EE 362 Electric and Magnetic Properties of Materials

1
EE 362 Electric and Magnetic Properties of
Materials

• Dr. Brian T. Hemmelman
• Chapter 3 Slides

2
The Formation of Energy Bands
In the infinite potential well or the hydrogen
atom we find that electrons can only occupy
specific energy levels.
? Probability density function of hydrogen
electron for n 1.
For two hydrogen atoms far apart we would have
3
Formation of Energy Bands
However, when we bring these close together their
wave functions (and thus probability density
functions overlap.
Thus, the electrons will interact with each other
(there is a chance they are in the same general
physical spot). This interaction causes the n1
energy level to split into two energy levels.
4
Formation of Energy Bands
This is related to the Pauli Exclusion Principle
which says no two electrons can occupy the same
quantum state (they cant have the same quantum
numbers). This for instance is why all electrons
in a single atom cant be in the 1s
shell. There is a spin up electron in the 1s
shell.Then a spin down electron in the 1s
shell.Then a spin up/down electron in the 2s
shell.Then a spin down/up electron in the 2s
shell So what happens when we put 5.01022 cm-3
Si atoms together? The n1 state must split into
(5.010222 electrons) states cm-3!
5
Formation of Energy Bands
Each of these energy levels only needs to be
separated by 10-19 eV of energy
though. Technically they are discrete energy
values but we usually consider them to just be a
continuous range of energies since they are all
so close together.
6
The Kronig-Penny Model
7
The Kronig-Penny Model
The net potential in a 1-Dimensional crystal
We can approximate this with the Kronig-Penny
Potential
8
The Kronig-Penny Model
Blochs Theorem says that if the potential is
periodic, then the solution must be periodic.
k ? wave number k is related to momentum
For E lt V0 (which applies to electrons in a
crystal) the solution of the Kronig-Penny model
can be found to be
This must be solved numerically or graphically.
What is important is that there are only certain
values of (?a) that yield real values of k in the
right side of the equation.
9
k-space
For a free particle k was
? Kinetic energy only
Note k is related to momentum! Conversely, we
can link electron energy to k
So for the free particle, there is a parabolic
relationship between E and k.
10
k-space
We often use an E versus k diagram to show the
relationship.
11
Back to the Kronig-Penny Model
In the Kronig-Penny model we want to look for
solutions that make k real
Only in the shaded regions does the left side of
the equation match the range ofcos(ka). Note
that k is changing in the vertical axis and
energy is changing in the horizontal axis (as ?
and E are related). This is basically a k versus
E diagram.
12
E versus k Diagram
We want E versus k, so just take the shaded
portions of the previous plot (that are the only
place where valid solutions exist) and flip the
axes.
13
Electrical Conduction
At absolute zero temperature all electrons in a
semiconductor are in their atomic bonds.
All of these electrons sit in their lowest energy
states since T 0. Thus, all lowest energy bands
are filled up with electrons and we have to fill
up the n1 band, the n2 band, etc.
14
Conduction and Valence Bands
The outer most band filled with electrons at T
0 is the valence band. As temperature is added
some electrons gain enough thermal energy to
break free of that bond and they go into the next
higher energy band called the conduction band.
Eg is the amount of energy needed to break free
of the atomic bond.
15
Conduction Electrons
Those electrons that break free are not bound to
any atom and are therefore free to move about.
They are thus called conduction electrons.
16
Drift Current
The motion of electrons due to an E-field
constitutes a drift current. If we had some
positive particles with volume density N (cm-3)
and an average drift velocity vd (cm/s) then the
drift current density would be
(q is the unit of charge, 1.610-19 Coulombs)
We could, instead of average drift velocity,
consider each particle individually.
where vi ? velocity of individual ith particle
If these charged particles are moving due to an
applied force then they are gaining energy. dE
F dx Fv dt F Applied Force dx
differential distance particle moves dE
increase in energy
17
Effective Mass
So the currents that flow will be related to the
velocities of the particles. Those velocities
indicate how well the particles move in the
material (crystal) because of the external
force. Specifically for the electrons in the
conduction band
e ? unit of charge
This vi relates to dE Fv dt, the force it
feels. However, this force is the net force
the electron feels which is both the external
force as well as the internal forces.
But you cant really measure/predict Fint so we
relate to what we can measure.
where m ? effective mass
18
Effective Mass
Let us reconsider the case of the free electron.
There we had
with m being the rest mass of the electron.
Taking the derivative with respect to k, we find
Recall that
Thus, the electron velocity is directly
proportional to the 1st derivative of E with
respect to k (the slope of the E vs. k diagram)!
19
Effective Mass
Now take the second derivative of E with respect
to k
Thus, the electron mass is inversely proportional
to the second derivative of E with respect to k
(the curvature of the E vs. k diagram)! This is
the situation for a free electron though. What
about electrons in a crystal?
20
Effective Mass
The free electron had a straight parabolic
relationship between E and k. In a crystal our
energy diagram is a little different.
Electrons is the bottom of a conduction band are
basically seeing a parabolic relationship as well.
21
Effective Mass
The general form of the equation for this
parabola can be written as
where C1 is the curvature parameter
Since E gt Ec we can see that C1 gt 0.Now take the
second derivative of E with respect to k.
comparing to free electron
Effective mass of electron in bottom of
conduction band
Since C1 gt 0 then m is also positive.
22
So What Is Nothing?
The drift current due to electrons in the valence
band can be written
Consider the totally full band of electrons.
Electrons have a velocity
However, the band is symmetric in k, and every
k-state is occupied, thus for every electron with
velocity v there is one with velocity v. Since
the band is totally full, the distribution of
electrons with respect to k cannot change no
matter how big the external field is, and thus
23
So What Is Nothing?
Since a totally full band yields no net drift
current we can now write
where
for the empty states!
This is the same thing as a negatively charged
particle with a negative effective mass. That is,
electrons at the top of the curved band act as
though they have negative effective mass. So we
instead consider these to be positively charged
particles with positive effective mass. These
positively charged particles we call holes.
24
The Hole
The physical reality is that a hole really
represents the absence of an electron in one of
the atomic bonds. We can picture this positively
charged hole moving through the crystal. Of
course, it is actually electrons moving from one
atomic bond to another, but as the electrons move
so to do the positions of the unfilled bonds. A
positive charge is now moving!
25
Metals, Insulators, and Semiconductors
26
Density of States
In semiconductors then that are above absolute
zero in temperature, we know that some of the
electrons will gain enough thermal energy to
break free of their atomic bonds and hop up to
the conduction band. Our goal then is to be able
to compute how many free electrons there actually
are (that can ultimately be turned into
current). There are two pieces of information
that we need to have to compute this (1) How
many quantum states are available at a particular
energy level?(2) What is the probability that a
quantum state at a particular energy level is
full? This could be viewed as computing how many
people live in an apartment building. First, we
need to know how many apartments there are on
each floor. Second, we need to know what is the
probability that an apartment on a given floor is
occupied.
27
Density of States
For each floor we would multiply the number of
apartments by the probability those apartments
are occupied. This will tell us how many people
are living on each floor. Then we can just add up
the results from all of the floors in the
building to determine the total number of people
in the building. In semiconductors, it is exactly
the same idea, except that we use more impressive
terminology! (1) The number of available quantum
states at each energy level (the number of
available apartments for the electrons) is given
by the Density of States function. (2) The
probability that a quantum state at a given
energy level is full (the probability that an
electron occupies that apartment) is given by the
Fermi-Dirac Probability Function
28
Density of States
The theory and derivation of the density of
states function is a bit abstract and confusing
for most peopleso in this case we will just skip
to the result!
29
The Fermi-Dirac Probability Function
Assume we are looking at the ith energy level
which has gi quantum states (or possible energy
spaces for electrons. We wish to put Ni
particles/electrons (Ni ? gi) into these gi
spaces (apartments).
There are gi locations we could put the first
particle.
30
The Fermi-Dirac Probability Function
That means there are then (gi-1) locations we
could put the 2nd particle.
Then there are (gi-2) locations for the 3rd
particle, and so on. So the total number of ways
of arranging Ni particles in the ith energy level
is
31
The Fermi-Dirac Probability Function
For instance, if gi 5 and Ni 3 5 spots for
the 1st particle (gi) 4 spots for the 2nd
particle (gi-1) 3 spots for the 3rd particle
(gi-(Ni-1)) (5 (3-1)) 3
There is one complication that we have to correct
for however
32
The Fermi-Dirac Probability Function
Here are all the ways to put 2 electrons in 3
states (6 3!/1!) But 1 3, 4 6, and 2 5
are essentially the same because the order of A
and B doesnt really matter.
33
The Fermi-Dirac Probability Function
So there are Ni! permutations that the particles
have between themselves that really dont count
as separate arrangements. Thus divide by Ni! to
go from permutations to combinations.
This is just for the ith energy level, so extend
over all i levels
34
The Fermi-Dirac Probability Function
W represents the total number of ways in which N
electrons can be arranged. The maximum value of
W (most probable distribution among the levels)
is found by varying Ni among the i energy levels
which still keeps the total number of particles
and total energy constant. The most probable
distribution turns out to be
fF(E) ? Fermi-Dirac distribution (Fermi
function)EF ? Fermi Energyk ? Boltzmanns
Constant 8.6210-5 eV/KkT 0.0259 eV _at_
300KfF(E) ? Probability of a quantum state at
energy E being filled
35
The Fermi-Dirac Probability Function
So what is the probability that a state
(apartment) is occupied by an electron?The
Fermi-Dirac distribution answers this for ½ spin
particles (electrons).
Lets analyze this function. Start with T 0
K.Then for E lt EF we have
For E gt EF we have
All states with energy E lt EF will be
occupied/filled (probability 1).All states
with energy E gt EF will be empty (probability
0).
36
The Fermi-Dirac Probability Function
What happens as T gt 0 K? Consider the point where
E EF. Here
This value will always be ½ regardless of the
temperature! As temperature increases, the Fermi
function spreads out.
So some states with energy E gt EF will be filled
when T gt 0 K.That is, thermal energy has popped
them out of there lowest energy state.
37
The Fermi-Dirac Probability Function
The Fermi function, fF(E) is the probability a
state is occupied by an electron. Thus, it should
be obvious that 1- fF(E) is the probability that
a state is not occupied by an electron (the state
is empty). Of course, saying that a state is
empty is the same thing as saying that a state is
occupied by a hole. So 1-fF(E) is the
probability a state is occupied by a hole!
38
The Boltzmann Approximation
Consider what happens to the Fermi-Dirac function
when (E EF) gtgt kT.
This is called the Maxwell-Boltzmann
Approximation and is considered valid for E EF
gt 3kT.