Microscopic validation of pore-boundary scattering, the low-temperature Seebeck anomaly, and the ZT ≈ 1 at 1000 K projection for porous silicon nanowire thermoelectrics

Randomly porous, heavily doped silicon nanowires reach ZT = 0.71 at 700 K, but the analytical model behind this result rests on an unvalidated pore-boundary scattering assumption, leaves the measured low-temperature Seebeck coefficient unexplained by the carrier-diffusion model, and extrapolates ZT ≈ 1 at 1000 K without experimental test. The problem asks for a predictive, microscopically validated account of phonon and charge-carrier transport in these nanowires over 300–1000 K: (i) spectral confirmation or correction of the assumed fully diffuse geometric boundary scattering length $l_b=4V/A_{pore}$ (with Matthiessen combination of scattering rates) for both phonons and carriers in the random porous geometry of ~4 nm Si crystallites; (ii) the quantitative physical origin of the low-temperature excess of the measured Seebeck coefficient over the diffusion model; and (iii) experimental validation or refutation of the projected $ZT\approx1$ at 1000 K, which requires measurement platforms stable above 700 K.

Background

Roughly half of primary energy worldwide is rejected as waste heat, and waste heat above 573 K carries the largest conversion potential. Thermoelectric generators convert heat directly into electricity; their efficiency is governed by the dimensionless figure of merit $ZT=S^2\sigma T/\kappa$, where $S$ is the Seebeck coefficient (the thermovoltage developed per unit temperature difference), $\sigma$ the electrical conductivity, $\kappa$ the thermal conductivity, and $T$ the absolute temperature. Silicon is abundant and CMOS-compatible, but bulk Si has $ZT\approx0.01$–$0.05$. Nanostructuring suppresses the lattice part of $\kappa$ without necessarily destroying $\sigma$: rough silicon nanowires (Hochbaum et al., 2008) and periodically perforated 'holey' silicon membranes (Tang et al., 2010) showed order-of-magnitude $\kappa$ reductions. Prior thermoelectric characterizations reached 900 K for bulk high-doped Si (Stranz et al., 2013) and 700 K for Si nanowires (Lee et al., 2016).

Yang et al. (2021) reported wafer-scale arrays of randomly porous silicon nanowires made by metal-assisted chemical etching (an etch catalyzed by a patterned metal film that leaves a nanowire forest) followed by post-doping, in which boron is diffused into the etched wires from a spin-on dopant source. Individual nanowires — porosity $\phi$ from 9% to 61%, Si crystallite size $\approx4$ nm, hole concentration up to $2.2\times10^{20}\ \mathrm{cm^{-3}}$ — had $\kappa$, $\sigma$, and $S$ measured on the same suspended microdevice from 300 to 700 K, giving $ZT=0.71$ at 700 K for the best sample ($\phi=46\%$). Their analytical model treats the lattice $\kappa$ with the Callaway–Holland model (a phenomenological phonon Boltzmann transport treatment that integrates mode-resolved relaxation times), $\sigma$ with the Boltzmann transport equation in the relaxation-time approximation (carrier scattering summarized by an energy-dependent relaxation time), and $S$ with a carrier-diffusion model; the individual scattering times are combined by Matthiessen's rule ($1/\tau_{total}=\sum_i 1/\tau_i$). For the pore walls, fully diffuse geometric boundary scattering is assumed with boundary scattering length $l_b=4V/A_{pore}$, where $V$ is the solid volume and $A_{pore}$ the pore surface area — an empirical $\kappa\propto V/A_{pore}$ scaling established by Yang et al. (2016) across Si films, wires, beams, and ribbons — and the same length is used for phonons and for charge carriers. The authors identify the microscopic scattering time for pore-boundary scattering as the biggest unknown in the modeling: the $l_b=4V/A_{pore}$ assumption has not been validated microscopically across the phonon spectrum for their random porous geometry. The diffusion model for $S$ reproduces the measured $S$ much better at high temperature than at low temperature, and the authors state that the reason for this behavior is unclear. Candidate mechanisms for such a low-temperature Seebeck excess include phonon drag (enhancement of $S$ by momentum transfer from out-of-equilibrium phonons to charge carriers) and energy filtering (preferential scattering of low-energy carriers at potential barriers, which raises $S$). Finally, because $\kappa$, $\sigma$, and $S$ are nearly temperature independent for the best nanowire, the model projects $ZT\approx1$ at $T=1000\ \mathrm{K}$; but the platinum/chromium heaters and thermometers of the suspended microdevice become unstable at the temperatures approaching 1000 K that the test requires, so this projection is untested.

Problem Statement

For randomly porous, heavily boron-doped silicon nanowires of the kind measured by Yang et al. (2021) — Si crystallite size $\approx4$ nm, porosity $\phi$ from 9% to 61%, hole concentration up to $2.2\times10^{20}\ \mathrm{cm^{-3}}$, thermoelectrically characterized from 300 to 700 K — establish a microscopically validated account of phonon and charge-carrier transport, replacing the assumed pore-boundary scattering description of the analytical model and testing its high-temperature extrapolation, specifically: (1) determine, by microscopic theory, atomistic simulation (e.g. phonon Boltzmann transport on the resolved random pore geometry, molecular dynamics, or phonon/electron Monte Carlo), or an equivalent spectrally resolved experiment, whether the fully diffuse geometric boundary scattering length $l_b=4V/A_{pore}$ entering Matthiessen's rule for both phonons and charge carriers quantitatively describes the pore-boundary-induced relaxation times across the phonon spectrum in this geometry, and identify the regimes (porosity, crystallite size, frequency, temperature) where it fails and what replaces it; (2) identify and quantify the physical origin of the discrepancy whereby the carrier-diffusion model reproduces the measured Seebeck coefficient much better at high temperature than at low temperature — including, but not limited to, boundary-limited phonon drag and energy filtering at crystallite and pore boundaries — such that the observed $S(T)$ from 300 to 700 K is reproduced within experimental uncertainty; and (3) experimentally validate or refute the projection $ZT\approx1$ at $T=1000\ \mathrm{K}$ for the best-performing porous Si nanowires, which requires measurement platforms (heater/thermometer materials, calibration, stability) operating on individual nanowires above 700 K. These three objectives jointly constitute one coherent deliverable and are all mandatory: a predictive, microscopically validated account of $\kappa(T)$, $\sigma(T)$, and $S(T)$ for porous Si nanowires over 300–1000 K; delivering objectives (1) and (2) alone does not resolve the task. Objective (3) is satisfied either by a high-temperature experimental result — a measurement of $ZT$ near $1000\ \mathrm{K}$ on a measurement platform validated to operate above 700 K — or, where such a platform is unavailable, by a quantified prediction of $ZT(1000\ \mathrm{K})$ with an explicit uncertainty budget built on the transport account of objectives (1) and (2) together with a documented high-temperature platform design and feasibility case. The device-level engineering directions raised in the same source (minimizing nanowire–electrode contact resistance for power output, and module-level integration, heat-exchanger, durability, and levelized-cost-of-electricity analyses) are recorded as context, not as part of this scientific completion boundary, because they are open-ended technology-development goals rather than questions with a reviewable scientific answer.

References

1. Lin Yang, Daihong Huh, Rui Ning, Vi Rapp, Yuqiang Zeng, Yunzhi Liu, Sucheol Ju, Yi Tao, Yue Jiang, Jihyun Beak, Juyoung Leem, Sumanjeet Kaur, Heon Lee, Xiaolin Zheng, Ravi S. Prasher. High thermoelectric figure of merit of porous Si nanowires from 300 to 700 K. Nature Communications 12, 3926 (2021). doi:10.1038/s41467-021-24208-3, https://doi.org/10.1038/s41467-021-24208-3
2. Lin Yang, Yang Yang, Qian Zhang, Yin Zhang, Youfei Jiang, Zhe Guan, Matthew Gerboth, Juekuan Yang, Yunfei Chen, D. Greg Walker, Terry T. Xu, Deyu Li. Thermal conductivity of individual silicon nanoribbons. Nanoscale 8, 17895–17901 (2016). doi:10.1039/C6NR06302K, https://doi.org/10.1039/C6NR06302K
3. Mohammad Ghanem, Philippe Dollfus, Raja Sen, Jelena Sjakste, Jerome Saint-Martin. Static and dynamic monte carlo simulations of phonon drag effects on thermoelectric properties in silicon nanostructures. Computational Materials Science 268, 114630 (2026). doi:10.1016/j.commatsci.2026.114630, https://doi.org/10.1016/j.commatsci.2026.114630
4. Raja Sen, Nathalie Vast, Jelena Sjakste. Role of dimensionality and size in controlling the drag Seebeck coefficient of doped silicon nanostructures: A fundamental understanding. Physical Review B 108, L060301 (2023). doi:10.1103/PhysRevB.108.L060301, https://doi.org/10.1103/PhysRevB.108.L060301
5. Rui Ning, Yuqiang Zeng, Vi Rapp, Buyi Zhang, Lin Yang, Ravi Prasher, Xiaolin Zheng. Thermoelectric performance of high aspect ratio double-sided silicon nanowire arrays. Journal of Applied Physics 135, 095001 (2024). doi:10.1063/5.0187590, https://doi.org/10.1063/5.0187590
6. Mykola Isaiev, Yuliia Mankovska, Vasyl Kuryliuk, David Lacroix. Thermal transport properties of nanoporous silicon with significant specific surface area. Applied Physics Letters 122, 172201 (2023). doi:10.1063/5.0148434, https://doi.org/10.1063/5.0148434
7. Han Wei, Yue Hu, Hua Bao. Influence of point defects and multiscale pores on the different phonon transport regimes. Communications Materials 4, 3 (2023). doi:10.1038/s43246-023-00330-1, https://doi.org/10.1038/s43246-023-00330-1
8. S. Aria Hosseini, Sarah Khanniche, P. Alex Greaney, Giuseppe Romano. Universal effective medium theory to predict the thermal conductivity in nanostructured materials. International Journal of Heat and Mass Transfer 183, 122040 (2022). doi:10.1016/j.ijheatmasstransfer.2021.122040, https://doi.org/10.1016/j.ijheatmasstransfer.2021.122040
9. Jyothi Sadhu, Hongxiang Tian, Jun Ma, Bruno Azeredo, Junhwan Kim, Karthik Balasundaram, Chen Zhang, Xiuling Li, P. M. Ferreira, S. Sinha. Quenched Phonon Drag in Silicon Nanowires Reveals Significant Effect in the Bulk at Room Temperature. Nano Letters 15, 3159–3165 (2015). doi:10.1021/acs.nanolett.5b00267, https://doi.org/10.1021/acs.nanolett.5b00267
10. Allon I. Hochbaum, Renkun Chen, Raul Diaz Delgado, Wenjie Liang, Erik C. Garnett, Mark Najarian, Arun Majumdar, Peidong Yang. Enhanced thermoelectric performance of rough silicon nanowires. Nature 451, 163–167 (2008). doi:10.1038/nature06381, https://doi.org/10.1038/nature06381
11. Jinyao Tang, Hung-Ta Wang, Dong Hyun Lee, Melissa Fardy, Ziyang Huo, Thomas P. Russell, Peidong Yang. Holey Silicon as an Efficient Thermoelectric Material. Nano Letters 10, 4279–4283 (2010). doi:10.1021/nl102931z, https://doi.org/10.1021/nl102931z
12. A. Stranz, J. Kähler, A. Waag, E. Peiner. Thermoelectric Properties of High-Doped Silicon from Room Temperature to 900 K. Journal of Electronic Materials 42, 2381–2387 (2013). doi:10.1007/s11664-013-2508-0, https://doi.org/10.1007/s11664-013-2508-0
13. Jaeho Lee, Woochul Lee, Jongwoo Lim, Yi Yu, Qiao Kong, Jeffrey J. Urban, Peidong Yang. Thermal Transport in Silicon Nanowires at High Temperature up to 700 K. Nano Letters 16, 4133–4140 (2016). doi:10.1021/acs.nanolett.6b00956, https://doi.org/10.1021/acs.nanolett.6b00956
14. Jose Manuel Sojo-Gordillo, Carolina Duque Sierra, Gerard Gadea Diez, Jaime Segura-Ruiz, Valentina Bonino, Marc Nuñez Eroles, Juan Carlos Gonzalez-Rosillo, Denise Estrada-Wiese, Marc Salleras, Luis Fonseca, Alex Morata, Albert Tarancón. Superior Thermoelectric Performance of SiGe Nanowires Epitaxially Integrated into Thermal Micro-Harvesters. Small 19, 2206399 (2023). doi:10.1002/smll.202206399, https://doi.org/10.1002/smll.202206399
15. Kelin Zhu, Haoxiong Cui, Zhe Chen, Ting Zhang, Jun Chen, Xiaolin Wu, Guoan Cheng, Ruiting Zheng, Chungang Guo. Excellent thermoelectric properties of single-crystalline thin-walled porous silicon. Surfaces and Interfaces 110448 (2026). doi:10.1016/j.surfin.2026.110448, https://doi.org/10.1016/j.surfin.2026.110448