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Calculating Band Structure
Nearly free electron• Assume plane wave solution for electrons
• Weak potential V(x)
• Brillouin zone edge
Tight binding method
• Electrons in local atomic states (bound states)
• Interatomic interactions >> lower potential
• Unbound states for electrons
• Energy Gap = difference between bound / unbound states
Crystal Field Splitting
• Group theory to determine crystalline symmetry
• Crystalline symmetry establishes relevant energy levels
• Field splitting of energy levels
However all approaches assume a crystal structures. Bands and energy gaps still exist without the need for crystalline structure. For these systems, Molecular Orbital theory is used.
Free Electron Model
• Energy bands consist of a large number of closely spaced energy levels.
• Free electron model assumes electrons are free to move within the
metal but are confined to the metal by potential barriers.
• This model is OK for metals, but does not work for semiconductors since
the effects of periodic potential have been ignored.
Kronig-Penny Model
• This model takes into account the effect of periodic arrangement of
electron energy levels as a function of lattice constant a
• As the lattice constant is reduced, there is an overlap of electron
wavefunctions that leads to splitting of energy levels consistent
with Pauli exclusion principle.
Energy bands for diamond versus lattice constant.
A further lowering of the
lattice constant causes the
energy bands to split again
Conduction / valence bands
Bound states
Conduction band states
Free electron model
Valence band states
Conduction / valence bands
Conduction band states
Lowest Unoccupied
Molecular Level
(LUMO)
Valence band states
Highest Occupied
Molecular Orbital
(HOMO)
Example band structures
Find:
Valence bands?
Conduction bands?
Energy Gap?
Highest Occupied Molecular
Level (HOMO)?
Lowest Unoccupied Molecular
Level (LUMO)?
Ge Si GaAs
Simple Energy Diagram
A simplified energy band diagram used to describe semiconductors. Shown
are the valence and conduction band as indicated by the valence band edge,
Ev, and the conduction band edge, Ec. The vacuum level, Evacuum, and the
electron affinity, , are also indicated on the figure.
Metals, Insulators and
Semiconductors
Possible energy band diagrams of a crystal. Shown are: a) a half filled band,
b) two overlapping bands, c) an almost full band separated by a small
bandgap from an almost empty band and d) a full band and an empty band
separated by a large bandgap.
Semiconductors
• Filled valence band (valence = 4, 3+5, 2+6)
• Insulator at zero temperature
Semiconductors Si, Ge
Filled p shells
4 valence electrons
Metals
Free electrons
Valence not 4
Binary system III-V: GaAs, InP, GaN, GaP
Binary II-VI: CdTe, ZnS,
Energy bands in Electric Field
Energy band diagram in the presence of a uniform electric field. Shown are
the upper almost-empty band and the lower almost-filled band. The tilt of
the bands is caused by an externally applied electric field.
Electrons travel down.
Holes travel up.
The effective massThe presence of the periodic potential, due to the atoms in the crystal without
the valence electrons, changes the properties of the electrons. Therefore, the
mass of the electron differs from the free electron mass, m0. Because of the
anisotropy of the effective mass and the presence of multiple equivalent band
minima, we define two types of effective mass: 1) the effective mass for density
of states calculations and 2) the effective mass for conductivity calculations.
Electron excited out of
valence bandTemperature
Light
Defect
…
Electron in conduction
band state
Empty state in valence
band (Hole = empty
state)
Motion of Electrons and Holes in Bands
Electrons - holes
Electron in conduction bandNOT localized
Hole in valence bandUsually less Mobile (higher
effective mass), but not always
Electron – hole pairs in different bands
large separation
Region Near Gap
In the region near the gap,
Local maximum / minimum
dE/dk = 0
effective mass m* = h2/(d2E/dk2)
Electrons
Minimum energy
Bottom of conduction band
Holes
Opposite E(k) derivative
“Opposite effective charge”
Top of valence band
kx
e(k)
Conduction
band
Valence
band
General Carrier Concentration
Gap
Conduction
band
Valence
band
Probability of hopping into state
n0 = (number of states / energy) * energy distribution
gc (E) = density of states
f (E) = energy distribution
Density of states
The density of states in a semiconductor equals the density per unit volume
and energy of the number of solutions to Schrödinger's equation.
Calculation of the number of states with wavenumber less than k
Fermi-surface (3-D)
• K-space
– Set of allowed k
vectors
• Fermi surface
– Electrons occupy
all kf2 states less
than Ef*2m/h
– kF ~ wavelength
of electron
wavefunction
2p/L
Allowed state
for k-vector
kx
ky
NLL
kFkF
3
63
3
2
3
)/2(
1
3
4pp
p
Area of sphere / k states in spheresVolume in lattice
Density of stateshttp://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm
Number of states:
Density in energy:
Kinetic energy of electron:
Density of states / energy:
In conduction band, Nc:
3
6 2
3
2 LN Fk
p
Different m*
in conduction and
valence band
Probability density functionsThe distribution or probability density functions describe the probability that
particles occupy the available energy levels in a given system. Of particular
interest is the probability density function of electrons, called the Fermi function.
The Fermi-Dirac distribution function, also called Fermi function, provides the
probability of occupancy of energy levels by Fermions. Fermions are half-
integer spin particles, which obey the Pauli exclusion principle.
Fermi-Dirac vs other distributions
Intrinsic: Ec – Ef = ½ Eg
High temperature:
Fermi ~ Boltzmann
Maxwell-Boltzmann:
Noninteracting particles
Bose-Einstein: Bosons
Carrier DensitiesThe density of occupied states per unit volume and energy, n(E), ), is simply
the product of the density of states in the conduction band, gc(E) and the
Fermi-Dirac probability function, f(E).
Since holes correspond to empty states in the valence band, the probability
of having a hole equals the probability that a particular state is not filled, so
that the hole density per unit energy, p(E), equals:
Carrier Densities
Product of density of states and distribution
-- defines accessible bands
-- within kT of Ef
Limiting Cases
0 K:
Non-degenerate semiconductors: semiconductors for which the Fermi
energy is at least 3kT away from either band edge.
Mass Action Law
The product of the electron and hole density equals the square of the
intrinsic carrier density for any non-degenerate semiconductor.
The mass action law is a powerful relation which enables to quickly
find the hole density if the electron density is known or vice versa
Doped SemiconductorAdd alternative element for electron/holes
Si valence = 4 P valence = 5 B valence=3
Si = Si = Si = Si =
== ==
Si = Si = Si = Si =
== ==
Si = Si = Si = Si =
== ==
Si = Si = Si = Si =
== === ===
Si = Si = Si = Si =
== ==
Si = Si = P = Si =
== ==
Si = Si = Si = Si =
== ==
Si = Si = Si = Si =
== === ===
Si = Si = Si = Si === ==
Si = Si -- B = Si === ==
Si = Si = Si = Si =
== ==
Si = Si = Si = Si =
== === ===
Pure Si
All electron paired
Insulator at T=0
Phosphorous
n-doped
Electron added to
conduction band
Boron
p-doped
Positive hole added to
conduction band
Hole
e-
Dopant Energy levels
Si
Eg=1.2eV
P 0.046eV
B 0.044eV
As 0.054eV
Cu 0.24eV
Cu 0.40eV
Cu 0.53eVAu 0.54eV
Au 0. 35eV
Au 0. 29eV
Energy required to donate hole
Energy required
to donate electronEasily ionized
= easily donate electrons
Large energy bad.Add scattering
Donate no carriers
Carrier concentration in thermal
equilibrium
• Carrier concentration vs. inverse temperature
1/T(K)
Thermally activated
Intrinsic carriers
N(carriers) = N(dopants)
Activation of dopants
Region of
Functional device
ne
Dopants and Fermi Level
• Free electron metal:
• Intrinsic semiconductor– n(electrons) = n(holes)
– Fermi energy = middle
• n-doped material – n(electrons) >> n(holes)
– Fermi level near conduction band
• p-doped materials– n(electrons) >> n(holes)
– Fermi level near conduction band
m
kkn F
FF
e2
,3
22
2
3
e
p
Ec
Ev
Ef
Ec
Ev
Ef
Ec
Ev
Ef
Fermi Energy is not material specific but depends on doping level and type
Mobility and Dopants
• Dopants destroy periodicity
– Scattering, lower mobility
e
1E14 1E15 1E16 1E17 1E18
Dopant Concentration (cm-3)
10000
1000
100
Mobility
(cm2/V-s)
GaAse
hSi
e