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A Semi-empirical Approach to Bulk Modulus of A^NB^8-N Semiconductors and Their Alloys

Mritunjai Kumar Pathak¹, Madhu Sudan Dutta¹, Sanjay Kumar Gorai², Ashok Kumar Gupta¹, Rajendra Nath Sinha¹ and Parmanand Mahto^{1* 1}Department of Physics, Vinoba Bhave University, Hazaribag, Jharkhand, India
²Department of Physics, Tata College, West Singhbhum, Jharkhand, India
^*Correspondence: Parmanand Mahto, Department of Physics, Vinoba Bhave University, Hazaribag, Jharkhand, India, Tel: pmahtovbu@rediffmail.com,

Received: 21-Feb-2023, Manuscript No. IPACRH-23-15742; Editor assigned: 23-Feb-2023, Pre QC No. IPACRH-23-15742 (PQ); Reviewed: 09-Mar-2023, QC No. IPACRH-23-15742; Revised: 21-Apr-2023, Manuscript No. IPACRH-23-15742 (R); Published: 28-Apr-2023, DOI: 10.21767/2572-4657.23.7.11

Introduction

Semiconductors have long been used widely in electronic industry and today there is hardly any field left without the application of semiconductors. This is because each type of semiconductors has some specific characteristics for an application. The range and horizon of application of semiconductors is broadened further by allowing them to tailor the material properties like lattice parameters, band gaps, bond length, iconicity hardness, bulk modulus etc. required for the new and desired applications [1-13]. For example, the dilute nitride alloys GaAs_1-xN_x and Ga_1-xIn_xAs_1-yN_y; dilute oxide alloys ZnS_1-xO_x and Zn_1-xCd_xTe_1-yO_y and some other such combinations B_xGa_1-xN, ZnO_1-xTe_x have been identified as Highly Mismatched Alloys (HMAs) which are the potential candidates for high efficiency solar cell. However, the stability of solar cells is of concern for their application in space, where they have to withstand bombardment of high energy particles like electrons, protons cosmic rays etc. that can cause severe damage to solar cells. Also in domestic, industrial and commercial applications the electronic appliances have to face mechanical stresses and electrical jerks due to voltage fluctuations.

Therefore, study on mechanical properties of these semiconductors and their alloys are of prime importance. Among these properties, focus has been made on bulk modulus, which has in many cases been correlated with material strength and hardness, and it is commonly accepted that material with larger bulk modulus are accepted to be harder ones. Of course, this is not always true and recently it has been found that even the materials with very high bulk moduli have low hardness. For example, Rhenium was predicted to have high bulk modulus (405 GPa), but its (Vickers’s) hardness was found to be 16.7 GPa, which is about five times lower than that of diamond [14]. Based on this common concept, C₃N₄ was initially proposed to be one of the hardest materials comparable with diamond as its bulk modulus was estimated to be 443 GP, but later on it was synthesized and its hardness was found to be even lower than diamond and BN [15]. Even then bulk modulus is considered as one of the important parameters to characterize physical properties, and for many applications it is used as indicator for materials strength and hardness. In the backdrop of the above scenario, a semi-empirical formula has been developed and presented in this study for bulk modulus of Group IV and IV-IV, II-VI and III-V semiconductors and their ternary and quaternary alloys.

Materials and Methods

Theoretical Methods

Brief review of earlier researches: On the theoretical front empirical tight binding method [16], effective bond orbital method [17,18], empirical self-consistent pseudopotential method [19] and density functional theory [20] have emerged as powerful tools for precise calculation of ground state properties of materials. Although various software is, available for these methods, they require profound understanding of the nature of chemical bonding in materials and also advanced computational facilities. Empirical methods easy to work and they also produced results quite in agreement with experimental values and also with those obtained from the advanced computational schemes. Therefore, a number of empirical methods have been developed by researchers for computing bulk modulus of semiconductors. Cohen developed the following bond length (d in Å) dependent relation for bulk modulus of semiconductors.

B=1761d^-3.5                                     (1)

Where;

B=Bulk modulus.

Later on an empirical parameter, λ was introduced in this formula to incorporate the effect of iconicity which stands as follows,

B=(1971-220 λ )/d^3.5                       (2)

Where;

λ=0,1 and 2 for group-IV and IV-IV, III-V and II-Vi respectively.

This formula was quite successful in estimating bulk modulus of III-V semiconductors, but a deviation of estimated values from experimental ones resulted in the range 10.1%-17.3% in case of II-VI systems. This relation was further extended to study non-octet semiconductors like Si₃N₄ for which following modified formula was suggested.

B=<N_C>(1971-220 λ )/4d^3.5            (3)

Where;

<N_C>=Stands for an average coordination number. Lam, et al., however, deduced an analytical relation of bulk modulus to the lattice parameters within the local density approximation and the pseudopotential as:

B=(1971/d^3.5)–408 (ΔZ)²/d⁴            (4)

Where;

ΔZ=0,1,2,3 for group-IV, III-V, II-VI and I-VII semiconductors respectively. But this formula was applicable to III-V semiconductors only. For II-VI systems they suggested some modification in it. Al-Douri, et al., also suggested a formula similar to that of Cohen.

B=(3000-100 λ) (a/2)^-3.5                  (5)

Where;

a=Lattice parameter. This formula estimated the values of bulk modulus which are consistent with the experimental results.

Neumann proposed a formula of bulk modulus in terms of lattice parameter ‘a’ and spectroscopically defined bond iconicity, f_i figured as:

B=b_oa^-m(1-g_of_i)                                (6)

Where;

b_o, m and g_o are constants and have the same value for all compounds. Al-Douri further proposed two formulae as:

B=(60- λ20). (E_gΓX/α_m)                   (7)

B=(30+ λ10) ((P_t^1/2/ E_gΓX)/3)         (8)

Where;

α_m=Metallicity (in eV); E_gΓX=Energy gap (in eV) along Γ–X. λ=0, 2 and 2.65 for group IV, III-V and II-VI semiconductors respectively.

P_t=Transition pressure.

Kamran, et al., however, suggested and also attempted to give theoretical basis to the following formula of bulk modulus: B=(1938.72-506.702 f_i)/d^3.5            (9)

Where;

f_i=Bond iconicity.

Results obtained using this formula, in general, fall within 19% of the experimental values. In the bond orbital method however, the bulk modulus was proposed as:

B=2*3^1/2(V₂ α_c³ + 7.8/d²)/3d³         (10)

 

Where;

V₂=Covalent energy, is given by,

V₂=-ηħ²/4π²md²                             (11 )

Here;

η=Overlap parameter,
α_c=Covalence of the bond and m is the mass of the electron.

This formula could give somewhat better results. In the ionic charge approach of bond orbital model, Verma proposed a d-3 dependence formula of bulk modulus.

A new formula proposed: Various formulae noted in the above sub-section clearly demonstrated the bond length dependence of bulk modulus along with some other parameters like bond ionicity, band gap, covalency, transition pressure, metallicity, covalent energy, bond charge etc. But the values predicted with different formulae differ widely among themselves and from the experimental ones. Also, the d-dependence power varies like d^-3.5, d^-4 and d³. This aspect, therefore, needs to be revisited. In this attempt, we begin with the basic definition of bulk modulus and proposed a semi empirical formula. From definition, bulk modulus, the property of a material showing its resistance to volume change on compression, is given by

B=-P/(dV/V)                                (12)

Where;

dV/V=Volume strain is dimensionless and the stress
P=F/A=E/V                                (13)

Here;

F=Force (restoring),
A=Area,
E=Energy
V=Volume.

The volume of the unit cell of a cubic crystal, V ∝ d³.

Where;

d=Bond length.

Therefore, bulk modulus should bear an inverse cubic dependence relation with bond length, i.e.

B ∝ E/d³                                 (14)

Here, bond energy E_b is proposed to be used in place of E. This is because the bond energy of a stronger bond must be higher and for such a bond the resistance against any deformation (volume) should be higher giving rise to higher bulk modulus. Accordingly, B is expected to bear a linear relationship with E_b/d³. With this assumption, we have applied the Universal Parameter Tight Binding (UPTB) theory to evaluate E_b and d for group–IV, IV-IV, III-V and II-VI semiconductors and some ternary and quaternary alloys. Then graphs were plotted for B versus E_b/d³ for group-IV and IV-IV, III-V and II-VI systems which were found to be linear, with the help of which the new relation has been proposed. But before arriving at the new relation, a brief introduction of UPTB theory appears plausible. In the UPTB formalism, the bond energy is expressed in terms of four main contributing energy terms as:

Where;

E_p=Promotional energy.
E_σ=Sigma bond formation energy.
E_o=Overlap energy.
E_m=Metallization energy.

These constituent energies are expressed in parameterized forms through the metallic, covalent, polar and hybrid energies, which are expressed in terms of atomic orbital energies. Actually, the metallic, covalent, polar and hybrid energies originate out of the non-zero matrix elements of the Hamiltonian between hybrids of the bond forming atoms.

The hybrid energy (expectation value) of a tetrahedral solid having sp³ hybrids is given by

H=Hamiltonian of the system.

These energy values differ somewhat in the solids from the corresponding values for free atoms. However, following Harrison’s prescription the free atom energy (Hartree-Fock term) values have been utilized in the present work. The metallic energy is given by

For tetrahedral solids, considering matrix elements between nearest neighbor atoms only, Froyen and Harrison proposed that;

Where;

where the subscript m is a quantum number, but m in the denominator represents the electron mass, d is the bond length (in Å ) η_ll'm and is a dimensionless coefficient with

In case of polar covalent solids, the hybrid polar energy is represented as:

Where;

c(a)=Cation (anion).

Therefore, following kraut and Harrison, a general expression of bond energy as given below was utilized.

Where;

V_1IV(c) and V_1IV(a) are the values of V₁ for the column-IV elements from the rows of the atoms corresponding to the cation and the anion respectively. In the equilibrium state, the bond energy is minimum. But among the various energy terms, only V₂ includes ‘d’. So E_b was minimized with respect to V₂. Then employing Newton-Raphson numerical technique, the self-consistent values of V₂, d and E_b were obtained. For selection of the overlap parameter η_o, initially, Harrison’s approach of using geometrical mean of η_o values corresponding to the respective homoploid semiconductor, i.e. for a compound A_iB_j with i and j denoting rows of elements A and B in the periodic table, tehffee ctive η_o is given by:

Where;

η_oi and η_oj=Values for η_o for the group-IV elements of the i^th and j^th rows respectively.

This mode of selection of η_o has earlier been used by Baranowski Talwar, et al. and Sasireka, et al., also, but in several cases the values of bond length deviated more from experimental values. In an attempt to improve the results, it is argued that elements exhibit periodic properties. So, each element of a period must have its own weightage in deciding the properties of a compound formed with the constituent elements. Therefore a weighted average of overlay parameters corresponding to the respective periods as given below was utilized in this study.

Where;

P_a(P_c)=Period number considering c-row as P=1.
η_o(η_c)=Overlap parameter of anion(cation).

This modi ication led to a two prong advantage irst the results of bond length improved and second, the number of iterations reduced to get self-consistency.

Results and Discussion

Using the method discussed, bond length and bond energy of group IV and IV-IV, III-V and II-VI semiconductors were calculated and are given in Tables 1-3 respectively.

Table 1: Bond length, d (in Å) and bond energy, E_b (in eV) of semiconductors belonging to group IV and IV-IV semiconductors.

Table 2: Bond length, d (in Å) and bond energy, E_b (in eV) of Gr. III-V semiconductors.

Table 3: Bond length, d (in Å) and bond energy, E_b (in eV) of group II-VI semiconductors.

Graphs are plotted for bulk modulus (experimental) against E_b/d³ for these systems for which linear graphs are obtained shown in Figures 1-3 respectively. Regression analysis gives high correlation of 0.996, 0.991 and 0.979 respectively for these systems.

Figure 1: Plot of B versus E_b/d³ for gr. IV and IV-IV semiconductors.

Figure 2: Plot of B versus E_b/d³ for III-V semiconductors.

Figure 3: Plot of B versus E_b/d³ for II-VI semiconductors.

With the help of these graphs, the following general relation is proposed four estimation of bulk modulus of the semiconductors

B=(48.23–7.39 λ^0.63547) E_b/d³ + (41.97 – 5.80 λ^2.19784)                   (26)

Where;

λ=|G_B-G_A|/2 with G_A and G_B being the group numbers of elements A and B respectively in the periodic table. The calculated values of bulk modulus along with the experimental and reported ones for group–IV and IV-IV, III-V and II-VI systems are shown in Tables 4-6 respectively.

Table 4: Bulk modulus B (in GPa) of gr. IV and IV-IV semiconductors.

Table 5: Bulk Modulus, B (in GPa) of III-V semiconductors.

Table 6: Bulk modulus, B (in GPa), of II-VI semiconductors.

In the above relation, λ takes care of the ionicity effect, because for λ=0, 1, 2 for group–IV and IV-IV, III-V and II-VI semiconductors for which the average bond iconicity increases along ((IV-IV) → (III-V) → (II-VI)). Hence the bulk modulus decreases in the sequence ((IV-IV) → (III-V) → (II-VI)). The calculated values of bulk modulus of the semiconductors studied are in reasonable agreement with the experimental and other reported values. It is interesting to note that the compounds involving elements from the carbon-row i.e. carbides, nitrides and oxides exhibit higher bulk moduli and their respective classes. This might be due to absence of core p-electrons in atoms of elements from C-row which allows them in deeper overlapping of orbitals in bond formation. Encouraged with these results, the general formula was applied as an extension to the ternary systems Cd_1-xZn_xS and In_1-xGa_xP having cation substitution, and ZnS_1-xSe_x and InP_1-xAs_x having anion substitution. Results are presented in Tables 7 and 8. The graphs showing variation of bulk modulus with composition are shown in Figures 4 and 5.

Table 7: Variation of Bulk Modulus (in GPa) with composition in ternary system Cd_1-xZn_xS and In_1-xGa_xP.

Table 8: Variation of bulk modulus (in GPa) with composition in ternary systems ZnS_1-xSe_x and InP_1-xAs_x.

Figure 4: Plot showing the variation of bulk modulus (in GPa) with composition in ternary systems Cd_1-xZn_xS and In_1-xGa_xP.

Figure 5: Plot showing the variation of bulk modulus (in GPa) with composition in ternary systems ZnS_1-xSe_x and InP_1-xAs_x.

It can be seen that B increases with x in Cd_1-xZn_xS and In_1-xGa_xP whereas it decreases with x in ZnS_1-xSe_x and InP_1-xAs_x. This is primarily because of ionic size effect. With decreasing ionic size in the former case (Cd⁺²=0.78 Å; Zn⁺²=0.6 Å; In⁺³=0.8 Å, Ga⁺³=0.62 Å) the bond length decreases, ionicity decreases and the bulk modulus increases, whereas in the latter case the increasing ionic size (S^-2=1.84 Å; Se^-2=1.98 Å; P^-3=2.12 Å; As^-3=2.22 Å) leads to opposite effect. The results also exhibit a small departure from the vegard's law. Therefore, bond length and bond energy of these systems were calculated using Vegard’s law. Such deviations from Vegard's law have also been reported in the lattice constants and bulk modulus in ZnS_xSe_1-x, lattice constants and band gaps in A_xZn_1-xO (A=Ca, Cd, Mg), B_xGa_1-xN and band gap in B_xIn_1-xN alloys.

The study is further extended to dilute oxide and diluted nitride quaternary HMAs Zn_1-xCd_xTe_1-yO_y, Cd_1-xZn_xTe_1-yO_y and Ga_1-xIn_xAs_1-yN_y, In_1-xGa_xAs_1-yN_y. Results are given in Tables 9 and 10.

Table 9: Variation of bond length, d (in Å), bond energy, E_b (in eV) and bulk modulus, B (in GPa) with composition in dilute oxide quaternary HMAs Zn_1-xCdxTe_1-yO_y and Cd_1-xZn_xTe_1-yO_y.

Table 10: Variation of bond length, d(in Å), bond energy, E_b (in eV) and bulk modulus, B (in GPa) with composition in dilute nitride quaternary HMAs Ga_1-xIn_xAs_1-yN_y and In_1-xGa_xAs_1-yN_y.

As expected the ionic-size (ionicity) effect is clearly re lected in these systems too. The results can be seen with y ranging from 0 to 1 for a given value of x and similarly with x ranging from 0 to 0.5 for a given value of y.

Conclusion

In consideration of the importance of bulk modulus of semiconductors for various applications including solar cells, a semi empirical formula for bulk modulus of semiconductors has been proposed involving bond energy and bond length. It has been applied to group IV and IV-IV, III-V and II-VI semiconductors and also to ternary and quaternary alloys. Trend of variation of bulk modulus with composition (alloy) is interesting to note. It is hoped that this study may shed light in better understanding of the trend of variation of various properties of semiconductor alloys especially in highly mismatched alloys.

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