EE 60556: Fundamentals of Semiconductors Lecture Note

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EE 60556 Fundamentals of SemiconductorsLecture
Note 12 (10/08/09)Free electron (or empty
lattice) band structure and various methods for
band structure calculation

• Outline
• Last class Kronig-Penney model (a simple QM
  treatment ? bandgap)
• Formation of free electron band structure in the
  reduced Brillouin zone (also see my notes)
• No practical usage in research topics but to
  understand how energy is folded into the first
  Brillouin zone and origin of energy degeneracy in
  band structures.
• Central equation ? bandgap near the zone edge
  (Kittel)
• Tight binding model, pseudopotential method
  (Kittel)
• k?p (handout)
• 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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Kronig-Penney model (a simple QM treatment)
xgt1, i.e. EgtU0
What happens when xgt U0? - Bound states still
exist since electrons can not escape the crystal.
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Why the band structure E(k) only covers part of
1st BZ? (again)
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Empty lattice energy bands (free electron) How
to construct E(k) in reduced zone
Recall that the reciprocal lattice is also
composed of repeating unit cells ? find the
equivalent G, X points etc. in every unit cell as
defined in the 1st BZ. (do not get confused with
1st BZ, 2nd BZ, etc, which is onion-like.)
The lowest energy state is at G0 point!

• Extended zone ? reduced zone
• When I fold the energy bands outside the 1st BZ,
  I am actually plotting E(k) of other unit cells
  in the equivalent region.
• For instance, in 1D crystal, I will fold all
  (Gi-Xi) bands into (G0-X0).
• While it is easy to do so from graphs for 1D, it
  is not for 3D. Better to use math to write down
  E(k) for each band in each cell.

Irreducible region of the BZ
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Empty lattice energy bands (free electron model)
In this entire semester, we use either K or G to
denote a reciprocal lattice vector
G0 X0
G-1 X-1
G1 X1

• Extended zone ? reduced zone
• For instance, in 1D crystal, I will fold all
  (Gi-Xi) bands into (G0-X0).
• While it is easy to do so from graphs for 1D, it
  is not for 3D. Better to use math to write down
  E(k) for each band in each cell.

Similarly we can apply this technique to 2D 3D
crystals
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Empty lattice energy bands (free electron model)
Band Structures
Irreducible region of the BZ
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Empty lattice energy bands (free electron model)
see my notes for details
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8 fold degenerate energy point 6 fold degenerate
Empty lattice energy bands
Practice 3D
Degeneracy means that there are more than one
quantum state (multiple k-values) corresponding
to the same energy level (single E value) Note
generally speaking, spin (another momentum) is
not considered in k. I.e. every k state can host
2 electrons (spin-up and spin-down).
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Review perturbation theory (see k.p handout also)

• First, solve a known potential (e.g. an isolated
  atom) problem (en, n are eigenvalues and
  eigenfunctions)
• Define new Hamiltonian (W is the perturbation)
• If the eigenvalues are non-degenerate, the first
  order eigenfunction correction is none and the
  first order energy correction is
• The second order energy correction and eigenvalue
  corrections are
• The total energy is

ket
bra
Bra-(c)-ket notation (Dirac notation) describes
normalized wavefunctions in QM. Bra is conjugate
transpose of ket and vice versa. An operator
always operates on the ket first and then
multiplies bra.
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Wave equation of electron in a periodic potential
central equation (general approach useful to
evaluate bandgap at the BZ boundary) its
usefulness is to see how the potential strength
links to the bandgap!
(K-Ch.7)
Similarly, the electron wavefunction must be able
to be expressed using its Fourier series.
The central equation a set of linear equations
as many as there are coeff. C. Most useful when
there are few terms of UG (shorter eqns) a few
Ck (a few eqns)
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The central equation a set of linear equations
as many as there are coeff. C. Most useful when
there are few terms of UG
Note that the size of the matrix depends on of
C(k), the more C(k), the larger the matrix, thus
the more accurate the solution. However, in
practice we can not deal with an infinitely large
matrix. We truncate it into 5x5 or even smaller
2x2 matrix to get a rough physical picture. We
have to be smart to choose the important
terms. Example next approximate solution near
the zone boundary
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Approximate solution at the zone boundary

• Assume the Fourier component UG of the potential
  energy are small in comparison with the kinetic
  energy of a free electron at the zone boundary.

• Since the kinetic energy of two waves with k
  -G/2 G/2 for an free electron are the same
  (degenerate), they are likely playing important
  role in the energy state at the zone boundary
  (perturbation theory says degenerate states
  without perturbation most likely will split under
  perturbation.).

• Next, we will retain only two coefficients C(G/2)
  C(-G/2) in the central eqn (? 2 eqns only in
  the set) and evaluate the approximate solution at
  the zone boundary. Assuming a crystal
  potential shown on the right (? 2 terms only in
  the summation).

k - G/2
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Approximate solution at the zone boundary
k - G/2
Take-home message Bandgap is proportional to the
potential strength.
Recall We can solve for the wavefunction at the
zone boundary. Not surprisingly, we get the
standing wave solutions
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Approximate solution near the zone boundary
Still retain two terms only in the wavefunction
Non-trivial solution requires
Two roots gt two bands
Take-home message the band structure is
parabolic near the bottom of C.B. and the top of
V.B.
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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)

• 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)
(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.