Prediction of Spectral Phonon Mean Free Path and Thermal Conductivity with Applications to Thermoelectrics and Thermal Management: A Review

Table 1

Analytical models of inverse relaxation time for different scattering processes.

Scattering process	Inverse relaxation time
Intrinsic Three-Phonon
𝑁 process
Herring^a	𝜏^−1𝐿,𝑁=𝐵𝐿𝜔^2𝑇^3, low 𝑇
	𝜏^−1𝑇,𝑁=𝐵𝑇𝜔𝑇^4, low 𝑇
	𝜏^−1𝐿,𝑁=𝐵^′𝐿𝜔^2𝑇, high 𝑇
	𝜏^−1𝑇,𝑁=𝐵^′𝑇𝜔𝑇, high 𝑇
Callaway^b	𝜏𝑁^−1=𝐵𝑁𝜔^2𝑇^3
𝑈 process
Klemens^c	𝜏𝑈^−1=𝐵𝑈𝜔^2𝑇^3−𝜃exp𝛼𝑇, low 𝑇
Klemens^d	𝜏𝑈^−1=𝐵𝑈𝜔𝑇^3−𝜃exp𝛼𝑇, low 𝑇
	𝜏𝑈^−1=𝐵^′𝜔^2𝑇, high 𝑇
Callaway^b	𝜏𝑈^−1=𝐵𝑈𝜔^2𝑇^3
Holland^e	𝜏^−1𝑇,𝑈=𝐵𝑇𝑈𝜔^2sinh(𝑥),𝜔1≤𝜔≤𝜔20,𝜔<𝜔1
Asen-Palmer et al.^f	𝜏^−1𝑇,𝑈=𝐵^′𝑇𝑈𝜔^2𝐶𝑇exp^𝑇𝑇
	𝜏^−1𝐿,𝑈=𝐵^′𝐿𝑈𝜔^2𝐶𝑇exp^𝐿𝑇
Boundary ^g	𝜏𝑏^−1=𝑣𝜆𝑙𝐹  𝑙: diameter, 𝑙=2𝑙1𝑙2/𝜋; 𝐹: surface roughmess
Impurity ^h	𝜏^−1im=𝜋2𝑔𝜔^2𝑉𝐷(𝜔)≈^0𝑔4𝜋𝑣^3𝜔^4∼𝐴𝜔^4

References: ^a[36], ^b[80], ^c[33], ^d[34], ^e[37], ^f[81], ^g[38–40], and ^h[35].