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\centerline{\large\bf MISFIT\ \ DISLOCATION\ \ WALLS\ \ IN\ \ FILMS    }
\centerline{\large\bf           WITH\ \ ALTERNATE\ \ LAYERS            }
\par\vspace{0.5cm}
\begin{center}
{\bf D.~Yu.~Baskov,\ \ M.~Yu.~Gutkin,\ \ I.~A.~Ovid'ko and A.~G.~Sheinerman}\\
\vspace{0.5cm}
Institute for Problems of Mechanical Engineering, Russian Academy of Sciences,\\
Bolshoj 61, Vasil.~Ostrov, St.~Petersburg, 199178 Russia                    \\
email: ovidko@def.ipme.ru
\end{center}
                                                       \par\bigskip\noindent
{\bf Abstract.}
We suggest a new mechanism for misfit strain accommodation in misfitting
multilayer films with alternate layers through the generation of misfit 
dislocation walls. The detailed analysis of the necessary conditions for their
appearance shows that, similar to one-layer films, misfit dislocation walls 
form above some critical film thicknesses which can be increased by varying 
the relation between thicknesses of adjoining film layers.
                                                                      \newpage
{\bf 1.~INTRODUCTION}
                                                         \par\bigskip\noindent
Solid films and composite multilayer coatings are widely used in contemporary
high technologies due to their unique electronic and optical properties 
(e.~g., [1--3]). The properties of films and coatings strongly depend on the 
elastic strains induced by the misfit of crystal lattices of different layers
and may degrade in the presence of defects. In single crystalline films, the 
most effective and wide-spread pathway for the accommodation of such misfit
strains lies in the generation of misfit dislocations (MDs) at or near 
interfaces~[4--7]. For thin one-layer films, MDs form commonly two-dimensional
ensembles in the interface plane. In this case, if the substrate is much 
thicker than the film, the critical film thickness above which MDs are likely
to nucleate depends on the misfit parameter only. For multilayer films on 
similar substrates, the system energy and the critical thickness depend also  
on the relations between the crystal lattice misfits and layer thicknesses. 
The extra degrees of freedom connected with the existence of 
different film layers lead to the opportunity of the formation of MD 
configurations which are not found in one-layer films. In particular, in 
multilayer systems MDs can be arranged into the arrays of dislocation dipoles,
with dislocations of opposite sign located at different interface boundaries. 
This situation has been extensively studied for the case of capped films 
(which are sandwiched between two substrate layers) (e.~g., [7-11]). In recent
work [12], another situation has been analyzed for the film consisting of 
alternate layers where MDs are situated either at the film/substrate interface
or at the interface nearest to the film/substrate one. 
\par
In the present paper, we will focus on the generation of the walls of complete
edge MDs at the interfaces of the film composed of alternate layers. The 
rotational mechanisms of misfit strain accommodation realized through the
formation of disclinations or dislocation walls in the case of one-layer 
film have already been observed experimentally [13--18] and analyzed 
theoretically [17--22]. It has been demonstrated that misfit disclinations
[13--22] or MD walls [19--22] can be generated at twin or grain boundaries, 
in single crystalline films on amorphous substrates and in nano- or 
polycrystalline films. In multilayer films, dislocation walls can be formed
by slip of dislocations from their sources. In the following, we will find 
the conditions for the formation of a first (individual) MD wall. For 
simplicity, we will suppose that dislocations composed the wall, are regularly
spaced and separated from each other by a pair of alternate film layers.
                                                               \par\bigskip
{\bf 2.~MISFIT DISLOCATION WALL IN MULTILAYER FILM}
                                                       \par\bigskip\noindent
Consider a system that consists of a semi-infinite substrate and a multilayer
film of thickness $H$ comprised of $N$ pairs of alternate layers $\alpha$ and
$\beta$ of thickness $h_1$ and $h_2$, respectively [$H=N(h_1+h_2)$] (Fig.~1).
The film and the substrate are supposed to be elastically isotropic solids and
have the same shear modulus $G$ and Poisson's ratio $\nu$. The misfit of the 
substrate, layers $\alpha$ and layers $\beta$ crystal lattices is assumed to 
be dilatational and characterized by the misfit parameters 
$f_1=(a_s-a_\alpha)/a_s$ and $f_2=(a_s-a_\beta)/a_s$ equal in 
the absence of MDs to the elastic strain within layers $\alpha$ and $\beta$, 
respectively. In the latter relation, $a_s$, $a_\alpha$ and $a_\beta$ denote 
the crystal lattice parameters of the substrate, layers $\alpha$ and layers 
$\beta$. Let the misfit parameters $f_1$ and $f_2$ be positive, and the lines
and Burgers vectors of the generated MDs be parallel to the film/substrate 
interface (Fig.~1).
\par
To determine the necessary conditions for the formation of an MD wall, we 
will use the standard technique comparing the energies of the system with and 
without the wall. The energy $W$ of the system with an MD wall extending from
the film/substrate interface to the free surface (Fig.~1a) per unit length of
dislocations can be presented as the sum of three terms:
\begin{equation}
                      W = W^f + W^w + W^{f-w}\;,                        % (1)
\end{equation}
where $W^f$ is the misfit strain energy, $W^w$ the self-energy of the MD wall
and $W^{f-w}$ the interaction energy of the misfit strain and the MD wall. 
When the MD wall is absent, the energy of the system is related to the misfit
strain only and equal to $W^f$. The necessary condition for the MD wall 
generation is that the system energy $W$ with the MD wall be smaller than the
system energy $W^f$ prior to the MD wall introduction, $\Delta W=W-W^f<0$, 
which yields
\begin{equation}
                          W^w + W^{f-w} < 0\;.                          % (2)
\end{equation}
The energies $W^w$ and $W^{f-w}$ appearing in (2) will be calculated in the 
next section.
                                                        \newpage
{\bf 3.~ENERGY OF MISFIT DISLOCATION WALL}
                                                        \par\bigskip\noindent
The self-energy $W^w$ of the MD wall shown in Fig.~1a can be written as
\begin{equation}
     W^w = \sum_{i=1}^N \left[W^d_i + \sum_{j=1}^{i-1} W^{d-d}_{i,j}\right]
         + NW^c\;,                                                       % (3)
\end{equation}
where $W^d_i$ is the self-energy of the $i$th MD (MDs are enumerated 
sequentially from $1$, for the MD nearest to the free surface, to $N$ for the 
MD at the film/substrate interface), $W^{d-d}_{i,j}$ is the interaction energy
of the $i$th and $j$th dislocations, and $W^c$ is the dislocation core energy.
The self-energies $W^d_i$ of MDs are given by~[23]
\begin{equation}
     W^d_i=\frac{Db^2}{2}\left(\ln\frac{2d_i}{r_0}-\frac{1}{2}\right),  % (4)
\end{equation}
where $b$ is the value of the dislocation Burgers vectors, 
$D=G/[2\pi(1-\nu)]$, $d_i=iH/N$ and $r_0$ is the cut-off radius of dislocation
elastic fields on its core; we put $r_0=b$. The energy $W^{d-d}_{i,j}$ 
follows from~[24] as
\begin{equation}
     W^{d-d}_{i,j} = -b\int_{r_0}^\infty
                       \sigma_{xy}^d(x=d_j-d_i,\,y,\,d_i)\,d\,y\;,      % (5)
\end{equation}
where $\sigma_{xy}^d(x,\,y,\,d_i)$ is the stress component of the $i$th 
dislocation lying at a distance $d_i$ from the free surface. The expression 
for $\sigma_{xy}^d(x,y,d_i)$ can be adapted from~[25]: 
\begin{equation}
     \sigma_{xy}^d(x,y,d_i) = Db
        \left\{ - \frac{y}{r_{-}^2} + \frac{2y(x-d_i)^2}{r_{-}^4} 
                + \frac{y}{r_{+}^2} + \frac{2y(d_i^2+4d_ix+x^2)}{r_{+}^4}
                + \frac{16d_iyx(x+d_i)^2}{r_{+}^6} \right\},            % (6)
\end{equation}
where $r^2_{\pm}=(x\pm d)^2+y^2$. It follows from (5), (6) and the condition 
$H/N\gg b$ that
\begin{equation}
     W^{d-d}_{i,j} = Db^2 \left( \ln\frac{d_i+d_j}{|d_i-d_j|}
                               - \frac{2d_id_j}{(d_i+d_j)^2} \right).   % (7)
\end{equation}
The dislocation core energy is approximated [25] as $W^c\approx Db^2/2$.
\par
Combining (3), (4) and (7), one obtains:
\begin{equation}
     W^w = \frac{Db^2}{2} 
              \left\{ \sum_{i=1}^N 
                 \left[ \ln\frac{2d_i}{b} 
                    + 2\sum_{j=1}^{i-1} \left( \ln\frac{d_i+d_j}{d_i-d_j}
                                             - \frac{2d_id_j}{(d_i+d_j)^2}
                                        \right) \right] 
                    + \frac{N}{2} \right\}.                             % (8)
\end{equation}
The energy  $W^{f-w}$ of the interaction of the MD wall with the misfit stress 
field is given by
\begin{equation}
     W^{f-w}=-b\sum_{i=1}^N\int_0^{d_i}\sigma_{yy}^f(x)\,d\,x,          % (9)
\end{equation}
where $\sigma_{yy}^f(x)$ is the misfit stress tensor component, which is given 
by [4,\ 26]
\begin{equation}
     \sigma_{yy}^f(x)=4\pi (1+\nu)D\sum_{k=0}^{N-1}
                  \left\{f_1[\Theta(x-x^{(2k+1)})-\Theta(x-x^{(2k+2)})]
                       + f_2[\Theta(x-x^{(2k)  })-\Theta(x-x^{(2k+1)})]\right\},
                                                                        % (10)
\end{equation}
where $\Theta(t)$ is the Heaviside function, $\Theta(t)=1$ for $t>0$ and 
$\Theta(t)=0$ for $t<0$; $x^{(2i+1)}=iH/N+h_2$, $x^{(2i)}=iH/N$, $i=0,...,N$.
Substitution of (10) to (9) gives
\begin{equation}
                    W^{f-w}=-2\pi(1+\nu)(N+1)DbHf_e\;,                  % (11)
\end{equation}
where the effective misfit $f_e=(f_1h_1+f_2h_2)/(h_1+h_2)$ is introduced. 
The energy change $\Delta W$ due to the MD wall shown in Fig.~1a is now given 
by the sum of (8) and (11). Taking into account (8) and (11), necessary 
condition (2) for the generation of the MD wall can be rewritten as
\begin{equation}
     \frac{b}{(N+1)H}\left\{\sum_{i=1}^N\left[\ln\frac{2d_i}{b}
                         + 2\sum_{j=1}^{i-1} 
                               \left(\ln\frac{d_i+d_j}{d_i-d_j}
                                   - \frac{2d_id_j}{(d_i+d_j)^2}\right)\right]
                         + \frac{N}{2}\right\} < 4\pi(1+\nu)f_e.        % (12)
\end{equation}
\par
The energy change $\Delta {\widetilde W}={\widetilde W}-W^f$ due to the 
generation of the MD wall shown in Fig.~1b, which is defined as the 
difference between the system energy ${\widetilde W}$ with such an MD wall and
the system energy $W^f$ without MD walls, is derived similarly to the 
expression for the $\Delta W$. The final result for $\Delta {\widetilde W} $ 
is
\begin{eqnarray}
     \Delta {\widetilde W}&=&\frac{Db^2}{2}\left\{\sum_{i=1}^N
                         \left[\ln\frac{2{\tilde d_i}}{b}+2\sum_{j=1}^{i-1}
         \left(\ln\frac{{\tilde d_i}+{\tilde d_j}}{{\tilde d_i}-{\tilde d_j}}
             - \frac{2{\tilde d_i}{\tilde d_j}}{({\tilde d_i}+{\tilde d_j})^2}
         \right)\right] + \frac{N}{2}                      
                                                            \right.\nonumber\\
                      & & \phantom{\frac{Db^2}{2}\left\{\right.}\left. 
                         - 4\pi(1+\nu)\;\frac{H}{b}
                            \left[(N+1)f_e - \frac{2f_1h_1}{h_1+h_2}\right]
                            \right\},                                   % (13)
\end{eqnarray}
where ${\widetilde d_i}=iH/N-h_1$. From (13) and the condition 
$\Delta {\widetilde W}<0$ it follows that
\begin{eqnarray}
     & & \frac{b}{(N+1)H}\left\{\sum_{i=1}^N
                        \left[\ln\frac{2 {\tilde d_i}}{b}
                            + 2\sum_{j=1}^{i-1} 
                                 \left(\ln\frac{{\tilde d_i}+{\tilde d_j}}
                                               {|{\tilde d_i}-{\tilde d_j}|}
                        - \frac{2{\tilde d_i}{\tilde d_j}}
                               {({\tilde d_i}+{\tilde d_j})^2}
                 \right)\right] + \frac{N}{2}\right\}
                                                     \nonumber\\
    &<& 4\pi(1+\nu)\left(1-\frac{2f_1h_1}{(N+1)(f_1h_1+f_2h_2)}\right)f_e\;.
                                                                        % (14)
\end{eqnarray}
\par
The plots $\Delta W$ and $\Delta{\widetilde W}$ in units of $Db^2/2$ against 
$H/b$ are shown in Fig.~2 for the case $h_2/h_1=2$, $N=5$ and 
$4\pi(1+\nu)f_e=0.15$. Curve 1 corresponds to the dependence of $\Delta W$ on
$H/b$, while curves 2 and 3 display the dependences of $\Delta{\widetilde W}$
on $H/b$ for the following two cases: $4\pi(1+\nu)f_1=0.1$ and $h_2/h_1=1$ 
(curve 2); and $4\pi(1+\nu)f_1=0  $ and $h_2/h_1=2$ (curve 3). One can 
conclude that the MD walls can nucleate in the film above some critical 
thicknesses $H_c$ and ${\widetilde H}_c$ corresponding to zero values of 
$\Delta W$ and $\Delta{\widetilde W}$, respectively. As follows from Fig.~2, 
if $f_1$ and $f_2$ as well as $h_1$ and $h_2$ are of the same order, 
$\Delta W$ is lower than $\Delta {\widetilde W}$ and $H_c<{\widetilde H}_c$ 
that could easy be predicted. For given $f_e$, $N$ and $H$, 
$\Delta {\widetilde W}$ and ${\widetilde H}_c$ decrease with increasing $f_1$ 
or decreasing the ratio $h_2/h_1$. At $H<H_c$ the MD walls shown both in 
Fig.~1a and in Fig.~1b (hereafter referred to as MD walls of first type and 
MD walls of second type, respectively) do not form. At 
$H_c<H<{\widetilde H}_c$ the generation of an MD wall of first type is 
possible. If $H>{\widetilde H}_c$, MD walls of both first and second type are 
favoured. 
\par
Figure 3 illustrates the plots of the critical thicknesses $H_c$ (curves 1 
and 2) and ${\widetilde H}_c$ (curves $1'$ and $2'$) against $4\pi(1+\nu)f_e$ 
for the case $4\pi(1+\nu)f_1=0.1$ and $h_2/h_1=1$. The plots of $H_c$ and 
${\widetilde H}_c$ (curves 1 and $1'$, respectively) shown for $N=5$ lie 
below the corresponding plots (curves 2 and $2'$) drawn for the case $N=10$. 
It is seen in Fig.~3 that both $H_c$ and ${\widetilde H}_c$ increase with $N$.
This result may seem paradoxical but it becomes clear if one takes in mind 
that within our model, the number of MDs in the wall increases automatically 
with $N$, thus giving rise to the energy (3). In fact, (3) contains terms 
which vary linearly with $N$ (dislocation self strain energies and core 
energies) as well as the interaction energy term which is proportional to 
$N^2$. On the other hand, the energy (11) of elastic interaction between MD 
walls and misfit stress varies linearly with $N$. As a result, the positive 
energy term (3) grow faster than negative one (11) as $N$ increases, and their
balance needs larger $H_c$ (or ${\widetilde H}_c$).
                                                                 \par\bigskip
{\bf 4.~CONCLUDING REMARKS}
                                                                  \par\bigskip
In this paper, we have determined the generation conditions for misfit 
dislocation walls in the multilayer film with alternate layers. Within our 
model, the film and the substrate are isotropic solids with equal elastic 
moduli, $G$ and $\nu$, and two dilatational misfits, $f_1$ and $f_2$, of 
adjoining crystal lattices. Misfit dislocations composed the walls have 
regularly been spaced with a separation of two film layers. 
\par
The principal results of our work are as follows:
\par
(i) misfit dislocation walls in a film with alternate layers can form in some 
range of parameters including film thickness $H$, misfits $f_1$ and $f_2$, 
the number $N$ of film layers and the ratio $h_2/h_1$ of adjoining layer 
thicknesses;
\par
(ii) for given $f_1$, $f_2$, $N$ and $h_2/h_1$, misfit dislocation walls can 
be generated above some critical thicknesses;
\par
(iii) the critical film thicknesses increase with the number $N$ of film 
layers;
\par
(iv) for large enough number $N$ of film layers, the critical thicknesses 
depend mainly on the effective misfit $f_e=(f_1h_1+f_2h_2)/(h_1+h_2)$ 
which can be varied (in particular, decreased) by varying the ratio $h_2/h_1$ 
of layer thicknesses.
\par
The results of this paper are important for technological applications of 
multilayer film/substrate composites. In particular, they can be used for estimating the structural and functional stability of real multilayer single crystal film/substrate composites as well as for describing possible accommodation processes in multilayer films with nanocrystalline layers of alternate composition obtained by thermal spray synthesis [27--29].
                                                                 \par\bigskip
{\bf 5.~ACKNOWLEDGEMENTS}
                                                                 \par\bigskip
This work was supported, in part, (for I.A.O.~and A.G.S.) by INTAS (grant 
99-0216), in part, (for M.Yu.G.~and I.A.O.) by the Volkswagen Foundation 
(research project 05019225) and, in part, (for I.A.O.) by the Russian 
Foundation of Basic Researches (grant 98-02-16075) and the Office of US Naval 
Research (grant N00014-99-1-0569). The support of the Russian Research Council
``Physics of Solid-State Nanostructures'' (grant 97-3006) and Russian 
Foundation of Basic Research (grant 000-01-00482) for M.Yu.G.~is also 
acknowledged.
                                                                     \newpage
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                                                                     \newpage 
{\bf FIGURE CAPTIONS}
                                                         \par\bigskip\noindent
Fig.~1. Misfit dislocation walls in  a multilayer film.
                                                         \par\bigskip\noindent
Fig.~2. Dependences of energy variations  $\Delta W$ (curve 1) and 
        $\Delta {\widetilde W}$ (curves 2 and 3) (in units of $Db^2/2$) due 
        to misfit dislocation walls on film thickness $H/b$. Curve 2 is drawn
        for $4\pi(1+\nu)f_1=0.1$ and $h_2/h_1=1$, while curve 3 corresponds to
        $4\pi(1+\nu)f_1=0$ and $h_2/h_1=2$.
                                                         \par\bigskip\noindent
Fig.~3. The critical thicknesses $H_c$ (curves 1 and 2) and ${\tilde H_c}$ 
        (curves $1'$ and $2'$) as functions of $4\pi(1+\nu)f_e$ shown for 
        $N=5$ (curves 1 and $1'$) and $N=10$ (curves 2 and $2'$).


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