**6. Mass transfer between clusters under dislocation-surface diffusion. Size distribution function**

Obtaining nanocrystals meeting the requirements raised to quantum dots by applying the conventional techniques, such as selective etching, growth at profiled substrate, chemical evaporation, condensation in glass matrices, crystallization under ultrahigh rate of cooling or annealing of amorphous matrices has not led to desirable results (Ledentsov *et al*., 1998; Pchelyakov *et al*., 2000). And only under the process of self-organization in semiconductor heteroepitaxial systems it occurs possible to form ideal heterostructures with quantum dots.

The technique of heteroepitaxial growing in the Stranski-Kastranov regime (Krastanow & Stranski, 1937) is the most widely used for obtaining quantum dots. In this case, layer-wise growth of a film is replaced, due to self-organization phenomena, by nucleation and following development of nanostructures in form of volume (3D) islands (Bimberg & Shchukin, 1999; Kern & Müller, 1998; Mo *et al.,* 1990). Islands with spatial limitation of charge carriers in all three directions are referred to as quantum dots. Quantum dots obtained in such a way have perfect crystalline structure, high quantum efficiency of radiation recombination and are characterized by enough high homogeneity in size (Aleksandrov *et al.,* 1974; Leonard *et al*., 1993; Moison *et al*., 1994; Ledentsov *et al*., 1996a, 1996b). Sizes of quantum dots can vary from several nanometers to several hundred nanometers. For example, size of quantum dots in heteroystems Ge-Si and InAs-GaAs lies within the interval from 10 nm to 100 nm, with concentration 1010÷1011 cm -2.

Much prominence is given in the literature to the size distribution function of islands, while this parameter of a system of quantum dots is of high importance in practical applications

technique at temperature 500ºС (Yakimov *et al*., 2007). The experimentally obtained histogram in Fig. 17,*а* corresponds to one layer of nanoclusters of Ge of a main size ~10.4 nm. One can see that for 0.6 *x* = theoretical results are well fitting the experimental data. For two layers of nanoclusters (with a mean size ~10.7 nm), cf. Fig. 17,*b*, ripening of nanoclusters is almost entirely determined by surface diffusion. The diffusion flow *Sj* constitutes about

It is of especial interest from the theoretical point of view to compare the computed dependences and experimentally obtained histograms illustrated in Fig. 18 (Shangjr Gwo *et al*., 2003). Nanoclusters of Со at Si3N4 substrate were obtained by applying the evaporation technique at room temperature with rate (0.3-1.2) ML/min. Histograms shown in Fig. 18

As opposed to heterostructures Ge/Si (001) and Ge/ SiO2 on the base of quantum dots of Ge, which are widely used in optoelectronics and microelectronics, Со is not semiconductor, and the system Со/Si3N4 is the model one for investigation of regularities of forming defect-

However, one can see from Figs. 16, 17, and 18 that the regularities of the Ostwald's ripening are the same both for the clusters of semiconductor, Ge, and for metallic clusters of Со. In both cases, irrespectively of metallic or semiconductor nature of clusters, ripening of them is governed by the combined mechanism of growth, i.e. the diffusion and the Wagner's ones, with predomination, in the resulting flow, of the flow *Sj* due to surface diffusion. It proves generality of the considered by us mechanism of cluster ripening, when the rate of growth of them is determined by the ratio of the diffusion flow, *Sj* , to the flow *ij*

**6. Mass transfer between clusters under dislocation-surface diffusion. Size** 

within the interval from 10 nm to 100 nm, with concentration 1010÷1011 cm -2.

Much prominence is given in the literature to the size distribution function of islands, while this parameter of a system of quantum dots is of high importance in practical applications

Obtaining nanocrystals meeting the requirements raised to quantum dots by applying the conventional techniques, such as selective etching, growth at profiled substrate, chemical evaporation, condensation in glass matrices, crystallization under ultrahigh rate of cooling or annealing of amorphous matrices has not led to desirable results (Ledentsov *et al*., 1998; Pchelyakov *et al*., 2000). And only under the process of self-organization in semiconductor heteroepitaxial systems it occurs possible to form ideal heterostructures with quantum dots. The technique of heteroepitaxial growing in the Stranski-Kastranov regime (Krastanow & Stranski, 1937) is the most widely used for obtaining quantum dots. In this case, layer-wise growth of a film is replaced, due to self-organization phenomena, by nucleation and following development of nanostructures in form of volume (3D) islands (Bimberg & Shchukin, 1999; Kern & Müller, 1998; Mo *et al.,* 1990). Islands with spatial limitation of charge carriers in all three directions are referred to as quantum dots. Quantum dots obtained in such a way have perfect crystalline structure, high quantum efficiency of radiation recombination and are characterized by enough high homogeneity in size (Aleksandrov *et al.,* 1974; Leonard *et al*., 1993; Moison *et al*., 1994; Ledentsov *et al*., 1996a, 1996b). Sizes of quantum dots can vary from several nanometers to several hundred nanometers. For example, size of quantum dots in heteroystems Ge-Si and InAs-GaAs lies

correspond to the following conditions: *а*) 0.1 ML Co; *b*) 0.17 ML Co; c) 0.36 ML Co.

90% of the total flow *j* ( *j j <sup>S</sup>* = 0.9 ) .

through the interface 'cluster-substrate'.

**distribution function** 

free nanoclusters.

(Bartelt *et al*., 1996; Goldfarb *et al*., 1997a, 1997b; Joyce *et al.,* 1998; Kamins *et al*., 1999; Ivanov-Omski *et al*., 2004; Antonov *et al*., 2005). In part, changing the form and sizes of islands, one can control their energy spectrum that is of great importance for practical applications of them. As the size distribution function becomes more homogeneous, as (for other equivalent conditions) the system of quantum dots becomes more attractive from the practical point of view.

Homogeneity of the size distribution function can be conveniently characterized by rootmean-square (*rms*) deviation, σ′ = *D* , where *D* – dispersion. As the size distribution function becomes narrower, as σ′ decreases. In this respect, the best size distribution functions have been obtained for island of germanium into heterosystem Ge/Si(001), where σ′ < 10% (Jian-hong Zhu *et al*., 1998).

Theoretical distributions corresponding to such magnitudes of dispersion *D* (or associated magnitudes of *rms*) have been obtained in papers (Vengrenovich *et al*., 2001b, 2005) in assumption that the main factor determining the form of the size distribution function of island film at later stages is the Ostwald's ripening. Computations have been carried out within the LSW theory, in assumption that dislocation diffusion is the limiting factor of the Ostwald's ripening. For that, the dislocation mechanism of growth of islands under the Ostwald's ripening is possible, if the flow of matter due to dislocation diffusion much exceeds the flow due to surface diffusion, i.e.

$$\left(D\_s^{(d)} Zd\left(\frac{d\mathbb{C}}{dR}\right)\_{R=r} \gg D\_s 2\pi r \left(\frac{d\mathbb{C}}{dR}\right)\_{R=r}\right) \tag{129}$$

where ( ) *<sup>d</sup> Ds* – the diffusion coefficient along dislocation grooves, *Ds* – the surface diffusion coefficient, *R r dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ – the concentration gradient at island surface, *<sup>d</sup>* – the width of dislocation groove, *d q* = 2 2 π , 2 2 *b* ≤ ≤ *q* 60*b* , where *b –* the Burgers vector, *Z* – the number of dislocation lines ending at the island base of radius *r* (*Z const* ≡ ) . For simplifying the computations, islands are considered as disk-like ones, with constant height *h*  (Vengrenovich *et al*., 2001b). General case, when both *h* and *r* are changed, is considered in paper (Vengrenovich *et al*., 2005).

Eq. (129) sets limitations on island sizes, which grow due to dislocation diffusion:

$$r \ll \frac{\text{Zd}}{2\pi} \frac{D\_s^{(d)}}{D\_s} \,. \tag{130}$$

If the condition Eq. (130) is violated, one must take into account in the resulting flow of matter, beside of the flow due to dislocation diffusion, the flow component caused by surface diffusion.

Under dislocation-surface diffusion one has:

$$
\dot{j} = \dot{j}\_d + \dot{j}\_s \,\tag{131}
$$

where *dj* – the flow to a particle due to diffusion along dislocations, *sj* – the flow due to surface diffusion, *dj* and *sj* are determined by the left and the right sides of Eq. (129), respectively.

Mass Transfer Between Clusters Under Ostwald's Ripening 145

Fig. 19 *а* illustrates the family of distributions computed following Eq. (137) with the step Δ = *x* 0.1 for magnitudes of *x* between zero and unity. One can see that as magnitude of *x* increases, as the maxima of the distributions decreases, and the magnitudes of *u*′ where *g*(*u*′) reaches maximum are shifted to the left, in direction of decreasing *u* . This shift is clearly observable in Fig. 19 *b*, where the same distributions normalized by their maxima are shown, so that ( ) max *g gu* ≡ ′ . For that, the magnitudes of *u*′ are determined from the

*a*

Fig. 19. Size distribution functions computed with the step 0.1 Δ*x* = (*а*); the same distributions, normalized by their maxima (enlarged version is in the inset), where

*a*

0,0 0,2 0,4 0,6 0,8 1,0

It must be noted that in the most cases experimental histograms are obtained as the dependences of the number of islands (share of islands) at the unit area on island's height, *h* . Theoretically, the choice of variable is arbitrary. For constant rate of change of island volume, it is of no importance, either *r* or *h* variable is constant. That is why, the

Fig. 20. Comparison with experiment (Neizvestnii *et al.,* 2001) 0.8 *x* = (*а*), 0.9 *x* = (*b*).

*g(u)/gmax*

0.0 0.2 0.4 0.6 0.8 1.0

42 22 2 2 3 ( 4 ) ( 4 2)4 3 12 9 0 *u u u x xu x x u x x* <sup>=</sup> ′ <sup>−</sup> − + −+ − + − = . (139)

0.0

0.2

0.4

0.6

0.2 0.4 0.6 0.8 1.0

0.8 0.9 1.0 0.0

*x=1*

*x=0*

*g(u)/gmax*

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

*u*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.9*

*u*

*x=1 x=0*

*b*

*b*

**6.1 Comparison with experimental data** 

0.0 0.2 0.4 0.6 0.8 1.0

*u*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.8*

*u*

*x=0.9 x=0.8 x=0.7 x=0.6 x=0.5 x=0.4 x=0.3 x=0.2 x=0.1 x=0*

following equation:

0.0

( ) max *g gu* ≡ ′ (*b*).

0.2

0.4

*g(u)*

0.0 0.2 0.4 0.6 0.8 1.0

*g(u)/gmax*

0.0 0.2 0.4 0.6 0.8 1.0

0.6

0.8

The rate of growth of isolated island under condition *h const* = is determined from equation:

$$\frac{d}{dt}(\pi r^2 \hbar) = j v\_m \,. \tag{132}$$

Substitution Eq. (131) in Eq. (132) and taking into account the magnitudes of *dj* and *sj* , as well as of the concentration gradient at the island boundary, one obtains:

$$\frac{dr}{dt} = \frac{\sigma}{2\pi\hbar kT \ln l} \frac{v\_{\infty}}{r^2} \left( D\_s^{(d)} Z d \frac{1}{r} + 2\pi D\_s \right) \left( \frac{r}{r\_k} - 1 \right). \tag{133}$$

Taking into account the ratio of flows,

$$\mathbf{x} = \frac{\dot{j}\_s}{\dot{j}} \mathbf{\hat{i}} \cdot \mathbf{1} - \mathbf{x} = \frac{\dot{j}\_d}{\dot{j}} \mathbf{\hat{i}} \cdot \frac{\dot{j}\_d}{\dot{j}\_s} = \frac{1 - \mathbf{x}}{\mathbf{x}} \mathbf{\hat{i}} \tag{134}$$

one can write Eq. (133) in the form:

$$\frac{dr}{dt} = \frac{\sigma \upsilon\_m^2 \mathbf{C}\_\alpha \mathbf{D}\_s^{(d)} \mathbf{Z} d}{\pi hkT \ln l} \frac{1}{r^3} \left( \frac{\mathbf{x}}{1-\mathbf{x}} \frac{r}{r\_\mathcal{g}} + \mathbf{1} \right) \left( \frac{r}{r\_k} - \mathbf{1} \right),\tag{135}$$

or:

$$\frac{dr}{dt} = \frac{\sigma \upsilon\_m^2 C\_\alpha D\_s}{hkT \ln l} \frac{1}{r^2} \left(\frac{1-\chi}{\chi} \frac{r\_\varrho}{r} + 1\right) \left(\frac{r}{r\_k} - 1\right). \tag{136}$$

Eqs. (135) and (136) describe the rate of growth of clusters under dislocation and surface diffusion with contributions ( *x* ) and ( 1 − *x* ) corresponding to the flows of them. Taking into account Eqs. (135) and (136) for the rate of growth and performing computations following the algorithm introduced in paper (Vengrenovich, 1982), one can represent the relative size distribution function of clusters, under assumption that mass transfer between clusters is realized due to dislocation-surface diffusion, in the form:

$$g'(u) = \frac{u^3(u^2+bu+c)^{\frac{D}{2}}}{\left(u-1\right)^K} \exp\left(\frac{F}{u-1}\right) \times \exp\left(\frac{E-Db}{\sqrt{c-\frac{b^2}{4}}} \tan^{-1}(\frac{u+b/2}{\sqrt{c-b^2/4}})\right),\tag{137}$$

Where

$$\begin{cases} D = \frac{3c^2 + \left(x^2 - 4x + 6b - 6\right)c + 6b^2 + \left(4x^2 - 16x + 14\right)b + 7x^2 - 28x + 19}{c^2 + \left(b + 1\right)2c + b^2 + 2b + 1}, \\\\ E = \frac{\left(3 - D\right)c + \left(D - 3\right)b^2 + \left(2b + 1\right)D + x^2 - 4x - 3}{2 + b}, \\\\ F = D\left(b + 1\right) - 3b - E, \\ K = 6 - D. \end{cases} \tag{138}$$

#### **6.1 Comparison with experimental data**

144 Mass Transfer - Advanced Aspects

The rate of growth of isolated island under condition *h const* = is determined from equation:

*m*

1 1 2 1

π

*j x j x*

<sup>1</sup> 1 1

1 1 1 1

*g k*

*k*

4

*E Db*

*c*

2 2

− ⎛ ⎞ <sup>−</sup> ⎜ ⎟

*s s*

= . (132)

. (133)

, (135)

. (136)

, (137)

(138)

<sup>−</sup> <sup>=</sup> , (134)

*k*

( ) <sup>2</sup>

*dt* π

2

*<sup>j</sup>* <sup>=</sup> , 1 *dj <sup>x</sup>*

*m d*

well as of the concentration gradient at the island boundary, one obtains:

2

2 ln

*sj <sup>x</sup>*

2 ( )

*d m s*

σ

π

σ

π

σ

2

*D*

*K*

( )( ) ( )

13 ,

<sup>⎪</sup> <sup>+</sup>

( )

*F Db b E*

= +− −

6 .

*K D*

⎪ = − ⎩

ln

diffusion with contributions ( *x* ) and ( 1 − *x* ) corresponding to the flows of them.

transfer between clusters is realized due to dislocation-surface diffusion, in the form:

3 2 <sup>2</sup> <sup>1</sup>

( )( ) ( )

2 2

3 3 21 43 , <sup>2</sup>

*b*

*Dc D b b D x x*

<sup>⎪</sup> − + − + + +−− <sup>=</sup>

2 2 2 2 2 2 2

<sup>⎧</sup> + −+− + + − + + − + <sup>⎪</sup> <sup>=</sup> <sup>⎪</sup> ++ ++ + <sup>⎪</sup>

*c x xb cb x x bx x*

*c b cb b*

( ) <sup>2</sup> <sup>2</sup> ( ) exp exp tan ( ) ( 1) <sup>1</sup> <sup>4</sup>

+ + ⎛ ⎞ <sup>+</sup> ′ <sup>=</sup> <sup>×</sup> ⎜ ⎟ <sup>−</sup> ⎝ ⎠ <sup>−</sup> <sup>−</sup> <sup>−</sup> ⎝ ⎠

*u u bu c F u b*

*u u b cb*

3 4 6 6 6 4 16 14 7 28 19 , 12 2 1

Taking into account the ratio of flows,

one can write Eq. (133) in the form:

*g u*

*D*

*E*

⎪

⎪

⎪

⎪ ⎨

or:

Where

*<sup>d</sup> r h jv*

Substitution Eq. (131) in Eq. (132) and taking into account the magnitudes of *dj* and *sj* , as

( )

<sup>∞</sup> ⎛ ⎞⎛ ⎞ <sup>=</sup> ⎜ ⎟ + − ⎜ ⎟ ⎝ ⎠⎝ ⎠

*<sup>j</sup>* − = , *<sup>d</sup>* <sup>1</sup> *s*

3

<sup>∞</sup> ⎛ ⎞⎛ ⎞ <sup>=</sup> ⎜ ⎟ + − ⎜ ⎟ <sup>−</sup> ⎝ ⎠⎝ ⎠

<sup>∞</sup> ⎛ ⎞ <sup>−</sup> ⎛ ⎞ <sup>=</sup> ⎜ ⎟ + − ⎜ ⎟ ⎝ ⎠⎝ ⎠

Eqs. (135) and (136) describe the rate of growth of clusters under dislocation and surface

Taking into account Eqs. (135) and (136) for the rate of growth and performing computations following the algorithm introduced in paper (Vengrenovich, 1982), one can represent the relative size distribution function of clusters, under assumption that mass

ln 1

2

*m s g*

*dr vCD х r r dt hkT l r х r r*

*dr v C D Zd xr r dt hkT l r х r r*

*dr v C <sup>r</sup> D Zd D dt hkT l r r r*

Fig. 19 *а* illustrates the family of distributions computed following Eq. (137) with the step Δ = *x* 0.1 for magnitudes of *x* between zero and unity. One can see that as magnitude of *x* increases, as the maxima of the distributions decreases, and the magnitudes of *u*′ where *g*(*u*′) reaches maximum are shifted to the left, in direction of decreasing *u* . This shift is clearly observable in Fig. 19 *b*, where the same distributions normalized by their maxima are shown, so that ( ) max *g gu* ≡ ′ . For that, the magnitudes of *u*′ are determined from the following equation:

$$\left. \left( \Im u^{4} - (\mathbf{x}^{2} - 4\mathbf{x})u^{2} + (\mathbf{x}^{2} - 4\mathbf{x} + \mathbf{2})4u^{2} - \Im \mathbf{x}^{2} + \mathbf{12}\mathbf{x} - \mathbf{9} \right|\_{\mathbf{u} = u'} = \mathbf{0} \,. \tag{139}$$

Fig. 19. Size distribution functions computed with the step 0.1 Δ*x* = (*а*); the same distributions, normalized by their maxima (enlarged version is in the inset), where ( ) max *g gu* ≡ ′ (*b*).

Fig. 20. Comparison with experiment (Neizvestnii *et al.,* 2001) 0.8 *x* = (*а*), 0.9 *x* = (*b*).

It must be noted that in the most cases experimental histograms are obtained as the dependences of the number of islands (share of islands) at the unit area on island's height, *h* . Theoretically, the choice of variable is arbitrary. For constant rate of change of island volume, it is of no importance, either *r* or *h* variable is constant. That is why, the

Mass Transfer Between Clusters Under Ostwald's Ripening 147

distribution function of clusters is described by the generalized Lifshitz-Slyozov-Wagner distribution (alloys, nanocomposites as nc*CdS* /polimer (Savchuk *et al*., 2010a, 2010b, 2010c) or the generalized Chakraverty-Wagner distribution (island films, heterostructures with

If in the resulting flow *j* the component *i V j j* << , it can lead to new mechanism of cluster growth under dislocation-matrix or dislocation-surface diffusion, for the each of which the specific size distribution function and the corresponding temporal dependences of *r* and

Comparison of the theoretically computed size distribution functions with experimentally

1. The introduced model of cluster ripening under simultaneous (combined) action of both the diffuse mechanism and the Wagner one is proved experimentally. Other of the considered models is also finds out experimental proof, *viz.* the case when one neglects the Wagner mechanism of growth and cluster ripening results from mixed dislocation-matrix and dislocation-surface diffusion. Thus, it is the most likelihood that, in practice, cluster growth follows to not only one isolated of the considered early mechanisms of growth, i.e. the diffusion mechanism or the Wagner one, but rather to the mixed (combined)

It also follows from the results of comparison of the computed and experimental data, that cluster growth under mixed (combined) dislocation-matrix or dislocation-surface diffusion is most probable than cluster growth under any of two mentioned mechanisms, if isolated. 2. In connection with intense development of nanotechnologies and related techniques for generating of nanostructures, the problem arises: in what framework is the LSW theory applied to analysis of nanosystems containing nanoclusters. The final answer on this question is now absent. Also, the main question concerning stability of nanosystems in respect to the Ostwald's ripening leaves opened. Nevertheless, it follows from the represented by us results of comparison of theoretical and experimental data, that in many cases the experimentally obtained histograms built for nanoparticles (nanoclusters) by many authors for various nanosystems are quite satisfactory fitted by the computed by us theoretical distributions (the generalized Lifshitz-Slyozov-Wagner distribution, the generalized Chakraverty-Wagner distribution etc.). In means that the developed by us LSW theory can be, in principle, be used for analysis of phase and structural transformations in nanosystems with nanophases. Of course, derived by us approach requires further

This investigation has been carried out under supporting the Ministry of Education and

Alechin A.P. (2004). Structural organization of a matter at surface in nanotechnology – a way to nanotechnology. *Russian Usp. Mod. Radioelectr.,* V. 5, (2004), pp. 118-122 Aleksandrov L.N., Lovyagin R.N., Pchelyakov O.P., Stenin S.I. (1974). Heteroepitaxy of

germanium thin films on silicon by ion sputtering. *J. Cryst., Growth*, V. 24-25, (1974),

obtained histograms leads to the following two main conclusions.

mechanism, when two mentioned limiting mechanisms act together.

investigations, both theoretical and experimental.

Science of Ukraine, grant No0110 U000190.

**8. Acknowledgement** 

pp. 298-301

**9. References** 

quantum dots, etc.)

*gr* are intrinsic.

distributions shown in Fig. 19 can be used also for comparison with experimentally obtained histograms, when the island height, *h* , is constant.

One of such comparisons is illustrated in Fig. 20 (*а – x* = 0.8 , *b – x* = 0.9 ). Experimentally obtained histogram normalized by unity on axes *uhh* ( *<sup>g</sup>* ) and ( ) max *gu g* corresponds to the height (on *h* ) distribution function in (Ge/ZnSe) (Neizvestnii *et al.,* 2001).

Fig. 21. Comparison of the experimentally obtained histograms with theoretically computed dependences (Vostokov *et al*., 2000) *x* = 0.2 (*а*), 0.4 *x* = (*b*)

In Fig. 21, the experimentally obtained histogram normalized in the same manner as in previous case, corresponds to the height distribution of islands of germanium (Ge/Si (001)) for the quantity of fall out of germanium 5.5 monolayers ( 5.5 *Ge d ML* = ) (Vostokov *et al*., 2000). Theoretical curves have been computed for *а – x* = 0.2 , and *b* – 0.4 *x* = . One can see that as *x* increases, as discrepancy between the experimentally obtained histogram and theoretically computed dependences increases also.
