Theory of low-energy electron diffraction for detailed structural determination of nanomaterials: Finite-size and disordered structures

We describe how a recent efficient theory of low-energy electron diffraction (LEED) enables the determination of finite-size and disordered nanostructures. Our cluster approach, called NANOLEED, speeds up the computation to scale as $n\log n$, rather than the usual $n^3$ or $n^2$, with $n$ the number of atoms, for example, thereby making nanostructures accessible. To illustrate this method’s capability to determine nanoscale structure, we apply it to calculate LEED intensities for Si nanowires of various lengths and thicknesses as well as for various deviations of these nanowires from the ideal Si bulk structure.

Figure 1

Examples of LEED patterns in reciprocal space for simple nanostructures made of Si atoms (sketched at far left), of both infinite size (left panels) and finite size (right panels): a string of Si atoms along (a) $\pm x$, (b) $\pm y$, and (c) $\pm z$; (d) a planar square grid; and (e) a simple cubic grid. The cones in the left-hand panels represent 1D Bragg conditions; they intersect Ewald spheres imposing energy conservation. The right-hand panels show two-dimensional patterns as would be seen in a standard LEED setup, as intensity vs parallel momentum; multiple-scattering intensities are calculated using our method for NANOLEED, using a Si-Si closest distance of $0.3\mathrm{nm}$, an electron energy of $100\mathrm{eV}$, and a damping of $4\mathrm{eV}$. In the right-hand panel of (c) only one profile is shown; the complete pattern is obtained by rotating this profile around the $z$ axis.

Figure 2

The influence of nanostructure size (left panels) on the LEED pattern (right panels) for three sample SiNWs. Assumed is an ideally terminated bulk Si structure with $[1,1,-2]$ growth direction along $\pm x$ and a (111) surface exposed toward positive $z$. From (a) to (c), the length (along $x$) increases and the width (along $y$) decreases as indicated.

Figure 3

Effect on LEED $I\text{−}V$ curves of a uniform 4% elongation (dashed lines) of a bulk-terminated SiNW relative to a bulklike geometry (full lines). The initial nanowire dimensions are $10.2\times 0.8\times 0.7{\mathrm{nm}}^3$, as in Fig. 2c. The wire thickness decreases with elongation to conserve volume. The beam labeling follows standard LEED practice for extended surfaces, such that the (0,0) beam represents specular reflection from the (111) upper surface of the wire (this beam or spot appears at the center of the right-hand patterns in Figs. 1, 2).

Figure 4

LEED patterns for H termination (top panels) and $\mathrm{Si}{\mathrm{H}}_3$ termination (bottom panels) of SiNWs with $[1,1,-2]$ growth direction. This termination is applied only to the upper (111) surface, as illustrated on the left.

Figure 5

Effect of a uniform 4% elongation of a $\mathrm{Si}{\mathrm{H}}_3$-terminated SiNW on LEED “rocking” scans at three different incident electron energies. The elongation is performed as in Fig. 3. The scans rotate the detector in the $xz$ plane of Fig. 4, with 0° along the $z$ axis marking specular reflection from the top of our SiNW.

Figure 6

Effect on $I\text{−}V$ curves of an outward displacement by $0.01\mathrm{nm}$ along the $z$ axis of the $\mathrm{Si}{\mathrm{H}}_3$ layer in the $\mathrm{Si}{\mathrm{H}}_3$-terminated SiNW of Fig. 4.

Figure 7

Effect on LEED intensities of “rounding” the relaxation (used in Fig. 6) of the topmost $\mathrm{Si}{\mathrm{H}}_3$ layer in our SiNW, as described in the text. Shown are “circular” scans around the long axis of the nanowire, with 90° along the $z$ axis marking specular reflection from the top facet of our SiNW.

Figure 8

Effect on LEED intensities of a small defect in our SiNW, as described in the text. Shown are azimuthal scans with fixed scattering angle calculated for $60\mathrm{eV}$, considering fixed polar angles of 45° and 60° with respect to the incident electron beam.