**Faebian Bastiman**

In this article I will explain what a reconstruction is and how it appears in a RHEED pattern without using terms like “Ewald sphere” and “reciprocal lattice rods”. To answer the question of what a reconstruction is, we must first consider why a surface reconstructs. Let’s use zinc blende GaAs(100) as an example.

The simplest lattice to imagine is a cubic lattice. The unit cell of a lattice is the smallest collection of atoms, that when repeated over and over again, can recreate the entire lattice. For a cubic lattice this is simply a cube with sides of length *a* and an atom on each corner (vertex). With 8 vertices it is easy to make the mistake that the unit cell comprises 8 atoms. Instead consider that each of those atoms overlaps into 7 neighbouring unit cells within a full lattice. Thus each vertex atom contributes 1/8 of an atom to each unit cell, and therefore each cubic unit cell comprises (on average) 1 atom.

The next two simplest lattices to image are the body centred cubic (bcc) and the face centred cubic (fcc). When comparing a bcc lattice to a cubic lattice, there exists an extra atom right at the centre of the cube. In this case a bcc unit cell comprises 2 atoms. Similarly a fcc unit cell has an extra atom on each face. In this case each of the 6 atoms on the faces are shared between 2 neighbouring unit cells. The fcc unit cell then comprises of 3 + 1 = 4 atoms.

Imagine a blue fcc lattice of As atoms and a green fcc lattice of Ga atoms. Now take the unit cells of each and place them side by side. Pick up the Ga fcc unit cell and push it into the As fcc unit it overlaps by 75% of its longest diagonal. You would have two cubes that would look like Figure 1a. If you remove any of the green atoms that are not inside the As fcc unit cell, and connect any of the nearest neighbouring As and Ga atoms with a red line: you have just created zinc blende GaAs!

a |
b |

Figure 1: (a) Two interleaved fcc unit cells (b) zinc blende unit cell |

Why the long winded explanation? Well on the one hand it is useful to know that a zinc blende unit cell can be considered as two interleaved fcc unit cells, since it saves us trying to image zinc blende without a framework. On the other hand, when you consider the {100} (i.e. 100 family of planes) you can simply consider the As (or Ga) fcc face.

*N.B. A quick aside on parentheses: (100) is the 100 plane, {100} are family of identical planes e.g. (100), (010), (001). Similarly [100] is the 100 direction, <100> are the family of directions. *

So consider the {100} planes, each plane is made up of multiple fcc faces tiled to make a pattern. During MBE growth GaAs is typically As terminated, so we will consider a surface made of As fcc faces repeated over and over. When you look at the {100} face of an fcc unit cell you see a 2D square consisting of 5 atoms (Figure 2a). However you can see that (in 2D) each of the corner atom overlap with the 3 neighbouring squares, and so each corner atoms only contributes ¼ of an atom to the square. When you add 4 x ¼ to the one in the centre of the face this gives 2 atoms per square. Since the length of the sides of the square is the lattice constant, *a*, you can say that the {100} plane of a zinc blende semiconductor has 2 atoms per *a*^{2} and in doing so can work out the atomic surface density. Similarly you can add up the atoms inside a zinc blende unit cell and say there are 8 x 1/8 + 6 x ½ + 4 = 8 atoms per a^{3} and work out the atomic volume density.

a |
b |

Figure 2 (a) As atoms of {100} plane aligned to <100> (b) As atoms of {100} plane aligned to <110> |

Consider the {100} plane of As atoms. The surface represents an abrupt end to the lattice. All the As atoms are bound to Ga atoms below, and would like to bind to Ga atoms above. However they cannot because there are no Ga atoms: the crystal is terminated at this plane. This means that every other As atom within the lattice is bonded to 4 Ga atoms, except the ones on the exposed surface. Or to put it a different way, all the As atoms in the crystal have 8 electrons in their outer shell except the surface atoms. With nothing else to bind to, the As atoms choose to bind to each other. The surface is the only point in the lattice where an As atom bonds to another As atom. The As atoms are then said to dimerise (i.e. create a dimer) and in so doing the surface is said to have reconstructed.

In fact not only do they always dimerise, they always dimerise in the same direction. They dimerise in the <-110> direction. They dimerise in this direction because their motion is restricted in the <110> direction by the Ga atoms below them and in order to dimerise the As atoms need to move closer to each other. If you take the plane described in Figure 2a and rotate it 45° you can identify a smaller square with 1 atom on each corner and sides with length *a*/√2 (Figure 2b). The square’s sides run in the <110> and <-110> directions. The row of blue As atoms at the top of Figure 2b also includes the position of the green Ga atoms in the plane below. Note the Ga atoms appear in the <110> direction. In this way the As atoms motion is restricted in the <110> direction and they can only move in the <-110> direction. Conversely when the surface is Ga terminated, the As atoms lay in the <-110> direction so Ga always dimerises in the <110> direction. Thus the two species create reconstructions that are at right angles to each other.

When the As atoms dimerise they create a feature on the surface that reapeats with half the periodicity (or occupies twice the space) of a single As atom. A reconstruction is therefore similar to a unit cell, in that it is the smallest feature that when repeated can recreate the entire surface. If one could freeze the {100} plane before it reconstructed it would look like Figure 3a. One can then draw a square around the smallest repeating pattern and note that the sides of the square are equal to *a*/√2. To save us using the term (*a*/√2 x *a*/√2) we call this (1 x 1). Once the surface has dimerised it looks like Figure 3b. Now because the As atoms have moved together the repeating pattern is a rectangle with lengths (2*a*/√2 x *a*/√2) or (2 x 1). This is then a (2 x 1) reconstruction.

a |
b |

Figure 3 (a) (1 x 1) reconstruction (b) (2 x 1) reconstruction |

The fact is GaAs does not form a (2 x 1) reconstruction. The force of coulomb repulsion prevents several As dimers aligning side-by-side in the <110> direction. In fact it is energetically favourable for only 2 As dimers to exist side-by-side and the next two are missing. This creates a pattern with a repeat period of (2 x 4). Since the Ga reconstruction appears at right angles to the As reconstruction, the Ga reconstruction is a (4 x 2). These are the main two reconstructions and along with c(4 x 4) they are the most commonly created and most useful reconstructions. They depend on both the As/Ga flux ratio and substrate temperature and can be used to create a static reconstruction map and to define when the As/Ga ratio = 1.

In order to do this you use RHEED. RHEED is a very useful tool to an MBE grower. It allows you to see a reciprocal space representation of the real space surface reconstruction. Though the fact is you do not need to know you are creating a reciprocal space reconstruction or even that RHEED pattern is in reciprocal space. In order to achieve a practical working knowledge of RHEED you simply need to know three things:

- the substrate surface acts like a diffraction grating to electrons
- when something gets bigger in real space, it gets smaller in reciprocal space
- when something is n times as far apart in real space, it is 1/n times as far apart in reciprocal space

The RHEED pattern can then be considered a diffraction pattern. Typically you would align the RHEED beam with the <110> and <-110> directions and you would observe the diffraction pattern created by each. First consider the surface of Figure 3a. Regardless of whether you view the surface in the <110> or the <-110> direction As atoms are evenly spaced. This would create a diffraction pattern consisting of only the primary (zero order) rods in both directions: a so called 1x (one-by) reconstruction. The RHEED pattern you would observe in each direction is shown diagrammatically in the figure. Since the real space (1 x 1) reconstruction spacing is proportional to *a*, the reciprocal space RHEED pattern spacing is proportional to 1/*a*. So when you look at a substrate with a bigger lattice constant, like InAs, the zero order RHEED pattern spacing is smaller.

Now consider Figure 3b. When you align the RHEED beam in the <-110> direction the pattern would be the same 1x, since the As atoms still possess the same spacing. However when you look in the <110> direction the pattern would appear as a 2x pattern. That is doubling the pattern in real space leads to a halving of the spacing between the RHEED pattern streaks in reciprocal space. Another way to think of it is a doubling of the number of streaks in the RHEED pattern. Note that the 2x occurs when you align the RHEED beam in the <110> direction (termed the <110> azimuth) and this indicates the atoms have rearranged (or moved) in the <-110> direction. This is because you are using the sample to create a diffraction grating at right angles to the electron beam.

So for the As rich (2 x 4) surface, the <110> azimuth is 2x, and the <-110> azimuth is 4x. Similarly for the Ga rich (4 x 2) surface, the <110> azimuth is 4x, and the <-110> azimuth is 2x.

Pingback: How to grow your first sample: (2×4)/(4×2) transition | Dr. Faebian Bastiman

Pingback: How to grow your first sample: As capping | Dr. Faebian Bastiman