3.5 Anharmonic effects

Anharmonic effects in solids go beyond the simple harmonic approximation. These effects lead to important phenomena like thermal expansion, phonon-phonon interactions, and multiphonon absorption. Understanding anharmonicity is crucial for accurately describing real materials' behavior.

Anharmonic potential energy models, like the Morse potential, account for asymmetry and non-equally spaced energy levels. This leads to thermal expansion, where materials grow with increasing temperature. Phonon-phonon interactions and multiphonon absorption processes further reveal the complexities of anharmonic effects in solids.

Anharmonic potential energy

• Anharmonic potential energy deviates from the harmonic approximation, which assumes a symmetric potential well
• Anharmonicity leads to various effects in solids, such as thermal expansion, phonon-phonon interactions, and multiphonon absorption
• Understanding anharmonic potential energy is crucial for accurately describing the behavior of real materials

Morse potential

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• Models the potential energy of a diatomic molecule more accurately than the harmonic potential
• Accounts for the dissociation of the molecule at large interatomic distances
• Includes terms that describe the asymmetry of the potential well and the anharmonicity of the vibrations
  • Potential energy: $V(r) = D_e [1 - e^{-a(r-r_e)}]^2$, where $D_e$ is the dissociation energy, $a$ is a parameter related to the well width, and $r_e$ is the equilibrium bond distance

Deviation from harmonic approximation

• Harmonic approximation assumes a symmetric potential well and equally spaced energy levels
• Anharmonicity leads to non-equally spaced energy levels and a shift in the equilibrium position
• Causes the vibrational frequency to depend on the amplitude of the vibration
  • Larger amplitudes result in lower frequencies due to the shallower potential at larger interatomic distances
• Introduces higher-order terms in the potential energy expansion, such as cubic and quartic terms

Thermal expansion

• Phenomenon where materials expand with increasing temperature due to anharmonic effects
• Occurs because the average interatomic distance increases with temperature
• Has important implications for the design and performance of materials in various applications

Asymmetric interatomic potential

• Interatomic potential is asymmetric, with a steeper repulsive wall at short distances and a shallower attractive well at larger distances
• Asymmetry leads to a net outward force on the atoms as the temperature increases
  • Atoms spend more time at larger interatomic distances due to the shallower potential well
• Results in an increase in the average interatomic distance and, consequently, thermal expansion

Volume vs temperature

• Volume of a solid increases with increasing temperature due to thermal expansion
• Relationship between volume and temperature is typically linear at low temperatures
  • $V(T) = V_0 [1 + \alpha (T - T_0)]$, where $V_0$ is the volume at a reference temperature $T_0$, and $\alpha$ is the volume expansion coefficient
• At higher temperatures, the volume-temperature relationship may deviate from linearity due to higher-order anharmonic effects

Linear and volume expansion coefficients

• Linear expansion coefficient $(\alpha_L)$ describes the relative change in length per unit change in temperature
  • $\alpha_L = \frac{1}{L} \frac{dL}{dT}$, where $L$ is the length of the material
• Volume expansion coefficient $(\alpha_V)$ describes the relative change in volume per unit change in temperature
  • $\alpha_V = \frac{1}{V} \frac{dV}{dT}$
• For isotropic materials, the volume expansion coefficient is approximately three times the linear expansion coefficient $(\alpha_V \approx 3\alpha_L)$

Phonon-phonon interactions

• Interactions between phonons that arise due to anharmonic terms in the potential energy
• Play a crucial role in determining the thermal properties of solids, such as thermal conductivity and thermal expansion
• Lead to phonon scattering, which limits the phonon mean free path and affects the thermal conductivity

Cubic and quartic terms

• Anharmonicity introduces higher-order terms in the potential energy expansion, such as cubic and quartic terms
• Cubic terms describe three-phonon interactions, where two phonons combine to create a third phonon or one phonon splits into two phonons
  • Responsible for thermal resistance and the finite thermal conductivity of solids at high temperatures
• Quartic terms describe four-phonon interactions, which are generally weaker than three-phonon interactions

Normal and umklapp processes

• Normal processes conserve the total phonon momentum, i.e., $\vec{q}_1 + \vec{q}_2 = \vec{q}_3$, where $\vec{q}_i$ are the phonon wave vectors
  • Do not directly contribute to thermal resistance, as they do not change the net phonon momentum
• Umklapp processes do not conserve the total phonon momentum, i.e., $\vec{q}_1 + \vec{q}_2 = \vec{q}_3 + \vec{G}$, where $\vec{G}$ is a reciprocal lattice vector
  • Result in a change of the net phonon momentum and directly contribute to thermal resistance
  • Dominate thermal resistance at high temperatures

Phonon scattering and lifetimes

• Phonon-phonon interactions lead to phonon scattering, which limits the phonon mean free path and lifetime
• Phonon lifetime $(\tau)$ is the average time between scattering events
  • Related to the phonon mean free path $(\Lambda)$ by $\Lambda = v_g \tau$, where $v_g$ is the group velocity of the phonon
• Scattering rates and lifetimes depend on the phonon frequency, temperature, and the strength of the anharmonic interactions
  • Higher temperatures and stronger anharmonicity lead to shorter phonon lifetimes and mean free paths

Thermal conductivity

• Measure of a material's ability to conduct heat
• Depends on the phonon mean free path, which is limited by phonon-phonon interactions and other scattering mechanisms
• Thermal conductivity decreases with increasing temperature due to increased phonon scattering

Phonon mean free path

• Average distance a phonon travels between scattering events
• Depends on the phonon frequency, temperature, and the strength of the anharmonic interactions
  • Higher temperatures and stronger anharmonicity lead to shorter mean free paths
• Limits the thermal conductivity of the material, as shorter mean free paths result in lower thermal conductivity

Thermal resistivity vs temperature

• Thermal resistivity $(\rho)$ is the inverse of thermal conductivity $(\kappa)$, i.e., $\rho = 1/\kappa$
• At low temperatures, thermal resistivity decreases with increasing temperature due to the increasing specific heat and phonon velocity
• At high temperatures, thermal resistivity increases with temperature due to increased phonon-phonon scattering (umklapp processes)
  • Results in a minimum in the thermal resistivity at an intermediate temperature

Umklapp process dominance

• Umklapp processes dominate the thermal resistance at high temperatures
• Probability of umklapp processes increases with temperature due to the increased phonon population and the availability of larger wave vectors
• Leads to a rapid increase in thermal resistivity and a decrease in thermal conductivity at high temperatures
  • Limits the maximum thermal conductivity achievable in materials

Multiphonon absorption

• Absorption of photons by a material through the simultaneous excitation of multiple phonons
• Occurs when the energy of the incident photon is greater than the fundamental phonon energy
• Allows the study of higher-order phonon modes and anharmonic effects in solids

Overtones and combination bands

• Overtones are multiphonon absorption processes where multiple phonons of the same mode are excited simultaneously
  • First overtone corresponds to the excitation of two phonons, second overtone to three phonons, and so on
• Combination bands involve the simultaneous excitation of phonons from different modes
  • Can provide information about the coupling between different phonon modes and the anharmonicity of the potential

Infrared and Raman spectroscopy

• Techniques used to study multiphonon absorption in solids
• Infrared spectroscopy measures the absorption of infrared photons by the material
  • Multiphonon absorption appears as weak absorption bands at higher frequencies than the fundamental phonon modes
• Raman spectroscopy measures the inelastic scattering of photons by phonons
  • Multiphonon processes appear as higher-order peaks in the Raman spectrum, shifted from the fundamental phonon frequencies

Frequency shifts and line broadening

• Anharmonicity leads to frequency shifts and line broadening of the multiphonon absorption bands
• Frequency shifts occur due to the anharmonic correction to the phonon frequencies
  • Overtones and combination bands appear at slightly different frequencies than the sum of the fundamental frequencies
• Line broadening occurs due to the decreased phonon lifetimes and the increased scattering rates associated with anharmonicity
  • Results in broader and less distinct absorption bands compared to the fundamental phonon modes