Introduction

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 $\sim E_c-0.55$ 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 t₂ 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 (Si₇₁H₆₀) with T_d symmetry centred on an atom, or 134 (Si₆₈H₆₆) atoms with D_3d symmetry and centred on the middle of a bond. The latter was used for trigonal defects like H$_{\rm BC}$, V₂ etc. Usually, the clusters were relaxed with the H atoms fixed. However, for vacancy-like defects, eg VP, V₂ 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 PtH₂. 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.