EE 362 Electric and Magnetic Properties of Materials

1
EE 362 Electric and Magnetic Properties of
Materials

• Dr. Brian T. Hemmelman
• Chapter 5 Slides

2
Carrier Transport (2)

• The two primary mechanisms of carrier transport
  in semiconductors
• Carrier Drift ? Due to an electric field
• Carrier Diffusion ? Due to a concentration
  gradient
• Let us begin by considering carrier drift which
  can be written as

Where ? is the volume charge density and vd is
the average drift velocity of the charge
carriers. If the volume charge density is due to
holes then we would have
Each hole moving due to the electric field would
be getting accelerated until it has a collision
with or is scattered by other particles in the
crystal.
3
Carrier Drift (3)
The average velocity of these positively charged
holes then can be written as
where ?p is the hole mobility and E is the
electric field. The hole mobility is essentially
an averaging effect of all the collisions and
accelerations and is a measure of how well (or
easily) the holes can move through the
crystal. Putting these two equations together we
have
Similarly, for electrons we find
The total drift current density then is the sum
of the electron and hole terms
4
Mobility (4)
The mobility of a charge carrier can also be
expressed as
where ?cx is the mean time between collisions for
carrier type x.
5
Mobility (5)

• The two primary mechanisms of scattering (or
  collisions) that occur are
• Phonon scattering, ?L phonons are quantized
  lattice vibrations (displacements) of the crystal
  atoms due to thermal energy. They disrupt the
  periodicity of the lattice.
• Ionized impurity scattering, ?I Intentional or
  unintentional impurities also disrupt the
  periodic potential seen by carriers and change
  the nature of electron/hole interactions.

The net mobility is then given as
6
Mobility in Silicon vs. Temperature and Doping
Concentration (6)
Electrons
Holes
7
Mobility vs. Doping Concentration _at_ T 300K for
Ge, Si, and GaAs (7)
8
Conductivity (8)
We have already derived an equation describing
the drift current density due to an applied
electric field. We can rewrite this to now
define conductivity.
The reciprocal of conductivity is defined as
resistivity.
9
Resistance (9)
For a semiconductor bar of length carrying
current I due to an applied potential V we can
write the drift current density and applied
electric field as
10
Resistance (10)
Substituting these expressions into our
conductivity equation yields
Solving this for V in terms of I, we find
Ohms Law!!
We should note that conductivity/resistivity
are primarily a function of the majority carrier
parameters.
11
Resistivity (11)
12
Carrier Diffusion (12)
Separate from the electrostatic issues related to
drift current, electrons and holes can also
create a net flow of charge in one direction due
strictly to differences in concentration. This
flow of charge due to concentration gradients is
called a diffusion current.
The total current density then can be written as
Or in 3D
13
Current Component Directions (13)
If we have a piece of semiconductor that is
subject to an applied electric field and has
electron and hole concentration gradients as
shown below, we can see the difference between
the particle flux and the corresponding current
component directions. The dashed arrows are
particle directions, and the solid arrows are
current directions.
14
Einstein Relationship (14)
If an electric field is present in a
semiconductor there will be band bending.
Remember, energy is related to potential through
the unit charge q
Perhaps there is non-uniform doping to create
this shift
(We could use any of the bands as a reference.
We are arbitrarily picking EFi.)
15
Einstein Relationship (15)
The electric field is related to the potential,
and thus the energy bands, as
Since no net current flows in equilibrium we can
also say
16
Einstein Relationship (16)
Since
We have then
So thus
But we also have
Setting these equal we find
17
The Hall Effect (17)
Charge moving (current) in a magnetic field feels
a force which acts on those charge carriers
causing them to be deflected in a semiconductor.
Both holes and electrons will be deflected in the
y direction for the setup below. In p-type
material (holes) we would see a buildup of
positive charge on the front face. For n-type we
would see a negative charge buildup. This will
induce an internal electric field along the
y-axis that ultimately balances out the magnetic
force.
18
The Hall Effect (18)
The net force between the magnetic field force
and the induced internal electric field force
must be zero, so we can write
The induced electric field is called the Hall
field, and the corresponding transverse voltage
(perpendicular to the applied voltage in the
x-direction) is called the Hall voltage. The
Hall voltage can be measured to determine if the
material is p-type or n-type and also compute the
carrier concentrations and mobilities. The Hall
voltage is
Since EH Ey we can then write
19
The Hall Effect (19)
For a p-type semiconductor, the drift velocity of
the holes is
Substituting this drift velocity into the Hall
voltage equation we get
Or solving for hole concentration, p, we find
where all the terms in the equation are
physically measurable quantities!
20
The Hall Effect (20)
For p-type semiconductor we could also write the
drift current density as
but,
so we substitute in and get
solving for mobility, ?, we find
The mobility can be found with readily-available,
easily measurable quantities!
21
The Hall Effect (21)
For n-type semiconductor we find analogous
expressions