where d i is the displacement vector of atom i, and Γ is the spring (or force) constant as the second-order derivatives of the potential. The equation of motion for the lattice vibration in terms of the displacement of atoms [ d (jl, t): displacement vector of the j -th atom in the l -th unit cell at time t ] is m j d 2 d ( j l , t ) d t 2 = − − ∑ ∑ j ′ l ′ Γ Γ ( j j ′ ′ l l ′ ) ⋅ ⋅ d ( j ′ l ′ , T ) , where m is the atomic mass and Γ is the force constant tensor. The atomic displacement is the summation over the normal modes [ s α : unit vector of mode α, ω p : angular frequency of wave, and κ p : wave vector]. Using this plane-wave displacement, the equation of motion becomes the eigenvalue equation M ω ω p 2 ( κ κ p , α α ) s α α ( κ κ p ) = D ( κ κ p ) s α α ( κ κ p ) , where M is the diagonal mass matrix and D is the harmonic dynamical matrix. Solving this eigenvalue equation gives the relation between the angular frequency ω p and the wave vector κ p, and this relation is called the phonon dispersion relation. Thus, the phonon dispersion relation is determined by matrices M and D, which depend on the atomic structure and the strength of interaction among constituent atoms (the stronger the interaction and the lighter the atoms, the higher is the phonon frequency and the larger is the slope dω p / d κ p). The Hamiltonian of phonon system with the harmonic approximation is H p = ∑ ∑ x 1 2 m p 2 ( x ) + 1 2 ∑ ∑ x , x ′ d i ( x ) D i j ( x − − x ′ ) d j ( x ′ ) , where D ij is the dynamical matrix element between atoms i and j, and d i (d j) is the displacement of i (j) atom, and p is momentum. From this and the solution to dispersion relation, the phonon annihilation operator for the quantum treatment is defined as b κ κ , α α = 1 N 1 / 2 ∑ ∑ κ κ p , α α e − − i ( κ κ p ⋅ ⋅ x ) s α α ( κ κ p ) ⋅ ⋅ [ ( m ω ω p , α α 2 ℏ ℏ ) 1 / 2 d ( x ) + i ( 1 2 ℏ ℏ m ω ω p , α α ) 1 / 2 p ( x ) ] , where N is the number of normal modes divided by α and ħ is the reduced Planck constant. The creation operator is the adjoint of the annihilation operator, b κ κ , α α † † = 1 N 1 / 2 ∑ ∑ κ κ p , α α e i ( κ κ p ⋅ ⋅ x ) s α α ( κ κ p ) ⋅ ⋅ [ ( m ω ω p , α α 2 ℏ ℏ ) 1 / 2 d ( x ) − − i ( 1 2 ℏ ℏ m ω ω p , α α ) 1 / 2 p ( x ) ] . The Hamiltonian in terms of b κ,α and b κ,α is H p = Σ κ, α ħω p,α [ b κ,α b κ,α + 1/2] and b κ,α b κ,α is the phonon number operator. The energy of quantum-harmonic oscillator is E p = Σ κ, α [ f p (κ, α) + 1/2] ħω p,α (κ p), and thus the quantum of phonon energy ħω p.