Download Lecture 5

The Ig Nobel Prizes are  “Booby Prizes”!
Electronic Band Structures
BW, Ch. 2; YC, Ch 2; & S, Ch. 2
~ Follow BW & YC + input from S & outside sources where appropriate
Band Structure  E(k)
• We are interested in the behavior of electronic energy
levels in crystalline materials (any, not just semiconductors!)
as a function of wavevector k or momentum p = ħk
• Group Theory: The math of symmetry that is
useful to simplify calculations of E(k).
• A Mathematical Subject! Detailed math coverage &
discussion will be kept to a minimum. Results &
PHYSICS will be emphasized over math!
• Many calculational methods exist:
– They are highly sophisticated & computational!
Basic knowledge assumed for this discussion:
1. You know that electron energies are quantized.
2. You’ve seen the simple Schrödinger Equation
(from quantum mechanics)
Note: If you are weak on this or need a review, please get
& read an undergraduate quantum mechanics book!
3. You are familiar with the crystal structures of some
simple crystals, such as those we recently discussed.
Note: If you are weak on this or need a review, please
get & read an undergraduate solid state book!
4. In a solid, the electronic energy levels form into
regions of allowed energy (bands) & forbidden energy (gaps).
See any undergraduate solid
state physics book!
• Overview: A qualitative/semi-quantitative discussion
now. Later, a detailed & quantitative discussion.
Electronic Energy Bands  E(k)
These describe the behavior of the electronic energy
levels in crystalline materials as function of
wavevector k or momentum p = ħk.
• Recall that for FREE electrons, the energies have
the simple form: E(k) = (p)2/2mo = (ħk)2/2mo
(mo= free electron mass)
• However, in crystalline solids: E(k) forms into
regions of allowed energies (bands) & regions of
forbidden energies (gaps).
• As we’ll see soon,
Often in solids, a good Approximation is:
E(k) = (ħk)2/2m*
m* is not mo!
m* = m*(k)  “effective mass”
However, the true E(k) are complicated!
• Accurate calculations of E(k) require
sophisticated quantum mechanics, AND they
also require sophisticated computational
methods to obtain them numerically.
Qualitative Picture of Bands in “r-Space”
(as functions of position in the solid)
One way to distinguish between solid types is by
how the electrons fill the bands & by the band gaps.
Electronic Energy Bands
Qualitative Picture
Allowed
band
Forbidden
gap
Allowed
band
Forbidden
gap
Allowed
band
Number of Atoms 
Electrons occupying
quantized energy
states around a
nucleus must obey the
Pauli Exclusion
Principle.
This prevents more
than 2 electrons (of
opposite spin) from
occupying the same
level.
Semiconductors, Insulators, Conductors
Full Band
Empty Band
All energy levels
are occupied
All energy levels
are empty
(contain electrons)
It can be shown that
Neither full nor empty bands participate
in electrical conduction.
• Consider the case of an Intrinsic Semiconductor (a
material with no impurities, so there are no acceptors or donors).
That is, there is no doping.
• For such a material, the highest energy band that is
completely filled with electrons (at T = 0 K) is called the highest
Valence Band.
• The next highest band is completely empty of electrons
(at T = 0K) & is called the lowest
Conduction Band.
• The Energy Difference between the bottom of
the Conduction Band & the top of the Valence
Band is called the Fundamental Band Gap.
(Sometimes also called “Band Gap” or “Bandgap”, or “Gap” )
• So, for such an Intrinsic Semiconductor, for energies near
the Band Gap, A Qualitative Picture of the Bands in “rSpace” (as functions of position ) looks like the figure below.
• Consider the Conduction
Process for this material.
Suppose that somehow, some
electrons are added to the
conduction band & some holes
are added to the valence band.
Electron Conduction
• In an applied electric field, the electrons in the conduction
band will move almost like free particles.
Hole Conduction
• Holes are treated similarly to positively charged particles in the valence
band. In an applied field, they behave almost like free particles.
Intrinsic Semiconductors
• Consider a nominally pure
semiconductor at T = 0 K.
• There are no electrons in
the conduction band
• At T > 0 K a small
fraction of electrons is
thermally excited into the
conduction band,
“leaving” the same
number of holes in the
valence band.
Calculated Si Bandstructure in k Space
GOALS
After this chapter, you should:
 Eg
1. Understand the underlying Physics
behind the existence of bands & gaps.
2. Understand how to interpret this
figure.
3. Have a rough, general idea about how
realistic bands are calculated.
4. Be able to calculate energy bands for
some simple models of a solid.
Note
Si has an indirect band gap!
Brief Quantum Mechanics (QM)
& Solid State (SS) Review
• QM results that I must assume that you know!
The Schrödinger Equation
(time independent; see next slide) describes electrons
– The solutions to the Schrödinger Equation result
in quantized (discrete) energy levels for electrons.
• SS results that I must assume that you know!
– Solutions to the Schrödinger Equation in a
periodic crystal give bands (allowed energy
regions) & gaps (forbidden energy regions).
Quantum Mechanics (QM)
• The Schrödinger Equation: (time independent)
Hψ = Eψ
This is a differential eigenvalue equation.
H  Hamiltonian operator for the system
(energy operator)
E  Energy eigenvalue, ψ  wavefunction
Particles are QM waves!
|ψ|2  probability density; ψ is a function of ALL
coordinates of ALL particles in the problem!
The Physics Behind E(k)
E(k)  Solutions to the Schrödinger
Equation for an electron in a solid.
QUESTIONS
Why (qualitatively) are there bands?
Why (qualitatively) are there gaps?
Bands & Gaps
• These can be understood from two
(very different) qualitative pictures!
• The two pictures are models (& Opposite
Limiting Cases) of the true situation.
• An electron in a perfectly periodic crystalline solid:
– The potential seen by this electron is perfectly periodic
The existence of this periodic
potential is  the cause of the
bands & the gaps!
Qualitative Picture #1
“A Physicist’s viewpoint”- The solid is looked at “collectively”
Almost Free Electrons (done in detail in Kittel’s Ch. 7!)
For free electrons: E(k) = (p)2/2mo = (ħk)2/2mo
Almost Free Electrons:
• Start with the free electron E(k), add small
(weakly perturbing) periodic potential V.
• This breaks up E(k) into bands (allowed
energies) & gaps (forbidden energy regions).
• Gaps: Occur at the k where the electron waves
(incident on atoms & scattered from atoms) undergo
constructive interference (Bragg reflections!)
Qualitative Picture #1
Forms the basis for REALISTIC bandstructure computational methods!
• Starting from the almost free electron viewpoint & adding a
high degree of sophistication & theoretical + computational rigor:
 Results in a method that works VERY WELL for
calculating E(k) for metals & semiconductors!
• An “alphabet soup” of computational techniques:
– OPW: Orthogonalized Plane Wave method
– APW: Augmented Plane Wave method
– ASW: Antisymmetric Spherical Wave method
– Many, many others
The Pseudopotential Method
(the modern method of choice!)
Qualitative Picture #2
“A Chemist’s viewpoint”
The solid is looked at as a collection of atoms & molecules.
Atomic / Molecular Electrons
• Atoms (with discrete energy levels) come
together to form the solid.
• Interactions between the electrons on
neighboring atoms cause the atomic energy
levels to split, hybridize, & broaden.
Quantum Chemistry!
• First approximation: Small interaction V! Occurs
in a periodic fashion (the interaction V is
periodic).
Qualitative Picture #2
Quantum Chemistry!
• First approximation: Small interaction V! Occurs
in a periodic fashion (interaction V is periodic).
• Groups of levels come together to form bands (& gaps).
• The bands E(k) retain much of the character of
their “parent” atomic levels (s-like & p-like bands, etc.)
Gaps:
• Also occur at the k where the electron
waves (incident on atoms & scattered from
atoms) undergo constructive interference
(Bragg reflections!)
Qualitative Picture #2
• Starting from the atomic / molecular electron
viewpoint & adding a high degree of sophistication
& theoretical & computational rigor
 Results in a method that works VERY WELL for
calculating E(k) (mainly the valence bands) for
insulators & semiconductors!
(Materials with covalent bonding!)
• An “alphabet soup” of computational techniques:
– LCAO: Linear Combination of Atomic Orbitals method
– LCMO: Linear Combination of Molecular Orbitals method
– The “Tightbinding” method & many others.
The Pseudopotential Method
(the modern method of choice!)
Bandstructure Theories in Crystalline Solids
 Pseudopotential Method 
 Tightbinding (LCAO) Method 
 Electronic Interaction 
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Semiconductors, 
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Metals
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Insulators
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Almost Free
Molecular 
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Electrons
Electrons 
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Isolated Atom,
Free
Atomic Electrons
Electrons