, c) given by equation (6). The final formula is

IðQÞiSF¼ Aj f ðQÞj2

Pi;j2�

Pj�Lj

n¼0

PiSFn sinc QjDid

ij þDiSFijn j

� ; ð9Þ

where we introduce the notation

DiSFijn ¼ nsignð�LÞd;

PiSFn ¼ Pðn; j�Lj; cÞ:

ð10Þ

Here, �L denotes l � l0 and P is the binomial distribution

function given by equation (6). To determine the n values

having non-zero probabilities, we use the mean value M(n, c)

and the standard deviation �(n, c) of the binomial distribution,

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Mðn; cÞ ¼ nc; �ðn; cÞ ¼ ncð1� cÞ: ð11Þ

In the calculation of the sumP

n it is sufficient to take only the

terms for which jn�Mðn; cÞj< 3�ðn; cÞ, which speeds up the

calculation. The same simplification is used for other defect

types.

Using the Debye formula for an ideal lattice [equation (8)],

the calculation for each pair of atoms i, j is performed in a

single step, while in the defects-included formula [equation

(10)], the corresponding number of steps is |�L| + 1, i.e. the

number of layers between atoms i, j. However, because of the

above simplification using equation (11), this number is

reduced if we decrease the defect concentration, so the whole

calculation procedure is only a few times longer than using the

Debye formula for an ideal lattice (for example, slightly less

than four times longer for c = 0.2). Thus, this calculation

method is much faster and also much more accurate than a

‘brute-force’ averaging of the intensities obtained from

different randomly chosen configurations of defects.

The intensity calculated for spherical Ge dots as a function

of the iSF concentration is given in Fig. 7(a). From the figure, it

is obvious the iSFs affect mainly the relative intensities of

individual diffraction maxima and change the ratio of these

intensities compared with an ideal lattice. The intensity of the

220 maximum is reduced, while 311 is enhanced as a result of

the defects; the intensity of the 400 peak is mostly reduced.

3.3. Extrinsic stacking faults

We use an analogous procedure for the derivation of the

formula for eSFs and we obtain (see Appendix A for details)

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Figure 7Diffracted intensity calculated for Ge NCs with defects using themodified Debye formula, as a function of the concentration of defects c.The crystal shape is spherical with radius R = 2.4 nm. The insets show theenlarged parts of the diffraction curves marked by squares. Their purposeis to compare the diffraction profiles obtained without defects and fordefects at concentration c = 0.2* (* denotes that the intensity isnormalized to the 111 peak). (a) Intrinsic stacking faults. (b) Extrinsicstacking faults. w denotes the peak that corresponds to the 102 peak ofthe wurtzite structure (shown by the black dashed line). (c) Twins.

Figure 6Diffracted intensities calculated for ideal Ge NCs using the Debyeformula, as a function of the size of the NCs. The crystal shape is sphericalwith radius R. (a) Calculated curves. (b) The curves normalized to thesame height of the first diffraction peak (111).

IðQÞeSF¼ Aj f ðQÞj2

Pi;j2�

Pj�Lj�2

n1¼0

P2

n2¼0

PeSFn1;n2

sinc QjDidij þDeSF

ijn1n2j

� ;

ð12Þ

where we introduce the notation

DeSFijn1n2¼ ð2n1 þ n2Þ d sign ð�LÞ;

PeSFn1;n2¼

Pðn1; j�Lj � 2; cÞPðn2; 2; cÞ; �L � 2,

Pðn2; 2; cÞ; �L< 2.

ð13Þ

The effect of eSFs in NCs on their diffraction curves is shown

in Fig. 7(b). For eSFs, the peak-height ratio also changes

compared with the ideal lattice and one additional peak

appears close to 2� = 38�. This peak corresponds to the 102

peak in the wurtzite lattice with the same lattice parameters. A

shift of the 111 maximum to smaller 2� angles, where the

wurtzite 002 peak occurs, is visible as well.

3.4. Twins

The derivation of the equation for twins is more compli-

cated and the details are given in Appendix A. For twins, the

total diffracted intensity is the sum of the intensities from NCs

containing different numbers of twin boundaries (TBs). Each

configuration of TBs occurs with a certain probability and

causes a certain shift of atoms with respect to the ideal atomic

positions. All probabilities are described by a (2 2|�L|)

matrix T, while the corresponding shifts are included in a

matrix F of the same size. The first row of the matrix T (and F)

is a vector which we denote by T+ (and F+), while the second

row we denote by T� (and F�). Thus, the symbols T+ (F+) and

T� (F�) denote row vectors of the shape 1 2|�L|.

The final formula for the diffracted intensity is

IðQÞTB¼ Aj f ðQÞj2

Pi;j2�

Pn3¼0;1

P2j�Lj

n4¼0

Tn3;n4sincðQjDid

ij þDTBijn3n4jÞ;

ð14Þ

where

DTBijn3n4¼ Fn3n4

� j�Lj�

sign ð�LÞd ð15Þ

and Tn3;n4is the n4th element of the probability vector T+ (or

T�) if n3 = 0 or n3 = 1, respectively. The same notation rule

applies for Fn3;n4.

The vectors T and F can be obtained by the following

recursive formulae:

Tjþ ¼ ½ð1� cÞTj�1

þ ; cTj�1� �;

Tj� ¼ ½cT

j�1þ ; ð1� cÞTj�1

� �;

Fjþ ¼ ½F

j�1þ þ 1;Fj�1

� þ 1�;

Fj� ¼ ½F

j�1þ � 1;Fj�1

� � 1�;

ð16Þ

where the index j denotes the matrix obtained for the jth

iteration and runs from 1 to |�L| � 1. The starting (j = 0)

vectors for the recursive relation are the 1 1 vectors

(scalars)

T0þ ¼ Pe;

T0� ¼ Po;

F0þ ¼ 1;

F0� ¼ �1;

ð17Þ

where Pe represents the probability that the first l00 layers [l00 =

min(l, l0)] have an even number of defective planes, i.e.

Pe ¼P2m<l00

m¼0

Pð2m; l00; cÞ; ð18Þ

while Po = 1 � Pe. Therefore, the number of elements of the

vectors T and F increases by a factor of 2 when �L

increases by 1, so the final number of vector elements is 2|�L|.

The matrix elements of T represent the probabilities of

different configurations of TBs occurring. The probabilities of

many configurations are very small and can be omitted from

the calculation, together with the corresponding phase-shift

values (the corresponding element of F). In this way, the

number of relevant matrix elements is reduced, which signif-

icantly speeds up the calculation procedure for a lower

concentration of TBs.

The diffraction curves calculated for NCs with twins are

given in Fig. 7(c). The presence of twins causes a change in the

peak-height ratios compared with an ideal lattice, as in the

case of stacking faults. However, the changes caused by twins

are less pronounced than those caused by stacking faults,

especially for the 311 peak.

3.5. Combination of different stacking faults

The equations for diffraction intensity given above are valid

if only one type of defect is present in the NCs. For the

calculation of diffraction intensity from NCs containing all

three types of defect, with different concentrations cI, cE and

cT for intrinsic and extrinsic SFs and twins, respectively, we use

a similar calculation procedure as for the calculation for twins.

The position of an atom in a lattice containing all types of

defect is

Rallðh; k; lÞ ¼ ha1 þ ka2 þ la3

þ

Pl

p¼0

ð�1ÞPpm¼0

CTm CI

p þ 2CEp � ðC

Ep þ 1Þ�pl þ 1

� �� l

d;

ð19Þ

where CI;E;Tp denote the coefficients of the matrix describing

the presence of defects for iSF, eSF and TB, respectively, and

�pl is the Kronecker symbol. The corresponding interatomic

distance is

Dallij ¼ Did

ij þ

Pl

p¼l0ð�1Þ

Ppm¼0

CTm

�CI

p þ 2CEp þ 1

��CI

p þ CEp � 1

�pl0 �

�1þ CE

p

�pl

�� j�Lj

d signð�LÞ:

ð20Þ

For deriving the final expression for diffracted intensity we

follow the same procedure as that used in Appendix A for the

case of twins, i.e. we average the sum given above step by step

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for each index p. Interatomic distances Dallij additionally

depend on coefficients CI;Ep , so we multiply the probabilities

obtained for twins with the probabilities of the presence of

intrinsic and extrinsic stacking faults, and add the corre-

sponding phases. The final expression is

IðQÞall¼ Aj f ðQÞj2

Pi;j2�

Pn3¼0;1

P24j�Lj

n4¼0

T 0n3;n4sinc

�QjDid

ij þDallijn3n4j;

ð21Þ

where

Dallijn3n4¼ ðF 0n3;n4

� j�LjÞd signð�LÞ;

Palln3;n4¼ P

allðn4Þ

;ð22Þ

while the probability matrix T 0n3;n4in this case depends on the

concentration of each type of defect separately (i.e. on cI, cE

and cT). The probability and phase matrices can be obtained

by the following recursive formulae:

T0jþ ¼ ½ð1� cTÞT

0j�1aþ ; cTT0j�1

a� �;

T0j� ¼ ½cTT0j�1aþ ; ð1� cTÞT

0j�1a� �;

F0jþ ¼ ½F0j�1aþ þ 1;F0j�1

a� þ 1�;

F0j� ¼ ½F0j�1aþ � 1;F0j�1

a� � 1�:

ð23Þ

However, T0j�1a are now vectors dependent additionally on cI

and cE as well as on cT. Similarly, the corresponding shift

vectors F0j�1a contain additional shifts due to the presence of

iSFs and eSFs. The vectors are given by

T0j�1a ¼ cIcETj�1

; cIqET0j�1 ; qIcET0j�1

; qIqET0j�1

� �;

F0j�1a ¼ F0j�1

3; F0j�1 1;F0j�1

2;F0j�1

� �;

ð24Þ

where qE,T = 1� cE,T, and the starting vectors for the recursive

formulae are

T0þ ¼ ½cEPe; qEPe�;

T0� ¼ ½cEPo; qEPo�;

F0þ ¼ ½2; 1�;

F0� ¼ �½2; 1�;

ð25Þ

where Pe and Po are given by equation (18). The diffraction

curve calculated for a combination of defects shown in Fig. 8 is

compared with those calculated for individual defect types and

for an ideal lattice. All curves show similar behaviour for the

same concentration of defects. After normalization to the 111

maximum, the presence of defects causes lowering of the 220

peak and raising of the 311 peak compared with the case

without defects. However, each defect type has its own char-

acteristic, so a precise measurement can resolve the presence

of different types of defects. These ‘characteristic’ properties

are tiny for small concentrations of defects, so very precise

measurement of the diffraction curves and the background

should be performed in order to resolve the defect type.

4. Analysis of diffraction curve measured on dots withdefects

In this section we apply the Debye-formula approach to the

investigation of defects in Ge NCs in (Ge+SiO2)/SiO2 multi-

layers after annealing at different temperatures. We perform a

full-profile fitting of the measured curves using the equations

for diffraction intensities derived in the previous section. The

fitting parameters are the concentration of defects c, the mean

NC radius R0, the lattice parameter a and the r.m.s. dispersion

of NC radii �R.

The calculation procedure is as follows. First, we calculate

the positions of atoms in an ideal lattice Rid. We assume a

spherical shape of the NCs with radius R, and we extract the

calculated positions of atoms occurring inside the given

sphere. We then apply the equation for diffraction intensity,

i.e. we calculate interatomic distances Did from the given

positions Rid, add the corresponding defect-caused shift

DiSF,eSF,TB and calculate the sum over all mutual atomic posi-

tions. For the calculation of diffraction from NCs with a given

distribution of their radii, we use a model of shells similar to

that given by Cervellino et al. (2003), assuming a log-normal

distribution of the radii with mean value R0 and r.m.s.

dispersion �R.

The presence of defects causes shifts in the atomic layers

with respect to the ideal lattice layers lying inside the given

sphere, so that the resulting shape of the NCs after the

introduction of defects can be different from the ideal case.

However, the maximal shift is 2d, since we use mod(n, 3)d

instead of nd in calculating DiSF,eSF,TB. This maximal shift can

be further reduced by noting that 2d has the same effect as

�d, and the maximal shift is actually d. This value is smaller

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Figure 8Comparison of diffracted intensity calculated for Ge NCs without defectsand with defects of different types: intrinsic, extrinsic, twins and theircombination (denoted by ‘all’). The concentration of each type of defectis c = 0.2. For the calculation of the diffraction for the combination of alldefects, the same concentration of each individual defect type is assumed,cI,E,T = 0.2/3, i.e. c = 0.2 in total. The crystal shape is spherical with radiusR = 2.4 nm. The intensities are normalized to the height of the firstdiffraction peak (111). The inserts show the enlarged parts of thediffraction curves marked by squares.

than the lattice constant, and thus the resulting shape of the

NC after the introduction of defects can be very well

approximated by the sphere given for the ideal lattice posi-

tions.

The experimentally measured curves, after subtraction of

the background caused by the SiO2 amorphous matrix

(described in detail in x2), are shown in Fig. 9, together with

the fitted curves. From the figure, it follows that the model of

the defects describes well the measured diffraction curves. In

the fitting procedure we first used equation (21) for a

combination of all defects and assumed the same probability

of all defect types (cI = cE = cT = c). We also performed a fit

using equations for each type of defect separately, i.e. equa-

tions (10), (12) and (14). However, all the results were very

similar. This behaviour was expected, since for a low

concentration of defects (c < 0.11, as found by our fit), all

theoretical curves have similar properties (see Fig. 8).

The values of the parameters obtained by the fitting

procedures are summarized in Table 1 for all annealing

temperatures. The results show an increase in NC size with

increasing annealing temperature, in good agreement with

results obtained by other techniques (Buljan et al., 2009a). The

concentration of defects also increases with annealing

temperature, which is consistent with the HRTEM observa-

tions presented in x2. Determination of the densities of

particular defect types was not possible from the experimental

data obtained so far, owing to a relatively high noise level and

a relatively small concentration of defects, which did not

enable the detection of the very small differences character-

istic of each defect type. However, the average concentrations

for each annealing temperature were successfully determined.

The increase in defect density can be linked to the crystal-

lization kinetics during annealing. Our in situ investigations of

annealing-temperature dependence of Ge crystallization in

the same system (not presented here) show that rapid crys-

tallization of Ge occurs at temperatures above 973 K; below

this temperature, crystallization is very slow. As a result of the

kinetic limitation of atomic ordering during rapid crystal-

lization at higher temperatures, the probability of defects

appearing increases with increasing crystallization tempera-

ture. Our data indicate that a lower annealing temperature

should be used for the formation of defect-free Ge NCs.

From the data presented in Table 1 it also follows that the

average lattice constant for all samples is smaller than the

lattice constant of bulk Ge. The lattice constant is smallest for

the AS sample, where the smallest NCs are formed. On

increasing the size of the NCs with annealing temperature, the

lattice constant increases toward the lattice constant of bulk

Ge. This behaviour corresponds to the well known depen-

dence of the mean lattice parameter of an NC on its size; the

NC lattice parameter differs from its bulk value because of the

influence of surface tension (see Cherian & Mahadevan, 2008;

Palosz, 2007), and this effect increases with decreasing size of

the NC. The change in lattice constant with NC size reported

here cannot be ascribed to the presence of plane defects. Plane

defects may cause shifts in the angular positions of the

diffraction maxima (Balogh et al., 2006), but this effect is

already included in our model and small angular shifts of the

diffraction maxima follow from our simulations shown in

Figs. 7 and 8.

The results obtained for the sample annealed at 1073 K are

shown in Fig. 10 in more detail. There are two important

features that should be noted when the results are compared

with those obtained assuming no defects (see Fig. 3). Firstly,

the simulated curve better describes the experimental data,

especially the 111 and 311 peaks, and secondly, the parameters

obtained by the fit are much more reliable than those obtained

assuming no defects. The size distribution obtained from the fit

assuming defects (shown in the inset) agrees very well with the

size distributions obtained by the other three techniques,

which was not the case for the results obtained assuming no

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Table 1Structural parameters of nanocrystals obtained from the fittingprocedure.

R0 is the mean NC radius, �R is the dispersion of the NC radii, c is the defectconcentration and a is the lattice constant.

Ta (K) AS 873 973 1073

R0 (nm) 1.1 (1) 1.50 (9) 1.73 (8) 2.78 (6)�R (nm) 0.7 (4) 0.6 (3) 0.52 (8) 0.34 (5)c 0 (0) 0 (0) 0.03 (2) 0.07 (2)a (A) 5.50 (3) 5.591 (8) 5.613 (7) 5.643 (4)

Figure 9Experimental diffraction curves after subtraction of the SiO2 background(points) and their best fits to the theory (lines). The fitting parameters(mean NC radius, defect concentration and lattice constant) are given inTable 1. Annealing temperatures are indicated above each spectrum.

defects. This confirms the necessity of introducing defects into

a description of the diffraction intensity.

5. Discussion

The theoretical description developed in this paper is based on

the Debye-formula approach, in which the diffracted intensity

is calculated as a double sum over all atoms in the NCs.

Therefore, this method is applicable in practice only to small

NCs, their maximum size limited to a few nanometres by the

computation time and/or the size of the memory stack. For

larger NCs in excess of 5 nm, a special numerical method must

be used in order to speed up the calculation (Cervellino et al.,

2003, 2005), based on a fit of the full diffraction profile

(Cervellino et al., 2005). In this method the scattered ampli-

tude is expressed as the sum of the contributions of various

reciprocal lattice points, each contribution being proportional

to the Fourier transform of the crystal shape function. In

several papers, the method has been extended to crystals with

various types of defect [see the review by Ungar (2007),

among others], but this approach fails for smaller crystals

where the contributions belonging to different reciprocal

lattice points overlap.

An alternative method for the investigation of the structure

of NCs is based on the atomic pair-distribution function (PDF)

obtained by a sine Fourier transform of the measured

diffraction curve (Billinge, 2007, 2008; Petkov, 2008; Petkov et

al., 2008). For the purpose of this paper the application of the

PDF method was not necessary, since we did not investigate

atomic ordering in the NCs, and we do not determine the

atomic structure of the NCs ‘ab initio’. Our method is suitable

for determination of the densities of individual defect types

occurring in NCs when the structure of the defects is known a

priori from HRTEM. Therefore, the high real-space resolution

achieved by the PDF method with hard X-rays is not necessary

for our task. Moreover, the ‘standard’ analysis of the

measured diffraction curve makes it possible to distinguish the

contribution of the NCs from the diffuse background stem-

ming from the amorphous matrix. On the other hand, the PDF

contains the distances of the atomic pairs in both the NCs and

the matrix, which makes its analysis more difficult. Never-

theless, a comparison of the Debye-function and PDF

methods is an important task which we will address in the near

future.

6. Conclusion

We have developed a new procedure for the calculation of

X-ray diffraction from randomly oriented diamond-like

nanocrystals containing randomly placed stacking faults and

twin boundaries. The method is based on a direct (analytic)

averaging of the scattered intensity over a statistical ensemble

of all defect types, configurations and sizes of nanocrystals. We

have applied the method to the analysis of diffraction curves

from Ge nanocrystals embedded in multilayers with amor-

phous SiO2 spacer layers, and we have determined the mean

size of the nanocrystals and the densities of the defects as

functions of the temperature of post-growth annealing. We

have demonstrated that both the mean size of the nanocrystals

and the mean densities of the defects increase with increasing

annealing temperature.

APPENDIX ADerivation of equations for diffracted intensities

In this section we derive the equations for the intensity

diffracted from NCs containing intrinsic SFs, extrinsic SFs and

twins. The general expression for the diffracted intensity in the

case of any defect structure is

IðQÞ ¼ Aj f ðQÞj2�� P

i;j2�

expð�iQ �DijÞ

�def

�O

; ð26Þ

where Dij = R(h,k,l) � R(h0,k0,l0), and h iO and h idef denote

averaging over lattice orientations and defect positions,

respectively. To simplify the notation, we denote l � l0 by �L,

and we replace dsign(�L) by d in the formula showing the

derivation procedure.

A1. Intrinsic

We derive the expression for the intensity diffracted from

NCs containing intrinsic SFs. The relative position of the

atoms i and j (Dij) can be written as

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Figure 10The fit (red line) of the experimental data (black circles) obtainedassuming the presence of defects in the NCs. The inset shows thedistribution of NC radii obtained by the fit, compared with thedistributions obtained by TEM, GISAXS and Raman methods. The fitshows much better agreement with the experimental data than the fitassuming no defects, shown in Fig. 3. Also, the obtained size distributionagrees better with the results of the three other techniques than the sizedistribution obtained assuming no defects.

DiSFij ¼ ha1 þ ka2 þ la3

þPl

m¼0

Cmd�

�h0a1 þ k0a2 þ l0a3 þ

Pl0m0¼0

Cmd

�

¼ Didij þ

Pl0m¼lþ1

Cmdsignð�LÞ; ð27Þ

where the symbols were introduced in x3. First, we perform

the averaging over different positions of the SFs in the NC,

� Pi;j2�

expð�iQ �DijÞ

�def

¼P

i;j2�

exp��iQ �Did

ij

�exp

��iQ �

Pl0m¼lþ1

Cmd�

def

¼P

i;j2�

exp��iQ �Did

ij

�Pð0; j�Lj; cÞ

þ Pð1; j�Lj; cÞ expð�iQ � dÞ þ Pð2; j�Lj; cÞ expð�2iQ � dÞ

þ � � � þ Pðj�Lj; j�Lj; cÞ expð�ij�LjQ � dÞ�; ð28Þ

where P(n, l, c) is the probability function given by

equation (6).

We obtained the last summation of equation (28) by

replacing the sumPl

m¼l0 Cmd with all its possible values (0d to

|�L|d) weighted by the corresponding probability function

P(n, l, c). From equation (26) we obtain


�� Pi;j2�

expð�iQ �DijÞ

�def

�O

¼ Aj f ðQÞj2� P

i;j2�

Pj�Lj

n¼0

Pðn; j�Lj; cÞ exp��iQ �

�Did

ij þ nd��

O

:

ð29Þ

This last equation represents a sum of exponential functions

multiplied by the probability function P(n, l, c); the averaging

of its individual terms over all lattice orientations yields the

final expression


Pi;j2�

Pj�Lj

n¼0

Pðn; j�Lj; cÞ

sinc�QjDid

ij þ nsignð�LÞdj�

¼ Aj f ðQÞj2P

i;j2�

Pj�Lj

n¼0

PiSFn sinc

�QjDid

ij þDiSFijn j; ð30Þ

identical to equation (9).

In the last expression, we used the notation

DiSFijn ¼ nsignð�LÞd;

PiSFn ¼ Pðn; j�Lj; cÞ:

ð31Þ

A2. Extrinsic

In the case of extrinsic SFs, the relative position of atoms i

and j (Dij) can be written as

DeSFij ¼ ha1 þ ka2 þ la3 þ 2

Pl

m¼0

Cmd� Cl

�

�h0a1 þ k0a2 þ l0a3 þ 2

Pl0m0¼0

Cmd� Cl0

�

¼ Didij þ

�2Pl0�1

m¼lþ1

Cm þ Cl þ Cl0

�d: ð32Þ

Deriving the expression for the diffracted intensity, we use the

same procedure as for the case of intrinsic SFs, i.e. we first

average over SF positions hidef and then over different lattice

orientations hiO. The averaging over SF positions was divided

into two parts, firstly averaging over all Cm 6¼ Cl;Cl0 (denoted

by hiCm), and secondly averaging over Cl and Cl0 (denoted by

hiCl;C0l).

Both types of averaging follow the same procedure as was

used for intrinsic SFs

� Pi;j2�

expð�iQ �DijÞ

�def

¼P

i;j2�

exp��iQ �Did

ij

�exp

��iQ �

�2Pl�1

m¼l0þ1

Cm þ Cl þ Cl0

�d

��def

¼P

i;j2�

exp��iQ �Did

ij

�exp

��iQ � 2

Pl�1

m¼l0þ1

Cmd

��Cm

�exp½�iQ � ðCl þ Cl0 Þd�

�Cl;C

0l

¼P

i;j2�

exp��iQ �Did

ij

Pj�Lj�2

n1¼0

Pðn1; j�Lj � 2; cÞ

expð�i2n1Q � dÞP2

n2¼0

Pðn2; 2; cÞ expð�in2Q � dÞ

¼P

i;j2�

Pj�Lj�2

n1¼0

P2

n2¼0

Pðn1; j�Lj � 2; cÞPðn2; 2; cÞ

expf�iQ � ½Didij þ ð2n1 þ n2Þd�g: ð33Þ

Similar to the case for intrinsic SFs, the total intensity is a sum

of exponential functions multiplied by the corresponding

probability functions:

IðQÞeSF¼ Aj f ðQÞj2

�� Pi;j2�

expð�iQ �DijÞ

�def

�O

¼ Aj f ðQÞj2P

i;j2�

Pj�Lj�2

n1¼0

P2

n2¼0

Pðn1; j�Lj � 2; cÞPðn2; 2; cÞ

sinc�QjDid

ij þ ð2n1 þ n2Þdsignð�LÞj�

¼ Aj f ðQÞj2P

i;j2�

Pj�Lj�2

n1¼0

P2

n2¼0

PeSFn1;n2

QjDidij þDeSF

ijn1n2j

� ; ð34Þ

where we have introduced the notation

DeSFijn1n2¼ ð2n1 þ n2Þdsignð�LÞ;

PeSFn1;n2¼

Pðn1; j�Lj � 2; cÞPðn2; 2; cÞ; �L � 2,

Pðn2; 2; cÞ; �L < 2.

ð35Þ

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A3. Twins

The interatomic distances DTBij in NCs with twin boundaries

can be obtained using a similar procedure as before:

DTBij ¼ Did

ij þ

� Pl0�1

p¼l

ð�1ÞPpm¼0

Cm � j�Lj

�d

¼ Didij þ ð�1ÞC0þC1þ��þCl0

�1þ Sðl0 þ 1Þ � j�Lj

�d; ð36Þ

where Sðl0 þ nÞ ¼�ð�1ÞCl0þn þ ð�1ÞCl0þnþCl0þnþ1 þ � � � þ

ð�1ÞCl0þnþCl0þnþ1þ��þCl�

is used to simplify notation. For the

derivation of the diffraction intensity we use the same

procedure as for stacking faults, i.e. averaging over positions of

defects, followed by averaging over lattice orientations.

The averaging over defect positions hidef ¼ hiC0...Cl0 ...Clis

performed in two steps. First, we average over C0 . . . C0l, where

we use the fact that ð�1ÞC0þC1þ��þCl0 = 1 if the sum in the

exponent is even, and �1 otherwise:� Pi;j2�

expð�iQ �DijÞ

�def

¼

�� Pi;j2�

expð�iQ �DijÞ

�C0...C0

l

�Cl0þ1...Cl

¼P

i;j2�

exp��iQ �

�Did

ij � j�Ljd��

Pe expf�iQ � d½1þ Sðl0 þ 1Þ�g

þ Po expfþiQ � d½1þ Sðl0 þ 1Þ�g�Cl0þ1...Cl

; ð37Þ

where Pe and Po denote the probabilities that the sum

C0 þ C1 þ � � � þ Cl0 is even or odd, given by equation (18).

Second, we average over other coefficients Cl0þ1 . . . Cl, where

we derive a recursive relation for the probability coefficients

and the corresponding phase shifts. Here we use the relation

hexp½iQ � dSðl0 þ nÞ�iCl0þn¼ c expf�iQ � d½1þ Sðl0 þ nþ 1Þ�g

þ q expfiQ � d½1þ Sðl0 þ nþ 1Þ�g; ð38Þ

where q = 1 � c.

Thus, we first average equation (38) over Cl0þ1, then over

Cl0þ2 and so on to Cl. After averaging over all Cm we obtain the

following formula for hidef:� Pi;j2�

expð�iQ �DijÞ

�def

¼

Pi;j2�

Pn3¼0;1

Pl

n4¼0

Tn3;n4exp½�iQ � dðFn3;n4

� j�LjÞ�; ð39Þ

where Tn3;n4is the n4th element of the probability vector T+

(or T�) if n3 = 0 (or n3 = 1), and Fn3;n4is the corresponding

element of the phase-shift vector for which the same notation

rule applies as for Tn3;n4. The vectors T and F can be

obtained by the following recursive formulae:

Tjþ ¼ ½ð1� cÞT

j�1þ ; cTj�1

� �;

Tj� ¼ ½cTj�1

þ ; ð1� cÞTj�1� �;

Fjþ ¼ ½F

j�1þ þ 1;Fj�1

� þ 1�;

Fj� ¼ ½F

j�1þ � 1;Fj�1

� � 1�;

ð40Þ

where the index j denotes the matrix obtained for the jth

iteration and runs from 1 to |�L| � 1. The starting (j = 0)

vectors for the recursive relation are the 1 1 vectors

(scalars)

T0þ ¼ Pe;

T0� ¼ Po;

F0þ ¼ 1;

F0� ¼ �1:

ð41Þ

The final expression for the intensity is obtained by averaging

over the lattice orientations in an analogous way to the

intrinsic and extrinsic SFs:

IðQÞTB¼ Aj f ðQÞj2

�� Pi;j2�

expð�iQ �DijÞ

�def

�O

¼ Aj f ðQÞj2P

i;j2�

Pn3¼0;1

P2j�Lj

n4¼0

Tn3;n4sinc

�QjDid

ij þDTBijn3n4j:

ð42Þ

Here,

DTBijn3n4¼ ðFn3n4

� j�LjÞsignð�LÞd: ð43Þ

This work was supported by the Ministry of Science,

Croatia. The authors are grateful to Medeja Gec for preparing

samples for TEM measurements and Aleksa Pavlesin for

assistance during the sample preparation. MB acknowledges

support from the National Foundation for Science, Higher

Education and Technological Development of the Republic

Croatia. VH acknowledges support from the Ministry of

Education of the Czech Republic (project No.

MSM0021620834) and from the NAMASTE project (the

Framework Seven EU programme). GD acknowledges the

support of the Slovenian Research Agency (grant No. P2-

0084).

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crystal structure of defect-containing semiconductor nanocrystals – an x-ray diffraction study

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