**5.1 Generalized Chakraverty-Wagner distribution for islands (clusters) of cylindrical form (** *h const* = **)**

The problem of determination of the size distribution function is analogous to the above considered problem for clusters of cupola-like form. Modeling the island film by disk-like

0.0 0.2 0.4 0.6 0.8 1.0

*u*

Fig. 13. Comparison of the dependence Eq. (96) with experimental histogram of nanoclusters *Ag* obtained by the molecular beam epitaxy technique at substrate *TiO*<sup>2</sup> (110) at room

Obtaining the heterostructures containing quantum dots of specified concentration, form, sizes and homogeneity is connected with considerable experimental difficulties. However, if even such structure has been obtained, its properties can change under the Ostwald's ripening. For that, as it has been shown above, the character of the size distribution function of clusters changes not only as a result of transition from one growth mechanism to another one, but also due to simultaneous action of such mechanisms (Sagalovich & Slyozov, 1987; Vengrenovich *et al*., 2006а, 2007a, 2008а, 2008b). Below we represent the results of investigation of the influence of cluster form on the size distribution function in semiconductor heterosystems with quantum dots. A heterosystem is considered as island

**5. Influence of form of nanoclusters in heterostructures on the size** 

film consisting of disk-like islands of cylindrical form, with height *h* (Fig. 14).

*h r* 

**5.1 Generalized Chakraverty-Wagner distribution for islands (clusters) of cylindrical** 

The problem of determination of the size distribution function is analogous to the above considered problem for clusters of cupola-like form. Modeling the island film by disk-like

*x=0.3*

0.0

*substrate* 

Fig. 14. Disc-like cluster of radius r and constant height h

0.2

0.4

0.6

*g(u)/gmax*

temperature

**form (** *h const* = **)** 

**distribution function** 

0.8

*r g /rk =1.625*

1.0

islands corresponds to heterostructure with more stable form of hut-clusters (Safonov & Trushin, 2007).

The rate of change of volume of cluster with constant height *h* (Fig. 14) is determined by the flow *j* of adatoms to (from) a cluster:

$$\frac{d}{dt}(\pi r^2 \text{h}) = j\nu\_{m'} \tag{105}$$

where υ*<sup>m</sup>* – adatom volume. From Eq. (105) one obtains:

$$\frac{dr}{dt} = \frac{1}{2\pi r \text{h}} j \nu\_m \,. \tag{106}$$

Following to (Vengrenovich *et al*., 2008а), the flow *j* consists of two parts:

$$\dot{j} = \dot{j}\_S + \dot{j}\_{i\text{-}\prime} \tag{107}$$

where *Sj* – the part of flow caused by surface diffusion, and *ij* – the part of flow of adatoms, which due to overcoming the potential barrier at the interface 'cluster-substrate' fall at cluster surface and, then, take part in formation of chemical connections (the Wagner mechanism of growth).

By definition, the diffusion part of a flow equals:

$$j\_S = 2\pi r D\_S (\frac{dC}{d\mathcal{R}})\_{R=r\text{ \textquotedblleft}R} \tag{108}$$

where *DS* – the surface diffusion coefficient, ( )*R r dC dR* <sup>=</sup> – concentration gradient at the interface 'cluster-substrate', which can be represented in the form (Chakraverty, 1967; Vengrenovich 1980a, 1980b; Vengrenovich *et al*., 2008а):

$$(\frac{d\mathbf{C}}{dR})\_{R=r} = \frac{\{\mathbf{C}\} - \mathbf{C}\_r}{\mathbf{1} \mathbf{n} \, l} \cdot \frac{\mathbf{1}}{r} \, \tag{109}$$

where *l* determines the distance from an island, ( ) *R lr* = , at which a mean concentration of adatoms at a substrate, *C* , is set around separate cluster of radius *r* ( *l* = 2, 3 ). Taking into account Eq. (109), one can rewrite Eq. (108) in the form:

$$j\_S = \frac{2\,\pi D\_S}{1\,\text{nJ}} \left( \left< \text{C} \right> - \text{C}\_r \right). \tag{110}$$

Concentration of adatoms at the cluster base, *Cr* , is determined by the Gibbs-Thomson equation:

$$\mathbf{C}\_r = \mathbf{C}\_\phi \exp\left(\Delta P \frac{\upsilon\_m}{kT}\right) \approx \mathbf{C}\_\phi (1 + \Delta P \frac{\upsilon\_m}{kT}) \,\tag{111}$$

where *C*∞ – the equilibrium concentration at temperature *T*, Δ*P* – the Laplacian pressure caused by island surface curvature, *k* – the Boltzmann constant. Pressure Δ*P* , in accordance with (Vengrenovich et al., 2008а), equals:

Mass Transfer Between Clusters Under Ostwald's Ripening 139

one obtains the formula for the rate of cluster growth under surface diffusion, with the share

*dr A x r r dt r xrr*

*dr B x r r*

ln

that corresponds to the Wagner mechanism of cluster growth with the share contribution *x*

Solving jointly Eq. (122) (or Eq. (123)) and Eq.(8) and applying the method derived in paper (Vengrenovich, 1982), one finds out the generalized relative size distribution function, *g u*′( ) , for disk-like clusters corresponding to the combined action of two mechanisms of growth,

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

2 4 10 8 4 ,

43 2

*xx xx <sup>B</sup> A*

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

432

For 1 *x* = , *B* = 28 9 , *D* = −17 9 , *C* = −2 3 , and Eq. (124) corresponds to the distribution

() ( ) ( ) <sup>28</sup> <sup>17</sup> <sup>2</sup> 9 9 2 3 1 2exp( ) <sup>1</sup> *gu u u u*

For 0 *x* = , *B* = 4 , *D* = −1 , 2 *C* = − , and Eq. (124) turns into the Wagner distribution

() ( ) <sup>4</sup> <sup>2</sup> 1 exp( ) <sup>1</sup> *gu u u*

However, for graphic representation of the size distribution function one must carry out

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

3 6 5 21 ,

2 3 2 1.

*A*

432

<sup>⎪</sup> =+ + ++ <sup>⎩</sup>

*Ax x x x*

2 22 ,

*A xxxx <sup>D</sup>*

<sup>1</sup> [1 ( ) ]( 1)

[1 ( ) ]( 1) <sup>1</sup> *g*

*g k*

*k*

<sup>2</sup> \* *<sup>C</sup> <sup>m</sup> <sup>B</sup> kT* β ∞συ= .

*<sup>B</sup> <sup>D</sup> C*

<sup>−</sup> <sup>=</sup> + − , (122)

*dt r x r r* <sup>=</sup> + − <sup>−</sup> , (123)

*u*

*u* ′ − − =− + − <sup>−</sup> . (126)

*g*(*u Q* ) = ⋅ *g*′(*u*) , (128)

*u* <sup>−</sup> ′ =− − <sup>−</sup> . (127)

, (125)

<sup>−</sup> ′ = − ++ <sup>−</sup> , (124)

\* 2

\*

<sup>2</sup> \*

*gu u u u x x*

2

*x x <sup>C</sup>*

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

*D CS m <sup>A</sup> hkT l* ∞συ= ,

i.e. the Wagner and the diffusion ones (Vengrenovich *et al*., 2010):

⎪

obtained in paper (Vengrenovich, 1980):

(Chakraverty, 1967; Vengrenovich, 1980a, 1980b):

computations following equation that is analogous to Eq. (28):

contribution ( ) 1 − *x* of the flow *ij* :

of the diffusion flow *Sj* , where

or:

where:

$$
\Delta P = \sigma \frac{dS}{dV} = \sigma \frac{2\pi \text{h} dr}{2\pi r \text{h} dr} = \frac{\sigma}{r} \,. \tag{112}
$$

Taking into account Eq. (112), *Cr* can be represented in the form:

$$\mathbf{C}\_r \approx \mathbf{C}\_\alpha \left( 1 + \frac{\sigma \upsilon\_m}{kT} \cdot \frac{1}{r} \right) \,\text{\,\,\,}\tag{113}$$

where σ – the specific magnitude of surface energy. A mean concentration of adatoms at surface, *C* , is determined, by analogy with Eq. (113), by the mean (or critical) cluster size *kr* :

$$
\left\langle \mathbf{C} \right\rangle \approx \mathbb{C}\_{\phi} \left( 1 + \frac{\sigma \upsilon\_{m}}{kT} \cdot \frac{1}{r\_{k}} \right) . \tag{114}
$$

Thus:

$$j\_S = \frac{2\pi D\_S C\_\alpha \sigma \upsilon\_m}{kT \ln l} (\frac{1}{r\_k} - \frac{1}{r}) = \frac{2\pi D\_S C\_\alpha \sigma \upsilon\_m}{kT \ln l} \frac{1}{r} (\frac{r}{r\_k} - 1) \cdot \tag{115}$$

In accordance with Wagner, the number of adatoms occurring in the unite of time at side surface of a cluster ( ) *h const* = is determined as:

$$\mathbf{j}\_1 = 2\pi r \mathbf{h} \beta \{\mathbf{C}\}\_{\prime} \tag{116}$$

and the number of adatoms leaving a cluster in the unite of time is:

<sup>2</sup> 2 *<sup>r</sup> j rh C* = π β, (117)

so that the resulting flow of atoms involved into forming chemical connections equals:

$$j\_i = j\_1 - j\_2 = 2\pi rh\beta (\{\mathbf{C}\} - \mathbf{C}\_r) = \frac{2\pi h\beta \mathbf{C}\_w \sigma \upsilon\_m}{kT} (\frac{r}{r\_k} - 1) \cdot \tag{118}$$

Substituting *Sj* and *ij* in Eq. (107), one obtains:

$$j = \frac{2\pi D\_S C\_\alpha \sigma \nu\_m^2}{kT \ln l} \cdot \frac{1}{r} (\frac{r}{r\_k} - 1) + \frac{2\pi \hbar \beta \mathcal{C}\_\alpha \sigma \nu\_m}{kT} (\frac{r}{r\_k} - 1) \cdot \tag{119}$$

Substituting Eq. (119) in Eq. (106), one finds out the rate of growth:

$$\frac{dr}{dt} = \frac{1}{2\pi rh} (\frac{2\pi D\_S C\_\alpha \sigma \upsilon\_m^2}{kT \ln l} \cdot \frac{1}{r} (\frac{r}{r\_k} - 1) + \frac{2\pi h\beta C\_\alpha \sigma \upsilon\_m}{kT} (\frac{r}{r\_k} - 1))\tag{120}$$

For the combined action of two mechanisms of growth, i.e. the diffusion and the Wagner ones, the rate of growth, *r* , will be dependent on the ratio of the flows *Sj* and *ij* . Designating, as previously, the shares of flows *Sj* and *ij* in general flow *j* , as *<sup>S</sup> xjj* = and 1 *<sup>i</sup>* − = *x jj* , respectively, so that the ratio of them equals:

$$\frac{\dot{j}\_S}{\dot{j}\_i} = \frac{\mathbf{x}}{1 - \mathbf{x}} \text{ \tag{121}$$

one obtains the formula for the rate of cluster growth under surface diffusion, with the share contribution ( ) 1 − *x* of the flow *ij* :

$$\frac{dr}{dt} = \frac{A}{r^2} \left[ 1 + (\frac{1-x}{x})\frac{r}{r\_g} \right] (\frac{r}{r\_k} - 1) \, \tag{122}$$

or:

138 Mass Transfer - Advanced Aspects

*dS hdr <sup>P</sup>*

 σ

<sup>1</sup> (1 ) *<sup>m</sup> C C <sup>r</sup> kT r* συ

A mean concentration of adatoms at surface, *C* , is determined, by analogy with Eq. (113),

<sup>1</sup> (1 ) *<sup>m</sup>*

2 2 1 1 <sup>1</sup> ( ) ( 1) ln ln *S m S m*

In accordance with Wagner, the number of adatoms occurring in the unite of time at side

<sup>2</sup> 2 *<sup>r</sup> j rh C* = π β

<sup>2</sup> 2( ) ( 1) *<sup>m</sup> i r*

*S m m*

∞ ∞

*D C r r h C <sup>j</sup> kT l r r kT r*

*dr D C r h C r dt rh kT l r r kT r*

so that the resulting flow of atoms involved into forming chemical connections equals:

*h C <sup>r</sup> j j j rh C C kT r*

π

*D C D C <sup>r</sup> <sup>j</sup> kT l r r kT l r r*

∞ ∞

<sup>1</sup>*j* = 2π β

*kT r* συ

*k k*

π β συ

( 1) ( 1) ln

( (1) (1)) 2 ln *S m m*

For the combined action of two mechanisms of growth, i.e. the diffusion and the Wagner ones, the rate of growth, *r* , will be dependent on the ratio of the flows *Sj* and *ij* . Designating, as previously, the shares of flows *Sj* and *ij* in general flow *j* , as *<sup>S</sup> xjj* = and

> 1 *S i*

*j x*

π β συ= ⋅ −+ − . (119)

*k k*

*k k*

∞ ∞ = ⋅ − + − (120)

π β συ

<sup>∞</sup> = −= − = − . (118)

*k*

 συ= − = − . (115)

σ

*C C*

συ

and the number of adatoms leaving a cluster in the unite of time is:

π β

 συ

Substituting Eq. (119) in Eq. (106), one finds out the rate of growth:

π

1 *<sup>i</sup>* − = *x jj* , respectively, so that the ratio of them equals:

<sup>2</sup> 2 2 1

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

 συ

Taking into account Eq. (112), *Cr* can be represented in the form:

– the specific magnitude of surface energy.

by the mean (or critical) cluster size *kr* :

*S*

surface of a cluster ( ) *h const* = is determined as:

π

1 2

π

π

Substituting *Sj* and *ij* in Eq. (107), one obtains:

where σ

Thus:

2 2

σ

Δ = = = . (112)

≈ <sup>∞</sup> + ⋅ , (113)

≈ <sup>∞</sup> + ⋅ . (114)

*rh C* , (116)

, (117)

*k*

*<sup>j</sup> <sup>x</sup>* <sup>=</sup> <sup>−</sup> , (121)

*dV rhdr r* π

π

$$\frac{dr}{dt} = \frac{B}{r} \mathbf{\dot{l}} + (\frac{\mathbf{x}}{1-\mathbf{x}}) \frac{r\_{\mathcal{S}}}{r} \mathbf{\dot{l}} (\frac{r}{r\_k} - 1) \,\,\,\,\tag{123}$$

that corresponds to the Wagner mechanism of cluster growth with the share contribution *x* of the diffusion flow *Sj* , where <sup>2</sup> \* ln *D CS m <sup>A</sup> hkT l* ∞συ = , <sup>2</sup> \* *<sup>C</sup> <sup>m</sup> <sup>B</sup> kT* β ∞συ = . Solving jointly Eq. (122) (or Eq. (123)) and Eq.(8) and applying the method derived in paper

(Vengrenovich, 1982), one finds out the generalized relative size distribution function, *g u*′( ) , for disk-like clusters corresponding to the combined action of two mechanisms of growth, i.e. the Wagner and the diffusion ones (Vengrenovich *et al*., 2010):

$$\log'(\mu) = \mu^2 \left(1 - \mu\right)^{-B} \left(\mu + \mathbf{x}^2 + \mathbf{x}\right)^D \exp(\frac{C}{1 - \mu})\,,\tag{124}$$

where:

$$\begin{cases} B = \frac{2\mathbf{x}^4 + 4\mathbf{x}^3 + 10\mathbf{x}^2 + 8\mathbf{x} + 4}{A}, \\ C = -\frac{2\mathbf{x}^2 + 2\mathbf{x} + 2}{A}, \\ D = -\frac{3\mathbf{x}^4 + 6\mathbf{x}^3 + 5\mathbf{x}^2 + 2\mathbf{x} + 1}{A}, \\ A = \mathbf{x}^4 + 2\mathbf{x}^3 + 3\mathbf{x}^2 + 2\mathbf{x} + 1. \end{cases} \tag{125}$$

For 1 *x* = , *B* = 28 9 , *D* = −17 9 , *C* = −2 3 , and Eq. (124) corresponds to the distribution obtained in paper (Vengrenovich, 1980):

$$\log\left(u\right)' = u^2 \left(1 - u\right)^{-2\xi} \left< u + 2 \right>^{-17} \Big| \exp(-\frac{2/3}{1 - u}) \,. \tag{126}$$

For 0 *x* = , *B* = 4 , *D* = −1 , 2 *C* = − , and Eq. (124) turns into the Wagner distribution (Chakraverty, 1967; Vengrenovich, 1980a, 1980b):

$$\log'(u) = u \left(1 - u\right)^{-4} \exp(-\frac{2}{1 - u})\,. \tag{127}$$

However, for graphic representation of the size distribution function one must carry out computations following equation that is analogous to Eq. (28):

$$\mathcal{g}(u) = \mathcal{Q} \cdot \mathcal{g}'(u) \, , \tag{128}$$

Mass Transfer Between Clusters Under Ostwald's Ripening 141

0.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.6*

*u*

Fig. 18. Comparison of dependence (128) with experimental histograms Co at Si3N4 obtained by evaporating at room temperature: (*a*) 0.1 ML Co, (*b*) 0.17 ML Co, (*c*) 0.36 ML Co (Shangjr

In other case that is illustrated in Fig. 17, the theoretical dependence Eq. (128) is compared with experimental histograms Ge/Si(001) obtained by the molecular-beam epitaxy

0.8

1.0

0.2

0.4

0.6

*g(u)/gmax*

Fig. 17. Comparison of dependence (128) with experimental histograms Ge at substrate Si obtained by molecular beam epitaxy with one (*a*) and two (*b*) layers of nano-clusters Ge

*a*

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.9*

*u*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.7*

*u*

*c*

*b*

*b*

*a*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.6*

*u*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.7*

*u*

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

0.0

0.0

Gwo *et al*., 2003)

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

(Yakimov *et al*., 2007)

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

where ( ) 1 2 0 *Q* π*h u g u du* <sup>Φ</sup> <sup>=</sup> ′ ∫ , Φ − the volume (mass) of disperse phase in the form of clusters.

#### **5.2 Discussion**

The dependences shown in Fig. 15,*а* correspond to the size distribution function Eq. (128) computed for various magnitudes of *x.* The limiting curves, for *x* = 0 and *x* = 1, correspond to the Wagner distribution (Wagner, 1961) and to the distribution obtained in papers (Vengrenovich, 1980a, 1980b), respectively. All other distributions within the interval 0 ≤ *x* < 1 describe the size distribution functions of clusters for the combined action of the Wagner and the diffusion mechanisms of growth. The same dependences normalized by their maxima are shown in Fig.15,*b*.

Fig. 15. Functions *g*(*u*) (*a*) and ( ( ) ) max *gu g* (*b*) computed following Eq. (128)

Fig. 16 illustrates comparison of experimental histogram of nanodots Ge/SiO2 (Kan *et al*., 2005), obtained by evaporating technique with following thermal annealing, with the theoretical dependence, Eq. (128), for 0.7 *x* = . A mean size of clusters is 5.6 nm. One can see satisfactory agreement of the theory and experimental data.

Fig. 16. Comparison of the dependence Eq. (128) with experimental histogram of nanodots Ge at substrate SiO2 obtained by evaporating with following thermal annealing (Kan et al., 2005)

The dependences shown in Fig. 15,*а* correspond to the size distribution function Eq. (128) computed for various magnitudes of *x.* The limiting curves, for *x* = 0 and *x* = 1, correspond to the Wagner distribution (Wagner, 1961) and to the distribution obtained in papers (Vengrenovich, 1980a, 1980b), respectively. All other distributions within the interval 0 ≤ *x* < 1 describe the size distribution functions of clusters for the combined action of the Wagner and the diffusion mechanisms of growth. The same dependences normalized by

> 0.0 0.2 0.4 0.6 0.8 1.0

*g(u)/gmax*

Fig. 16 illustrates comparison of experimental histogram of nanodots Ge/SiO2 (Kan *et al*., 2005), obtained by evaporating technique with following thermal annealing, with the theoretical dependence, Eq. (128), for 0.7 *x* = . A mean size of clusters is 5.6 nm. One can see

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.7*

*u*

Fig. 16. Comparison of the dependence Eq. (128) with experimental histogram of nanodots Ge at substrate SiO2 obtained by evaporating with following thermal annealing (Kan et al., 2005)

*<sup>u</sup>* 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 15. Functions *g*(*u*) (*a*) and ( ( ) ) max *gu g* (*b*) computed following Eq. (128)

*a*

*x=1 x=0.9 x=0.8 x=0.7 x=0.6 x=0.5 x=0.4 x=0.3*

*u*

satisfactory agreement of the theory and experimental data.

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

, Φ − the volume (mass) of disperse phase in the form of clusters.

0.0 0.2 0.4 0.6 0.8 1.0

*x=0 x=1*

*b*

where

*Q*

**5.2 Discussion** 

0.0

0.4

0.8

1.2

*g(u)*

1.6

2.0

( )

*h u g u du*

their maxima are shown in Fig.15,*b*.

*x=0.2 x=0.1 x=0*

′ ∫

1 2 0

<sup>Φ</sup> <sup>=</sup>

π

Fig. 17. Comparison of dependence (128) with experimental histograms Ge at substrate Si obtained by molecular beam epitaxy with one (*a*) and two (*b*) layers of nano-clusters Ge (Yakimov *et al*., 2007)

Fig. 18. Comparison of dependence (128) with experimental histograms Co at Si3N4 obtained by evaporating at room temperature: (*a*) 0.1 ML Co, (*b*) 0.17 ML Co, (*c*) 0.36 ML Co (Shangjr Gwo *et al*., 2003)

In other case that is illustrated in Fig. 17, the theoretical dependence Eq. (128) is compared with experimental histograms Ge/Si(001) obtained by the molecular-beam epitaxy

Mass Transfer Between Clusters Under Ostwald's Ripening 143

(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

Homogeneity of the size distribution function can be conveniently characterized by root-

functions have been obtained for island of germanium into heterosystem Ge/Si(001), where

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

> ( ) 2 *<sup>d</sup> s s*

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

*r*

*dC dC D Zd D r*

*R r R r*

= =

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ – the concentration gradient at island surface, *<sup>d</sup>* – the width of

π

*dR dR*

where ( ) *<sup>d</sup> Ds* – the diffusion coefficient along dislocation grooves, *Ds* – the surface diffusion

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

2

*Zd D*

π*D*

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

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),

( )

*d s s*

′ = *D* , where *D* – dispersion. As the size distribution

′ decreases. In this respect, the best size distribution

⎛⎞ ⎛⎞ >> ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ (129)

, 2 2 *b* ≤ ≤ *q* 60*b* , where *b –* the Burgers vector, *Z* – the number

<< . (130)

*d s jj j* = + , (131)

σ

σ

view.

σ

coefficient,

surface diffusion.

respectively.

mean-square (*rms*) deviation,

function becomes narrower, as

′ < 10% (Jian-hong Zhu *et al*., 1998).

exceeds the flow due to surface diffusion, i.e.

*R r*

π

*dC dR* <sup>=</sup>

dislocation groove, *d q* = 2 2

paper (Vengrenovich *et al*., 2005).

Under dislocation-surface diffusion one has:

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

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

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 defectfree nanoclusters.

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* through the interface 'cluster-substrate'.
