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Review of Semiconductor Physics
Solid-state physics
The daunting task of solid state physics
• Quantum mechanics gives us the fundamental equation
• The equations are only analytically solvable for a handful of special cases
• One cannot solve the equations for more than two bodies!
• Solid-state physics is about many-body problems
There are 5 × 1022 atoms/cm3 in Si
Si atom: 1s22s22p63s23p2
Core: Nueclear + 1s22s22p6, Valence electrons: 3s23p2
We’ll come back to this later
Each particle is in the potential of all the other particles, which depends on
their positions, which must be solved from the equation…
You have an equation with ~1023 unknowns to solve. Mission impossible!
• Solid state physic is all about approximations.
Review of Semiconductor Physics
Crystal structures
If we assume the atomic cores have known and fixed positions, we only need to solve
the equations for the valence electrons. Life much easier!
Static lattice approximation
• Justification
• Related/similar approximation: Born-Oppenheimer
Crystal structures
If you shine X-ray on a piece of solid, very likely you’ll have a diffraction pattern.
Remember Bragg?
That means periodicity in the structure.
Review of Semiconductor Physics
Crystal structures
Bravais Lattices
A mathematical concept:
• No boundary or surface
• No real (physical) thing – just points, hence no defects
• No motion
Unit cells (or primitive unit cells) -- The smallest unit that repeats itself.
Fig. 4.1
Fig. 4.2
Crystal structure = lattice + basis
Honeycomb
Simple cubic
From Geim & McDonald, Phys Today Aug 2007, 35.
Lattices
Conventional & primitive unit cells
BCC
How many atoms in the
conventional unit cell?
BCC & FCC are Bravais Lattices.
FCC
U. K. Mishra & J. Singh, Semiconductor Device Physics and Design
E-book available on line thru UT Lib.
Fast production of e-books. The caption is NOT for this figure.
Try not to be confused when reading fast generated books/papers nowadays.
Bragg refraction and the reciprocal lattice
• Bragg refraction
• Definition of the reciprocal lattice
• 1D, 2D, and 3D
The 1D & 2D situations are not just mathematical practice or fun, they can
be real in this nano age…
• BCC & FCC are reciprocal lattices of each other
4
4
4
4
4
4
• Miller indices
Referring to the origin of the reciprocal lattice’s definition, i.e, Bragg refraction,
a reciprocal lattice vector G actually represents a plane in the real space
z
y
x
(100)
(200)
Easier way to get the indices:
Reciprocals of the intercepts
• Wigner-Seitz primitive unit cell and first Brillouin zone
The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that
are closer to that lattice point than to any of the other lattice points.
The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby
(closest) lattice points. At the midpoint of each line, another line (or a plane, in 3D) is
drawn normal to each of the first set of lines.
1D case
2D case
3D case: BCC
The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
1D
2D
3D:
Recall that the reciprocal lattice of FCC is BCC.
4
4
4/a
Why is FCC so important?
4
Why is FCC so important?
It’s the lattice of Si and many III-V semiconductors.
Si: diamond, a = 5.4 Å
GaAs: zincblende
Crystal structure = lattice + basis
Modern VLSI technology uses the (100) surface of Si.
Which plane is (100)? Which is (111)?