Revealing quantum effects in highly conductive δ-layer systems

The continued downscaling of field-effect transistors faces technological and fundamental challenges, motivating exploration of beyond-Moore and beyond-CMOS systems that require highly conductive yet strongly confined material platforms. Atomically precise planar δ-doped layers in semiconductors (e.g., Si:P) enable ultrahigh current densities and precise electrostatic control at the nanoscale. Recent ARPES studies reported shallow sub-bands near the Fermi level (including a third, 3Γ, sub-band) that standard closed-system or periodic-boundary theoretical treatments failed to explain without ad hoc modifications (e.g., artificially increased dielectric constant). Moreover, closed-system methods cannot directly compute current from quantum flux and often rely on semiclassical approximations. The research question addressed here is whether a fully quantum-mechanical open-system approach can (i) explain the origin of the shallow conducting sub-bands observed experimentally and (ii) accurately predict macroscopic conductive properties such as sheet resistance across a wide range of δ-doping profiles.

Experimental ARPES work (e.g., Mazzola et al., Holt et al.) observed quantized conduction and valence states, including three sub-bands (1Γ, 2Γ, 3Γ) in high-density Si:P δ-layers, and an A-manifold for thicker layers, findings that strained previous theories unless the Si dielectric constant was unrealistically adjusted. Prior computational studies employed effective mass, tight-binding, or DFT with either closed (Dirichlet) boundaries or periodic boundaries. However, these approaches generally do not yield current directly from quantum flux; periodic approaches assume unit transmission in ideal wires and neglect impurity/roughness scattering, while closed systems often require Drude-Sommerfeld-type assumptions (j proportional to n). Thus, there is a need for open-system treatments that capture transmission, scattering, and self-consistent electrostatics without double-counting screening.

The study employs a charge self-consistent open-system non-equilibrium Green's function (NEGF) Keldysh formalism solving the coupled Poisson–Schrödinger equations in a 2D cross-section (xz-plane) with open boundary conditions modeling semi-infinite source/drain contacts. The device channel (length L = 50 nm along x) comprises a lightly doped Si cap, a highly P-doped δ-layer, and a lightly doped Si body; the y-direction is treated as infinite with plane-wave solutions, enabling analytical integration over k_y into effective 2D Fermi-Dirac distributions. Simulations are performed at T = 4 K. Typical parameters include: Si:P δ-layer sheet donor densities around ND = 1.0 × 10^14 cm^-2 with thickness t = 0.2 nm (monolayer approximation; nearest-neighbor Si spacing ≈ 0.23 nm), and Si cap/body acceptor doping NA = 1.0 × 10^17 cm^-3. The framework computes total DOS, LDOS(E,z), and transmission per mode T(E) via standard NEGF relations (mode representation). To validate macroscopic transport, a heuristic elastic defect scattering model is included: geometry scattering is inherently captured by the quantum treatment; additional defect scattering (vacancies, dislocations, impurities) is introduced via coherence-breaking scatterers with linear density ν = N/L per mode, yielding an effective transmission along the channel. Inelastic scattering is neglected (phase relaxation length exceeds mean free path at low T in Si:P δ-layers). Numerically, the Contact Block Reduction (CBR) method and an open-system predictor–corrector scheme with Anderson mixing are used. The electronic structure uses a single-band (Γ-valley) effective mass approximation in a jellium Poisson model with Hartree plus exchange-correlation (Perdew–Zunger). Bulk Si parameters are used (mt = 0.98 m, ml = 0.19 m, εs = 11.7). No fitting parameters are used in the self-consistent calculations; the sheet conductance model contains a single parameter (linear defect density).

• The open-system NEGF approach explains the shallow conducting sub-bands observed by ARPES in Si:P δ-layers, including the 3Γ sub-band at high sheet doping, without artificial adjustment of the dielectric constant.
• DOS and LDOS reveal two occupied sub-bands (1Γ and 2Γ) at approximately −190 meV and −20 meV for ND ≈ 1.0 × 10^14 cm^-2 and t = 0.2 nm; these occupied states are largely independent of the δ-layer depth D from the surfaces, consistent with ARPES.
• LDOS(E,z) shows spatially distinct electron layers: the lowest-energy mode is centered at the δ-plane (z = 0), while higher-energy occupied modes reside off-center (≈ ±1 nm), forming a characteristic "quantum menorah" pattern.
• Sub-band count and energy splitting depend strongly on δ-doping profile: • At fixed thickness t, increasing sheet doping increases both the number of conducting modes and their energy splitting (progression from 1Γ to 1Γ+2Γ to 1Γ+2Γ+3Γ). • At fixed sheet doping, increasing thickness t increases the number of modes but decreases their energy splitting, especially between 1Γ and 2Γ for thicker layers (e.g., t ≈ 4 nm), suggesting previously reported <50 meV splittings occur for relatively thick (>5 nm) δ-layers.
• Current sharing among sub-bands differs from electron population: for t = 0.2 nm and ND = 1.2 × 10^14 cm^-2, electrons distribute ≈90% (1Γ) and 10% (2Γ) while current contributions are ~50% each; for t = 4 nm at the same sheet doping, electron distribution ≈35% (1Γ), 45% (2Γ), 20% (3Γ) with current contributions ≈10%, 25%, and 65%, respectively.
• Effective electron cloud thickness exhibits two regimes versus sheet doping and δ-layer thickness: for ND ≤ 10^13 cm^-2 (weak confinement), thickness is set by doping and nearly independent of t; for ND ≥ 5 × 10^13 cm^-2 (strong confinement), thickness is governed by t and becomes nearly independent of doping.
• The approach reproduces measured sheet resistance values across multiple experimental datasets for a wide range of donor densities when including elastic defect scattering.
• The sheet resistance depends significantly on δ-layer thickness at fixed sheet charge density; sharper (thinner, higher-density) profiles lead to increased quantum-mechanical resistivity compared with semiclassical j ~ n expectations.
• Strong deviations from semiclassical j ∝ n are found at high donor densities, underscoring the necessity of fully quantum open-system treatments.

By employing an open-system NEGF framework that directly computes transmission and quantum flux, the study resolves key discrepancies between experiments and prior closed-system theories. It accounts for the observed shallow sub-bands (including 3Γ) without ad hoc dielectric adjustments and explains their relative insensitivity to δ-layer depth while capturing strong dependence on sheet doping and thickness. Spatial separation of occupied states into distinct electron layers with different average energies clarifies how current can be carried predominantly by higher-energy, outer sub-bands even when the lowest-energy sub-band holds most electrons. The identification of weak and strong confinement regimes (set by doping versus thickness) provides design rules for tuning electron cloud thickness. The significant dependence of sheet resistance on thickness at fixed sheet density, and the demonstrated deviation from j ~ n, highlight the impact of quantum confinement and mode-dependent transmission on macroscopic transport. These insights are relevant for engineering δ-doped platforms in quantum devices, ultra-scaled transistors, and for selective energy filtering concepts in thermoelectrics.

An open-system quantum transport treatment of semiconductor δ-layer systems reveals a quantized sub-band structure (“quantum menorah”) with spatially separated electron layers having distinct average energies. The model explains shallow sub-bands seen in ARPES and reproduces experimentally measured sheet resistances over wide doping ranges. Two central effects are established: (1) highly nonlinear dependence of electron cloud thickness on the δ-doping profile, and (2) increased quantum-mechanical resistivity for sharper δ-layer profiles. The ability to selectively filter carriers by energy, combined with high conductivity, suggests opportunities for thermoelectric device concepts. The findings emphasize the necessity of fully quantum, open-system treatments for highly confined conductive systems and inform the design of qubits, ultra-scaled FETs, and beyond-CMOS devices.

• Inelastic scattering is neglected based on low-temperature conditions where phase relaxation length exceeds mean free path; results may differ at higher temperatures or inelastic-dominated regimes.
• Single-band (Γ-valley) effective mass approximation is used to reduce computational burden; multi-valley effects and full bandstructure could refine quantitative details for thicker or different δ-layer configurations.
• Jellium approximation for dopant charge and use of bulk Si parameters may omit atomistic disorder and valley splitting nuances present in tight-binding/DFT.
• Defect scattering is modeled heuristically via a linear density of coherence-breaking scatterers; detailed microscopic scattering mechanisms (e.g., roughness, impurity distributions) are not explicitly resolved.