Understanding RHEED patterns

Diffraction patterns produced by RHEED, like diffraction patterns from many other techniques, are best understood through the Ewald Sphere formalism. The Ewald Sphere is a geometric construction expressing two facts about the diffracted electrons: total momentum is conserved, and momentum transfer can occur only with wavevec- tors of the reciprocal lattice. To construct an Ewald sphere, draw the reciprocal lat- tice and put the tip of the incoming wavevector at the origin of the reciprocal lattice. Then, draw a sphere around the origin of the incoming wavevector with radius given by the magnitude of that wavevector. Diffraction will occur where the sphere inter- sects the reciprocal lattice. The sphere (technically a shell) enforces the energy conser- vation condition, since all points on the sphere that intersect points on the reciprocal lattice are separated from the origin of the lattice by a reciprocal lattice vector.17

The standard Ewald Sphere construction works well for diffraction from a three- dimensional lattice. In RHEED, however, only the surface is sampled because the out-of-plane wavevector is very small. Because the momentum in the out-of-plane di- rection is low, there is little penetration. As a result, the three-dimensional reciprocal lattice is reduced to a two-dimensional reciprocal net from the surface and is contin- uous in the third dimension, forming a series of one-dimensional rods. Diffraction oc-

17_{The point at the origin of the reciprocal lattice, for example, corresponds to the trivial re-}

ciprocal lattice vector indicating no scattering. In RHEED, this spot corresponds to specular reflection.

curs when these rods intersect the Ewald sphere [180].

Each of these rods intersects the Ewald sphere in exactly two points. One of the points lies below the “shadow” plane of the sample and is not seen on the diffraction image. Hence the intersection of each rod with the Ewald Sphere produces one reflec- tion in the diffraction pattern. Multiple rods corresponding to diffraction by wavevec- tors of the same magnitude will intersect the Ewald sphere in the same plane, thus defining a circle of points where the rod intersects the sphere. These circles are known as Laue18 _{circles. The width of the reciprocal rods increases with disorder, blurring}

the RHEED image from points into streaks. Because the electron wavelength is very small, the surface only needs to be locally ordered to give a diffraction pattern. In particular, a polycrystalline material with small grain sizes will give a diffraction pat- tern that is a sum of the intensities of the different patterns from each orientation.

This formalism describes coherent, elastic scattering: energy is conserved and the direction of the incident wavevector plays a crucial role in determining the diffrac- tion pattern. Both of these assumptions can fail. The most important example is a two-step scattering process that produces distinctive “Kikuchi lines” in diffraction pat- terns[178]. In this process, the incident electrons interact with the surface and scatter diffusely. Typically, this collision is nearly elastic. When these electrons are emitted from the surface, they are emitted over a wide range of angles. These scattered elec- trons can then undergo diffraction the same way as an incident electron, but because

Figure 3.8: A graphical representation of the Ewald Sphere technique used for RHEED. Reprinted figure with permission from Ref. [180]. Copyright 1983 by the American Physical Society.

they have a range of directions, the diffraction condition is poorly represented by the Ewald sphere picture where the initial wavevector is emphasized. The intensity of Kikuchi lines depends strongly on the surface quality because they are broadened by terraces and steps on the surface[177].

For some applications, including the identification of Kikuchi lines, it is important to know how much the electron wave diffracts when it enters the crystal. Typically, an electron penetrating into the bulk is able to lower its energy by a quantity known as the inner potential. This potential plays an effect analogous to the index of refrac- tion in photon optics and can produce bending of Kikuchi lines at low emission angles and even total internal reflection for certain lines.19 For diamond, the inner potential is estimated to be between 15 eV[181] and 18 eV[182].

The small-angle Kikuchi line pattern in produced for scattering off of a given recip- rocal lattice vector can be calculated.20 Lines in θand ϕare given by:

h|a∗₁|ϕ+l|a∗₃| ( θ+ eV E )1/2 = G 2 2K (3.3.1)

Here, h andl are the indices of the surface reciprocal lattice. G2 is the magnitude of the corresponding reciprocal lattice vector. a∗₁ and a∗₃ are the primitive reciprocal lattice vectors in the direction of the beam and in the out-of-plane direction.21 _{E is}

19_{See chapter 7 and especially equation 7.4 of Ichimiya[177].}

20_{See example 7.1.1 in [177].}

the energy of the incoming electron andeV is the energy of an electron in the inner potential. Free software is available to calculate the expected Kikuchi lines from a given crystal surface[183].

The above equation often predicts many more Kikuchi lines than are typically seen in a sample. This formalism can be extended to incorporate surface wave resonant conditions that describe which lines are favored by resonance as well as the widths of these lines. Also, both constructive and destructive interference are possible: distinc- tive “dark” bands (sometimes called “defect” lines) are also Kikuchi lines.