Standard Silicon

Silicon is the most abundant element in the Earth’s crust, lead only by oxygen. It constitutes 25.7% of the mass of the crust. Si can exist in different crystal phases but in the context of solar cells, it is used in its diamond structure. This form of silicon can also be named c-Si. It will be further studied in the following points.

Crystalline structure of silicon

The space group of the Silicon in diamond configuration has a space group Fd¯3m (cubic)

The Si has the followings lattice parameters.

Table 6: Theoretical values of the lattice parameters from Materials Project. They are obtained from ab initio calculation [33]

The Si atom Wyckoff position is described as :

Table 7: Theoretical values of the Wyckoff position from Materials Project [33]

Polyhedral representation.

"Ball and Stick" representation.

"Stick" representation.

The Si atom Wyckoff position is described as :

Figure 17: Silicon unit cell representations obtained with Vesta software and CIF data from Materials Project of mp-149 Si.

Stability of standard silicon

Silicon has known isotopes that range from 22-Si to 44-Si. Among them, only 3 are stable : Si^28, Si^29 and Si^30.

Silicon has a (high) melting point of 1414°C (1687 K) and a boiling point of 3265°C (3538K). These temperatures are among the second highest for metalloids and nonmetals, only surpassed by boron. High melting temperatures implies strong bonds in the crystal structure. This means a lot more energy is required to break silicium covalent bonds compared to other metalloids and nonmetals. [34]

The difference phases Si can be found in are described in its phase diagram on Fig. 18.

Figure 18: Temperature–Pressure phase diagram of Si determined by Clapeyron equation. Si-I is the stable diamond structure, Si-II is the β-Sn structure [35]

Band structure and refractive index

Fig. 19 shows a part of the band structure of silicon, it is the band along the (100) and (111) directions. For the band gap, the fundamental band gap is the value of the indirect band, so the difference between the maximum of the valence band and the minimum of the conduction band shown as Eg on Fig. 19. And it has a value of 1.1 eV. The optical band gap is characterised by the smallest direct band gap, which corresponds to E1. This value can be found by looking at the absorption coefficient of silicon at room temperature on Fig. 20 and we find a value for E1 of 3.05 eV. The value E2 corresponds to to the maximum absorption of the silicon and has a value of 4.3 eV. [36]

Since silicon has an indirect band gap, its absorption spectrum is more temperature dependent than a material with a direct band gap. The reason is that at low temperature, there are fewer phonons and therefore the probability that a phonon and a photon are simultaneously absorbed to create a transition is lower. Also, if the photon energy of light is close to the band gap, the photon can penetrate much further before being absorbed in an indirect band gap material than in a direct band gap material. [37]

Figure 19: Band gap of Silicon [36]

Figure 20: Absorption coefficient of Silicon at 300 K [36]

The refractive index of the silicon is given by the same formulas used in the previous section with the perovskite. Fig. 21 shows the refractive index (n) and extinction coefficient (k) as a function of eV and wavelength. It shows that the maximum n is situated at 371 nm and has a value of 6.83. So at this value, the phase velocity of light in the medium is the lower.

Figure 21: Refractive index and extinction coefficient of silicon as a function of wavelength [38]

Vibrational properties (phonons)

The vibrational properties are given by the phonon dispersion on Fig. 22 where the density of state of our material can be analysed. They are useful to determine the thermal and mechanical properties of our material. We notice an important peak in the density of states graph at a frequency of about 470 cm−1 . This peak means that at this frequency, silicon has a higher thermal conductivity. Since thermal conductivity is related to heat transport by phonons, a higher density of phonons will include a higher probability of heat transport and therefore a better thermal conductivity.

Figure 22: Density of state of silicon