On the electronic and vibrational dimensionality of nanometer-scale silicon structures

Abstract: We discuss the problem of assessing the electronic and vibrational dimensionality of a semiconductor nanostructure: How thin and/or wide must a nanostructure be in order to induce electron and phonon confinement? Clarifying the physical justification for common criteria found in the literature, we view the electron coherence length (defined as the electron and phonon inelastic mean free path) as their field of view' and argue (or, better yet,speculate') that this sets the important length scale. Considering the example of Si nanosheets at room temperature, and drawing from results found in the literature, we estimate that the critical length below which electrons are subject to quantum confinement is of the order of (or smaller than) 8 nm, when their coherence length is determined by energy losses to phonons and remote phonons in gated structures. On the contrary, no single length-scale can be given for phonons: Taking their coherence length as determined by scattering with electrons and anharmonic three-phonon processes, short wavelength acoustic and optical phonons may be confined only by structures as small as 10 nm. Long-wavelength acoustic phonons, instead, may exhibit a coherence length of the order of 1 micrometer, so that they may be confined over much larger distances.

• The paper presents a new criterion for determining electronic and vibrational dimensionality in Silicon Nano structures, arguing that coherence length is the key parameter over confinement length.
• The paper's results provide quantitative insights for Si nanosheets with certain geometries (1.6 nm thickness and 12 nm width), suggesting that these dimensions support a 2DEG.
• The authors argue that both phonon dimensionality and electron dimensionality, in silicon nanostructures, control the overall phonon dimension.

The question of when a semiconductor nanostructure should be treated as electronically or vibrationally low-dimensional is usually settled by appeal to intuition: compare the confinement length to a mean free path, a de Broglie wavelength, an effective Bohr radius, or the thermal energy $k_{\rm B}T$. As Fischetti, Nielsen, and Chen point out in "On the electronic and vibrational dimensionality of nanometer-scale silicon structures" (2608.18277), these criteria are applied inconsistently across the literature — some authors treat electrons as confined (1D) in Si nanowires up to 30 nm wide (2608.18277), while others treat them as bulk 3D particles in InAs nanowires of comparable cross section despite the much longer mean free path in InAs. The paper's contribution is to place the criterion on a firmer physical footing and to supply quantitative estimates for the case of greatest technological urgency: Si nanosheets 1.6 nm thick and 12 nm wide in double-gate structures with SiO$_2$/HfO$_2$ stacks.

Coherence length as the relevant scale

The authors argue that the correct length scale is not the elastic mean free path but the phase-coherence (dephasing) length $\lambda_\phi$, set by inelastic processes — electron-phonon scattering for electrons; anharmonic three-phonon decay and phonon absorption by electrons for phonons. Elastic scattering at rough interfaces or ionized impurities does not destroy phase information except after a configuration average. The physical picture is that of a "field of view": a particle can only form a standing wave between confining boundaries if it retains phase memory long enough to travel between them coherently.

This heuristic is supported by a toy calculation: an electron in a hard-wall box perturbed by a time-dependent potential mimicking electron-phonon coupling acquires a collisional broadening $\Delta E_\mu = \hbar/\tau_{\phi,\mu}$. Requiring this broadening to remain smaller than the unperturbed subband energy yields $\lambda_\phi > 2L/\pi \sim L$, equivalently that the rate at which the electron strikes the walls must exceed its decoherence rate. The same conclusion emerges from a spectral-function analysis: if the imaginary part of the self-energy approaches the subband spacing, the discrete spectrum merges into a renormalized continuum indistinguishable from bulk. Notably, this rehabilitates the common $k_{\rm B}T$ criterion, but only with $k_{\rm B}T$ replaced by the collisional broadening ${\rm Im}\,\Sigma_0$. The authors are candid that this treatment is an "educated guess" rather than a rigorous derivation; a fully self-consistent NEGF/DFT (or GW) treatment with complete inelastic electron-phonon interaction would be required but is described as daunting.

Electron dimensionality in Si nanosheets

Applying the criterion to a 1.6 nm-thick (100) Si nanosheet, using scattering rates from the authors' companion transport study, gives electron-phonon rates of roughly $3\times10^{13}$ s$_2$0 (clamped boundary conditions, CBCs) to $_2$1 s$_2$2 (free-standing boundary conditions, FSBCs). With a ground-state group velocity along the quantization direction of about $_2$3 cm/s, the coherence length is 3–8 nm — comfortably larger than the sheet thickness, so quantization along $_2$4 holds and the electrons form a 2DEG. Bulk-3D scattering rates would give similar or longer coherence lengths, reinforcing the conclusion.

A critical thickness estimate follows from assuming the electrical quantum limit, where rates scale as $_2$5: equating the decoherence rate to the wall-hitting frequency yields

$_2$6

in rough agreement with independent estimates (~7–8 nm) from requiring the unprimed/primed subband splittings to exceed $_2$7. The analogous critical width for a 1.6 nm-thick sheet comes out near 8 nm as well (about 6 nm for (011)-oriented edges). Since the 12 nm width exceeds this, and since remote interfacial polar phonon (IPP) scattering pushes rates above $_2$8–$_2$9 s$_2$0, the authors conclude that electrons in such nanosheets constitute a 2DEG, not a 1DEG — a direct rebuttal of studies treating comparable or larger structures as fully confined.

A corollary concerns transport formalism itself. Projecting the coherence length onto the transport direction gives values of order 0.1–0.3 nm near the source and ~0.2 nm for hot carriers near the drain under high bias. Electrons therefore lose phase memory over distances far shorter than any channel length, so semiclassical transport suffices. The authors advance the deliberately controversial claim that purely ballistic transport at room temperature is effectively unreachable in conventional devices, because irreversible Coulomb interactions — including with contact electrons — join phonons in destroying coherence; they note no direct observation of quantum-interference signatures in conventional room-temperature devices, in contrast to cryogenic GaAs experiments.

Phonon dimensionality: no single scale

For phonons the situation is qualitatively different because the coherence length depends strongly on wavelength and branch. Boundary scattering, though dissipative after configuration averaging, remains elastic and is excluded from the dephasing budget; only three-phonon anharmonic decay and absorption by electrons count. The latter term scales with the free-electron density, which acts as the bottleneck — without carriers, phonons cannot decay through this channel.

Combining experimental acoustic-phonon lifetimes in freestanding Si membranes (tens to hundreds of ps even at high frequency), Debye-model estimates of the confined phonon densities ($_2$1 cm$_2$2, $_2$3 cm$_2$4), and branch-resolved electron-phonon rates, the authors obtain effective coherence lengths for long-wavelength acoustic phonons ranging from 50 nm to as much as 10 μm. Anharmonic decay dominates over electron-induced decay, increasingly so at device operating temperatures (80–350 K), since the phonon density grows as $_2$5 and the three-phonon rate faster than $_2$6, while electron-acoustic-phonon rates grow only linearly. The consequence is stark: short-wavelength acoustic and optical phonons require structures below roughly 10 nm to be confined, whereas long-wavelength acoustic modes may be confined laterally in sheets narrower than ~50 nm, making some branches effectively one-dimensional. The authors propose treating this mixed regime by analogy with finite-barrier quantum wells, with an effective barrier height set by the acoustic mismatch at the Si/insulator interface.

Limitations and open questions

The paper is explicit about its own status. The central criterion — coherence length exceeding confinement length — is argued qualitatively and via toy models, not derived rigorously; the intermediate regime (coherence length within a factor of order unity of the confinement length) is left deliberately unresolved. Several quantitative steps rest on scaling assumptions: electron-phonon rates assumed to scale as $_2$7 or $_2$8, phonon densities estimated within a Debye approximation acknowledged to underestimate the true density because dispersion flattening is ignored, and 1D coherence lengths extrapolated from nanowire calculations without accounting for phonon confinement along the thickness. The self-consistency question — whether to use 2D-confined or bulk coherence lengths — is resolved by an internal-consistency conjecture rather than proof. Whether boundary scattering contributes any genuine dephasing beyond configuration averaging, including claimed multi-phonon contributions to Kapitza conductance, is left open, as is the full NEGF-level treatment of the crossover between confined and bulk spectra.

Conclusion

The paper establishes that the electronic and vibrational dimensionalities of a nanostructure are separate questions with different answers. For electrons in Si nanosheets at 300 K, a single critical scale of approximately 8 nm (or smaller once IPP scattering is included) governs both thickness and width, so realistic 1.6 nm × 12 nm devices host a 2DEG treated semiclassically along the channel. For phonons, no single scale exists: optical and short-wavelength acoustic modes confine below ~10 nm, while long-wavelength acoustic modes remain coherent over micrometers and can be confined over far larger dimensions. These estimates are specific to Si at room temperature, but the qualitative framework — inelastic coherence length as the decisive scale — should transfer to other semiconductor nanostructures, and the paper makes clear that putting that framework on rigorous footing remains an open problem.