Origin of ultralow thermal conductivity in amorphous Si thin films investigated using nanoindentation, 3ω method, and phonon transport analysis

The origin of the ultralow thermal conductivity in amorphous Si thin films was investigated by comparing their phonon transport properties with those of single-crystal Si. The group velocity and thermal conductivity were measured at 300 K using nanoindentation and the 3ω method, respectively. The phonon mean free path (MFP) and phonon frequency were determined using the measured properties and models. The scattering in the disordered structure of amorphous Si thin films caused a significant decrease in the phonon MFP with an increase in the phonon frequency, leading to ultralow thermal conductivity. However, the group velocity was unaffected by the disordered structure.

][8] Thus, amorphous materials can be regarded as ultimate materials with reduced grain sizes.][14][15] However, owing to the complex structure of amorphous materials, experimental investigation of the origin of the ultralow thermal conductivity (by analyzing phonon transport properties, such as group velocity, phonon mean free path (MFP), and phonon frequency) is challenging.Minnich et al. measured the phonon MFPs of Si using thermal conductivity spectroscopy and obtained good agreement with first-principles calculations. 16)egner et al. analyzed the breakdown in diffusive phonon transport caused by high-frequency surface temperature modulation to determine the contributions of phonons to the thermal conductivity in crystalline and amorphous silicon, which depend on their mean free path. 17)][20][21] In addition, the phonon frequency in singlecrystal Si was estimated using the combined method and modeling. 22)Nanoindentation is frequently used to measure the elastic modulus and hardness of structural materials, 23,24) and group velocity can be estimated from the elastic modulus.Therefore, the combined method has the potential to measure the phonon transport properties of amorphous materials, which can be used to elucidate the origin of their ultralow thermal conductivity.
In the present study, sputter-deposited Si thin films were used as amorphous materials because the material properties of Si have been well documented, and comparison is easy.[30][31][32] The phonon transport properties of the films were obtained using a combination of nanoindentation and 3ω method.To elucidate the origin of the ultralow thermal conductivity in amorphous Si thin films, we compared the phonon transport properties of amorphous Si with those of single-crystal Si reported in our previous studies and extracted the distinctive properties of amorphous Si. 20,22) This method can be applied not only to thermoelectric materials but also to materials with disordered structures as a new analysis method.Furthermore, it can be used to control the thermal properties of amorphous materials.
Amorphous Si thin films were deposited on polished alumina substrates (AO-2525, Furuuchi Chemical; dimensions: 25 mm × 25 mm × 1.0 mm) using a pressure gradient sputtering system (PGS, Kenix) without substrate heating.A 50-mm diameter high-purity (99.9%) poly-Si target (Furuuchi Chemical) was used for the deposition.The distance between the target and the substrate was set to 200 mm.The sputter deposition was performed using highpurity (99.995%)Ar gas at a pressure of 0.6 Pa and a radio frequency power of 30 W. The deposition time was 6 h, and the thickness of the resulting film (as measured using a Bruker DektakXT profilometer) was approximately 400 nm.The structural characteristics of the films were studied using X-ray diffraction (XRD; Mini Flex 600, Rigaku) and Raman spectroscopy (XploRA: HORIBA).The Cu Kα radiation (λ = 0.154 nm) was used for XRD measurements, and an Ar + laser beam excitation at 514.5 nm was used to obtain the Raman spectrum.The group velocity of the amorphous Si thin films was estimated at approximately 300 K by nanoindentation (ENTNEXUS; ELIONIX) using a Berkovich indenter operating in the continuous stiffness mode.The indentation depth was 10% of the film thickness.The nanoindentation process is described in detail in our previous reports. 18,19)[35] Figure 1 shows the structural characteristics of the Si thin films.The XRD pattern in Fig. 1(a) shows only the peaks corresponding to the alumina substrate, suggesting that the Si thin films exhibit low crystallinity or an amorphous phase.The Raman spectrum of the Si thin films in Fig. 1(b) shows a broad peak centered at approximately 480 cm −1 , which corresponds to amorphous Si.Thus, the XRD and Raman results indicate that the as-deposited Si thin films are amorphous.
Figure 2 shows the group velocities of the amorphous Si thin films (measured at approximately 300 K) compared with that of single-crystal Si.The longitudinal group velocity v L and transverse group velocity v T were estimated using the Young's moduli and shear moduli, respectively, obtained via nanoindentation.The average group velocity v ave is expressed as 3/v ave 3 = 1/v L 3 + 2/v T 3 .The longitudinal, transverse, and average group velocities of the amorphous Si thin films are 8254, 5159, and 5604 m s −1 , respectively.These values are approximately 5% lower than those of the single-crystal Si film.Therefore, the group velocities are largely unaffected by the disordered structure of the amorphous film.
Table I shows the thermal conductivities of the amorphous Si thin films measured at approximately 300 K compared with those of previously reported amorphous Si thin films and single-crystal Si.The thermal conductivity reported in this study corresponds to the lattice thermal conductivity because the amorphous and single-crystal Si thin films were undoped (i.e. the electronic thermal conductivity is negligible).][38] In contrast, the thermal conductivity of single-crystal Si obtained in our previous study was 132 W/(m•K). 22)Therefore, the amorphous structure reduces the thermal conductivity to approximately 1/40 that of single-crystal Si.These results demonstrate that the disordered structure has little effect on the group velocity but significantly affects the thermal conductivity.
To investigate the phonon transport in amorphous Si thin films, we estimated the phonon MFPs at approximately 300 K for different scattering mechanisms based on the phonon transport properties of single-crystal Si, as shown in Fig. 3. Based on Matthiessen's rule, the effective phonon MFP Λ eff is expressed as: 39) where Λ um , Λ dis , Λ imp , and Λ b are the phonon MFPs corresponding to the Umklapp, disorder, impurity, and boundary scattering mechanisms, respectively.Using the    ), the Λ eff of amorphous Si thin films was estimated to be 1.1 nm (based on the phonon gas model (Λ eff = 3κ l /Cv ave )).The specific heat C of amorphous Si is 1.6 × 10 6 J/(m 3 •K). 40)We assume that the Λ um of amorphous Si is the same as that of single-crystal Si because Umklapp scattering is inherent to the material.Impurity and boundary scattering were not considered because the amorphous Si thin films are undoped and do not contain crystal grains.From the values of Λ eff (1.1 nm) and Λ um of single-crystal Si (40 nm), Λ dis of amorphous Si thin films was found to be 1.1 nm.This result indicates that disorder scattering is more dominant than Umklapp scattering in amorphous Si thin films.
To obtain further insight into the phonon transport in amorphous Si thin films, the phonon frequencies were calculated at 300 K.For Umklapp scattering, the relaxation time τ um was determined using the semi-empirical equation: 41,42) A T, Here, A um is approximated from the relation: where the Grüneisen parameter (γ) is assumed to be equal to 2, m is the average mass of a single atom (m = 4.65 × 10 −26 kg), and θ is the Debye temperature (θ = 674 K).Therefore, Λ um was calculated using the equation: For disorder scattering, the relaxation time τ dis was modeled using the equation: 13)

B
, 5 where B dis is a constant coefficient that incorporates the effect of temperature.The scaling exponent n for amorphous materials is 2-4. 12,43,44)A value of 2 corresponds to anharmonic scattering, 45) whereas a value of 4 corresponds to Rayleigh-type scattering from point defects. 46)Therefore, Λ dis was calculated using the equation: We used the "minimum thermal conductivity" model proposed by Cahill et al. to determine the value of B dis . 47)When the thermal conductivity reaches its minimum value, the minimum phonon MFP corresponds to half the Debye wavelength at the cutoff frequency. 48)In this case, the minimum phonon MFP is calculated to be 0.9 nm using the Debye wavelength of 1.8 nm, indicating that the Λ dis of 1.1 nm is very close to the minimum phonon MFP.Using the values of average group velocity (5607 m s −1 ), minimum phonon MFP (0.9 nm), and cutoff frequency (88.6 THz) in Eq. ( 6), the values of B dis were determined to be 1.Considering the above results, the ultralow thermal conductivity of the amorphous Si thin films can be attributed to the significant decrease in the phonon MFP with an increase in the phonon frequency.
To summarize, the origin of the ultralow thermal conductivity in amorphous materials was investigated by preparing sputter-deposited amorphous Si thin films.The presence of an amorphous phase was confirmed using XRD and Raman spectroscopy.The group velocity and thermal conductivity were determined at 300 K using nanoindentation and the 3ω method, respectively.The phonon MFPs were determined using the measured group velocities and thermal conductivities.Furthermore, the contributions of different scattering components to the effective phonon MFP were determined to estimate the phonon frequency.The ultralow thermal conductivity of the amorphous Si thin films is attributed to the significant decrease in the phonon MFP with the increase in the phonon frequency due to scattering in the disordered structure of amorphous Si.This method can also be used to control the thermal properties of materials with different disordered structures.

2 ×
10 15 at n = 2 and 9.1 × 10 42 at n = 4.The relationship between the phonon MFP and the phonon frequency for different scattering mechanisms is shown in Fig. 4. The phonon frequency of Umklapp scattering is 54 THz at a phonon MFP of 40 nm.On the contrary, the phonon frequencies of disorder scattering are 79.8MHz at n = 2 and 85.2 MHz at n = 4.The phonon frequency of the amorphous structure is 1.5 times higher than that of the single-crystal structure with only Umklapp scattering and very close to the cutoff frequency.

Fig. 3 . 4 .
Fig. 3. Phonon mean free paths of the amorphous Si thin films.Fig. 4. Relationship between the phonon mean free path and phonon frequency of the amorphous Si thin films.

Table I .
Thermal conductivity values of the amorphous Si thin-film in this work, previously reported amorphous Si films, and single-crystal Si.
©2023The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd