Diffusion coefficients

Diffusion coefficients of the surface layers are calculated to investigate transport properties of the Va(111),Va(001) and Va(011) samples. The coefficients are found from the particle trajectories, $\vec r_{i,\mu}(t)$, by calculating the average mean square displacement $R_{l,\mu}^2$ (See Fig 5.22):

$$R_{l,\mu}^2=\left< \frac{1}{n_l} \sum_{i \in l } [ \vec r_{i,\mu}(t+\tau) - \vec r_{i,\mu}(\tau)]^2 \right >$$ (6.10)

where $\mu=x,y,z$ is a coordinate index, the sum includes atoms in the layer $l$, and the angular brackets denote averaging over time from the origin ($\tau$). The diffusion coefficients $D_{l,\mu}$ are calculated separately in the $x,y$ and $z$ directions, according to Einstein relation for each layer:

$$D_{l,\mu}=\lim_{t \rightarrow \infty } \frac{R^2_{l,\mu}}{2t}$$ (6.11)

Figure: Mean square displacement $R_{l,z}^2$ in the $z$ direction of an atom in the surface layer of Va(111) vs. time. Note the increased mobility of particles with temperature.

Figure: Diffusion coefficients of $Va(001)$ as a function of layer number at temperature T=2200 K.

The diffusion coefficients are larger at the surface region.(See 5.23) The mobility of the atoms increases with elevation of the sample temperature and converges to the liquid bulk values. These observations correlate with the structural variations in the surface region exhibited in the pair correlation functions, the structural order parameters, and the local density profiles. The diffusion coefficients of the first crystal layer of the Va(011) as a function of temperature are shown in Figs. 5.24.

Figure: Diffusion coefficients of the surface layer of Va(011) vs. temperature in different directions. Note an anisotropy of the in-plane diffusion coefficients : $D_x~>~D_y$

The diffusion coefficients are different in different directions $D_x > D_y$ for Va(011) and Va(111). In the course of diffusion an atom jumps from one point (its current position) to another one (the nearest vacant place on the lattice), the distance between these two points is termed as jump distance. This jump distance is larger along the $x$-direction than in the $y$-direction, because the nearest-neighbor distance is larger in the $x$-direction than in the $y$, i.e. $a_x>a_y$ (See Table 5.1). Hence the diffusion coefficients are larger in the $x$-direction $D_x > D_y$. The diffusion coefficient along the $z$-direction is smaller than along the $x$ and $y$ directions. That difference can again be explained by the fact that $a_z <a_x,~a_y$ (See Table 5.1), and therefore, the jump distance is the smallest along the $z$ direction, thus $D_z < D_x,~D_y$.

The diffusion coefficients of the least packed Va(111) surface are the largest one. The diffusion coefficients of the Va(001) are smaller than the Va(111) ones, yet they are larger than the diffusion coefficients of the close packed Va(011) surface, which are close to zero even at the elevated temperatures (See Fig 5.25).

Figure: In plane diffusion coefficients as a function of temperature for the surface layer of Va(111),Va(001) and Va(011) calculated for the $y$ direction).