Analysis of lattice thermal conductivity and phonon-phonon scattering relaxation rate: Application to Ge

Abstract

The three-phonon scattering relaxation rates and their temperature exponents have been analysed in the frame of Guthrie's classification of the phonon-phonon scattering events as class I and class H events and as a result of this, a new expression T3p ~ = = (BN, t + Bu, i e-~ (BN, n q-B'O, IIe-~ TraIl(T) for the three phonon scattering relaxation rates has been proposed for the first time to calculate the lattice thermal conductivity of a sample. Using the expression proposed above, the lattice thermal conductivity of Ge has been analysed in the temperature range 2 --1000K and result obtained shows a very good agreement with the experimental data, The percentage contributions due to three-phonon normal and umklapp processes are also reported. The role of four phonon processes is also included at high temperatures, To estimate an approximate value of the scattering strength and the phonon conductivity, the analytical expression is also obtained in the frame of the expression proposed above for r3~1.The phonon-phonon scattering relaxation rate has been studied by a number of workers [1-13] due to its very important role in the lattice thermal conductivity, and it has been found that the three-phonon scattering relaxation rate involves a complicated dependence on the phonon frequency and temperature due to the complicated structure of the Brillouin Zone and the strong temperature dependence of the distribution function. As a result of this, even at present we lack an exact analytical expression for it. However, for practical purposes, it has been expressed by simple relations [1-13] as a function of the phonon frequency and temperature. It is also found that the phonon-phonon scattering processes can be divided into two groups; normal processes (N processes) in which momentum is conserved, and umklapp processes (U processes) in which momentum is not conserved. The roles of N and U processes have been studied by a number of worker [14][15][16][17][18][19][20][21][22] by calculating the phonon conductivities of different samples.Recently, Guthrie [7, 8] studied the three-phonon scattering relaxation rate by dividing phonon-phonon scattering events into two classes : class I events, in which the carrier phonon is annihilated by combination and class II events, in which the carrier phonon is annihilated by splitting. Following Guthrie, the three-phonon scattering relaxation rate t~lh can be expressed as t:~h = %~h(ClaSS I) + z3~(class II)(1)

Analysis of the lattice thermal conductivity and phonon-phonon scattering relaxation rate: Application to Mg2Ge and Mg2Si

“…As a result, Dubey [33] proposed a new expression for Z3phT" for transverse phonons -1 ~ (BTN, I + Bxu,~e-~ O~Tmx't(x) (14) ~3ph, T because the contribution due to class II events is not possible for Z3ph,-1 T for transverse phonons. Similarly, the expression for z3ph,-1 L for longitudinal phonons is given by [33] --1…”

“…In the lack of an expression for the exact value of re(T), to minimise the uncertainty, Dubey [33] suggested the use of the average value of its maximum and minimum values, which is more realistic compared to the use of the maximum value as suggested by Joshi and Verma [3]. Thus, the expression for re(T) used by Dubey [33] (9) for class II events.…”

“…Recently, considering the role of the three-phonon N-and U-processes, and following the Guthrie classification of the phonon-phonon scattering events, Dubey [33 ] studied the lattice thermal conductivity of a sample by proposing a new expression for the three-phonon scattering relaxation rate as or. Thermal Anal.…”

“…In view of Eqs (10) and (11), Dubey [33] expressed the scattering -1 for class I events and z -1 for class II events as relaxation rates z3vh, ~ 3ph, ii…”

“…Solid He [41] and LiF [53], solid HD [54] are exceptions [4l, 53,54] to this. Following the earlier work of Verma et al [55], Dubey [33] used a better dispersion relation q = (co~v) (I + re) 2) to replace q into co in eqn. (20), where r is a constant and depends on the dispersion curve of the sample under study.…”