EE 60556: Fundamentals of Semiconductors Lecture Note

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EE 60556 Fundamentals of SemiconductorsLecture
Note 13 (10/13/09)Midterm exam review, Tight
binding model and electron effective mass in a
crystal

• Outline
• Last class post midterm review, central equation
  ? bandstrucutre near the zone edge (Kittel)
• Tight binding model, pseudopotential method
  (Kittel ch.9, optional)
• k?p (handout optional)
• Effective mass, group velocity, bandgap, direct
  or indirect
• Crystal momentum of an electron (Kittel p.173)
• Note that (1) no theoretical model can calculate
  bandgap accurately, but they do a reasonable job
  at calculating E-k diagram.
• (2) After brief discussion of band structure
  calculation, we will focus on how to read a band
  structure (goal of this course)!

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Methods to calculate bandstructures
Midterm exam 1 review 45 minutes

• Solve the Schrodinger Equation and apply the
  Block theorem
• Simplify the complicated crystal potential to
  something solvable. E.g. Kronig-Penney model.
• Treat the complicated crystal potential as a sum
  of a simpler potential (solvable Schrodinger Eqn)
  and potential perturbation. E.g. near
  free-electron model (plane waves perturbation),
  k.p perturbation theory and tight-binding model
  (atomic orbitals perturbation).
• Numerical methods, e.g. density function theory
  and quantum Monte Carlo
• Etc. For more information, see the following
  websites for instance,
• http//en.wikipedia.org/wiki/Density_functional_th
  eory
• http//en.wikipedia.org/wiki/Electron_configuratio
  n

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Tight-binding model (K- Ch.9)

• Assuming electron energy ltlt crystal potential
  energy, thus the atomic orbitals (s-state,
  p-state etc.) describing electron energy in an
  isolated atom remains to be the backbone of the
  electron states. The influence of other atoms to
  the total crystal potential is treated as a small
  perturbation.
• Generally, only contributions from the nearest
  neighbor atoms are considered.

?E
The stronger the interactions between atoms, the
wider the spread of the band (?E)
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Tight-binding model let us take a look at the
atomic orbitals
It is also called the LCAO approximation linear
combination of atomic orbitals
The rest 1/(4p)1/2 (p/4)1/2 (p/4)1/2 (p/4)1/2
Angular part of the wavefunctions
Bond directions along lt111gt in Si/GaAs
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Tight-binding model nature of the resultant C.B
and V.B.
Conduction band
Valence band
C.B s-like (direct semi) ? isotropic more
indirect ? more p ? anisotropic V.B. p-like
Generally at very low energy so that often not
shown in band structure
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Tight-binding method (K- Ch.7, optical reading)

• Electron total wavefunction sum of one
  eigenfunction (atomic orbital of an isolated
  atom, e.g. s-state) of all atoms
• Assuming proper Ckj so that the wavefunction
  satisfies the Block Theorem (assuming single atom
  basis and N atoms in the crystal)
• Energy with the first order correction
• (H is the hamiltonian of the crystal)
• Rewrite
• Obtain an energy band stemming from this orbital

Atomic orbital at atom m
Energy associated with this atomic orbital at an
isolated atom
Energy associated with interatomic interaction
between the nearest neighbors (which generally
decays exponentially with interatomic distance,
therefore only nearest neighbors are considered).
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Tight-binding method (K- Ch.7, optional reading)
(10)
Energy band is spread within 2?6 12?. The
stronger the interatomic interaction, the wider
the energy band. ? the higher the curvature ? the
smaller the effective mass. (what is an effective
mass? Next slide!)
?E
?E
Can you derive the effective mass tensor for
these two crystals? (practice see next slide!)
Tight-binding method is quite good for the inner
electrons of atoms, but not often a good
description of the conduction electrons. It is
used to describe approximately the d bands of the
transition metals and the valence bands of
diamond-like and inert gas crystals.
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Now we can derive a bandstructure (E-k diagram),
what information can we read from it? Effective
mass, group velocity. Let us also revisit the
physical meaning of k the wave vector
Kittel p.173 significance of k 1) the phase
factor in Bloch function, 2) electron crystal
momentum (not its total momentum) and used in the
conservation laws that govern collision processes
in crystals.
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Effective Mass

• Parabolic approximation
• Bigger the curvature, smaller effective mass
• light holes and heavy holes

Electrons in crystal obey the Newtons law when
the external force and its wave vector are used.
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Effective Mass

• Importance the motion of electrons in a crystal
  can be visualized and described in a
  quasi-classical manner!
• electron billiard ball Newtonian mechanics
  because m accounts for the effect of crystal
  forces and quantum mechanical properties!

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4-fold degenerate at the bottom of Ge conduction
band
6-fold degenerate
Constant energy surface in conduction band