\documentstyle[12pt]{article}
\textheight 24cm
\textwidth 17cm
\oddsidemargin -0.1cm
\topmargin -2cm
%\pagestyle{empty}
\renewcommand{\baselinestretch}{1.5}
\newcommand{\eps}{\mbox{$\varepsilon$}}
\newcommand{\del}{\mbox{$\Delta$}}
\newcommand{\bm}[1]{\mbox{\boldmath$#1$}}
\newcommand{\p}{\mbox{$\partial$}}
\begin{document}
\thispagestyle{empty}
\centerline{\large\bf  NANO-ISLANDS ON PLASTICALLY DEFORMED}
\centerline{\large\bf  SUBSTRATES WITH DISCLINATIONS}
\par\bigskip
\begin{center}
{\bf I.A. Ovid'ko and A.G. Sheinerman}\\
Institute for Problems of Mechanical Engineering, Russian Academy of Sciences,\\
St.~Petersburg, 199178 Russia\\
email: ovidko@def.ipme.ru
\end{center}
\par\bigskip\noindent
{\bf Abstract.}
Spatial arrangements of nano-islands deposited on plastically
deformed substrates containing wedge disclinations (defects of rotational
type) are theoretically examined. It is shown that the elastic interaction
between the nano-islands and the disclinations is capable of causing
preferable spatial positions of nano-islands. The parameters of the
substrate/nano-island systems are  theoretically revealed
that control  spatial positions
of nano-islands on disclinated substrates.
\newpage\noindent
{\bf 1. INTRODUCTION}
\par\bigskip\noindent
The strain-driven formation of spatially ordered ensembles of nano-islands recently attracted tremendous attention motivated by the interest to their wide applications in nanotechnologies; see, e.g. ~[1-14]. Of special importance are applications of self-assembled semiconductor nano-islands (quantum dots) exhibiting unique functional properties exploited in electronic and optoelectronic applications. In doing so, desired, from an applications viewpoint, spatial order of nano-island ensembles comes into play owing to the elastic interaction between nano-islands     (see reviews~[1-3]). Recently, a promising technological method for fabrication of spatially ordered ensembles of quantum dots has been suggested and theoretically examined in papers~[15,\ 16], which is the formation of nano-islands (quantum dots) on free surfaces of substrates with internal interphase or low-angle grain boundaries. This method exploits the idea to control spatial organization of arrays of nano-islands by means of the elastic interaction between nano-islands and ordered networks of dislocations at the internal boundaries.
\par
We think that the method under consideration can be treated as a partial case of the general technological approach which deals with a human-controlled modification of substrates in order to fabricate films and nano-island ensembles (on substrates) with desired functional properties. In this context, a promising method to modify substrates, which is another partial case of the general approach discussed, is plastic deformation of substrates. Actually, plastic deformation processes are capable of creating internal sources of stresses in a substrate that influence spatial positions of nano-islands on the substrate free surface due to elastic interaction between the nano-islands and the stress sources. In doing so, the stress sources~--- defect configurations~--- in plastically deformed solids often form spatially ordered structures which, therefore, can give rise to the formation of spatially ordered ensembles of nano-islands on the free surfaces of pre-deformed substrates. The main aim of this paper is to theoretically examine the behavior of nano-islands on plastically deformed substrates that contain disclinations (defect of rotational type) generated under mechanical load.
\par\bigskip\noindent
{\bf 2. FORMATION OF DISCLINATIONS IN PLASTICALLY DEFORMED SUBSTRATES}
\par\bigskip\noindent
Dislocation walls (low-angle grain boundaries), which have a rather regular spatial arrangement, are often formed in crystalline solids under plastic deformation~[17,\ 18]. If such deformation-induced dislocation walls are generated in a deformed substrate, they can cause spatial ordering of nano-islands deposited on the substrate. In order to quantitatively examine this phenomenon, we should specify stress fields created by dislocation walls on the substrate free surface. The deformation-induced dislocation walls are, in general, ragged  (Fig.~1). That is, the "end" dislocation belonging to a dislocation wall and being closest to the substrate free surface is distant by $H$ $(>0)$ from the free surface (Fig.~1). The "end" dislocation of a ragged dislocation wall serves as a stress source of disclination type, or, in short, a disclination~[18]. The stress field created by the disclination on the substrate free surface dominates over any other contributions to the stress field created by a dislocation wall on the free surface. In these circumstances, the effects of deformation-induced dislocation walls on spatial arrangement of nano-islands on the substrate free surface, in the first approximation, can be described as those associated with stress fields of the disclinations. 
\par
In the following sections of this paper we will theoretically examine the influence of  the disclinations on the characteristics of nano-island ensembles on the substrate free surface with some assumptions (that simplify the theoretical analysis) taken into consideration. In particular, the disclinations are assumed to be identical and form regular orthogonal rows equally distant from the free surface. The consideration under simplifying assumptions will allow us to reveal the key specific features of nano-island ensembles on disclinated substrates  and will  serve as a basis for further, more detailed examinations of nano-islands deposited on plastically deformed substrates.
\par
Finally, it should be noted that  the distance between disclination rows and the substrate free surface can be regulated in the following two-step  manner. At the first step, the substrate is plastically deformed, in which case low-angle grain boundaries are formed (Fig.~2a). At the second step, a new layer of a technologically controlled thickness is deposited on the deformed substrate  (Fig.~2b). The aforesaid method allows one to synthesize a composite substrate with the controlled minimum thickness between disclinations and the substrate free surface.
\par\bigskip\noindent
3. {\bf NANO-ISLANDS ON DISCLINATED SUBSTRATES. MODEL}
\par\bigskip\noindent
Let us consider a model composite system consisting of an elastically isotropic semi-infinite substrate~1 and elastically isotropic pyramid-like nano-islands~2 (Fig.~3). The shear modulus $G$ and the Poisson ratio $\nu$ are assumed to be identical for the substrate and the nano-islands. For definiteness, hereafter we confine our examination to the situation with cubic crystal lattices of the substrate and the nano-islands. In these circumstances, the interphase boundaries between the substrate and the nano-islands are characterized by the two-dimensional dilatation misfit $f=2(a_1-a_2)/(a_1+a_2)$, where $a_1$ and $a_2$ are the crystal lattice parameters of the substrate and the nano-islands, respectively. The disclination ensemble in the substrate is modeled by the two infinite orthogonal rows of periodically arranged disclinations (Fig.~3) characterized by a strength $\omega$ and distant by $H$ from the substrate free surface. The spacing between neighboring parallel disclinations (period) is designated by $p$.
\par
Let us calculate equilibrium positions of nano-islands on a disclinated substrate. In general, equilibrium positions of nano-islands on a substrate are defined as those corresponding to the minimum elastic energy density of a system. The energy density includes both the terms depending on the spatial positions of  nano-islands and the terms independent on such positions. The former terms are related to both interaction of nano-islands with stress fields induced by stress sources in the substrate and island/island interaction. In this paper we focus our consideration on disclinations as stresses sources in the substrate, that influence spatial positions of nano-islands. In doing so, in the first approximation, we consider the situation where the spacing  between neighboring nano-islands  is close to that between neighboring disclinations. This situation occurs when adatoms  (composing the nano-island phase) have high mobility at the substrate free surface; they easily move to those sites at the disclination-stressed surface that are favorable for nucleation of nano-islands. In the situation discussed, the island/island interaction is negligible because nano-islands create short-range stress fields. Also, in our first approximation analysis we will neglect the interaction of disclinations with nano-island-induced stress fields.
\par
With the aforesaid assumptions, equilibrium positions of nano-islands are those corresponding to a minimum of the elastic energy $W$ of each nano-island. The energy $W$ can be written as follows~[16]:
\begin{equation}
 W=w_0 R(c/l)V,
 \end{equation}
where $w_0$ denotes the elastic energy density (per unit volume) of a nano-island, which does not 
take into account its relaxation\footnote {In other words, $w_0$ is the energy density of a        nano-island treated as a  fragment of a thin film.}; $V$ is the nano-island volume; and $R(c/l)$ is the shape 
factor depending on the ratio $c/l$ of the nano-island height to its base length.
\par
To calculate $w_0$, we need formulae for strain fields created by two rows of disclinations (Fig.~3) on the substrate free surface. These formulae will be derived in the next section.
\par\bigskip\noindent
{\bf 4. DISLOCATION-INDUCED STRAINS ON SUBSTRATE FREE SURFACE}
\par\bigskip\noindent
Let us calculate strains induced by disclination rows (Fig.~3) at the substrate free surface. To do so, 
we will use the following known~[18] formula for the stress created by a wedge disclination with 
strength $\omega$ and the disclination line core distant by $H$ from the free surface (Fig.~4):
\begin{eqnarray}
\sigma_{yy}(x,\ y)&=&\frac{G\omega}{2\pi(1-\nu)}
\left\{\frac{1}{2}\frac{(x+H)^2+y^2}{(x-H)^2+y^2}+
\frac{(x+H)^2}{(x+H)^2+y^2}-\frac{(x-H)^2}{(x-H)^2+y^2}\right.\nonumber\\
&+&\left.\frac{2H[(x-H)^2(2H-x)+y^2(2H-3x)]}{[(x-H)^2+y^2]^2}\right\}.
\end{eqnarray}
There is one non-zero component, $\eps_{yy}(y)$, of the tensor of strain created by such a disclination 
at the free surface, which entails from formula (2)  and Hooke's law as follows:
\begin{equation}
\eps_{yy}(y)=\frac{\sigma_{yy}(0,\,y)}{2(1+\nu)G}=
\frac{\omega}{\pi(1-\nu^2)}\frac{H^2}{y^2+H^2}.
\end{equation}
\par
Now let us consider  strains, $\eps^{(1)}1_{yy}(y)$ and $\eps^{(2)}_{zz}(z)$, created by rows of 
disclinations with strength $\omega$ and line cores ($x=0$, $y=np$) and ($x=0$, $z=np$), where 
$n$ is integer (Fig.~3). The strains in question can be calculated as the sums of the strains (given by 
formula (3)) created by isolated disclinations belonging to the orthogonal rows. In doing so, we get:
\begin{eqnarray}
&& \eps_{yy}^{(1)}(y)=\frac{\omega H^2}{\pi(1-\nu^2)}\sum_{n=-\infty}^{\infty} 
\frac{1}{(y-np)^2+H^2}=\frac{\omega}{1-\nu^2}q(y/p),\\
&& \eps_{zz}^{(2)}(z)=\eps_{yy}^{(1)}(z)=\eps^{ar}(z),
\end{eqnarray}
where
\begin{equation}
q(t)=\frac{H}{p}\frac{\sinh(2\pi H/p)}{\cosh(2\pi H/p)-\cos(2\pi t)}.
\end{equation}
\par
For illustration, the dependences $q(y/p)$ are presented in Fig.~5, for various values of the parameter 
$H/p$. The minimums and  the maximums of the function $q(y/p)$ are located at the points 
$y=p(n+1/2)$ and $y=np$, respectively. The maximum and the minimum values of the function 
$q(t)$ are respectively as follows:
\begin{eqnarray}
&& q_{min}=\frac{H}{p}\tanh\frac{\pi H}{p},\\
&& q_{max}=\frac{H}{p}\coth\frac{\pi H}{p}.
\end{eqnarray}
In the limiting case with $H/p \ll 1$, we have $q_{min}\approx 0$ and $q_{max}\approx 1/\pi$.
\par\bigskip\noindent
{\bf 5. ELASTIC ENERGY AND EQUILIBRIUM POSITIONS OF NANO-ISLANDS}
\par\bigskip\noindent
In order to calculate $W$ and analyze its minimums (corresponding to equilibrium positions of nano-islands), first, let us calculate the elastic energy density $w_0$ of nano-islands, that does not take into 
account their relaxation. In the discussed situation with an in-plane stressed solid (nano-islands are treated 
as fragments of a thin film), we have:
\begin{equation}
w_0=\frac{G}{1-\nu}[(\eps_{yy}^f(y))^2+
2\nu \eps_{yy}^f(y)
\eps_{zz}^f(z)+
(\eps_{zz}^f(z))^2],
\end{equation}
where $\eps^f_{yy}(y)=\eps^{ar}(y)+f$ and $\eps^f_{zz}(z)=\eps^{ar}(z)+f$.
\par
The surfaces $W(y,\,z)=w_0R(c/l)V$ (in units of $G\omega^2R(c/l)V/[(1-\nu^2)^2(1-\nu)]$) are 
shown in Fig.~6, for various values of the parameters $p/H$ and $(1-\nu^2)f/\omega$. From Fig.~6 it 
follows that the points of minimums of $w_0(y,\,z)$ (and, therefore, $W(y,\,z)$) are located under 
diagonal lines of the disclination network at the lines $y=z \pm np$. These points are located either 
under the nodes of the disclination network (Fig.~6a) or under other points of diagonal lines 
(Fig.~6b) depending on values of parameters $p/H$ and $(1-\nu^2)f/\omega$. From Fig.~6 it follows 
that  the rate of change of the nano-island energy $W(x,\,y)$ at large values of $H/p$ is low, which 
results in a low value of the force that acts on the nano-island. This increases the probability of 
the nano-islands to be shifted from their equilibrium positions.
\par
From the condition $y_{eq}=\pm z_{eq}-np$ and formulae (4)--(6), we find that 
$\eps^{ar}(y_{eq})=\eps^{ar}(z_{eq})$ and, as a corollary, the function $w_0$ at the points 
($y_{eq}$, $z_{eq}$) of its minimums can be written as follows:
\begin{equation}
w_0(y_{eq},\, z_{eq})=2G\frac{1+\nu}{1-\nu}(\eps^{ar}(y_{eq})+f)^2=
2G\frac{1+\nu}{1-\nu}\frac{\omega^2}{(1-\nu^2)^2}
\left(q(y_{eq}/p)+\frac{(1-\nu^2)f}{\omega}\right)^2.
\end{equation}
Formula (10) gives the conditions at which nano-islands are located at points under the centers or nodes of  the 
disclination network cells (squares) or at some intermediate points under diagonal lines of the 
disclination network. So, there are the following situations depending on the parameters $H/p$ and 
$(1-\nu^2)f/\omega$:
\par
(i) $q_{min}+f(1-\nu^2)/\omega>0$, that is, $f/\omega>0$ or ($f/\omega<0$ and 
$\gamma>1/q_{min}$), where $\gamma=\omega/[f(1-\nu^2)]$. In this situation, the points of 
minimum of $w_0$ coincide with the points $y=p(n+/12)$; the equilibrium positions of the     nano-islands are located under the centers of the disclination network cells.
\par
(ii) $q_{max}+f(1-\nu^2)/\omega<0$, that is, $f/\omega<0$ and $\gamma<1/q_{max}$. In this 
situation, the points of minimum of $w_0$ coincide with the points $y=np$  of maximum of the 
function $q(y,\,p)$; the equilibrium positions of the nano-islands are located under the nodes of the 
disclination network cells.
\par
(iii) $f/\omega<0$ and $1/q_{max}<\gamma<1/q_{min}$. In this situation, the equilibrium positions 
of nano-islands are located under the points of the diagonal lines of the disclination network, which 
satisfy the condition $q(y_0)=-f(1-\nu^2)/\omega$.
\par
In Fig.~7 the state diagram of the system under consideration is shown in the coordinates ($H/p$, $\gamma$). The regions I (with $\gamma>1/q_{min}(H/p)$), II (with 
$1/q_{max}(H/p)<\gamma<1/q_{min}(H/p)$) and III (with $\gamma<1/q_{max}(H/p)$)
on the diagram correspond to the situations (i), (ii) and (iii), respectively.
\par
The situations (i) and (ii), with nano-islands homogeneously distributed over the substrate free 
surface occur  either at low values, $|\omega|<\omega_{c1}=(1-\nu^2)f/q_{max}$, of the disclination 
strength modulus $|\omega|$ or at high values, $|\omega|>\omega_{c2}=(1-\nu^2)f/q_{min}$, of 
$|\omega|$; see regions I and III in Fig.~7. The critical values of $\omega_{c1}$ and $\omega_{c2}$ 
decrease with increase of $H/p$. In doing so,  regions I and III, which are characterized by  a 
homogeneous distribution of nano-islands, expand. In the limiting case with $H/p \to 0$, we have:
$\omega_{c1} \to \pi (1-\nu^2)|f|$ and $\omega_{c2} \to \infty$. That is, the distribution of     nano-islands is homogeneous only at low values of $|\omega|$ ($<\omega_{c1}$). For $H/p\gg 1$, we 
have: $\omega_{c1}\to 0$ and $\omega_{c2}\to 0$, in which case the distribution of nano-islands is 
homogeneous at any values of disclination strength $\omega$.
\par\bigskip\noindent
{\bf 6. CONCLUDING REMARKS}
\par\bigskip\noindent
In this paper we have theoretically described the influence of disclinations (that terminate dislocation walls) in plastically 
deformed substrates on the spatial arrangement of nano-islands deposited  on the substrate. The 
influence in question results from the elastic interaction between the disclinations and the        nano-islands. In the first approximation, deformation-induced disclination ensembles in substrates have 
been modeled as regular orthogonal rows of identical  wedge disclinations equally distant from the 
substrate free surface (Fig.~3). In the framework of the model,  it has been shown that nano-islands 
deposited on a disclinated substrate have preferable equilibrium spatial positions caused by interaction of the nano-islands with the stress 
fields of the disclinations. So, there are the following preferable positions of nano-islands  depending 
on the parameters of the substrate/nano-island system: equilibrium positions of nano-islands are 
located under the  (i) centers, (ii) nodes, (iii) diagonal lines of the disclination network cells (see 
Fig.~6). The aforesaid potentially allows one to design and fabricate nano-island ensembles with 
desired spatial arrangements (and, therefore, functional characteristics) using technologically 
controlled and/or influenced parameters (misfit parameter, distance between disclinations, etc.) of substrate/nano-island systems as an input.
\par\bigskip\noindent
{\bf ACKNOWLEDGEMENTS}
\par\bigskip\noindent
This work was supported by INTAS (grant 99-01216), the Office of US Naval Research (grant N00014-99-1-0569) and, in part (for I.A.O.), by the Russian Foundation of Basic Researches (grant 98-02-16075) and the Volkswagen Foundation (research project 05019225).
\newpage\noindent
{\bf REFERENCES}
\par\bigskip\noindent
[1] N. N. Ledentsov, V. M. Ustinov, V. A Shchukin, P. S. Kop'ev,
Zh. I. Alferov, D. Bimberg
\par// Semiconductors
{\bf 32} (1998) 343.\par\noindent
[2] V. A. Shchukin and D. Bimberg //
   Rev. Mod. Phys. {\bf 71} (1999) 1125 .\par\noindent
[3] V.M. Ustinov, In: {\it Nanostructured Films and Coatings}, 
NATO ARW Series
(eds. G.M.
\par Chow, I.A. Ovid'ko and T. Tsakalakos) (Kluwer, Dordrecht 2000) 
p.~41.\par\noindent
[4] L. Goldstein, F. Glas, J.Y. Marzin, M.N. Charasse and G. Le Roux //
 Appl. Phys. Lett.\par {\bf 47} (1985) 1099.\par\noindent
[5] G.E. Cirlin, N.K. Polyakov, V.N. Petrov, V.A. Egorov, D.V. Denisov, 
B.V. Volovik, V.M.
\par Ustinov, Zh.I. Alfer4ov, N.N. Ledentsov, R. Heitz, D. Bimberg, 
N.D. Zakharov, P. Werner 
\par
and U. G\"{o}sele, 
Mater. Phys. Mech. {\bf 1} (2000) 15.\par\noindent
[6] J. Eaglesham and M. Cerullo //
 Phys. Rev. Lett. {\bf 64} (1990) 1943.\par\noindent
[7] Y.-W. Mo, D.E. Savage, B.S. Shwartzentruber and M.G. Lagally //
 Phys. Rev. Lett. {\bf 65}
\par
(1990) 1020.\par\noindent
[8] M. Krishnamurthy, J. S. Drucker and J.A. Venables //
  J. Appl. Phys. {\bf 69} (1991) 6461.\par\noindent
[9] P. Shittenhelm, M. Gail, J. Brunner, J. F. N\"{u}tzel and G. Abstreiter //
  Appl. Phys. Lett.
\par
{\bf 67} (1995) 1292.\par\noindent
[10] T.I. Kamins, E.C. Carr, R.S. Williams and S.I. Rosner //
  J. Appl. Phys. {\bf 81} (1997) 211.\par\noindent
[11] S.A. Chaparro, J. Drucker, Y. Zhang, D. Chandrasekhar,
M.R. McCartney and D.J.
\par\ 
Smith //
Phys. Rev. Lett. {\bf 83} (1999).1199.\par\noindent
[12] S.A. Chaparro, Y. Zhang, J. Drucker and D.J. Smith //
  J. Appl. Phys. {\bf 87} (2000) 2245.\par\noindent
[13] J. Tersoff, C. Teichert and M.G. Lagally //
 Phys. Rev. Lett. {\bf 76} (1996) 1675.\par\noindent
[14] T.T. Ngo, P.M. Petroff, H. Sakaki and J.L. Merz //
 Phys. Rev. B{\bf 53} (1996) 9618.\par\noindent
[15] A.E. Romanov, P.M. Petroff and J.S. Speck //
 Appl. Phys. Lett. {\bf 74} (1999) 2280.\par\noindent
[16] A. Bourret //  Surf. Sci. {\bf 432} (1999) 37.\par\noindent
[17] J.P. Hirth  and J. Lothe {\it Theory of Dislocations} (Wiley:
New York, 1982).\par\noindent
[18] A.E. Romanov and V.I. Vladimirov, In: 
{\it Dislocations in Solids}, vol. 9, edited by F.R.N. Nabarro 
(North Holland, Amsterdam, 1992) p.~191.
\newpage
\centerline{\bf Figure Captions}
\par\bigskip\noindent
Fig. 1. Dislocation walls in plastically deformed substrate are terminated by stress sources of disclination type.\\
Fig. 2. Formation of disclinations in substrate. (a) Plastic deformation of substrate. (b) Deposition of layer on plastically deformed substrate.\\
Fig. 3. Nano-islands on substrate with disclination network consisting of two infinite orthogonal rows of wedge disclinations.\\
Fig. 4. Wedge disclination in semi-infinite elastic medium.\\
Fig. 5. Dependences of parameter $q$ on non-dimensional coordinate $y/p$, for $H/p=1$, $H/p=0.3$, and $H/p=0.1$ (from top to bottom).\\
Fig. 6. Energy $W$ (in units of $G\omega^2R(c/l)V/[(1-\nu^2)^2(1-\nu)]$) accumulated in nano-islands as a function of nano-island coordinates $x$ and $y$, for (a) $(1-\nu^2)f/\omega=1$,  $H/p=0.1$; (b) $(1-\nu^2)f/\omega=-1$,  $H/p=0.2$; and (c) $(1-\nu^2)f/\omega=0.15$,  $H/p=0.1$.\\
Fig. 7. Dependences of parameters $q_{max}$ and $q_{min}$ on $H/p$.
\end{document}

