Transcript Lecture2 - Bama.ua.edu

Lecture 2
PH 4891/581, Jan. 9, 2009
This file has parts of Lecture 2.
First, here is the link to Dr. Spears’ website on
semiconductor physics. Look at the chapter on
semiconductor crystal structures, which has some
general material on crystals in general. However, the
phrase “basis vector” is used incorrectly, illustrating the
risks of getting information from the internet, even from
well-known persons.
Rotational symmetries
2D lattices:determined by primitive translation vectors a1 and a2
Square has 4-fold (90º = 360º / 4) rotation symmetry
a2
90º
a1
Rectangle does not:
Rotational symmetry of lattices
Rotate these in PowerPoint to determine rotational symmetry group
Square lattice:
Rectangular lattice:
Oblique lattice
ALL lattices have 180º symmetry!
“Systems” (different rotational
symmetries) in 2D
• Oblique (0º, 180º rotation only)
• Square (0º, 90º, 180º, 270º, 4 mirrors)
• Rectangular (0º, 180º, 2 mirrors)
• Hexagonal (0º, 60º, 120º, 180º, 240º ,
300º, 6 mirrors)
Not done! Rectangular system has
2 possible lattices
Rectangular system (symmetries are 0º,
180º rotations & 2 mirrors)
Has rectangular lattice:
Add atoms at centers of
rectangles:
Same symmetries!
“Centered rectangular lattice”
So there are 5 “Bravais lattices” in 2D.
So the 4 2D symmetry systems
have a total of 5 Bravais lattices
• Oblique (2-fold rotation only)
– One lattice (“oblique”)
• Square (4-fold rotations, 4 mirrors)
– One lattice (“square”)
• Rectangular 2-fold, 2 mirrors)
– TWO lattices: rectangular
and
– centered rectangular
• Hexagonal (6-fold rotations, 6
mirrors) -- One lattice (“hexagonal”)