EE 362 Electric and Magnetic Properties of Materials

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EE 362 Electric and Magnetic Properties of
Materials

• Dr. Brian T. Hemmelman
• Chapter 7 Slides

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P-N Junctions (2)
A p-n junction is nothing more than a section of
p-type semiconductor next to a section of n-type
semiconductor in the same crystal substrate. In
Chapter 5 we saw how we could make resistors out
of either p-type or n-type semiconductor. It
perhaps seems odd then that if we put p and n in
series that they do not act like two resistors in
series. To understand why this is, we need to go
back down to the atomic level and look at what is
happening. First we will look at things
qualitatively to get the big picture and then we
can go back for the quantitative view with
equations and numbers. To begin with, let us
imagine we were able to pick up and place a piece
of p-type material next to the n-type material.
The interface between the two sections of doped
material is called the metallurgical junction.
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In The Beginning (3)
The p-type material has a high concentration of
holes while the n-type material has a low
concentration of holes, and the n-type material
has a high concentration of electrons while the
p-type material has a low concentration of
electrons. Therefore there is a concentration
gradient for both electrons and holes and they
will begin to diffuse across the junction in
opposite directions.
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The Space-Charge Region (4)
As this diffusion occurs we can expect that some
of the free electrons diffusing from the n-type
material will encounter some of the free holes
diffusing from the p-type material, and they will
therefore recombine
These free electrons and holes came from the
dopants. For each hole there is a corresponding,
negatively-charged acceptor ion left fixed in the
crystal. For each electron there is a
corresponding, positively-charge acceptor ion
left fixed in the crystal.
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The Space-Charge Region (5)
As more free carriers diffuse across and
recombine there are more and more fixed dopant
ions left behind. These dopant ions begin to
form a wall of charge all with the same sign on
each side of the metallurgical junction. The
organized charge at the junction gives rise to
the formation of an electric field between the
two walls. This electric field exerts a force on
the other free holes and electrons on the p-side
and n-side respectively that starts to oppose
further diffusion of charge carriers.
Eventually after enough free charges have
diffused across the junction, the electric field
becomes strong enough to completely oppose and
balance the diffusion processes. Around the
metallurgical junction then we do not see any
significant free electrons and holes (because
they have recombined)
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The Space-Charge Region (6)
This area of no free charge is called the
space-charge region or depletion region. As
there is no free charge in the depletion region
we should note that this area is now very much
like an insulator.
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The Space-Charge Region (7)
The electric field in the space charge region
then will act as a finite potential barrier to
the free charge carriers. Even though the
diffusion and drift mechanisms are balancing each
other out, that doesnt mean that there arent
any electrons and holes moving across the
depletion zone. Rather, there is an equilibrium
so that the net flow is zero. A simple viewpoint
would be to picture that for each hole that
diffuses across from the p-side to the n-side
(high concentration to low concentration) there
is also a hole from the n-side that will drift
across to the p-side (repelled away from the wall
of positive ions on the right side of the
depletion zone and attracted to the wall of
negative ions on the left side).
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Reverse Bias (8)
Now consider what happens when we apply a reverse
bias to the pn junction.
We can view this as negative charges flowing into
the p-region and helping to recombine with more
of the free holes and positive charges flowing
into the n-region to recombine with more of the
free electrons. Of course the more direct
mechanism is that the electric field generated
across the pn junction by the externally applied
voltage is in the same direction as the internal
electric created by the dopant ions.
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Reverse Bias (9)
The end result is that the depletion region gets
wider (and thus the width of the finite potential
barrier has increased).
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Forward Bias (10)
On the other hand, when we apply a forward bias,
our externally applied electric field is in the
opposite direction as the internal field in the
depletion region. Thus the net electric field
decreases as does the width of the depletion
region (and therefore the width of the finite
potential barrier).
With a smaller potential barrier we would expect
it to be easier for particles to get from one
side to the other. And moving charge particles
equals current!
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The pn Junction in Thermal Equilibrium (11)
With no externally applied voltage the Fermi
level in the semiconductor must be constant.
Because an electric field exists in the
space-charge region there is a corresponding
built-in voltage, Vbi.
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The Built-In Voltage (12)
We know that n0 can be written as (among other
things)
The potential in the n-region is ?Fn
Thus,
Solving for ?Fn we get
Similarly, in the p-region we get
Thus the built-in voltage is
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Internal Electric Field (13)
Using Poissons Equation in 1-D we can write
where ?(x) ? electric potential and ? ? volume
charge density.
Our volume charge densities in the depletion
region are
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Internal Electric Field (14)
We can find the internal electric field by
integrating Poissons Equation then.
Determining the electric field as a function of
position on just the p-side we have
But at x -xp the electric field is zero, E 0,
so we can find the constant c1 to be
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Internal Electric Field (15)
Our final equations then for the electric field
on the p-side (and n-side) are then
As there are no discontinuities in the electric
field we can set these two equations equal to
each other x 0.
This simply says that the area of the - charge
box and the charge box on Slide 13 are
equal. In other words, the total negative charge
equals the total positive charge.
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Internal Potential (16)
The internal potential can now be found by
integrating the electric field.
Potential is defined with respect to reference
point. That is, it is the potential difference
that matters so we can pick our reference
point. Let ?(x) 0 at x -xp. Then our
constant is found to be
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Internal Potential (17)
The built-in voltage then can be determined by
evaluating the potential function at x xn where
we reach the edge of the charge distribution.
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Putting It All Together (so far) (18)
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Space Charge Width (19)
We can now look at the physical dimensions of the
space charge region and how much it extends into
the p and n material. We know
Substituting back into the equation for Vbi on
Slide 17 we have
If we had instead solved for xn in terms of xp
above we can find
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Space Charge Width (20)
Adding xp and xn together we find the total space
charge (or depletion) width.
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Reverse Applied Bias (21)
In reverse bias the potential we apply increases
the total electric field in the depletion region
(there is the internal electric field between the
dopant ions plus our applied field). Thus we
should expect even more band bending on the
energy band diagram.
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Reverse Applied Bias (22)
The electrons occupying the conduction band
states in the n-type material on the right see a
step potential to their left. We know that
particles with energy less than the height of the
barrier will be reflected by the barrier. Very
few electrons occupy states with energy above the
height of the barrier and have a chance to make
it across.
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Reverse Applied Bias (23)
In reverse bias there is an even bigger step
potential and thus we can understand why very few
electrons can move from the n-side to the p-side.
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Reverse Applied Bias (24)
Quantitatively we can recalculate the new space
charge width, which will increase to
We can also recalculate the electric field
everywhere. However, we know it will be
increased in magnitude everywhere and the most
important value is simply the maximum value of
the electric field, which will occur at x 0.
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Junction Capacitance (25)
As we change the applied voltage, we change the
width of the depletion region. In reverse bias,
for instance, we increase the depletion width
which means we have uncovered even more dopant
ions. Note the dopant ions are fixed and cannot
move. These dopant ions then act just like
charge built-up on two capacitor plates! As we
change the applied voltage, we change the number
of uncovered dopant ions or essentially the
charge on our capacitor plates.
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Junction Capacitance (26)
In this case we define a junction capacitance
in terms of not the absolute charge that builds
up on the plates because of an absolute voltage
(because we have an initial amount of depletion
region charge even at zero applied voltage), but
rather in terms of the change in charge per
change in voltage
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Junction Capacitance (27)
Since (looking at the n-side)
then,
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Example 7.5 (28)
We have a silicon p-n junction at 300 K that has
been doped with Na 1016cm-3 on the p-side and
Nd 1015cm-3 on the n-side. It has a reverse
bias voltage of VR 5 Volts. Compute the
junction capacitance.
If we assume a cross-sectional area of (0.1mm)2
10-4 cm2 then we get an overall capacitance of
0.366 pF.
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One-Sided Junction (29)
Let one dopant concentration be significantly
larger than the other, e.g. Na gtgt Nd. This is
called a pn junction.
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One-Sided Junction (30)
The expression for the junction capacitance of a
one-sided junction reduces to