Multiscale approach to the electronic structure of doped semiconductor surfaces

The inclusion of the global effects of semiconductor doping poses a unique challenge for first-principles simulations, because the typically low concentration of dopants renders an explicit treatment intractable. Furthermore, the width of the space-charge region (SCR) at charged surfaces often exceeds realistic supercell dimensions. Here, we present a multiscale technique that fully addresses these difficulties. It is based on the introduction of a charged sheet, mimicking the SCR-related field, along with free charge which mimics the bulk charge reservoir, such that the system is neutral overall. These augment a slab comprising “pseudoatoms” possessing a fractional nuclear charge matching the bulk doping concentration. Self-consistency is reached by imposing charge conservation and Fermi level equilibration between the bulk, treated semiclassically, and the electronic states of the slab, which are treated quantum-mechanically. The method, called CREST—the charge-reservoir electrostatic sheet technique—can be used with standard electronic structure codes. We validate CREST using a simple tight-binding model, which allows for comparison of its results with calculations encompassing the full SCR explicitly. Specifically, we show that CREST successfully predicts scenarios spanning the range from no to full Fermi level pinning. We then employ it with density functional theory, obtaining insight into the doping dependence of the electronic structures of the metallic “clean-cleaved” Si(111) surface and its semiconducting $(2\times 1)$ reconstructions.

Figure 1

Explanation of the charge-reservoir electrostatic sheet technique (CREST). See text for details and explanation of terms. (a) Schematic description of a SCR at a doped semiconductor surface. (b) The physical system is bisected at $z_d$ into two oppositely charged parts. (c) The semi-infinite part of the system is “moved left,” creating a parallel capacitor with an intermediate vacuum “layer.” (d) Description of a typical slab system after contraction of the SCR to a charged sheet. The idealized $U_l$ or electrostatic energy is shown as a blue curve, denoted by $U_i\left(z\right)$, and $U_l$ resulting from macroscopic averaging (see Sec. 2b) of the electrostatic energy from a typical DFT or other atomistic calculation is shown as a red curve, denoted by $U_a\left(z\right)$ [slightly offset from $U_i\left(z\right)$ to make differentiation easier]. Also shown is the linear curve extrapolated from $U_a\left(z\right)$ in the vacuum between the charged sheet and the slab's virtual surface [$U^{\mathrm{vac}}\left(z\right)$; red dashed line].

Figure 2

A slab system similar to the system in Fig. 1, but fully passivated on both sides. Key values are shown (see text for details).

Figure 3

Illustrative example of an $n$-doped system, calculated using our 1-D TB code ($N_{\mathrm{dop}}=1.25\times {10}^{17}{\text{cm}}^{-3}$, $N_{ss}=2.5\times {10}^{11}{\text{cm}}^{-2}$) with a slab of 300 sites (600 Å), thick enough to include the full SCR: (a) The site-projected DOS (PDOS) showing the valence band (VB) and the conduction band (CB) with the gap between them. $E_F$ is indicated by the horizontal white line. The bands are bent by the field in the SCR. The acceptor surface state resides on the right side of the slab. As it is invisible relative to the color scale, its position is marked on the figure. (b) Column plot showing the net charge per area associated with each “atom” in the TB treatment. In the shaded region on the right (adjacent the surface), charge magnitudes are scaled down by a factor of 200.

Figure 4

Comparison of “full-SCR” (blue dashed line), “confined-SCR” (green dash-dotted line), and CREST-corrected (red solid line) calculations of the system of Fig. 3. (a) Electrostatic energy $U_l$ computed from the full-SCR system and from thin slabs containing 30, 40, or 50 TB sites (denoted by “30”, “40”, or “50”, respectively). The sets of curves are offset 0.04 eV from each other for clarity, with the same reference full-SCR result appearing in each set. Red round markers indicate the positions of $(z_d,U_d)$ in each case, with the red dotted lines (denoted by CREST-E.M. in the legend) extending to the left of $z_d$ showing $U_l$ in the effective-mass bulk, obtained by solving Eqs. (2) and (3) numerically within CREST. (b) Total DOS in the vicinity of $E_{VBM}$. Results from the full system and from the 30-site slab are shown. Vertical lines indicate the Fermi level position in each calculation. The defect level inside the gap is enhanced by a factor of ${10}^4$. DOS curves are broadened by convolution with a Gaussian function with a standard deviation of 0.01 eV.

Figure 5

Results from confined-SCR (green lines, triangles) and CREST (red lines, circles) calculations using our 1-D TB code, relative to the reference full-SCR calculation: (a) Fermi level and (b) energy of the defect state, $E_d$. The horizontal line marks 0, the reference value. Energies are referenced to $E_{VBM}$ in each respective calculation. Panel (c) shows the values for $\Delta {\phi }_b$ resulting from the CREST calculations. Bulk doping: $n$ type, $N_{\mathrm{dop}}=1.25\times {10}^{17}{\text{cm}}^{-3}$.

Figure 6

CREST corrections to $E_F$ values in thin-slab TB simulations of SCR-possessing systems. Energies are referenced to $E_{VBM}$ in each respective calculation. All doping densities shown are $n$ type, and the surface state is a deep acceptor, approximately 0.1 eV above $E_{VBM}$.

Figure 7

Development of the plane-averaged long-range electrostatic energy, $U_{lr}$, as obtained from FHI-AIMS, in CREST simulations with slabs comprising 26 (solid lines), 20 (dashed lines), and 14 (dash-dotted lines) Si layers, of the highly doped (HDn) CC (blue lines), PB (green lines), and NB (red lines) surfaces. Curves are aligned horizontally to $z_{\mathrm{surf}}$, the position of the topmost surface atom in each calculation, and vertically to $U_{lr}$ in the vacuum outside (to the right of) the real surface examined. For clarity, the curves for the PB surface are offset upward by 3.5 eV, and for the NB surface by 7 eV.

Figure 8

Surface band structures of the metallic, clean-cleaved Si(111) ($1\times 1$) surface [(a) and (b)], and of the semiconducting, positively buckled $\pi$-chain Si(111) ($2\times 1$) surface [(c) and (d)]. Results from both CREST-corrected [(a) and (c)] and confined-SCR [(b) and (d)] calculations are shown. Within each panel, different doping levels are compared: intrinsic (blue dashed lines), moderately $n$ doped (MDn; $N_{\mathrm{dop}}={10}^{17}{\text{cm}}^{-3}$; green dot-dashed lines), and heavily $n$ doped (HDn; $N_{\mathrm{dop}}=2\times {10}^{19}{\text{cm}}^{-3}$; red solid lines). $E_F$ values are marked by straight horizontal lines. The surface bands are the freestanding (curved) lines. The “bulklike” bands (see text) are shaded in, with blue indicating regions where bands exist only in the intrinsic/MDn cases, light red where bands exist only in the HDn case, and dark gray where bands exist at all doping densities. The edges are marked with the respective lines for clarity. Energies are referenced to the maximum of the highest occupied bulklike band, $E_{VBM}$, in each respective surface band diagram. Panel (e) describes the paths taken in $k$ space.