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phys. stat. sol. (b) 245, No. 4, 678–680 (2008) / DOI 10.1002/pssb.200743335 p s sbasic solid state physics

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Lattice energy of zinc blende (AIIIBV and AIIBVI) solids

A. S. Verma* and D. Sharma

Department of Physics, B.S.A. College, Mathura 281004, India

Received 15 August 2007, revised 25 October 2007, accepted 20 November 2007 Published online 8 January 2008

PACS 63.10.+a, 63.20.–e * Corresponding author: e-mail ajay_phy@rediffmail.com, Phone: +91 565 2423417, Mob.: +91 9412884655

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction A considerable amount of experimen-tal and theoretical work has been done during the last few years on the structural, mechanical and optical properties of zinc blende (AIIIBV and AIIBVI) semiconductors [1–4]. Modern computational methods have made it possible to study the structural, mechanical and optical properties of a wide variety of molecules and solids in great detail. There are however, instances where this level of detail either cannot be easily attained because of the complexity of the system or is not needed, as when studying broad trends in the behavior of a large set of systems. Empirical concepts such as valence, empirical radii, electronegativity, ionicity and plasmon energy are then useful [5, 6]. These concepts are directly associated with the character of the chemical bond and thus provide means for explaining and classify-ing many basic properties of molecules and solids. Any change in crystallographic environment of an atom is re-lated to core electrons via the valence electrons. The change in wave function that occurs for the outer electrons usually means a displacement of electric charge in the va-lence shell so that the interaction between valence, shell, and core electrons is changed. This leads to a change in binding energy of the inner electron and to a shift in the position of the absorption edge. The present authors [7–10] have recently calculated the electronic, mechanical and optical properties with the help of the ionic charge theory of solids. This is due to the fact that the ionic charge depends on the number of va-

lence electrons, which changes when a metal forms a com-pound. Therefore, we thought it would be of interest to give an alternative explanation for lattice energy of zinc blende (AIIIBV and AIIBVI) structured solids. 2 Theory, results and discussion On the basis of the Born–Haber cycle, Ladd and Lee [11] experimentally estimated the lattice energy (U) of ionic crystals. Different theoretical models, based on the Born–Haber theory, have been proposed by several other researchers [12, 13] and the plasmon oscillations theory of solids has been used by Reddy et al. [14, 15] for the calculation of U of the binary semiconductors and later used by Kumar et al. [16], in the case of AIIBVI and AIIIBV semiconductors. According to Kumar et al. [16], the lattice energy (U) of AIIBVI and AIIIBV semiconductors may be expressed as

U = 421.224 + 27.940 (ħωp) – 0.178 (ħωp)2 . (1a)

The energy of a quantum of plasma oscillations of the va-lence electrons in both metal and compound is given by the relation [19]

p= 28.8 ( / ) ,Z Wω σ�

(1b)

where Z is the effective number of valence electrons taking part in the plasma oscillations, σ is the specific gravity and W is the molecular weight. Equation (1b) is valid for free electrons but it is also applicable for semiconductors and

In this paper we present an expression relating the lattice en-ergy (U in kcal/mol) for the AIIIBV and AIIBVI semiconductors with the product of ionic charges (Z1Z2) and nearest-neighbor distance d (Å). The lattice energy of these compounds exhibit a linear relationship when plotted on a log– log scale against

the nearest-neighbor distance d (Å), but fall on different straight lines according to the ionic charge product of the compounds. A fairly good agreement has been found between the observed and calculated values of the lattice energy for AIIIBV and AIIBVI semiconductors.

phys. stat. sol. (b) 245, No. 4 (2008) 679

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2.85

2.90

3.00

0.14 0.19 0.24

ZnSZnSe

ZnTeCdS CdSe

AlAsAlP

AlSb

BP

B As

InP

log √ d

log

U(k

cal/m

ol)

2.95

A B

III V

II VI

A B

Semiconductors

Semiconductors

↓

↑Z Z1 2 = 4

↑

Z Z1 2 = 9

↓

Figure 1 In the plot of log U and log ,d AIIIBV semiconductors lie on a line nearly parallel to the line for AIIBVI semiconductors, which depends upon the product of ionic charges. In this figure all lattice energy values are taken from [16]. insulators, up to a first approximation. Because, plasmon energy (ħωp) depends on the number of valence electrons and ionic charge also depends on the number of valence electrons, which changes when a metal forms a compounds. The lattice energy of AIIIBV and AIIBVI semiconductors ex-hibit a linear relationship when plotted against nearest-neighbor distance, but fall on different straight lines ac-cording to the ionic charge product of the compounds, which is presented in Fig. 1. We observe that in the plot of lattice energy and nearest-neighbor distance; the AIIIBV semiconductors lie on lines nearly parallel to the line for the AIIBVI semiconductors. From Fig. 1 it is quite obvious that the lattice energy trends in these compounds decreases with increasing nearest-neighbor distance and fall on dif-ferent straight lines according to the ionic charge product of the compounds. We are of the view that the lattice en-ergy (U) of AIIIBV and AIIBVI semiconductors can be evalu-ated using their ionic charge by following relation:

0.015

1 21255 ( ) / ,U Z Z d=

(2)

where Z1 and Z2 are the ionic charge on the cation and an-ion, respectively, and d is the nearest-neighbor distance in Å. A detailed discussion of lattice energy for these materi-als has been given elsewhere [11–18] and will not be pre-sented here. Using Eq. (2) the lattice energy for AIIBVI and AIIIBV semiconductors has been calculated. The results are presented in Table 1. The calculated values are in fair agreement with the values reported by previous studies [16, 17]. 3 Conclusion We come to the conclusion that the product of ionic charges of any compound is a key param-eter for calculating the electronic, optical and mechanical properties. The lattice energy of these materials is in-versely related to the nearest-neighbor distance and di-rectly depends on the product of ionic charges. From Fig. 1, we observe that the data points for the AIIIBV semiconduc-

Table 1 In this table we present the values of lattice energy (U in kcal/mol) for zinc blende (AIIIBV and AIIBVI) semiconductors. The value of the product of ionic charges (Z

1Z2) = 4 for AIIBVI and

(Z1Z

2) = 9 for AIIIBV semiconductors.

solids d [7] U [17] U [16] U

[this work]

ZnS 2.34 838 838 ZnSe 2.46 836 818 817 ZnTe 2.64 815 795 789 CdS 2.52 798 807 CdSe 2.62 795 778 792 CdTe 2.81 766 756 764 HgS 2.53 806 HgSe 2.63 790 HgTe 2.80 766 AlN 1.87 969 949 AlP 2.36 837 844 AlAs 2.43 817 832 AlSb 2.66 771 795 GaN 1.88 949 946 GaP 2.36 834 844 GaAs 2.45 808 829 GaSb 2.65 763 797 InN 2.08 884 899 InP 2.54 795 814 InAs 2.61 779 803 InSb 2.81 748 774 BN 1.55 999 1042 BP 1.94 944 931 BAs 2.04 911 908 BSb 2.24 867 TiN 2.11 893 TiP 2.49 822 TiAs 2.58 808 TiSb 2.75 782

tors fall on a line nearly parallel to the line for the AIIBVI semiconductors, which means that ionic binding dominates all these compounds. It is also noteworthy that the pro-posed empirical relation is simpler widely applicable and values are in better agreement with experiment data as compared to the empirical relation proposed by previous researchers [16, 17]. We have been reasonably successful in calculating the lattice energy using the product of ionic charges and nearest-neighbor distance of the materials for zinc blende crystals. It is natural to say that this model can easily be extended to chalcopyrite crystals for which work is in progress and will be appearing in a forthcoming paper. Hence, it is possible to predict the order of physical proper-ties of semi conducting and metallic compounds from their ionic charges.

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680 A. S. Verma and D. Sharma: Lattice energy of zinc blende (AIIIBV and AIIBVI) solids

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