Prof. Jasprit Singh
Fall 2001
EECS 320
GSI: As noted in the class the GSI for this course is Manish Hemkar
(mhemkar@engin.umich.edu).
Homework 1
This homework is due on Thursday September 20.
Problem 1: An Aluminum interconnect is used in an IC. The parameters
for the interconnect are:
Length
Area
Atomic mass
Density
valence
mobility
=
=
=
=
=
=
1 mm
10 m 5m
27
2:7 gcm;3
3
50 cm2 =V:s
Calculate the resistance of the interconnect.
Problem 2: Last week Motorola announced the ability to produce a 12 inch
diameter GaAs layer grown on a 12 inch silicon substrate.
Calculate the number of Si atoms that t on a 12 inch line along the (110)
direction.
Problem 3: Calculate the wavelengths and wavevectors for an electron and
a photon with energies of:
(i) 1.0 eV;
(ii) 1.0 keV;
The dierence in the values for an electron and a photon is very important
in understanding optoelectronics.
Problem 4: Calculate the number of allowed electron states between energies 1.0 and 1.1 eV for a volume 10;4 cm3 . The energy-momentum relation
is
2 2
E = 1:0 eV + h2mk ; m0 = 9:1 10;31 kg
0
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Problem 5: Plot the Fermi function for energies between 1.0 eV and 1.2
eV. Assume that EF = 1:1 eV . Plot the function for a temperature of (i) 77 K
and (ii) 300 K.
Also plot the Boltzmann function (which is an approximation to the Fermi
function) on the same plots.
Examine how well the Boltzmann approximation works (or does not work).
SOME IMPORTANT ISSUES DISCUSSED THIS WEEK
A classical model for electron current in solids
Information processing devices that have been responsible for the modern
information age are based on one common theme. If an input is applied to
them a well dened output results. The input may be due to an electrical
signal or an electromagnetic wave or a pressure wave etc. The output may
be a current pulse or a voltage pulse or light etc. This input-output relation is
then exploited to design digital switches, ampliers, memory devices, oscillators,
lasers etc. In order to understand and then exploit the input-output relation
we need to understand how electrons behave inside the material forming the
electronic device.
All electronic and optoelectronic devices based on semiconductors exploit
special properties electrons have inside semiconductors. The neutrons and protons that are present do not participate directly in any physical process although they are essential and, along with core electrons, provide the chemistry
that causes the material to be a semiconductor. The neutrons and protons are
immobile while the electrons being very light particles in comparison are \free"
to move around.
To understand the properties of semiconductors and their devices, we must
understand the properties of electrons inside semiconductors. In particular, we
should be able to understand two aspects of the electronic properties: i) what
is the energy, momentum, position, etc. of the electron inside a semiconductor;
and ii) how do electrons respond to an external perturbation such as an electric
eld, magnetic eld, electromagnetic eld, etc. In free space we know that
electrons obey the classical \free" electron equations:
2
E = 2pm
0
2
dp = F = ;e (E + v H)
ext
dt
where m0 , E , p, Fext , and ;e are the electron mass, energy, momentum, force,
and charge. The external forces include the eects of an electric eld E and a
magnetic eld H. Using such classical equations we are able to describe a number
of important properties of electrons. For example, we can use these equations
to design a cathode ray tube (like a TV or computer screen) by predicting how
electrons will strike the screen pixels. However, a number of important things
cannot be understood with these classical equations. We cannot understand,
for example, why metals like aluminum or copper have such a high conductivity
while materials like glass or silicon or diamond have such poor conductivity. In
fact we cannot understand any signicant aspect of semiconductor technology.
To develop this understanding we need to use quantum mechanics. Once we
know some basic properties of electrons in semiconductors through quantum
mechanics we can then develop some simple classical looking.
In an electrical engineer's world all materials are divided into three categories:
(i) Metals: These are materials which have a very high conductivity or low
resistivity to current ow; (ii) Insulators: These are materials which have very
poor conductivity; and (iii) Semiconductors: These are materials whose conductivity falls in between metals and insulators. Each kind of material mentioned
above nds important uses. For example, in an extension cord used to bring
electricity from an outlet to say a hair dryer, the inner core is made of copper |
a metal | while the outer cover is made from an insulator. The metal carries
the current due to its low resistivity while the insulator prevents us from getting
a shock.
An important law that is widely used in electrical engineering is Ohm's law
which tells us how current I ows in a material when a potential V is applied.
The law has the form
I = VR
This law was introduced by Ohm on empirical basis i.e on the basis of observation with no fundamental explanation. Let us express Ohm's law in some
other equivalent forms. If A is the area of the sample and L its length, the
resistance of the sample is
R = AL = 1 AL
where and are known as the resistivity and conductivity of the material.
We may write the current as
I = JA
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where J is the current density. We may also write the electric eld applied to
the sample as
E = VL
Ohm's law now becomes
J = E
Let us proceed further along this path. We have by the denition of current
density
J = nev
where n is the density of the carriers carrying current, e is their charge, and v
is the average velocity with which the carriers are moving in the direction of
the eld. We may also dene a relation between velocity of the carriers and the
applied eld through a denition
v = E
where is called the mobility of the carriers. With this denition we get
= ne
All of the equations we have given above are just dierent ways of expressing
Ohm's law. Even though we have introduced carrier density and charge and
mobility we have not derived Ohm's law. The rst attempt at deriving Ohm's
law was due to Drude at the turn of the 20th Century.
A Microscopic Picture to Describe Current Flow
As noted above Drude was the rst person to attempt to explain current
ow on a fundamental level. At this time there was no knowledge of quantum
mechanics and Drude built a model based on the chemistry and classical physics
known at that time. He exploited his understanding of atoms, electrons and
nuclei.
The picture of a solid used by Drude is shown schematically in Fig. 2.1 of
the text. Solids are a collection of atoms. Each atom has a nucleus around
which electrons are present. The total number of electrons in each atom are Za
which is the atomic number of the atom. Out of these electrons, Zc are valence
electrons in the outermost shell of the atom. These electrons are weakly bound
to the atoms and are capable of carrying current when a eld is applied. The
remaining electrons are core electrons which are strongly bound to the nucleus
and don't participate in current ow.
We are now interested in the behavior of a very large number of negatively
charged electrons moving through positively charged xed ions. To appreciate
the enormity of the problem let us examine the number density of electrons
involved. An element contains 6.022 1023 atoms per mole (the Avogadro's
number). If is the density of the material, the number of moles per unit
volume are =A, where A is the atomic mass. We now assume
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that the number of electrons that are free to conduct current is Zc where Zc
are the number of electrons in the outermost shell (i.e., the valence of the element) of the atom as shown in Fig. 2.1. The electron density for the conduction
electrons is now
n = 6:022 1023 ZAc
For most materials, this number is 1023 cm;3 ! There is an enormously large
density of free conduction electrons in the material. One can dene an average
radius rs of a spherical volume per electron by
4rs3 = 1
3
n 3 1=3
or
rs = 4n
This radius is 1-2 A for most materials!
Drude wanted to understand why the conductivity of dierent materials is
dierent. For this purpose he used the model of atoms described above. He
made the following assumptions for his model:
A solid is made up of a collection of atoms.
Each atom has electrons around the nucleus as shown in Fig. 2.1.
The electrons can be divided into two categories | electrons that are core
electrons and electrons that are in the outer shell of the atom and are known as
valence electrons.
The core electrons are strongly bound to the atom and don't participate
in any current conduction.
The outer shell valence electrons are weakly bound to the nucleus and in
the solid become free to carry current.
As the electrons move in the presence of an applied electric eld they
scatter from the nuclei and reach a terminal velocity (much like rain drops
falling through air).
An outcome of the Drude model is that the conductivity of any material is
proportional to the valence electron per volume that are in the material. While
this model worked for some metals, it failed to explain why conductivity of
materials like diamond (C) or silicon (Si) is so small. It also failed to explain
a number of other important experimental observations that were made shortly
after Drude introduced his model. Some of these observations are:
In some materials current appears to be carried by positively charged
particles!
Conductivity of some materials is changed by orders of magnitude by
introduction of tiny amounts of (say one part in a million or less) impurities.
In some materials Ohm's law is not obeyed at all.
These and many other observations could simply not be reconciled with Drude's
or any other model based on classical physics. We now know that to understand
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these observations we have to invoke quantum mechanics. Before doing so let
us understand some structural aspects of solids.
Crystalline Materials
Since all solid state devices are made from materials that are solids, we
need to understand the properties of electrons in these solids. Solids can be
characterized as:
i)Amorphous: In these materials the atoms are arranged in a random fashion
with no long range order between them.
ii)Polycrystalline: In these the atoms are arranged in precise order over
regions called grains. The size of these grains range from 0:1 ; 5:0m. From
one grain to the next there is no order in the arrangement of the atoms.
iii)Crystalline: In these there is a complete order in the arrangement of
atoms. To dene the precise order we use the lattice and the basis. The lattice
is a mathematical abstraction describing a periodic arrangement of points in
space. The basis tells us about the atom or atoms that are to be attached to
each lattice point to form the crystal.
All the devices we will discuss in EECS 320 are made from crystals.
To describe a crystal structure we need to dene a lattice and a basis. The
lattice is described by a set of unit vectors from which a periodic array of points
in space are generated. A basis consists of one or more atoms that are placed
on every lattice site to generate the crystal.
All semiconductors of interest to us have an underlying face centered cubic
(fcc) lattice. The edge of the cube has a length a which is called the lattice constant of the material. Every semiconductor has its own unique lattice constant.
The basis for all of the semiconductors consists of two atoms. Their positions
are:
(0; 0; 0); (a=4; a=4; a=4)
where the rst atom is referred to a lattice point.
The volume of a unit cell (smallest volume which when repeated generates
the entire crystal) is a3 =4.
Going Beyond Classical Physics
The inability to explain things like conductivity on a microscopic level forces
us to invoke quantum mechanics. The general rule of thumb for when we need
quantum mechanics is the following:
Find the smallest distance over which the potential energy changes appreciably. Let us say this is d.
Find the de Broglie wavelength of the particle dened as
= h=p
where h is the Planck's constant and p is the particle momentum. If d we
need quantum mechanics.
When we need to describe how electrons behave inside atoms, or inside
solids we need to use the wave description of particles because the potential
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energy changes appreciably over 1
A. However, this does not mean that we
will abandon all classical equations. Our approach in EECS 320 will be the
following:
We will use quantum mechanics to understand the properties of electrons
inside solids.
Once we have a sucient understanding we will incorporate our understanding into a more physical eective classical picture.
We will study all our device behavior using the simpler classical description.
In our initial understanding where we use quantum mechanics we need to
use the Schroedinger equation to describe the properties of electrons. To establish familiarity with quantum mechanics we briey examine some simple
manifestations of quantum eects. The eects we review are the following:
The uncertainty principle: According to this principle it is not possible to
measure certain quantities with complete accuracy simultaneously. For example, we cannot measure the position and momentum (or the energy and the time
duration of the particle in that energy state) of a particle with complete accuracy. In wave phenomena this uncertainty is well known and quite intuitive.
When we think of waves, we know that there is some fuzziness in describing
where the wave is.
In most physical problems that we come across, we don't feel the importance
of the uncertainty principle. Like other manifestations of quantum mechanics,
this is felt at levels where our senses don't work. Ofcourse this is why it took
so long for physicists to discover quantum mechanics.
Tunneling: An important outcome of quantum mechanics or wave mechanics of particles is that particles can tunnel into forbidden regions. In these
forbidden regions the energy of the particles is smaller than the potential energy. Thus classical physics says that the particle cannot enter these regions.
However in quantum mechanics where the particle is described as a wave, it is
seen that tunneling occurs through the forbidden regions.
Tunneling plays a very important role in semiconductor devices as we shall
see later.
Quantization of particle energies: Another important manifestation of
quantum mechanics is that for many potential energy proles, the energies of
a particle are not continuous but are discrete. Examples are a quantum well
(a problem often known as the \particle in a box" problem) and electron in an
atom.
The free electron problem:
The easiest problem to address in quantum mechanics is the free electron
problem where the potential energy is zero or constant in space. In this case
the solutions of the Schroedinger equation are very simple and are simply given
by
(x; y; z ) = p1 ei(kx x+iky y+kz z)
V
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The electron has momentum given by
px = hkx ; py = h ky ; pz = hkz
Also in this plane wave picture the electron wave has the wavelength given by
= 2
k
An extremely important outcome of the free electron problem is that we can
derive a simple analytical form for the density of states for the electrons. The
density of states N (E ) gives us the number of allowed states per energy per
volume around the energy E . Thus if we have a volume V for the region of
interest, and we are interested in nding the number of allowed states between
energies E and E + dE this number is given by
number = N (E ) dE V
When we talk about allowed states this means the solutions for the electron
energies consistent with Schroedinger equation Thus for the free electron problem this means all allowed values of kx ; ky ; kz . To nd the density of states we
simply need to count all possible k-values consistent with the boundary conditions.
Electrons in atoms:
In the case of an electron around a proton (the hydrogen atom problem) we
have discrete energy levels denoted by atomic physics notation of 1s; 2s; 2p:::.
These energy levels correspond to the electron being bound to the proton. Measured with respect to the vacuum energy where the electron is free from the
proton, the bound energies are negative. In between the allowed energies there
is a forbidden energy region or bandgap. Thus in an atom we have for the electronic energies a series of allowed energies separated by bandgaps. This picture
gets somewhat modied when we discuss electrons in solids (which are collection
of atoms) but this general concept of allowed energies and gaps is still valid as
we will discuss later in this class.
TOPICS TO BE COVERED NEXT WEEK
Next week we will discuss the following topics:
Eective masses and concept of holes
Bandstructure of some semiconductors
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