Theoretical investigation of the electronic and optical properties of pseudocubic Si3P4, Ge3P4 and Sn3P4

doi:10.1364/OPEX.14.000710

« Show journal navigation

Theoretical investigation of the electronic and optical properties of pseudocubic Si₃P₄, Ge₃P₄ and Sn₃P₄

Ming Xu, Songyou Wang, Gang Yin, Liangyao Chen, and Yu Jia »View Author Affiliations

Optics Express, Vol. 14, Issue 2, pp. 710-716 (2006)
http://dx.doi.org/10.1364/OPEX.14.000710

View Full Text Article

Acrobat PDF (200 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. Computational method
3. Results and discussion
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (15)
Cited By
Figures (5)
Metrics

Abstract

Group-IV phosphides are relatively unknown materials as compared to the Group-IV carbide. In this work, we detailed the first principles calculations of the electronic and optical properties of the pseudocubic M₃P₄ (M=Si, Ge, Sn) using the density function theory (DFT). Results are in good agreement with those previous works. Furthermore, the optical constants, such as the dielectric function, energy loss function and effective number of valence electrons are calculated and presented in the study.

1. Introduction

IV-V compounds have attracted great attentions in the last decade due to their important applications. C₃N₄, Si₃N₄ and Ge₃N₄ are well known for their high bulk moduli and wide band gaps to be used as the high-performance engineering materials [1–3

M. L. Cohen, “Predicting useful materials,” Science 261, 307(1993). [CrossRef] [PubMed]

]. Recently, Feng et al investigated the structural properties and bulk moduli of C3P4, Si₃P₄, Ge₃P₄ and Sn₃P₄ by using the first principles calculations successfully to predict that the most stable phase of those compounds is the pseudocubic structure [4–9

A. T. L. Lim, Y. P. Feng, and J. C. Zheng, “Stability of Hypothetical Carbon Phosphide Solids,” Int. J. Mod. Phys. B 16, 1101 (2002). [CrossRef]

]. Their results show that the bulk modulus decreases as the group-IV atom changed from C to Sn and it is much smaller than that of the corresponding Nitrides. For Bulk M₃N₄, however, there is still lack of the experimental data for the crystalline M₃N₄. In this work, we made continuous efforts to study the structural and optical properties of bulk M₃N₄ in consideration of their stable phase.

The electronic structures and optical properties of pseudocubic M₃N₄ are studied based on the first-principles density function theory [10

P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev. 136, B864 (1964). [CrossRef]

, 11

W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev. 140, A1133 (1965). [CrossRef]

] with the generalized gradient approximation (GGA) [12

J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865 (1996). [CrossRef] [PubMed]

]. We first carefully calculated the electronic structures, because the optical properties depend on both the inter- and intra-band transitions. Afterwards, we present the optical properties of pseudocubic M₃N₄ in our work, including the dielectric function, electron energy loss function and effective number of the valence electrons. The paper is organized as follows: A brief description of the calculation method is given in Sec. II; the results for optimized structural parameters, band structures, dielectric properties and some discussions are shown in Sec. III; and a summary is finally drawn in Sec. IV.

2. Computational method

First-principles total-energy calculations were performed using the CASTEP code [13

M. Segall, P. Lindan, M. Probert, C. Pickard, P. Hasnip, S. Clark, and M. Payne, “First-principles simulation: ideas, illustrations and the CASTEP code,” J. Phys. Condens. Matter 14, 2717(2002). [CrossRef]

]. The generalized gradient approximation for exchange-correlation effects and the Vanderbilt ultrasoft pseudopotentials were used in the calculations. The maximum plane-wave cutoff energy was taken as 300eV and the numerical integration of the Brillouin zone was performed using a 5×5×5 Monkhorst-Pack k-point sampling procedure.

3. Results and discussion

The structural properties, including the crystal structure, lattice constant and bulk modulus, are very important to understand other properties of the material fundamentally. Previous works showed that the pseudocubic Si₃P₄, Ge₃P₄ and Sn₃P₄ all have the lower energy than other structures in the study [6–9

M. Huang, Y. P. Feng, A. T. L. Lim, and J. C. Zheng, “Structural and electronic properties of Si₃P₄ ,” Phys. Rev. B 69,054112 (2004). [CrossRef]

]. The pseudocubic phase (Fig. 1), which is a type of the defective zinc-blend structure, belongs to the P4̅2m space group and is both energetically and mechanically stable when M₃N₄ compounds are formed.

Fig. 1. Ball-stick model of pseudocubic-M₃N₄

In this work, we first optimized the geometrical structure of M₃P₄ compounds in the ground states. In table 1, we present the calculated lattice constants (a and c), the bulk modulus (B₀) and bond length as compared to the theoretical results in previous works. Our results of the lattice constant agree well with other theoretical results within the difference of about 1% and there is no big difference between a and c, especially for Si₃P₄ and Ge₃P₄,. Therefore, this pseudocubic phase shows little anisotropic properties though atoms are missed on the planes of the ideal zinc-blend structure along the c-axis.

Table 1 Calculated equilibrium structure parameters and properties of the pseudocubic M₃N₄

		Si₃P₄	Ge₃P₄	Sn₃P₄
Lattice constant (Å)
This work	a	5.038	5.142	5.531
This work	c	5.038	5.142	5.504
Previous work (Ref.6)	a	5.027
Previous work (Ref.6)	c	4.998
Bulk modulus (GPa)
This work		74.28	61.83	47.51
Previous work B₀		77(Ref. 6)	62(Ref. 8)	36(Ref.8)
Bond length(Å)		2.26	2.32	2.51

By applying the small strains onto the equilibrium lattice, the bulk moduli of M₃N₄ are obtained by fitting the calculated total energy vs volume data to Murnaghan equation of states. The calculated bulk moduli given in table 1 are close to that of previous works. The small differences maybe come from the parameters selected in our calculations. For example, we used the exchange correlation potential of the PBE form of GGA, while the work of Ref. 6 selected the PW91 form.

The calculated band structures of the pseudocubic M₃N₄ are plotted in Fig. 2. The results show that they all have the small and indirect band gaps. The indirect band gap of Si₃P₄ occurs between the A and Γ points corresponding to the valence and conduction bands, respectively. The tops of the valence bands of the Ge₃P₄ or Sn₃P₄ structures, however, are located at the points that are near the Γ point along the Γ-M direction.

Fig. 2. Calculated band structure for pseudocubic-M₃N₄. (a) Si₃P₄, (b) Ge₃P₄ and (c) Sn₃P₄

The values of calculated band gaps of Si₃P₄, Ge₃P₄ and Sn₃P₄ are 0.33eV, 0.17eV and 0.83eV, respectively. Since the GGA method may underestimate the band gap and there is no experiment carried out for these compounds, the band gap calculated by GW quasiparticle theory may give more accurate value [14

M. S. Hybertsen and S. G. Louie, “Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies,” Phys. Rev. B 34, 5390 (1986). [CrossRef]

, 15

M. P. Surh, S. G. Louie, and M. L. Cohen, “Quasiparticle energies for cubic BN, BP, and BAs,” Phys. Rev. B 43, 9126 (1991). [CrossRef]

]. Anyway, the results show that all three compounds are indirect band gap semiconductors and the band gap of Sn₃P₄ is wider than the others.

The gross features of the density of states (DOS) from M₃N₄ compounds are quite similar, so we only show the result for Si₃P₄ as an example in Fig. 3. It can be seen that there are three regions in the energy range of the valence band. In the first region, the band with the energy ranging from -14 eV to - 9 eV is mainly from the 3s states of P and Si sites. In the second region, the band with the energy ranging from -9 eV to - 6 eV is hybridized by the 3s and 3p states. The third region of the valence band ranging from -6eV to 0 eV, combined with the conduction band above the energy level of 0 eV, is mostly originated from the 3p states of the P and Si sites.

Fig. 3. Calculated total and partial density of states for pseudocubic- Si₃P₄

It is well known that the linear electronic response of a system to an external electromagnetic field can be measured by the complex dielectric function ε(ω)=ε₁(ω)+iε₂(ω). The imaginary part of the dielectric function ε₂(ω) can be obtained from the electronic structure calculations by using the matrix elements between the occupied and unoccupied wave functions.

In terms of the electric-dipole approximation, the imaginary part of the dielectric functions can be calculated by the following formula:

ε_{2} (ω) = \frac{8 π^{2} e^{2}}{ω^{2} m^{2} V} \sum_{c, v} \sum_{k} {∣ c, k ∣ \hat{e} ∙ p | v, k ∣}^{2} δ (E_{c} (k) - E_{v} (k) - ħ ω)

(1)

where c and v represent the conduction and valence bands, respectively; ∣n,k> is the eigenstate obtained from band structure calculations; p is the momentum operator; ê is the external field vector; and ω is the frequency of incident photons.

The real part ε₁(ω) can be derived from the ε₂(ω) by the Kramer-Kroning transformation. Furthermore, other optical properties like the energy loss function L(ω) can be derived from ε₁(ω) and ε₂(ω) as

L (ω) = \frac{ε_{2} (ω)}{(ε_{1}^{2} (ω) + ε_{2}^{2} (ω))}

(2)

We calculated both of the longitudinal and transverse dielectric functions to see that these two functions are almost identical within the difference of about 1%. It indicates that the IVV compounds with the pseudocubic phase are not strongly anisotropic though they are structurally asymmetric. Therefore, only the longitudinal optical properties of M₃N₄ are taken into consideration in the study.

The comparison of the calculated optical properties for the three pseudocubic compounds is given in Fig. 4. The imaginary dielectric function ε₂(ω) represents the optical absorption of the material. Each spectrum has a prominent absorption peak. They are located at about 4.0, 3.9 and 3.5 eV photon energy, respectively. The imaginary part ε₂(ω) of Si₃P₄ around 4.0 eV is mainly attributed to the transitions at the Γ point from 0.41 eV to 4.05 eV and 4.28 eV to 0.09 eV. The ε₂(ω) peak of Ge₃P₄ is located at about 3.9 eV, corresponding to the transitions from 0.17 eV to -3.9 eV and 4.1 eV to -0.1eV at the Γ point. For the Sn₃P₄ structure, the peak occurs at about 3.5 eV and is due to the main part of transitions from 1.6 eV to -1.9eV at the Γ point. It is noted that the peak shown in the ε₂(ω) spectra is not just due to the single interband transition since the multiple direct and indirect transitions will happen around the same energy peak positions.

Fig. 4. Dielectric functions (a) and energy-loss function (b) of M₃N₄

The real dielectric function ε₁(ω) is obtained from ε²(ω) by the Kramers-Kronig transformation. In the zero frequency limit, ε₁(ω) is the static dielectric constant, excluding any contribution from the lattice vibrations. The static dielectric constants for the three compounds are 14.66, 18.03 and 15.32, respectively. These values seem larger than that of c-(Si, Ge)₃N₄ in the cubic spinal structure [16

W. Y. Ching, Shang-Di Mo, and Lizhi Ouyang, “Electronic and optical properties of the cubic spinel phase of c-Si₃N₄, c-Ge₃N₄, c-SiGe₂N₄, and c-GeSi₂N₄ ,” Phys. Rev. B 63, 245110(2001). [CrossRef]

The energy loss function is an important optical parameter to show the peak often associated with the plasma frequency and described as the plasma resonance [17

S. Saha, T. P. Sinha, and A. Mookerjee, “Electronic structure, chemical bonding, and optical properties of paraelectric BaTiO₃ ,” Phys. Rev. B 62, 8828 (2000). [CrossRef]

]. Above and below the plasma frequency the material will have the dielectric and metallic properties, respectively. The calculated peak of L(ω) of Si₃P₄ is located at about 18 eV, while those of Ge₃P₄ and Sn₃P₄ are shifted to lower energies at about 17.5 eV and 15 eV, respectively.

Since the plasma resonance is intrinsically arisen from the collected oscillations of free electrons, the effective number n_eff (ω _m ) of valence electrons per atom that contributes to the interband transitions can be calculated in terms of the expression as

n_{eff} (ω_{m}) = \frac{2 m ε_{0}}{π e^{2}} \int_{0}^{ω_{m}} ω ε_{2} (ω) dω

(3)

where ω_m denotes the upper limit of integration, and the quantities m and e are the electron mass and charge, respectively. The results are shown in Fig. 5. The effective number of valence electrons of M₃N₄ decreases with increasing atomic number. The saturated n_eff(ωm) above 10 eV is close to the average number of the valence electrons per atom in M₃N₄. The discrepancies of peak positions in the energy loss function [see Fig. 4(b)] partly attribute to the different saturated values of n_eff for the Si₃P₄, Ge₃P₄ and Sn₃P₄ structures.

Fig.5. The calculated effective number of valence electrons (n_eff ) participating in the interband optical transitions of M₃N₄

4. Conclusion

The electronic structures and optical properties of the pseudocubic Si₃P₄, Ge₃P₄ and Sn₃P₄ have been studied by using the first principles density functional method. The energy band and density of states are calculated as compared to that of previous works. Results are in good agreement with early ones to show that the pseudocubic Si₃P₄, Ge₃P₄ and Sn₃P₄ all have indirect band gaps. The bulk modulus of M₃N₄ is lower than that of M₃N₄. The complex dielectric and energy loss functions are also calculated based on the band structure. The effective number of the valence electrons of M₃N₄, which contributes to the interband optical transitions, decreases with increasing atomic number. The results show that the electronic and optical properties of the M₃N₄ compounds having the pseudocubic phase are nearly isotropic.

Acknowledgments

This work was partially supported by the Natural Science Foundation of China (Grant No. 60578046) and the Shanghai Science and Technology Commission (Grand No. 05PJ14016).

References and links

1.	M. L. Cohen, “Predicting useful materials,” Science 261, 307(1993). [CrossRef] [PubMed]
2.	J. L. He, L. C. Guo, D. L. Yu, R. P. Liu, Y. J. Tian, and H. T. Wang, “Hardness of cubic spinel Si₃N₄ ,” Appl. Phys. Lett. 85, 5571(2004). [CrossRef]
3.	B. Molina and L. E. Sansores, “Electronic structure of Ge₃N₄ possible structures,” Int. J. Quantum Chem. 80, 249 (2000). [CrossRef]
4.	A. T. L. Lim, Y. P. Feng, and J. C. Zheng, “Stability of Hypothetical Carbon Phosphide Solids,” Int. J. Mod. Phys. B 16, 1101 (2002). [CrossRef]
5.	A. T. L. Lim, Y. P. Feng, and J. C. Zheng, “Interesting electronic and structural properties of C₃P₄ ,” Mater. Sci. Eng. B 99, 527 (2003). [CrossRef]
6.	M. Huang, Y. P. Feng, A. T. L. Lim, and J. C. Zheng, “Structural and electronic properties of Si₃P₄ ,” Phys. Rev. B 69,054112 (2004). [CrossRef]
7.	M. Huang and Y. P. Feng, “Further study on structural and electronic properties of silicon phosphide compounds with 3:4 stoichiometry,” Comput. Mater. Sci. 30, 371 (2004). [CrossRef]
8.	M. Huang and Y. P. Feng, Phys. Rev. B 70,184116 (2004); [CrossRef]
9.	M. Huang and Y. P. Feng, “Theoretical prediction of the structure and properties of Sn₃N₄ ,” J. Appl. Phys. 96, 4015 (2004). [CrossRef]
10.	P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev. 136, B864 (1964). [CrossRef]
11.	W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev. 140, A1133 (1965). [CrossRef]
12.	J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865 (1996). [CrossRef] [PubMed]
13.	M. Segall, P. Lindan, M. Probert, C. Pickard, P. Hasnip, S. Clark, and M. Payne, “First-principles simulation: ideas, illustrations and the CASTEP code,” J. Phys. Condens. Matter 14, 2717(2002). [CrossRef]
14.	M. S. Hybertsen and S. G. Louie, “Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies,” Phys. Rev. B 34, 5390 (1986). [CrossRef]
15.	M. P. Surh, S. G. Louie, and M. L. Cohen, “Quasiparticle energies for cubic BN, BP, and BAs,” Phys. Rev. B 43, 9126 (1991). [CrossRef]
16.	W. Y. Ching, Shang-Di Mo, and Lizhi Ouyang, “Electronic and optical properties of the cubic spinel phase of c-Si₃N₄, c-Ge₃N₄, c-SiGe₂N₄, and c-GeSi₂N₄ ,” Phys. Rev. B 63, 245110(2001). [CrossRef]
17.	S. Saha, T. P. Sinha, and A. Mookerjee, “Electronic structure, chemical bonding, and optical properties of paraelectric BaTiO₃ ,” Phys. Rev. B 62, 8828 (2000). [CrossRef]

OCIS Codes
(160.0160) Materials : Materials
(300.6470) Spectroscopy : Spectroscopy, semiconductors

ToC Category:
Materials

Citation
Ming Xu, Songyou Wang, Gang Yin, Liangyao Chen, and Yu Jia, "Theoretical investigation of the electronic and optical properties of pseudocubic Si₃P₄, Ge₃P₄ and Sn₃P₄," Opt. Express 14, 710-716 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-710

Sort: Journal | Reset

References

M. L. Cohen, "Predicting useful materials," Science 261, 307(1993). [CrossRef] [PubMed]
J. L. He, L. C. Guo, D. L. Yu, R. P. Liu, Y. J. Tian, and H. T. Wang, "Hardness of cubic spinel Si₃N₄," Appl. Phys. Lett. 85, 5571(2004). [CrossRef]
B. Molina and L. E. Sansores, "Electronic structure of Ge₃N₄ possible structures," Int. J. Quantum Chem. 80, 249 (2000). [CrossRef]
A. T. L. Lim, Y. P. Feng, and J. C. Zheng, "Stability of hypothetical carbon phosphide solids," Int. J. Mod. Phys. B 16, 1101 (2002). [CrossRef]
A. T. L. Lim, Y. P. Feng, and J. C. Zheng, "Interesting electronic and structural properties of C₃P₄," Mater. Sci. Eng. B 99, 527 (2003). [CrossRef]
M. Huang, Y. P. Feng, A. T. L. Lim, and J. C. Zheng, "Structural and electronic properties of Si₃P₄," Phys. Rev. B 69, 054112 (2004). [CrossRef]
M. Huang and Y. P. Feng, "Stability and electronic properties of Sn₃P₄," Phys. Rev. B 70, 184116 (2004) [CrossRef]
M. Huang and Y. P. Feng, "Theoretical prediction of the structure and properties of Sn₃N₄," J. Appl. Phys. 96, 4015 (2004). [CrossRef]
P. Hohenberg and W. Kohn, "Inhomogeneous electron gas," Phys. Rev. 136, B864 (1964). [CrossRef]
W. Kohn and L. J. Sham, "Self-consistent equations including exchange and correlation effects," Phys. Rev. 140, A1133 (1965). [CrossRef]
J. P. Perdew, K. Burke, and M. Ernzerhof, "Generalized gradient approximation made simple," Phys. Rev. Lett. 77, 3865 (1996). [CrossRef] [PubMed]
M. Segall, P. Lindan, M. Probert, C. Pickard, P. Hasnip, S. Clark, and M. Payne, "First-principles simulation: ideas, illustrations and the CASTEP code," J. Phys. Condens. Matter 14, 2717(2002). [CrossRef]
M. S. Hybertsen and S. G. Louie, "Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies," Phys. Rev. B 34, 5390 (1986). [CrossRef]
M. P. Surh, S. G. Louie, and M. L. Cohen, "Quasiparticle energies for cubic BN, BP, and BAs," Phys. Rev. B 43, 9126 (1991). [CrossRef]
S. Saha, T. P. Sinha, and A. Mookerjee, "Electronic structure, chemical bonding, and optical properties of paraelectric BaTiO₃," Phys. Rev. B 62, 8828 (2000). [CrossRef]

Appl. Phys. Lett.

J. L. He, L. C. Guo, D. L. Yu, R. P. Liu, Y. J. Tian, and H. T. Wang, "Hardness of cubic spinel Si₃N₄," Appl. Phys. Lett. 85, 5571(2004). [CrossRef]

Int. J. Mod. Phys.

A. T. L. Lim, Y. P. Feng, and J. C. Zheng, "Stability of hypothetical carbon phosphide solids," Int. J. Mod. Phys. B 16, 1101 (2002). [CrossRef]

Int. J. Quantum Chem.

B. Molina and L. E. Sansores, "Electronic structure of Ge₃N₄ possible structures," Int. J. Quantum Chem. 80, 249 (2000). [CrossRef]

J. Appl. Phys.

M. Huang and Y. P. Feng, "Theoretical prediction of the structure and properties of Sn₃N₄," J. Appl. Phys. 96, 4015 (2004). [CrossRef]

J. Phys. Condens. Matter

M. Segall, P. Lindan, M. Probert, C. Pickard, P. Hasnip, S. Clark, and M. Payne, "First-principles simulation: ideas, illustrations and the CASTEP code," J. Phys. Condens. Matter 14, 2717(2002). [CrossRef]

Mater. Sci. Eng. B

A. T. L. Lim, Y. P. Feng, and J. C. Zheng, "Interesting electronic and structural properties of C₃P₄," Mater. Sci. Eng. B 99, 527 (2003). [CrossRef]

Phys. Rev.

P. Hohenberg and W. Kohn, "Inhomogeneous electron gas," Phys. Rev. 136, B864 (1964). [CrossRef]
W. Kohn and L. J. Sham, "Self-consistent equations including exchange and correlation effects," Phys. Rev. 140, A1133 (1965). [CrossRef]

Phys. Rev. B

M. Huang, Y. P. Feng, A. T. L. Lim, and J. C. Zheng, "Structural and electronic properties of Si₃P₄," Phys. Rev. B 69, 054112 (2004). [CrossRef]
M. Huang and Y. P. Feng, "Stability and electronic properties of Sn₃P₄," Phys. Rev. B 70, 184116 (2004) [CrossRef]
M. S. Hybertsen and S. G. Louie, "Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies," Phys. Rev. B 34, 5390 (1986). [CrossRef]
M. P. Surh, S. G. Louie, and M. L. Cohen, "Quasiparticle energies for cubic BN, BP, and BAs," Phys. Rev. B 43, 9126 (1991). [CrossRef]
S. Saha, T. P. Sinha, and A. Mookerjee, "Electronic structure, chemical bonding, and optical properties of paraelectric BaTiO₃," Phys. Rev. B 62, 8828 (2000). [CrossRef]

Phys. Rev. Lett.

J. P. Perdew, K. Burke, and M. Ernzerhof, "Generalized gradient approximation made simple," Phys. Rev. Lett. 77, 3865 (1996). [CrossRef] [PubMed]

Science

M. L. Cohen, "Predicting useful materials," Science 261, 307(1993). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper