The distribution of the intensity in reciprocal space scattered in a GISAXS experiment can be calculated by the distorted-wave Born approximation (DWBA). In this approach one divides the sample into two parts – a non-disturbed system and the disturbance. The scattering from the non-disturbed system is calculated exactly (i.e. using the multiple-scattering dynamical theory), whereas the disturbance scatters only kinematically. This approach is very frequently used; however its validity has to be discussed and confirmed in any particular system. Generally speaking, the DWBA approach is applicable if multiple scattering from the disturbance can be neglected. In the case of quantum dots arranged in a three-dimensional matrix embedded in a semi-infinite medium, one usually considers the medium as the non-disturbed system and the ensemble of the quantum dots as the disturbance.
In the following we assume that the dots are fully buried in an amorphous semi-infinite substrate with an ideally flat surface (i.e. the influence of the surface roughness is neglected). The reciprocal-space distribution of the wave scattered from the substrate exhibits an infinitely narrow rod-like maximum along the surface normal (crystal truncation rod, CTR) and the intensity distribution along the CTR is determined by the specular reflectivity of the substrate.
In the following, we neglect this wave and consider only the wave scattered from the dots. The reciprocal-space distribution of the wave scattered from the dots is
In this formula
is a constant,
is the difference in the electron densities of the dot material and the surrounding matrix,
are position vectors of the dots,
is the scattering vector (the difference of the wavevectors of the scattered and incident beams),
is the complex scattering vector corrected to refraction at the vacuum–substrate interface (for details see Renaud et al.
is the Fourier transformation of the shape function
of a dot occurring in position
; the shape function is unity in the dot volume and zero outside it.
are the Fresnel transmittivities of the substrate surface corresponding to the primary and scattered waves, respectively; the factor
exhibits a maximum (so-called Yoneda wing) if the incidence angle
and/or the exit angle
equal the critical angle
of total external reflection.
brackets in equation (1)
denote the averaging over the positions and shapes of the quantum dots. In order to calculate this averaging, one has to assume how the dot sizes are connected with their positions. In the literature, two limiting approaches can be found (Renaud et al.
). The decoupling approximation (DA) assumes that the sizes of the dots are not statistically correlated with their positions (Guinier, 1963
). Strictly speaking, this approximation is valid only in very diluted systems; usually it is reasonable to assume that the distance between larger dots is on average larger than between smaller dots. The local monodisperse approximation (LMA) assumes that the sample is divided into domains, each domain containing dots of a given size and given distribution of the distances (Pedersen, 1994
; Renaud et al.
). In each domain one calculates the average over the dot positions and finally the averaging over the domains is carried out. In this paper we will restrict ourselves to the DA only.
Within the DA, the averaging indicated in equation (1)
is straightforward. After some algebra one obtains
Here we have denoted
this function equals the number
of the QDs if we neglect the imaginary part of the scattering vector
. The function
is the correlation function of the dot positions; the averaging here is performed only over the dot positions. In the following we denote
The main goal of this paper is to formulate physically relevant models of the positions of the quantum dots, from which we can calculate the correlation function
. As we emphasized in §1
, both SRO and LRO approaches are used. Within SRO, the position of a given dot is affected only by the positions of the neighbouring dots, while LRO assumes that the dots randomly deviate from pre-defined periodic ideal dot positions. In the following, we will derive the correlation functions for one-dimensional and two-dimensional dot arrays arranged within the SRO and LRO models, and finally we present the correlation function of a three-dimensional dot ensemble; for this case we will use a combination of the SRO and LRO models.
3.1. One-dimensional SRO model
Let us start with a one-dimensional chain of quantum dots along the
axis and we index the dots by the integer index
. The position of the dot with index
with respect to the origin is denoted by
which can be expressed as a sum of random connection vectors
or in terms of basis vectors
of one-dimensional ideal (undisturbed) lattice and deviation vectors
denotes the total deviation of a dot with index
from its ideal position. The mean value of the connection vectors
. We assume that
are statistically independent.
A direct calculation of equation (4)
yields the one-dimensional correlation function in the form
and we have neglected absorption effects, so
. Absorption will be introduced in the three-dimensional model. In equation (7)
denotes the number of the coherently irradiated dots; if this number is very large (i.e.
if the mean dot distance is much smaller than the size of the coherently irradiated sample surface), one can use the limiting expression for
contains the undisturbed positions of the dots, while the function
depends on the statistical distribution of the deviation vectors
. We have assumed that the components
of the random deviation
are normally distributed with zero mean and root mean square (r.m.s.) dispersion
Fig. 2 presents examples of the calculated correlation functions for
. In panel (a
) of this figure we plotted the values of
axis parallel to
. The correlation function exhibits maxima (satellites) in the points
is an integer (satellite order). The full width at half-maximum (FWHM) of the zero satellite is
; in the limiting case in equation (7)
the central peak is infinitely narrow (
-like). For finite
, the central maximum is accompanied by tiny fringes with the period of
. Since the degree of coherence of the primary beam usually continuously decreases from unity to zero, these fringes are not observed and in the following they are removed by averaging the correlation function over various N
’s. This averaging does not affect the shape of the non-zero satellites. The FWHMs of the non-zero satellites depend almost quadratically on the satellite order
) displays the one-dimensional correlation function
as a function of two components
of the scattering vector. In the reciprocal
plane the correlation function exhibits a streak along the
axis, with increasing
the streaks become broader and weaker. Here we have neglected refraction and absorption to keep the focus on the ordering properties. Thus, for this case,
. Refraction and absorption effects will be introduced later in three-dimensional models.
3.2. One-dimensional LRO model
A one-dimensional system of QDs can be described by an LRO model if the positions of QDs fluctuate independently around their pre-defined
(ideal) positions. Thus, within the LRO model, the position
th dot can be expressed as
where random vectors
describe the deviation of the dot from its ideal position. Within the SRO model, the position of the dot with index
was defined with respect to the position of the dot with index
, so the total deviation from the undisturbed position increases with
. Thus, the main difference between SRO and LRO models is the total deviation vector of the dot
with respect to the origin:
for the SRO model while
for the LRO model.
Assuming that vectors
are statistically independent we obtain the correlation function for the LRO model,
are defined in equation (8)
Fig. 3 compares the correlation function of one-dimensional chains of QDs arranged in LRO and SRO models. Analogously to the SRO model we assumed that the random deviations
have zero average values and their components are normally distributed, while different components of
are statistically independent. In contrast to the SRO model, the widths of the correlation peaks in the LRO do not depend on the r.m.s. deviation
and they are inversely proportional to the size
of the coherently irradiated chain. By increasing the disorder in the dot positions, the diffuse part of the correlation function between the maxima increases.
Comparison of correlation functions of one-dimensional SRO and LRO models calculated with the same parameters as in Fig. 2.
3.3. Two-dimensional models
The construction of a physically sound two-dimensional SRO model is not a straightforward task. One possible approach (the IPM; Eads & Millane, 2001
) assumes that each dot is labelled by two indexes
and its position vector can be written as
two types of the connection vectors
are assumed with the mean values
Therefore, the IPM assumes that the dots occupy the points of a disordered two-dimensional lattice with the lattice vectors
After simple calculation we obtain the following expression for the two-dimensional correlation function,
are the one-dimensional correlation functions described in equation (7)
, in which the functions
are replaced by
) shows the positions of the dots generated randomly using the IPM and normal distribution of the deviations
; in the simulation we used the values
nm. The corresponding correlation function is plotted in Fig. 4(b
). The satellite maxima of the correlation function lie in the points of a lattice reciprocal to the lattice generated by the vectors
; the FWHMs of the maxima increase with the satellite orders.
Figure 4 (a) Positions of the dots randomly generated using the two-dimensional ideal paracrystal model (IPM). The inset displays the histogram of the nearest dot distances, the pair of short black lines denote the vectors . The parameters of the correlation are (more ...)
The IPM is not fully applicable if the dots are created by a self-organization process resulting in a random lattice, since the IPM assumes the existence of an a priori
defined ideal lattice with the basis vectors
. This is illustrated in Fig. 5, where we have plotted the positions of the dots generated randomly assuming that the random nearest dot distances obey the Gamma distribution with the given mean
and given r.m.s. dispersion
. The simulation has been carried out using the Monte Carlo (MC) accept–reject sampling method described by Robert & Casella (2004
Figure 5 (a) Positions of quantum dots randomly generated using a given distribution of nearest distance and an accept–reject method; we used the same mean distance and the r.m.s. deviation as in Fig. 4. The inset shows the actual distribution (more ...)
Comparing Figs. 4(a) and 5(a) it is obvious that, in contrast to the IPM, the array of randomly generated dots does not exhibit any pre-defined lattice directions, in spite of the fact that the distributions of the nearest dot distances are very similar (see the insets in Figs. 4
a and 5
a). The correlation function of the randomly generated array of dots is isotropic (see Fig. 5
b) and no distinct satellite maxima in reciprocal-lattice points are visible.
In Fig. 6 we compare the radial profile
of this correlation function with the radial profile of the correlation function
(plotted in Fig. 4
) averaged over all azimuthal directions of the vector
. The dashed line denotes the azimuthally averaged function
which was calculated for the same value
nm as that used by the MC simulations in Fig. 5(a
); obviously the maxima in this correlation function are much narrower than those following from the MC simulation. In order to get a good match of both radial correlation functions, we have to increase the
value of the IPM model to
nm (unbroken line). From Fig. 6 it follows that the correlation function of the IPM azimuthally averaged over all directions of the scattering vector
is a good approximation of the correlation function of a two-dimensional SRO model generated by an MC simulation, in which the directions of the connection vectors
are isotropically distributed; however, one has to use an approximately two times larger r.m.s. dispersion of the dot distances in the IPM model.
Figure 6 The radial correlation function of the two-dimensional SRO model obtained by numerical Monte Carlo method (dots) using nm, and the azimuthally averaged correlation functions of the IPM model with nm (dashed line) and 4 nm (unbroken (more ...)
3.4. Three-dimensional models
In the previous sections we constructed the one- and two-dimensional SRO models as well as an LRO model of the positions of quantum dots and we calculated the corresponding correlation function. The next step, i.e. the definition of a three-dimensional model, depends much on the mechanism of the ordering of the quantum dots during their nucleation and growth. In the following, we formulate three various three-dimensional models realized by different experimental recipes and compare the theoretical descriptions with experimental results.
For all systems we assume that the quantum dots create a disordered three-dimensional lattice with the averaged basis vector
. Each dot is labelled by three indexes
and its position is given by
are the random displacement vectors, describing the deviation of the dot position from the ideal position from the origin corresponding to the basis vectors
The SRO and LRO models differ in the definition of the displacement vectors as was shown in §§3.1
The geometry used for the description and modelling of GISAXS intensity distributions is schematically shown in Fig. 7. The primary X-ray beam lies in the
plane (plane of incidence) and makes a small angle
(angle of incidence) with the
axis (Fig. 7
). All experimental GISAXS maps were taken with
= 0.2°, i.e.
very close to the critical angle
of total external reflection. In the actual experimental arrangement the detector plane was perpendicular to the primary beam; however, for the sake of simplicity we calculate the intensity distribution in the reciprocal
plane perpendicular to the sample surface. The distortion of the intensity map due to the angle
of the detector plane with the
plane is negligible.
lie in the plane parallel to the substrate (
plane), while the direction of the vector
corresponds to the direction of the correlation of the positions of the dots belonging to different periods of the multilayer. The
] corresponds to the multilayer period. Thus, the coordinates of the basis vectors
The choice of the basis vectors is based on the growth process of the samples. The diffusion and growth properties are usually similar in the plane parallel to the substrate, while they are different in the growth direction (assumed perpendicular to the substrate). However, the models developed are generally valid for any choice of the basis vectors. We will use two configurations in the simulations of GISAXS intensity distributions, namely assuming that (i) the probing beam is parallel (
, Fig. 7
) and (ii) perpendicular (
, Fig. 7
) to the common plane of
and the surface normal.
The absorption effects are included in the three-dimensional model via
the imaginary part of the complex scattering vector
. For the chosen geometry,
only for the
, while the parallel components are real and equal to those in vacuum
. To keep the formulas as simple as possible, we neglect the absorption in the distances comparable to the deviations
of the dots from their ideal positions. Then, the functions
defined in the previous section contain only the real part
of the scattering vector
The total intensity [equation (1)
] in the three-dimensional case is given by
is the number of the dots along the basis vector
is given by the product of three one-dimensional correlation functions. The functions
are defined in §3
In the three-dimensional models discussed later we will treat separately the
components of the random vectors
to have the generally valid formulas. This is necessary because deviations around ideal positions are not necessarily isotropic; their r.m.s. deviations may be different in different directions, for example in the case of nucleation on pre-patterned substrates. Another reason is that the ordering type may be different for different components of the same basis vector (SRO or LRO), as in the case of the multilayer stack which is described by the LRO model, while the basis vector
is not perpendicular to the multilayer surface. All these cases will be shown in the specific models given below. Thus we deal in total with three components of three deviation vectors (nine in total), and we assume that the components
are statistically independent with zero means and r.m.s. dispersions
. Therefore the functions
) can be written as a product of three components:
The components are given by
In the following we consider three specific cases (models) differing in the type of QD ordering.
3.5. Model 1
Model 1 describes a system of QDs with the same type of ordering along all three average basis vectors [
]. If the QD positions along all basis vectors obey SRO ordering, this model is suitable for the description of QD systems formed by a self-assembly process with no external constraints. Such systems may be realized by arrays of QDs formed by self-ordered growth in thick homogeneous layers (Buljan, Pinto et al.
) or in multilayers where the layer sequence can be described by the SRO model.
The correlation function
for this case is a generalization of the two-dimensional SRO ideal paracrystal model, i.e.
it is a product of three one-dimensional SRO correlation functions:
are given by equation (7)
differs slightly from
because absorption effects are included in it via
the imaginary part of the
component of the scattering vector
Using correlation function
and equation (20)
, we have simulated the two-dimensional GISAXS intensity distributions. The simulations are shown in Fig. 8. The simulations are performed for various parameters of the disorder. The QDs are assumed to be spherical and arranged in a rhombohedral lattice with the basis vectors given in Table 1, along with the parameters of the disorder and dot sizes. Two types of intensity sheets (indicated by the lines in Fig. 8
) may be distinguished in the GISAXS simulations shown in Fig. 8. The first type are the sheets (streaks) placed parallel to the
axis. These sheets are the consequence of the correlation of the QD positions within the plane parallel to the substrate (in-plane correlation). They become broader and weaker with increasing
, and their FWHMs also increase with growing in-plane components of the in-plane disorder, i.e.
. This is visible in Figs. 8(a
). The effect of the increase in the vertical component of the in-plane disorder [
] causes a decrease in intensity and a lateral broadening of the sheets with an increase in
(see Figs. 8
Sets of parameters (P1–P12) used for the simulations of the GISAXS intensity maps
The second type of sheets are the tilted ones. They appear as a result of the correlation in the QD positions corresponding to different layers. The influence of the increase in the lateral [
] and vertical [
] disorder on this type of sheet is illustrated in Figs. 8(g
) and 8(j
), respectively. The increase in the lateral disorder causes a broadening and weakening of the correlation peaks in the
direction, while the increase in the vertical disorder broadens the sheets along
. In summary, for model 1, in which all the disorder components are described by SRO, all correlation peaks broaden with the increase in the degree of disorder.
The simulations shown in Fig. 8 are obtained for the perpendicular geometry with no averaging of the azimuthal directions of
. As stated previously (see §3.3
), this case may be successfully used for systems where some pre-defined direction of the basis vectors exists. But, for systems with no pre-defined direction or with domains randomly rotated around the normal to the surface, the azimuthal averaging (over all rotations of basis vectors around the
axis) should be performed (see Fig. 7). An example showing simulation of the azimuthally averaged intensity distribution (using the parameter set P8) is shown in Fig. 9. The influences of the parameters on the peak profiles follow the same rules as in the non-averaged system (Fig. 8).
Simulation of two-dimensional GISAXS intensity map obtained using model 1 and azimuthal averaging for the set P8 of the disorder parameters. The intensity scale is the same as in Fig. 8.
An example of the application of this model to self-assembly of Ge quantum dots in continuous thick Al
film is shown in the next section.
3.6. Example 1: self-assembly of Ge quantum dots in an alumina matrix
Here we present an example showing the application of model 1 for the description of Ge QD lattices produced by magnetron sputtering deposition of a continuous Ge+Al
layer at 773 K on a flat substrate. Owing to the elevated deposition temperature QDs form during the layer growth. The dots formed during the deposition affect the shape of the growing surface, which incites a self-organization process during the layer growth. The result of the deposition is the formation of domains of QDs that are ordered in a three-dimensional tetragonal lattice.
The formed dot lattice is schematically presented in Figs. 10(a
) while the experimentally measured STEM cross section of the film is shown in Fig. 10(c
). The domains are randomly rotated with respect to the surface normal. More details about the origins of self-assembly in this kind of film are given in Buljan, Pinto et al.
). The nature of the deposition process indicates that the ordering in all directions can be described by the SRO model: the substrate used for the deposition is isotropic and flat and it actually does not influence significantly the QD ordering. On the other hand, a single continuous film is deposited, so there is also no reason for a long-range ordering in a direction perpendicular to the surface, which would be the case for a regular multilayer.
Experimentally measured and simulated GISAXS maps of this sample are shown in Figs. 10(d
) and 10(e
), respectively. The positions of the lateral maxima in the measured map do not depend on the azimuthal direction of the primary X-ray beam. This means that the regular ordering appears in domains that are randomly azimuthally rotated. The same follows from the STEM images of the film. Therefore, the simulation of the experimentally measured map was performed by averaging of equation (20)
over all azimuthal orientations of the basis vectors. The parameters used for the simulation are given in Table 2. In the fitting procedure of the GISAXS data we assumed that some components of the r.m.s. deviations
are equal because of the sample symmetry, i.e.
The indexes L and V in equation (26)
are used to describe disorder of the longitudinal (parallel to the substrate) and vertical (perpendicular to the substrate) components of the basis vectors, respectively. The first index refers to the basis vector described, and the second one to the deviation vector. Thus, σLV
describes the vertical deviation of the in-plane basis vectors a
. Model 1 is valid for this sample, since the parameters obtained are in good agreement with those from STEM, which are also given in Table 2.
Sets of parameters obtained by fitting the experimentally measured GISAXS intensity maps and determined from STEM cross sections for examples 1–3
3.7. Model 2
Model 2 describes a three-dimensional QD array where the QDs are ordered according to the long-range-order model along the basis vector
, and the short-range ordering occurs in the other directions. This model is suitable for the description of QDs arranged in a multilayer, where the long-range ordering along
is induced by a process defining ‘ideal’, i.e.
non-disturbed, positions of the dots. Such a process may be ion beam irradiation of a multilayer (Buljan, Bogdanović-Radović et al.
), or regular patterning of the substrate in one direction. In Buljan, Bogdanović-Radović et al.
) we have shown the ordering of the positions of Ge quantum dots in a (Ge+SiO
multilayer achieved by a post-growth irradiation of a multilayer by ion beam. The points where the tracks of individual ions cross the Ge-rich layers represent the ideal positions of the Ge quantum dots. Therefore, the position of the
-th dot can be expressed by equation (15)
are the random lateral displacements of the dots obeying the SRO model, and the random displacements
are defined with respect to the ‘ideal’ positions
. In the multilayer sample mentioned above, the vertical component
equals the multilayer period and the direction of the basis vector
is defined by the direction of the irradiating ions.
In this case, the correlation function equals a product of two one-dimensional SRO correlation functions and one one-dimensional LRO correlation function,
are given by equation (7)
is the one-dimensional correlation function of the LRO model including absorption [see also equation (12)
Fig. 11 shows simulated GISAXS maps obtained for the same sets of the disorder parameters P1–P12 as in model 1.
Figure 11 Simulations of GISAXS intensity distribution maps obtained from QD lattices described by model 2. The simulations show the dependence of the intensity distribution on the degree of disorder. (a)–(c) Influence of . (d)–(f) Influence of (more ...)
The properties of the lateral correlation sheets (stemming from the in-plane correlations) are the same as those for model 1: the sheets broaden in the
direction with the increase in
. However, the properties of the correlation sheets coming from the ordering along
are different from those shown for model 1. The most important feature is the width of these sheets, which is constant in the direction perpendicular to the direction of
. The increase in the disorder parameters
causes a decrease in their intensities in the directions of
, respectively, but the widths remain constant. This feature is a consequence of the LRO model assumed along
However, the width of the sheets increases with decreasing
. This effect is illustrated in Fig. 12.
Figure 12 Simulations of two-dimensional GISAXS intensity maps obtained using model 2 with various values of indicated in the figure, and the set of disorder parameters P8 given in Table 1. The intensity scale is the same as in Fig. 11.
Azimuthal averaging for all three basis vectors in the systems described by model 2 (see Fig. 13
) is not common, since the LRO model assumes the existence of a pre-defined direction (given by the basis vector
, in our case). However, within this model, the azimuthal averaging can be carried out with respect to the basis vectors
only. Therefore, the QDs make LRO-ordered chains along
, but the ordering of the chains in the plane parallel to the substrate should be averaged over all azimuthal orientations of
. This case is shown in Fig. 13(b
). The lateral sheets parallel to the
axis, visible in Fig. 13, are the consequence of the in-plane correlations of the QD positions. The width of these sheets is slightly broader when compared with the non-averaged case (see Fig. 11
). This is expected because we ‘see’ different projections of basis vectors
due to the azimuthal averaging.
Figure 13 Simulations of two-dimensional GISAXS intensity maps obtained with model 2 and azimuthal averaging for the set of disorder parameters P8. (a) The azimuthal directions of all vectors are included in the azimuthal averaging; (b) only the azimuthal directions (more ...)
The application of model 2 to the analysis of GISAXS maps experimentally measured on the ordered QD array produced by ion beam irradiation is given in the next section.
3.8. Example 2: quantum dot lattices formed by ion beam irradiation
An example of a QD arrangement that can be described by model 2 is a (Ge+SiO
multilayer irradiated by oxygen ions and subsequently annealed. Owing to the ion beam irradiation, QDs are formed along the traces of individual ions (Buljan, Bogdanović-Radović et al.
). We choose the basis vector
to be directed along the traces. The positions of the traces in the lateral
plane can be described by the SRO model and for the description of the lateral positions of the traces we use the basis vectors
. The total intensity is obtained after azimuthal averaging of the basis vectors
, while the third basis vector
is kept fixed. The schematical view of the QD arrangement is shown in Figs. 14(a
), while the STEM image of the film cross section is shown in Fig. 14(c
). The GISAXS maps of the same system measured in parallel
geometries are shown in Figs. 14(d
) and 14(e
In the perpendicular configuration, the sheets stemming from the ordering along
are perpendicular to
they are tilted; the tilt angle with respect to the surface normal equals the angle of
with the surface. In the parallel configuration, the sheets are parallel to
The simulations of the measured GISAXS maps are shown in Figs. 14(f
). The simulations are performed using the azimuthal averaging of the basis vectors
(see Fig. 13
). We have fitted the model parameters to the experimentally measured GISAXS maps. The resulting parameters are in very good agreement with those obtained from the STEM image (see the numerical values in Table 2).
3.9. Model 3
Model 3 is designed for the description of QD arrays where QDs are long-range ordered along a direction different from the direction of any basis vector (say in the
direction), while the ordering in all other directions obeys the SRO model. Thus, the arrangement along the basis vectors is of a ‘mixed’ nature – the lateral components of the random displacements
obey the SRO model, while the vertical components
are arranged according to the LRO model. Model 3 is applicable if the dots occur in a multilayer, where the vertical periodicity of the multilayer imposes the ‘ideal’ vertical components of the dot position vectors. The position vector of a dot with indexes
The correlation function for this model is
are given by
Here we have denoted
The simulations of the GISAXS maps for various disorder degrees are shown in Fig. 15. The behaviour of the sheets caused by the in-plane ordering is the same as in models 1 and 2. However, the width of the sheets corresponding to the correlation of the positions in different layers is different. In accordance with the ‘mixed’ nature of the correlation function [equation (33)
], the width of the streaks increases along
. However the width in the
direction is constant, but the intensity decreases if the disorder parameter
increases. However, if
is sufficiently small, models 1 and 3 yield very similar results.
The influence of the azimuthal averaging on the results of model 3 is shown in Fig. 16. Similarly to model 1, azimuthal averaging makes the GISAXS intensity distribution symmetric with respect to the
= 0 axis. Also, it is not sensitive to the azimuthal orientation of the probing beam with respect to the sample. This is expected due to averaging over all possible azimuthal orientations. The peaks visible in Fig. 16 are broader than for the non-averaged case (Fig. 15).
Simulated GISAXS intensity map from the QD lattices described by model 3 after azimuthal averaging. The set P8 of the disorder parameters is used. The intensity scale is the same as in Fig. 15.
An example showing the application of this model to a QD lattice produced by self-ordered growth on a flat substrate is given in the next section.
3.10. Example 3: quantum dot lattices formed by self-assembly on a flat substrate
Here we show the application of model 3 for the simulation of the GISAXS maps of QD lattices formed by self-assembled growth of Ge QDs in an amorphous SiO
matrix. The samples are produced by magnetron sputtering of 20 (Ge+SiO
bilayers on a flat Si(111) substrate (Buljan, Desnica et al.
). The multilayer is periodic, i.e.
the vertical distances of the QDs follow the long-range-ordering model. The deposition was performed at an elevated substrate temperature making possible the self-assembly of the dots. The resulting lattices of quantum dots have a rhombohedral face-centred-cubic-like structure. The ordered regions appear in domains randomly rotated around the surface normal. The arrangement of the QDs in a domain is schematically shown in Figs. 17(a
), while the STEM measurement of the film cross section is shown in Fig. 17(c
). The measured GISAXS intensity distribution is shown in Fig. 17(d
). The intensity distributions are not sensitive to the azimuthal direction of the X-ray probing beam (see Fig. 7).
The analysis of the measured maps is performed using model 3. Additionally we have performed an azimuthal averaging of the calculated intensity, to include the effect of randomly oriented domains. The parameters of the QD lattices and sizes of the QDs, obtained by fitting of the measured GISAXS maps to the theoretical maps, are shown in Table 2. Examples of measured and simulated GISAXS maps, using the parameters obtained by the fit, are shown in Figs. 17(d) and 17(e), respectively.
Model 3 has also been successfully applied to the description of the ordering of Ge quantum dots deposited on rippled Si substrates (Buljan, Grenzer, Keller et al.
), SiGe multilayers (Pinto et al.
) and ordering of Ge QDs in an Al
matrix (Buljan, Radić et al.