Fig. 1: Si:P δ-layers, ARPES measurements, and the sub-band structure.

a Schematic of the δ-layer sample and b the symmetric doping potential centred around the position of the P dopant plane. The strong potential created by the P atoms confines the conduction band, developing a 2D electronic structure which has been understood to be comprised of two states, labelled 1Γ (red) and 2Γ (blue). c The Brillouin Zone (BZ) of silicon showing the directions relevant for this work. The blue dashed lines show the 2D BZ, and the pink shading and lines show a 2D slice through the 3D BZ of silicon at the relevant value of k_z for this work (darkest shade: 1st bulk BZ). The high symmetry directions k_x, k_y (red axes), and k_xy (black axis) are also indicated. d Tight-binding calculations (following ref. ¹⁰) showing the 1Γ and 2Γ states dispersing across E_F (with E_F adjusted such that the minimum of the dispersion matches with the data in panels (f) and (g)) along the k_xy and k_x (or k_y) directions (blue and green markers, respectively). e ARPES acquisition of a ‘dummy’ Si sample (i.e., Si bulk plus 2 nm epitaxial overlayer, but without the P dopant δ-layer), showing the absence of any states around E_F. f ARPES acquisition along the k_xy direction showing only two states. A momentum distribution curve (MDC) at the E_F shows no presence of additional states. g ARPES measurement along the k_x (or k_y) direction showing the existence of three states, (marked with blue, red, and yellow lines to serve as a guide for the eye). As discussed in the main text, the agreement between the TB calculation (d) and the ARPES data in panels (f) and (g) is unsatisfactory. h An energy-dependent curve integrated in the range k_∥ = 0 ± 0.015 Å⁻¹, further confirming the occupied nature of the 3rd state. All ARPES measurements are performed using hν = 36 eV^13,15.