The values of κ for the corresponding GM6001 film thicknesses 100, 300, and 400 nm at 300 K increased gradually to approximately 0.52, approximately 1.85, and approximately 3.51 W/m · K, respectively. We also found that the thermal conductivities of the films were 1.7 to 11.5 times lower than that of bulk Fe3O4 (approximately 6 W/m · K) [17]. It has been well understood that the significant reduction in the thermal conductivity of the thin films (100 to 400 nm in thickness) compared to the bulk materials could be due to the enhanced phonon-boundary scattering in thin films predicted previously by Callaway [18]. In addition, we added the theoretical calculation results of Callaway’s model in the same figure

(solid line in Figure 5a,b).

The results predicted by the Callaway model agree reasonably well with the experimental data, including the results for bulk Fe3O4. We can thus confirm that the significant reduction in the thermal conductivity for nanoscale thin films is principally a result of phonon-boundary EPZ015938 scattering. In the following section, the calculation model is discussed in detail. Figure 5 Temperature-dependent conductivities of three Fe 3 O 4 films and a simple theoretical calculation based on the Callaway model. (a, b) Measured thermal conductivities of 100-, 300-, and 400-nm-thick Fe3O4 thin films at temperatures of 20 to 300 K using the 3-ω method, including the thermal conductivity of bulk materials. The solid line denotes thermal conductivity of bulk materials from the

theoretical Callaway model, which includes the this website effect of the impurity, Umklapp process, boundary scattering with film grain size, and film thickness. To determine the temperature dependence of the thermal conductivity, κ(T), in Fe3O4 thin films quantitatively, we performed a theoretical calculation (i.e., fitting) based on the relaxation time model using the following expression predicted by Callaway in 1959 [18]: Immune system (2) where ω is the phonon frequency, k B is the Boltzman constant, ℏ is the reduced Planck constant, x denotes the dimensionless parameter, x = ℏω/k B T, θ D is the Debye temperature, T is the absolute temperature, and c is the velocity of sound. The total combined phonon scattering rate (relaxation time, τ c) is given by (3) where d 1 is the grain size of the thin films (approximately 13.2, approximately 86, approximately 230 nm for the 100-, 300-, and 400-nm-thick films, respectively, from the AFM measurements shown in Figure 1), A and B are independent parameters of temperature and fitting, respectively, and c is the sound velocity, which is highly dependent on the direction of movement of phonons (average c = 2,500 m/s) [17]. To add the film thickness in Equation 3, we modified the phonon scattering rate given as (4) where d 2 is the corresponding film thickness. For the Fe3O4 films, we estimated that the values of A and B in Equation 4 were numerically optimized as approximately 8.