Physics PY4118
Physics of Semiconductor Devices
Crystalography
Taken mostly from: Crystal2.ppt
http://www.ems.psu.edu/~ryba/coursework/zhong%20shan%20da%2
0xue%20-%20course%20materials/class%20slides/
Coláiste na hOllscoile Corcaigh, Éire University
College Cork, Ireland
ROINN NA FISICE Department of Physics
2.1
Symmetry?
This is actually really important for some
semiconductor devices, especially:
Inversion Symmetry:
This is (not) required for:
Second harmonic generation
The electro-optic effect
Piezo-electric effect
etc.
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
3.2
ELEMENTS OF SYMMETRY
Each of the unit cells of the 14 Bravais lattices has one or
more types of symmetry properties, such as inversion,
reflection or rotation,etc.
SYMMETRY
INVERSION
REFLECTION
ROTATION
3
Lattice goes into itself through
Symmetry without translation
Operation
Element
Inversion
Point
Reflection
Plane
Rotation
Axis
Rotoinversion
Axes
4
Reflection Plane
A plane in a cell such that, when a mirror reflection in this
plane is performed, the cell remains invariant.
5
Rotation Axis
90°
120°
180°
This is an axis such that, if the cell is rotated around it
through some angles, the cell remains invariant.
The axis is called n-fold if the angle of rotation is 2π/n.
6
Stereographic projections of symmetry groups
Types of pure rotation symmetry
Rotation 1, 2, 3, 4, 6
Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Draw point group diagrams (stereographic projections)
symmetry elements
equivalent points
7
Stereographic projections of symmetry groups
Types of pure rotation symmetry
Rotation 1, 2, 3, 4, 6
Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Draw point group diagrams (stereographic projections)
symmetry elements
equivalent points
8
Stereographic projections of symmetry groups
Types of pure rotation symmetry
Rotation 1, 2, 3, 4, 6
Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Draw point group diagrams (stereographic projections)
symmetry elements
equivalent points
9
Stereographic projections of symmetry groups
Types of pure rotation symmetry
Rotation 1, 2, 3, 4, 6
Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Draw point group diagrams (stereographic projections)
All objects,
structures with
i symmetry are
centric
symmetry elements
equivalent points
10
Stereographic projections of symmetry groups
Types of pure rotation symmetry
Rotation 1, 2, 3, 4, 6
Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Draw point group diagrams (stereographic projections)
symmetry elements
equivalent points
11
Stereographic projections of symmetry groups
Types of pure rotation symmetry
Rotation 1, 2, 3, 4, 6
Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Draw point group diagrams (stereographic projections)
symmetry elements
equivalent points
12
Stereographic projections of symmetry groups
More than one rotation axis - point group 222
symmetry elements
equivalent points
13
Stereographic projections of symmetry groups
More than one rotation axis - point group 222
symmetry elements
equivalent points
14
Stereographic projections of symmetry groups
More than one rotation axis - point group 222
symmetry elements
equivalent points
orthorhombic
15
Stereographic projections of symmetry groups
More than one rotation axis - point group 222
[100]
16
Stereographic projections of symmetry groups
More than one rotation axis - point group 222
[010]
[100]
17
Stereographic projections of symmetry groups
More than one rotation axis - point group 222
[001]
[010]
[001]
[100]
[010]
[100]
18
Stereographic projections of symmetry groups
Rotation + mirrors - point group 4mm
[001]
19
Stereographic projections of symmetry groups
Rotation + mirrors - point group 4mm
[100]
20
Stereographic projections of symmetry groups
Rotation + mirrors - point group 4mm
[001]
[110]
[010]
[100]
[110]
21
Stereographic projections of symmetry groups
Rotation + mirrors - point group 4mm
symmetry elements
equivalent points
tetragonal
22
Stereographic projections of symmetry groups
Rotation + mirrors - point group 2/m
[010]
23
Stereographic projections of symmetry groups
Rotation + mirrors - point group 2/m
symmetry elements
equivalent points
monoclinic
24
And here are the 32 point groups
System
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Cubic
Hexagonal
Trigonal
Point groups
1, 1
2, m, 2/m
222, mm2, 2/m 2/m 2/m
4, 4, 4/m, 422, 42m, 4mm, 4/m 2/m 2/m
23, 2/m 3, 432, 43m, 4/m 3 2/m
6, 6, 6/m, 622, 62m, 6mm, 6/m 2/m 2/m
3, 3, 32, 3m, 3 2/m
25