## Generalized Pseudo-Unit-Cell model for long-wavelength optical phonons of multinary mixed crystals: Application to
_{x}B |

Optics Express, Vol. 21, Issue 10, pp. 11715-11727 (2013)

http://dx.doi.org/10.1364/OE.21.011715

Acrobat PDF (1352 KB)

### Abstract

Long-wavelength optical phonons in multinary mixed crystals are studied based on the Pseudo-Unit-Cell model. A unitary matrix method is developed to calculate the eigenfrequencies of optical phonons in multinary mixed crystals. The analytical expressions of oscillator strengths and dielectric constants of the multinary mixed crystals are obtained as a function of the phonon frequencies. The results indicate that the composition dependence of oscillator strengths shows clearly the phonon-mode behaviors of the mixed crystals. The theory and calculation method can be applied to any type of multinary mixed crystals. It is found that there is a composition independent point for the dielectric constant of quaternary mixed crystals.

© 2013 OSA

## 1. Introduction

## 2. Model of long-wavelength optical phonons in multinary mixed crystals

3. I. F. Chang and S. S. Mitra, “Long wavelength optical phonons in mixed crystals,” Adv. Phys. **20**, 359–404 (1971) [CrossRef] .

*A*

_{1x1}⋯

*A*

_{ixi}⋯

*A*

_{nxn}

*B*

_{1y1}⋯

*B*

_{jyj}⋯

*B*

_{mym}, where

*x*and

_{i}*y*are the molar fractions of the ions. The

_{j}*A*ions are cations, and the

_{i}*B*ions are anions. The lattice of the mixed crystal consists of two sublattices. The

_{i}*A*ions are distributed in one of them randomly, while the

_{i}*B*ions are distributed in another one randomly. The nearest neighbors of an

_{j}*A*ion are always

_{i}*B*ions, and verse versa. The probability of finding a

_{j}*B*ion around

_{j}*A*is proportional to its molar fraction

_{i}*y*. Their distributions follow the principle of statistics, and so on for other ions with the opposite polarity.

_{j}## 3. Numerical calculations for *Hg*_{1−x}*Mn*_{x}Te_{1−y}*Se*_{y}

_{x}Te

_{y}

*y*composition ratio range, the high frequency dielectric constants have a linear dependence on the ratio

*x*. While in the large

*y*range, the dependence relation becomes nonlinear and it decreases as

*x*increases. The static dielectric constant has a similar variation trend.

## 4. Composition independence of dielectric constant in *A*_{x}B_{1−x}*C*_{y}D_{1−y} mixed crystals

_{x}B

_{y}D

*Hg*

_{1−x}

*Mn*

_{x}Te_{1−y}

*Se*by using analytical expressions given by Eqs.(37)–(40). The results are

_{y}*x*

_{0}= 0.6705,

*y*

_{0}= 0.1514 for the high frequency dielectric constant and

*x′*

_{0}= 0.6674,

*y′*

_{0}= 0.1899 for the static dielectric constant. It is found that the analytical results match the points well in Fig. 5.

*Zn*

_{x}Mg_{1−x}

*Se*

_{y}Te_{1−y}and

*Ga*

_{x}In_{1−x}

*N*

_{y}P_{1−y}using the method introduced in the former sections. In the calculations the second neighboring force constants are neglected because the numerical calculations of

*Hg*

_{1−x}

*Mn*

_{x}Te_{1−y}

*Se*indicate that the static dielectric constants are independent of the second neighboring force constants. The corresponding parameters are shown in Table. 2, and the results are shown in Fig. 6 and Fig. 7. Numerical calculations indicate that there is an invariant point for high frequency dielectric constant of

_{y}*Zn*

_{x}Mg_{1−x}

*Se*

_{y}Te_{1−y}, and there is also an invariant point for static dielectric constant of

*Ga*

_{x}In_{1−x}

*N*

_{y}P_{1−y}. According to Eqs.(38) and (40), the value of invariant point for high frequency dielectric constant of

*Zn*

_{x}Mg_{1−x}

*Se*

_{y}Te_{1−y}is

*x*

_{0}= 0.7781, and that for the static dielectric constant of

*Ga*

_{x}In_{1−x}

*N*

_{y}P_{1−y}is

*x′*

_{0}= 0.3326. While, invariant point for static dielectric constant of

*Zn*

_{x}Mg_{1−x}

*Se*

_{y}Te_{1−y}and invariant point for high frequency dielectric constant of

*Ga*

_{x}In_{1−x}

*N*

_{y}P_{1−y}do not exist.

## 5. Conclusion

## Acknowledgment

## References and links

1. | I. F. Chang, Ph.D. dissertation, University of Rhode Island, (1968). |

2. | I. F. Chang and S. S. Mitra, “Application of a modified Random-Element-Isodisplacement model to long-wavelength optic phonons of mixed crystals,” Phys. Rev. |

3. | I. F. Chang and S. S. Mitra, “Long wavelength optical phonons in mixed crystals,” Adv. Phys. |

4. | R. S. Zheng and M. Matsuura, “Electron-phonon interaction in mixed crystals”, Phys. Rev. B |

5. | R. S. Zheng and T. Taguchi, “Theory of long-wavelength optical lattice vibrations in multinary mixed crystals: Application to group-III nitride alloys,” Phys. Rev. B |

6. | R. S. Zheng, “Properties of optical phonons in multinary phosphide mixed crystals,” J. Shenzhen University (Science and Engineering) , |

7. | F. Yang and R. S. Zheng, “Properties of optical phonons in Zn1-x-yMgyBexSe quaternary mixed crystal,” Solid State Commun. |

8. | M. A. Abid, H. Abu. Hassan, Z. Hassan, S. S. Ng, S. K. Mohd. Bakhori, and N. H. Abd. Raof, “Experimental investigation of long-wavelength optical lattice vibrations in quaternary AlxInyGa1-x-yN alloys and comparison with results from the pseudo-unit cell model,” Physica B |

9. | M. Romcevic and N. Romcevic, “Phonons in multicomponent alloys,” J. Alloys Compd. |

10. | L. Genzel, T. P. Martin, and C. H. Perry, “Model for long - wavelength optical-phonon modes of mixed crystals,” Phys. Status Solidi B |

11. | M. Born and K. Huang, |

12. | E. Oh, R. G. Alonso, J. Miotkowski, and A. K. Ramdas, “Raman scattering from vibrational and electronic excitations in a II–VI quaternary compound: Cd1-x-yZnxMnyTe,” Phys. Rev. B |

13. | M. Grynberg, R. Le Toullec, and M. Balkanski, “Dielectric function in HgTe between 8 and 300K,” Phys. Rev. B |

14. | A. Manabe and A. Mitsuishi, “Far-infrared reflection spectra of HgSe,” Solid State Commun. |

15. | W. Gebicki and W. Nazarewicz, “Long-wavelength optical phonons in MgxHg1-xTe mixed crystals,” Phys. Status Solidi B |

16. | M. Romcevic, V. A. Kulbachinskii, N. Romcevic, P. D. Maryanchuk, and L. A. Churilov, “Optical properties of Hg1-xMnxTe1-ySey,” Infrared Phys. Technol. |

17. | H. Makino, H. Sasaki, J. H. Chang, and T. Yao, “Raman investigation of Zn1-xMgxSe1-yTey quaternary alloys grown by molecular beam epitaxy,” J. Cryst. Growth |

18. | D. Huang, C. Jin, D. Wang, X. Liu, J. Wang, and X. Wang, “Crystal structure and Raman scattering in Zn1-xMgxSe alloys,” Appl. Phys. Lett. |

19. | C. S. Yang, W. C. Chou, D. M. Chen, C. S. Ro, and J. L. Shen, “Lattice vibration of ZnSe1-xTex epilayers grown by molecular-beam epitaxy,” Phys. Rev. B |

20. | O. Madelung, |

**OCIS Codes**

(160.0160) Materials : Materials

(160.4760) Materials : Optical properties

**ToC Category:**

Materials

**History**

Original Manuscript: March 22, 2013

Revised Manuscript: April 18, 2013

Manuscript Accepted: April 23, 2013

Published: May 6, 2013

**Citation**

Zhenfeng Liao, Jingzhen Li, Ruisheng Zheng, Xiaowei Lu, and Hongyi Chen, "Generalized Pseudo-Unit-Cell model for long-wavelength optical phonons of multinary
mixed crystals: Application to A_{x}B_{1-x}C_{y}D_{1-y} type mixed
crystals," Opt. Express **21**, 11715-11727 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11715

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### References

- I. F. Chang, Ph.D. dissertation, University of Rhode Island, (1968).
- I. F. Chang and S. S. Mitra, “Application of a modified Random-Element-Isodisplacement model to long-wavelength optic phonons of mixed crystals,” Phys. Rev. 172, 924–933 (1968). [CrossRef]
- I. F. Chang and S. S. Mitra, “Long wavelength optical phonons in mixed crystals,” Adv. Phys. 20, 359–404 (1971). [CrossRef]
- R. S. Zheng and M. Matsuura, “Electron-phonon interaction in mixed crystals”, Phys. Rev. B 59, 15422–15429 (1999). [CrossRef]
- R. S. Zheng and T. Taguchi, “Theory of long-wavelength optical lattice vibrations in multinary mixed crystals: Application to group-III nitride alloys,” Phys. Rev. B 66, 075327 (2002). [CrossRef]
- R. S. Zheng, “Properties of optical phonons in multinary phosphide mixed crystals,” J. Shenzhen University (Science and Engineering), 23(1), 10–15 (2006).
- F. Yang and R. S. Zheng, “Properties of optical phonons in Zn1-x-yMgyBexSe quaternary mixed crystal,” Solid State Commun. 141, 555–558 (2007). [CrossRef]
- M. A. Abid, H. Abu. Hassan, Z. Hassan, S. S. Ng, S. K. Mohd. Bakhori, and N. H. Abd. Raof, “Experimental investigation of long-wavelength optical lattice vibrations in quaternary AlxInyGa1-x-yN alloys and comparison with results from the pseudo-unit cell model,” Physica B 406, 1379–1384 (2011). [CrossRef]
- M. Romcevic and N. Romcevic, “Phonons in multicomponent alloys,” J. Alloys Compd. 416, 64–71 (2006). [CrossRef]
- L. Genzel, T. P. Martin, and C. H. Perry, “Model for long - wavelength optical-phonon modes of mixed crystals,” Phys. Status Solidi B 62, 83–92 (1974). [CrossRef]
- M. Born and K. Huang, Dynamical Theory of Crystal Lattices, (Oxford University, 1954).
- E. Oh, R. G. Alonso, J. Miotkowski, and A. K. Ramdas, “Raman scattering from vibrational and electronic excitations in a II–VI quaternary compound: Cd1-x-yZnxMnyTe,” Phys. Rev. B 45, 10934–10341 (1992). [CrossRef]
- M. Grynberg, R. Le Toullec, and M. Balkanski, “Dielectric function in HgTe between 8 and 300K,” Phys. Rev. B 9, 517–526 (1974). [CrossRef]
- A. Manabe and A. Mitsuishi, “Far-infrared reflection spectra of HgSe,” Solid State Commun. 16, 743–745 (1975). [CrossRef]
- W. Gebicki and W. Nazarewicz, “Long-wavelength optical phonons in MgxHg1-xTe mixed crystals,” Phys. Status Solidi B 80, 307–311 (1977). [CrossRef]
- M. Romcevic, V. A. Kulbachinskii, N. Romcevic, P. D. Maryanchuk, and L. A. Churilov, “Optical properties of Hg1-xMnxTe1-ySey,” Infrared Phys. Technol. 46, 379–387 (2005). [CrossRef]
- H. Makino, H. Sasaki, J. H. Chang, and T. Yao, “Raman investigation of Zn1-xMgxSe1-yTey quaternary alloys grown by molecular beam epitaxy,” J. Cryst. Growth 214–215, 359–363 (2000). [CrossRef]
- D. Huang, C. Jin, D. Wang, X. Liu, J. Wang, and X. Wang, “Crystal structure and Raman scattering in Zn1-xMgxSe alloys,” Appl. Phys. Lett. 67, 3611–3613 (1995). [CrossRef]
- C. S. Yang, W. C. Chou, D. M. Chen, C. S. Ro, and J. L. Shen, “Lattice vibration of ZnSe1-xTex epilayers grown by molecular-beam epitaxy,” Phys. Rev. B 59, 8128–8131 (1999). [CrossRef]
- O. Madelung, Semiconductors-Basic Data, 2nd revised ed., (Springer, 1996). [CrossRef]

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