Reflection High Energy Electron Difraction (RHEED) on Si

Drag mouse to rotate 3D reciprocal lattice in the left window. Nodes of the lattice crossed by the XY plane passed through the (000) point are painted in the green color. The nodes with there indexes are ploted in the right window too. The right window shows the electron difraction picture observed in experiment. You can see angles of rotation around X and Y axes (in degrees) in the Status bar.
RHEED on GaAs.

Electron difraction on solid state crystals

De Broglie wave length of electron l (mesured in angstrom [A]) is determined by its energy e (mesured in electron volt [V]) by
l = 12.2 / (e [V])^1/2 [A].
For high energy electrons with e = 25kV we got l = 0.076A therefore
l << a_Si , a_GaAs
where a_Si = 5.43A is the Si lattice period and a_GaAs = 5.65 is the GaAs one.
Electron wave vector k is determined by its pulse p as k = 2p/h p, where h is the Planck constant. We have too |k| = 1/l. Therefore
|k| >> g_o = 1/a.
Electron difraction on solid state crystals is determined by the Bragg difraction low
k' - k = g,
where k is the incident electron wave vector, k' is the difracting one, g is a reciprocal lattice vector. For elastic electron scattering |k| = |k'|.

For the Simple Cubic Bravais lattice (see 3D Solid State Crystal models) reciprocal lattice is the Simple Cubic too. Therefore g is
g_hkl = g_o (hx + ky + lz),
where h,k,l are integers and x,y,z are orthogonal unit vectors.
Fig. You see in Fig. that if |k| >> |g| (i.e. scattering angle a is small) then g lays in the plane perpendicular to k. For small a we get
a ~ tg a = |g_hkl| / |k| = |g_hkl| l = r_hkl / L ,
r_hkl = Ll |g_hkl| .
Therefore difraction picture coinsides really with cross-section of the reciprocal lattice by the plane perpendicular to falling electron beam.
As since Si lattice is not the Simple Cubic one, then the next reflections are forbidden
|h| + |k| = 2n + 1,
|k| + |l| = 2n + 1,
|h| + |l| = 2n + 1,

h = 2, k = 2, l = 2,

hk0 |h| + |k| <> 4n,
h0l |h| + |l| <> 4n,
0kl |k| + |l| <> 4n.
The GaAs lattice is similar to the Si one but is made of different atoms, therefore the last "four" reflections are allowed but weak.

We used L = 43cm and the screen diameter D = 7cm. It corresponds to the real RHEED difractometer in the "Katun" molecular beam epitaxy system. Free sources

E-notes Authors: Evgeny Demidov, Yury Drozdov IPM RAS.
updated 29 Jan 2001