Conductivity and size quantization effects in semiconductor $$\delta$$-layer systems

Abstract

We present an open-system quantum-mechanical 3D real-space study of the conduction band structure and conductive properties of two semiconductor systems, interesting for their beyond-Moore and quantum computing applications: phosphorus $$\delta$$ δ -layers and P $$\delta$$ δ -layer tunnel junctions in silicon. In order to evaluate size quantization effects on the conductivity, we consider two principal cases: nanoscale finite-width structures, used in transistors, and infinitely-wide structures, electrical properties of which are typically known experimentally. For devices widths $$W<10$$ W < 10 nm, quantization effects are strong and it is shown that the number of propagating modes determines not only the conductivity, but the distinctive spatial distribution of the current-carrying electron states. For $$W>10$$ W > 10 nm, the quantization effects practically vanish and the conductivity tends to the infinitely-wide device values. For tunnel junctions, two distinct conductivity regimes are predicted due to the strong conduction band quantization.

Multi‐Scale Modeling of Tunneling in Nanoscale Atomically Precise Si:P Tunnel Junctions

“…This value obtained is consistent with previously published data, which had very large error bars and were calculated using a quasi‐1D (Q1D) WKB model (100 ± 50 meV) [ 29 ] (Figure 4a, orange). Another model of similar junctions was performed using effective mass NEGF and gave a value for the barrier height in a Si:P tunnel junction as ≈80 meV [ 51 ] (Figure 4a, black). A first‐approximation of the barrier height taken as the difference between the Fermi energy in the monolayer doped terminals and the bulk silicon conduction band minimum in the barrier is ≈90 meV [ 35 ] (Figure 4a, red).…”

“…Comparison of different modeling techniques to determine the tunneling resistance in atomically abrupt tunnel junctions. a) Calculated values of the barrier height V 0 in Si:P tunnel junctions from a quasi‐1D WKB fit [ 29 ] (orange, uncertainty indicated by the shaded oval), extracted from bandstructure variations [ 35 ] (red circle), from effective mass NEGF model [ 51 ] (black circle), and the value from this work (blue cross, uncertainty within marker size). b) Calculations of the lateral seam width in Si:P tunnel junctions using a quasi‐1D WKB fit [ 29 ] (orange, uncertainty indicated by the shaded oval) and from this work (blue cross, uncertainty within marker size).…”

Uncovering anisotropic effects of electric high-moment dipoles on the tunneling current in δ-layer tunnel junctions

“…We also emphasize that in our calculations there is no fitting parameters, we use only the standard values of electron effective masses and dielectric constant for silicon. Finally, the slight differences between our computed tunneling resistances and experimental measurements can also be accounted for by the following reasons: i) Certain variations in the width, thickness, and doping density of the δ -layer (note that a doping density of N D = 1.0 × 10 14 was assumed in 29 , while a higher doping density, N D = 2 × 10 14 , was employed in 38 ); ii) The possible presence of impurities and/or defects near the tunnel gap.…”

“…Fig. 9 shows our computed tunneling resistance from our previous work 29 for an effective width of the δ -layer of 7 nm. The figure also includes the resistance measurements and tight-binding calculations for tunnel junctions of δ -layer width of 7 nm from M. Donnelly et al 38 .…”

“…In the open-system NEGF framework, the computational device consists of a semi-infinite source and drain, in contact with the channel of length L, which is composed of a lightly doped Si body and cap and two very thin, highly P doped layers (referred to as left and right δ -layers) separated by an intrinsic gap of length L gap . To avoid the boundary effects between the source and drain contacts and size quantization effects 29 , the total channel length is chosen to be L = 30.0 nm + L gap , while the device height is set to H = 8 nm and the device width is set to W = 15.0 nm, with an effective width of the δ -layer of 13.0 nm. In our analyses, we have considered a thickness of δ -layer of t = 1.0 nm and a sheet doping density of N D = 1.0 × 10 14 cm −2 (N…”