Thermally Induced Crossover from 2D to 1D Behavior in an Array of Atomic Wires: Silicon Dangling-Bond Solitons in Si(553)-Au

The self-assembly of submonolayer amounts of Au on the densely stepped Si(553) surface creates an array of closely spaced “atomic wires” separated by 1.5 nm. At low temperature, charge transfer between the terraces and the row of silicon dangling bonds at the step edges leads to a charge-ordered state within the row of dangling bonds with $\times 3$ periodicity. Interactions between the dangling bonds lead to their ordering into a fully two-dimensional (2D) array with centered registry between adjacent steps. We show that as the temperature is raised, soliton defects are created within each step edge. The concentration of solitons rises with increasing temperature and eventually destroys the 2D order by decoupling the step edges, reducing the effective dimensionality of the system to 1D. This crossover from higher to lower dimensionality is unexpected and, indeed, opposite to the behavior in other systems.

Figure 1

Ground-state structure and low-energy excitation of the Si(553)-Au atomic wire system. The underlying substrate consists of Si(111) terraces separated by steps. Each terrace contains a dimerized double row of Au atoms (gold) and a row of Si dangling bonds (gray spheres) at the step edge. The electron occupancy of these dangling bonds creates a ground state with tripled periodicity: for every two saturated dangling bonds (SDBs, small spheres) there is a third, unsaturated dangling bond (UDB, large spheres). At finite temperatures, some of these UDBs (blue) become mobile and hop to adjacent sites (red). This excitation creates a soliton-antisoliton pair that can subsequently dissociate.

Figure 2

SPA-LEED pattern of Si(553)-Au at an electron energy of 150 eV and temperatures (a) 60 K and (b) 180 K. The $\times 2$ streaks between the rows of sharp integer-order spots arise from dimerized Au double rows on the (111)-oriented terraces. The rows of elongated spots at $\times 3$ positions indicate the tripled periodicity and long-range order of the UDBs at the Si step edge. The intensity of the $\times 2$ streak is nearly unaffected at higher temperature, while the $\times 3$ features fade away. In (a) the unit cells of the Au (blue) and Si (green) sublattices as well as the directions of the line profiles ${\mathrm{LP}}_{\parallel }$ and ${\mathrm{LP}}_{\perp }$ (Fig. 3) are indicated (for more details see [28]). (c) Surface structural model showing Si step-edge atoms (gray) and Au atoms (gold). The unit cells are depicted with the same color coding as in (a).

Figure 3

FWHM of the $\times 3$ diffraction spots (red data points) as a function of temperature in (a) the $[\overline{3}\overline{3}10]$ direction and (b) the $[1\overline{1}0]$ direction, as indicated in Fig. 2. Results from Monte Carlo simulations based on DFT interactions are shown in (b) by blue squares for the (2,2,1) configuration and triangles for the (2,2,0) configuration. The result from the analytical model in Eq. (1) is shown in black. The increase of the FWHM in (a) indicates the loss of interwire correlation, while in (b) it indicates a decreasing correlation length along the steps. At low temperatures the FWHM is constant in the $[\overline{3}\overline{3}10]$ direction up to $\sim 90\mathrm{K}$. Insets in (a),(b): Line profiles for both directions at various temperatures (shifted for better visibility). (c) Structural model of creation and separation of a soliton-antisoliton pair. Charge is transferred from an SDB to a UDB, generating a hop of the UDB to a neighboring site and creating a soliton-antisoliton pair. If this pair separates then a phase-shifted domain with $\times 3$ periodicity is formed.

Figure 4

Temperature dependence of the exponent $\nu$ of the generalized Lorentzian of Eq. (2) in the $[1\overline{1}0]$ direction. The exponent drops from 1.5 at $T_{-}=93\mathrm{K}$ to 1.0 at $T_{+}=128\mathrm{K}$, indicating a crossover from 2D to 1D behavior. Insets: Experimental and fitted spot profiles at 60 K and at 128 K. At 60 K the profile is best fit by $\nu =3/2$ (2D behavior), while at $T_{+}$ the best fit is $\nu =1$ (1D behavior).