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 Inter-atom interactions

Band gap 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

Semiconductors Si, Ge
Binary system III-V: GaAs, InP, GaN, GaP
Filled p shells
Binary II-VI: CdTe, ZnS,
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
e(k)
In the region near the gap,

Local maximum / minimum

dE/dk = 0 Conduction
band

effective mass m* = h2/(d2E/dk2)

Electrons kx
Minimum energy
Bottom of conduction band

Holes
Opposite E(k) derivative
“Opposite effective charge” Valence
Top of valence band 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 Allowed state
for k-vector
• K-space
– Set of allowed k
vectors
• Fermi surface
– Electrons occupy
all kf2 states less kx
than Ef*2m/h
– kF ~ wavelength 2p/L
of electron
wavefunction

Volume in lattice
Area of sphere / k states in spheres
 4pk F 3  1 
  kF 3
 3  (2p / L)3   6p 2 L3  N
  
 Density of states
http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm

kF 3
Number of states: 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 half-
integer 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 valence = 4 P valence = 5 B valence=3
=
=
=

=
=

=
Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Hole
=
=

=
=
Si = Si = Si = Si = Si = Si = P = Si = Si = Si -- B = Si =
=

=
=

=
=
Si = Si = Si = Si = Si = Si = Si = Si = e- Si = Si = Si = Si =
=

=
=

=
=
Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si =
=
=
=

=
=

=
Pure Si
Phosphorous Boron
n-doped p-doped
All electron paired
Electron added to Positive hole added to
Insulator at T=0
conduction band conduction band
 Dopant Energy levels
P 0.046eV
As 0.054eV

Easily ionized Energy required

= easily donate electrons to donate electron
Si
Eg=1.2eV
Au 0.54eV
Cu 0.53eV Large energy bad.
Add scattering
Cu 0.40eV Donate no carriers
Au 0. 35eV

Au 0. 29eV
Cu 0.24eV

B 0.044eV

Energy required to donate hole

 Carrier concentration in thermal
equilibrium

• Carrier concentration vs. inverse temperature

Region of
Thermally activated
Intrinsic carriers
Functional device

ne
N(carriers) = N(dopants)
Activation of dopants

1/T(K)
 Dopants and Fermi Level

 kF 3  2
 kF
2
• Free electron metal: ne   , e 
 3p 2  F
  2m

• Intrinsic semiconductor Ec
– n(electrons) = n(holes)
– Fermi energy = middle Ef
Ev

Ec
• n-doped material Ef
– n(electrons) >> n(holes)
– Fermi level near conduction band
Ev

• p-doped materials Ec
– n(electrons) >> n(holes)
– Fermi level near conduction band 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 e
GaAs
Mobility e
(cm2/V-s) 1000
h
Si
100
1E14 1E15 1E16 1E17 1E18

Dopant Concentration (cm-3)

 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