Limits to Metallic Conduction in Atomic-Scale Quasi-One-Dimensional Silicon Wires

The recent observation of ultralow resistivity in highly doped, atomic-scale silicon wires has sparked interest in what limits conduction in these quasi-1D systems. Here we present electron transport measurements of gated $\mathrm{Si}:\mathrm{P}$ wires of widths 4.6 and 1.5 nm. At 4.6 nm we find an electron mobility, ${\mu }_{\mathrm{el}}\simeq 60{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$, in excellent agreement with that of macroscopic Hall bars. Metallic conduction persists to millikelvin temperatures where we observe Gaussian conductance fluctuations of order $\delta G\sim e^2/h$. In thinner wires (1.5 nm), metallic conduction breaks down at $G\lesssim e^2/h$, where localization of carriers leads to Coulomb blockade. Metallic behavior is explained by the large carrier densities in $\mathrm{Si}:\mathrm{P}$ $\delta$-doped systems, allowing the occupation of all six valleys of the silicon conduction band, enhancing the number of 1D channels and hence the localization length.

Figure 1

(a) and (b) Overview STM images ($-2.0\mathrm{V}$, 100 pA) of two atomic-scale wires ($W1$) and ($W2$) with two contacts ($S$, $V_1$) and ($D$, $V_2$) on either end, allowing four-terminal ($4T$) resistance measurements. In-plane gates, $G_1$ and $G_2$, allow us to tune the electron density. (c),(d) Atomic resolution images show the alignment along $⟨110⟩$. The lithographic widths, $w=4.6$ ($W1$) and $w=1.5\mathrm{nm}$ ($W2$), correspond to six and two dimer rows (DR) of the $\mathrm{Si}(001)-(2\times 1)$ surface reconstruction. (e),(f) $4T$ $IV$ characteristics at $T=4.2\mathrm{K}$ and for varying gate bias, $V_G$.

Figure 2

$4T$ differential conductance $G_W$ of two atomic-scale wires $W1$ (a) and $W2$ (b). Upper panel: $G_W$ as a function of gate voltage $V_G$, and $4T$ voltage drop,$V_{4T}$. Lower panel: Conductance, $G_W(V_{4T}=0\hspace{0.17em}\mathrm{V})$ with respect to $G_0\sim e^2/h$ (black dashed lines). The dashed blue line in (a) shows a linear fit to extract electron mobility. Coulomb blockade oscillations (b) are observed in the 1.5 nm wide wire ($W2$).

Figure 3

Atomistic modeling of the electronic structure at $T=4.2\mathrm{K}$. (a) Number of conducting modes $N_{\mathrm{eff}}$ (diamonds) as a function of wire width, compared to a simple analytical estimate (black dashed line, see text). (b)–(d) Modeling of electron localization due to dopant disorder. (b) Example supercell configurations used to calculate (c) band structure and density of states. (d) Charge density $|\psi {|}^{2}dr$ [1] based on a repetition of the $L=10\mathrm{nm}$ cell in (b).

Figure 4

(a) Differential conductance at millikelvin temperatures, comparing the 4.6 nm ($W1$) and the 1.5 nm ($W2$) wide wires. (b) and (c) Conductance histograms, $P(G_W)$, of both wires, indicating metallic and insulating behavior, respectively. The solid blue lines are fits to a normal [Eq. (4)] and a log-normal [Eq. (5)] distribution, expected in the metallic and insulating regimes, respectively.