Suppression of Midinfrared Plasma Resonance Due to Quantum Confinement in δ -Doped Silicon

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:$\ensuremath{\delta}$ layers---atomically thin sheets of phosphorus dopants in silicon that induce electronic properties beyond traditional doping. Previously it was shown that P:$\ensuremath{\delta}$ layers produce a distinct Drude tail feature in ellipsometry measurements. However, the ellipsometric spectra could not be properly fit by modeling the $\ensuremath{\delta}$ layer as a 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:$\ensuremath{\delta}$ 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 $\ensuremath{\delta}$ layer are allowed to diffuse out from the plane of the layer, destroying its well-confined two-dimensional character that is critical to its distinctive electronic properties.