Revealing quantum effects in highly conductive δ-layer systems

The continued downscaling of field-effect transistors faces technological and fundamental challenges, motivating the search for alternative, energy-efficient computing systems. Both conventional CMOS and many new transistor technologies require high drive current, low off current, and scalability. At the material level, this translates to the need for a highly conductive, highly confined material system allowing precise current control. High conductivity is crucial for fast transistor charging/discharging, while high electrostatic confinement in nanoswitches is needed due to high nanodevice density and the need for individual channel control. Atomically precise techniques for producing planar dopant-based structures in semiconductors have led to interest in δ-layer structures, where dopant atoms form mono- or multi-atomic layers with densities exceeding solid solubility limits. These structures exhibit very high current densities, showing potential for beyond-Moore applications. Si:P δ-layer systems, consisting of a thin, highly phosphorus-doped 2D sheet embedded in lightly doped silicon, have been actively studied for decades for applications in quantum computing and advanced microelectronic devices. Recent angle-resolved photoemission spectroscopy (ARPES) measurements revealed shallow conductive states determining the system's conductive properties, showing conduction and valence band quantization near the Fermi level and the presence of three sub-bands (1Γ, 2Γ, and 3Γ) that existing theoretical models couldn't fully explain without adjusting the silicon dielectric constant. Previous computational studies using closed-system or periodic boundary conditions couldn't directly extract conductive properties from the quantum-mechanical flux, necessitating additional approximations like the Drude-Sommerfeld model. This paper addresses these limitations by employing an open-system approach.

Extensive research has been conducted on the electronic structure and conductive properties of Si:P δ-layer systems. Early studies utilized effective mass, tight-binding, or density functional theory formalisms, often employing either closed-system approaches with Dirichlet boundary conditions or periodic boundary conditions. However, these methods lacked the ability to directly compute conductive properties from the quantum-mechanical flux, requiring additional classical or semi-classical approximations. The use of periodic boundary conditions, while allowing conductivity extraction in idealized cases, neglected the influence of charged impurities and interface roughness. The limitations of these previous approaches highlight the need for an open-system treatment capable of directly calculating conductive properties and simulating more complex structures.

This study employs an open-system non-equilibrium Green's function (NEGF) Keldysh formalism to systematically investigate the conductive properties of semiconductor δ-layer systems at low temperatures. The computational model consists of semi-infinite source and drain regions (represented by NEGF open boundary conditions) contacting a channel composed of lightly doped Si cap, a highly P-doped layer, and a lightly doped Si body. The source and drain extend the channel to infinity along the x-axis, maintaining the same z-axis properties. The y-direction is assumed infinite with a flat electrostatic potential, allowing for a 2D solution of the Poisson-open-Schrödinger equation. The Schrödinger equation is analytically integrated over the y-axis momentum, resulting in effective 2D Fermi-Dirac distribution functions. All simulations assume a temperature of 4K. The density of states (DOS) is investigated as a function of layer depth (D) from the Si cap/body surfaces, focusing on symmetric configurations. The doping density in the δ-layer is given in cm⁻², and a constant acceptor doping density of 1.0 x 10¹⁷ cm⁻³ in the Si body/cap is assumed. The open-system quantum-mechanical treatment provides a continuous DOS function, with sharp peaks corresponding to propagation modes due to z-axis confinement. The analysis also includes the local density of states (LDOS), revealing the spatial distribution of electrons. A heuristic elastic defect scattering model is incorporated to account for meso- and macroscopic scale effects, neglecting inelastic scattering due to the longer phase-relaxation length compared to the mean free path at low temperatures. Two types of elastic scattering are considered: geometry scattering (accounted for in the charge self-consistent quantum transport framework) and defect scattering (simulated via abstract coherence-breaking scatterers). The effective transmission function is modified to incorporate the defect scattering. The charge self-consistent solution is obtained using the "jellium" approximation, solving the Poisson equation and the single-band (Γ-valley) effective mass Schrödinger equation with open-system boundary conditions. The Contact Block Reduction method and the open-system predictor-corrector method augmented with Anderson mixing are used for efficient implementation. Standard values for electron effective masses and silicon dielectric constant are employed, with no fitting parameters used in charge-self-consistent calculations. Sheet resistance calculations include a single parameter, the linear defect density.

The open-system quantum-mechanical treatment revealed a quantized conduction sub-band structure termed "quantum menorah." This structure consists of spatially separated layers of free electrons with distinct average energies. The occupied states (1Γ and 2Γ) are largely independent of the δ-layer depth but strongly depend on dopant density. Unoccupied states above the Fermi level are strongly influenced by δ-layer depth. The LDOS shows that the lowest energy mode is centered around the δ-layer, while higher energy modes are located off-center. The number of conduction sub-bands and their energy splitting are strongly influenced by both δ-layer thickness (t) and doping density (ND). For a fixed thickness, increasing doping increases the number of conducting modes and energy splitting. Conversely, for a fixed sheet doping, increasing thickness increases the number of modes but decreases energy splitting. The current is carried relatively equally among conductive sub-bands, even when more carriers occupy lower-energy sub-bands. As confinement weakens, higher-energy sub-bands contribute more significantly to the current. The effective electron cloud thickness shows highly nonlinear dependence on sheet doping density and δ-layer thickness. At low doping densities, thickness is independent of δ-layer thickness and determined by doping, while at high densities, thickness depends only on δ-layer thickness. The calculated sheet resistance values accurately reproduce experimental data from various groups across a range of δ-layer donor densities. This is achieved by incorporating a heuristic elastic defect scattering model.

The findings address the limitations of previous models by providing a direct and accurate calculation of the conductive properties of δ-layer systems. The "quantum menorah" structure and the identification of spatially distinct electron layers with different average energies provide a novel understanding of the quantum effects in these systems. The significant influence of δ-layer thickness on sheet resistance for a fixed charge density highlights the importance of a fully quantum-mechanical treatment. The agreement between the simulated sheet resistance and experimental data validates the accuracy and applicability of the open-system approach. These results have implications for designing future thermoelectric systems where selective charge carrier filtering could lead to high figures of merit, as well as for qubits and ultra-scaled devices where understanding the electron cloud thickness and its dependence on the doping profile is crucial.

This paper presents an open-system quantum transport treatment for semiconductor δ-layer systems, revealing a quantized sub-band structure. This approach successfully explains the origin of shallow conducting sub-bands and accurately predicts sheet resistance values, highlighting the importance of a fully quantum-mechanical treatment for highly confined and conductive systems. The spatially separated electron layers with distinct energies open possibilities for novel thermoelectric applications. Future research could explore other materials and doping profiles, and extend the model to include inelastic scattering effects for a more comprehensive understanding.

The study primarily focuses on low-temperature behavior, neglecting inelastic scattering effects which might become important at higher temperatures. The model employs a single-band effective mass approximation, potentially simplifying some band structure details. The heuristic defect scattering model is a simplification of a complex physical phenomenon, and more realistic defect models could improve accuracy. The computational cost of the open-system approach limits the size and complexity of structures that can be modeled effectively.