EE 362 Electric and Magnetic Properties of Materials

1
EE 362 Electric and Magnetic Properties of
Materials

• Dr. Brian T. Hemmelman
• Chapter 8 Slides

2
Assumptions (2)

• Abrupt Junction There is a clear dividing line
  between the p-type material and the n-type
  material.
• Abrupt Space Charge Region The edge of the
  space charge region are well defined and abrupt.
• Boltzmann Approximation Applies The doping
  concentrations used in both materials are low
  enough that EF gt 3kT away from the conduction and
  valence band edges.
• Low-Level Injection The quantity of minority
  holes and electrons injected across the pn
  junction are small enough to be low-level
• Total current through the pn structure is
  constant We do not violate Kirchoffs Current
  Law the amount of current that flows into one
  terminal is that same amount of current that
  flows out the other terminal and all parts in
  between.
• Each electron and hole current component is
  continuous No black holes or antimatter
  allowed!
• The electron and hole currents are constant in
  the depletion region No recombination in the
  depletion region

3
Boundary Conditions (3)
The built-in potential for the junction was found
to be
Lets rearrange this a bit.
4
Boundary Conditions (4)
Under complete ionization though we have
where nn0 is the electron majority carrier
concentration on the n-side and np0 is the
electron minority carrier concentration on the
p-side. Thus, for thermal equilibrium we have
5
Boundary Conditions in Forward Bias (5)
When we apply a positive voltage (from p to n) we
generate an electric field that opposes the
internal electric field. Therefore the energy
bands are less curved.
6
Boundary Conditions in Forward Bias (6)
Intuitively we know that should lead to more
charge and current due to spillover (think of
our stack of cannonballs now able to get over the
band-bending). Quantitatively, now becomes
np is the forward biased electron concentration
on the p-side.
So
Similarly,
These are the concentrations of minority carriers
at the edge of the space charge region.
7
Boundary Conditions in Forward Bias (7)
In forward bias we have increased the quantity of
minority charge carriers on both sides of the
depletion region because we are injecting
electrons from the n-side to the p-side (where
they are now the minority carrier) and we are
injecting holes from the p-side to the n-side
(where they are now the minority carrier).
8
Minority Carrier Distribution (8)
We have found what the concentrations were at the
edge of the space charge region. What are they
elsewhere? Note, we have created additional
minority charges above the thermal equilibrium
value. This is just like the generation of
minority charge carriers we studied in Chapter 6.
The difference is that instead of
photogeneration, they are being generated by
injection across the space charge region. Thus,
we can use everyones favorite Ambipolar
Transport Equation!
More specifically, we have in the n-region
where
(excess minority holes on n-side)
9
Boundary Conditions for Excess Minority Carrier
Distribution (9)
Now, the applied electric field exists primarily
across the space charge region only (and not very
much outside of it). Thus E ? 0, and we also
have g 0. If we are looking at steady-state
(after the voltage was applied) then
So
10
Boundary Conditions for Excess Minority Carrier
Distribution (10)
In a 1-dimensional case this is just an ordinary
derivative, and so we can rewrite this as
The general solutions to these equations are
We can conclude that both A and D must be zero in
order to keep the excess minority carrier
concentration from going to infinity.
11
Boundary Conditions for Excess Minority Carrier
Distribution (11)
We can also determine additional boundary
conditions for the problem.
12
Minority Carrier Distribution (12)
Applying these additional boundary conditions
lets us completely solve the Ambipolar Transport
Equations to find the minority carrier
distributions on the n-side and the p-side.
Note ?pn(x) ? 0 as x ? ? and ?np(x) ? 0 as x ? -?
13
p-n Junction Current (13)

• Now, go back to our assumptions
• Total current through the junction is constant.
• Electron and hole current components are constant
  through the depletion region.
• Thus,

14
p-n Junction Current (14)
What are these two values, Jp(xn) and
Jn(-xp)? They are the value of the diffusion
currents at the edge of the depletion
region! Recall that the electric field outside of
the depletion region is 0 (or almost 0).
Therefore we will not have any significant drift
of excess minority electrons on the p-side or of
excess minority holes on the n-side. The values
of these two diffusion currents at the edge of
the depletion region are found as
15
p-n Junction Current (15)
However, for uniformly doped regions both pn0 and
np0 are constants so
Using the formulae for ?pn(x) and ?np(x) derived
on Slide 12 we find the above equations will
evaluate to
16
p-n Junction Current (16)
Since the current components through the
depletion region are constant (no recombination)
we can extend these two values across the
depletion region and add the two together to get
the total current density.
17
p-n Junction Current (17)
All the terms out front that are not dependent on
the applied voltage we can rename as the
parameter called the ideal reverse saturation
current density
Thus, we can rewrite the pn junction diode
equation into the familiar form
18
p-n Junction Current (18)
Vt is the thermal voltage which is 0.0259 Volts
at T 300K. For Va Vt the exponential term
will dominate and we can rewrite the current
density as
If we multiply by the cross-sectional area of the
diode we then get the I-V relationship for a
diode.
19
Diode Current Components Outside the Depletion
Region (19)
We computed the diffusion currents at the edge of
the depletion region to get the diode equation.
We can compute the minority carrier diffusion
current components throughout the rest of the
diode as well. They are
20
Diode Current Components Outside the Depletion
Region (20)
21
Another Real-World Example (21)

• When you break a bone, there are two parts that
  can begin to regrow or heal.
• The first is the periosteum cells (the bones
  fibrous covering). The innermost periosteum
  cells have the power of osteogenesis (bone
  formation).
• After a fracture, these cells get turned on and
  begin to divide. The daughter cells turn into
  osteoblasts.
• Osteoblasts are the cells that make the collagen
  fibers of bone (collagen fibers are based on
  proteins).
• Apatite crystals then condense out of the blood
  serum onto the fibers.

22
Another Real-World Example (22)
23
Another Real-World Example (23)

• The other tissue that forms new bone is marrow.
• Its cells dedifferentiate and form a blastema,
  filling the central part of the fracture.
• The blastema cells then turn to cartilage cells
  and then into more osteoblasts (same sequence as
  regeneration of salamander limbs).

24
Another Real-World Example (24)

• There is a third growth process unique to bone,
  Wolffs Law Bone responds to stress by growing
  into whatever shape meets the demands of its
  environment.
• When bone is bent, one side is compressed and the
  other is stretched. Extra bone grows to shore up
  the compressed side and some is absorbed from the
  stretched side.

25
Another Real-World Example (25)

• This occurs because something stimulates the
  periosteum to grow new bone at a surface where
  there is compressional stress, and dissolves bone
  where there is tensional stress.
• What is something that reacts to stress/strain?
• Consider piezoelectricity (like the piezoelectric
  buzzer elements you can buy).

26
Another Real-World Example (26)

• In a piezoelectric material, a small amount of
  electrons are freed up from bonds under stress
  and migrate toward the side of compression (the
  charge on the inside of the curve is negative).
• Once a steady-state stress is achived the charge
  buildup disappears.
• When you reduce the stress, an equal an opposite
  pulse occurs.

27
Another Real-World Example (27)

• This is almost what happens in bone, but not
  quite. There isnt as much of a rebound signal
  when stress is released.
• Regeneration studies show the build-up of
  negative charge is what is the growth stimulation
  signal. The removal of the rebound charge is
  what keeps bone growth from happening on the
  tensile side.

28
A Biological pn Junction (28)

• Collagen and apatite are the keys!
• Collagen exhibits n-type behavior.
• Apatite is p-type.
• Collagen is the piezoelectric material
• Mechanical stress produces a piezoelectric signal
  from the collagen.
• That signal has two phases for stress and release.

29
A Biological pn Junction (29)

• The signal is rectified by the pn junction
  between apatite and collagen.
• Its strength indicates how much stress is there.
• Its polarity tells the cells which direction the
  stress is.
• Osteogenic cells in the negative potential area
  are stimulated to grow more bone.

30
A Biological pn Junction (30)
31
Diffusion Resistance (31)
Our ideal diode I-V relationship is
ID ? Diode Current IS ? Reverse Saturation Current
The small-signal incremental conductance is just
the slope of the DC I-V curve.
32
Diffusion Resistance (32)
The small-signal incremental resistance then is
As long as Va is somewhat large then the -1
term is negligible and thus
33
Diffusion Capacitance (33)
As you increase or decrease the applied voltage,
the excess carriers will increase or decrease
slightly. This is like the buildup of charge on
two capacitor plates.
34
Small Signal Models (34)
Small-signal equivalent circuit
Complete small-signal equivalent circuit