At any finite temperature, atoms in a crystal oscillate around their mechanical equilibrium positions. Quantizing these collective normal modes transforms continuous lattice vibrations into discrete bosonic quasiparticles called phonons. Phonons govern thermal conductivity, limit electron mobility through scattering, and mediate fundamental interactions across solid-state physics.
1. Classical Dynamics of 1D Diatomic Lattices
Consider a one-dimensional chain composed of two distinct atoms of masses (M_1) and (M_2) ((M_1 > M_2)) connected by nearest-neighbor springs with spring constant (C), with lattice parameter (a). Let (u_n) and (v_n) denote the displacements of the two atoms in the (n)-th unit cell. The coupled classical equations of motion are:
[
M_1 rac{d^2 u_n}{dt^2} = C (v_n + v_{n-1} – 2u_n), quad M_2 rac{d^2 v_n}{dt^2} = C (u_{n+1} + u_n – 2v_n)
]
Seeking harmonic traveling wave solutions (u_n = u_0 e^{i(k n a – omega t)}) and (v_n = v_0 e^{i(k n a – omega t)}), substitution into the dynamical matrix yields the secular determinant:
[
begin{vmatrix}
2C – M_1 omega^2 & -C(1 + e^{-i k a})
-C(1 + e^{i k a}) & 2C – M_2 omega^2
nd{vmatrix} = 0
]
Solving this quadratic equation in (omega^2) yields two distinct dispersion branches:
[
omega^2(k) = C left( rac{1}{M_1} + rac{1}{M_2}
ight) pm C sqrt{left( rac{1}{M_1} + rac{1}{M_2}
ight)^2 – rac{4 sin^2(ka/2)}{M_1 M_2}}
]
This yields two fundamental physical branches within the first Brillouin zone ((|k| le pi/a)):
- Acoustic Branch ((-)): At long wavelengths ((k o 0)), (omega approx v_s |k|), where (v_s = sqrt{rac{C a^2}{2(M_1 + M_2)}}) is the longitudinal speed of sound. Both atoms in the unit cell oscillate in phase with equal displacement ((u_0 approx v_0)).
- Optical Branch ((+)): At (k o 0), the frequency remains finite: (omega_0 = sqrt{2C left( rac{1}{M_1} + rac{1}{M_2}
ight)}). The two atoms oscillate precisely (180^circ) out of phase ((M_1 u_0 + M_2 v_0 = 0)). In ionic or polar semiconductors (e.g., GaAs, InP), this antiphase motion generates a microscopic fluctuating electric dipole that couples strongly to infrared electromagnetic radiation.
2. Second Quantization of Lattice Vibrations
Treating the harmonic lattice in the canonical Hamiltonian framework, the total vibrational energy is diagonalized using bosonic creation ((hat{a}_{mathbf{q},s}^dagger)) and annihilation ((hat{a}_{mathbf{q},s})) operators for each wavevector (mathbf{q}) and polarization branch (s):
[
hat{H}_{ ext{ph}} = sum_{mathbf{q}, s} hbar omega_s(mathbf{q}) left( hat{a}_{mathbf{q},s}^dagger hat{a}_{mathbf{q},s} + rac{1}{2}
ight)
]
Because phonons are bosons, their thermal occupation number at temperature (T) obeys the Planck-Bose-Einstein distribution:
[
langle n_{mathbf{q},s}
angle = rac{1}{e^{hbar omega_s(mathbf{q}) / k_B T} – 1}
]
3. Phonon Density of States (DOS) and Heat Capacity
The total vibrational energy is evaluated by integrating over the phonon density of states (g(omega)):
[
U(T) = int_0^{omega_{max}} g(omega) hbar omega left( rac{1}{e^{hbar omega / k_B T} – 1} + rac{1}{2}
ight) domega
]
Two fundamental approximations describe (g(omega)):
- Debye Model (Acoustic Phonons): Assumes a linear dispersion (omega = v_s q) up to a cutoff frequency (omega_D) defined by total mode conservation ((int_0^{omega_D} g(omega) domega = 3N)). In 3D, the DOS is quadratic:
[
g_D(omega) = rac{3V}{2pi^2 v_s^3} omega^2 = rac{9N}{omega_D^3} omega^2, quad omega le omega_D
]
At low temperatures ((T ll Theta_D = hbar omega_D / k_B)), this yields the classic Debye (T^3) law for lattice heat capacity:
[
C_v = rac{partial U}{partial T} = rac{12pi^4}{5} N k_B left( rac{T}{Theta_D}
ight)^3
] - Einstein Model (Optical Phonons): Approximates dispersionless optical phonons as independent oscillators of identical frequency (omega_E): (g_E(omega) = 3N delta(omega – omega_E)). It captures the exponential freeze-out of optical phonon modes at low temperatures ((C_v sim e^{-hbar omega_E / k_B T})).
