\documentstyle[preprint,aps]{revtex}
\tighten
\begin{document}

\title{  Theory of Si Donor-Acceptor  Complexes in GaAs}
\author{ R. Jones}

\address{
Department of Physics, University of Exeter, Exeter, EX44QL, UK.  }
\author{ S. \"Oberg}
\address{
Department of Mathematics, University of Lule\aa, Lule\aa,
S95187, Sweden}

\maketitle
\bigskip 

\begin{center} 
PACS NOs: 
78.50.Ge,  73.20.Hb 
\end{center}
\bigskip 

\begin{abstract}
Local  density functional   cluster  calculations are  carried out  on
Si$_{\rm Ga}$-As-X  complexes where X=  Li, Be, Cu,  Zn and a vacancy.
We find that  that the donor  and acceptor  move {\it away}  from each
other contrary  to  a simple argument  based on  a Coulomb  attraction
between   them.  This  result explains  the  insensitivity   of the Si
related  local vibrational modes to  the identity of  the impurity and
their similarity to the case when X is a vacancy.
\end{abstract}
\pacs{    63.20.Pw 61.70.Tm}

The activation efficiency of  Si as a donor in  heavily doped GaAs  is
currently of great interest  and  for this  reason the structure   and
properties of defects where the donor activity  is reduced through the
formation  donor-acceptor complexes are of   interest.  We shall  show
that these structures are much  more complicated than at first  sight.
It might be thought  that an ionized donor  and acceptor both lying at
neighboring Ga  sites would move towards each  other because  they are
subject to a  mutual attractive Coulomb   force, even though  they are
subject to the  constraints imposed by the  bonds to  their neighbors.
It is however one of the aims  of this paper to  show that this simple
picture is  entirely wrong  and the actual    movements are much  more
subtle.

The local vibrational  modes (LVMs) of  a number of  Si donor-acceptor
complexes in GaAs are well  known \cite{newman,newman1}. Of particular
interest here   are  the  cases when  the   acceptor  lies on   the Ga
sublattice  and is a  next-to-nearest neighbor of Si$_{\rm Ga}$. Their
LVMs are given in the Table for  five cases: when  the acceptor is Zn,
Cu, Be,  Li and a  vacancy. One surprising feature  immediately stands
out: the  Si related LVMs are relatively  insensitive to  the chemical
identity of  the acceptor   and  generally the  triplet  mode  of  the
isolated  donor  at 384 cm$^{-1}$  is split  into a low  lying doublet
around 376 cm$^{-1}$ and a higher lying  singlet around 400 cm$^{-1}$.
This splitting suggests   that the impurity  (or vacancy)   leads to a
reduction in the length of  one of the Si-As  bonds, whereas the three
other Si-As bonds  are slightly lengthened. This  would then lead to a
defect with  approximate $C_{3v}$   symmetry possessing a   Si related
vibrational doublet lying    below a  singlet.  It   might further  be
reasoned that  the  compressed Si-As bond is  the  one where As  is  a
neighbor of the  acceptor. However,  this cannot  be correct.   In the
first  place the singlet  mode would then be  sensitive to the type of
impurity:  a large atom   like  Zn or  Cu   would  not give the   same
compression   as   a  small    one  like   Li    and  especially   the
vacancy.  Secondly, in the case  of Li there is   bound to be coupling
between the modes associated with the  Li-As and Si-As bonds where the
As atom  is bonded to the two  impurities.  This would then imply that
the Si-related mode would shift in frequency when $^{7}$Li is replaced
by $^6$Li.  But the Table makes it  clear that there is no discernible
coupling  between  the Si  and  Li  modes and the  two  atoms  must be
dynamically  independent.  Indeed, it is   more reasonable to draw the
conclusion  that  in every  case either the  impurity   or Si, or both
atoms, move away from their lattice sites leading to an {\it increase}
in the   Si-impurity   distance so  that   the  Si  related  modes are
independent of the identity  of the impurity  and are then similar  to
the Si-As-V$_{\rm Ga}$   case.  We shall   show that first  principles
theoretical   calculations   support  this    surprising picture.  The
situation is less clear in the  case of Be$_{\rm Ga}$-As-Si$_{\rm Ga}$
as only two  Si-related LVMs have been  observed \cite{mohad} and even
the Be related modes lying close to the LVM of  Be$_{\rm Ga}^-$ at 482
cm$^{-1}$ \cite{laith} have so far remained undetected.

Here   we examine the  structure  and   dynamical properties of  these
acceptor-donor  complexes    using   first-principles  local   density
functional cluster  theory.   The  method  has been described   before
\cite{jones,briddon}. The wave-function  is expanded in a  combination
of $s$, $p$ and  $d$ Gaussian  orbitals centered  on atoms as  well as
bond centers.  $d$-orbitals are used for  Zn and Cu but not otherwise.
The  charge density is  also expanded in  Gaussian functions at atomic
and bond-center sites.   Pseudopotentials of ref.  \cite{BHS} are also
used. The self-consistent  energy was found as  well as  the forces on
all the  atoms  which  were allowed to    move until  equilibrium  was
achieved by a conjugate gradient scheme.  Previously \cite{briddon} we
have shown that the method gives Ga-As bond  lengths within 2\% of the
experimental value.  The double  derivatives between central  atoms of
the cluster can   be    found  and  used  to  fit   a   Musgrave-Pople
potential. This short range potential gives the highest phonon mode at
$\Gamma$  equal to 293  cm$^{-1}$ but there is  no splitting of the TO
and LO modes as the potential does not include  any long range Coulomb
field.  This potential is used to  describe interactions between atoms
in clusters  containing  impurities but the second  energy derivatives
between the impurities     and  their neighbors  were  then   directly
determined by the {\it ab initio} method.

The central Ga  atom of a  H-terminated 131 atom tetrahedral  cluster,
Ga$_{31}$As$_{40}$H$_{60}$, was replaced  by   Si$^+$  and the   whole
cluster relaxed.  The four Si-As bonds were found to be equal to 2.404
\AA~and  the triplet $^{28}$Si LVM occurred  at 352 cm$^{-1}$. This is
to be  compared  with the  experimental  frequency  of  384  cm$^{-1}$
\cite{spitzer}.  This value is close to  that found using a smaller 71
atom cluster \cite{jones4}.    The isotope shift  when  $^{30}$Si  was
substituted is 9 cm$^{-1}$ which is close to the experimental value of
11 cm$^{-1}$ given in Table 1.

A   Si$^+_{\rm As}$ defect  analysed in  a  similar sized cluster, but
centered on  As, gave the four Si-Ga  lengths to  be 2.326 \AA~and the
shortening of these bonds implies a higher triplet vibrational mode at
388  cm$^{-1}$  ($^{28}$Si). The experimental  value  is 399 cm$^{-1}$
\cite{allred}. We note  that the Si-As bond is  longer than  the Si-Ga
one. This  would not be the  case if the atomic  radii given by Slater
\cite{slater} were used to determine these  bond lengths. In that case
the   Si-As   and  Si-Ga  bonds  would    be   about  2.36   and  2.47
\AA~\cite{comme}  and if these    lengths were used to  determine  the
vibration frequencies, then the Si$_{\rm Ga}$ LVM would be higher than
that of Si$_{\rm  As}$ contrary to experiment.   Using a local density
functional supercell method,   Saito et  al \cite{saito}  gave   Si-As
lengths of 2.41  \AA~in close agreement  with those found  here and an
LVM at 370 cm$^{-1}$.   Northrup  \cite{northrup} also using  a  plane
wave supercell approach found Si-As and Si-Ga lengths of 2.43 and 2.39
\AA~respectively.  Thus all the  theoretical methods assign the longer
length   to Si-As.   X-ray lattice   parameter measurements in  p-type
liquid  phase epitaxial  layers  and  n-type melt-grown specimens  are
complicated to interpret because of the presence of more than one type
of  defect.   A pronounced  lattice contraction has   been observed in
strongly p-type samples \cite{fewster}, which was almost linear in the
hole concentration,  and   much less  strain was  recorded   in n-type
samples.  A straight  forward interpretation would  simply be that the
Si-As bond is longer than  the Si-Ga one  but there are complications.
It is now known  \cite{newman1} that Si$_{\rm Ga}$  is also present in
the p-type  material  and  compensates the  Si acceptor.    Hence, the
presence of large lattice strain in the p-type samples could either be
due  to Si$_{\rm Ga}$ or  to Si$_{\rm As}$.  In n-type material, there
are several defects  introduced along with   Si$_{\rm Ga}$ which  make
interpretation of the  X-ray data much  more difficult.  Thus  further
work is necessary  to establish the relative  strains introduced by Si
donors and acceptors.

We now consider the donor-acceptor Si$_{\rm Ga}$-As-X complexes, where
X  is  Cu$^{--}$  and   Zn$^{-}$.  For  these   calculations, 86  atom
stoichiometric       clusters   were    used    with   the composition
Ga$_{22}$As$_{22}$H$_{42}$ and  centered on the    middle of a   bond.
These clusters have the advantage that they are neutral for X equal to
Zn, and singly negatively charged   when X is   Cu.  The initial  Si-X
direction is along [011], and the  common As neighbor lies along [111]
from Si$_{\rm  Ga}$.   All the  complexes have  $C_{1h}$ symmetry so a
pair of Si-As bonds, as  well as a pair  of  X-As bonds, are equal  in
length.
 
Consider  first the case when  X= Cu$^{--}$. Cu  is known to  act as a
double acceptor  which can  be  passivated by  H \cite{johnson}  or Li
\cite{egilsson}.  We found that Cu$^{--}$  lies on site but repels its
four  neighbors  outward so  that  the Cu-As  bonds have   length 2.69
\AA. There  is an occupied triplet $t_2$  level rather low in the gap.
For the Si-As-Cu  complex, Cu was  displaced nearly parallel to [111],
away  from Si,  by  0.61 \AA~ whereas  Si  remained very close  to its
lattice site. The  Cu atom has almost moved  into an interstitial site
leaving a  vacancy behind.  The   four Cu-As bonds  were 2.599, 2.599,
2.633 \AA~and   a   longer bond of  2.753   \AA  ~with the   common As
atom. Thus although Cu has moved a great  distance along [111], it has
also repelled its three other neighbors.  Three Si-As bonds were close
together at 2.401, 2.401 and 2.416 \AA~whereas the Si bond with the As
atom nearest  to Cu is  significantly shorter at  2.341 \AA.  Thus, in
this case the breaking  of the Cu-As bond has  allowed the As  atom to
move towards Si. We might  then anticipate similar  Si related LVMs in
the vacancy case.  The ordering of Si  bonds also suggests that the Si
vibrational  doublet will lie beneath  the singlet as indeed observed.
The impurity has moved away from Si in spite of the fact that there is
a  Coulomb attraction  between them  which  might have been thought to
bring the two impurities closer together.  Clearly the tendency for Si
to pull in its  neighbors together with  the tendency for Cu  to repel
its neighbors  outwards  outweighs  the  effect  of   this  attractive
interaction.  The calculated LVMs consist of a Si related near doublet
at   366   and 371   cm$^{-1}$   and   a  singlet    at 415  cm$^{-1}$
($^{28}$Si). The ordering  and location of the modes  are close to the
experimental results at 374, 376 and 399 cm$^{-1}$ \cite{spitzer}.

When X is Zn, the shift of Zn from its site was  much less marked.  It
still moved 0.15 \AA, but along [011] this time, and  away from the Si
stom which  again remained close  to its lattice  site. The four Si-As
bonds were then 2.395, 2.388,  2.388 \AA~and a  shorter bond with  the
common As atom with Zn of 2.295 \AA.  The four Zn-As bonds were 2.685,
2.685,  2.692 and 2.699   \AA~with the As   atom  nearest to Si.   The
frequencies are given in Table 1 and comprise a doublet around 376 and
379 cm$^{-1}$ and  a higher singlet at  451 cm$^{-1}$ ($^{28}$Si). The
experimental results \cite{allred1} (Table 1)  show the same structure
of a low lying doublet  around 378 and  382 and a higher lying singlet
at 395 cm$^{-1}$.  It  is clear that  we have underestimated the shift
of the Zn atom away from Si and this  has led to  the singlet being 55
cm$^{-1}$ too high. Nevertheless, the Si and Zn atoms have moved apart
contrary to an argument based solely on Coulomb effects.

We now consider the two light impurities  Li and Be which each possess
an  LVM when present    as  isolated acceptors.  These   defects  were
investigated  in 131 atom  clusters where  all  the atoms except the H
terminators were allowed to move.

An  isolated   ionized Li$^{--}_{\rm  Ga}$   acceptor possessed  Li-As
lengths  of 2.372 \AA~and the triplet  LVM occurred at 427.5 cm$^{-1}$
for $^6$Li and 401.4 cm$^{-1}$ for $^7$Li. The experimental values are
482  and  450  cm$^{-1}$   respectively  \cite{chen,theis}.  We   then
investigated the  donor-acceptor complex.  This time the Li$^{--}_{\rm
Ga}$ ion remained close to its lattice site  and pulled in the four As
neighbors  making bonds of lengths   2.295,  2.273, 2.273  and a  {\it
longer} bond of 2.356 \AA~with the As atom  bonded to Si.  The Si atom
moved 0.14 \AA~along [0\=1\=1] {\it away} from the Li atom so that its
bonds were 2.396, 2.396, 2.328 and a long bond, 2.414 \AA, with the As
atom bonded to   Li. The short bond occurs   now with a As  atom along
[1\=1\=1].  This     distortion was not  found   in   the  heavy metal
acceptors. Indeed,  the   roles  of Si   and X    seem to  have   been
interchanged as now X remains  close to its  lattice site and Si moves
away from its own site.  The ordering of Si  bond lengths suggests the
Si doublet will  again  lie below that  of the  singlet which  is then
associated with  the short Si-As bond  which  is far  away from the Li
acceptor. This then  would imply  that the  singlet mode is  decoupled
from the Li  modes and be quite unperturbed  then the isotopic mass of
Li is changed.  The six LVMs are given in the  Table.  There are three
Li related modes  at 498, 492 and  440 cm$^{-1}$ with  the lowest mode
associated with the longer  Li-As bond where As is  also bonded to Si.
The shifts in the frequencies when  $^7$Li is replaced with $^6$Li are
also reproduced: the  three Si modes   are hardly affected  but the Li
ones increase  by  about 38   cm$^{-1}$ compared  to the  experimental
result of 33 cm$^{-1}$.  This shows that the two impurities are almost
decoupled although the   lowest Li-mode at  440  shifts slightly  with
$^{30}$Si. Unfortunately, there are  no measurements on $^{30}$Si. The
three  Si related   modes at  365,  379  and  433  ($^{28}$Si)  are in
reasonable agreement  with  the doublet-singlet structure  observed at
374, 379  and 405  cm$^{-1}$.  Thus  although the  distortion with the
light acceptor Li   is different in detail   from those  of  the heavy
metallic acceptors, the essential result is that the Li-Si distance is
greater than  one  would have expected  from a   simple idea involving
Coulomb attraction between the impurities.

We  now investigate the case   of Be.  A substitutional $^9$Be$^-_{\rm
Ga}$ ion has  Be-As bonds  of 2.245 \AA~  and a  triplet mode at   464
cm$^{-1}$ in  good  agreement  with  the  experimental value   at  484
cm$^{-1}$  \cite{laith}.   For the pair, the   Be ion shifted slightly
along [\=111] leading to bond lengths  of 2.265, 2.254, 2.254 \AA~ and
a longer bond with the As atom bonded  to Si of  length 2.370 \AA. The
Si  ion moved slightly along  [100] {\it closer}  to this As atom. The
Si-As lengths were 2.384 \AA~with this As atom and 2.368, 2.387, 2.387
\AA~with the other  three As atoms. This distortion  is similar to the
case  of Li above:  the shortest Si bond is   with an As atom furthest
away from   Be.  The Be-Si   distance  at 4.016 \AA~is   slightly {\it
greater}  than the distance between Ga  atoms (3.97 \AA).  The LVMs of
the Be  and Si ions are  given in the Table.  Two of  the Be modes lie
close  to the  modes of  the  isolated acceptor,  but  the third mode,
associated with the longer Be-As bond, lies at a  much lower value and
below the Si  related LVMs.  This is seen  by examining the case  of a
hypothetical  $^{10}$Be impurity.  The Si LVMs  are again split into a
low lying doublet  and a high frequency singlet.   In this  case there
are only  two observed modes  at  381 and 388  cm$^{-1}$ \cite{mohad}.
However, the intensity of these  lines was very small.  It may be that
the diffusion constant of  Be is so low  that few pairs are formed and
the other modes might have  weak absorption strengths.  The Be-As bond
length  is shorter than that  of Si-As and  hence the strain due to Be
should  be greater than that   due  to Si$_{\rm  Ga}$. However,  X-ray
rocking  curves in molecular beam   epitaxial grown Be doped material,
show that the strain  is greater in n-type  GaAs:Si,Be than in  p-type
GaAs:Si,Be  samples  \cite{mohad}.  Only   Si$_{\rm Ga}$  and Si$_{\rm
Ga}$-As-Be$_{\rm Ga}$  pairs were found  in the p-type sample, but the
LVM  studies \cite{mohad} show  the presence of Si-X,  Si-Y as well as
Si$_{\rm As}$ in the n-type sample \cite{mohad}.  The last, as we have
discussed  above,  is expected to    gives  rise to  a  large  lattice
contraction.  Si-X    is  now thought     to be V$_{\rm  Ga}$-Si$_{\rm
As}$-As$_{\rm  Ga}$ \cite{mcquaid} whereas  Si-Y  is believed to be  a
Si$_{\rm   Ga}$-V$_{\rm Ga}$ complex  \cite{ono}   which we shall  now
discuss.

The (Si$_{\rm Ga}$-V$_{\rm  Ga}$)$^{--}$ defect was  investigated in a
130  atom cluster centered on  Si$_{\rm Ga}$.  Upon relaxation, the Si
atom moved slightly  away from its site  and possessed bonds of length
2.431, 2.427, 2.427 and a  short bond of 2.394 \AA~with  one of the As
atoms bordering the vacancy. This  atom moved towards the vacant  site
but the other  three As atoms surrounding the  vacancy moved away from
it.  The LVMs in the Table show  the doublet at  354, 359 is below the
singlet   at    372 cm$^{-1}$   ($^{28}$Si)    in agreement   with the
experimental ones.
 
Our  conclusions are that the  doublet-singlet  ordering of Si related
LVMs observed in Si-As-X defects results  from a short range repulsion
between X  and the Si  atom.   This is  counter-intuitive and probably
arises because if the impurity  moved closer to  Si, then the electron
densities on Si and X  overlap, leading to an  increase in energy.  It
is then misleading to treat the Si$^+$ and X$^-$ as point charges.

We  thank  Ron. Newman for  many   stimulating discussions  during the
course of the work.  We also thank John Northrup for communicating his
values of the  Si donor and acceptor  lengths. Finally, S. \"O. thanks
the   Swedish  National  Scientific  Research   Council for  financial
support.

\begin{references}

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J. C. Slater, {\it Quantum Theory of Molecules and Solids II},
McGraw Hill, New York, 1965, page 320.
\bibitem{comme} 
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\end{references}

\begin{table}
\caption{  Frequencies, cm$^{-1}$,  of Si$_{\rm Ga}$-As-X  Complexes in GaAs}
\begin{tabular}{lccr}
Defect & Calculated  & Experimental  & Ref. \\
\hline
$^{28}$Si$_{\rm Ga}$ & 352 &384   &   \cite{spitzer}  \\
$^{29}$Si$_{\rm Ga}$ & 347 & 379   &  \cite{spitzer}  \\
$^{30}$Si$_{\rm Ga}$ & 343 & 373    &  \cite{spitzer}  \\
$^{28}$Si$_{\rm As}$ & 388 & 399   &   \cite{allred}\\
$^{28}$Si$_{\rm Ga}$-$^{64}$Cu$_{\rm Ga}$ & 366, 371, 415& 374, 376 399
& \cite{spitzer}  \\
$^{28}$Si$_{\rm Ga}$-$^{65}$Zn$_{\rm Ga}$ &  376, 379, 451 &378, 382 395 &
\cite{allred1}\\
$^{28}$Si$_{\rm Ga}$-$^{7}$Li$_{\rm Ga}$ & 365, 379, 433, 440, 492, 498  &   374, 379, 405, 438, 447, 454
 & \cite{chen}, \cite{theis} \\
$^{28}$Si$_{\rm Ga}$-$^{6}$Li$_{\rm Ga}$ &  365, 379, 434, 471, 529, 536 &374, 379, 405, 476, 480, 487
 &   \cite{chen}, \cite{theis}\\
$^{30}$Si$_{\rm Ga}$-$^{7}$Li$_{\rm Ga}$ & 356, 367, 422, 438, 492, 498  & 
 &  \\
$^{28}$Si$_{\rm Ga}$-$^{9}$Be$_{\rm Ga}$ &  363, 390, 396,
408, 474,485 & 381, 388 & \cite{mohad}  \\
$^{28}$Si$_{\rm Ga}$-$^{10}$Be$_{\rm Ga}$ &    349, 390, 394, 408, 452,
462& &  \\
$^{30}$Si$_{\rm Ga}$-$^{9}$Be$_{\rm Ga}$ &    362, 380, 387, 397, 474, 485
& &  \\
$^{28}$Si$_{\rm Ga}$-V$_{\rm Ga}$ &  354, 359, 372 &367, 368, 398   &   \cite{ono} \\
$^{30}$Si$_{\rm Ga}$-V$_{\rm Ga}$ &  344, 349, 362 &   &  \\
\end{tabular}
\label{table1}
\end{table}

\end{document}
