# Silicon Carbide Analysis

2.1 Introduction

The discussion has been done in chapter1 about the nature crystal structure of Silicon Carbide and its properties was under the equilibrium condition, which means the location of each atom is restricted at its lattice site. The vibration of atoms about their equilibrium sites will be considered in this chapter. The fundamental reason of this vibration is thermal and the lowest achievable energy (i.e. zero-point energy). The forces acting on atoms tend to always make them return to the equilibrium position. The relation between atomic forces and thermal motion can be called lattice dynamic. The atomic force at equilibrium is described by the following formula:

F=-∇ ⃑ U=0 where U is binding energy "potential

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The first three lower branches are called "acoustic branches" and they are classified by their polarization as T_1A , T_2A and LA which can be count as one longitudinal (LA) and two transverse acoustics (T_1A, T_2A). The acoustic branches are characterized by the relation ω → 0 when q → 0, with atoms vibrating in phase. The other upper branches above the first 3 branches are called the "optical branches", which can also be classified as longitudinal or transverse, and they are characterized by the relation ω ≠ 0 when q → 0, with atoms moving out of

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Thermal or any external forces causes the atoms to oscillate. The quantum or the collective of vibrational energy that originate due to the oscillating atoms within a lattice structure is known as phonon. Phonon does not carry any momentum because of the relative motion of the atoms and the motion of their center of mass. However, for practical purpose we can describe that phonon is neither a wave nor a particle "quasi-particle" thus, it has momentum ℏq [33]. Phonons play a major role in understanding materials and their physical properties of condensed matter such as specific heat and thermal conductivity.

Moreover, the average number of phonons in the q^thmode in thermal equilibrium obeys Bose-Einstein distribution which given by:

( n) ̅_q= 1/(exp(ℏω(q)/(k_B T))-1) (2 4)

Where k_B is Boltzmann's constant. According to this formula, we can conclude the fowling:

1- At absolute zero T=0 , it can be clearly seen that there are no phonons in a crystal.

2- At low temperatures ℏω≫k_B T, the average number of phonons