Electrons and Holes
ECE 2204
Intrinsic Carrier Concentration
• Intrinsic carriers are the free electrons and holes
that are generated when one or more chemical
bonds between the Si atoms break.
▫ The intrinsic carrier concentration, the number of
broken bonds per cubic centimeter, is given by:
EG

32
2 kT
ni  B T e
12
Electron and Hole Concentration
• When a bond breaks:
▫ One electron is freed and can wander through the
crystal, but can’t leave the crystal. Hence, it is called a
nearly free electron.
 A nearly free electron is also known as a conduction
electron.
▫ One hole is generated and is temporarily tied between
the two atoms with the broken bond
 A hole is the lack of an electron to fill the outer shell of the
two Si atoms that were sharing the now ‘free’ electron.
A bond can break when a Si atom absorbs enough energy from phonons
(packets of heat) to equal the bandgap energy. Or, if it absorbs a photon that
has as much or more energy than the bandgap energy, the atom can use the
energy to break one of the bonds with another Si atom.
Electron and Hole Concentrations
• For intrinsic (pure) semiconductors:
ni  n  p
2
ni  n  p
▫ The electron concentration, n, is the number of free
electrons per cubic centimeter
▫ The hole concentration, p, is the number of outer shell
electrons that are missing because a chemical bond between
Si atoms broke.
Electrons and Holes
• All electrons with the same energy and momentum are
considered to be equal.
• All holes with the same energy and momentum are
considered to be equal.
▫ However, this is not true in magnetic semiconductors, such
as MnGaAs, where spin up electrons are different from spin
down electrons and similarly spin up holes are different
from spin-down holes.
 We aren’t going to worry about this exception.
▫ Magnetic semiconductors are being investigated as an
alternative to Si.
 The technology is called spintronics.
Energy Diagram
An energy diagram is a schematic drawing of the positions of the bottom of the
conduction band and the top of the valence band as a function of distance.
distance
Direct Semiconductor
In this energy vs. momentum
diagram (also known as an E-k
diagram), the lowest point in the
conduction band is directly above
the highest point on the valence
band (the curves below 0 eV).
The difference between these two
points is the bandgap energy, Eg.
It is also the minimum energy
required to take a valence electron
(an outer shell electron) and put it
into the conduction band (break a
bond).
From www.ioffe.ru
If the free electron drops down
into the valence band, energy
must be released using some
particle that has no momentum.
The only particle that meets this
requirement is a photon (a packet
of light).
Indirect Semiconductor
In this E-k diagram, the lowest
point in the conduction band is
not directly above the highest
point on the valence band.
The difference between these
two points is still the bandgap
energy, Eg.
If the free electron drops down
from the lowest point in the
conduction band in the highest
point in the valence band,
energy must be released using
some particle that has
momentum. The particle that
meets this requirement is a
phonon (a packet of heat).
Si
Modified from www.ioffe.ru
The probability that Si will emit
light is approximately 1 in 1015
transitions of an electron from
the conduction band to the
valence band. For most direct
semiconductors, the probability
is greater than 99 times out of
100.
Bandgap Energy vs. Temperature
What is the temperature dependence
of ni?
EG

32
2 kT
ni  B T e
12
B depends on the properties of the semiconductor – how heavy are the
electrons and holes and how many places are available in the
conduction band for electrons and in the valence band for holes. The
value of B has a very small temperature dependence.
Values for B
Material
B (K-3 – cm-6)
Si
1.08 x 1031
Ge
2.31 x 1030
GaAs
1.27 x 1029
Trends in intrinsic semiconductors
• ni increases with increasing temperature. Thus, the
electrons and holes concentration in an intrinsic
semiconductor increases with increasing
temperature.
▫ The curves stop when every atom has one broken
bond. This is the temperature where the
semiconductor is about to melt.
• Almost always, smaller the bandgap energy, the
larger ni is at a particular temperature.
▫ As discussed in the Semiconductor Material slides, Eg
decreases as one moves down the periodic table.