Download Physics 361 Principles of Modern Physics

Physics 361
Principles of Modern Physics
Lecture 25
Bonds, Bands, and Semiconductors
This Lecture
• Hybridization
• Solid State Physics -- Lattices and Bands
• Fermi-Dirac Statistics
• Semiconductors, Impurities, and Doping
• Law of Mass Action and Doping
Next Lecture
• Semiconductor Devices
• Bose-Einstein Statistics
• Lasers
Hybridization – hybrid sp orbitals
For the atoms in the third and fourth group – they contain at least one empty p
orbital. For such atoms, it does not cost much energy to take an electron in an
s orbital and move it to the vacant p
orbital.
The physical reason for this is that the
two s electrons will be feeling a relatively
strong electrostatic repulsion because
they occupy the same orbital. By moving one
of these electrons to an empty p orbital relieves this electrostatic repulsion, so
costs very little energy.
After this promotion, we now have two additional unpaired orbitals for making
molecular bonds.
When both s and p orbitals form bonds together, a bit of the s and p orbitals are
present in each atomic orbital that creates the molecular orbitals.
Hybrid sp orbitals
Each hybrid orbital contains a linear combination of the s and p orbitals.
For example, for the sp2 case we have three orbitals:
Hybrid sp orbitals and bonding with carbon
Ethene is H2C=CH2
It has three sigma
bonds in order to
connect all the atoms.
The remaining pz
orbitals form pi bonds, which makes the carbon atoms double-bonded. In this way,
all four electrons in each carbon is used in bonding.
Other examples are methane (CH4) which has four single sp3 bonds to hydrogen
atoms.
Also, Ethyne (HC≡CH):
Combining large numbers of atoms together
As we combine atoms together
in large groups, their coupling forces
new orbitals to form, which break
degeneracies.
As we form new orbitals, we always
conserve the number of orbitals.
As the number of orbitals approaches
a very large number for a macroscopic
piece of matter, the new mixed orbitals
form a nearly continuous band of states.
Band Theory
For example, consider atoms
containing s and p orbitals.
As the atoms are brought together,
they form bands.
At the top of the bands are the
highest energy antibonding states.
At the bottom, are the lowest
energy bonding states.
There can be energy gaps where
there are no states.
Filling of Bands
To have electrical conduction,
we require electrons to be in states
near to ones which are empty.
Thus, we only have conduction
when bands are only partially filled.
To determine conducting
properties, we just need to count
the electrons in the material!
As electrons are added to the
material, the fall to the lowest
energy level. (Just like a ball rolling
down a hill.)
Conductors, semiconductors, and insulators
For a material to be a good
conductor, we need a partially filled
band.
First, there need to be electrons in a
band for them to conduct a current.
Second we need adjacent empty
states for the energized electrons to
inhabit. This gives the electrons
some wiggle room in order to
accelerate.
Compare to a classical charged
particle in an electric field. As the
particle experiences a force, the
energy state of the particle changes.
If there are no energy states available
for the particle to inhabit – it can’t
accelerate!!
Force
Insulators and semiconductors only have filled
bands and empty bands – neither of which
contribute conductivity at low fields or
temperature.
Conductors
For the material at right, this could
represent the bands for a group 1
material like Na.
In that case, we have one valence
electron for each 3s orbital.
If we have N atoms in a chunk of material,
there will be N 3s electrons that can
contribute to the s-band. The s-band
must have N orbitals in the band, since
orbital number is conserved. So we
conclude that the s-band can hold 2N
electrons (because each of its orbitals can
hold a spin up and a spin down electron).
We conclude that the top s-band for a
typical group-1 metal must be only half
filled.
This makes such a material a conductor.
Conductors – and complicated bands
We might think that a typical group
2 material like Be or Mg would be
an insulator.
However, bands can mix in
complicated ways as the atoms
approach each other.
When bands overlap with each
other in energy, we can obtain a
partially filled band.
For Be and Mg, we have p and s
bands which overlap in energy, so
neither band is completely filled.
Therefore, they are both metals.
Insulators and Semiconductors
These are materials have completely filled
bands.
Good examples of these are the group 4
materials. Carbon, Silicon, and
Germanium. They all form the “diamond”
lattice structure and have the same
characteristic bands made from sp3
orbitals – shown at right.
As the atoms are brought close together,
the bonding and antibonding bands split
and a gap opens.
For Si and Ge, the band gap is a little
larger than 1 eV, but for C (diamond) with
a closer lattice spacing, the bandgap is
larger (~7 eV).
All these materials are what are called
band insulators – having filled bands.
However, as an insulators band is reduced
to close to 1 eV, one typically calls such
insulators semiconductors.
Semiconductors
To understand why a small bandgap
materials is a semiconductor, we
need to understand how to fill
states occupied by fermions (like
electrons) at a finite temperature.
Fermions fill states according to the
Fermi-Dirac distribution function
fFD (E) =
1
(E-EF )/kBT
e
What this function tells +1
us is the
probability to find an energy state
at that energy filled.
This function is plotted at right for
two different temperatures.
The Fermi-Dirac Function & Orbital States
Energy
Let’s understand what the FD
function tells us.
EF
Let’s first consider very low
temperature near absolute zero.
In this case the FD function is a step
function.
If we have a bunch of discrete
orbitals, the FD function tells us the
probability they will be filled.
At low temperature, the electrons
will fill the lowest energy states
possible, without the multiplefilling of states (except for an
electron spin degeneracy) –
because they are fermions.
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
The Fermi-Dirac Function & Orbital States for
T>0
Energy
At larger temperatures there are a
larger number of statistical
fluctuations which can cause
electrons to be thermally excited to
higher-energy states.
EF
This only happens close to the
Fermi energy because low lying
states require too much energy
compared to the thermal
fluctuations (kBT) to reach an
unoccupied level.
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Density of states
larger chunk
band
The widths of the bands are
determined by the proximity of the
atomic orbitals.
small chunk
The number of actual orbitals in a
band is proportional to the total
number of the orbitals – that is, the
amount of material in the chunk of
matter.
Over a small relevant range of energy,
we will be interested in knowing the
number of orbitals per unit volume of
material. This is the density of states
.
g(E)
The density of electrons per unit
volume per unit energy is given by
the product g(E) fFD (E).
fFD (E) =
1
e(E-EF )/kBT +1
The Fermi-Dirac Function & Insulators
For insulators with large band gaps,
the FD function is almost
completely 1 in the highest
occupied band (“valence” band)
and zero in the lowest unoccupied
(“conduction” band).
So the number of electrons per unit
volume in the conduction band is
approximately zero, and the
number of empty states in the
valence band is approximately zero
as well.
Thus for a large band gap material,
there is no wiggle room for
electrons to accelerate within any
of the bands.
fFD (E) =
1
e(E-EF )/kBT +1
The Fermi-Dirac Function & Semiconductors
For semiconductors, as the band
narrows we get some excited
electrons from the valence band to
the conduction band.
Thus for a small band-gap material,
there is some wiggle room for
electrons to accelerate within the
bands.
The excited electron in the
conduction band can now be
accelerated and contribute to the
conductivity.
The excited electron also leaves
behind an empty state in the valence
band – this is called a hole – and it
also contributes to the conductivity.
e-
fFD (E) =
1
e(E-EF )/kBT +1
The Fermi-Dirac Function & Semiconductors
As the temperature increases (to
lowest order), we expect more
excited charge carriers and thus
more electrical conductivity.
e-
fFD (E) =
1
e(E-EF )/kBT +1
Holes
Conduction
The empty states in the valence
band act like positively charged
particles with an effective mass.
Energy
They will contribute to the
conduction very much like the
excited electron and we must
consider both their contributions.
EF
h+
Valence
Let’s now look at conductivity as a
function of carrier density.
e-
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Charge Carrier Motion in a
Conductor
The zig-zag black line
represents the motion
of a charge carrier in a
conductor
The net drift speed is small
The sharp changes in
direction are due to
collisions
The net motion of
electrons is opposite
the direction of the
electric field
Charge Carrier Motion in a
Conductor
The drift speed Vd is small
(~1meter/hour)
The fermi speed Vf is very
fast! Due to quantum
mechanics of electrons.
(~106 meters/second)
Charge rearrangements
and signals travel near
speed of light
(~108 meters/second)

vf
23
Written as a vector and a current density we have
j = nqvd
24
Ohms law
Ohm’s law just says that there is a
proportional relation between the applied
electric field in a conductor and the
current that flows.
j =sE
Using Newton’s second law, we have
F = -eE = me a
Assume electrons accelerate for an
average time t . So,
at = vd = -eEt / me
Inserting this into the current density,
ne 2t
je = nqvd =
E
me
which is Ohm’s law.
For holes we get a similar expression.
pe 2t
jh =
E
mh
E
-e
F
Relaxation time
The relaxation time t is present in the
Ohm’s law relations,
ne 2t
je = nqvd =
E
me
pe 2t
jh =
E
mh
This time is the average time between
collisions and is given by the Fermi
velocity (v f) and the mean free path ( l )
between collisions as,
t=
l
vf
(Note: The relaxation time can be different
for electrons and holes.)
E
-e
F
Conduction in Semiconductors
Energy
The carriers near the band edges are the
most important for understanding
conduction within a semiconductor.
Thus, we can estimate the density of DE
carriers (per unit volume) in the bands
within a small energy range given by
DE = kBT is
e-
EF
n » g(EC ) fFD (EC )DE » N(me ) fFD (EC )
DE
EC
h+
EV
p » g(EV ) fFD (EV )DE » N(mh ) fFD (EV )
where EC and EV are the band edges and
N(mh ) and N(me ) are the orbitals/volume.
Materials with roughly the same effective
mass have roughly the same density of
states near the band edges. So the number
of states near band edges are roughly the
same for most materials, and the FD
function dominates the determination of
the number of carriers.
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Ohms law for semiconductors
For semiconductors, the total current is
the sum due to the electrons and the
holes
jTotal = je + jh
The two forms we derived earlier for
these are:
2
ne t
je =
E
me
pe 2t
jh =
E
mh
The carrier densities are typically the most
sensitive to temperature and band gap
sizes. The carrier densities can vary by
many orders of magnitude. Thus, a great
deal of focus in semiconductors is on the
carrier densities.
E
-e
F
Intrinsic Semiconductors
Generally, EF lies close to the
middle of a pure semiconductor.
The reason for this is due to the fact
that for every excited electron we DE
need to have a left-over hole –
so ni = pi (where the subscript
denotes intrinsic).
This is considered the intrinsic
behavior of semiconductors.
Before moving on to extrinsic
behavior, we will look at another
way to excite electron-hole pairs
with lower energy – these pairs are
called “excitons”.
DE
Energy
e-
EC
EF
h+
EV
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Excitons
Since electrons and holes have
opposite charge, they can be bound
into a hydrogen-like state. As with
hydrogen, this energy is negative,
so it reduces the energy barrier for
forming a hole-electron pair – and
thus increases the probability for
their formation.
Energy
ee-
bound
h+
The binding energy is much
reduced (to less than 1 eV)
compared to the 13.6 eV ground
state of hydrogen due to dielectric
screening and the effective reduced
mass of the system.
Since the hole-electron pair is
bound (and must move together), it
does not contribute to the net
electrical current since it has a total
charge of zero. Thus, we can ignore
excitons when considering
electrical conduction.
EC
h+
EF
EV
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Impurities in Semiconductors
Intrinsic
An intrinsic semiconductor will have
zero net charge. This is even the
case when we have thermally
excited electron and hole carriers,
because n = p .
If we insert a neutral impurity to a
semiconducting material, the overall
charge of the material will be
constant, but the material can
disrupt the carrier charge balance.
Well engineered impurities for
semiconductors in technological
applications are designed to easily
ionize when they are imbedded
within the semiconductor.
Impurity atom with extra electron
Impurities in Semiconductors -2
Since the impurity can ionize easily
in the semiconductor, the impurity
will either give up an electron to the
rest of the semiconductor (called “ndoping” or “electron doping”) or
take in an electron from the rest of
the semiconductor (called “pdoping” or “hole doping”).
Impurity atom with extra electron
Impurities which give up electrons Extra electron excited to conduction band (n-doping)
(like the one at right) are called
“donors”. They give the electron to
the conduction band.
Impurities which take in an electron
are “acceptors”. They take electrons
from the valence band and thus
create a hole.
+
Impurities in Semiconductors -3
An example of an acceptor is shown at
right.
For silicon, the most common acceptor
impurity used is boron. This is a group
3 element that can also form sp3
hybrid orbitals and bond in the
diamond structure with silicon – so it
can easily replace a silicon atom in the
lattice. However boron has one fewer
electron than Si, so it cannot fill all of
its sigma bonding orbitals. Thus it likes
to take one from the valence band of
the Si lattice to form a hole.
In a similar way, donor impurities (ndopants) are taken from the group 5
elements (such as arsenic or
phosphorus).
Impurity acceptor atom
Electron taken from valence band (p-doping)
-
Doped Semiconductors
Impurity states can be represented
on the band diagram as located
within the band. If an impurity state
is close to the conduction band, it
can be easily ionized to form an
electron in the conduction band
This will increase the conductivity.
It will also increase the electron
density such that
n > p.
This will also increase the Fermi
level, EF .
All of this holds for hole doping if the
impurities have levels located here.
In that case EF is driven downwards,
Energy
e-
e-
e-
EC
EF
EV
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Doped Semiconductors
Impurity states can be represented
on the band diagram as located
within the band. If an impurity state
is close to the conduction band, it
can be easily ionized to form an
electron in the conduction band
This will increase the conductivity.
It will also increase the electron
density such that
n > p.
This will also increase the Fermi
level, EF .
All of this holds for hole doping if the
impurities have levels located here.
In that case EF is driven downwards.
Energy
e-
e-
e-
EC
EF
EV
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Physical argument for
decrease in hole density
As we dope with donors, the
electron density increases while the
hole density must decrease – if
Fermi energy is to increase.
Energy
e-
e-
e-
EC
EF
The decrease in holes is due to
recombination (“hole-electron
annihilation”).
Increased density of electrons means probability for holes to
recombine increases – so their density decreases. This is the “Law of
Mass Action”.
The Law of Mass Action makes doping consistent with a shift up and
down of the Fermi energy.
EV
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1
Law of Mass Action
When doping, the product of the
densities of electrons and holes is a
constant in equilibrium.
np = n
2
i
So in equilibrium, we can change the
balance between carriers, but not
their product.
For a material, the product is a
function of only temperature.
Energy
e-
e-
e-
EC
EF
EV
Probability 1
fFD (E) =
1
e(E-EF )/kBT +1