Generalized Pseudo-Unit-Cell model for long-wavelength optical phonons of multinary mixed crystals: Application to AxB<sub>1-</sub>xCyD<sub>1-y</sub> type mixed crystals

Zhenfeng Liao; Jingzhen Li; Ruisheng Zheng; Xiaowei Lu; Hongyi Chen

doi:10.1364/OE.21.011715

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−yMg_yBe_xSe 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_xIn_yGa_1−x−yN 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_{ix_i} ⋯ A_{nx_n} B_1y₁ ⋯ B_{jy_j} ⋯ B_{my_m}, where x_i and y_j are the molar fractions of the ions. The A_i ions are cations, and the B_i ions are anions. The lattice of the mixed crystal consists of two sublattices. The A_i ions are distributed in one of them randomly, while the B_j ions are distributed in another one randomly. The nearest neighbors of an A_i ion are always B_j ions, and verse versa. The probability of finding a B_j ion around A_i is proportional to its molar fraction y_j. 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.

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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_i, the other ion is a pseudoion consists of B_j 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]:

\begin{matrix} m_{A_{i}} {\ddot{u}}_{A_{i}} = - \sum_{j = 1}^{m} y_{j} F_{A_{i} B_{j}} (u_{A_{i}} - u_{B_{j}}) - \sum_{\begin{array}{l} k = 1 \\ k \neq i \end{array}}^{n} x_{k} F_{A_{i} A_{k}} (u_{A_{i}} - u_{A_{k}}) \\ + E_{loc} \sum_{j = 1}^{m} y_{j} e_{A_{i} B_{j}} (1 \leq i \leq n), \end{matrix}

\begin{matrix} m_{B_{j}} {\ddot{u}}_{B_{j}} = - \sum_{i = 1}^{n} x_{i} F_{A_{i} B_{j}} (u_{B_{j}} - u_{A_{i}}) - \sum_{\begin{array}{l} l = 1 \\ l \neq j \end{array}}^{m} y_{l} F_{B_{j} B_{l}} (u_{B_{j}} - u_{B_{l}}) \\ - E_{loc} \sum_{i = 1}^{n} x_{i} e_{A_{i} B_{j}} (1 \leq j \leq m), \end{matrix}

where m, e and u with subscripts A_i, B_j are respectively, the masses, effective charges and displacements of the ions A_i and B_j. F_{A_iB_j} is the nearest-neighbor force constant of the (ij) pair of ions and E_loc is the local electric field. F_{A_iA_k} and F_{B_jB_l} 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:

\sum_{i}^{n} x_{i} e_{A_{i}} - \sum_{j}^{m} y_{j} e_{B_{j}} = 0,

where we use the shortening

e_{A_{i}} = \sum_{j = 1}^{m} y_{j} e_{A_{i} B_{j}},

e_{B_{j}} = \sum_{i = 1}^{n} x_{i} e_{A_{i} B_{j}} .

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:

\begin{array}{l} m_{B_{m}} m_{A_{i}} ({\ddot{u}}_{A_{i}} - {\ddot{u}}_{B_{m}}) & = & - m_{B_{m}} \sum_{j = 1}^{m} y_{j} F_{A_{i} B_{j}} (u_{A_{i}} - u_{B_{j}}) - m_{B_{m}} \sum_{\begin{array}{l} k = 1 \\ k \neq i \end{array}}^{n} x_{k} F_{A_{i} A_{k}} (u_{A_{i}} - u_{A_{k}}) \\ + m_{A_{i}} \sum_{k = 1}^{n} x_{k} F_{A_{k} B_{m}} (u_{B_{m}} - u_{A_{k}}) + m_{A_{i}} \sum_{l = 1}^{m - 1} y_{l} F_{B_{m} B_{l}} (u_{B_{m}} - u_{B_{l}}) \\ + m_{B_{m}} e_{A_{i}} E_{loc} + m_{A_{i}} e_{B_{m}} E_{loc} (1 \leq i \leq n), \end{array}

\begin{array}{l} m_{B_{m}} m_{B_{j}} ({\ddot{u}}_{B_{j}} - {\ddot{u}}_{B_{m}}) & = & = m_{B_{m}} \sum_{l = 1}^{n} x_{i} F_{A_{i} B_{j}} (u_{B_{j}} - u_{A_{i}}) - m_{B_{m}} \sum_{\begin{array}{l} l = 1 \\ l \neq j \end{array}}^{m} y_{l} F_{B_{j} B_{l}} (u_{B_{j}} - u_{B_{l}}) \\ + m_{B_{j}} \sum_{i = 1}^{n} x_{i} F_{A_{i} B_{m}} (u_{B_{m}} - u_{A_{i}}) + m_{B_{j}} \sum_{l = 1}^{m - 1} y_{l} F_{B_{m} B_{l}} (u_{B_{m}} - u_{B_{l}}) \\ - m_{B_{m}} e_{B_{j}} E_{loc} + m_{B_{j}} e_{B_{m}} E_{loc} (1 \leq j \leq m - 1) . \end{array}

Since in the crystal the local field does not equal to the macroscopic field, according to the basic theory of electrodynamics, the local electric field E_loc is given by the famous Lorentz relation

E_{loc} = E + \frac{P}{3 ε_{0}},

where E is the macroscopic electric field and ε₀ is the permittivity of vacuum, and P is the polarization field. The polarization field is given by

P = n_{e} (\sum_{i}^{n} x_{i} e_{A_{i}} u_{A_{i}} - \sum_{j}^{m} y_{j} e_{B_{j}} u_{B_{j}}) + n_{e} (\sum_{i}^{n} x_{i} α_{A_{i}} + \sum_{j}^{m} y_{j} α_{B_{j}}) E_{loc},

where n_e is the number of cation-anion pairs per unit volume, α_{A_iB_j} = α_{A_i} + α_{A_j} 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,

S_{i} = u_{A_{i}} - u_{B_{m}} (1 \leq i \leq n),

S_{n + j} = u_{B_{j}} - u_{B_{m}} (1 \leq j \leq m - 1) .

We assume that the motion of atoms is harmonic with the time dependence of 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 = ε₀E + 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:

E_{loc} = \frac{ζ n_{e}}{3 ε_{0} (1 - ζ Γ)} (\sum_{i}^{n} x_{i} e_{A_{i}} S_{i} - \sum_{j}^{m - 1} y_{j} e_{B_{j}} S_{n + j}),

where ζ = 1 for TO phonons and ζ = −2 for LO phonons, Γ represents the effect of the polarizability of ions in mixed crystal,

Γ = \frac{n_{e}}{3 ε_{0}} (\sum_{i}^{n} \sum_{j}^{m} x_{i} y_{j} α_{A_{i} B_{j}}) .

Substitute (12) into Eq.(6) and Eq.(7) we can get a matrix equation as follows,

[U] [S] = 0,

where [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:

P = (ε (ω) - 1) ε_{0} E,

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,

ε (\infty) = \frac{1 + 2 Γ}{(1 - Γ)} .

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,

\frac{ε (ω)}{ε (\infty)} = \frac{(ω_{L 1}^{2} - ω^{2}) (ω_{L 2}^{2} - ω^{2}) \dots (ω_{L n + m - 1}^{2} - ω^{2})}{(ω_{T 1}^{2} - ω^{2}) (ω_{T 2}^{2} - ω^{2}) \dots (ω_{T n + m - 1}^{2} - ω^{2})} .

As introduced by Genzel, Martin and Perry [10], we can define the oscillator strengths f_i(1 ≤ i ≤ n + m − 1) for each phonon modes by the following equation:

ε (ω) = ε (\infty) + \sum_{i = 1}^{n + m - 1} \frac{f_{i} ω_{T i}^{2}}{(ω_{T i}^{2} - ω^{2})} .

By performing the partial fraction decomposition, we obtain the oscillator strengths as follows,

f_{i} = ε (\infty) \prod_{j = 1}^{n + m - 1} (ω_{L j}^{2} - ω_{T i}^{2}) \cdot {(ω_{T i}^{2} \prod_{j = 1, j \neq i}^{n + m - 1} (ω_{T j}^{2} - ω_{T i}^{2}))}^{- 1} .

As is stated above, the theory in Zheng’s paper has been extended to the general case. The generalized theory and results can be applied to any type of mixed crystals. Zheng’s theory can be included as a special case.

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_j = 1, respectively, thus x_i′ = 0(i′ ≠ i), y_j′ = 0(j′ ≠ j), then the parameters will be deduced as

\frac{n_{e} α_{A_{i} B_{j}}}{3 ε_{0}} = \frac{ε_{i j} (\infty) - 1}{ε_{i j} (\infty) + 2},

\frac{n_{e} e_{A_{i} B_{j}}^{2}}{3 ε_{0}} = \frac{3 m_{i} m_{j}}{m_{i} + m_{j}} \frac{(ω_{L i j}^{2} - ω_{T i j}^{2}) ε_{i j} (\infty)}{{(ε_{i j} (\infty) + 2)}^{2}},

F_{A_{i} B_{j}} = \frac{m_{i} m_{j}}{m_{i} + m_{j}} \frac{ε_{i j} (0) + 2}{ε_{i j} (\infty) + 2} ω_{T i j}^{2},

where ε_ij(∞) (ε_ij(0)) and ω_Lij (ω_Tij) are the high frequency (static) dielectric constant and eigenfrequency of LO (TO) phonon of the (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,

F_{A_{k} A_{i}} (A_{k} u_{A_{i} B_{j}}) = m_{A_{k}} ω_{I}^{2} (A_{k} u_{A_{i} B_{j}}) - F_{A_{k} B_{j}},

F_{B_{l} B_{j}} (B_{l} u_{A_{i} B_{j}}) = m_{B_{l}} ω_{I}^{2} (B_{l} u_{A_{i} B_{j}}) - F_{A_{i} B_{l}} .

The second neighboring force constants above are obtained in the binary crystals. We assume that the second neighboring force constant between atom A_k and A_i (B_l and B_j) linearly depends on the atoms’ compositions in mixed crystal, thus the force constants are as followings:

F_{A_{k} A_{i}} = \sum_{j = 1}^{m} \frac{x_{k}}{x_{k} + x_{i}} y_{j} F_{A_{k} A_{i}} (A_{k} u_{A_{i} B_{j}}) + \sum_{j = 1}^{m} \frac{x_{i}}{x_{k} + x_{i}} y_{j} F_{A_{i} A_{k}} (A_{i} u_{A_{k} B_{j}})

F_{B_{l} B_{j}} = \sum_{i = 1}^{n} \frac{y_{l}}{y_{j} + y_{l}} x_{i} F_{B_{l} B_{j}} (B_{l} u_{A_{i} B_{j}}) + \sum_{i = 1}^{n} \frac{y_{j}}{y_{j} + y_{l}} x_{i} F_{B_{j} B_{l}} (B_{j} u_{A_{i} B_{l}})

3. Numerical calculations for Hg_1−xMn_xTe_1−ySe_y

As an example, we applied the theory to A_xB₁₋_xC_yD_1−y type mixed crystals. A specific material is Hg_1−xMn_xTe_1−ySe_y. 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.

Table 1. Parameters of binary crystals corresponding to quaternary mixed crystal Hg_1−xMn_xTe_1−ySe_y, the unit of the phonon frequencies is cm⁻¹, the parameters are taken from Refs. [12–16].

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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_TO₁ ≤ ω_LO₁ ≤ ω_TO₂ ≤ ω_LO₂ ≤ ω_TO₃ ≤ ω_LO₃. 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.

Fig. 2 Concentration dependence of LO and TO phonon frequencies of the quaternary mixed crystal Hg_1−xMn_xTe_1−ySe_y.

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Fig. 3 Concentration dependence of oscillator strengths of the quaternary mixed crystal Hg_1−xMn_xTe_1−ySe_y.

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Fig. 4 Concentration dependence of dielectric constants of the quaternary mixed crystal Hg_1−xMn_xTe_1−ySe_y.

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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 Hg_1−xMn_xTe_1−ySe_y.

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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_xB_1−xC_yD_1−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 (\infty) = \frac{1 + 2 Γ}{(1 - Γ)} = Φ (Γ),

where

Γ = \frac{n_{e}}{3 ε_{0}} (\sum_{i}^{n} \sum_{j}^{m} x_{i} y_{j} α_{A_{i} B_{j}}),

represents the weighed sum of the binary crystal polarizability.

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_xB_1−xC_xD_1−y type quaternary mixed crystals, the Γ parameter becomes

Γ = \frac{n_{e}}{3 ε_{0}} (x y α_{a c} + (1 - x) y α_{b c} + x (1 - y) α_{a d} + (1 - x) (1 - y) α_{b d}),

and it can be rearranged as

Γ = \frac{n_{e}}{3 ε_{0}} (x [y α_{a c} + (1 - y) α_{a d} - y α_{b c} - (1 - y) α_{b d}] + y α_{b c} + α_{b d} - y α_{b d}),

and

Γ = \frac{n_{e}}{3 ε_{0}} (y [x α_{a c} + (1 - x) α_{b c} - x α_{a d} - (1 - x) α_{b d}] + x α_{a d} - x α_{b d} + α_{b d}) .

If Γ is independent of the composition x (or y), then we can let the coefficients in the square brackets to be zero, that is,

y_{0} α_{a c} + (1 - y_{0}) α_{a d} = y_{0} α_{b c} + (1 - y_{0}) α_{b d},

or

x_{0} α_{a c} + (1 - x_{0}) α_{b c} = x_{0} α_{a d} + (1 - x_{0}) α_{b d} .

Quaternary mixed crystal can be regarded as consisting of two pseudoions, one pseudoion is a group of ion 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 = y₀, 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 y₀ (or x₀) is obtained as follows:

y_{0} = \frac{(α_{b d} - α_{a d})}{(α_{a c} + α_{b d} - α_{b c} - α_{a d})},

x_{0} = \frac{(α_{b d} - α_{b c})}{(α_{a c} + α_{b d} - α_{b c} - α_{a d})} .

If the value of y₀ (or x₀) 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,

\begin{array}{l} \frac{ε (\infty) - 1}{ε (\infty) + 2} & = & x y \frac{ε_{a c} (\infty) - 1}{ε_{a c} (\infty) + 2} + x (1 - y) \frac{ε_{a d} (\infty) - 1}{ε_{a d} (\infty) + 2} \\ + (1 - x) y \frac{ε_{b c} (\infty) - 1}{ε_{b c} (\infty) + 2} + (1 - x) (1 - y) \frac{ε_{b d} (\infty) - 1}{ε_{b d} (\infty) + 2} . \end{array}

Dielectric constant of mixed crystals can be calculated based on above equations. The value of the invariant points can be expressed by ε_ij(∞) as follows,

y_{0} = {(\frac{(ε_{b d} (\infty) + 2) (ε_{a d} (\infty) + 2) (ε_{b c} (\infty) - ε_{a c} (\infty))}{(ε_{a c} (\infty) + 2) (ε_{b c} (\infty) + 2) (ε_{a d} (\infty) - ε_{b d} (\infty))} + 1)}^{- 1},

x_{0} = {(1 + \frac{(ε_{b d} (\infty) + 2) (ε_{b c} (\infty) + 2) (ε_{a d} (\infty) - ε_{a c} (\infty))}{(ε_{a c} (\infty) + 2) (ε_{a d} (\infty) + 2) (ε_{b c} (\infty) - ε_{b d} (\infty))})}^{- 1} .

The numerical results also show the existences of invariant points in the static dielectric constant. The value of the invariant point in the static dielectric constant can be also calculated analytically by Eqs.(37) and (38) with ε_ij(∞) instead of ε_ij(0), that is,

y_{0}^{'} = {(\frac{(ε_{b d} (0) + 2) (ε_{a d} (0) + 2) (ε_{b c} (0) - ε_{a c} (0))}{(ε_{a c} (0) + 2) (ε_{b c} (0) + 2) (ε_{a d} (0) - ε_{b d} (0))} + 1)}^{- 1},

x_{0}^{'} = {(1 + \frac{(ε_{b d} (0) + 2) (ε_{b c} (0) + 2) (ε_{a d} (0) - ε_{a c} (0))}{(ε_{a c} (0) + 2) (ε_{a d} (0) + 2) (ε_{b c} (0) - ε_{b d} (0))})}^{- 1} .

Based on the parameters given in Table. 1, we can calculate the value of the invariant points of Hg_1−xMn_xTe_1−ySe_y 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 Zn_xMg_1−xSe_yTe_1−y and Ga_xIn_1−xN_yP_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−xMn_xTe_1−ySe_y 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 Zn_xMg_1−xSe_yTe_1−y, and there is also an invariant point for static dielectric constant of Ga_xIn_1−xN_yP_1−y. According to Eqs.(38) and (40), the value of invariant point for high frequency dielectric constant of Zn_xMg_1−xSe_yTe_1−y is x₀ = 0.7781, and that for the static dielectric constant of Ga_xIn_1−xN_yP_1−y is x′₀ = 0.3326. While, invariant point for static dielectric constant of Zn_xMg_1−xSe_yTe_1−y and invariant point for high frequency dielectric constant of Ga_xIn_1−xN_yP_1−y do not exist.

Table 2. Parameters of binary crystals corresponding to mixed crystals Zn_xMg_1−xSe_yTe_1−y and Ga_xIn_1−xN_yP_1−y. The parameters are taken from Refs. [17–20].

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Fig. 6 Concentration dependence of dielectric constants of the quaternary mixed crystal Zn_xMg_1−xSe_yTe_1−y.

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Fig. 7 Concentration dependence of dielectric constants of the quaternary mixed crystal Ga_xIn_1−xN_yP_1−y.

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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

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

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Material	ω_TO	ω_LO	ε(0)	ε(∞)	ω_I(Hg)	ω_I(Mn)	ω_I(Te)	ω_I(Se)
HgTe	119	133	8.74	7		186		157
HgSe	132	173	25.6	15		218	114
MnTe	185	216	11	8.07	124			151
MnSe	195	218	7.5	6	162		126

Material	ZnSe	ZnTe	MgSe	MgTe	GaN	GaP	InN	InP
ω_TO(cm⁻¹)	205	180	237	160	559	355	447	303.3
ω_LO(cm⁻¹)	250	206	312	270	746	367	586	344.5
ε(∞)	5.85	6.64	2.981	2.139	5.35	9.11	8.4	9.61
ε(0)	8.7	8.7	5.167	6.091	8.9	11.11	14.4	12.56

Material	ω_TO	ω_LO	ε(0)	ε(∞)	ω_I(Hg)	ω_I(Mn)	ω_I(Te)	ω_I(Se)
HgTe	119	133	8.74	7		186		157
HgSe	132	173	25.6	15		218	114
MnTe	185	216	11	8.07	124			151
MnSe	195	218	7.5	6	162		126

Material	ZnSe	ZnTe	MgSe	MgTe	GaN	GaP	InN	InP
ω_TO(cm⁻¹)	205	180	237	160	559	355	447	303.3
ω_LO(cm⁻¹)	250	206	312	270	746	367	586	344.5
ε(∞)	5.85	6.64	2.981	2.139	5.35	9.11	8.4	9.61
ε(0)	8.7	8.7	5.167	6.091	8.9	11.11	14.4	12.56

Generalized Pseudo-Unit-Cell model for long-wavelength optical phonons of multinary mixed crystals: Application to A_xB_1-_xC_yD_1-y type mixed crystals

Abstract

1. Introduction

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

3. Numerical calculations for Hg_1−xMn_xTe_1−ySe_y

4. Composition independence of dielectric constant in A_xB_1−xC_yD_1−y mixed crystals

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (2)

Equations (40)

Optics Express

Abstract

1. Introduction

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

3. Numerical calculations for Hg1−xMnxTe1−ySey

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

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (2)

Equations (40)

Optics Express

3. Numerical calculations for Hg_1−xMn_xTe_1−ySe_y

4. Composition independence of dielectric constant in A_xB_1−xC_yD_1−y mixed crystals