Carrier Densities: Electrons
•To find density of electrons in semiconductor:
•Find density of available states for electrons.
•Find probability that each of these states is
occupied.
• Density of occupied states per unit volume
and energy, n(E), is given by the product of the
density of states in the conduction band, gc(E)
and the Fermi-Dirac probability function, f(E).
Carrier Densities: Holes
• Holes = empty states in the valence band.
•Probability of having a hole = probability
that a particular state is not filled.
•Hole density per unit energy, p(E), equals:
• gv(E) is the density of states in the valence
band.
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Carrier Densities
•To obtain density of carriers, integrate the
density of carriers per unit energy over all
possible energies within a band.
•Approximate solution: use simple
particle-in-a box model, where one
assumes that the particle is free to move
within the material.
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The carrier density (electrons) in a
semiconductor:
Where gc(E) is the density of states in the
conduction band and f(E) is the Fermi function.
Carrier Density Integral
Density of states, gc(E), Density per unit energy, n(E), Probability of
occupancy, f(E). Carrier density, no, equals the crosshatched area.
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Carrier density at zero Kelvin
At T = 0 K, f(E) = 1 for all E < EF
f(E) = 0 for all E > EF.
and integration yields:
This expression can be used to approximate the carrier density in
heavily degenerate semiconductors provided kT << (EF - Ec) > 0
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Non-degenerate Semiconductors
• Non-degenerate: if only a small number of
donor and/or acceptor atoms have been
introduced into it.
•Concentration of impurity atoms are small in
comparison to that of the host atoms.
•Impurity atoms are spaced far apart from each
other, such that there is no interaction among
donor electrons or acceptor holes.
•The product of the electron and hole density of
a non-degenerate semiconductor is always equal
to the square of its intrinsic carrier density,
whether the semiconductor is intrinsic or
extrinsic.
Degenerate Semiconductors
•Degenerate: one that has been doped to such high levels
that the dopant atoms are an appreciable fraction of the
host atoms.
•Impurity atoms are close enough to allow their donor
electrons (or acceptor holes) to interact.
•Single discrete donor/acceptor energy levels will split
into a band of energies due to interaction.
• May cause overlap with the bottom of the conduction
band or top of valence band.
• Overlap occurs when the impurity concentration
becomes comparable with the effective density of states.
• For donors, when the concentration of electrons in the
conduction band exceeds Nc, then the Fermi level will lie
within the conduction band.
•Overlapping in degenerate semiconductor will make it
behave more like a conductor than a semiconductor.
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Non-degenerate semiconductors
•Non-degenerate : if EF ≥ 3kT from EC or EV.
•This allows the fFD(E) to be replaced by fMB(E) - a
simple exponential function.
•The carrier density integral can then be solved
analytically yielding:
Non-degenerate semiconductors
where Nc is the effective density of states in the
conduction band. The Fermi energy, EF, is obtained
from:
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•M-B distribution applies to non-interacting
particles, which can be distinguished from each
other.
•Provides the probability of occupancy for noninteracting particles at low densities, e.g. atoms
in an ideal gas.
•The Maxwell-Boltzmann distribution function
is given by:
Probability of occupancy vs energy of the F-D,
B-E and M-B distribution. Assumes EF = 0
All almost equal for large E (a few kT beyond
EF). F-D = 100% for E ~ a few kT below EF.
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Similarly for holes, one can approximate the hole
density integral as:
where Nv is the effective density of states in the
valence band. The Fermi energy, EF, is obtained
from:
Example: Calculate the effective densities of states in the conduction and valence bands
of germanium, silicon and gallium arsenide at 300 K. The effective density of states in
the conduction band of germanium equals:
where the effective mass for density of states was used (Appendix 3). Similarly
one finds the effective densities for silicon and gallium arsenide and those of the
valence band:
Note that the effective density of states is temperature dependent and can be
obtain from:
where Nc(300 K) is the effective density of states at 300 K.
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Degenerate Semiconductors
A useful approximate expression of carrier
density applicable to degenerate semiconductors
was obtained by Joyce and Dixon and is given by:
for electrons
for holes.
Intrinsic semiconductors
• Semiconductors which do not contain
impurities.
• Contain electrons and holes of equal
density (intrinsic carrier density, symbol ni)
• n = p = ni
• Thermal activation of an electron from
the valence band to the conduction band
yields a free electron in the conduction
band as well as a free hole in the valence
band.
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Intrinsic Carrier Density
•Intrinsic semiconductors - usually nondegenerate.
• Use expressions for the electron and hole
densities in non-degenerate semiconductors.
•Fermi energy of intrinsic material = Ei.
• Relations between the intrinsic carrier
density and the intrinsic Fermi energy:
Intrinsic Carrier Density
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Intrinsic semiconductors …
•To eliminate Ei from both equations, multiply
both equations and take square root.
• Provides expression for the intrinsic carrier
density as a function of the effective density of
states in the conduction and valence band, and
the bandgap energy Eg = Ec - Ev.
Intrinsic semiconductors …
•Temperature dependence of ni is dominated
by the exponential dependence on the energy
bandgap.
• Also there is temperature dependence of the
effective densities of states and that of the
energy bandgap.
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Intrinsic carrier density versus temperature
Mass action law
•Product of electron and hole densities in a nondegenerate semiconductor equals the square of the
intrinsic carrier density.
•This is referred to as the mass action law.
•It enables one to find no if po is known and vice versa.
•Only valid for non-degenerate semiconductors in
thermal equilibrium.
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Intrinsic Fermi energy (Ei)
• Intrinsic Fermi energy - typically close to the midgap
energy, half way between the conduction and valence
band edge.
• Ei can be obtained form equations for the intrinsic
electron and hole density:
• Ei can also be expressed as a function of the effective
masses of the electrons and holes in the semiconductor:
Dividing above expression for electron density by the
one for the intrinsic density allows to write the carrier
densities as a function of the intrinsic density and the
intrinsic Fermi energy, Ei, or:
Similarly,
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The same relations can also be rewritten to
obtain the Fermi energy from either carrier
density, namely:
Doped (Extrinsic) Semiconductors
•Contain impurity atoms incorporated into the
crystal structure of the semiconductor.
• Unintentional due to lack of control during
the growth of the semiconductor, or on
purpose to provide free carriers in the
semiconductor.
• Free carriers are generated when these
impurities give off electrons to the conduction
band in which case they are called donors.
•If they give off holes to the valence band, they
are called acceptors.
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Donor Impurity Atom – donates an
electron to the conduction band without
creating a hole in the valence band;
resulting material is an n-type
semiconductor
Acceptor Impurity Atom – generates a hole
in the valence band without generating an
electron in the conduction band; resulting
material is an p-type semiconductor
Dopant Atoms and Band Gap Energy Levels
• When dopant atoms are introduced into a
semiconductor crystal, new donor and/or acceptor levels
are created in the band gap.
•Donor level Ed and acceptor level Ea created
in the band gap by doping.
• Electrons from impurity atoms will not be restricted to
the energy levels allowed for the host atoms.
•Can reside in energy levels forbidden to the electrons of
the host atoms.
• Adding impurity atoms to an intrinsic semiconductor to
form an extrinsic semiconductor creates new energy
levels within the band gap of the semiconductor.
•The new energy levels in the band gap can be occupied
by extra electrons or extra holes from the impurity
atoms.
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Donor Energy Band
Diagram
Acceptor Energy
Band Diagram
Doped Semiconductors
•The donor energy level is filled prior to
ionization.
•Ionization causes the donor to be emptied,
yielding an electron in the conduction band and
a positively charged donor ion.
•The acceptor energy level is empty prior to
ionization.
•Ionization of the acceptor corresponds to the
empty acceptor level being filled by an electron
from the filled valence band.
•This is equivalent to a hole given off by the
acceptor atom to the valence band.
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electron
hole
Ionization of a shallow donor
Ed = Donor Ionization Energy
Ionization of a shallow acceptor
Ea = Acceptor Ionization energy
•A doped semiconductor will contain free
carriers if the impurities are ionized.
•Shallow impurities - ones which require little
energy to ionize (≤kT).
• Deep impurities require energies much larger
than the thermal energy to ionize so that only a
fraction of the impurities present in the
semiconductor contribute to free carriers.
•Deep impurities located ≥ 5kT away from either
band edge, are very unlikely to ionize.
•Such impurities make recombination centers
called traps., in which electrons and holes fall
and annihilate each other.
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At room temperature:
T = 300K
Boltzmann constant (k):
=1.380658 x 10-23 J/K
= 8.617385 x 10-5 eV/K
kT = 0.02586 eV ~ 25.9 meV
Ionization
• A process by which a free charge carrier
from an impurity atom is released in an
extrinsic semiconductor.
• Donor impurity atoms donates electrons
to the conduction band.
• Acceptor atom accepts an electron from
the valence band.
•Ionization Energy: Energy needed to
elevate the donor electron into the
conduction band or a valence band
electron to an acceptor level.
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Ionization
• Intrinsic Semiconductor.
• Impurity atoms are introduced.
• Creates donor and acceptor energy
levels within the forbidden band.
• Donor levels - occupied by donor
electrons from donor impurity atoms.
• Acceptor levels - occupied by holes from
acceptor impurity atoms.
•If supplied with enough energy, electrons
in donor level can be to jump into the
conduction band and participate in
conduction.
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•Similarly, valence band electrons can be
excited to leave the valence band and jump
into an acceptor level.
• Equivalent to a hole from the acceptor level
entering the valence band.
• Electrons excited from the valence band into
acceptor levels also contribute to conduction
as moving holes in the valence band.
•“Ionization” = process of exciting an electron
from the donor level to the conduction band or an electron from the valence band to an
acceptor level.
•The energy needed is known as 'ionization
energy'.
Complete Ionization
• Complete ionization - When all donor or
acceptor atoms in a doped semiconductor
have become ionized.
• All donor atoms – positively charged.
•Acceptor atoms = negatively charged.
• Usually occurs at room temperature.
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Freeze-Out
• Freeze-out: when no donor/acceptor atoms are
ionized.
• All donor/acceptor atoms neutral. Occurs at 0 K.
• Between 0 K and room temperature, partial
ionization.
• All energy states: below Ef are full, above Ef are
empty.
Shallow impurities
•Shallow impurities: small ionization energies.
•Readily ionize, free carrier density = the
impurity concentration.
•Shallow donors: electron density = donor
concentration.
•Shallow acceptors: hole density = acceptor
concentration.
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Compensated Semiconductor
• Semiconductor which contains both shallow donors and
shallow acceptors.
• Equal amounts of donor and acceptor atoms
compensate each other, yielding no free carriers.
•Electrons given off by the donor atoms fall into the
acceptor state.
•Ionizes the acceptor atoms without yielding a free
electron or hole.
•Resulting carrier density in compensated material, ≅
difference between the donor and acceptor
concentration.
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