Reflection High Energy
Electron Diffraction
Wei-Li Chen
11/15/2007
RHEED
• Reflection High Energy Electron Diffraction,
RHEED, is an important real time analytical tool
to monitor growth front condition.
Substrate thermal cleaning monitoring
Controlling initial growth stage
Monitoring surface structure and growth dynamics
Growth rate measurement

• The small incident angle makes it sensitive to
the structure of top monolayers.
Bravais Lattice
• A Bravais lattice is an infinite arrays of discrete
points with an arrangement and orientation that
appears exactly the same, from whichever of the
points the array is viewed.
• A Bravais lattice consists of all points with
position vectors R of the form
R  n1 a1  n2 a2  n3 a3
where a1 , a2 , a3 are ant three vectors not all
in the same plane, and ni s are integers.
Crystal Structure
• The atom group, which builds complete crystal
structure by translational operations is called
crystallographic unit cell
• Crystal structure = Lattice + Basis
z
Unit Cell Geometry
c
In General:
a ≠ b ≠ c and
α≠ß≠γ
c 
a
O

b
zo

b
c
y
xo
a
x
b
x
(a) A parallelepiped is chosen to describe geometry of
a unit cell. We line the x, y and z axes with the edges of
the parallelepiped taking lower-left rear corner as the
(b) Identification
Crystal
Lattice
Basis
a
a
90
Unit cell
Unit cell
(a)
(c)
(b)
Basis placement in unit cell
(d)
(0,0)
y
(1/2,1/2)
x
Fig. 1.70: (a) A simple square lattice. The unit cell is a square with a
side a. (b) Basis has two atoms. (c) Crystal = Lattice + Basis. The
unit cell is a simple square with two atoms. (d) Placement of basis
atoms in the crystal unit cell.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
UNIT CELL GEOMETRY
CUBIC SYSTEM
a = b = c 90°
Many metals, Al, Cu, Fe, Pb. Many ceramics and
semiconductors, NaCl, CsCl, LiF, Si, GaAs
Simple cubic
Body centered
cubic
Face centered
cubic
TETRAGONAL SYSTEM
a = b - c  == =90°
Body centered
tetragonal
Simple
tetragonal
In, Sn, Barium Titanate, TiO2
ORTHORHOMBIC SYSTEM
a - b - c  = = =90°
S, U, Pl, Ga (<30°C), Iodine, Cementite
(Fe3C), Sodium Sulfate
Simple
orthorhombic
Body centered
orthorhombic
Base centered
orthorhombic
Face centered
orthorhombic
HEXAGONAL SYSTEM
a = b - c  = = 90° ; = 120°
RHOMBOHEDRAL SYSTEM
a = b = c  =  =  - 90°
Cadmium, Magnesium, Zinc,
Graphite
Arsenic, Boron, Bismuth, Antimony, Mercury
(<-39°C)
Hexagonal
Rhombohedral
MONOCLINIC SYSTEM
a - b - c  =  = 90° ;  - 90°
TRICLINIC SYSTEM
a - b - c  -  -  - 90°
Selenium, Phosphorus
Potassium dicromate
Lithium Sulfate
Tin Fluoride
Simple
monoclinic
Base centered
monoclinic
Triclinic
Fig. 1.71: The seven crystal systems (unit cell geometries) and fourteen Bravais
lattices.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Reciprocal Lattice
The set of all wave vectors K that yield plane waves
with the periodicit y of a given Bravais lattice is known
as its reciprocal lattice.
 ei K r  R   ei K r
 ei K R  1
The three primitive vectors of the reciprocal space :
b1  2π
a2  a3

a1  a2  a3

; b2  2π
a3  a1

a1  a2  a3

; b3  2π
a1  a2

a1  a2  a3

Miller Indices
h k l 
h k l 
h k l
hkl
Plane
Vector in Bravais lattice
Crystal symmetry equivalent plane sets
Crystal symmetry equivalent vector sets
x1 , x2 , x3 are plane intersecs of three coordinate s
1 1 1
h:k :l  : :
x1 x2 x3
plane h k l   reciprocal vector h k l 
z
z intercept at 
b
Miller Indices (hk) :
1
1
1/ 1
2
c
x intercept at a/2
1

(210)
y
a
Unit cell
x
y intercept at b
(a) Identification of a plane in a crystal
z
(010)
(010)
z
(010)
(010)
(010)
y
y
x
x
(110)
(001)
(100)
z
(111)
z
(111)
(110)
y
x
-z
(b) Various planes in the cubic lattice
-y
y
x
Fig. 1.40: Labelling of crystal planes and typical examples in the
cubic lattice
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Bragg’s and von Laue’s Formula
Bragg condition 2dsin   n


d cos   d cos  '  d  nˆ  nˆ '  m  d  kˆ  kˆ'  2m


R is the lattice vector  R  k  k '  2m or e i k- k'  R  1
Recall the definition k of the reciprocal lattice vector e i K R  1
 
 k-k'  K is a lattice vector in reciprocal space.
1
Alternativ ely, k  k  K  k  Kˆ  K
2
The component of the incident w ave vector k along the
reciprocal lattice vector K must be half the length of K
K is represent as G in Kittel' s book 
The Edwald Construction
k  G
The Geometry of RHEED
• The geometry of RHEED is quite simple, Fig. 1.
An accelerated electron beam (5 – 100 keV) is
incident on the surface with a glancing angle (<
3 deg) and is reflected. The high energy of the
electrons would result in high penetration depth.
However, because of the glancing angle of
incidence, a few atomic layers are only probed.
This is the reason of the high surface sensitivity
of RHEED. Upon reflection, electrons diffract,
forming a diffraction pattern that depends on the
structure and the morphology of the probed
surface.
 usually smaller th an 3o
The amplitude of scattering
f 0 (s) 
Edwald' s constructi on
2me e
 (r ) exp[ i (k  ki )  r ]d r
h 2 V
 k  k - k i  G  ha   k b  l c 
where h, k, l are integers and
r is the position v ector
k and ki are diffracted and incident w ave vectors.
This is a fourier tr ansform of the real crystal space.
Constructi ve interferen ce happens at the direction
where  k matches with discrete displaceme nt in
the reciprocal space.
a   2
b  2
c   2
bc
a b c
,
ca
a b c
,
ab
a bc
“Molecular Beam Epitaxy “ edited by R. F. C. Farrow
Fourier Transform
   
1
 r  
f s exp  i s  r d s

2 
f s    r exp i s  r d r
3
s  k  k'  2
s  G  K 
2

sin  
4sin 

Ideal smooth surface
real smooth surface
3D clusters
polycrystal
Powder or Polycrystal
diffraction spots  rings
Rotation of crystal
Growth Calibration by RHEED Oscillation
The reflection intensity of the
specular point is related to the
roughness of the surface, which
changes periodically with the
accumulation of film thickness.
“Molecular Beam Epitaxy “ edited by R. F. C. Farrow
GaN RHEED Oscillation
Growth rate reduction due to
thermal decomposition
Surface Reconstruction
• In order to minimize the energy of the near-surface
region of the crystal, the atoms rearrange themselves in
a regular fashion which exhibits long range order. Each
ordered arrangement of the near surface region is known
as a surface reconstruction.
• Surface reconstruction reflects the stoichiometry of the
growth process and influences the growth mechanism.
• RHEED is used to monitor the surface reconstruction of
the growth front since it is sensitive to atomic layers near
the surface.
• Usually surface reconstruction is affected by substrate
temperature, impinging fluxes, III/V flux ratio, and the
existence of surfacants.
“Molecular Beam Epitaxy”
edited by R. F. C. Farrow
“Molecular Beam Epitaxy “ edited by R. F. C. Farrow
Polarity of GaN c-plane surface
GaN Surface Reconstruction
2-fold
3-fold
As as a Surfacant