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.

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 crosssection 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
Enotes
Authors: Evgeny Demidov, Yury Drozdov IPM RAS.
updated 29 Jan 2001