Simulated doping of Si from first principles using pseudoatoms

Semiconductor doping is a process of fundamental importance to semiconductor physics and solid-state electronics, but cannot be explicitly simulated from first principles due to the huge system size needed for most doping scenarios. We examine the efficacy of the simulation of doping in silicon by the inclusion of “pseudoatoms” with fractional nuclear charge, introduced via specially constructed pseudopotentials. These provide a net charge carrier concentration matching an arbitrarily chosen doping level, at no increase of the computational cost. By extending this approach to consider minute deviations from the integer charge, we demonstrate that the electron Fermi level can be set to any value within the forbidden gap, at minimal perturbation of the electronic structure. Beyond the bulk scenario, we successfully simulate the development of the space-charge region in a heavily doped p-n junction and the doping dependence of the work function of the hydrogen-passivated (semiconducting) Si(111) surface.

Figure 1

Comparison of the DFT-calculated (solid blue line) and effective-mass-calculated [using Eq. (1), dashed red line] Fermi level position in the bulk as a function of the nominal doping density for (a) $p$- and (b) $n$-type doping.

Figure 2

Density of states (DOS) curves for bulk Si, in the vicinity of the forbidden gap, obtained from an eight-atom unit cell with various levels of effective doping, from heavily (2×10${}^{19}$ cm${}^{-3}$) $p$ doped (blue) to heavily (2×10${}^{19}$ cm${}^{-3}$) $n$ doped (red). Fermi level positions are shown as vertical lines. Also shown is a reference calculation of intrinsic Si (black, dashed), obtained from a two-atom diamond-cubic unit cell. $E$ $=$ 0 eV denotes the valence band maximum (VBM) of this reference calculation.

Figure 3

Electrostatic energy along a heavily doped (2×10${}^{19}$ cm${}^{-3}$) p-n junction: PBE-based DFT result with uniformly modified atoms on either side of the junction (solid blue line); theoretical model result based on Eqs. (3, 4, 5) (dashed red line); PBE-based DFT result with only the central atom of each region of the junction modified (dot-dashed green line). The central dotted vertical line marks the interface position, with the $n$-doped region on the right. The other two dotted lines are the midpoints of each region ($p$, $n$) in the calculated cell. Note a difference of sign in the model results with respect to Eqs. (3, 4, 5), owing to the difference between potential and electron energy.

Figure 4

Fermi level as a function of the per-atom doping density, in the nondegenerate doping regime, of an H-terminated 26-Si-layer slab (solid black line), an H-terminated 20-layer slab (dotted blue line), a Cl-terminated 26-layer slab (dot-dashed red line), and bulk Si (dashed black line). For convenience, both $n$- and $p$-doping regimes are shown, with 0 representing the CBE for the $n$-type regime (negative $E_F$ values) and the VBE for the $p$-type regime (positive $E_F$ values).

Figure 5

(a) Work-function development vs Fermi level derived from doped slab and bulk calculations. For convenience, both $n$- and $p$-doping regimes are shown, with 0 representing the CBE for the $n$-type regime (negative $E_F$ values) and the VBE for the $p$-type regime (positive $E_F$ values). Solid lines are used in nondegenerate doping regimes; dashed lines in degenerate doping regimes; dotted lines mark linear fits to nondegenerate regimes (slopes found were $-$1.007 and $-$1.002 for $n$- and $p$-type regimes, respectively). Also shown are zoom-ins for the degenerate $n$ (b) and $p$ (c) regimes of the H-terminated (solid black line) and Cl-terminated (dot-dashed red line) 26-Si-layer slab, along with the linear fit to the H-terminated slab's nondegenerate regime (dotted black line).