CONDENSED MATTER PHYSICS
Theoretical Conspectus
(9 credit points)
The course is studied in twelve week of 7th semester. The number of credit points – 9. It consists
of 240 working hours. Studying Condensed matter physics, due attention should be paid to the
theory and experiment coordination. During the first five weeks lectures and workshops taking
place. 6 - 8 weeks intended for laboratory works. Problem-based learning takes place from 9 to 11
weeks. In week 12 takes place a written exam. The schedule provided hours for consultation.
1. CRYSTALS AND CRYSTAL STRUCTURES
1. Bravais lattices
2. Miller indices, crystal planes and directions
3. Common semiconductor crystal structures
4. Growth of semiconductor crystals
Solid materials are classified by the way the atoms are arranged within the solid. Materials in which
atoms are placed at random are called amorphous. Materials in which atoms are placed in a highly
ordered structure are called crystalline. Poly-crystalline materials are materials with a high degree of
short-range order and no long-range order. These materials consist of small crystalline regions with
random orientation called grains, separated by grain boundaries. Crystals naturally form as liquid
material cools down, since the close proximity of atoms lowers their energy. However, since
crystallization typically occurs in multiple locations simultaneously, one finds that the polycrystalline structure is quite common except for materials such as glass which tend to be amorphous.
Crystalline silicon dioxide does occur in the form of quartz but only if the temperature and pressure
promote crystal formation.
Of primary interest in this text are crystalline semiconductors in which atoms are placed in a highly
ordered structure. Crystals are categorized by their crystal structure and the underlying lattice. While
some crystals have a single atom placed at each lattice point, most crystals have a combination of
atoms associated with each lattice point. This combination of atoms is also called the basis.
The classification of lattices, the common semiconductor crystal structures and the growth of singlecrystal semiconductors are discussed in the following sections.
Bravais lattices
The Bravais lattices are the distinct lattice types, which when repeated can fill the whole space. The
lattice can therefore be generated by three unit vectors,
and a set of integers k, l and m
so that each lattice point, identified by a vector , can be obtained from:
The construction of the lattice points based on a set of unit vectors is illustrated by Figure below:
In three dimensions there are fourteen
distinct Bravais lattices. The fourteen
lattices of three-dimensional crystals
are classified as shown in Table below,
where a1, a2 and a3 are the magnitudes
of the unit vectors defining the
traditional unit cell.
The cubic lattices are an important
subset of these fourteen Bravais lattices
since a large number of semiconductors
are cubic. The three cubic Bravais
lattices are the simple cubic lattice, the
body-centred cubic lattice and the face-centred cubic lattice. Since all unit vectors identifying the
traditional unit cell have the same size, the crystal structure is completely defined by a single number.
This number is the lattice constant, a.
Miller
indices, crystal planes and directions
Crystal planes of a crystal are characterized by their Miller indices. The Miller indices are defined as
the smallest possible integers, which have the same ratios as the inverse of the intersections of a given
plane with a set of axis defined by the unit vectors of that crystal. The intersections between the plane
and the axis occur at p, q, and r. The
corresponding Miller indices are
therefore
, where A is an integer
chosen such that the Miller indices are the
smallest possible integers. It should be
noted that the resulting Miller indices are
the same for all parallel planes of atoms
in a crystal.
Common semiconductor crystal
structures
The most common crystal structure
among frequently used semiconductors is
the diamond lattice. Each atom in the
diamond lattice has a covalent bond with four adjacent atoms, which together form a tetrahedron.
This lattice can also be formed from two face-centred-cubic lattices, which are displaced along the
body diagonal of the larger cube by one quarter of that body diagonal. The diamond lattice therefore
is a face-centred-cubic lattice with a basis containing two identical atoms.
2. DENSITY OF ENERGY STATES
The Fermi function gives the probability of occupying an available energy state, but this must be
factored by the number of available energy states to determine how many electrons would reach the
conduction band. This density of states is the electron density of states, but there are differences in its
implications for conductors and semiconductors. For the conductor, the density of states can be
considered to start at the bottom of the valence band and fill up to the Fermi level, but since the
conduction band and valence band overlap, the Fermi level is in the conduction band so there are
plenty of electrons available for conduction. In the case of the semiconductor, the density of states is
of the same form, but the density of states for conduction electrons begins at the top of the gap.
Electron Energy Density
The behavior of electrons
solids depends upon the
distribution of energy
among the electrons.
Since
electrons
are
fermions, the distribution
function is the Fermi-Dirac
distribution
in
This distribution determines the probability that a given energy state will be occupied, but must be
multiplied by the density of states function to weight the probability by the number of states
available at a given energy.
The determination of how many ways there are to obtain energy in an incremental energy range
dE can be approached with the quantum mechanical particle in box. The energy for an infinite
walled box is
.
Treating the "quantum numbers" n as a space such that a
given set of n values determines a point in that space, you
can argue that the number of possible states is proportional
to the "volume" in "n-space".
It is convenient to define a radius R in n-space:
The energy can be expressed in terms of R and vice versa.
..
The n-space associated with the particle-in-a-box involves only positive values of n, so the
volume must be divided by 8. It then must be multiplied by 2 to account for the two possible spin
values of the electron. The number of values is then
.
The number of states per unit volume is
.
The final density of states as a function of energy is then the derivative of this population with
respect to energy
.
This represents the number of electron states per unit volume per unit energy at energy E. This
energy density is a factor in many of the electrical properties of solids. Note that the result is
independent of the dimension L which was chosen above, showing that the expression can be
applied to the bulk material.
3. CARRIER DISTRIBUTION FUNCTION
1. Fermi – Dirac distribution function
2. Impurity distribution functions
3. Other distribution functions and comparison
The distribution or probability density functions describe the probability with which one can expect
particles to occupy the available energy levels in a given system. Of particular interest is the
probability density function of electrons, called the Fermi function. The derivation of such probability
density functions belongs in a statistical thermodynamics course. However, given the importance of
the Fermi distribution function, we will carefully examine an example as well as the characteristics
of this function. Other distribution functions such as the impurity distribution functions, the BoseEinstein distribution function and the Maxwell Boltzmann distribution are also provided.
Fermi-Dirac distribution 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. The Pauli exclusion principle postulates that only one Fermion can occupy
a single quantum state. Therefore, as Fermions are added to an energy band, they will fill the available
states in an energy band just like water fills a bucket. The states with the lowest energy are filled first,
followed by the next higher ones. At absolute zero temperature (T = 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, one finds that the transition between completely filled states and
completely empty states is gradual rather than abrupt.
Electrons are Fermions. Therefore, the Fermi function provides the probability that an energy level
at energy, E, in thermal
equilibrium with a large
system, is occupied by an
electron. The system is
characterized
by
its
temperature, T, and its
Fermi energy, EF. The
Fermi function is given
by:
.
This function is plotted in
Figure 1 for different
temperatures.
The Fermi function has a
Figure 1. The Fermi function at three different temperatures
value of one for energies,
which are more than a few times
kT below the Fermi energy. It equals 1/2 if the energy equals the Fermi energy and decreases
exponentially for energies which are a few times kT larger than the Fermi energy. While at T = 0 K
the Fermi function equals a step function, the transition is more gradual at finite temperatures and
more so at higher temperatures.
Impurity distribution functions
The distribution function of impurities differs from the Fermi-Dirac distribution function although
the particles involved are Fermions. The difference is due to the fact that an ionized donor energy
level still contains one electron, which can have either spin (spin up or spin down). The donor energy
level cannot be empty since this would leave a doubly positively charged atom, which would have an
energy different from the donor energy. The distribution function for donors therefore differs from
the Fermi function and is given by:
.
The distribution function for acceptors differs also because of the different possible ways to occupy
the acceptor level. The neutral acceptor contains no electrons. The ionized acceptor contains one
electron, which can have either spin, while the doubly negatively charged state is not allowed since
this would require a different energy. This restriction would yield a factor of 2 in front of the
exponential term. In addition, one finds that most commonly used semiconductors have a two-fold
degenerate valence band, which causes this factor to increase to four, yielding:
.
Other distribution functions and comparison
Other distribution functions include the Bose-Einstein distribution and the Maxwell-Boltzmann
distribution. These are briefly discussed below and compared to the Fermi-Dirac distribution
function.
The Bose-Einstein distribution function applies to bosons. Bosons are particles with integer spin and
include photons, phonons and a large number of atoms. Bosons do not obey the Pauli exclusion
principle so that any number can occupy one energy level. The Bose-Einstein distribution function is
given by:
.
This function is only defined for E > EF.
The Maxwell Boltzmann applies to non-interacting particles, which can be distinguished from each
other. This distribution function is also called the classical distribution function since it provides the
probability of occupancy for non-interacting particles at low densities. Atoms in an ideal gas form a
typical example of such particles. The Maxwell-Boltzmann distribution function is given by:
.
A plot of the three distribution functions, the Fermi-Dirac distribution, the Maxwell-Boltzmann
distribution and the Bose-Einstein distribution is shown in Figure.
All three functions are almost equal for large energies (more than a few kT beyond the Fermi energy).
The Fermi-Dirac distribution reaches a maximum of 100% for energies, which are a few kT below
the Fermi energy, while the Bose-Einstein distribution diverges at the Fermi energy and has no
validity for energies below the Fermi energy.
4. HEAT CAPACITY
1.
2.
3.
4.
Specific heat
Law of Dulong and Petit
Einstein model
Debye model
The specific heat is the amount of heat per unit mass required to raise the temperature by one
degree Celsius. The relationship between heat and temperature change is usually expressed in the
form shown below where c is the specific heat. The relationship does not apply if a phase change
is encountered, because the heat added or removed during a phase change does not change the
temperature.
The specific heat of water is 1 calorie/gram °C = 4.186 joule/gram °C which is higher than any
other common substance. As a result, water plays a very important role in temperature regulation.
The specific heat per gram for water is much higher than that for a metal, as described in the
water-metal example. For most purposes, it is more meaningful to compare the molar specific
heats of substances.
The molar specific heats of most solids at room temperature and above are nearly constant, in
agreement with the Law of Dulong and Petit. At lower temperatures the specific heats drop as
quantum processes become significant. The low temperature behavior is described by the
Einstein-Debye model of
Law of Dulong and Petit
The difference of specific heat is mainly because it is expressed as energy per unit mass; if you
express it as energy per mole, they are very similar. It is in fact that similarity of the molar specific
heats of metals which is the subject of the Law of Dulong and Petit. The similarity can be
accounted for by applying equipartition of energy to the atoms of the solids.
From just the translational degrees of freedom you get 3kT/2 of energy per atom. Energy added
to solids takes the form of atomic vibrations and that contributes three additional degrees of
freedom and a total energy per atom of 3kT. The specific heat at constant volume should be just
the rate of change with temperature (temperature derivative) of that energy.
Einstein's Contribution to Specific Heat Theory
The Law of Dulong and Petit assumed that Maxwell-Boltzmann statistics and equipartition of
energy could be applied even at low temperatures. Einstein recognized that for a quantum
harmonic oscillator at energies less than kT, the Einstein-Bose statistics must be applied. This
was the same conclusion that was drawn about blackbody radiation. The statistical distribution
of energy in the vibration states gives average energy:
where this frequency is the frequency of a quantum vibrator.
The derivative of this gives:
In the Einstein treatment, the appropriate frequency in the expression had to be determined
called the Einstein temperature. Although the general match with experiment was reasonable, it
was not exact. Debye advanced the treatment by treating the quantum oscillators as collective
modes in the solid which are now called "phonons".
Debye's Contribution to Specific Heat Theory
Einstein's oscillator treatment of specific heat gave qualitative agreement with experiment and
gave the correct high temperature limit (the Law of Dulong and Petit). The quantitative fit to
experiment was improved by Debye's recognition that there were a maximum number of modes
of vibration in a solid. He pictured the vibrations as standing wave modes in the crystal, similar
to the electromagnetic modes in a cavity which successfully explained blackbody radiation. The
density of states for these modes, which are called "phonons", is of the same form as the photon
density of states in a cavity.
To impose a finite limit on the number of modes in the solid, Debye used a maximum allowed
phonon frequency now called the Debye frequency. In the treatment of specific heat, we define a
Debye temperature by
.
For low temperatures, Debye's treatment led to a specific heat
.
The dependence upon the cube of the temperature agreed with experimental results for nonmetals,
and for metals when the electron specific heat was taken into account. The measurement of the
low temperature specific heat variation with temperature has led to tabulation of the Debye
temperatures for a number of solid materials. The full expression for the Debye specific heat must
be evaluated by numerical procedures. It has the correct limiting values at both high and low
temperatures.
Debye Specific Heat
The specific heat expression which arises from Debye theory can be obtained by taking the
derivative of the energy expression above.
.
This expression may be evaluated numerically for a given temperature by computer routines.
The data for silver shown at left is from Meyers.
It shows that the specific heat fits the Debye
model at both low and high temperatures.
Since the Debye specific heat expression can be evaluated as a function of temperature and gives
a theoretical curve which has a specific form as a function of T/TD, the specific heats of different
substances should overlap if plotted as a function of this ratio. At left below, the specific heats of
four substances are plotted as a function of temperature and they look very different. But if they
are scaled to T/TD, they look very similar and are very close to the Debye theory.
5. BAND THEORY OF SOLIDS
A useful way to visualize the difference between conductors, insulators and semiconductors is to
plot the available energies for electrons in the materials. Instead of having discrete energies as in
the case of free atoms, the available energy states form bands. Crucial to the conduction process
is whether or not there are electrons in the conduction band. In insulators the electrons in the
valence band are separated by a large gap from the conduction band, in conductors like metals the
valence band overlaps the conduction band, and in semiconductors there is a small enough gap
between the valence and conduction bands that thermal or other excitations can bridge the gap.
With such a small gap, the presence of a small percentage of a doping material can increase
conductivity dramatically.
An important parameter in the band theory is the Fermi level, the top of the available electron
energy levels at low temperatures. The position of the Fermi level with the relation to the
conduction band is a crucial factor in determining electrical properties.
Energy Bands for Solids
Energy Bands Comments
Insulator Energy Bands
Most solid substances are insulators, and in terms of the band theory of solids this implies that there
is a large forbidden gap between the energies of the valence
electrons and the energy at which the electrons can move freely
through the material (the conduction band).
Glass is an insulating material which may be transparent to
visible
light for reasons closely correlated with its nature as an
electrical insulator. The visible light photons do not have
enough
quantum energy to bridge the band gap and get the electrons up
to an
available energy level in the conduction band. The visible
properties of glass can also give some insight into the effects of
"doping" on the properties of solids. A very small percentage of
impurity atoms in the glass can give it color by providing
specific available energy levels which absorb certain colors of
visible
light. The ruby mineral (corundum) is aluminum oxide with a
small
amount (about 0.05%) of chromium which gives it its
characteristic pink or red color by absorbing green and blue
light.
While the doping of insulators can dramatically change their
optical
properties, it is not enough to overcome the large band gap to
make
them good conductors of electricity. However, the doping of
semiconductors has a much more dramatic effect on their electrical conductivity and is the basis
for solid state electronics.
Semiconductor Energy Bands
For intrinsic semiconductors like silicon and germanium, the
Fermi level is essentially halfway between the valence and
conduction bands. Although no conduction occurs at 0 K, at
higher temperatures a finite number of electrons can reach the
conduction band and provide some current. In doped
semiconductors, extra energy levels are added.
The increase in conductivity with temperature can be modeled
terms of the Fermi function, which allows one to calculate the
population of the conduction band
Conductor Energy Bands
In terms of the band theory of solids, metals are unique as
good conductors of electricity. This can be seen to be a
result of their valence electrons being essentially free. In
band theory, this is depicted as an overlap of the valence
band and the conduction band so that at least a fraction of
valence electrons can move through the material.
in
the
the
6. CHARGE CARRIERS IN SEMICONDUCTORS
1.
2.
3.
4.
Equilibrium Distribution of Electrons and Holes
The n0 and p0 Equations
The Intrinsic carrier Concentration
The Intrinsic Fermi-Level Position
Current is the rate at which charge flows. In a semiconductor two types of charge carrier, the electron
and the hole, can contribute to a current. Since the current in a semiconductor is determined largely
by the number of electrons in the conduction band and the number of holes in the valence hand, an
important characteristic of the semiconductor is the density of these charge carriers. The density of
electrons and holes is related to the density of states function and the Fermi distribution function, both
of which we have considered. A qualitative discussion of these relationships will be followed by a
more rigorous mathematical derivation of the thermal-equilibrium concentration of electrons and
holes.
Equilibrium Distribution of Electrons and Holes.
The distribution (with respect to energy) of electrons in the conduction band is given by the density
of allowed quantum states times the probability that a state is occupied by an electron. This statement
is written in equation form as
n(E) = g(E)fF(E)
where fF(E) is the Fermi-Dirac probability function and g(E) is the density of quantum states in the
conduction band. The total electron concentration per unit volume in the conduction band is then
found by integrating this Equation over the entire conduction-band energy.
Similarly, the distribution (with respect to energy) of holes in the valence bend is the density
of allowed quantum states in the valence hand multiplied by the probability that a state is not occupied
by an electron. The total hole concentration per unit volume is found by integrating this function over
the entire valence-band energy.
To find the thermal-equilibrium electron and hole concentrations, we need to determine the position
of the Fermi energy EF with respect to the bottom of the conduction-band energy Ec and the top of
the valence-band energy Ev. To address this question, we will initially consider an intrinsic
semiconductor. An ideal intrinsic semiconductor is a pure semiconductor with no impurity atoms and
no lattice defects in the crystal (e.g., pure silicon). For an intrinsic semiconductor at T = 0 K, all
energy states in the valence band are filled with electrons and all energy states in the conduction band
are empty of electrons. The Fermi energy must, therefore, be somewhere between Ec and Ev. (The
Fermi energy does not need to correspond to an allowed energy.)
As the temperature begins to increase above 0 K, the valence electrons will gain thermal energy. A
few electrons in the valence band may gain sufficient energy to jump to the conduction band. As an
electron jumps from the valence band to the conduction band, an empty state, or hole, is created in
the valence band. In an intrinsic semiconductor, then, electrons and holes are created in pairs by the
thermal energy so that the number of electrons in the conduction band is equal to the number of holes
in the valence band.
The n0 and p0 Equations.
We have argued that the Fermi energy for an intrinsic semiconductor is near mid gap. In deriving the
equations for the thermal-equilibrium concentration of electrons n0 and the thermal-equilibrium
concentration of holes p0, we will not be quite so restrictive. We will see later that, in particular
situations, the Fermi energy can deviate from this midgap energy. We will assume initially, however,
that the Fermi level remains within the band gap energy.
Thermal-equilibrium electron concentration in the conduction band can be written as
n0 = Nc exp[(-(Ec – EF))/kT].
The parameter Nc is called the effective density of states function in the conduction band. If we were
to assume that m* = mo, then the value of the effective density of states function at T = 300 K is Nc
= 2.5 x 10 · 19 cm-3, which is the order of magnitude of Nc for most semiconductors. If the effective
mass of the electron is larger or smaller than mo, then the value of the effective density of states
function changes accordingly, but is still of the same order of magnitude. The thermal equilibrium
concentrations of electrons in the conduction band of holes in the valence band are directly related to
the effective density of states constants and to the Fermi energy level.
The Intrinsic Carrier Concentration.
For an intrinsic semiconductor, the concentration of electrons in the conduction band is equal to the
concentration of holes in the valence band. We may denote ni and pi as the electron and hole
concentrations, respectively, in the intrinsic semiconductor. These parameters are usually referred to
as the intrinsic electron concentration and intrinsic hole concentration. However, ni = pi, so normally
we simply use the parameter ni as the intrinsic carrier concentration, which refers to either the intrinsic
electron or hole concentration.
The Fermi energy level for the intrinsic semiconductor is called the intrinsic Fermi energy. To the
intrinsic semiconductor, and then we can write
ni² = Nv Nc exp[(-(Ec – EF))/kT] = Nv Nc exp[-Eg /kT].
where Eg is the band gap energy. For a given semiconductor material at a constant temperature, the
value of ni is a constant, and independent of the Fermi energy. The intrinsic carrier concentration for
silicon at T = 300 K may be calculated by using the effective density of states function values. The
value of ni calculated for Eg = 1.12 eV is ni = 6.95 x 10 · 9 cm-1. The commonly accepted value of ni
for silicon at T = 300 K is approximately 1.5 x 10 · 10 cm-1. This discrepancy may arise from several
sources. First, the values of the effective masses are determined at a low temperature where the
cyclotron resonance experiments are performed. Since the effective mass is an experimentally
determined parameter, and since the effective mass is a measure of how well a particle moves in a
crystal, this parameter may be a slight function of temperature. Next, the density of states function
for a semiconductor was obtained by generalizing the model of an electron in a three-dimensional
infinite potential well. This theoretical function may also not agree exactly with experiment.
However, the difference between the theoretical value and the experimental value of ni is
approximately a factor of 2, which, in many cases, is not significant. The intrinsic carrier
concentration is a very strong function of temperature.
The Intrinsic Fermi-Level Position.
We have qualitatively argued that the Fermi energy level is located near the center of the forbidden
band gap for the intrinsic semiconductor. We can calculate the intrinsic Fermi-level position. It may
be written as
EFi =1/2(Ec + Ev) + 1/2 kT ln(Nv /Nc ),
or
EFi =1/2(Ec + Ev) + 3/4 kT ln(m*p / m *n ).
The first term, 1/2(Ec + Ev), is the energy exactly midway between Ec and Ec , or the mid gap energy.
If the electron and hole effective masses are equal so that m*p = m *n , then the intrinsic Fermi level
is exactly in the center of the band gap. If m*p > m *n the intrinsic Fermi level is slightly above the
center, and if m*p < m *n it is slightly below the center of the band gap. The density of states function
is directly related to the carrier effective mass; thus a larger effective mass means a larger density of
states function. The intrinsic Fermi level must shift away from the band with the larger density of
states in order to maintain equal numbers of electrons and holes.
7. DOPANT ATOMS AND ENERGY LEVELS
The intrinsic semiconductor may be an interesting material, but the real power of semiconductors is
realized by adding small, controlled amounts of specific dopant, or impurity, atoms. This doping
process can greatly alter the electrical characteristics of the semiconductor. The doped semiconductor,
called an extrinsic material, is the primary reason we can fabricate the various semiconductor devices.
Qualitative Description.
We were discussed the covalent bonding of silicon and considered the simple two-dimensional
representation of the single-crystal silicon lattice.
Now consider adding a group V element, such as phosphorus, as a substitution impurity. The group
V element has five valence electrons. Four of these will contribute to the covalent bonding with the
silicon atoms, leaving the fifth more loosely hound to the phosphorus atom. We refer to the fifth
valence electron as a donor electron. The phosphorus atom without the donor electron is positively
charged.
At very low temperatures, the donor electron is bound to the phosphorus atom. However it should
seem clear that the energy required to elevate the donor electron into the conduction band is
considerably less than that for the electrons involved in the covalent bonding.
If a small amount of energy, such as thermal energy, is added to the donor electron, it can be elevated
into the conduction band, leaving behind a positively charged phosphorus ion. The electron in the
conduction band can now move through the crystal generating a current, while the positively charged
ion is fixed in the crystal.
This type of impurity atom donates an electron to the conduction band and so is called a donor
impurity atom. The donor impurity atoms add electrons to the conduction band without creating holes
in the valence band. The resulting material is referred to as an n-type semiconductor (n for the
negatively charged electron).
Now consider adding a group III element, such as boron, as a substitution impurity to silicon.
The group III element has three valence electrons, which are all taken up in the covalent bonding.
One covalent bonding position appears to be empty.
If an electron were to occupy this "empty" position, its energy would have to be greater than that of
the valence electrons, since the net charge state of the boron atom would now be negative. However,
the electron occupying this "empty" position does not have sufficient energy to him in the conduction
band, so its energy is far smaller than the conduction-band energy.
The "empty" position associated with the boron atom becomes occupied, and other valence electron
positions become vacated. These other vacated electron positions can be thought of as holes in the
semiconductor material. The hole can move through the crystal generating a current, while the
negatively charged boron atom is fixed in the crystal.
The group III atom accepts an electron from the valence band and so is referred to as an acceptor
impurity atom. The acceptor atom can generate holes in the valence hand without generating electrons
in the conduction band. This type of semiconductor material is referred to as a p-type material (p for
the positively charged hole).
The pure single-crystal semiconductor material is called an intrinsic material. Adding controlled
amounts of dopant atoms, either donors or acceptors, creates a material called an intrinsic
semiconductor.
An extrinsic semiconductor will have either a preponderance of electrons (n type) or a preponderance
of holes (p type).
8. THE EXTRINSIC SEMICONDUCTOR
1. Equilibrium Distribution of Electrons and Holes
2. The n0p0 Product
3. Degenerate and Nondegenerate Semiconductor
We defined an intrinsic semiconductor as a material with no impurity atoms pres in the crystal. An
extrinsic semiconductor is defined as a semiconductor in controlled amounts of specific dopant or
impurity atoms have been added so that thermal-equilibrium electron and hole concentrations are
different from the intrinsic carrier concentration. One type of carrier will predominate in an extrinsic
semiconductor.
Equilibrium Distribution of Electrons and Holes.
Adding donor or acceptor impurity atoms to a semiconductor will change the distribution of electrons
and holes in the material. Since the Fermi energy is related to the distribution function, the Fermi
energy will change as dopant atoms are added. If the Fermi energy changes from near the mid gap
value, the density of electrons in the conduction band and the density of holes in the valence hand
will change. When EF > EFi, the electron concentration is larger than the hole concentration, and when
EF < EFi the hole concentration is larger than the electron concentration. When the density of electrons
is greater than the density of holes, the semiconductor is n type; donor impurity atoms have been
added. When the density of holes is greater than the density of electrons, the semiconductor is p type;
acceptor impurity atoms have been added. The Fermi energy level in a semiconductor changes as the
electron and hole concentrations change and, again, the Fermi energy changes as donor or acceptor
impurities are added.
The expressions previously derived for the thermal-equilibrium concentration of electrons and holes
are again given as
n0 = Nc exp[(-(Ec – EF))/kT] and p0 = Nv exp[(-(EF – Ev))/kT].
The Fermi energy may vary through the band gap energy, which will then change the values of n0
and p0.
Since n0 > p0, the semiconductor is n type. In an n-type semiconductor, electrons are referred to as
the majority carrier and holes as the minority carrier. Similarly, in a p-type semiconductor where p0
> n0, holes are the majority carrier and electrons are the minority carrier.
We may derive another form of the equations for the thermal-equilibrium concentrations of electrons
and holes. The thermal-equilibrium electron concentration can be written as
n0 = ni exp[(EF – EFi)/kT].
Similarly, we will obtain
p0 = ni exp[(-(EF – EFi))/kT].
As we will see, the Fermi level changes when donors and acceptors are added, but these Equations
show that, as the Fermi level changes from the intrinsic Fermi level, n0 and p0 change from the ni
value. If EF > EFi, then we will have n0 > ni and p0 < ni. One characteristic of an n-type semiconductor
is that EF > EFi so that n0 > p0. Similarly, in a p-type semiconductor, EF < EFi, SO that p0 > ni and n0
< ni, thus p0 > n0.
The n0 p0 Product.
We may take the product of the general expressions for n0 and p0. The result may be written as
n0 p0 = Nv Nc exp[-Eg /kT].
The values of n0 and p0 are not necessarily equal. So we have that, for the semiconductor in thermal
equilibrium,
n0 p0 = ni².
This equation states that the product of n0 and p0 is always a constant for a given semiconductor
material at a given temperature. Although this equation seems very simple, it is one of the
fundamental principles of semiconductors in thermal equilibrium. It is important to keep in mind that
this Equation was derived using the Boltzmann approximation. If the Boltzmann approximation is
not valid, then likewise, this Equation is not valid.
An extrinsic semiconductor in thermal equilibrium does not, strictly speaking, contain an intrinsic
carrier concentration, although some thermally generated carriers are present. The intrinsic electron
and hole carrier concentrations are modified by the donor or acceptor impurities. However, we may
think of the intrinsic concentration ni simply as a parameter of the semiconductor material.
Degenerate and Nondegenerate Semiconductors.
In our discussion of adding dopant atoms to a semiconductor, we have implicitly assumed that the
concentration of dopant atoms added is small when compared to the density of host or semiconductor
atoms. The small number of impurity atoms is spread far enough apart so that there is no interaction
between donor electrons, for example, in an n-type material. We have assumed that the impurities
introduce discrete, noninteracting donor energy states in the n-type semiconductor and discrete,
noninteracting acceptor states in the p-type semiconductor. These types of semiconductors are
referred to as nondegenetate semiconductors.
If the impurity concentration increases, the distance between the impurity atoms decreases and a point
will he reached when donor electrons, for example, will begin to interact with each other. When this
occurs, the single discrete donor energy will split into a hand of energies. As the donor concentration
further increases, the band of donor states widens and may overlap the bottom of the conduction band.
This overlap occurs when the donor concentration becomes comparable with the effective density of
states. When the concentration of electrons in the conduction band exceeds the density of states Nc,
the Fermi energy lies within the conduction bud. This type of semiconductor is called a degenerate
n-type semiconductor.
In a similar way, as the acceptor doping concentration increases in a p-type semiconductor, the
discrete acceptor energy states will split into a band of energy and may overlap the top of the valence
band. The Fermi energy will lie in the valence band when the concentration of holes exceeds the
density of states Nv. This type of semiconductor is called a degenerate p-type semiconductor.
The energy states below are mostly filled with electrons and the energy states above EF are mostly
empty. In the degenerate n-type semiconductor, the states between EF and Ec , are mostly fill with
electrons; thus, the electron concentration in the conduction band is very large. Similarly, in the
degenerate p-type semiconductor, the energy states between Ev and EF are mostly empty; thus, the
hole concentration in the valence band is very large.
9. CHARGE NEUTRALITY
1. Compensated Semiconductors
2. Equilibrium Electron and Hole Concentrations
In thermal equilibrium, the semiconductor crystal is electrically neutral. The electrons are distributed
among the various energy states, creating negative and positive charges, but the net charge density is
zero. This charge-neutrality condition is used to determine the thermal-equilibrium electron and hole
concentrations as a function of the impurity doping concentration. We will define a compensated
semiconductor and then determine the electron and hole concentrations as a function of the donor and
acceptor concentrations.
Compensated Semiconductors.
A compensated semiconductor is one that contains both donor and acceptor impurity atoms in the
same region. A compensated semiconductor can he formed, for example, by diffusing acceptor
impurities into an n-type material, or by diffusing donor impurities into a p-type material. An n-type
compensated semiconductor occurs when Nd > Na, and a p-type compensated semiconductor occurs
when Na > Nd.
If Na = Nd, we have a completely compensated semiconductor that has, as we will show, the
characteristics of an intrinsic material. Compensated semiconductors are created quite naturally
during device fabrication.
Equilibrium Electron and Hole Concentrations.
The charge neutrality condition is expressed by equating the density of negative charges to the density
of positive charges. We then have
n0 + N -a = p0 + N+d
or
n0 + (Na – pa) = p0 + (Nd – nd),
where n0 and p0 are the thermal-equilibrium concentrations of electrons and holes in the conduction
band and valence band, respectively. The parameter nd is the concentration of electrons in the donor
energy states, so N+d = (Nd – nd), is the concentration of positively charged donor states. Similarly, pa
is the concentration of holes in the acceptor states, so N -a= (Na – pa) is the concentration of negatively
charged acceptor states. We have expressions for n0, p0, nd and pa in terms of the Fermi energy and
temperature.
If we assume complete ionization, nd and pa are both zero, and Equation becomes
n0 + Na = p0 +Nd.
10. POSITION OF FERMI ENERGY LEVEL
1. Mathematical Derivation
2. Variation of EF with Doping Concentration and Temperature
3. Relevance of the Fermi energy
We discussed qualitatively how the electron and hole concentrations change as the Fermi energy level
moves through the band gap energy. Then we calculated the electron and hole concentrations as a
function of donor and acceptor impurity concentrations. We can now determine the position of the
Fermi energy level as a function of the doping concentrations and as a function of temperature.
Mathematical Derivation.
The position of the Fermi energy level within the band gap can be determined by using the equations
already developed for the thermal-equilibrium electron and hole concentrations. If we assume the
Boltzmann approximation to be valid, then we have n0 = Nv exp[-(Ec – EF)/kT]. We can solve for Ec
– EF from this equation and obtain
Ec – EF = kT ln (Nc/n0).
If we consider an n-type semiconductor in which Nd >> ni, then n0 ≈ Nd , so that
Ec – EF = kT ln (Nc/Nd).
The distance between the bottom of the conduction band and the Fermi energy is a logarithmic
function of the donor concentration. As the donor concentration increases, the Fermi level moves
closer to the conduction band. Conversely, if the Fermi level moves closer to the conduction band,
then the electron concentration in the conduction band is increasing. We may note that if we have a
compensated semiconductor, then the Nd term is simply replaced by Nd –Na, or the net effective donor
concentration.
Variation of EF with Doping Concentration and Temperature.
We may plot the position of the Fermi energy level as a function of the doping concentration. As the
doping levels increase, the Fermi energy level moves closer to the conduction band for the n-type
material and closer to the valence band for the p-type material. Keep in mind that the equations for
the Fermi energy level that we have derived assume that the Boltzmann approximation is valid.
The intrinsic carrier concentration ni, is a strong function of temperature, so that EF is a function of
temperature also. As the temperature increases, ni increases, and EF moves closer to the intrinsic
Fermi level. At high temperature, the semiconductor material begins to lose its extrinsic
characteristics and begins to behave more like an intrinsic semiconductor. At the very low
temperature, freeze-out occurs; the Boltzmann approximation is no longer valid and the equations we
derived for the Fermi-level position no longer apply. At the low temperature where freeze-out occurs,
the Fermi level goes above Ed for the n-type material and below Ea for the p-type material. At absolute
zero degrees, all energy states below EF are full and all energy states above EF are empty.
Relevance of the Fermi Energy.
We have been calculating the position of the Fermi energy level as a function of doping concentrations
and temperature. This analysis may seem somewhat arbitrary and fictitious. However, these relations
do become significant later in our discussion of pn junctions and the other semiconductor devices we
consider. An important point is that, in thermal equilibrium, the Fermi energy level is a constant
throughout a system.
11. CARRIER DRIFT
1. Drift Current Density
2. Mobility Effects
3. Conductivity
Any motion of free carriers in a semiconductor leads to a current. This motion can be caused by an
electric field due to an externally applied voltage, since the carriers are charged particles. We will
refer to this transport mechanism as carrier drift. In addition, carriers also move from regions where
the carrier density is high to regions where the carrier density is low. This carrier transport mechanism
is due to the thermal energy and the associated random motion of the carriers. We will refer to this
transport mechanism as carrier diffusion. The total current in a semiconductor equals the sum of the
drift and the diffusion current.
As one applies an electric field to a semiconductor, the electrostatic force causes the carriers to first
accelerate and then reach a constant average velocity, v, due to collisions with impurities and lattice
vibrations. The ratio of the velocity to the applied field is called the mobility. The velocity saturates
at high electric fields reaching the saturation velocity. Additional scattering occurs when carriers flow
at the surface of a semiconductor, resulting in a lower mobility due to surface or interface scattering
mechanisms.
Diffusion of carriers is obtained by creating a carrier density gradient. Such gradient can be obtained
by varying the doping density in a semiconductor or by applying a thermal gradient.
Both carrier transport mechanisms are related since the same particles and scattering mechanisms are
involved. This leads to a relationship between the mobility and the diffusion constant called the
Einstein relation.
The electron mobility and hole mobility have a similar doping dependence: For low doping
concentrations, the mobility is almost constant and is primarily limited by phonon scattering. At
higher doping concentrations, the mobility decreases due to ionized impurity scattering with the
ionized doping atoms. The actual mobility also depends on the type of dopant.
Note that the mobility is linked to the total number of ionized impurities or the sum of the donor and
acceptor densities. The free carrier density is to first order related to the difference between the donor
and acceptor concentration. The minority carrier mobility also depends on the total impurity density.
The minority-carrier mobility can be approximated by the majority-carrier mobility in a material with
the same number of impurities.
The conductivity of a material is defined as the current density divided by the applied electric field.
Since the current density equals the product of the charge of the mobile carriers, their density and
velocity, it can be expressed as a function of the electric field using the mobility. To include the
contribution of electrons as well as holes to the conductivity, we add the current density due to holes
to that of the electrons, or:
.
The conductivity due to electrons and holes is then obtained from:
.
The resistivity is defined as the inverse of the conductivity, namely:
.
12. THE HALL EFFECT
The Hall Effect refers to the potential difference (Hall voltage) on the opposite sides of an
electrical conductor through which an electric current is flowing, created by a magnetic field
applied to the current. Edwin Hall discovered this effect in 1879.
The ratio of the voltage created to the product of the amount of current and the magnetic field
divided by the element thickness is known as the Hall coefficient. It is a characteristic of the
material from which the conductor is made, as its value depends on the type, number and properties
of the charge carriers that constitute the current.
The Hall effect comes about due to the nature of the current flow in a
conductor. Current consists of the movement of many small chargecarrying "particles" (typically, but not always, electrons). These charges
experience a force, called the Lorentz Force, when a magnetic field is
present that is not parallel to their motion. When such a magnetic field is
absent, the charges follow an approximately straight, 'line of sight' path.
However, when a perpendicular magnetic field is applied, their path is
curved so that moving charges accumulate on one face of the material.
This leaves equal and opposite charges exposed on the other face, where
there is a dearth of mobile charges. The result is an asymmetric
distribution of charge density across the hall element that is
perpendicular to both the 'line of sight' path and the applied magnetic
field. The separation of charge establishes an electric field that opposes
the migration of further charge, so a steady electrical potential builds up
for as long as the current is flowing.
For a simple metal where there is only one type of charge carrier (electrons) the Hall voltage VH is
given by
The Hall coefficient is defined as
where I is the current across the plate length, B is the magnetic flux density, d is the depth of the
plate, e is the electron charge, and n is the charge carrier density of the carrier electrons.
As a result, the Hall effect is very useful as a means to measure both the carrier density and the
magnetic field.
One very important feature of the Hall effect is that it differentiates between positive charges
moving in one direction and negative charges moving in the opposite. The Hall effect offered the
first real proof that electric currents in metals are carried by moving electrons, not by protons. The
Hall effect also showed that in some substances (especially semiconductors), it is more appropriate
to think of the current as positive “holes” moving rather than negative electrons.
Hall effect in Semiconductors
When a current carrying semiconductor is kept in a magnetic field, the carriers of the semiconductor
experience a force in a direction perpendicular to the magnetic field and current field, this is called
Hall effect in semiconductors. The simple formula for the Hall coefficient given above becomes
more complex in semiconductors where the carriers are generally both electrons and holes which
may be present in different concentrations and have different mobility.
In physics, electron mobility (or simply, mobility), is a quantity relating the drift velocity of
electrons to the applied electric field across a material, according to the formula:
,
where
is the drift velocity,
is the applied electric field,
is the mobility.
It is the application for electrons of the more general phenomenon of electrical mobility of charged
particles in a fluid under an applied electric field.
In semiconductors, mobility can also apply to holes as well as electrons.
For moderate magnetic fields the Hall coefficient is
,
where n is the electron concentration, p the hole concentration,
the electron mobility ,
the
hole mobility and the electronic charge.
For large applied fields the simpler expression analogous to that for a single carrier type holds.
.
13. THERMO-ELECTRIC EFFECTS IN SEMICONDUCTORS
The temperature dependence of the current in a semiconductor can be included by generalizing the
drift-diffusion current equation. The proportionality constant between the current density and the
temperature gradient is the product of the conductivity, and the thermo-electric power. The Peltier
coefficient is related to the
thermo-electric power by:
. If electrons and
holes are present in the
semiconductor one has to
include the effect of both
when calculating the Peltier
coefficient:
.
The
resulting
Peltier
coefficient as a function of
temperature for silicon is
shown in the figure below:
Peltier coefficient for p-type
(top curve) and n-type
(bottom curve) silicon as a
function of temperature. The
doping density equals 1014 cm-3. The Peltier coefficient is positive for p-type silicon and negative
for n-type silicon at low temperature. The semiconductor becomes intrinsic at high temperature.
Given that the mobility of electrons is higher than that of holes, the Peltier coefficient of intrinsic
silicon is negative.
14. THE P-N JUNCTION
1. Basic Structure of pn Junction
2. Zero Applied Bias
3. Forward and Reverse Applied Bias
Basic Structure of pn Junction
A p-n junction consists of two semiconductor
regions with opposite doping type as shown in
Figure 1. The region on the left is p-type with
an acceptor density Na, while the region on the
right is n-type with a donor density Nd. The
dopants are assumed to be shallow, so that the
electron (hole) density in the n-type (p-type)
region is approximately equal to the donor
Figure 1: Cross-section of a p-n junction
(acceptor) density.
We will assume, unless stated otherwise, that the
doped regions are uniformly doped and that the transition between the two regions is abrupt. We will
refer to this structure as an abrupt p-n junction. Frequently we will deal with p-n junctions in which
one side is distinctly higher-doped than the other. We will find that in such a case only the low-doped
region needs to be considered, since it primarily determines the device characteristics. We will refer
to such a structure as a one-sided p-n junction. The junction is biased with a voltage Va as shown in
Figure 1. We will call the junction forward-biased if a positive voltage is applied to the p-doped
region and reversed-biased if a negative voltage is applied to the p-doped region. The contact to the
p-type region is also called the anode, while the contact to the n-type region is called the cathode, in
reference to the anions or positive carriers and cations or negative carriers in each of these regions.
Zero Applied Bias
The principle of operation will be explained using an experiment, which is in principle possible but
not necessarily executable in practice. We imagine that one can bring both semiconductor regions
together, aligning both the conduction and valence band energies of each region. This yields the socalled flatband diagram shown in Figure 2.
Figure 2 : Energy band diagram of a p-n junction (a) before and (b) after merging the n-type and ptype regions
Note that this does not automatically align the Fermi energies, EF,n and EF,p. Also, note that this
flatband diagram is not an equilibrium diagram since both electrons and holes can lower their energy
by crossing the junction. A motion of electrons and holes is therefore expected before thermal
equilibrium is obtained. The diagram shown in Figure 2 (b) is called a flatband diagram. This name
refers to the horizontal band edges. It also implies that there is no field and no net charge in the
semiconductor.
Thermal equilibrium
To reach thermal equilibrium, electrons/holes close to the metallurgical junction diffuse
across the junction into the p-type/n-type region where hardly any electrons/holes are present. This
process leaves the ionized donors (acceptors) behind, creating a region around the junction, which is
depleted of mobile carriers. We call this region the depletion region, extending from x = -xp to x = xn.
The charge due to the ionized donors and acceptors causes an electric field, which in turn causes a
drift of carriers in the opposite direction. The diffusion of carriers continues until the drift current
balances the diffusion current, thereby reaching thermal equilibrium as indicated by a constant Fermi
energy. This situation is shown in Figure 3.
While in thermal equilibrium no external
voltage is applied between the n-type and
ptype material, there is an internal potential,
which is caused by the work function
difference between the n-type and p-type
semiconductors. This potential equals the
built-in potential, which will be further
discussed in the next section.
The built-in potential in a semiconductor
equals the potential across the depletion
region in thermal equilibrium. Since
thermal equilibrium implies that the Fermi
energy is constant throughout the p-n
diode, the built-in potential equals the
Figure 3 : Energy band diagram of a
difference between the Fermi energies, EFn
and
p-n junction in thermal equilibrium
EFp, divided by the electronic charge. It also
equals the sum of the bulk potentials of
each
region, since the bulk potential quantifies the distance between the Fermi energy and the intrinsic
energy. This yields the following expression for the built-in potential.
.
Forward and reverse
bias.
We now consider a p-n
diode with an applied bias
voltage, Va. A forward
bias
corresponds
to
applying
a
positive
voltage to the anode (the
p-type region) relative to
the cathode (the n-type
region). A reverse bias
corresponds to a negative
voltage applied to the
Figure 4: Energy band diagram of a p-n junction
cathode. Both bias modes are
under reverse and forward bias
illustrated with Figure 4. The applied
voltage is proportional to the difference
between the Fermi energy in the n-type and p-type quasi-neutral regions.
As a negative voltage is applied, the potential across the semiconductor increases and so does the
depletion layer width. As a positive voltage is applied, the potential across the semiconductor
decreases and with it the depletion layer width. The total potential across the semiconductor equals
the built-in potential minus the applied voltage, or:
.
PROBLEM
In preparing the transport of semiconductor wafers for chip production have
been mixed their labels, i.e. was unclear semiconductor wafers type and level of
doping. Must propose and justify the best way to restore factory settings of
semiconductor wafers. Practically realize their proposed approach and identify
the necessary parameters of the semiconductor wafers.
Topics related to the problem:














Intrinsic and Extrinsic semiconductors.
Semiconductor doping.
The Intrinsic carrier Concentration.
The n0 and p0 Equations.
The Intrinsic Fermi-Level Position.
Carrier mobility.
n - type semiconductor and its physical properties.
p - type semiconductor and its physical properties.
Equilibrium Electron and Hole Concentrations.
Position of Fermi energy level as a function of impurity doping concentration.
Impurity semiconductor conductivity.
Impurity semiconductor conductivity dependence on temperature.
Hall effect.
Hall effect temperature dependence.
GLOSSARY OF IMPORTANT TERMS
binary semiconductor - a two-element compound semiconductor, such as gallium arsenide (GaAs).
covalent bonding - the bonding between atoms in which valence electrons are shared.
diamond lattice - the atomic crystal structure of silicon, for example, in which each atom has four
nearest neighbours in a tetrahedral configuration.
doping - the process of adding specific types of atoms to a semiconductor to favourably alter the
electrical characteristics.
elemental semiconductor - a semiconductor composed of a single species of atom, such as silicon
or germanium.
epitaxial layer - a thin, single-crystal layer of material formed on the surface of a substrate.
ion implantation - one particular process of doping a semiconductor.
lattice -the periodic arrangement of atoms in a crystal.
Miller indices - the set of integers used to describe a crystal plane.
primitive cell - the smallest unit cell that can be repeated to form a lattice.
substrate - a semiconductor wafer or other material used as the starting material for further
semiconductor processing, such as epitaxial growth or diffusion.
unit cell - a small volume of a crystal that can he used to reproduce the entire crystal.
acceptor atoms - impurity atoms added to a semiconductor to create a p-type material charge carrier
The electron and/or hole that moves inside the semiconductor and gives rise to electrical currents.
compensated semiconductor - a semiconductor that contains both donors and the same
semiconductor region.
complete ionization - the condition when all donor atoms are positively up their donor electrons and
all acceptor atoms are negatively charged by accepting electron
degenerate semiconductor - a semiconductor whose electron concentration or hole concentration is
greater than the effective density of states, so that the Fermi level is in the condition band (n type) or
in the valence band (p type).
donor atoms - impurity atoms added to a semiconductor to create an n-type material.
effective density of states - the parameter Nc , which results from integrating the density of quantum
states times the Fermi function over the conduction-band energy and the parameter Nv, which results
from integrating the density of quantum states times [1- f(E)] over the valence-band energy.
extrinsic semiconductor - a semiconductor in which controlled amounts of donors an acceptors have
been added so that the electron and hole concentrations change from the intrinsic carrier concentration
and a preponderance of either electrons (n type) or holes (p type) is created.
freeze-out - the condition that occurs in a semiconductor when the temperature is 1ower and the
donors and acceptors become neutrally charged. The electron and hole concentration become very
small.
intrinsic carrier concentration ni - the electron concentration in the conduction band and the hole
concentration in the valence band (equal values) in an intrinsic semiconductor.
intrinsic Fermi level EFi - the position of the Fermi level in an intrinsic semiconductor.
intrinsic semiconductor - a pure semiconductor material with no impurity atoms and no lattice
defects in the crystal.
nondegenerate semiconductor - a semiconductor in which a relatively small number of donors
and/or acceptors have been added so that discrete, no interacting donor states and/or discrete, no
interacting acceptor states are introduced.
conductivity - a material parameter related to carrier drift; quantitatively, the ratio of drift current
density to electric field.
diffusion - the process whereby particles flow from a region of high concentration to a region of low
concentration.
diffusion current - the current that results from the diffusion of charged particles.
drift - the process whereby charged particles move while under the influence of an electric field.
drift current - the current that results from the drift of charged particles.
drift velocity - the average velocity of charged particles in the presence of an electric field.
Hall voltage - the voltage induced across a semiconductor in a Hall effect measurement.
ionized impurity scattering - the interaction between a charged carrier and an ionized impurity
center.
lattice scattering - the interaction between a charged carrier and a thermally vibrating lattice atom.
mobility - the parameter relating carrier drift velocity and electric field.
resistivity - the reciprocal of conductivity; a material parameter that is a measure of the resistance to
current.
INTERACTIVE DEMONSTARTION
(Si, Ge, GaA lattice)
http://jas.eng.buffalo.edu/education/solid/unitCell/home.html
http://www.cmmp.ucl.ac.uk/~ijf/3c25/crystals_small.html
http://www.cmmp.ucl.ac.uk/~ijf/3c25/bravais.gif
http://www.cmmp.ucl.ac.uk/~ijf/3c25/surfex.html
(Bravais lattices)
Applets in Semiconductor, Microelectronics and Photonics
(Generation, recombination)
http://jas.eng.buffalo.edu/education/semicon/recombination/indirect.html
(Fermi-Level, Fermi function)
http://jas.eng.buffalo.edu/education/semicon/fermi/functionAndStates/functionAndStates.html
(Position of Fermi energy level and a function of impurity doping concentration)
http://jas.eng.buffalo.edu/education/semicon/fermi/bandAndLevel/fermi.html
(Position of Fermi energy level and a function of impurity doping concentration Si)
http://jas.eng.buffalo.edu/education/semicon/fermi/levelAndDOS/index.html
(p – n junction)
Applets in Semiconductor, Microelectronics and Photonics
Applets in Semiconductor, Microelectronics and Photonics
Applets in Semiconductor, Microelectronics and Photonics
http://jas.eng.buffalo.edu/education/pn/biasedPN/index.html
Applets in Semiconductor, Microelectronics and Photonics
(p – n junction technology process)
PN Diode Applet
(MOS structure)
http://jas.eng.buffalo.edu/education/mos/mosCap/biasBand10.html
http://www.effects.ru/fieldDN0/33.html
(Hall effect)
http://www.magnet.fsu.edu/education/tutorials/java/halleffect/index.html
EXAM
There will be final exam. Exam will consist of several problems and questions. The exam will be
closed book. An equation sheet and the physical parameters and constants those are required in the
solutions will be provided. Exam will be planned to take one hour, but you will be given 2 hours to
complete it in an attempt to minimize time pressure. The format of your exam solutions should be the
same as that used for the homework assignments: units must be shown explicitly and your work must
be readable. You will need a calculator for the exam.
QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
Why do solids occur in the form of a crystal?
How do we classify the different crystals?
How many Bravais lattices are there in two dimensions? How many in three dimensions?
List the three cubic Bravais lattices.
Describe the procedure for finding the volume density of atoms in a crystal.
Describe the procedure for finding the packing density of lattice in a crystal.
Describe the procedure for obtaining the Miller indices that describe a plane in a crystal.
Sketch a lattice plane given the Miller indices.
Describe the procedure for finding the surface density of atoms on a particular plane in a crystallattice.
10. Descibe the Diamond crystal structure.
11. Describe the type of bond (interaction) between atoms.
12. Describe various defects in a crystal.
13. Describe the experimental determination of structure.
14. Density of States
15. Fermi – Dirac distribution function
16. Impurity distribution functions
17. Other distribution functions and comparison
18. Specific heat
19. Law of Dulong and Petit
20. Einstein model
21. Debye model
22. Equilibrium Distribution of Electrons and Holes
23. The n0 and p0 Equations
24. The Intrinsic carrier Concentration
25. The Intrinsic Fermi-Level Position
26. Dopant atoms and energy levels
27. Equilibrium Distribution of Electrons and Holes
28. The n0p0 Product
29. Degenerate and Nondegenerate Semiconductors
30. Charge neutrality
31. Compensated Semiconductors
32. Equilibrium Electron and Hole Concentrations
33. Position of Fermi energy level as a function of impurity doping concentration
34. Drift Current Density
35. Mobility Effects
36. Conductivity
37. Hall effect
38. Thermo-electric effects in semiconductors
39. p – n junction
Reading list
There are many books available on Condensed Matter Physics. The following two books are
recommended since they cover most of the material at a suitable level:
Donald A. Neaven. Semiconductor Physics and Devices, 3rd edition.
H. M. Rosenberg. The Solid State. 3rd edition.
There are a number of other books worth consulting. Some aspects of the course are treated well
by
J. R. Hook and H. E. Hall. Solid State Physics. 2nd edition.
A. P. Sutton. Electronic Structure of Materials.
M. de Podesta. Understanding the Properties of Matter.
Advanced texts for continued studies are
C. Kittel. Introduction to Solid State Physics. 7th edition
J. S. Blakemore. Solid State Physics, 2nd edition
You can find more information on the web at:
http://su.lt/fic-gamtos-moksl/fizikos-katedra/mokomosios-knygos-paskait-konspektai