Effect of phonon-boundary scattering on phonon-drag factor in Seebeck coefficient of Si wire

Introduction of Si nanostructures into Si-based thermoelectric devices has been widely studied in order to enhance their performance by reducing their thermal conductivity through the suppression of phonon transport.^1–4 Several studies reported the reduction of thermal conductivity in a low dimensional structure owing to the promotion of the boundary scattering in phonon transport.^5–8 Moreover, Martin et al. reported that the surface roughness strongly affects the thermal conductivity in thin Si nanowires, which originates from the phonon-surface scattering.⁹ The phonon-scattering at the boundaries primarily depends on the surface roughness and the incident phonon wavelength. Diffusive scattering occurs when the phonons are reflected by a boundary that has surface roughness comparable to or higher than the phonon wavelength. This dependency is characterized by the specularity parameter, p, expressed as^10,11

where η is the rms roughness, k is the phonon wavenumber, and θ is the incident angle with respect to the surface normal. The specularity indicates the probability of the phonon being specularly reflected at the boundary surface. The impact of the surface roughness on the phonon-boundary scattering has been studied in the scope of thermal transport.^10–14 

On the other hand, the Seebeck coefficient S is also influenced by the phonon transport through the phonon-drag effect, which originates from the momentum transfer to electrons from phonons flowing under a temperature gradient. In general, the Seebeck coefficient of a semiconductor comprises a phonon-drag factor S_ph as well as an electronic factor S_e originating from carrier transport.^15,16 Geballe and Sadhu reported that S_ph is significantly observed in bulk Si at room temperature.^17,18 In addition, we previously reported that in quite thin Si-on-insulator (SOI) layers with thickness below 10 nm, S_ph is not influenced by the thickness.^19,20 However, S_ph is likely affected by reducing the dimensions of the layer into wire.^21,22 The phonon-boundary scattering seems to have less impact on S_ph in ultrathin Si layers, and its relation to S_ph in different dimensions of Si is still not clear. In the present study, we investigate the Seebeck coefficient of Si wires with micro/nanometer dimensions and discuss the influence of the sample dimensions on its phonon-drag component from the viewpoint of the specularity.

The Si wires discussed in this study were fabricated in SOI wafer consisting of a top Si layer, a SiO₂ layer, and a p-type Si substrate with the same process as the previous investigation.²² Si wires were patterned in the top Si layer of SOI wafer by a lithography method with a thickness of 30 nm, a length of 1 mm, and a width of 10 μm, 5 μm, and 1 μm. The p-type wires were prepared by thermal diffusion of B atoms. The carrier concentration of the B-doped wire was determined to be 3.6 × 10¹⁷ cm⁻³ by the Hall effect measurement. Al electrodes were patterned at both ends of the wire.

The Seebeck coefficient was measured by a probe measurement system equipped with an infrared camera, which is explained in the previous report.²² The Seebeck coefficient was evaluated by giving a temperature gradient to the sample using a resistive heater near room temperature. The temperature difference was observed by an infrared camera placed above the sample. A couple of probes were directly attached to the Al electrode to measure the thermoelectromotive force (TEMF). The Seebeck coefficient was evaluated from the slope of the linear relationship between the measured TEMF, ΔV = V_H − V_L, and the temperature difference, ΔT = T_H − T_L by S = −ΔV/ΔT, where the subscripts H and L indicate the high- and low-temperature region. The uncertainty in the measured Seebeck coefficient evaluated from the deviation between the data and the linearly fitted line was ±0.11 mV/K. The measurement was performed for several single wires with the same dimensions in order to confirm the repeatability of the measurement. These measured values are included in Fig. 1, as an error bar.

Figure 1 shows the measured Seebeck coefficient of the p-type Si wire as a function of wire width. The blue dotted line shows the linear fitted and the red broken line indicates the calculated S_e of bulk Si with a carrier concentration of 3.6 × 10¹⁷ cm⁻³.^20,23 The arrow at the right axis corresponds to the reported value of the bulk Si with a carrier concentration of ∼10¹⁵ cm⁻³.¹⁷ From Fig. 1, the Seebeck coefficient is clearly observed to be higher than the calculated S_e, which indicates the contribution of the phonon-drag effect. Moreover, the Seebeck coefficient decreases with a decrease in the wire width, and the widest wire has the Seebeck coefficient that is close to the value of bulk Si despite its small thickness of 30 nm. This result seems strange since the phonon-drag contribution in the Si wires is affected by their width even though they are on a micrometer order.

Since the decrease in the Seebeck coefficient is considered to originate from the enhancement of phonon-boundary scattering, the discrepancy between the case in thin layer and wire can be qualitatively explained by considering that the boundary scattering of phonons is strongly related to the roughness on the wire surface. In order to clarify this, the specularity characterizing the phonon-boundary scattering is determined by the surface roughness of the boundary using Eq. (1). The top surface roughness and sidewall surface roughness of the wire were observed by using atomic force microscopy (AFM) and are shown in Fig. 2. The average rms roughness of the top surface was evaluated to be 0.4 ± 0.1 nm. On the other hand, the sidewall surface roughness was observed by constructing a contour diagram of the wire edge shown in Fig. 2(b) from the top view. The rms roughness of the sidewall surface was estimated to be 1.4 ± 0.1 nm.

It has been reported that the phonons contributing to the phonon-drag factor are spectrally distinct from phonons contributing to the thermal conductivity.²⁴ The phonon modes, which significantly contribute to the phonon drag, have smaller wavenumbers compared with the phonon modes contributing to the thermal conductivity. By calculating the phonon participating in intravalley electron–phonon scattering in Si, as shown in Fig. 3, the typical wavenumber of phonons, which mainly contribute to the phonon drag, k_{phonon drag}, is estimated to be 7 × 10⁶ cm⁻¹. That is, this phonon wavenumber is smaller than the major contributor in the thermal transport k_{thermal conductivity}, which is estimated to be ∼4 × 10⁷ at 300 K.¹³ This result indicates that the high phonon wavelength is mainly responsible for the phonon-drag effect since the wavenumber of electrons in the Si conduction valley is small and they interact mostly with small-wavenumber phonons.

Figure 4 shows the calculated specularity of the top and sidewall surfaces by using Eq. (1) with an assumption of θ = 0° (normal incidence) as a function of phonon wavenumber. The specularity of the top and sidewall surfaces for the phonon modes participating in the phonon-drag is obtained to be 0.73 and 0.02, respectively. These results indicate that the boundary scattering at the top surface is much less effective than the sidewall surface in the range of phonon wavenumber contributing to the phonon-drag effect. On the other hand, in the range of phonon wavenumber, which participates in the thermal conductivity, the specularity is quite low p ≈ 0 (i.e., diffusive scattering). This leads to a significant reduction in the thermal conductivity due to phonon boundary scattering.

By considering the specularity, the phonon-boundary scattering rate is defined by¹² 

where v is the phonon velocity, and l is the length normal to the boundary surface, which corresponds to the wire width and thickness for sidewall and top surfaces, respectively. The total phonon-scattering rate $τtotal_thickness−1$ and $τtotal_width−1$ were calculated by adding the phonon-boundary scattering and the phonon–phonon scattering $τph−1$ including the phonon–phonon Umklapp scattering $τU−1$ and the normal scattering $τN−1$ calculated by^20,25

where ℏ is the reduced Planck constant, γ is the Gr$ü$neisen constant, ω is the angular phonon frequency, T is the absolute temperature, H is the Debye temperature, M is the average mass of Si, k_B is the Boltzmann constant, and V is the volume per atom of Si. Equations (3) and (4) show that $τph−1$ has a dependency on temperature. Hence, a higher temperature decreases the mean free time of phonons, thus leading to the phonon-drag effect. However, although the average temperature increased to 370 K when the temperature gradient was applied during the measurement, the relationship between the TEMF and temperature difference had a good linear relation. Therefore, the Seebeck coefficient is constant, and the phonon-drag effect is preserved, at least, up to 370 K. The calculated mean free time is shown in Fig. 5, as a function of length. In this calculation, it was assumed that the roughness at the Si/SiO₂ interface is identical to the roughness of the top surface of the Si wire. From Fig. 5, it is found that τ_{total_width} is affected by the wire width in the range below 10 μm. This is consistent with the experimental results, where the Seebeck coefficient decreases with a decrease in the wire width below 10 μm. On the other hand, the thickness affects τ_{total_thickness} at a smaller length, in the range below 1 μm, which is qualitatively consistent with the experimental result that the influence of τ_{B_thickness} on τ_{total_thickness} at a smaller length than that the influence of τ_{B_width} on τ_{total_width}.

However, it is quantitatively discrepant from the fact that the Si wire as thin as 30 nm has the Seebeck coefficient value close to the bulk Si. From Eq. (2),

Therefore, the straight line of τ_B in Fig. 5 shifts vertically in accordance with the term,

That is, with an increase in the β value, the critical length, where the influence of τ_B begins to be significant, becomes small. Figure 6 shows the calculated β as a function of specularity. In this figure, with increasing p, β increases monotonously. Above p = 6, the gradient of the β–p graph also increases significantly, and then, an abrupt change in β appears at the high specularity value (p > 0.9). This result indicates that the critical length significantly decreases even if the evaluation of rms roughness fluctuates slightly to be smaller at p = 0.73, which is more sensitive compared with the case of p = 0.02. This is a possible reason for the quantitative discrepancy between the experimental and theoretical critical lengths in the thickness direction. Thus, it is clarified that the roughness on the wire surface is an important factor for the phonon-drag effect in the Si Seebeck coefficient. Hence, for preserving the phonon-drag contribution to the Seebeck coefficient in Si nanowires, the wire surface should be quite smooth.

We have fabricated micro/nanometer-scaled Si wires containing the phonon-drag contribution to their Seebeck coefficient. The size dependency of the phonon-drag factor of the Seebeck coefficient was clearly observed, which is due to the enhancement of the phonon-boundary scattering. The calculation of the specularity of the wire surface reveals that the boundary scattering at the top surface is much less effective than the sidewall surface in the range of phonon wavenumber contributing to the phonon drag. Thus, the roughness of the wire surface is an important parameter for preserving the phonon-drag contribution to the nanostructured Si Seebeck coefficient while the thermal conductivity is reduced by nanostructuring.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors thank the Ministry of Research, Technology and Higher Education, the Republic of Indonesia, for the RISET-Pro scholarship. This work was financially supported in part by JST CREST Grant No. JPMJCR19Q5, Japan.