First-principle calculations of effective mass of silicon crystal with vacancy defects

Crystalline silicon adopts the structure of diamond with 8 Si atoms in unit cell. The supercell containing 216 Si atoms with 3 × 3 × 3 model is used to produce defects in crystal so that the interaction between the defect and its image could be neglected. The crystalline structure with monovacancy, divacancy and hexavacancy ring defects can be obtained by removing one, two or six Si atoms from the supercell, respectively. The optimized configuration with the lowest energy is selected to be further studied.

All calculations are performed by using the VASP [20, 21] (Vienna ab initio Simulation Package) within the projector augmented-wave (PAW) approach [22]. The ground state of the electronic structure is described within density functional theory (DFT) using the generalized gradient approximation (GGA) with Perdew-Wang (PW91) exchange correlation function [23]. The energy cutoff for expansion of wave functions and potentials is 300 eV. Monkhorst-Pack special k-point method is used with a grid of 2 × 2 × 2 [24]. The entire systems are relaxed by conjugate gradient method until the force on each atom is less than 0.01 eV.

Firstly, the geometry of silicon crystal with a defect is optimized. The relaxed structures are shown in Fig. 1. The distances between Si–Si atoms (d_Si–Si) around the monovacancy defect are all equal to 3.86 Å before relaxation. After relaxation, four distances between Si–Si atoms decrease to 3.53 Å, while the other two decrease to 3.03 Å. Obviously, the structure around the monovacancy defect distorts. It is typically called as the Jahn-Teller effect which influences the electronic properties of silicon crystal with monovacancy defect. This is in agreement with the previous report [25]. With respect to divacancy defect, it is found that the optimized structure relates to the initial structure. By fine tuning the positions of six Si atoms around the divacancy defect, two different configurations with almost degeneration energy are obtained finally. The difference lies in the six Si–Si distances that are formed between two Si atoms in every three Si atoms around the defect. For the first configuration, four Si–Si distances d_Si–Si are 3.49 Å, while the other two are 2.80 Å, as shown in Fig. 1b, which is named as LP divacancy defect. However, for the second configuration, four Si–Si distances d_Si–Si are 3.10 Å, and the other two are 3.62 Å, as shown in Fig. 1c, which is named as RB divacancy defect. From the structure point of view, local distortion could be observed for the two configurations with divacancy defects, and thus lower the structural symmetry. This indicates the appearance of the Jahn-Teller effect. From the energy point of view, the total energy of the silicon crystal with LP divacancy defect is only by 3 meV lower than that with RB divacancy defect, in accord with the reports by Pesola et al. [10] and Makhov et al. [26]. Therefore, the small energy difference shows that the two configurations could both exist in realistic conditions. With regard to hexavacancy ring defect, the Si–Si distances around the defect all decrease from 3.84 Åto 2.74 Å after relaxation, as shown in Fig. 1d.

Next, we will investigate the electronic structures of perfect silicon and silicon with defects. The band structure of the perfect silicon crystal is shown in Fig. 2 (Fermi level is set to be 0 eV, and the same hereinafter). The high symmetry k-points of Brillouin zone along the band paths in Fig. 2 are G(0, 0, 0), F(0, 0.5, 0), Q(0, 0.5, 0.5) and Z(0, 0, 0.5). The results clearly show that silicon crystal is a semiconductor with indirect band gap. The valence band maximum (VBM) locates at G point, while conduction band minimum (CBM) locates at F point. The band that includes the CBM is drawn with blue solid rhombus, which is marked with band “1”. Another band that includes the VBM is drawn with red hollow rhombus, which is marked with band “2”. The band “2” is twofold degenerate because the spin-orbital coupling is not considered in our calculations. Moreover, it is evidently found that the energy bands along G-F-Q and G-Z-Q pathways are symmetrical due to the cubic symmetry in a perfect silicon crystal.

The band structure of silicon crystal with monovacancy defect is shown in Fig. 3. It is found that the band including VBM (band “2”) exhibits the characteristics of localized state and the corresponding charge density mainly concentrates around the vacancy defect by band decomposed charge density analysis. In contrast to the band structure of perfect crystal, two evident defect bands stand above the Fermi level in the crystalline silicon with monovacancy defect, which are marked with band “3” (the curve drawn with purple hollow triangle) and band “4” (green solid triangle). The two defect bands are also related with the localized states and have the asymmetric characteristics. The bands along G-F-Q and G-Z-Q pathways and the band degeneracy are quite different. It indicates that the localized cubic symmetry has changed. Combined the localized structure with monovacancy defect, it can be understood that the change of localized symmetry results from the Jahn-Teller distortion.

The band structures with divacancy defects are shown in Fig. 4 and Fig. 5. Similar to the case of monovacancy defect, the band with VBM (band “2”) corresponds to the localized state, and two clear defect bands (band “3” and “4”) occur above the Fermi level. The dispersion of the two bands relates to their pathways, and shows the asymmetric character along G-F-Q and G-Z-Q pathways. Similar to the case of monovacancy, the reason results from the Jahn-Teller distortion, regardless of LP divacancy defect or RB divacancy defect, thus lowering the original cubic symmetry.

Fig. 6 shows the band structure of silicon crystal with hexavacancy defect. It is different from the cases of monovacancy and divacancy defects. Firstly, the band with VBM corresponds to extended state instead of localized state by band decomposed charge density analysis. Secondly, the defect bands (band “3” and “4”) above the Fermi level are very close to conduction band. Finally, the band structure exhibits high symmetry. In other words, the bands along G-F-Q and G-Z-Q pathways are symmetrical. Moreover, it is worthwhile to note that the Jahn-Teller distortion is not found in crystalline silicon with hexavacancy defect. Therefore, it can be confirmed that the asymmetry of band relates to the Jahn-Teller distortion in the silicon crystals with monovacancy and divacancy defects.

As silicon is an important material in solar cells and electronic devices, the electronic conductivity is very important to its realistic applications. The theory of effective mass can be regarded as a direct method to understand the electronic properties of a material. As a consequence, the related calculations of effective mass for silicon crystal with vacancy defect can be used to study the influence of defects on its conductivity. The effective mass of an electron or a hole can be obtained by the following formula [27, 28]: me(ℏ)∗=h2d2E(k)dk2−1 $$ m_{e(\hbar )}^*\; = \;{h^2}\;{\left( {\frac{{{d^2}E(k)}}{{d{k^2}}}} \right)^{ - 1}} $$

where ħ is reduced Planck’s constant, k is wave vector, E(k) is electronic energy for the wave vector.

The effective mass of electron near the CBM is considered firstly. It should be noted that the bottom of conduction band in our calculations is the lowest unoccupied band, which does not include the defect states, corresponding to the band “1” in all of the band structures. The calculated effective masses are shown in Table 1. For a perfect silicon crystal, the effective mass at F point strongly depends on the direction. The effective mass along FQ direction is 0.197 m₀ (m₀ is free electron mass), which is much smaller than the effective mass of 0.95 m₀ along FG direction, in agreement with the results reported by Wang et al. [29]. Moreover, the effective mass at F point (0, 0.5, 0) is equal to that at Z point (0, 0, 0.5). It means that the effective masses at F point along FQ and FG directions are equal to each other, as well as at Z point along ZQ and ZG directions. It originates from the symmetry of the bands along G-F-Q and G-Z-Q pathways, showing the characteristics of cubic symmetry in a perfect silicon crystal. From Table 1, it can be seen that the electron effective masses at point F and point Z are almost equivalent when vacancy defects exist in crystalline silicon. Specially, the effective mass of electron at point F slightly increases along FG direction when the monovacancy defect and LP divacancy defect are induced, while it is almost unchanged along FQ direction. On the contrary, for the silicon crystal with RB divacancy defect and hexavacancy defect, the effective mass of electron at point F along FG direction decreases a little, whereas it increases along FQ direction. Particularly, the effective mass of electron increases greatly from 0.197 m₀ in the perfect crystal to 1.236 m₀ in the crystal with hexavacancy defect.

Table 1

The calculated electron effective masses for the bottom of the conduction bands (band ”1” shown in all energy band structures) in crystalline silicon with or without the vacancy defects.

Next, the effective masses of a hole near the VBM are calculated and listed in Table 2. Since the bands containing VBM for the crystal with monovacancy and divacancy defects are apparently localized, the hole effective masses calculations are carried out for the bands with non-localized states. Therefore, the band “2” is used to calculate the hole effective masses in the perfect crystal and the crystal with hexavacancy defect, while the first band related to non-localized state under the band “2” is used in the crystal with monovacancy and divacancy defects. For the bands that are considered, however, only the effective masses of holes at G and Q points are calculated because the tops of the band locate at G and Q points. For a perfect crystal, the effective masses at G point along GF and GZ directions are equivalent, as well as the effective masses at Q point along QF and QZ directions. It shows that the effective masses at G and Q points along two different pathways are symmetrical and the difference between them is very small. When vacancy defects are introduced, the symmetry of effective mass of holes retains and the values are somewhat larger than those in a perfect crystal.

Table 2

The calculated hole effective masses for the top of the valence bands (the band related to the non-localized states) in crystalline silicon with or without the vacancy defects.

The effective mass of an electron or a hole of the defect bands and the localized bands near the VBM in a crystal with various kinds of vacancy defects are shown in Table 3. The positive value corresponds to the effective mass of an electron while the negative value corresponds to the effective mass of a hole. The effective mass of an electron is only calculated with respect to the unoccupied defect bands. In contrast, the effective masses of holes are calculated at the tops of localized bands near the VBM. The results for monovacancy defect show that the effective masses of holes near the top of band are somewhat smaller. For instance, the effective masses of holes at point G along GF and GZ directions are 0.180 m₀ and 0.311 m₀, respectively. Similarly, the effective masses of electron at point Q along QF and QZ, corresponding to the bottom of interval band “4”, are 0.338 m₀ and 0.542 m₀. For the crystal with divacancy, regardless of LP or RB, the effective masses of holes at VBM strongly depend on directions. For example, the effective masses of holes at point G along QF and QZ directions in the crystal with LP divacancy defect are 1.593 m₀ and 0.429 m₀, respectively. Likewise, the effective masses of holes at point F along FG and FQ directions in the crystal with LP divacancy defect are 0.447 m₀ and 0.999 m₀, and 0.841 m₀ and 0.378 m₀ at point Z along ZQ and ZG directions, respectively. Compared with the crystal with LP divacancy defect, the difference is more obvious in the crystal with RB divacancy defect. The effective masses of holes at point Z along ZQ and ZG directions are 9.982 m₀ and 0.479 m₀, respectively. For the crystal with hexavacancy defect, either defect band “3” or band “4” have many valleys. Furthermore, the energies for these two bands are very close to each other. Therefore, the calculated values of electron effective mass are very abundant. It can be seen from the calculated results that teh effective masses have apparently symmetric characteristics with respect to Q point. It further shows that the band of the crystal with hexavacancy is symmetric, in accord with the previous energy band analysis.

Table 3

The calculated electron (hole) effective masses for the defect states and the bands near the top of the valence band (the non-localized states) in crystalline silicon with the vacancy defects (negative values refer to hole effective masses, and the positive ones to electron effective masses).