Ionization of a P-doped Si(111) nanofilm using two-dimensional periodic boundary conditions

We examine the ionization of a P dopant in a Si(111) nanofilm using first-principles electronic structure calculations with 2D periodic boundary conditions. The electrostatic divergence of a charged periodic system is resolved by defining an electrostatic reference potential along the confined direction. After ionization, there is an overall electrostatic potential drop of the system. A nanofilm with larger periodicity can reduce the potential drop by screening the P ion, and leads to a smaller ionization energy. We compare the ionization energy calculated for the P-doped Si nanofilm with a P-doped Si nanocrystal and a P-doped Si(110) nanowire. As dimensionality decreases, quantum confinement tends to lower the ionization energy by raising the defect level. However, lower dimensionality also reduces screening after P ionization. This leads to a larger electrostatic potential drop and offsets the effect of quantum confinement on the ionization energy.

Figure 1

(a) The self-consistent potential $V_{\mathrm{SCF}}$ plotted from the ${\text{SiH}}_4^{+}$ position ($z=0$) to the top of the simulation box. The variation of $V_{\mathrm{SCF}}$ approaches the electrostatic potential of a uniformly charged sheet $V_{\mathrm{plate}}$ for sufficiently large $z$. $V_{\mathrm{plate}}=4\pi \sigma z+2Z_{\mathrm{net}}{E}_{\mathrm{self}}$, where $\sigma =\frac{1}{L_x{L}_y}$ is the surface charge density, $Z_{\mathrm{net}}=1$, and $E_{\mathrm{self}}$ is defined as in Eq. (4). The second term in $V_{\mathrm{plate}}$ corresponds to our choice of the electrostatic reference potential. (b) The total energy $E_{\mathrm{tot}}$ of ${\text{SiH}}_4^{+}$ plotted as a function of the height of the simulation cell $L_z$ ($\blacksquare , L_x=L_y$ fixed at 30 ${\text{a}}_B$) and the periodicity $L_x=L_y=L$ ($•, L_z$ fixed at 15 ${\text{a}}_B$). (c) The variation of the eigenvalue spectrum of ${\text{SiH}}_4^{+}$ with the periodicity $L$. The spectrum for $L=\infty$ corresponds to the result from a nonperiodic confined calculation.

Figure 2

The atomic geometry of a P-doped Si nanocrystal, Si(110) nanowire, and Si(111) nanofilm. The three structures are oriented such that the (111) direction points upwards, and the (110) direction points into the paper. The Si atoms are colored grey, and the cyan atoms represent the H passivation. The large yellow atoms at the centers of the structures are the P substitutional dopants.

Figure 3

(a) The ionization energy $IE$ of the P-doped Si(111) nanofilm as a function of the periodicity $L$. The $•$ curve uses a total energy approach. The eigenvalue of the defect level $|{\varepsilon }_d|$ when the system is neutral ($\blacksquare$) is plotted for comparison. (b) $IE$ exhibits an exponential decay with $L$. This is verified by a linear dependence when the $•$ curve is plotted on a logarithm-linear scale.

Figure 4

(a) A contour plot of the P defect wave function ${\left|\mathrm{\Psi }\right|}^2$. The red contour corresponds to the value 0.005 ${\AA }^{-3}$. (b) The variation of ${\left|\mathrm{\Psi }\right|}^2$ is plotted against $z$ (after integrating with respect to $x$ and $y$). The blue dot on the $x$ axis denotes the P location, and the two red dots correspond to the $z$ coordinates of the H passivation. (c) The average variation of ${\left|\mathrm{\Psi }\right|}^2$ against the radial distance $r$ (in cylindrical coordinates with $\theta$ and $z$ dependence integrated out). (d) ${\left|\mathrm{\Psi }\right|}^{2}r$ is plotted against $r$ (solid line). Its integral (dashed line) shows that $\mathrm{\Psi }$ is normalized.

Figure 5

The density of states of the P-doped Si(111) nanofilm before and after ionization of the P dopant. There is an electrostatic potential drop of the system as a whole after ionization. The drop is larger for a nanofilm with a smaller periodicity (a) $L=21.1$ Å compared to the nanofilm with (b) $L=94.4$ Å. The three vertical dashed lines denote (from left to right) the energies of the valence-band maximum, the defect level, and the conduction-band minimum, respectively.

Figure 6

(a) The ionization energy $IE$ versus nanofilm periodicity $L$. $IE$ is calculated from either first-principles (quantum) or classical simulations. (b) The $IE$(classical) results can be fitted to an exponential function of the form $IE\left(\infty \right)+\left(\mathrm{\Delta }\right)exp(-L/\lambda )$. $\mathrm{\Delta }$ (red solid curves) and $\lambda$ (blue dashed curves) depend on the thickness $t$ and the dielectric constant $K$ of the dielectric slab used in the classical simulations. $K$ is along the $x$ axis. The numbers within the plot correspond to different values of $t$. The classical curve plotted in (a) corresponds to $t=9.7$ Å and $K=3.8$.

Figure 7

Comparison of the ionization energy $IE$ ($•$) between different P-doped Si structures: a Si nanocrystal (0D), a Si(110) nanowire (1D), a Si(111) nanofilm (2D), and a Si bulk crystal (3D). The P-doped nanostructures are illustrated in Fig. 2. The defect levels $|{\varepsilon }_d|$ of the P-doped Si structures ($\blacksquare$) are plotted as well.