Local density profile

A local density profile $\rho(z)$ is defined as the average number of atoms in a slice of width $\Delta z$, which is parallel to the solid substrate. The proper choice of the width $\Delta z$ of a slice is a trade-off between two factors. First, a very small width results in too few particles in each slice, and therefore one observes large statistical errors and data scattering. Second, a very large width of a slice will not show the actual dependence of the properties on the distance from the surface. Hence a balance between those two requirements must be achieved. To facilitate the presentation we use $\rho(z)$ represented by a continuous function (See Fig. 5.10) defined according to Chen et al.[79] :

$$\rho(z)=\left<\frac{1}{\sqrt{2\pi \Delta z}} \sum_{i} exp(-\frac{(z-z_i)^2}{2\Delta z})\right >$$ (6.3)

where $z_i$ is $z$ coordinate of atom $i$, with $z=0$ set at the bottom of the non-fixed slab, and the angular brackets indicate time average. We use $\Delta z~ =~0.1a_0/2\sqrt3$, where $a_0$ is the lattice parameter.

Figure: Density profile of $Va(001)$ at T=$2200$ K

The premelting of the crystal surface exhibits itself in the loss of long-range order. This transition can be examined by monitoring the layer-by-layer modulation of the density profile of the system at various temperatures up to the melting point $T_m$. Sets of the plots of the local density profiles for the samples Va(111),Va(001), and Va(011) are shown in Figs. 5.11-5.13, respectively. At low temperatures the density profile $\rho(z)$ consists of a series of sharp, well resolved peaks. The atoms are packed in the layers with constant density in each layer and virtually no atoms in between these parallel layers. As the temperature is increased the effective width of the each layer becomes broader due to the enhanced atomic vibration, and the position of the peaks move to larger values of $z$ due to thermal expansion.

At the temperatures close to the melting point the atomic vibration becomes so large, especially in the first layer, that disorder sets in, with atomic migration taking place between the layers, as evidenced by the fact that the minima of $\rho(z)$ between two peaks rise to non-zero values. This is a reminiscent of a liquid-like structure (which is also observed in the plane radial distribution function). Hence, the system crosses over to a state of ``premelting'' at elevated temperatures. At some temperature ( $T^* \simeq~ 1000~K$ for the Va(111), $T^*\simeq~ 1500~K$ for the Va(001), $T^*\simeq~ 2200~K$ for the Va(011)) the density of the topmost layer becomes slightly lower than that of the underlying layers. The loss of density is compensated by appearance of atoms on top of the first surface layer. These atoms are called adatoms and the additional surface layer is termed as an adlayer. As the temperature is elevated atoms from the deeper subsurface layers start to diffuse toward the adlayer. The distinction between the layers becomes blurred. The generation of adatom-vacancy pairs induces disorder and converts the topmost layers into a thin quasiliquid film.

Figure: Density profile across $Va(111)$ along the $z$ direction perpendicular to the surface at various temperatures as indicated.

Figure: Density profile across $Va(001)$ along the $z$ direction perpendicular to the surface at various temperatures, as indicated.

Figure: Density profile across $Va(011)$ along the $z$ direction perpendicular to the surface at various temperatures, as indicated.

A general conclusion is made concerning the formation of an adlayer relying on analysis of the temperature dependence of the density profiles; the formation begins first on the least packed surface Va(111), thereafter at higher temperature a quasiliquid film appears on the Va(001) face, which has an intermediate density between the Va(111) and Va(011) faces. Finally, surface premelting occurs on the close packed face Va(011) at a temperature which is very close to the melting point $T_m$, and practically all adatoms come from the first surface layer.

These results are in good agreement with the of investigation of surface premelting of fcc metals Al [78], Ni [79], and Cu [80]. It was found that an adlayer appears first on the least packed (011) surface, then at higher temperature on the (001) face, and finally the onset of disorder is observed on the close packed (111) surface at a temperature close the melting temperature.