Scaling Theory Put into Practice: First-Principles Modeling of Transport in Doped Silicon Nanowires

We combine the ideas of scaling theory and universal conductance fluctuations with density-functional theory to analyze the conductance properties of doped silicon nanowires. Specifically, we study the crossover from ballistic to diffusive transport in boron or phosphorus doped Si nanowires by computing the mean free path, sample-averaged conductance $⟨G⟩$, and sample-to-sample variations $\mathrm{std}(G)$ as a function of energy, doping density, wire length, and the radial dopant profile. Our main findings are (i) the main trends can be predicted quantitatively based on the scattering properties of single dopants, (ii) the sample-to-sample fluctuations depend on energy but not on doping density, thereby displaying a degree of universality, and (iii) in the diffusive regime the analytical predictions of the Dorokhov-Mello-Pereyra-Kumar theory are in good agreement with our ab initio calculations.

Figure 1

Transmissions through wires with a single B (left) or P (right) dopant atom placed at the different radial positions shown in the inset. Left: energies relative to the valence band maximum, $E_v$; right: energies relative to the conduction band minimum, $E_c$. The band gap is 2.84 eV.

Figure 2

Energy dependent average conductance calculated at $L=10\mathrm{nm}$ (circles) and $L=100\mathrm{nm}$ (dots) with the GF method. Solid lines follow from Eq. (1). Left inset: Resistance vs wire length at energies $E-E_c=0.01$, 0.06, and 0.16 eV (indicated with arrows in the main frame; same behavior is seen in the entire energy range). Solid lines are obtained using Eq. (1). Right inset: $⟨\ln T⟩$ vs length at the same energies, where $T=G/(2e^2/h)$ is the transmission. Solid lines are linear fits in the interval $150\mathrm{nm}<L<200\mathrm{nm}$ from which the localization length is determined.

Figure 3

Sample standard deviation, $\mathrm{std}(G)$ vs normalized length $L/l_e$ with average dopant-dopant separation $d=19\mathrm{nm}$ (dots), $d=10\mathrm{nm}$ (squares) and $d=5\mathrm{nm}$ (circles). (a)–(c) correspond to the energies $E-E_v=-0.12$, $-0.20$, $-0.30\mathrm{eV}$ in the valence band for B-doped wires, (d)–(f) correspond to the energies $E-E_c=0.06$, 0.16, 0.26 eV in the conduction band for P-doped wires. The solid lines are analytical solutions to the DMPK equation of Ref. 23 which equals the UCF value $0.73e^2/h$ at $L=0$. The small arrows mark the maximum $\mathrm{std}(G)$, $s$, estimated from the single-dopant transmissions.

Figure 4

Maximal standard deviation (a) for P-doped wires ($d=10\mathrm{nm}$), with a homogeneous radial distribution (squares) and a pure surface doped wire (circles). The solid lines show the standard deviation, $s$, among the single-dopant transmissions and the pristine wire, and the dashed line marks the UCF value $0.73e^2/h$. Panel (b) shows the MFP vs energy for the two radial distributions (circles and squares). The lines represent values obtained from the single-dopant transmissions.