Classical-to-Quantum Crossover in 2D TMD Field-Effect Transistors: A First-Principles Study via Sub-10 nm Channel Scaling Beyond the Boltzmann Tyranny

TL;DR

The paper tackles the challenge of sub-10 nm scaling in monolayer TMD FETs by examining the quantum–classical crossover between tunneling and thermionic transport. Using a first-principles framework that combines VASP-based DFT, NEGF-DFT transport (NanoDCAL), and an effective gate model within the Landauer formalism, it characterizes how channel length and temperature govern transport, OFF/ON currents, and subthreshold swing. A Bohr correspondence-based analysis shows that the Landauer current asymptotically matches Richardson thermionic emission in the long-channel limit, and the authors introduce a quantum–classical competition parameter ζ to quantify the crossover. The study finds that tunnel-dominated transport persists for short channels (≤9 nm) with a sensitivity to J_OFF, while longer channels (≈10 nm) approach thermionic behavior with SS nearing the Boltzmann tyranny limit adjusted by gate efficiency; a near-optimal channel length around 10 nm emerges for robust 2D FET performance, with clear operating-temperature and gate-voltage windows identified for practical devices.

Abstract

Scaling field-effect transistors (FETs) into the sub-10-nm regime fundamentally alters the transport mechanism, challenging long-standing design rules. This study investigates monolayer TMD FETs with channel lengths from 12 nm to 3 nm, quantifying the competition between semiclassical thermionic current and quantum tunneling. We show that quantum transport, as described by the Landauer formula, asymptotically approaches classical thermionic emission in the long-channel and high-temperature limit, in accordance with Richardson law. A competition parameter $ζ$ cleanly delineates the semiclassical-to-quantum transition, and two characteristic temperatures emerge: $T_{op}$ (minimizing $J_{OFF}$ and $T_{c}$ (thermionic onset). For $L_{ch}<9$ nm, $T_{op}<300$ K and $J_{OFF}$ is tunneling-dominated; the 3 nm device remains tunneling-dominated up to 500 K and achieves a subthreshold swing overcoming Boltzmann tyranny via the steep slope of $τ(E)$. However, the short-channel effect also generates leakage current and makes the transistor difficult to turn off. For $L_{ch} \geq 9$ nm, $T_{op}>300$ K and $J_{OFF}$ is thermionic-dominated, and the subthreshold swing approaches Boltzmann tyranny scaled by $α_{in}}$. Consequently, the ideal channel length for 2D FETs is $L_{ch} \approx 10$ nm. These results provide criteria for selecting the optimal operating temperature and gate-voltage windows in miniaturizing 2D FETs, and pinpoint the crossover at which quantum tunneling current becomes comparable to semiclassical thermionic emission.

Figures (7)

• Figure 1: (a) Schematic of the Pt–WSe$_2$–Pt nanojunction. (b) Transmission coefficients $\tau(E)$ computed from NEGF-DFT (NanoDCAL) for $L_{\mathrm{ch}} = 3, 6, 9,$ and $12$ nm on linear (left axis) and $\log_{10}$ (right axis) scales. (c) Schematic of the gate architecture of the nanojunction as a field-effect transistor with an EOT of 8 Å (dielectric constant 3.9). (d) The upper panel shows the transmission band gap $(E_V , E_C)$ (vertical blue bars) shifted by the gate voltage $V_g$, with the reference energy being the chemical potential $\mu = 0$ at $V_g=0$ (the black horizontal line). The lower panel illustrates the correlation between $V_g$ and $V_{\mathrm{G}}^{\mathrm{eff}}(V_g)$. Refer to Ref.[FET-AlN].
• Figure 2: (a) presents $J/T$ on a $\log_{10}$ scale as a function of temperature ranging from 100 to 500 K plotted against a $1/T$ scale. Note that here $J = J_{\text{min}}$ denotes the current density at $V_g \approx 0.9$ V, which causes the chemical potential $\mu(V_g)$ to align with the center of the transmission band gap. Symbols for each $L_{\mathrm{ch}}$ are color-encoded by $\zeta=(I_{SC}-I_{QM})/({I_{SC}+I_{QM}})$, illustrating the competition between quantum tunneling and thermionic emission currents. The slopes of the SC regime (red) correspond to the work function $W = \chi - \mu(V_g)$. (b) displays $\tau_{\text{min}}$ (black solid circles) on the left axis and $J_{\text{min}}$ on the right axis for temperatures of $T=100$ K (blue open circles) and $T=300$ K (red open squares). (c) displays the integrands, $\left[{(f^{R}-f^{L})\tau(E)}\right ]$, of the Landauer formula as a function of $E$ for $T=100$ K (black line) on the left axis and $T=300$ K (red line) on the right axis.
• Figure 3: Contour plots of the quantum-classical competition parameter $\zeta$ are presented over a range of temperature (100–500 K) and gate voltage $V_g$ (–1.5 to 1.5 V) for Pt–WSe$_2$–Pt thermoelectric junctions at $V_{ds}=50$ mV with channel lengths $\mathrm{L}_{\text{ch}}=$ (a) 3 nm, (b) 6 nm, (c) 9 nm, and (d) 12 nm. The rivalry between quantum transport in blue and semi-classical thermionic current in red is represented by $-1 \leq \zeta \leq 1$.
• Figure 4: (a) shows the transmission coefficient $\tau(E)$ (right vertical axis) as a function of energy with a band gap and illustrates how the gate voltage $V_g$ (left vertical axis), associated with an effective gate voltage $V_{\text{G}}^{\text{eff}}$ (top horizontal axis), shifts the chemical potential $\mu(V_g)$ (bottom horizontal axis) relative to the band gap. (b) shows the current density $J[\mu(V_g)]$ for Pt–WSe$_2$–Pt junctions with channel lengths $L{\mathrm{ch}} = 3$ nm (blue dashed line), 6 nm (green dash–dotted line), 9 nm (orange dotted line), and 12 nm (red solid line), evaluated at $V_{ds} = 50$ mV and $T = 300$ K. (c) shows the subthreshold swing, SS$[\mu(V_g)]$, plotted on a $\log_{10}$ scale on the left vertical axis for the same set of junctions, using the color and line-style convention of panel (b). The right vertical axis displays the normalized SS, $\text{SS}\times(\alpha_{\text{in}}/\text{BT})$, in linear scale. The black arrow together with the gray horizontal line indicates the Boltzmann tyranny (BT).
• Figure 5: Three-dimensional surface plots of the current density $\log_{10}J(V_g, T)$ for Pt–WSe$_2$–Pt nanojunctions as a function of gate voltage $V_g$ (from $–1.5$ to $1.5$ V) and temperature $T$ (from $100$ to $500$ K), evaluated at a fixed drain-source bias of $V_{\mathrm{ds}} = 50$ mV. Results are presented for channel lengths of (a) 3 nm, (b) 6 nm, (c) 9 nm, and (d) 12 nm.