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Vacancy in diamond

In the present section, we show the results concerning the energy levels induced by a vacancy in diamond. The tight binding model has been used to describe the interactions between the atoms of carbon. In this model, the eigenvalues and the eigenstates of the Hamiltonian are obtained by direct diagonalization, from which the density of states can be derived.

The density of electronic states (DOS) for diamond has been calculated with 60 special k-points, and is shown in figure (10.1). The zero of the energies is arbitrarily chosen at the lowest unoccupied state. The first broad pick is associated with the s states. The sharp pick at about -16 eV arises from the s-pinteraction, and the third pick is connected to the p states. The calculation of the energy difference between the highest occupied and the lowest unoccupied states in pure diamond yields $\sim$ 5.8 eV, a value close to the measured energy gap in diamond of 5.48 eV [117], and $\sim$ 24.1 eV for the width of the valence-band, in excellent agreement with experiment (24.3 eV) [117]. The features of the DOS of diamond are therefore well reproduced by the present tight binding model.


  
Figure 10.1: Density of electronic states (DOS) for diamond, calculated with 60 special k-points.
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\centerline{\epsfxsize=12.0cm \epsfbox{diamdos.eps}}
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For the calculations of the electronic structure of the vacancy in diamond, samples of 63 and 215 carbon atoms were used, in which a vacancy was created by extracting one carbon atom from a 64-atom and a 216-atoms supercells respectively. The big sample was used to unsure that no interaction between defects of neighboring supercells (due to the periodic boundary conditions applied) can spread the energy levels calculated by the tight binding model. The fully relaxed geometry is obtained by slowly quenching the structure at a temperature of 300 K, by tight binding molecular dynamics. This dynamical process has been chosen here over a conjugate gradient relaxation (or other static relaxations) to reproduce the dynamical Jahn-Teller effect observed in diamond.

It has to be noted that only neutral vacancy is considered here. The reason is that charged vacancies cannot be properly descibed by the present tight binding model, since a ``global'' self-consistency is required for this purpose, and not only the local one used here. Moreover, we have seen above that the energy levels induced by a neutral vacancy V0 are similar to those induced by a positively charged vacancy V+, not in their magnitude but in their relative position. Therefore, the results on neutral vacancy can provide significant insights on doped diamond, where charged vacancy should be prefered. It is found that the tetrahedral symmetry around the vacancy is broken by the relaxation of the atoms adjacent to the defect. This distortion is associated with a Jahn-Teller effect, so that the dangling bonds around the vacancy are bent to pair the atoms along the [110] direction. The tetragonal distortion occuring here can be seen in figure (10.2), where the atoms surrounding the vacancy are shown before and after the dynamical relaxation.

It is also found that the net motion of the atoms surrounding the vacancy is outward, in contrast to the vacancy in silicon, for example. These atoms therefore move more into the plane of their three nearest neighbors, and the bonding becomes more sp2-like than sp3-like, as can be explained by the relative stability of graphite over diamond. This preference is certainely in contrast to other elemental semiconductors such as silicon.


  
Figure 10.2: Positions of the nearest neighbors the vacancy before (yellow) and after (red) the dynamical relaxation, The green atom marks the position of the vacancy.
\begin{figure}\centerline{\epsfxsize=12.0cm \epsfbox{vac63atoms.ps}}
\end{figure}

The energy levels induced by the presence of a vacancy in the diamond lattice has also been calculated by the tight binding model, using 60 special k-points. The results are shown in figure (10.3), together with the diamond DOS for direct comparison. A clear pick appears in the band gap, at about 1.2 eV from the bottom of the conduction band. Calculatons show that the states correaponding to this pick are occupied by two electrons. Moreover, at the edge of the conduction band, unoccupied states are also induced by the presence of the vacancy. This is in good agreement with the model based on group theory explained before. Therefore, with the tight binding model used here, it is shown that neutral vacancies are a source of n-type defects. It has to be noticed that the results obtained with the big sample containing 215 atoms are very similar to those shown above. There is, therefore, no interactions between vacancies of neighboring supercells in the case of the small sample.


  
Figure 10.3: Density of electronic states (DOS) for vacancy in diamond (solid line), compared to the DOS of diamond (dashed line), calculated with 60 special k-points.
\begin{figure}\centerline{\epsfxsize=12.0cm \epsfbox{vac63dos.eps}}
\end{figure}

We can therefore predicte the role played by these defects in the case of boron doped p-type diamond. In such a material, the vacancy can be ionized by the compensation of free hole (formed by the partial ionization of boron acceptors) by electrons. The B- ions can therefore interact with the positively charged vacancy V+ to create neutral complexes. This process is plausible since the energy levels induced by the vacancy are located above the fundamental level of boron.


next up previous contents
Next: The <100> split-interstitial Up: Electronic structure of defects Previous: Electronic structure of defects
David Saada
2000-06-22