Spin-polarized local density functional cluster calculations are
carried out on substitutional Ni and Ni-H 
complexes. We find that
Ni 
undergoes a Jahn-Teller distortion along 
with Ni moving slightly along the cube axis. The distorted state
gives 
and 
levels in ascending order of energies
within the gap in agreement with experiment. Several candidate
NiH defects are investigated: the lowest energy structure consists
of a substitutional Ni atom together with two H atoms at anti-bonding
sites to two Si neighbors of Ni. This gives H related vibrational
modes and a spin-polarized charge density close to those reported for
Pt-H .
The vacancy model of substitutional transition metal (TM) impurities
on the far-right of the periodic table [1, 2], considers
that the d levels of the TM impurity lie deep in the valence band of
Si and couple weakly with the 
levels arising from the 
symmetry of the ideal vacancy. The important implication is then that
the gap-levels and structure of the defect are largely determined by
those of a perturbed vacancy [3, 4, 5, 6].
For Pt , Pd and Ni , the gap levels contain three
electrons and according to the Jahn-Teller theorem, a distortion will
occur for an S=1/2 state, leading to a splitting of these levels and
a lowering of energy. Deep level transient spectroscopy shows that
Ni possesses acceptor and donor activation energies at 
and 
eV respectively [7]. Electron
paramagnetic resonance (EPR) measurements on Ni [8] and
Pt [2, 9] reveal that the defect has 
symmetry and furthermore, uniaxial stress measurements on the EPR
lines have shown [9, 10] that the gap levels split
into a configuration 
.
This ordering of levels is the reverse to that found for the
negatively charged vacancy. Further evidence for the vacancy model
has been suggested recently when it was found that Pt-H complexes can
be created by high temperature in-diffusion of the impurities
[11, 12, 13, 14]. Two localized vibrational
modes (LVMs) attributed to Si-H have been observed which undergo small
shifts in mixed H-D implantation or when the charge state of the
defect is changed. These results establish the presence of more than
one H in the defect. The first suggestion was that the H atoms bond
to two of the four Si atoms surrounding the vacancy. It is the
purpose of this work to investigate this model.
As far as we have been aware, there have been no ab initio
calculations of the Jahn-Teller distortion or the splitting of the
levels, nor of the complexes with H. We report here local
density functional spin-polarized calculations on substitutional
nickel, Ni, and Ni-H complexes using large (up to 133 atoms)
H-terminated clusters. A key ingredient is that we allow the atoms
surrounding the defect to move and this enables us to investigate in
detail the Jahn-Teller distortion due to Ni 
and to obtain the
structure and the LVMs of the Ni-H defect. The LVMs are found by
two methods. In the first we numerically differentiated the forces on
H and the surrounding atoms and these, together with a previously
determined potential, were used to construct the dynamical matrix of
the cluster. This gives quasi-harmonic LVMs. In the second method
[16], the energies corresponding to displacing the H atoms
from their equilibrium sites were found and the Schrödinger equation
for the oscillator solved numerically using these energies as a
potential in accordance with the Born-Oppenheimer approximation. The
two methods give rather similar results.
The ab initio method has been described previously [17]
and it has been adapted for the present study to deal with d
orbitals and spin-polarization effects [18, 19]. We
investigate the case of Ni since spin-orbit coupling is less important
in this case than for Pd and Pt. The electronic wavefunctions of the
cluster were expanded in atom-sited Gaussian orbitals. Seven
independent Gaussian s, p, d orbitals with different exponents were
placed on the Ni atom, two on each H atom and four on the central four
Si atoms. The remaining basis consisted of fixed linear combinations
of the s and p orbitals with different exponents and these are
centered on the other atoms. The charge density was fitted to a
linear combination of s-Gaussian functions: 14, 5 and 3 sited on Ni,
Si and H respectively. Ni and Si pseudopotentials were taken from
Ref. [20]. We have shown that the method describes small
molecules containing Ni very accurately: the bond lengths of
Ni(CO) 
for example are found to be within 1% of their
experimental values. The cluster approach for H defects in bulk Si
locates LVMs to within about 100 cm 
[21].
We investigated Ni embedded in a 71 atom cluster,
NiSi 
H 
. The Ni atom was placed along [100] so that the
symmetry of the cluster is . Three close-by levels for each
spin lay in the gap region and these are occupied with three electrons
according to fermi-statistics corresponding to 0K, i.e. the lowest
was filled, the next filled in the up-spin case only and the others
were empty. The self-consistent energy and forces acting on the
central 17 atoms were then found for this S=1/2 configuration, and
these atoms were allowed to move to minimize the energy of the cluster
using a conjugate gradient algorithm. The Ni atom moved closer to the
substitutional site while the surrounding Si atoms moved outwards from
their lattice positions. The splitting between the -derived
levels gradually decreased and eventually we were unable to obtain a
self-consistent solution. This arises because during the course to
self-consistency, the three close-by gap levels, the uppermost of
which is empty, cross over and if, for example, the state falls
below the 
state, the charge density and hence potential changes
discontinuously. On the next iteration, the level ordering often
reverses and the process never converges. This problem of charge
sloshing is well-known and often occurs for close-by levels. It can
be overcome by occupying the gap-levels with fermi-statistics
corresponding to a finite temperature larger than the splitting of the
levels. This spreads out the electrons among the three gap
levels and the discontinuity in the charge density arising from
cross-over is reduced. However, this almost eliminates the Jahn-Teller
driving force for the Ni atom to lie off-site. For the
finite-temperature calculation the lowest energy configuration
corresponded to a structure where the Ni atom is essentially on site
but the four surrounding Si atoms were displaced outwards from their
lattice sites by 0.270 Å and two of them were further displaced
along [100] by 0.006 Å. Second shell Si atoms were displaced by
0.07 Å. The three gap levels were split by only 0.005 eV
and the highest occupied spin-up level was 0.6 eV below the
corresponding (but empty) spin-down level. Thus the departure from
symmetry is extremely small. The ordering of the -derived
gap levels is is the
same as that found experimentally.
The problem of obtaining self-consistent solutions can now be overcome
by restricting the electronic configuration to be . Then during the passage to
self-consistency, there is no discontinuity in the charge density but
the energy levels that are occupied at each stage are not necessarily
in ascending order. However, towards the end of the self-consistency
cycle the energy levels ordered correctly and the final energy
obtained was lower than that obtained by the imposition of a
temperature. The Ni atom was then moved along [100] and the
equilibrium structure was found to correspond to a displacement of
0.13 Å from the lattice site. The two Si atoms with cartesian
components along [100] were displaced almost radially by (0.15, 
0.25, 0.22) Å and moved away from each other, whereas the
other two Si atoms with components along [ 
00] were
displaced (-0.07, 0.16, 
0.13) Å so that they approached
each other. Curiously, the four Ni-Si bonds are almost identical in
length at 2.645 Å. This distortion is similar to that found
experimentally for the negatively charged vacancy [22].
Suppose now, we occupy the derived states according to
. This is the
configuration found experimentally for the negatively charged vacancy
[22] but not for Ni . Then the self-consistent cluster
energy is higher than the previous configuration by 0.017 eV. We note
that although we have shown that the distortion has a lower
energy than it leaves open the possibility that other
symmetries, e. g. 
have even lower energies as suggested in
Ref. [15]. The slight distortion from to
symmetry is consistent with the ease of reorientation of Pt around the
central lattice site. This motion occurs for temperatures above 12K or
even when the slightest stress is imposed on the crystal at 2K
[9].
The pseudo-wave function of the -derived levels all possess a
nodal surface lying between the TM impurity and the surrounding Si
atoms and peaks on either side of the Si atoms along 
. This suggests that H atoms will lie either between Si and
Ni in the configuration suggested previously
[10, 11, 13, 14], or outside at anti-bonding (AB)
sites as shown in Fig. 1. The effect of H was investigated in a 133
atom negatively charged cluster NiSi 
H 
where 19 central
atoms were relaxed. There were no problems with self-consistency in
the present case. For the first defect, the H atoms repelled Ni along
[100] from its lattice site by 0.20 Å and formed Si-H bonds of
length 1.508 Å with a H-H separation of 1.517 Å. The LVMs due to H
were calculated to lie at 2477 and 2511 cm . These values are
far from the experimental ones for PtH
[10, 11, 12] and strongly suggest that this model
is incorrect. The strong coupling between the H atoms occurs because
there is so little room for both Ni and H within the vacancy. The
second defect (Fig. 1) when relaxed yielded an energy 0.08 eV below
the first structure and the Ni atom moved 0.3 Å along [100] towards
the H atoms and the Si-H lengths became 1.557 Å. The separation
between the H atoms was 7.3 Å. The vibrational modes are given in
the Table. The two H related modes are slightly split, by 7 cm ,
with the higher mode being 
. Now, uniaxial stress splitting
experiments on PtH [12] show that the upper and lower
modes also have and 
symmetries respectively. The stretch
modes are close to those reported for PtH but the bend modes have
not yet been detected.
Anharmonic effects are known to be important for H modes
[16, 23, 24] and we investigated their influence in the
following way: the two modes at 2009.9 and 2003.2 cm represent
and vibrational modes in which the H atoms move either in
phase or out of phase with each other. The energies necessary to
displace the H atoms by an amount r, parallel to the Si-H bonds,
were evaluated for each of these modes. This energy contains only even
powers of r in the mode whereas it contains both even and odd
powers in the mode. The Schrödinger equation for the
oscillator was solved numerically [16] and the and
frequencies were then found to be 2037 and 2060 cm and
are separated by 23 cm . However, now the mode lies
above the mode contrary to experiment. Thus even though the
separation of the H atoms is very large, 7.3 Å, the two modes are
split and their ordering reversed by these anharmonic effects by an
amount twice as large as that observed for Pt-H . However, there
are other anharmonic terms which should be considered. For example,
terms where 
and 
are
the changes in the equilibrium Si 
-H lengths and angles
respectively. These mix the stretch and bend modes and are only
present in the mode where 
. These have the effect of
decreasing the mean Si-H length hence pushing up the frequency.
Such a term then might well displace the mode above that of
.
We investigated several other models of the defect but all gave
energies above that of the AB structure. In addition, their
vibrational modes were far from the observed values and often, the
mode lay higher than that of the . We suggest then that
the H atoms lie at AB sites in the Pt case especially as its larger
size makes it even less likely that H would lie inside.
Further arguments against the configuration in which H atoms are
inside the vacancy come from the comparison with LVMs assigned to the
vacancy-hydrogen, VH , and H 
complexes. The former are
observed at 2122 and 2144 cm [25]. The separation
between the H modes here, 23 cm , is larger than that
observed in PtH where it might be expected that the TM squeezes
the H atoms together, increasing their interaction and the mode
splitting. In the case of H , the AB sited H atom gave an LVM at
1838 cm [21] rather close to the stretch mode of the
PtH complex, whereas the other H-LVM, attributed to H-Si stretch
where Si has a tetrahedral environment, has a frequency at 2061.5
cm - somewhat higher than that found for Pt-H . Recently,
Uftring et al [26] analyzed the anisotropic hyperfine
parameter in PtH and concluded that the Pt-H distance is about
4.5 Å which is close to the calculated Ni-H length of 4.28 Å in
the AB sited model.
The wave-function for the highest occupied level in NiH 
(AB) has
symmetry and vanishes in the (011) plane containing the
two H atoms. This level occurs around mid-gap in our calculations and
the next lower level is very close to the valence band. The positions
of these levels is only approximate but they suggest that the observed
EPR signals are due to PtH complexes, for it is known that when
lies above 
eV, then the defect is not paramagnetic.
There must be a second acceptor level corresponding to the filling of
around eV and leading to diamagnetic
NiH 
. Uftring et al [26] also concluded that the
defect has two acceptor levels as the effect of illumination on the
paramagnetic complex is easily understood to arise from the capture of
photo-generated holes by PtH . In addition, they suggested that
the Pt hyperfine data is consistent with the H atoms lying in the
nodal plane of the highest occupied level. For NiH , the contour
plots of the pseudo-wavefunctions suggest that the isotropic hyperfine
interaction with H would vanish leaving only an anisotropic one.
However, spin-polarization causes a difference in all the spin-up and
spin-down valence wave-functions and results in a small polarized
charge density of -0.007 a.u. at each proton. This is only -2% of
the charge density of a H atom in vacuo. Its magnitude is within a
factor of three found experimentally for PtH
[10, 11, 12, 14]. It is unclear whether this
discrepancy is due to calculational errors or differences between Ni
and Pt.
In conclusion, the calculations show that the substitutional Ni
defect with symmetry is unstable against a displacement along
the axis. The gap level is split into
and levels in ascending energy. Ni can complex
with two H atoms at AB sites and act as a double acceptor. It gives
two H-LVMs around 2000 cm . The polarization charge density is
very small and negative at the H nuclei and we suggest that PtH
complexes assume the same structure.
S. Ö. thanks the Swedish National Scientific Research Council, for financial support. R. J. thanks G. D. Watkins, M. J. Stavola and S. Uftring for useful discussions. The authors also thank the HPCI for computer time on the T3D where some of the calculations were performed.