next up previous
Next: Quantum confinement Up: Diamond and graphite Previous: Lonsdaleite

Electronic structure of diamond, lonsdaleite and amorphous carbon

The irreducible Brilliouin zones of the hexagonal primitive lattice, relevant for the hexagonal diamond and the fcc Brilliouin zone, relevant for cubic diamond is drawn in Fig.2.7. In this Figure a few symmetry points of the fcc BZ are projected onto the hexagonal primitive BZ with the assumption that bondlengths and bond angles are preserved in going from the hexagonal to cubic structure. The cubic diamond structure has two atoms in the unit cell, while the hexagonal has four. Therefore, the hexagonal structure has twice as many bands as the cubic structure at any $k$- point in the BZ.

The band structures and densities of states of cubic and hexagonal diamonds calculated by the local density approximation (LDA) method are shown on Figs.2.8 and 2.9. It is seen that the spectrum of the density of states of cubic diamond consists of the valence and conduction bands separated by an energy of 5.5 eV. The valence band is fully occupied, leaving the conduction band empty. Thus diamond is typical of the group IV semiconductors. The band gap is indirect because the wave vector at which the valence band is a maximum ($\Gamma$) does not coincide with the wave vector where the conduction band is a minimum ($L$).

There is a considerable similarity between the cubic and hexagonal bands. The band gap for hexagonal diamond remains indirect as the conduction band minimum at the $K$-point occurs well below the $\Gamma$-minimum. The LDA calculations shows the band gap of the hexagonal structure to be about 4.5 eV [12].

Figure 2.7: Irreducible Brillouin zones of the hexagonal primitive lattice and the fcc lattice. Wave vectors of the irreducible fcc BZ have been projected onto the hexagonal BZ.
\begin{figure}\centerline{\epsfxsize=12.0cm \epsfbox{}}\end{figure}

Figure 2.8: Band structures of cubic and hexagonal diamonds, taken from [12]
\begin{figure}\centerline{\epsfxsize=14.0cm \epsfbox{}}\end{figure}

Figure 2.9: Comparison of the DOS of cubic and hexagonal diamonds, taken from ref. [12]
\begin{figure}\centerline{\epsfxsize=11.0cm \epsfbox{}}\end{figure}

Amorphous carbon can form the large number of different bonding types. Hence, the electronic structure of amorphous carbon is governed by the relative importance of three and fourfold sites. A purely four-fold coordinated model of amorphous carbon [13] predicts only $sp^3$ bonding to occur which gives a large gap in the electronic density of states. The electronic structure predicted by this model is similar to a broadened diamond-carbon density of states. It is now clear that this is not the correct model for diamond-like amorphous carbon and later tight-binding calculations [6,7] (see Chapter 6) have found states which close the gap and have been associated to 3-fold coordinated atoms exhibiting $sp^2(\pi)$ bonding. The total number of states in the gap, which appear due to $sp^3$ bonds, increase when $sp^2$ orbitals are introduced into the simulation. However, some models [14] produced from the Tersoff potential have a significant density of states near the Fermi level. This is in contradiction to experimental and ab initio (see Chapter 7) calculations [15] which show only a small density of states at the Fermi level.

Figure 2.10: The electronic density of states of a-C, taken from ref. [16]
\begin{figure}\centerline{\epsfxsize=8.0cm \epsfbox{}}\end{figure}

The electronic structure of amorphous carbon simulations performed by ab initio method [16] is shown on Fig.2.10. The part of the density of states corresponding to $sp^3(\sigma)$ bonding is very similar to a broadened diamond-like electronic structure. Most of the states around the Fermi level are found to be $\pi$-like in nature leaving no band gap. Therefore the optical properties of amorphous carbon will be dominated by the $sp^2(\pi)$ bonded sites. There are however, relatively few of these (less than 10% of the atoms in the sample are 3-fold coordinated). They are not found to be clustered together as some earlier models of amorphous carbon predicted [17]. Instead they are found either bonded to three 4-fold coordinated atoms leaving a single electron in a delocalized p-like orbital, or rather often to two 4-fold atoms and another 3-fold site. This structure is similar to an ab initio calculation on diamond-like amorphous carbon performed by Drabold [18], where 3-fold sites are found to group in pairs. Due to the lack of clustering of $sp^2$ sites, it follows that it is the intermediate range correlations of the $sp^2$ sites which will have profound effects on the optical spectrum.

next up previous
Next: Quantum confinement Up: Diamond and graphite Previous: Lonsdaleite
sorkin anastasy 2006-10-30