A Simple Approach to Compute Interatomic Force Constant for Mono and Diatomic Semiconductors

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1. Introduction

Obtaining a mathematical relation of interatomic Hooke’s force constant to bulk modulus and interatomic distance in solids, especially in semiconductors, is very important as a relation connecting macroscopic and microscopic quantities that helps in understanding the interaction between atoms and calculating energies and other quantities. Dedicated bond force constant and bulk modulus of C*n* fullerenes (*n* = 20, 28, 36, 50, 60) are computed using density functional theory (DFT) and finite element analysis (FEA) by Peon *et al*. The bond force constants predicted by DFT are then used as an input for finite element analysis (FEA) of the fullerene [1]. Santiago-Perez *et al*. obtained the linear chain equations that should be used to obtain the phonon dispersion relations along high symmetry directions from the 3D bulk problem [2]. Other work by Kot *et al*. found that there is a well-defined relationship between the geometric characteristics of the mass spring model systems and physical properties of the modeled materials [3], and so Dolocan *et al*. presented theoretical expressions relating the cohesive energy to bulk modulus, the force constant and the lattice constant applicable to solids with a variety of crystal structures, by assuming, under nearest-neighbor approximation, the interatomic force to be a polynomial of 2^{nd} degree
$F=-\beta x+\gamma {x}^{2}$ [4].

In this paper, it has been assumed that Hooke’s force is the only affecting interatomic force under nearest-neighbor approximation (
$F=-\beta U$ ), which was used to find a mathematical expression for solid crystals of diatomic primitive cell, including those of mono-atomic one automatically, relating interatomic Hooke’s force constant (*β*) to bulk modulus and interatomic distance by using two methods, linear statistical fitting with computer programming and mathematical analysis. We used the vibrational dispersion relation of a linear lattice of diatomic primitive cell and Data of some semiconductors (C (diamond), Si, Ge, SiC, ZnS, ZnTe, CdS and CdTe).

In spite of existing other forces than Hooke’s one and which may be of other atoms than the neighbors, but that do not affect much the process of calculating energies and other quantities as was shown in Einstein and Debye theories for specific heat in solids [5] [6], we tried to simplify the mathematical processing and, at the same time, obtain an accurate mathematical relation as far as possible.

In order to get the mathematical relation that we seek, we considered that the interatomic forces in a linear lattice are subject to Hooke’s law with small atomic deflections as (
$F=-\beta U$ ), where *F* is the Hooke’s force, *U* is small deflection and (*β*) is the force constant, with neglecting other effects to facilitate the process mathematically for scientific purposes [7].

2. Theoretical Work

As we know, the two functions of the dispersion relation which describe linear vibrations of a diatomic primitive cell as shown in Figure 1 (the mono-atomic primitive cell is automatically included) are [8].

The acoustic branch

${\omega}_{ac}^{2}=\beta \left[\left(\frac{1}{M}+\frac{1}{m}\right)-\sqrt{{\left(\frac{1}{M}+\frac{1}{m}\right)}^{2}-\frac{4}{mM}{\mathrm{sin}}^{2}\left(\frac{qa}{2}\right)}\right],$ (1)

The optical branch

${\omega}_{op}^{2}=\beta \left[\left(\frac{1}{M}+\frac{1}{m}\right)+\sqrt{{\left(\frac{1}{M}+\frac{1}{m}\right)}^{2}-\frac{4}{mM}{\mathrm{sin}}^{2}\left(\frac{qa}{2}\right)}\right],$ (2)

where *m* and *M* are the masses of both atoms in the primitive cell (
$M\ge m$ ) and *a* is the interatomic distance. The possible deflections of two atoms are

$U\left(l\right)={U}_{m}{\text{e}}^{i\left(q\cdot la-\omega t\right)}$ (for the bigger mass), (3)

$u\left(l\right)={u}_{m}{\text{e}}^{i\left(q\cdot la-\omega t\right)}$ (for the smaller mass), (4)

where *U _{m}* and

Figure 2 shows the two acoustic and optical branches functions which are separated by the prohibited *ω*-zone.

Where the allowed values of *q* (wave number) are given as follows:

$q=l\frac{2\pi}{Na}$, *N* is the number of primitive cells in the crystal,
$l=\pm 1,\pm 2,\pm 3,\cdots $

This means that all the solution points are within the first two Brillouin zones, so, if the number of atoms in each primitive cell is three, then all the solutions are within the first three Brillouin zones [7].

Figure 1. A linear solid lattice of diatomic primitive cell.

Figure 2. The two curves of dispersion relation [9].

Figure 3. Approximation of dispersion relation by a straight line *ω = sq* beside *q* = 0 [9].

We proceeded from the acoustic branch of the dispersion relation in a linear solid lattice of diatomic cell (Equation (1)) and the fact that where *q* approaches zero, it approximately takes the following form [6]:

$\omega =sq$, (5)

where *s* is the wave (sound) speed via the lattice, as shown in Figure 3.

2.1. Linearstatistical Fitting

We should make some mathematical simplifications in Equation (1) (the acoustic branch) to remove the function dependence on the physical characteristics as follows.

Equation (1) may be rewritten under the following form:

${\omega}^{2}=\frac{\beta}{m}\left[\left(\frac{m}{M}+1\right)-\sqrt{\frac{{m}^{2}}{{M}^{2}}+1+\frac{2m}{M}\mathrm{cos}aq}\right],$ (6)

or

$\sqrt{\frac{m}{\beta}}\omega =\sqrt{\left(\frac{m}{M}+1\right)-\sqrt{\frac{{m}^{2}}{{M}^{2}}+1+\frac{2m}{M}\mathrm{cos}aq}},$ (7)

assuming that $x=aq$, $p=\frac{M}{m}$ ( $M\ge m$, then $p\ge 1$ )and $\varpi =\sqrt{\frac{m}{\beta}}\omega $, then

$\varpi =\sqrt{\left(1+\frac{1}{p}\right)-\sqrt{1+\frac{1}{{p}^{2}}+\frac{2}{p}\mathrm{cos}x}}.$ (8)

By using a VISUAL BASIC program that generates 50
$\left(\varpi ,x\right)$ points beside
$x=0$ via substituting with values of *x* in Equation (8) and uses them for linear statistical fitting as
$\varpi ={\eta}_{0}+{\eta}_{1}x$ after calculating each of
${\eta}_{0}$ and
${\eta}_{1}$, we can reach the sought relation.

When running the program which needs to enter the values of *p* = *M*/*m* for each matter, the
${\eta}_{1}$ values are as Table 1 shows.

Table 1. Estimated values of ${\eta}_{1}$ for the used matters.

Because that the curve $\varpi \left(x\right)$ passes through the origin (0, 0), so ${\eta}_{0}\cong 0$ and ${\eta}_{1}$ is the tangent slope of linear fitting function $\varpi ={\eta}_{1}x$ besides $x=0$.

$\varpi ={\eta}_{1}x\Rightarrow \sqrt{\frac{m}{\underset{\_}{\beta}}}\underset{\_}{\omega}={\eta}_{1}aq,$ (9)

where $\underset{\_}{\omega}$ and $\underset{\_}{\beta}$ are calculated by linear fitting process, then

$\underset{\_}{\omega}=a{\eta}_{1}\sqrt{\frac{\underset{\_}{\beta}}{m}}q,$ (10)

which we compare with Equation (5), so

$\underset{\_}{\omega}=a{\eta}_{1}\sqrt{\frac{\underset{\_}{\beta}}{m}}q\cong sq$, then (11)

$\beta =\frac{m{s}^{2}}{{a}^{2}{\eta}_{1}^{2}}.$ (12)

2.2. Mathematical Analyzing Proof

Proceeding from Equation (8), which is

$\varpi =\sqrt{\left(1+\frac{1}{p}\right)-\sqrt{1+\frac{1}{{p}^{2}}+\frac{2}{p}\mathrm{cos}x}}.$

A first estimation of the force constant
$\beta $ may be derived from simple consideration. Indeed, as *x* approaches zero,
$\mathrm{cos}\theta \cong 1-\frac{{x}^{2}}{2}$, so, Equation (8) can be rewritten under the following form

${\varpi}^{2}=\left(1+\frac{1}{p}\right)-\sqrt{1+\frac{1}{{p}^{2}}+\frac{2}{p}\left(1-\frac{{x}^{2}}{2}\right)}$, around $x=0$ (13)

or ${\varpi}^{2}=\left(1+\frac{1}{p}\right)\left[1-\sqrt{1-\frac{{x}^{2}}{p{\left(1+1/p\right)}^{2}}}\right]$, (14)

for values of *x* around zero, we have
$\frac{{x}^{2}}{p{\left(1+1/p\right)}^{2}}\ll 1$, and, consequently

(as ${\left(1+y\right)}^{1/2}\cong 1+\frac{1}{2}y$ as $y\cong 0$ ), Equation (14) reduces to the simple form

${\varpi}^{2}=\frac{{x}^{2}}{2\left(p+1\right)},$ (15)

as $x=aq$, $p=\frac{M}{m}$ ( $M\ge m$, then $p\ge 1$ )and $\varpi =\sqrt{\frac{m}{\beta}}\omega $, then from Equation 15, one can obtain the following expression of $\omega $.

$\omega =\left(\sqrt{\frac{\beta}{2\left(M+m\right)}}a\right)q$ (16)

According to Equation (5), we get

$s=\left(\sqrt{\frac{\beta}{2\left(M+m\right)}}a\right),$ (17)

then $\beta =\frac{4\stackrel{\xaf}{m}{s}^{2}}{{a}^{2}}$, (18)

where $\stackrel{\xaf}{m}=\frac{m+M}{2}$.

3. Comparison and Discussion

Comparing the estimated force constant and the analytically extracted one, we get the results shown in Table 2 in which the last column shows the calculated values of $\beta $ according to Equation 18 and we can observe a good concordance between these values and those given by the fitting process according to Equation 12 (in the adjacent column) and, so, as Figure 4 shows.

We can now introduce the interatomic force constant, *β*, via the following relation

$\beta =\frac{4\stackrel{\xaf}{m}{s}^{2}}{{a}^{2}},$

Table 2. Comparison between the estimated force constant (Equation (12)) and the analytically extracted one (Equation (18)).

$s=\sqrt{\frac{B}{\rho}}=\sqrt{\frac{B{v}_{c}}{m+M}}$ ^{,}
$\rho $ is the density [12], (19)

${v}_{c}$ is the primitive cell volume, such as ${v}_{c}=\frac{{a}^{3}}{\underset{\_}{n}}$ [12], (20)

where $\underset{\_}{n}$ is the number of atoms in a primary cell; Figure 5 shows the difference between primary cell and primitive cell in B. C. C. structure as an example.

Substituting from Equation (20) in Equation (19) then in Equation (18), we get

$\beta =\frac{2}{\underset{\_}{n}}Ba,$ (21)

which is the relation we are looking for.

As an example, for F. C. C. Structure, as $\underset{\_}{n}=4$, $\beta =\frac{1}{2}Ba$.

It is worth noting that a similar relation has been shown to be valid for solids with a variety of crystal structures [4].

Comparing Equation (12) and Equation (18), we find

$\frac{m{s}^{2}}{{a}^{2}{\eta}_{1}^{2}}=\frac{4\stackrel{\xaf}{m}{s}^{2}}{{a}^{2}},$ (22)

Figure 4. Comparison between the estimated *β* and the analytically extracted *β*.

Figure 5. Primary cell and primitive cell in B. C. C. structure [13].

then $\frac{m{s}^{2}}{{a}^{2}{\eta}_{1}^{2}}=\frac{4\stackrel{\xaf}{m}{s}^{2}}{{a}^{2}},$ (23)

which gives us the exact values of the slope ${\eta}_{1}$ for the matters used in this work as shown in Table 3, and can be compared to their estimated values in Table 1 which is shown in Figure 6.

As direct results from the obtained force constant relation, we can calculate the maximum acoustic angular frequency ${\left({\omega}_{ac}\right)}_{\mathrm{max}}$, the minimum optical angular frequency ${\left({\omega}_{op}\right)}_{\mathrm{min}}$ and the maximum optical angular frequency ${\left({\omega}_{op}\right)}_{\mathrm{max}}$, which can easily be measured experimentally.

From Figure 2, we can see that

$\beta =\frac{1}{2}M{\left({\omega}_{ac}\right)}_{\mathrm{max}}^{2}=\frac{1}{2}m{\left({\omega}_{op}\right)}_{\mathrm{min}}^{2}=\frac{1}{2}\mu {\left({\omega}_{op}\right)}_{\mathrm{max}}^{2},$ (24)

where $\mu $ is the reduced mass $\left(\frac{1}{\mu}=\frac{1}{M}+\frac{1}{m}\right)$.

By substituting with $\beta =\frac{2\left(M+m\right){s}^{2}}{{a}^{2}}$, we can deduce that

Table 3. Comparison between the extracted ${\eta}_{1}$ and the estimated ${\eta}_{1}$ for the used matters.

Figure 6. Comparison between the estimated ${\eta}_{1}$ and the analytically extracted ${\eta}_{1}$.

Table 4. Values of ${\left({\omega}_{ac}\right)}_{\mathrm{max}}$, ${\left({\omega}_{op}\right)}_{\mathrm{min}}$ and ${\left({\omega}_{op}\right)}_{\mathrm{max}}$ according to the resulted relation of $\beta $, ( $\beta =\frac{4\stackrel{\xaf}{m}{s}^{2}}{{a}^{2}}$, where $\stackrel{\xaf}{m}=\frac{M+m}{2}$ ).

${\left({\omega}_{ac}\right)}_{\mathrm{max}}=2\sqrt{\left(1+\frac{1}{p}\right)}\cdot \frac{s}{a},$ (25)

${\left({\omega}_{op}\right)}_{\mathrm{min}}=2\sqrt{\left(1+p\right)}\cdot \frac{s}{a},$ (26)

and ${\left({\omega}_{op}\right)}_{\mathrm{max}}=\frac{2\left(1+p\right)}{\sqrt{p}}\cdot \frac{s}{a},$ (27)

which Table 4 shows their values for the substances of this work.

4. Conclusions

The simple approach, detailed in this work, based on linear fitting by computer programming and mathematical analysis, permitted computation of the constant force (CF) as a linear function of the bulk modulus and interatomic distance parameters, under the assumption that only interactions between first nearest neighbors are considered. The two derived relations from the two above-mentioned methods have led to similar results for some mono- and diatomic molecules of the semiconductor type. Furthermore, these two relations enabled us to express the interatomic force constant as proportional to the bulk modulus and the interatomic distance.

The importance of deriving this relation is that many physical quantities are linked to the force constant, the bulk modulus and the interatomic distance and so, it could be a contribution to increase crystal structure understanding and may be used in many other theoretical works.

Moreover, this resulted mathematical relation makes it possible to calculate interatomic Hooke’s force constant for solid matters in terms of bulk modulus and interatomic distance, and so to calculate bulk modulus in terms of the others, which can help understanding the relationship between sub-fusion and fusion energies with fusion temperature.

The results of this work will be used in an attempt to extract mathematical relations between correlation energy, sub-fusion energy and fusion energy with temperature, which will clarify many physical issues and provide broader research possibilities.

References

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