Hypothetical sp3 Allotropes of Silicon, Bct Type, with 3, 5, and 6 Atoms

1. Crystal Structure

The carbon allotropes, Bct-C₄, have been widely studied as an interesting structure. They contain 8 atoms in their tetragonal unit cell [4]. Variant structures derived from this model include, like these, a ring formed by a triangle, pentagon, or hexagon of three, five, or six silicon atoms, centered in the tetragonal cell (bct-Si_n, n = 3, 5, 6), bridged with other rings located at the nertices of the tetrahedron; and at the center of the latter, in the case of a triangular or pentagonal ring (Si₃, Si₅). The unit cell is composed of 6, 10, and 12 silicon atoms, the latter in bct-Si _{3, 5, 6}. (Figure 1).

Figure 1. .

Figure 1. .

Figure 2. .

Figure 2. .

Figure 3. .

Figure 3. .

2. Calculation Model

2.1. Electronic Properties

The free-moving electrons of solids, based on the density electrons, were established by the free-electron gas model of Drude and Sommerfeld [4]^, [5] .

N_A is Avogadro's number or constant, Z indicates the valence electron number, A is the atomic weight of the chemical element, ρ_m is the mass density (mass divided by volume), and n_e is the conduction electrons or the number of free electrons. This is the basic equation for calculating their concentration.

For a unit cell, we can simplify the equation to:

In this equation, we substitute the mass density, ρm, for nºA, which is the number of atoms that make up the unit cell [ 6 ].

The effective or dynamic mass of the electron is defined by the formula:

where m* is the effective mass of electrons, σ is the electron conductivity, e is the charge of an electron, and τ is the relaxation time, which is the average time between consecutive collisions of an electron. It is an average value, since not all collisions occur at regular intervals.

There is a relationship between this effective or dynamic mass of free-moving electrons, m*, and the forbidden energy gap or band gap ^{[ 7 ]}.

And between these and electronic mobility^{[ 7 ]} , which measures the efficiency of semiconductors.

Table 1. Basic electronic properties of sp3 bct type silicon allotropes - Si_{3, 5, 6}.

Table 1. Basic electronic properties of sp3 bct type silicon allotropes - Si_{3, 5, 6}.

Table 2. Electronic and optical energy band gaps of some elementary semiconductors and the binary compound, GaAs.

Table 2. Electronic and optical energy band gaps of some elementary semiconductors and the binary compound, GaAs.

Table 3. Basic optical properties of sp3 bct type silicon allotropes - Si _{3, 5, 6}.

Table 3. Basic optical properties of sp3 bct type silicon allotropes - Si _{3, 5, 6}.

2.2. Optical Properties

To calculate the basic nonlinear optical properties of any elemental allotrope, we must start from the basic equation to calculate the value of the forbidden energy optical band gap. The basic formula for this gap is:

where E₀ = 2E_g ^opt, is the energy of a simple oscillator.^{[ 8 ]} This formula is based on the Wemple-DiDomenico dispersion relation. The optical band gap is formed in materials that form a high exciton energy. In these, a photon has enough energy to form a bound electron-hole pair or exciton; but not enough to separate the two. It is in this situation that the optical band gap is formed, with lower energy than the transport band. In the optical band gap, photon absorption occurs; while in the transport band gap, non-bound electron-hole pairs are formed. In the vast majority of inorganic semiconductors, the electron-hole attraction that is formed is very small, so the electronic and optical band gaps are almost equal ^{[ 9 ]}. In organic semiconductors and single-walled nanotubes, on the other hand, this distinction between both optical and electronic bands can be greater. We have thus, that in inorganics, generally:

There is a relationship between the linear refractive index, obtained ( n₀ ), and the optical band gap energy, E_g^{[ 10 ]}.

Thus, as a relation, with the non-linear refractive index, n₂^{[ 10 ]} :

The time-averaged Poynting vector is related to the electric field amplitude ^{[ 12 ]} , by:

Finally, the Kerr refractive index will be obtained from the sum of the two refractive indices, linear and non-linear; as well as the electric field E₀ ^{[ 12 ]} :