For atomically thin (2D) semiconductor channels thinner than about 4 nm, finite-element electrostatics showed that the natural scale length Λ\Lambda controlling short-channel effects is set by the physical gate-oxide thickness, Λ≈γ tox\Lambda \approx \gamma\, t_{ox} with γ≈4/3\gamma \approx 4/3 for symmetric double gates, rather than by the classical expression Λ∼ϵchtchtox/ϵox\Lambda \sim \sqrt{\epsilon_{ch} t_{ch} t_{ox}/\epsilon_{ox}}. That theory was established only in an idealized limit: subthreshold operation with no mobile channel charge, Ohmic contacts, quasi-static electrostatics, and two extreme contact geometries. The open problem is how this scale-length picture — the proportionality constant γ\gamma, the tch≈4t_{ch} \approx 4 nm crossover below which Λ\Lambda saturates, the minimum channel length Lmin≈3.5ΛL_{min} \approx 3.5\Lambda, and the resulting requirement of gate oxides thinner than about 1.1 nm for sub-5 nm channels — generalizes to realistic self-consistent conditions: finite carrier charge and screening under bias, and contacts beyond the Ohmic assumption and the two extreme geometries, with oxide thinning bounded by the source's remark that it "will be eventually limited by the gate leakage current". A direct successor theory (2022) extended the framework to channel extensions and junction boundary coupling but stayed in the charge-free subthreshold regime; experimental sub-nm-equivalent-oxide-thickness dielectrics with low leakage now exist, but the self-consistent finite-charge scale-length theory remains absent.

Background

A field-effect transistor (FET) is a switch in which a gate electrode controls current flow through a semiconductor channel; in modern chips the channel is a few tens of nanometers long. When the channel length LL becomes comparable to an electrostatic decay length, the drain electrode electrostatically lowers the potential barrier near the source even without the gate's consent. These short-channel effects (SCEs) include drain-induced barrier lowering (DIBL, the shift of the threshold voltage with drain bias), subthreshold-swing degradation (the subthreshold swing SS is the gate voltage needed per decade of current reduction in the off state), and threshold-voltage roll-off. Classical electrostatic theory summarizes the tolerance to scaling in a natural scale length Λ\Lambda: the channel can be shortened only to a small multiple of Λ\Lambda (about 3Λ3\Lambda–3.5Λ3.5\Lambda) before SCEs become unacceptable, and DIBL grows roughly as exp⁡(−L/Λ)\exp(-L/\Lambda). For silicon-on-insulator and double-gate transistors, standard models (Yan, Suzuki, Frank and co-workers in the 1990s) give Λ∼ϵchtchtox/ϵox\Lambda \sim \sqrt{\epsilon_{ch} t_{ch} t_{ox}/\epsilon_{ox}}, where tcht_{ch} is the semiconductor channel thickness, toxt_{ox} the gate-oxide thickness, and ϵch\epsilon_{ch}, ϵox\epsilon_{ox} the dielectric permittivities; thinner bodies and thinner oxides were therefore the historical levers. Two-dimensional (2D) semiconductors such as monolayer MoS2_2 (thickness ≈0.6\approx 0.6 nm) invert the usual hierarchy: the channel is intrinsically thinner than the gate oxide, tch<toxt_{ch} < t_{ox}, an extreme not covered by the classical formulas. Fringing fields through the oxide (fringing-induced barrier lowering, FIBL) then dominate the drain's influence on the channel barrier. Suryavanshi, English, Wong, and Pop (arXiv:2105.10791, 2021; the work stayed an arXiv preprint) used finite-element solutions of Laplace's equation to show that for double-gate (DG) 2D FETs with tch<4t_{ch} < 4 nm the scale length saturates at Λ≈γ tox\Lambda \approx \gamma\, t_{ox} with γ≈4/3\gamma \approx 4/3, independent of the oxide permittivity ϵox\epsilon_{ox} and of tcht_{ch}, so the classical square-root expression fails and only the physical oxide thickness (not the effective oxide thickness, EOT — the SiO2_2-equivalent thickness that accounts for a dielectric's permittivity) matters. In asymmetric single-gate (SG) devices, fringing through the surrounding dielectric adds a second-order correction, but Λ\Lambda is ultimately bounded below by the DG value. Comparing with double-gated monolayer-MoS2_2 experimental data (self-aligned 10 nm top-gate and double-gate devices from the same group), the work extracted Lmin≈3.5ΛL_{min} \approx 3.5\Lambda and concluded that a 5 nm-channel 2D FET with negligible SCEs requires tox<1.1t_{ox} < 1.1 nm — a thickness at which gate leakage through the oxide becomes the binding constraint. The theory, however, was derived with deliberately idealized assumptions: quasi-static electrostatics solved in the subthreshold regime with zero mobile charge in the channel (no self-consistent carrier screening and no quantum capacitance — the finite channel density of states that limits how much charge the gate can induce), Ohmic source/drain contacts, and only two extreme contact architectures (edge 'tall' contacts covering the oxide sidewall, and 'short' contacts touching only the channel). A successor paper by Gilardi, Bennett, Yoon, Pop, Wong, and Mitra (IEEE Trans. Electron Devices 69, 5302, 2022) extended the classical scale-length theory to low-dimensional FETs by introducing boundary-coupling parameters that capture the electrostatic role of source/drain channel extensions, reducing subthreshold-slope prediction errors from more than a factor of two to under 5% — but it remains a subthreshold, charge-free electrostatics theory.

Problem Statement

Establish how the electrostatic scale-length theory of two-dimensional field-effect transistors — the linear law Λ≈γ tox\Lambda \approx \gamma\, t_{ox} with γ≈4/3\gamma \approx 4/3 for symmetric double gates, the channel-thickness crossover at tch≈4t_{ch} \approx 4 nm below which Λ\Lambda saturates and becomes independent of ϵox\epsilon_{ox} and tcht_{ch}, and the minimum channel length Lmin≈3.5ΛL_{min} \approx 3.5\Lambda — generalizes beyond the idealized limit in which it was derived (subthreshold operation, zero mobile channel charge, Ohmic contacts, quasi-static electrostatics, and the two extreme short/tall contact geometries). Specifically, the assumptions to be relaxed are the zero-mobile-charge (subthreshold) assumption and the Ohmic-contact and two-extreme-geometry assumptions, while quasi-static electrostatics may be retained; these three relaxations — finite mobile-carrier charge (with quantum capacitance and screening under gate and drain bias, treated self-consistently), contacts beyond Ohmic injection, and contact geometries beyond the two extreme short/tall cases — are jointly required, so a result that relaxes only one of them is partial. Within that scope, determine how all three of γ\gamma, the ≈4\approx 4 nm crossover thickness, and LminL_{min} shift for realistic 2D FETs: a complete result reports the generalized dependence of Λ\Lambda — equivalently of γ\gamma, the crossover thickness, and LminL_{min} — on charge and contact conditions over a stated regime of validity, or shows that no such generalized dependence exists, whereas extracting the shifts for a single device or geometry alone does not complete the task. Alternatively, demonstrate that the linear picture fails — that is, that the linear form Λ≈γ tox\Lambda \approx \gamma\, t_{ox} or the saturation and crossover themselves cease to hold — and identify the replacement relation; a result in which the linear form survives while γ\gamma or the crossover thickness merely acquires a charge or contact dependence stays in the affirmative branch and must report those shifts. The generalized framework must remain valid in the oxide-thickness design regime the source identifies: the generalized Λ(tox)\Lambda(t_{ox}) must stay finite and must reduce to the source's charge-free law Λ≈(4/3) tox\Lambda \approx (4/3)\,t_{ox} in the zero-charge limit; a 5 nm channel requires tox≲1.1t_{ox} \lesssim 1.1 nm, and continued oxide thinning toward that limit "will be eventually limited by the gate leakage current".

References

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