The transition metals Au, Ag, Pt and Pd form substitutional defects
in Si with
a simple energy level structure. They possess donor or
(0/+) levels around *E*_{v}+ 0.35 eV and, in the case of Au and Ag,
acceptor or (-/0)
levels at
eV.
Pt and Pd have
(-/0) levels around *E*_{c}-0.23 eV and double donor (+/++) levels
around
*E*_{v}+0.1 eV.

From a theoretical viewpoint, the energy levels are understood to rise from
the
*t*_{2} manifold lying deep in the gap and occupied by 3 electrons for neutral
Au and Ag and
2 for Pt and Pd.
Most calculations do not give the correct Si band gap and there is then a
problem in
calculating these levels.
We show here that using a combination of first principles theory and an
empirical correction to the level position, the donor and acceptor
levels can be found to within about 0.2 eV.
The method we use is described below and applied to a number of deep level
defects as well as to Ag, Au, Pd
and Pt impurities. We also investigate the effect of bringing H atoms up to
the
impurity and compare these with experimental results
using capacitance transient spectroscopic techniques on Si doped
with Ag, Au, Pd and Pt into which hydrogen has been introduced.

A density functional cluster scheme, applied to large H-terminated
clusters, is used to first deduce the structure of the defect [#!aimpro!#],
but an
extension to this
procedure must be sought if the electrical levels are required. The (0/+)
level with respect to *E*_{v} is the difference between
the ionisation energy of the defect and that of bulk Si. If the wavefunction
of
the defect is localised within the cluster, then in principle the ionisation
energy of the defect can be calculated
by the cluster method. However, as the valence band wavefunctions are always
extended
throughout the
cluster and affected by the surface, the bulk ionisation energy
cannot
be calculated by the method. To circumvent this problem we compare the
ionisation energy of the
defect, *I*_{d}, with that of a standard defect, *I*_{s} (taken to be C_{i}),
with localised
levels.
The (0/+) level is then the sum of *I*_{s}- *I*_{d} and the (0/+) level of
C_{i}, *i.e.* 0.28 eV.
In the same way the electron affinities can be used to determine the acceptor
level relative to that of C_{i}.

By comparing the ionisation energies of defects in this way, we can
eliminate a systematic shift in the calculated levels caused by
the cluster surface. This follows as the wavefunction for the
ionised defect will
decay exponentially away from the centre of the cluster, and the shift in the
level position caused by the surface, to first order, then depends only on
the asymptotic part of the
wavefunction.
Clearly, this decay is
related to the depth of
the level from the band edges and the shift would be the same
for the standard defect if the latter possessed the same energy level
and the same total charge within the core.
In the same way, the error in estimating the
Hubbard *U*-parameter would be the same as in the standard defect if
*U* depended only on the
asymptotic part of the wavefunction.
Thus we expect that the
error in estimates of the levels to increase with the separation
between
the defect level and that of the standard.
In practice, the ionisation energies and electron affinities are calculated
from the Kohn-Sham levels by relaxing
the transition state corresponding to an additional
1/2 electron or hole. In this way the effect of a change in
structure between the ionised and neutral defects is
treated to first order. This technique supersedes an
earlier
one
based on scaling
the band gap although the two methods give rather similar results
[#!icds97-1!#].

The clusters used either contained 131 atoms (Si_{71}H_{60}) with
*T*_{d} symmetry centred on an
atom, or 134 (Si_{68}H_{66}) atoms with *D*_{3d} symmetry and centred on
the middle of a bond.
The latter was used for trigonal defects like H,
V_{2} etc.
Usually, the
clusters were relaxed with
the H atoms fixed. However, for vacancy-like defects, eg VP, V_{2} and VO, the
clusters were prerelaxed to force a
strong reconstruction across the Si dangling bonds and all atoms were
then relaxed.
Second acceptor levels of defects can also be found by comparing
the second electron affinities with that of PtH_{2}. The (-/-) level of
this defect is believed to lie between 0.045 eV and 0.1 eV
below *E*_{c} and we assume a value
of 0.07 eV.