Suppression of mid-infrared plasma resonance due to quantum confinement in delta-doped silicon

TL;DR

This paper develops a quantum mechanical model explaining the suppression of plasma resonance in delta-doped silicon layers, highlighting the effects of charge confinement and anisotropy on plasma behavior.

Contribution

It introduces a physically accurate description of delta-doped layers that accounts for quantum confinement and anisotropy, explaining the absence of plasma resonance features.

Findings

Quantum confinement suppresses plasma resonance in delta-doped silicon.

Charge density profile and anisotropy are key to plasma resonance suppression.

Resonance reappears when dopant atoms diffuse out of the confined layer.

Abstract

The classical Drude model provides an accurate description of the plasma resonance of three-dimensional materials, but only partially explains two-dimensional systems where quantum mechanical effects dominate such as P:$\delta$-layers - atomically thin sheets of phosphorus dopants in silicon that induce novel electronic properties beyond traditional doping. Previously it was shown that P:$\delta$-layers produce a distinct Drude tail feature in ellipsometry measurements. However, the ellipsometric spectra could not be properly fit by modeling the $\delta$-layer as discrete layer of classical Drude metal. In particular, even for large broadening corresponding to extremely short relaxation times, a plasma resonance feature was anticipated but not evident in the experimental data. In this work, we develop a physically accurate description of this system, which reveals a general approach to designing thin films with intentionally suppressed plasma resonances. Our model takes into account the strong charge density confinement and resulting quantum mechanical description of a P:$\delta$-layer. We show that the absence of a plasma resonance feature results from a combination of two factors: i), the sharply varying charge density profile due to strong confinement in the direction of growth; and ii), the effective mass and relaxation time anisotropy due to valley degeneracy. The plasma resonance reappears when the atoms composing the $\delta$-layer are allowed to diffuse out from the plane of the layer, destroying its well-confined two-dimensional character that is critical to its novel electronic properties.