Quantum-continuum calculation of the surface states and electrical response of silicon in solution

A wide range of electrochemical reactions of practical importance occur at the interface between a semiconductor and an electrolyte. We present an embedded density-functional theory method using the recently released self-consistent continuum solvation (SCCS) approach to study these interfaces. In this model, a quantum description of the surface is incorporated into a continuum representation of the bending of the bands within the electrode. The model is applied to understand the electrical response of silicon electrodes in solution, providing microscopic insights into the low-voltage region, where surface states determine the electrification of the semiconductor electrode.

Figure 1

(a) Atomic-level view of a semiconductor–solution interface. (b) Electrostatic profile across the semiconductor–solution interface, showing the bending of the electronic bands on the semiconductor side, described by an extended Mott-Schottky layer, and the electrical double layer on the solution side, represented by a Helmholtz-Stern layer in series with a Gouy-Chapman layer of oppositely charged ions.

Figure 2

(a) Partition of a rutile ${\mathrm{SiO}}_2$(110)-electrolyte interface into three regions. Region I represents the continuum bulk semiconductor section, Region II the quantum surface of the semiconductor, and Region III the continuum electrolyte solution. Both Regions I and III extend infinitely. The colors indicate the changing dielectric constant in the simulation; red corresponds a dielectric constant of $\sim 3.9$ for ${\mathrm{SiO}}_2$ and blue corresponds to a dielectric constant of $\sim 78$ for water at room temperature. (b) Profile of the electrostatic potential across the ${\mathrm{SiO}}_2$-electrolyte interface with the dotted horizontal line representing the Fermi level ${\epsilon }_{\mathrm{F}}$ of the slab. Having set the potential to zero at the boundaries of the cell, the flatband potential ${\mathrm{\Phi }}_{\mathrm{FB}}$ equals the negative of the Fermi level.

Figure 3

(a) The potential of a charged slab with planes of countercharge on each side, creating a potential drop. The dotted line represents the electrostatic potential $\overline{\phi }$ of the charged slab subtracted from that of a slab with zero charge as shown in Fig. 2. (b) A cutoff value $z_{\mathrm{c}}$ corresponding to the inflection of the potential $\overline{\phi }$ is determined. To the left of this cutoff a Mott-Schottky extrapolation is applied, as shown by the new dotted line. By examining several different charge distributions, the specific distribution where the Fermi levels match is found. The width of the depletion region is shortened here for illustrative purposes and would normally extend for several nanometers.

Figure 4

Lateral and top views of representative surface terminations for silicon: (a) Si(110) with oxygen (O*), (b) Si(110) with a hydroxyl group (O* + H*), (c) Si(110) with two oxygens absorbed into the surface layers (2O*), (d) Si(110) with an oxygen absorbed into the second layer with and adsorbed hydroxyl group (2O* + H*), (e) Si(110) with a ${\mathrm{SiO}}_4$ tetrahedron terminated by a hydrogen (4O* + H*), (f) Rutile ${\mathrm{SiO}}_2$(110), (g) Cristobalite ${\mathrm{SiO}}_2$(100), (h) Cristobalite ${\mathrm{SiO}}_2$(110), (i) Quartz ${\mathrm{SiO}}_2$(1000), and (j) Quartz ${\mathrm{SiO}}_2\left(11\overline{2}0\right)$.

Figure 5

(a) The total charge versus voltage curves for Si(110) structures. (b) The total charge versus potential curves for ${\mathrm{SiO}}_2$ structures. The lines correspond to the fitted trends of an empirical model that consists of an ideal Mott-Schottky semiconductor in series with a linear capacitor representing the surface states.