The lattice thermal conductivity of rough silicon nanowires is strongly suppressed below the smooth-boundary prediction, an effect attributed since 2008–2009 to phonon–surface roughness scattering and exploited for thermoelectrics. Martin, Aksamija, Pop, and Ravaioli [Phys. Rev. Lett. 102, 125503 (2009)] computed roughness-limited conductivity from a perturbative, semiclassical scattering rate built on a Gaussian autocovariance of the rough surface, and themselves flagged its unverified regimes: diameters below roughly 20 nm and low temperatures (where an explicit quantum treatment may be needed), and strong roughness Δ/L>1\Delta/L > 1 (where the Gaussian spectrum model may fail). Later work contested rather than confirmed the theory: exact atomistic calculations find the perturbative rates overpredicted by an order of magnitude, measured roughness spectra are non-Gaussian power laws, and the mechanism of the ultralow conductivity was still described as debatable in 2020. The open problem is a quantitatively validated account of roughness-limited phonon transport — a reconciled κ(T,D,Δ,L)\kappa(T, D, \Delta, L) prediction across the flagged regimes, or an established alternative mechanism — consistent with the available exact atomistic calculations and the measured conductivities.

Background

Phonons are the quantized lattice vibrations that carry heat in crystalline silicon; at room temperature and below, the lattice (phonon) contribution dominates the thermal conductivity κ of a silicon nanowire (NW), a single-crystal wire with diameter D of a few to a few hundred nanometers. When D is comparable to or smaller than the phonon mean free path (hundreds of nanometers in bulk Si at 300 K), collisions with the wire surface limit heat flow. For a perfectly diffuse boundary, every phonon is re-emitted with randomized direction, giving the Casimir limit, κ∝D\kappa \propto D, the classical upper bound on boundary-limited conductivity. Rough silicon nanowires made by chemical etching show κ far below this limit — as low as a few W/(m·K) versus 150 W/(m·K) in bulk — which made them celebrated thermoelectric materials (Hochbaum et al., Nature 2008; Boukai et al., Nature 2008) and made explaining the suppression a central question of nanoscale thermal transport.

Real rough surfaces are characterized statistically: the root-mean-square height fluctuation Δ (typically 0.5–5 nm for etched Si NWs) and a correlation length L along the wire axis (typically 1–10 nm), encoded in the autocovariance function of the height profile, whose Fourier transform gives the roughness power spectrum via the Wiener–Khinchin theorem. Martin, Aksamija, Pop, and Ravaioli [Phys. Rev. Lett. 102, 125503 (2009)] modeled the rough boundary as a small perturbation of the NW cross-section and computed, to first order in the perturbation, a phonon–surface roughness scattering rate proportional to that power spectrum, inserted it into a semiclassical Boltzmann transport calculation of κ(T,D,Δ,L)\kappa(T, D, \Delta, L) for Si NWs down to D≈20D \approx 20 nm, and reproduced the then-measured ultralow κ with Δ of a few nanometers. The authors stated the theory's validity limits explicitly: perturbation theory requires the boundary perturbation to remain small compared with the phonon energy, so for low-temperature phonons and diameters below about 20 nm "explicit quantum treatment may be required"; for strong roughness Δ/L>1\Delta/L > 1 the Gaussian autocovariance model may break down; reliable measurements of Δ and L for very small NWs were scarce; and the treatment assumed purely elastic boundary scattering with no phonon emission into the surrounding oxide or vacuum.

Subsequent work fragmented rather than settled these questions. Carrete et al. [Phys. Rev. B 84, 075403 (2011)] solved roughness scattering exactly (non-perturbatively) for atomistic, atomic-height roughness and found the perturbative approach overpredicts scattering rates by roughly an order of magnitude, while the Casimir limit remains a good approximation even at 2.22 nm diameter for shallow roughness. Luisier [J. Appl. Phys. 110, 074510 (2011)] performed fully quantum atomistic transport (atomistic force constants with a wave-function matching formalism) for rough sub-5-nm Si NWs, attributing the κ degradation mainly to phonon band-structure changes rather than to a scattering rate at all. Lim et al. [Nano Lett. 12, 2475 (2012)] measured the actual height power spectra of individual roughened wires and found power-law (not Gaussian) spectra, with κ correlated to the spectra and sitting below the Casimir limit for reasons the authors state remain unknown. Phonon Monte Carlo with exponentially correlated roughness (Maurer et al., Appl. Phys. Lett. 106, 133108 (2015)) and non-equilibrium Green's function (NEGF) quantum-transport calculations (Abhinav and Muttalib, J. Phys. Commun. 3, 105010 (2019)) each reproduce parts of the data with yet further model choices, and Yang et al. [Chin. Phys. B 29, 086502 (2020)] argue the ultralow κ "can not be explained by the classical phonon–surface scattering mechanism", proposing phonon scattering by surface bond-order imperfection instead. No published work reconciles these competing accounts into a validated theory spanning the regimes the 2009 authors flagged.

Problem Statement

Establish a quantitatively validated account of surface-roughness-limited phonon heat conduction in thin silicon nanowires in the regimes that Martin et al. (Phys. Rev. Lett. 102, 125503, 2009) themselves flagged as unverified — diameters below roughly 20 nm, low temperatures (below the 25–350 K range over which the 2009 theory was numerically validated), and strong roughness Δ/L>1\Delta/L > 1 with non-Gaussian roughness spectra. Concretely: (i) resolve the quantitative discrepancy between the perturbative roughness scattering rates of the 2009 theory and exact atomistic calculations, which find those rates overpredicted by about an order of magnitude (Carrete et al., 2011); (ii) determine whether a single scattering-based mechanism — with the measured power-law roughness spectra rather than a Gaussian autocovariance — or an alternative mechanism such as surface bond-order imperfection and band-structure modification (Luisier, 2011; Yang et al., 2020) dominates the observed suppression; and (iii) deliver a predictive description of κ\kappa across those regimes that is quantitatively consistent with the published measurements on individual rough wires (Hochbaum et al., 2008; Li et al., 2003; Lim et al., 2012) and, where the same regimes are covered, with the exact atomistic calculations (Carrete et al., 2011); the wire-array measurements (Feser et al., 2012) and the 0.3–5 K corrugated-wire conductance data (Blanc et al., 2013) provide important context but are not treated as strict acceptance criteria, since they involve array coupling and coherent wave effects beyond the source theory's single-wire elastic-scattering setting. The single overarching objective is a validated theory of the ultralow thermal conductivity of rough Si nanowires; items (i)–(iii) are all mandatory and jointly supporting results of that one objective, and the completion boundary follows the alternative stated in item (ii): if (ii) finds that a scattering-based mechanism dominates, (iii) takes the form of a predictive κ(T,D,Δ,L)\kappa(T, D, \Delta, L) description; if (ii) instead establishes an alternative mechanism, (iii) takes the form of a predictive description in the variables appropriate to that mechanism, quantitatively accounting for the same observations. Item (i) is mandatory in both branches; in the alternative-mechanism branch it reduces to establishing quantitatively why the perturbative roughness-scattering rate is incorrect. Here "quantitatively validated" means agreement with the independent exact calculations and the measurements within their stated uncertainties.

References

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