Momentum-Space Imaging of Ultra-Thin Electron Liquids in δ-Doped Silicon

Abstract

Two-dimensional dopant layers (δ-layers) in semiconductors provide the high-mobility electron liquids (2DELs) needed for nanoscale quantum-electronic devices. Key parameters such as carrier densities, effective masses, and confinement thicknesses for 2DELs have traditionally been extracted from quantum magnetotransport. In principle, the parameters are immediately readable from the one-electron spectral function that can be measured by angle-resolved photoemission spectroscopy (ARPES). Here, buried 2DEL δ-layers in silicon are measured with soft X-ray (SX) ARPES to obtain detailed information about their filled conduction bands and extract device-relevant properties. This study takes advantage of the larger probing depth and photon energy range of SX-ARPES relative to vacuum ultraviolet (VUV) ARPES to accurately measure the δ-layer electronic confinement. The measurements are made on ambient-exposed samples and yield extremely thin (< 1 nm) and dense (≈10¹⁴ cm^-2 ) 2DELs. Critically, this method is used to show that δ-layers of arsenic exhibit better electronic confinement than δ-layers of phosphorus fabricated under identical conditions.

Keywords: 2DEG; ARPES; arsenic in silicon; delta layer; silicon; soft X-ray angle-resolved photoelectron spectroscopy (soft X-ray ARPES).

Figure 1

Sample schematic and the evolution of the silicon conduction valleys vs δ‐layer confinement. a) Schematic representation of our δ‐layer samples, with a native oxide that forms due to ambient exposure. The silicon overgrowth thickness, z, is indicated, as is the electronic thickness of the δ‐layer, δz. The δ‐layer creates an approximately V‐shaped potential well in the plane perpendicular to the δ‐layer, which quantizes the out‐of‐plane and in‐plane conduction valleys into a series of subbands denoted as nΓ and nΔ, respectively. The tic‐marks on the depth axis indicate 1 nm steps. b) Evolution of the silicon conduction valleys from 3D (six degenerate, ellipsoidal valleys) to c) 2D (4 Δ‐valleys + 2 Γ‐valleys). d) Plot of the transverse (δk _T) versus longitudinal (δk _L) extent of the k_x (diamonds, green region), k _y (squares, green region) and k _z valleys (circular markers, purple region). The filled and hollow markers represent data from 2 and 3 nm deep δ‐layers respectively. The solid black line indicates the expected valley morphology for bulk, 3D silicon, whose gradient is equal to the mass anisotropy of silicon. The colored background represents the eccentricity of the ellipsoid, which spans from zero (blue) to one (yellow).

Figure 2

Fermi‐surface measurements of (upper row) phosphorous and (lower row) arsenic δ‐layers with 2 nm silicon overgrowth. a,e) Schematic representations of the measured six conduction valleys of silicon embedded within the bulk fcc Brillouin zone, indicating 2D behavior. Fermi surface slices for the phosphorous and arsenic δ‐layer samples are shown along the following planes: b,f) k _x‐k _z and c,g) k _x‐k _y through the zone center (Γ), d,h) k _x‐k _y through the center of the upper k _z valley (see also the color‐coded slices on panels a,b). Fermi surfaces are integrated from −50 meV to E _F. In panel (d,h), the 1Γ and 2Γ states are denoted, based on the fits acquired in Figure 4.

Figure 3

Extracting δ‐layer confinement from the longitudinal span of the Δ‐ and Γ‐valleys. a) Visual representation of Equation (1), which is applied to extract the δ‐layer confinement, δz, from the longitudinal extent of the Δ‐ and Γ‐valleys. δk _∞ represents the longitudinal FWHM of the Δ‐valley along the k _y‐axis, which is broadened by the intrinsic mean free path (MFP), λ; δk _z is the longitudinal FWHM of the Γ‐valley along the k _z‐axis, which includes both the MFP and confinement broadening. The line profile shows the Lorentzian deconvolution process (see Supporting Information for more details), allowing the δ‐layer confinement to be extracted from the k _z response of the Γ‐valley; a reasonable fit (black) to the data (red) can be achieved by convolving the green and blue contributions. b) k _x‐k _y Fermi surface and c) k _F vs k _y for the 2 nm arsenic δ‐layer sample, where $k_y^{\prime }=k_y-$0.94 Å^–1. d–g) k _x‐k _z Fermi surfaces (hν = 350 – 410 eV, integrated from −50 meV to E _F) for 2 and 3 nm phosphorous and arsenic δ‐layers, as indicated. White dots indicate the cusps of the parabola in k _x at each value of k _z, where $k_{\mathrm{z}}^{\prime }=k_{\mathrm{z}}-$10.18 Å^–1. h–k) Plots of k _F as a function of k _z extracted from panels (a–d), whereas the triangle data points are taken from the same valley, but at a higher photon energy (≈ 820 eV). The inset of (c) and (k) indicates the valley that is probed; the green, pink, and blue arrows in panels (b,c) and (d,h) offer a guide to the eye. The best fit line is shown in black, with the shaded areas indicating the 1σ fit confidence.

Figure 4

Analysis of the conduction band quantization for phosphorous and arsenic δ‐layers. a) Measured and calculated energies of the δ‐layer subbands. b) Measured and calculated electron density for each δ‐layer subband from the fits acquired in (c–f). For the theory, the percentages denote the occupation of each δ‐layer subband; when the degeneracy is accounted for, each one of the six valleys have an equal occupancy of ≈16.6%. c–f) SX‐ARPES measured Γ band dispersions for 2 and 3 nm phosphorous and arsenic δ‐layers at hν = 380 eV (which corresponds to the centroid of the Γ valley in Figure 3(a)). The purple and blue parabolas are the fits to the ARPES data, showing that the 1 and 2Γ states can be deconvolved. The line profile above each image shows the momentum distribution curves taken at (1); the line profile to the right, labeled (2), shows the corresponding fit to the energy distribution curve.