## 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 [1–3]. 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 [4–6]. 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 *Zn*_{1−x−y}*Mg _{y}Be_{x}Se* quaternary mixed crystal and found that the numerical result of phonon mode strengths agrees with experimental data well [7]. Abid and his co-authors made experimental investigations into the long-wavelength optical phonons of

*Al*

_{x}In_{y}Ga_{1−x−y}

*N*mixed crystal and compared the experimental data with Zheng’s theory, they claimed their measurement shows good agreement with Zheng’s work [8]. In the very recent year, M. Romcevic and N. Romcevic developed a general method to calculate the eigenfre-quencies of the optical phonons [9]. 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] 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]. 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 *A*_{1x1} ⋯ *A*_{ixi} ⋯ *A*_{nxn} *B*_{1y1} ⋯ *B*_{jyj} ⋯ *B*_{mym}, where *x _{i}* and

*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}In the PUC model, the effective unit cell of the mixed polar crystal consists of two ions with opposite polarities. In this paper, one of the ions is a pseudoion which is similar to combination of *A _{i}* ions weighted with their molar fractions

*x*, the other ion is a pseudoion consists of

_{i}*B*ions in the same way. From our point of view, the mixed crystal is considered as a pseudo-binary crystal here. Based on Pseudo-Unit-Cell model, the equations of lattice motion can be written as followings [9]:

_{j}*m*,

*e*and

**u**with subscripts

*A*,

_{i}*B*are respectively, the masses, effective charges and displacements of the ions

_{j}*A*and

_{i}*B*.

_{j}*F*

_{AiBj}is the nearest-neighbor force constant of the (

*ij*) pair of ions and

**E**

*is the local electric field.*

_{loc}*F*

_{AiAk}and

*F*

_{BjBl}are the second neighboring force constants between the ions with the same polarity. Because the crystal as whole is without charge, thus due to charge neutral condition, there is the following relation: where we use the shortening In Eqs.(1) and Eqs.(2), only

*n*+

*m*− 1 equations are independent because there is no external force exerting on the crystal and the mass center of the ions is static, thus the degree of freedom diminishes to

*n*+

*m*− 1, then we do a transformation as follows:

**E**

*is given by the famous Lorentz relation where*

_{loc}**E**is the macroscopic electric field and

*ε*

_{0}is the permittivity of vacuum, and

**P**is the polarization field. The polarization field is given by

*n*is the number of cation-anion pairs per unit volume,

_{e}*α*

_{AiBj}=

*α*

_{Ai}+

*α*

_{Aj}is the polarizability of the (

*ij*) pair of ions. From the formulae above, we can see that the polarization is related not only to the displacements of each ion but also to the polarizability of each ion. In order to simplify the mathematical derivations, the new variables are introduced as follows,

*exp*(

*iωt*), according to Zheng’s paper [5], the transverse vibration modes of the lattice have nothing to do with the external electric field, and the longitudinal modes are coupled with the external field. Then the eigenfrequencies and eigenstates of the LO (TO) phonons can be calculated simply by setting

**D**=

*ε*

_{0}

**E**+

**P**= 0 (

**E**= 0) in the above equations, respectively. Considering the Lorentz relation, combine Eq.(9), Eq.(10) and Eq.(11), then the local field with respect to the new variables is obtained as follows:

*ζ*= 1 for TO phonons and

*ζ*= −2 for LO phonons, Γ represents the effect of the polarizability of ions in mixed crystal,

*U*] is a (

*n*+

*m*−1)×(

*n*+

*m*−1) matrix whose elements are the coefficients of the linear equations with respect to the new variables, and [

*S*] is a column vector. From the solvability of the equation, the determinant of the coefficient matrix should be zero. Thus we can calculate the eigenfrequency of the transverse and longitudinal mode optical lattice vibrations, respectively. Considering the following relation: let the frequency go to infinity, when the lattice vibrations have no effect on the dielectric constant, the high frequency dielectric constant was obtained as follows, By using the similar method proposed by Zheng, we obtain the results that have the same form as those obtained in his paper, but our results are general and can be applied to any type of mixed crystals. The main result is the dielectric function which indicate the dispersion relation of the material,

*f*(1 ≤

_{i}*i*≤

*n*+

*m*− 1) for each phonon modes by the following equation:

For further calculations of the eigenfrequencies of TO and LO modes, we need to determine the microscopic parameters that appeared in the above equations. In the PUC model, the microscopic parameters can be expressed in terms of macroscopic measurable physical parameters in the framework of well-known Born-Huang procedure [11] without any adjusting parameter. By setting *x _{i}* = 1 and

*y*= 1, respectively, thus

_{j}*x*= 0(

_{i′}*i′*≠

*i*),

*y*= 0(

_{j′}*j′*≠

*j*), then the parameters will be deduced as

*ε*(∞) (

_{ij}*ε*(0)) and

_{ij}*ω*(

_{Lij}*ω*) are the high frequency (static) dielectric constant and eigenfrequency of LO (TO) phonon of the (

_{Tij}*ij*) end binary crystal, respectively. Now all the parameters have been determined except for the second neighboring force constants. In order to determine the second neighboring force constants, we need more experimental parameters. As is already mentioned in other literatures [3, 9], the second neighboring force constant is related to the corresponding impurity mode. The relation between the frequency of impurity mode and the second neighboring force constant is as follows,

*A*and

_{k}*A*(

_{i}*B*and

_{l}*B*) linearly depends on the atoms’ compositions in mixed crystal, thus the force constants are as followings:

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

_{x}Te

_{y}

As an example, we applied the theory to *A _{x}B*

_{1−}

_{x}C_{y}D_{1−y}type mixed crystals. A specific material is

*Hg*

_{1−x}

*Mn*

_{x}Te_{1−y}

*Se*. It is an important semimagnetic semiconductor material, it has been widely used in the applications of infrared photodiodes and light emitting diodes. We calculate the eigenfrequency of the mixed crystal and the results can be illustrated by three-dimensional figures. We also calculate the dielectric constant in high frequency as well as in low frequency. The oscillator strength is also an important result which indicates the phonon mode behaviors clearly. In this example, we make systematic investigation into the optical properties related to the lattice vibrations. The data used for the calculation are shown in Table. 1.

_{y}In the present paper we plot the 3D figures of phonon frequencies, dielectric constants and oscillator strengths which vary as the composition ratio *x* and *y*. In above figures, we can get a basic understanding on the behavior of the phonons. Clearly we can see six curve surfaces in the Fig. 2, each represents the eigenfrequency of optical phonon. Any of the planes will not cross another except in the boundaries. This obey the rule *ω _{TO}*

_{1}≤

*ω*

_{LO}_{1}≤

*ω*

_{TO}_{2}≤

*ω*

_{LO}_{2}≤

*ω*

_{TO}_{3}≤

*ω*

_{LO}_{3}. Figure 3 indicates that the oscillator strengths vary at different range of

*x*and

*y*, and in some regions, the oscillator strength trends to zero. According to Fig. 4, the static dielectric constant is always lager than the high frequency dielectric constant with the same composition ratios. This because in high frequency case, the lattice vibration has no effect on the dielectric constant and the dielectric constant is enlarged in static condition by the effect of polarization that is related to the displacement of ions.

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.

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 *A*_{x}B_{1−x}*C*_{y}D_{1−y} mixed crystals

_{x}B

_{y}D

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,

From the formula above we can know that the dielectric constant is just related to the polarizability of the atoms and the composition ratios. In the case of *A _{x}B*

_{1−x}

*C*

_{x}D_{1−y}type quaternary mixed crystals, the Γ parameter becomes

*x*(or

*y*), then we can let the coefficients in the square brackets to be zero, that is,

*A*and

*B*weighed with their ratios

*x*and 1 −

*x*, we call it

*E*. The other one is a group of ion

*B*and

*D*weighed with their ratios

*y*and 1 −

*y*, we call it

*F*. Eq.(32) indicates that when the composition ratio

*y*=

*y*

_{0}, the polarization effect will not change as the ratio

*x*. This means that the polarizability between different ions keep balance. The explanation of Eq.(33) is similar to that of Eq.(32). From Eqs.(32) and (33) the value of

*y*

_{0}(or

*x*

_{0}) is obtained as follows:

*y*

_{0}(or

*x*

_{0}) is between 0 and 1, the invariant point does exist. Otherwise, there is no invariant point in the whole composition range. Since the polarizability can be expressed in terms of the high frequency dielectric constants of binary crystals, thus within the whole

*x*and

*y*ranges, the following relation holds,

*ε*(∞) as follows,

_{ij}*ε*(∞) instead of

_{ij}*ε*(0), that is,

_{ij}Based on the parameters given in Table. 1, we can calculate the value of the invariant points of *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.

We also calculated the dielectric constants of mixed crystals *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.

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]