OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11715–11727
« Show journal navigation

Generalized Pseudo-Unit-Cell model for long-wavelength optical phonons of multinary mixed crystals: Application to AxB1-xCyD1-y type mixed crystals

Zhenfeng Liao, Jingzhen Li, Ruisheng Zheng, Xiaowei Lu, and Hongyi Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11715-11727 (2013)
http://dx.doi.org/10.1364/OE.21.011715


View Full Text Article

Acrobat PDF (1352 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

In recent years, more and more attentions have been focused on mixed crystal materials. Mixed crystals have been used to fabricate optoelectronic devices. They offer more flexible choices for consecutive layers in heterostructures and quantum wells with desirable lattice constants and band offsets. The optical properties of ternary mixed crystals have been studied thoroughly by lots of authors. Genzel and Chang had conducted a serial of investigations on ternary mixed crystals [1

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

3

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

]. They put forward a model called Modified-Random-Element-Isodisplacement Model (MREI). The model turned out to be effective in calculating the eigenfrequency of optical phonons when it was applied to ternary mixed crystals. Many authors conducted research on quaternary mixed crystals using various models and obtained some results. Zheng and Taguchi have published their papers on optical phonons of multinary mixed crystals [4

4. R. S. Zheng and M. Matsuura, “Electron-phonon interaction in mixed crystals”, Phys. Rev. B 59, 15422–15429 (1999) [CrossRef] .

6

6. R. S. Zheng, “Properties of optical phonons in multinary phosphide mixed crystals,” J. Shenzhen University (Science and Engineering) , 23(1), 10–15 (2006).

]. Their theory and method can be applied to the case in which one sublattice consists of different kinds of ions, that is to say purely anion solution or purely cation solution. In their paper, the dielectric constants and the oscillator strengths are obtained as a function of element concentrations. Yang applied Zheng’s theory to Zn1−xyMgyBexSe quaternary mixed crystal and found that the numerical result of phonon mode strengths agrees with experimental data well [7

7. 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] .

]. Abid and his co-authors made experimental investigations into the long-wavelength optical phonons of AlxInyGa1−xyN mixed crystal and compared the experimental data with Zheng’s theory, they claimed their measurement shows good agreement with Zheng’s work [8

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 406, 1379–1384 (2011) [CrossRef] .

]. In the very recent year, M. Romcevic and N. Romcevic developed a general method to calculate the eigenfre-quencies of the optical phonons [9

9. M. Romcevic and N. Romcevic, “Phonons in multicomponent alloys,” J. Alloys Compd. 416, 64–71 (2006) [CrossRef] .

]. Their theory can be applied to any mixed crystals, while they did not give the analytical expressions of dielectric function and oscillator strengths. The generalization of the theory to any type of multinary mixed crystals is significant because it can not only give a fundamental understanding of the physics of the multinary compounds, but also provide a useful tool to analyze the physical phenomena of the compounds qualitatively or quantitatively. In the present paper, we extend the Pseudo-Unit-Cell (PUC) model [3

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

] to any multinary mixed crystals. The eigenfrequencies of optical phonons can be calculated with a unitary matrix method and the analytical expressions of dielectric constants as well as oscillator strengths are given as a function of the concentrations.

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

There are various theoretical models used to study the optical properties of mixed crystals. Among them the MREI Model is the most widely used and successful one. The advantage of this model is not only simple enough for application, but also that it can give predictions that agree with experiments well. The PUC model is another good theoretical model also proposed by Chang and Mitra [3

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

]. Many researchers have found the validity of the models when they were applied to the ternary mixed crystals. The PUC model simplifies the concept of many-body and random problem greatly, the Lagrangian of the system is easy to obtain. The PUC model can be regarded as a special case of the MREI model at the long-wavelength limit. From other perspective, we can also use Newtonian equations for this model. It would be easier to get the solution in the general case. In the present work, we start directly from the PUC model and extend it to the general case. As shown in Fig. 1, we consider the general type of mixed crystal A1x1AixiAnxn B1y1BjyjBmym, where xi and yj are the molar fractions of the ions. The Ai ions are cations, and the Bi ions are anions. The lattice of the mixed crystal consists of two sublattices. The Ai ions are distributed in one of them randomly, while the Bj ions are distributed in another one randomly. The nearest neighbors of an Ai ion are always Bj ions, and verse versa. The probability of finding a Bj ion around Ai is proportional to its molar fraction yj. Their distributions follow the principle of statistics, and so on for other ions with the opposite polarity.

Fig. 1 Schematic of a multinary mixed crystal.

3. Numerical calculations for Hg1−xMnxTe1−ySey

Table 1. Parameters of binary crystals corresponding to quaternary mixed crystal Hg1−xMnxTe1−ySey, the unit of the phonon frequencies is cm−1, the parameters are taken from Refs. [1216].

table-icon
View This Table
| View All Tables

Fig. 2 Concentration dependence of LO and TO phonon frequencies of the quaternary mixed crystal Hg1−xMnxTe1−ySey.
Fig. 3 Concentration dependence of oscillator strengths of the quaternary mixed crystal Hg1−xMnxTe1−ySey.
Fig. 4 Concentration dependence of dielectric constants of the quaternary mixed crystal Hg1−xMnxTe1−ySey.

Based on the present theory, one can make a detail investigation into the dielectric constant as a function of the composition ratio. The Fig. 5 indicate that in small 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.

Fig. 5 Concentration dependence of dielectric constants of the quaternary mixed crystal Hg1−xMnxTe1−ySey.

From the numerical results, we discover some very interesting properties of the material. As is shown in Fig. 5 evidently, all the curves of different y (or x) values cross a fixed point. That means when the composition ratio x equals about 0.67, the value of high frequency dielectric constant is independent of the composition ratio y. As the composition ratio y equals about 0.15, the value of the high frequency dielectric constant is independent of the composition ratio x. As composition ratio x equals about 0.67, the value of static dielectric constant is independent of the composition ratio y, and as composition ratio y equals about 0.19, the constant is independent of the composition ratio x. This novel property has not been mentioned by other authors. We can call these points as invariant points or composition independent points of dielectric constant. It should be noted that the invariant point for high frequency dielectric constant does not equal to that of static dielectric constant.

4. Composition independence of dielectric constant in AxB1−xCyD1−y mixed crystals

Theoretically, we can prove analytically the existence of the invariant points with respect to the composition ratios. According to the present theory, the high frequency dielectric constant is independent of the lattice vibration,
ε()=1+2Γ(1Γ)=Φ(Γ),
(27)
where
Γ=ne3ε0(injmxiyjαAiBj),
(28)
represents the weighed sum of the binary crystal polarizability.

Based on the parameters given in Table. 1, we can calculate the value of the invariant points of Hg1−xMnxTe1−ySey by using analytical expressions given by Eqs.(37)(40). The results are x0 = 0.6705, y0 = 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.

We also calculated the dielectric constants of mixed crystals ZnxMg1−xSeyTe1−y and GaxIn1−xNyP1−y using the method introduced in the former sections. In the calculations the second neighboring force constants are neglected because the numerical calculations of Hg1−xMnxTe1−ySey 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 ZnxMg1−xSeyTe1−y, and there is also an invariant point for static dielectric constant of GaxIn1−xNyP1−y. According to Eqs.(38) and (40), the value of invariant point for high frequency dielectric constant of ZnxMg1−xSeyTe1−y is x0 = 0.7781, and that for the static dielectric constant of GaxIn1−xNyP1−y is x′0 = 0.3326. While, invariant point for static dielectric constant of ZnxMg1−xSeyTe1−y and invariant point for high frequency dielectric constant of GaxIn1−xNyP1−y do not exist.

Table 2. Parameters of binary crystals corresponding to mixed crystals ZnxMg1−xSeyTe1−y and GaxIn1−xNyP1−y. The parameters are taken from Refs. [1720].

table-icon
View This Table
| View All Tables
Fig. 6 Concentration dependence of dielectric constants of the quaternary mixed crystal ZnxMg1−xSeyTe1−y.
Fig. 7 Concentration dependence of dielectric constants of the quaternary mixed crystal GaxIn1−xNyP1−y.

Dielectric constant is a very important physical parameter in material science and optoelectronics. The dielectric constants should be measured precisely when the materials are used to fabricate electronic or optical devices. In some cases, It is needed that two materials on both sides of the interface have the same dielectric constant. In this case, light propagating in the materials will not be reflected or deflected by the interface. Thus the composition independent property of dielectric constants of quaternary mixed crystals may have potential applications in optoelectronics and optical engineering. Based on our formula, experimental researchers can calculate the high frequency and static dielectric constants easily and can predict whether there is an invariant point in the dielectric constant. All the parameters needed are the high frequency and static dielectric constants of the corresponding end binary crystals.

5. Conclusion

In this paper, the PUC model is extended to calculate the phonon frequencies of any type of multinary mixed crystals. The dielectric constants and oscillator strengths are obtained as a function of the composition ratios. The mode behaviors of the quaternary mixed crystals can be investigated through the oscillator strengths. The novel concentration independent property for the dielectric constants of quaternary mixed crystal is discovered by the present study. Analytical expressions of the composition independent points are obtained in the framework of generalized PUC model. Some typical quaternary mixed crystals are calculated as examples.

Acknowledgment

We acknowledge the National Natural Science Foundation of China (Grant No. 61027014), and we also acknowledge the financial support from the State Key Program of National Natural Science of China (Grant No. 61136001).

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. 172, 924–933 (1968) [CrossRef] .

3.

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

4.

R. S. Zheng and M. Matsuura, “Electron-phonon interaction in mixed crystals”, Phys. Rev. B 59, 15422–15429 (1999) [CrossRef] .

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 66, 075327 (2002) [CrossRef] .

6.

R. S. Zheng, “Properties of optical phonons in multinary phosphide mixed crystals,” J. Shenzhen University (Science and Engineering) , 23(1), 10–15 (2006).

7.

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] .

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 406, 1379–1384 (2011) [CrossRef] .

9.

M. Romcevic and N. Romcevic, “Phonons in multicomponent alloys,” J. Alloys Compd. 416, 64–71 (2006) [CrossRef] .

10.

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] .

11.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices, (Oxford University, 1954).

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 45, 10934–10341 (1992) [CrossRef] .

13.

M. Grynberg, R. Le Toullec, and M. Balkanski, “Dielectric function in HgTe between 8 and 300K,” Phys. Rev. B 9, 517–526 (1974) [CrossRef] .

14.

A. Manabe and A. Mitsuishi, “Far-infrared reflection spectra of HgSe,” Solid State Commun. 16, 743–745 (1975) [CrossRef] .

15.

W. Gebicki and W. Nazarewicz, “Long-wavelength optical phonons in MgxHg1-xTe mixed crystals,” Phys. Status Solidi B 80, 307–311 (1977) [CrossRef] .

16.

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] .

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 214–215, 359–363 (2000) [CrossRef] .

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. 67, 3611–3613 (1995) [CrossRef] .

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 59, 8128–8131 (1999) [CrossRef] .

20.

O. Madelung, Semiconductors-Basic Data, 2nd revised ed., (Springer, 1996) [CrossRef] .

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 AxB1-xCyD1-y type mixed crystals," Opt. Express 21, 11715-11727 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11715


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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. 172, 924–933 (1968). [CrossRef]
  3. I. F. Chang and S. S. Mitra, “Long wavelength optical phonons in mixed crystals,” Adv. Phys. 20, 359–404 (1971). [CrossRef]
  4. R. S. Zheng and M. Matsuura, “Electron-phonon interaction in mixed crystals”, Phys. Rev. B 59, 15422–15429 (1999). [CrossRef]
  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 66, 075327 (2002). [CrossRef]
  6. R. S. Zheng, “Properties of optical phonons in multinary phosphide mixed crystals,” J. Shenzhen University (Science and Engineering), 23(1), 10–15 (2006).
  7. 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]
  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 406, 1379–1384 (2011). [CrossRef]
  9. M. Romcevic and N. Romcevic, “Phonons in multicomponent alloys,” J. Alloys Compd. 416, 64–71 (2006). [CrossRef]
  10. 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]
  11. M. Born and K. Huang, Dynamical Theory of Crystal Lattices, (Oxford University, 1954).
  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 45, 10934–10341 (1992). [CrossRef]
  13. M. Grynberg, R. Le Toullec, and M. Balkanski, “Dielectric function in HgTe between 8 and 300K,” Phys. Rev. B 9, 517–526 (1974). [CrossRef]
  14. A. Manabe and A. Mitsuishi, “Far-infrared reflection spectra of HgSe,” Solid State Commun. 16, 743–745 (1975). [CrossRef]
  15. W. Gebicki and W. Nazarewicz, “Long-wavelength optical phonons in MgxHg1-xTe mixed crystals,” Phys. Status Solidi B 80, 307–311 (1977). [CrossRef]
  16. 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]
  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 214–215, 359–363 (2000). [CrossRef]
  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. 67, 3611–3613 (1995). [CrossRef]
  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 59, 8128–8131 (1999). [CrossRef]
  20. O. Madelung, Semiconductors-Basic Data, 2nd revised ed., (Springer, 1996). [CrossRef]

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.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited