Charge Carriers in Semiconductors
• At T = 0K, a semiconductor is an insulator with no free
charge carriers.
• At T > 0K, some electrons in the valence band are excited
to the conduction band, where they are free to move about.
• The empty state in the valence band left behind by a
promoted electron is referred to as a hole.
• Ec – bottom of conduction band.
• Ev – top of valence band.
• At 0K, valence band of a semiconductor is completely
filled with electrons while its conduction band is empty.
• Electrons in the valence band cannot conduct electrical
current because there are no available states to transfer to.
• This causes a semiconductor to behave like an insulator at
0 deg K.
• At T > 0K, some electrons in the valence band of
the semiconductor gain enough energy to
overcome the band gap and jump up to the
conduction band, becoming free to conduct
electricity.
• The empty state left behind by the excited
electron in the valence band is known as a 'hole.'
• Holes in the valence band, like electrons in the
conduction band, are also free to move about.
• In a semiconductor, both electrons in the
conduction band and holes in the valence band
can be utilized to conduct electricity, which is why
both of them are also known as 'charge carriers'.
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Intrinsic Semiconductors
• Pure semiconductor materials, no impurity atoms.
• Charge carriers are solely the electrons and holes
created in pairs by thermal energy.
• Electrons in the valence band gain thermal energy
and are promoted into the conduction band as
temperature increases, leaving behind
corresponding holes in the valence band: “electron-hole” pair.
• The number of electrons in the conduction band
equals the number of holes in the valence band.
• The band gap determines how temperature
variation affects conductivity.
• Intrinsic conduction: process that results from the
band structure of a pure element or compound.
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Intrinsic Carrier Density versus Temperature in GaAs, Si and Ge
Intrinsic Semiconductors….
• One-to-one correspondence between the electrons in
the conduction band and the holes in the valence band
in an intrinsic semiconductor.
• In intrinsic semiconductors, density of intrinsic
electrons (ni) = density of intrinsic holes (pi).
•Conductivity σ of an intrinsic semiconductor is:
σ = Neqe(µe+µh) and Ne = N0 exp(-Eg/2kT),
where Ne = number of conduction electrons,
qe = electron charge,
µe = mobility of the conduction electrons,
µh = mobility of the valence holes.
N0 = material constant.
Thus, σ = σ0 exp(-Eg/2kT) where σ0 = N0qe(µe+µh).
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Intrinsic Semiconductors….
• σ = σ0 exp(-Eg/2kT) OR lnσ = lnσ0 -Eg/2kT
• Plot of ln σ versus 1/T is a straight line whose slope is Eg/2k.
•Band gap of an intrinsic semiconductor may be
determined experimentally by observing how its
conductivity varies with temperature.
•The mobility of both electrons and holes decreases
linearly with an increase in temperature
• Number of mobile charge carriers increases
exponentially with an increase in temperature.
• With increasing temperature, the exponential increase
in the number of carriers is a more dominating factor than
the linear decrease in carrier mobility, so the conductivity
of an intrinsic semiconductor always increases as
temperature increases.
•Ohm’s law: I = V/R where R= electrical resistance.
•R depends on resistivity (ρ), an intrinsic property of
the material. R = ρl/A and σ = 1/ρ.
• σ = electrical conductivity.
•Electric field E = V/l.
• Ohm's law: j = σ E where j = I/A = current density.
• Conductivity:107 (Ω-m) typical of metals, 10-20 (Ωm) for good electrical insulators and in the range 10-6
to 104 (Ω-m) for semiconductors.
• Mobility (µ): a quantity relating dependence of drift
velocity of charge carriers to the applied electric field.
σ = n |e| µe + p |e| µh
where p is the hole concentration and µh the hole mobility.
Electrons move much faster than holes: µe > µh
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Extrinsic Semiconductors
¾Semiconductors doped with foreign atoms to
alter their intrinsic electron and hole
concentrations.
¾Doped semiconductors: Semiconductors, which
contain impurities (foreign atoms) incorporated
into the crystal structure of the semiconductor.
¾Sources of impurities: Either (a) Added on
purpose to provide free carriers in the
semiconductor OR (b) Incorporated
unintentionally, due to lack of control during the
growth of the semiconductor.
¾Free carriers are generated when impurity atoms
give off electrons or holes.
Donors
Donor Impurity Atom – donates an electron to
the conduction band without creating a hole in
the valence band; resulting material is an ntype semiconductor.
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.
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Acceptors
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.
The acceptor energy 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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Dopant Atoms and Band Gap Energy Levels
• Addition of impurity atoms to an intrinsic
semiconductor to form an extrinsic semiconductor
basically creates new energy levels within the band
gap of the semiconductor.
•Donor atoms: the extra electrons loosely bound to
the donor atoms are not restricted to the energy
levels allowed for the host atoms. They occupy
energy levels forbidden to the electrons of the host
atoms. This energy level is the “donor level” Ed,
located in the upper half of the forbidden band.
• Acceptor atoms: Similarly,holes from the acceptor
atoms occupy “acceptor level” Ea, located in the
lower half of the forbidden band.
E
E
Ec
Ed
Ec
Ei
Ei
Ev
Ea
Ev
Ionization of a shallow donor
Ionization of a shallow donor
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Ionization
• Is a process by which free charge carriers (electrons and
holes) are produced in an extrinsic semiconductor.
Ionization Energy – energy needed to move a donor
electron from the donor level (Ed) to the conduction band.
Or energy needed to move a valence band electron from
the valence band to the acceptor level (Ea).
• An extrinsic semiconductor will only have free carriers if
the impurity atoms are ionized.
• Shallow impurities: those whose ionization energies are ≤
kT. Deep impurities require larger energies to ionize.
• If ionization energy > 5kT very minimal ionization will
occur.
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Complete Ionization
-When all donor and acceptor atoms have become ionized.
-Occurs at about room temperature for extrinsic semiconductors.
-At absolute zero, the opposite of complete ionization occurs, i.e.,
freeze-out.
-Freeze-out: when all donor and acceptor atoms are neutrally
charged, i.e., no donor electrons are elevated into the conduction
band and no electrons from the valence band are elevated into the
acceptor level.
-Between absolute zero and room temperature – various levels of
partial ionization of donor/acceptor atoms exist.
Ionization Energies
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Temperature Dependence
•Extrinsic semiconductors exhibit conductivities that
behave differently with temperature variation.
•Conductivity of an extrinsic semiconductor depends on
the type of impurity atoms that it has been doped with.
Conductivity of Extrinsic Semiconductors
- Conduction in extrinsic semiconductors occurs at
lower temperatures than for intrinsic semiconductors.
- Conductivities can be vastly increased (no or po
>> ni )
- Semiconductor becomes either n-type or p-type.
- N-type semiconductor: extrinsic with more shallow
donor impurity atoms.
- P-type semiconductor: extrinsic with more shallow
acceptor impurity atoms.
-Compensated semiconductor: extrinsic with equal
amounts of shallow donors and acceptors. No net free
charge carriers.
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Carrier Distribution Functions
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• Distribution or probability density functions describe the probability with which one can
expect particles to occupy the available energy
levels in a given system.
• Our interest is in probability density function
of electrons, called the Fermi function.
• Probability density functions derived in
statistical thermodynamics.
• Other distribution functions: Bose-Einstein
distribution function, Maxwell Boltzmann
distribution.
• Fermi-Dirac distribution function provides the
probability of occupancy of energy levels by
Fermions.
•As Fermions are added to an energy band,
they will fill the available states in an energy
band.
•Electrons are Fermions. F-D function
provides the probability that an energy level at
energy, E, in thermal equilibrium with a large
system, is occupied by an electron.
• States with the lowest energy are filled first,
followed by the next higher ones.
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•At 0 K, the energy levels are all filled up to a
maximum energy, which we call the Fermi
level.
• No states above the Fermi level are filled.
•At higher temperature, transition between
completely filled states and completely empty
states is gradual rather than abrupt.
• The system: characterized by its temperature
T and Fermi energy EF.
•At 0K, electrons are distributed in the lowest
possible energy levels.
• At higher temperature, electrons gain energy
and transfer to higher energy levels.
•Change in occupancy of the energy levels as
temperature increases is quantified by the FermiDirac Distribution Function f(E).
•It gives probability that an energy level is
occupied by an electron at absolute temperature
T.
The Fermi-Dirac Distribution:
f (E) =
1
1 + (exp{[ E − E f ] / kT })
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Fermi-Dirac Distribution Function ƒ(E)
- describes the statistical behavior of electrons among
available energy states
ƒ(E):
- gives the probability that an allowed energy state is
occupied by an electron
- ratio of filled to total quantum states at any energy E
where:
N(E) = number of particles per unit volume per unit
energy;
g(E) = number of quantum states per unit volume per
unit energy
The F-D function at three different temperatures
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•The function = 1 for energies, which are
more than a few times kT below EF.
• The function = 1/2 for E = EF.
• The function decreases exponentially for
energies which are a few times kT larger
than EF.
• At T = 0 K, the function is a step
function, the transition is more gradual at
finite temperatures and more so at higher
temperatures.
Fermi-Dirac Distribution Function at T = 0 K
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Fermi-Dirac Distribution Function at T > 0K
Density of states
• Finding # of available states at each energy
helps calculate the density of carriers in a
semiconductor.
• Number of electrons at each energy - then
obtained by multiplying the number of
states with the probability that a state is
occupied by an electron.
• Number of energy levels – very large,
depends on semiconductor size.
• Just calculate the number of states per
unit energy and per unit volume.
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Calculation of the Density of States
• Density of states in a semiconductor equals
the density per unit volume and energy of the
number of solutions to Schrödinger's
equation.
• Assume semiconductor can be modeled as
an infinite quantum well in which electrons
with effective mass, m*, are free to move. The
energy in the well is set to zero.
•The semiconductor is assumed a cube with
side L.
Calculation of the Density of States
• Solve Schrödinger's wave equation.
•Solution where V(x) = 0 are sine and cosine
functions:
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Schrödinger equation: Total energy (classical) E =
K.E. + PE
Define wavefunction, Ψ, multiply through to convert
expression into a wave equation:
Introduce operator
Apply to a plane wave to provide the square of the
momentum, p:
k = wavenumber = 2π /λ.
Replace p2 by this operator. Yields the timeindependent Schrödinger equation:
Where V(x) = 0, solutions are sine and cosine
functions:
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A and B - constants to be determined.
Ψ= 0 at the infinite barriers of the well.
Ψ = 0 at x = 0 so that only sine functions can
be valid solutions.
Ψ = 0 at x = L
Yields possible values for the wavenumber,
kx.
Similar results for y and z direction.
Each possible solution ≡ a cube in k-space
with size nπ/L.
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•Calculate volume of 1/8 a sphere with radius
k then divide it by the volume corresponding
to a single solution.
•Gives total number of solutions with a
different value for kx, ky and kz and with a
magnitude of the wavevector less than k.
Results:
Factor of two - two possible spins of each
solution.
Use chain rule to obtain the density per unit
energy:
The kinetic energy E of a particle with mass m*
is related to the wavenumber, k, by:
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The density of states per unit volume and per
unit energy, g(E), is:
The density of states is zero at the bottom of
the well as well as for negative energies
Density of States for electrons in Semiconductor
•Same analysis for electrons in a semiconductor.
• Use effective mass of electrons to account for
effect of periodic potential on the electron.
• Minimum energy = energy at the bottom of
the conduction band, Ec.
• Below EC, density of states is zero.
• Density of states for electrons in the
conduction band is given by:
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The density of states for holes in the valence
band is given by:
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).
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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.
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.
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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
Non-degenerate semiconductors
•Non-degenerate semiconductor one with EF at least
3kT away from either band edge.
•This definition allows the fFD(E) to be replaced by
fMB(E) - a simple exponential function.
•The carrier density integral can then be solved
analytically yielding:
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Non-degenerate semiconductors
where Nc is the effective density of states in the
conduction band. The Fermi energy, EF, is obtained
from:
•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:
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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.
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:
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Degenerate Semiconductors
A useful approximate expression applicable to degenerate semiconductors
was obtained by Joyce and Dixon and is given by:
for electrons and by:
for holes.
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