Dependence of Ionicity and Thermal Expansion Coefficient on Valence Electron Density in A^{II}B^{IV}C_{2}^{V} Chalcopyrite Semiconductors
Amar BAHADUR^{1*} and Madhukar MISHRA^{2}
^{1 }Department
of Physics, Kamla Nehru Institute of Physical and
Social Sciences, Sultanpur 228 118 (U.P.),
^{2 }Department
of Physics, Birla Institute of Technology and
Science, Pilani  333 031 (Rajasthan),
Email(s): (1) abr.phys@gmail.com; (2) madhukar@bitspilani.ac.in
^{* }Corresponding author: Phone: +91 9451431428
Abstract
A striking correlation has been found to exist between the free electron density parameter, average bond length, homoplar energy gap, heteropolar energy gap, ionicity and thermal expansion coefficient for A^{II}B^{IV}C_{2}^{V} chalcopyrite semiconductors. The estimated values of these parameters are in good agreement with the available experimental values and theoretical findings. The electron density parameter data is the only one input data to estimate all above properties.
Keywords
Electron density parameter; Bond ionicity; Thermal expansion coefficient; Ternary chalcopyrite semiconductors.
Introduction
In recent years, ternary chalcopyrite semiconductors have attracted considerable attention because of their potential applications in the field of light emitting diodes, nonlinear optics, photovoltaic devices and solar cells [14]. The solid solutions of these semiconductors have been used in electrooptic devices [57]. Their mixed crystals are being used for fabrication of detectors, lasers and integrated optic devices such as switches, modulators and filters etc. These chalcopyrites have many other practical applications in the areas of fiber optics, sensors and communication devices. In spite of their promising applications, some physical properties of these compounds have still not been significantly investigated. Frequent attempts have been made at understanding the crystal ionicity of these compounds. Phillips [8], Van Vechten [910], Levine [11] and other researchers [1213] have developed various theories and calculated crystal ionicity for the case of simple compounds. Phillips and Van Vechten have calculated the homopolar and heteropolar contribution to the chemical bond in the binary crystals. Chemla [14] and several workers [1516] have extended this theory to some complex crystals, neglecting the effect of noble metal d electrons. Levine [17] has extended Phillips and Van Vechten (PVV) theory of bond ionicity in case of various types of complex compounds considering the effect of d core electrons. It is clear form the Levine’s modifications and as well as PVV theory that the homopolar energy gap depends upon the nearest neighbor distance, while heteropolar energy gap is a function of nearest neighbor distance and the number of valence electron taking part in bond formation. Kumar et al. [1819] have calculated these parameters in terms of plasmon energy because plasmon energy depends directly on the effective number of valence electrons in a compound. In all the above investigations, the ionicity has been evaluated in terms of nearest neighbor distance, valence and plasmon energy.
In recent years, various electronic and mechanical properties of ternary tetrahedral semiconductors have been explained using plasma oscillation theory [2022], which leads to the fact that these physical properties depend directly on effective number of valance electrons and the density of the conduction electrons. The plasmon energy is related to the effective number of valence electrons as,
_{} (1)
where_{}is the effective number of valence electrons, _{} is the charge, _{}is the mass of electron and_{}is the plasmon energy is given by relation [23] as:
_{} (2)
where, _{}is effective number of electrons taking part in plasma oscillation, _{} is the density and _{}is the molecular weight of compound. This equation is valid for free electron model but to a fairly good approximation it can also be used for semiconductors and insulators. The valence electron density is described by electron density parameter _{}which also depends upon the density of the conduction electrons and effective number of valence electrons according to relation [24],
_{} (3)
We, therefore, explore in the current work a new method for correlating electron density parameter with the crystal ionicity and thermal expansion coefficient for ternary chalcopyrite semiconductors. In the proposed approach, electron density parameter is the only input required for computation of the average bond length, homopolar energy gap, heteropolar energy gap, ionicity and thermal expansion coefficient.
Theory and Calculation
Average Bond Length, Homopolar Energy Gap, Heteropolar Energy Gap And Ionicity
The average bond length (d) is also related to the effective number of valence electrons, so there must be some correlation between bond length and electron density parameter. Using eqn. (1) and (3), we can express number of free electrons in terms of r_{s}_{ }as
_{} (4)
In tetrahedral semiconductors, the effective number of valence electrons N_{e,XY} can also be expressed in terms of individual bond properties as
_{} (5)
where,
_{}and_{}
are
the number of valence electrons of the atom X and Y respectively,
in XY compounds.
The N_{CX}_{ }, N_{CY}_{ } and
N_{CZ}_{ }are
coordination number of atoms and _{} is bond
volume. For A^{II}B^{IV}C_{2}^{V}semiconductors,
N_{CA }= N_{CB }=
_{} (6)
_{} (7)
The average energy gap_{}can be separated into the homopolar and heteropolar parts according to following relation:
_{} (8)
which yield following relation for bond ionicity (f_{i}):
_{} (9)
where, _{},_{}are homopolar energy gap, ionic gap, respectively, given by relations as:
_{} (10)
_{} (11)
where d_{XY} is distance between atoms X and Y, K_{s} is Thomas Fermi momentum, _{}, _{}is ThomasFermi screening factor which is related to the effective number of free electrons in compound and b is prescreening constant.
With the help of eqn. (6)  (7) and (10), homopolar part of energy of individual bonds can be expressed in terms of free electron density parameter as:
_{} (12)
_{} (13)
The physical meaning of Eq. (11) is that _{} is given by the difference between the screened Coulomb potentials of atoms _{}and _{} having core charges _{} and _{}. These potentials are to be evaluated at the covalent radii _{}. Only a small part of the electrons are in the bond, the rest screen the ion cores, reducing their charge by the Thomas Fermi screening factor _{}, which affects the chemical trend in a compound. This screening factor, as well as the bond length, is related to the effective number of free electrons in a compound. Also, the electron density parameter directly depends upon the effective number of valence electrons. Thus, there must be some correlation between the physical processes which involve the ionic contribution _{} to the average energy gap _{} and the electron density parameter _{}.
The expression for Thomas Fermi momentum is given by relation as:
_{} (14)
where, _{}is Bohr radius (0.529_{}) and _{}, the Fermi wave vector given by relation as: _{}. Using eqn. (4), the Fermi wave vector can be expressed in terms of _{} as: _{}and accordingly Fermi momentum becomes
_{} (15)
Using the values of Eq. (15), the heteropolar energy can be expressed in terms of _{}as
_{} (16)
here, _{}, is 3 and 1, for _{} and _{} bond, respectively, in A^{II}B^{IV}C_{2}^{V} semiconductors. The avergae value of prescreening constant (b) has been taken as 1.3966 (Ref. 17) and 2.4516 (Ref. 17) for _{} and _{} bond, respectively. Using these values, we obtain expressions for heteropolar energy gap for _{} and _{} bond as:
_{} (17)
_{} (18)
The ionicity of _{} and _{} bonds in A^{II}B^{IV}C_{2}^{V} semiconductors have been calculated using Eq. (9). The ionicity of both bonds (_{} and _{}) exhibit a linear relationship when plotted against electron density parameter. In fig. 1, we observe that in plot of ionicty and electron density parameter, the _{} bond lie on line nearly parallel to the _{} bond and the ionicity trends in these bonds increases with increasing electron density parameter. Based on above discussion, we propose the following equation for the calcualation of the ionicity of ternary chalcopyrite semiconductors,
_{} (19)
where _{}and_{}are constants, The numerical values of these constants for _{} and _{} bonds are 0.4014 and 0.1651 and 0.265 and 0.365 respectively.
Table 1. Properties of A^{II}C^{V} bond in A^{II}B^{IV}C_{2}^{V} compounds
Compounds 
Electron density parameter 
Average bond length 
Homopolar energy gap 
Heteropolar energy gap 
Ionicity_{} 


Present work 
Levine [17] 


_{} 
_{} 
_{} 
_{} 


ZnSiP_{2} 
2.12 
2.38 
4.653 
4.557 
0.4895 
0.438 

ZnGeP_{2} 
2.13 
2.39 
4.605 
4.515 
0.4902 
0.442 

ZnSnP_{2} 
2.15 
2.42 
4.459 
4.388 
0.4920 
0.455 

ZnSiAs_{2} 
2.20 
2.46 
4.245 
4.200 
0.4948 
0.436 

ZnGeAs_{2} 
2.21 
2.48 
4.168 
4.133 
0.4957 
0.422 

ZnSnAs_{2} 
2.24 
2.51 
4.061 
4.038 
0.4971 
0.450 

CdSiP_{2} 
2.28 
2.56 
3.854 
3.852 
0.4998 
0.539 

CdGeP_{2} 
2.28 
2.56 
3.862 
3.859 
0.4997 
0.532 

CdSnP_{2} 
2.31 
2.59 
3.755 
3.762 
0.5011 
0.536 

CdSiAs_{2} 
2.35 
2.64 
3.572 
3.596 
0.5034 
0.553 

CdGeAs_{2} 
2.35 
2.64 
3.565 
3.590 
0.5035 
0.549 

CdSnAs_{2} 
2.38 
2.67 
3.473 
3.505 
0.5047 
0.553 

Table 2. Properties of B^{IV}C^{V} bond in A^{II}B^{IV}C_{2}^{V} compounds
Compounds 
Electron density parameter 
Average bond length 
Homopolar energy gap 
Heteropolar energy gap 
Ionicity_{} 


Present work 
Levine [17] 


_{} 
_{} 
_{} 
_{} 


ZnSiP_{2} 
1.85 
2.25 
5.304 
2.705 
0.2064 
0.177 

ZnGeP_{2} 
1.90 
2.32 
4.926 
2.532 
0.2090 
0.219 

ZnSnP_{2} 
2.04 
2.49 
4.156 
2.171 
0.2143 
0.298 

ZnSiAs_{2} 
1.93 
2.35 
4.779 
2.464 
0.2100 
0.220 

ZnGeAs_{2} 
1.97 
2.41 
4.501 
2.334 
0.2119 
0.182 

ZnSnAs_{2} 
2.10 
2.56 
3.870 
2.033 
0.2163 
0.135 

CdSiP_{2} 
1.84 
2.25 
5.346 
2.724 
0.2061 
0.191 

CdGeP_{2} 
1.91 
2.33 
4.909 
2.524 
0.2091 
0.231 

CdSnP_{2} 
2.04 
2.49 
4.152 
2.169 
0.2144 
0.298 

CdSiAs_{2} 
1.92 
2.35 
4.789 
2.469 
0.2099 
0.234 

CdGeAs_{2} 
1.98 
2.42 
4.436 
2.304 
0.2124 
0.199 

CdSnAs_{2} 
2.11 
2.57 
3.813 
2.006 
0.2167 
0.148 

Generalizing Eqs. (6)(7), (12)(13), (17), (18) and (19), the expression for average bond length, homoploar, heteropolar energy gap and ionicity can be written as:
_{} (20)
_{} (21)
_{} (22)
_{} (23)
where _{}and _{} are constants depending upon bonds of compound. Form these Eqs., it follows that bond properties can be evaluated form free electron density parameter. The free electron density parameter has been calculated from eqn. (3). Using Eq. (6)(7), (12)(13), (17), (18) and (19), we have calculated the average bond length, homopolar gap, heteropolar energy gap and bond ionicity for _{} and _{} bond in A^{II}B^{IV}C_{2}^{V} semiconductors and presented in Table1 and 2, respectively. For comparison, other estimates of these parameters are also presented. In most of compounds, our calculated values are in good agreement with the earlier reported values [17].
Figure 1. Plot of ionicity (f_{i}) and electron density parameter (r_{s}) for A^{II}B^{IV}C_{2}^{V} chalcopyrite semiconductors. ● corresponds to the IIV bond and ♦ corresponds to IVV bond in A^{II}B^{IV}C_{2}^{V} chalcopyrite semiconductors. In this figure all data are taken from calculated values, which are presented in Table 1 and 2
Thermal Expansion Coefficient
Based on thermal expansion coefficient data, Neumann [25] has proposed following expression for the average thermal expansion coefficient for binary tetrahedral semiconductors:
_{} (24)
where A is constant, T_{m} is melting temperature, d is the bond length. The value of A is 0.021 for all tetrahedrally coordinated compounds (A^{IV}, A^{III}B^{V}, A^{II}B^{VI}) as estimated from a hard sphere model based on the diamond structure. The value _{} is equal to the bond length of diamond i.e._{}. The ternary chalcopyrites of general composition A^{II}B^{IV}C_{2}^{V} can be considered as similar to those of A^{II}B^{IV }and A^{II}B^{VI}. Thus, Eq. (24) can also be reasonably used to describe the thermal expansion coefficient of the ternary chalcopyrite. From Eq. (6) and (7), we get following relation between_{}and_{}as:
_{} (25)
For A^{II}B^{IV}C_{2}^{V} chalcopyrite semiconductors, the values of B and_{}are 16.1 (_{}) and 1.573_{}, respectively [25]. Using Eq. (25), we have calculated the thermal expansion coefficient for A^{II}B^{IV}C_{2}^{V}, ternary chalcopyrite semiconductors and presented in Table 3 along with experimental data and theoretical findings.
Table 3. Thermal expansion coefficient of A^{II}B^{IV}C_{2}^{V} compounds
Compounds 
T_{m}(K) [2526] 
α_{L} = (10^{6}K^{1}) 

Present work Eq. (25) 
Ref. [25] 
Exp. [25, 27] 

ZnSiP_{2} 
1643 
6.15 

6.3 
ZnGeP_{2} 
1298 
8.52 

6.7 
ZnSnP_{2} 
1203 
6.57 

3.2 
ZnSiAs_{2} 
1369 
5.99 
5.3 

ZnGeAs_{2} 
1148 
7.62 
6.7 
1.0 
ZnSnAs_{2} 
1048 
5.80 
4.5 
2.3 
CdSiP_{2} 
1393 
5.83 
7.0 
7.2 
CdGeP_{2} 
1073 
9.00 

6.1 
CdSnP_{2} 
843 
10.5 
4.9 

CdSiAs_{2} 
1120 
6.16 
5.0 

CdGeAs_{2} 
938 
8.19 

6.0 
CdSnAs_{2} 
871 
5.56 

4.7 
Summary and Conclusion
The present relation may be considered to be first attempt to obtain simple correlations between free electron density parameter, average bond length, homopolar energy gap, heteropolar energy gap, ionicity and thermal expansion coefficient for ternary chalcopyrite semiconductors. These equations can be considered advantageous over others in the sense that it relates the electronic and thermal properties of semiconducting compounds with the free electron density parameter. The proposed relations yield not only satisfactory results, but also a comparison with the standard data provides a direct and precise check of the validity. Therefore, it is possible to predict the order of homopolar and heteropolar energy gap and the ionicity of semiconducting compounds from their free electron density parameter. The reasonable agreement between our calculated and previously known values of α_{L} indicates that the proposed relations are both useful and accurate for estimating thermal properties for binary tetrahedral semiconductors. Hence, we conclude from present analysis that the ionicity and thermal properties in semiconducting compounds can be evaluated from density of valence electrons. In the proposed approach, the calculation is simple, fast and more accurate. The only information needed is electron density parameter; no other experimental values are required. It is natural to say that present approach can easily be extended to the other more complex crystals.
Acknowledgement
We are thankful to Dr. A.K. Srivastava (H.O.D., Department of Physics and Electronics,
K.N.I.P.S.S., Sultanpur, U.P.,
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