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.
A further lowering of the
lattice constant causes the
energy bands to split again
Energy bands for diamond versus lattice constant.
Formation of Bands
Periodic potential
Band gap
Inter-atom interactions
Many more states
Conduction / valence bands
Free electron model
Conduction band states
Valence band states
Bound states
Conduction / valence bands
Conduction band states
Lowest Unoccupied
Molecular Level
(LUMO)
Valence band states
Highest Occupied
Molecular Orbital
(HOMO)
Electrons fill from bottom up
Semiconductor = filled valence band
Example band structures
Ge
Si
GaAs
Find:
Valence bands?
Conduction bands?
Energy Gap?
Highest Occupied Molecular
Level (HOMO)?
Lowest Unoccupied Molecular
Level (LUMO)?
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
Metals
Free electrons
Valence not 4
Binary system III-V: GaAs, InP, GaN, GaP
Binary II-VI: CdTe, ZnS,
Semiconductors Si, Ge
Filled p shells
4 valence electrons
Eg Temperature Dependence
Eg Doping Dependence
Doping, N, introduces impurity bands that lower the bandgap.
Energy bands in Electric Field
Electrons travel down.
Holes travel up.
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.
The effective mass
The 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.
Motion of Electrons and Holes in Bands
Electron excited out of
valence band
Temperature
Light
Defect
…
Electron in conduction
band state
Empty state in valence
band (Hole = empty
state)
Electrons - holes
Electron in conduction band
NOT localized
Hole in valence band
Usually 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
e(k)
Conduction
band
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
Valence
band
General Carrier Concentration
Probability of hopping into state
n0 = (number of states / energy) * energy distribution
Conduction
band
Gap
gc (E) = density of states
f (E) = energy distribution
Valence
band
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)
ky
• K-space
Allowed state
for k-vector
– Set of allowed k
vectors
• Fermi surface
– Electrons occupy
all kf2 states less
than Ef*2m/h
– kF ~ wavelength
of electron
wavefunction
kx
2p/L
Volume in lattice
Area of sphere / k states in spheres
 4pk F 3 

1


 3  (2p / L)3  



kF 3
6p
2
L3  N
Density of states
http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm
Number of states:
kF 3
N  2 6p 2 L3
Density in energy:
Kinetic energy of electron:
Density of states / energy:
In conduction band, Nc:
Different m*
in conduction and
valence band
Density of States in 1, 2 and 3D
Probability density functions
The 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 halfinteger spin particles, which obey the Pauli exclusion principle.
Fermi-Dirac vs other distributions
Maxwell-Boltzmann:
Noninteracting particles
Bose-Einstein: Bosons
Intrinsic: Ec – Ef = ½ Eg
High temperature:
Fermi ~ Boltzmann
Carrier Densities
The 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
Carrier Densities
Electrons
Holes
Limiting Cases
0 K:
Non-degenerate semiconductors: semiconductors for which the Fermi
energy is at least 3kT away from either band edge.
Intrinsic Semiconductor
Intrinsic semiconductors are usually non-degenerate
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 Semiconductor
Add alternative element for electron/holes
=
=
=
=
=
=
=
=
=
=
=
=
=
Si = Si = Si = Si =
=
=
=
=
=
Si = Si = Si = Si =
=
Si = Si = Si = Si =
=
=
=
B valence=3
=
P valence = 5
=
Si valence = 4
=
=
=
=
=
=
=
=
=
=
Si = Si -- B = Si =
=
Si = Si = P = Si =
=
Si = Si = Si = Si =
=
=
Si = Si = Si = Si =
=
=
=
=
=
Phosphorous
n-doped
Electron added to
conduction band
=
=
=
=
=
All electron paired
Insulator at T=0
Si = Si = Si = Si =
=
=
=
=
=
Pure Si
=
=
Si = Si = Si = Si =
=
=
=
Si = Si = Si = Si =
=
Si = Si = Si = Si = e-
=
Si = Si = Si = Si =
Boron
p-doped
Positive hole added to
conduction band
Hole
Dopant Energy levels
P 0.046eV
As 0.054eV
Energy required
to donate electron
Easily ionized
= easily donate electrons
Si
Eg=1.2eV
Cu 0.53eV
Cu 0.40eV
Cu 0.24eV
Au 0.54eV
Large energy bad.
Add scattering
Donate no carriers
Au 0. 35eV
Au 0. 29eV
B 0.044eV
Energy required to donate hole
Carrier concentration in thermal
equilibrium
• Carrier concentration vs. inverse temperature
Thermally activated
Intrinsic carriers
Region of
Functional device
ne
N(carriers) = N(dopants)
Activation of dopants
1/T(K)
Dopants and Fermi Level
2
2
 kF 3 
 kF

• Free electron metal: ne  
,
e

 3p 2  F
2m


• Intrinsic semiconductor
–
–
n(electrons) = n(holes)
Fermi energy = middle
Ec
Ef
Ev
• n-doped material
–
–
n(electrons) >> n(holes)
Fermi level near conduction band
• p-doped materials
–
–
n(electrons) >> n(holes)
Fermi level near conduction band
Ec
Ef
Ev
Ec
Ef
Ev
Fermi Energy is not material specific but depends on doping level and type
Mobility and Dopants
• Dopants destroy periodicity
e
– Scattering, lower mobility
10000
Mobility
(cm2/V-s) 1000
e
GaAs
e
h
Si
100
1E14
1E15
1E16
1E17
Dopant Concentration (cm-3)
1E18
Doping / Implantation
Implants:
(1)NBL (isolation)
(2) Deep n (Collector)
(3) Base well (p)
(4) Emitter (n)
(5) Base contact
• Simple bipolar transistor = 5 implants
• Complicated CMOS circuit >12