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\begin{document}

\runninghead{Dynamical effects and vacancy motion in silicon at high temperature}{Dynamical effects and vacancy motion in silicon at high temperature}

\normalsize\textlineskip
\thispagestyle{empty}
\setcounter{page}{1}

\copyrightheading{}			%{Vol. 0, No. 0 (1993) 000--000}

\vspace*{0.88truein}

\fpage{1}
\centerline{\bf DYNAMICAL EFFECTS AND VACANCY MOTION}
\vspace*{0.035truein}
\centerline{\bf IN SILICON AT HIGH TEMPERATURE}
\vspace*{0.37truein}
\centerline{\footnotesize ENRICO SMARGIASSI}
\vspace*{0.015truein} \centerline{\footnotesize\it Centre Europ\'een
de Calcul Atomique et Mol\'eculaire,}
\baselineskip=10pt
\centerline{\footnotesize\it ENS Lyon, 69364 Lyon CEDEX 07,
France}
\vspace*{10pt}
\centerline{\normalsize and}
\vspace*{10pt}
\centerline{\footnotesize ROBERTO CAR}
\vspace*{0.015truein}
\centerline{\footnotesize\it Departement de Physique, University of Geneva}
\baselineskip=10pt \centerline{\footnotesize\it Quai E. Ansermet,
Geneva, Switzerland}
\vspace*{0.225truein}
\publisher{(received date)}{(revised date)}

\vspace*{0.21truein}
\abstracts{We report a first-principles Molecular Dynamics investigation of the
atomic motion in Silicon in the presence of an artificially created
vacancy at high temperature ($T\ge 1200 K$). We observe that atomic
diffusion events are affected by strong dynamical correlations. At
temperatures close to the melting point we discover characteristic
premelting phenomena which involve simultaneous jumps of several atoms
and introduce a large amount of disorder in the structure.}{}{}

%\vspace*{10pt}
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\section{Introduction}	%) A SECTION HEADING
\vspace*{-0.5pt}
\noindent

Atomic diffusion in semiconductors governs the mobility of dopant
impurities and is at the basis of several important processes for the
fabrication of electronic devices\cite{1}. Not surprisingly this
field has been the object of intense study over the last 30 years. In
spite of this, even for the case of a well-known material like silicon
and a relatively simple and basic process like self-diffusion
(i.e. the diffusion of host atoms at regular lattice sites) the debate
on which physical mechanisms are responsible for atomic motion has
lasted for a long time. Competitors for this process are three basic
mechanisms, namely vacancy, self-interstitial and and intrinsic
concerted exchange. Experimental studies can give only indirect
evidence on the microscopic mechanisms responsible for
self-diffusion. On the other hand, systematic {\sl ab-initio} total
energy theoretical studies have been limited for a long time to $T=0$
energetics\cite{2,3,4} due to their large
requirements in terms of computer resources. They indicate that all
the above three mechanisms should have activation energies within the
experimental range\cite{1,5} of 4 -- 5 eV. More recently,
rate theory and lattice dynamics calculations for the formation and
migration entropies have
appeared\cite{6,7}. However, before drawing any
conclusion, finite $T$ calculations have to be performed consistently
for all the mechanisms. In fact, dynamical effects are not accounted
for by the above theoretical techniques and can only be studied using
Molecular Dynamics (MD) or similar methods. Extensive MD studies of
defect motion carried out on model substances like the
``Lennard-Jonesium'', where important dynamical correlations were
found at high temperature\cite{8}, suggest that finite-$T$
calculations are indeed essential for a proper understanding of
diffusion.

\par

The study of defects in real materials has been hindered for a long
time by the difficulty of finding sufficiently reliable interatomic
potentials, on which the entire machinery of MD relies. Finding a good
interatomic potential for point defects is notoriously difficult, as
one has to deal with different chemical environments at the same time:
the bulk far from the defect (in particular the vibrational spectrum
has to be described accurately) and the defect neighborhood where the
local electronic structure can be quite different. This has proven to
be difficult even for simple metals\cite{9}. One expects this
problem to be especially acute in semiconductors, where the
possibility of extensive rebonding phenomena casts doubts on the
reliability of any few-body potential fitted to bulk properties. To
make matters worse, in silicon the concept itself of vacancy as simple
pointlike defect may need to be substantially modified. Indeed, in
order to explain the high entropy of diffusion, it has been suggested
that, at high temperature, vacancies and self-interstitials should
extend over several lattice sites, forming kinds of disordered local
zones\cite{10}.

\par

In this paper we study in detail, with Molecular Dynamics techniques,
atomic diffusion events induced in silicon at high temperature by an
artificially rather than thermally created vacancy. To overcome the
above difficulties with classical interatomic potentials we dispensed
from them altogether and adopt the technique of of first-principles
MD\cite{11}, which has already been successfully applied to model
other diffusion processes, like hydrogen diffusion in crystalline and
amorphous silicon\cite{12}. Within this approach the forces acting on
the atoms arise directly from the Born-Oppenheimer (BO) electronic
ground-state. This is crucial to describe accurately the subtle
chemical effects resulting from bond breaking and forming process in
covalent materials.

\textheight=7.8truein
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\section{Method}
\noindent

The present simulations were performed at fixed volume using a
periodically repeated simple cubic cell containing 63 atoms and a
neutral vacancy. We used a norm-conserving pseudopotential\cite{13}
with only s-nonlocality and expanded the electronic wavefunctions in
plane waves with a cutoff of 12 Ry. We performed a number of $T=0$
calculations to test the reliability of our model and of our chosen
parameters, in particular cell size and k-point sampling; tests done
using a cell containing 127 atoms did not show appreciably different
results. The resulting relaxed vacancy at $T=0$ has $D_{2d}$ symmetry,
in agreement with low temperature experiments\cite{14}. The distortion
consists mainly of a pairing and an inward breathing displacement of
the neighboring atoms, in agreement with other
calculations\cite{15}. The magnitude of both these displacements is of
the order of 0.5 a.u. The distortion around the vacancy changes only
slightly by including p-nonlocality and/or by doubling the size of the
cell. The computed vacancy formation energy is $E_f = 2.7$ eV
($E_f=3.2$ eV when also p-nonlocality is included), to be compared
with a fully converged value of $E_f = 4.1$ eV\cite{16}. Similarly,
with our choice of parameters, the vacancy migration energy $E_m$,
i.e. the energy difference between the fully relaxed split- and
equilibrium-vacancy configurations, is well converged and equal to 0.3
eV (experimentally\cite{14} $E_m \approx$ 0.4 eV). We conclude that
our simulations provide a realistic description of the atomic motion
induced by a vacancy\fnm{a}. \fnt{a}{Our calculations are performed
at fixed lattice parameter ($a =$ 10.263 a.u., appropriate to the
$T=0$ crystal), while experiments are performed at constant
pressure. We have explicitly checked that expanding the lattice
parameter to a value appropriate to $T=1400$ K does not change
appreciably $E_m$.}

\par

The equations of motion were integrated with a time step of 0.484 fs.
The ionic temperature was controlled by a Nos\'e
thermostat\cite{17}. An electronic ``mass'' parameter $\mu = $
3500 a.u. was used in order to speed up the time evolution by allowing
a larger time step. The price to pay for this is that some energy
transfer from ions to electrons occurs. One could put another Nos\'e
thermostat on the electrons so as to keep them close to the
instantaneous BO surface. We used the simpler method of quenching the
electrons on the ground state systematically every 500 time steps;
this was found to be sufficient to keep the system close enough to the
BO surface. We spanned the temperature range accessed in
self-diffusion experiments with runs at $T=$ 1200, 1400, and 1600
K. To allow for equilibration an initial period of 2 ps was excluded
from all our measurements; we observed that after that period the
calculated power spectra of the system dynamics had stabilized,
indicating that at least approximate equipartition had been reached
between the ionic degrees of freedom. The observation times were
13.55, 13.06, and 7.74 ps at $T=$ 1200, 1400, and 1600 K,
respectively.



\section{Results}
\noindent

Diffusion events are easily identified by plotting the time evolution
of the instantaneous distances of atoms from their original
equilibrium lattice sites. A schematic illustration of the most common
diffusion events is given in Fig. 1,
\begin{figure}[htbp]
\vspace*{5.5truecm} %ORIGINAL SIZE=1.6TRUEIN x 100% - 0.2TRUEIN
\fcaption{Dynamical behavior of atoms in a diamond lattice in the
presence of a vacancy. A: thermal vibration. B: first-neighbor
jump. C: Double atomic jump. D: Second-neighbor jump.}
\end{figure}
along with the illustration of how these events appear in such a plot. 
The plots for the actual runs are reported in Fig. 2, 
\begin{figure}[htbp]
\vspace*{10.5truecm}		%ORIGINAL SIZE=1.6TRUEIN x 100% - 0.2TRUEIN
\fcaption{Instantaneous distances of the atoms from
their original lattice positions. The arrows indicate first, second,
and third neighbor distances in the perfect crystal.}
\end{figure}
where the four panels refer to $T=$ 1200, 1400, 1600, and 1750 K,
respectively\fnm{b}. \fnt{b}{A videotape of the atomic motion in part
of the 1200 K run exists. Further details may be obtained by
contacting the author for correspondence (ES).} The plots at $T=$ 1200
and 1400 K report only segments of significantly longer runs. For
temperatures up to 1600 K most atoms simply vibrate around their
equilibrium positions and only occasional diffusion events occur. At
$T=$ 1200 and 1400 K, these consist of first-neighbor jumps (case b in
Fig. 1), while at $T=$ 1600 K second- (case d) and third-neighbor
jumps are also detected. A qualitatively different picture appears at
$T=$ 1750 K where all the atoms simultaneously display diffusive
behavior after a very short time. This is a collective phenomenon
nucleated by the presence of a vacancy. The temperature of 1750 K is
higher than the experimental melting temperature ($T_M=$ 1685 K) and
it is also quite higher than the melting temperature calculated for a
similar model of Si\cite{18}.

\par

At low temperature we expect to observe a sequence of uncorrelated
first-neighbor vacancy jumps. Our results show that already at $T=$
1200 K this picture needs modifications. We indeed observed a tendency
of the jumps to hump together in relatively short time spans of few ps
followed by relatively long periods of several ps where no jumps are
detected. Sometimes two jumps were so close (i.e. the time delay
$\delta < 1/\Omega_{Debye}$) to be better characterized as a double
jump (case c). A detailed analysis of these events showed that the
covalent bond between the two atoms performing the double jump was not
broken at any time, so that arguably the energy needed to activate the
process was not much higher than needed for a simple jump.

\par

We have not observed enough jumps to be able to check quantitatively
if the distribution of time delays between jumps is appropriate to a
series of random jumps, but we found qualitatively that there is a
preference of the jumps to happen at time delays corresponding to the
acoustic peak in the phonon spectrum. These facts suggest the presence
of dynamical or memory effects, which are further made evident by
considering the geometry of a typical sequence of jumps such as the
ones reported in Fig. 3,
\begin{figure}[htbp]
\vspace*{11truecm}		%ORIGINAL SIZE=1.6TRUEIN x 100% - 0.2TRUEIN
\fcaption{Atomic trajectories projected on a (110)
plane at $T=1400~K$ in the time interval 2.7 ps $<t<$ 8.5 ps.}
\end{figure}
showing two characteristic atomic chains of
the diamond lattice. The atoms of the upper chain simply oscillate
around their equilibrium positions, while those in the lower of the
lower chain occasionally jump into into an empty first-neighbor site,
due to a vacancy moving from left to right along the chain. Only after
performing the seven jumps shown in the picture the vacancy began
moving along a different chain. A similar tendency of the vacancy to
perform several jumps along the same atomic chain was systematically
observed both at $T=$ 1200 and at $T=$ 1400 K. At the same time we
noticed a definite reduction of the rate of return jumps compared to
the value of 1/4 expected for a simple random walk.

\par

Dynamical effects are even more pronounced at $T=$ 1600 K, where, in
addition to nearest-neighbor jumps, we also observed second- and
third-neighbor jumps. These processes are characterized by larger
activation barriers but are favored entropically. Some of these events
are associated to the concerted motion of several atoms close to the
vacancy, which create temporarily a disordered local zone. One process
is schematically illustrated in Fig. 4:
\begin{figure}[htbp]
\vspace*{6.5truecm}		%ORIGINAL SIZE=1.6TRUEIN x 100% - 0.2TRUEIN
\fcaption{Schematic representation of a many-atom
diffusion process. Full dots: atoms in their perfect lattice
positions. Empty dots: metastable state ($\theta \approx$ tetrahedral
bond angle). ``V'' indicates the vacant site before and after the
process.}
\end{figure}
it involves mainly two
first-neighbors and one second-neighbor of the vacancy. Initially
these atoms move away from their crystalline sites and form the
intermediate state indicated in the picture by the open circles. This
state lives for more than 1 ps during which the vacancy does not exist
anymore as a point defect but is spread over several lattice
sites. Interestingly, local tetrahedral coordination is maintained on
average by the atoms involved in this process, indicating strong
rebonding effects. Again, this rebonding process appear essential to
keep the energy of this structure low enough to allow it to be
observed in an MD simulation and it is unlikely to be properly
described by any classical interatomic potential. After approximately
2 ps the disordered fluctuation decays and one recovers the
crystalline lattice plus a point-like vacancy. As a result one atom
has moved to a first-neighbor site, another to a second-neighbor, and
and a third to a third-neighbor, while the vacancy still occupies its
original position.

\par

An event of this kind can be considered as a dynamical premelting
fluctuation and can provide a realistic model of the extended vacancy
concept originally put forward by Seeger and Chik\cite{10} to
explain the high entropy of diffusion in silicon. Premelting effects
associated to bulk point defects have been experimentally
detected\cite{19} in the case of lead and are likely to be a
common phenomenon in nature. It would be very interesting if the
existence of premelting fluctuations at point defects in silicon could
also find direct experimental confirmation. For instance, in the case
of vacancies, this could possibly be done with positron annihilation
experiments, although up to $T\sim$ 1500 K no evidence was
found\cite{20} for such effects in silicon using this
technique.

\par

Premelting events may lead to an increase of diffusion at high
temperature. They certainly require a large fluctuation in the local
energy to happen but at high temperatures this is more than offset by
the gain in entropy, hence close to $T_m$ the slope of the Arrhenius
plot should bend upwards. In Fig. 5
\begin{figure}[htbp]
\vspace*{6.5truecm}		%ORIGINAL SIZE=1.6TRUEIN x 100% - 0.2TRUEIN
\fcaption{Arrhenius plot of the diffusion coefficient
$D_v$. Asterisks: MD results. The error bars are statistical error
estimates. Dotted line: simple rate theory (see text). The continuous
line has a slope of 0.9 eV.}
\end{figure}
we report the diffusivity $D_V$ as measured in our simulation using
the Einstein relation $D_V = \lim_{t\to\infty} \sum_I \langle |{\bf
R}_I (t)-{\bf R}_I (0)|^2 \rangle$, where the sum runs over all the
atoms in the cell. We also report in the same figure a simple rate
theory estimate of $D_V$: this has been obtained using $T=0$
calculations to compute the activation barrier ($E_m=0.3$ eV) and the
jump attempt frequency ($\nu = $ 20 THz) of the vacancy. These values
should be taken with a grain of salt: we have only few data points
affected by large error bars, due to limited statistics, and in
addition some finite-size effects can be present. Still, the
calculated slope of the Arrhenius plot at high temperature ($0.9\pm
0.4$ eV) is larger than the value of rate theory, which is exact in
the low-temperature limit, and a similar effect occurs in the
pre-exponential coefficient. Provided that premelting fluctuations are
not restricted only to a narrow temperature range below $T_m$ such an
effect should be observable.

\section{Conclusions}
\noindent

Our MD simulations of the atomic motion of the in crystalline silicon
at high temperature reveal the presence of characteristic dynamical
effects associated with a vacancy in a covalent tetrahedral
network. Within our approach, we observe diffusion events at
sufficiently high temperature. We find that dynamical correlations are
important. In addition near the melting point we observe occasional
simultaneous jumps of several atoms, which introduce locally a
significant amount of disorder and may be considered as a dynamic
realization of the extended vacancy concept. This leads to a high
diffusion entropy and might originate a deviation from a a simple
Arrhenius law in the temperature dependence of the diffusion
coefficient. When these value for $D_V$ are combined with the
formation free-energy of the vacancy, which determines the thermal
equilibrium concentration of defects, one obtains the vacancy
contribution to self-diffusion. This has been discussed
elsewhere\cite{16}.

\nonumsection{Acknowledgments}
\noindent

We benefited from stimulating discussions with P. Bl\"ochl,
G.~L. Chiarotti, S.~T. Pantelides and M. Parrinello. This work has
been supported in part by the International School for Advanced
Studies (SISSA) - Centro Interuniversitario del Nord-Est per il
Calcolo Automatico (CINECA) collaborative project, under the
sponsorship of the Italian Ministry for Public Education, by the
European ARO grant N. DAJA45-89-C-0025 and by the Italian National
Research Council (CNR) grant N. 89.0006.69.

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\end{document}


