Module 4A: Diode Basics
Up to now, we have studied a number of circuit components – the resistor, the capacitor, the inductor,
the ideal voltage source, and the ideal current source. We have considered both DC and AC sources,
and used the mainstays of circuit analysis – V, I, P and R – in order to more completely understand what
happens when these devices are connected together.
Along the way we’ve introduced a number of mathematical ideas to help us. These include techniques
for solving the ever-present simultaneous equations that result from Kirchhoff’s Laws, some simple
techniques to solve (or avoid having to solve) differential equations, which are ubiquitous in RC and RL
circuits due to the presence of derivatives in our fundamental equations for capacitors and inductors,
and techniques for dealing with complex numbers which arise (as a “lesser evil” to the differential
equations that would otherwise result) from our study of AC circuits.
In this module we’ll learn about a completely new device, the diode, which is relatively simple in
operation. However, in introducing the diode we will be introduced to some very interesting materials
physics. Finally, in Module 4B we will discover that despite the diode’s relatively simple operation, it is
in fact a highly versatile component.
While resistors, capacitors and inductors represented the state of the art back in the 19th century,
diodes and other semiconductor devices based on the same essential physics we’ll study here have
fueled most of the progress in electronics – radio, television, radar, consumer electronics, computers
and even some laser technology - for the past half century. Resistors, capacitors and inductors made it
possible for circuits to carry energy from one place to another; the humble diode and its more
sophisticated descendants have made it possible for circuits to carry information as well.
I.
What is a Diode?
As we did with other components, I will start by developing a model for an “ideal” diode - whose only
flaw is the fact that it doesn’t actually exist - and gradually introduce the complications that accompany
real diodes. Fortunately, though the ideal diode model is never exactly satisfied by real diodes, in many
cases the ideal diode model is close enough to reality that it is useful for circuit
modeling.
Simply put, a diode is a device which conducts electricity in one direction, but not in
the other direction. Its symbol (right) is a very suggestive arrow. Going by our standard
of “conventional current flow,” from positive to negative, the diode acts like an ideal conductor if
current flows in the direction of the arrow (known as forward bias), and acts like an ideal insulator if
current attempts to flow against the arrow’s pointing direction (known as reverse bias). In other words,
it is somewhat like a “variable resistor” with R = 0 for current flowing in one direction, and R  ∞ for
current flowing the other way. (If we assume electron flow, then the electrons will flow “upstream,”
from the point of the arrow towards its tail. But in this course we have been assuming conventional
flow, and we will continue to do so. Therefore, remember that current, as we’ve been using it, will only
flow in the direction of the arrow.)
Circuit analysis with an ideal diode is relatively simple. If the symmetry of the circuit makes it easy to
determine the direction of current flow through the diode, you replace the diode with its appropriate
limit (short circuit for forward bias, open circuit for reverse bias) and solve the circuit.
This is analogous to the way we dealt with t = 0 and t  ∞ limits for capacitors and inductors, by
removing them altogether and replacing them with shorts or opens as appropriate!
Now, if it is not possible to determine the direction of current flow a priori (as often happens when
multiple sources are present), there is a two-step process to follow:
-Assume that the current flow is in the forward-biased direction, replace the diode with a short
circuit, and solve the circuit accordingly.
-If the current turns out negative (i.e. reverse bias), you must re-do the analysis by replacing the
diode with an open circuit (which sets the current in the diode’s loop to zero).
If more than one diode is present, you will have to take some care to try different combinations of
forward and reverse bias until you find a solution which is self-consistent.
Start by assuming that all diodes are in forward bias (and assign your current directions appropriately).
Then solve for all the currents: if they’re all positive, you’re good to go, but if any are negative it implies
those diodes are in reverse bias. So assume reverse bias for those diodes and repeat your analysis.
But now you need to check for consistency: you’re not done until all your assumed forward-biased
diodes must turn out to actually be in forward bias (i.e. all the currents are in fact in the correct
direction) and all your assumed reverse-bias diodes must turn out to actually be in reverse bias (i.e. the
currents are zero, and the “downstream” side of the diode is at a higher potential than the “upstream”
side).
II.
History
Historically, the first device that exhibited diode properties
was something known as the cat’s whisker. In this device,
a very thin wire was positioned so that it barely made
contact with a crystal, usually of galena (lead sulfide). In
this configuration, the cat’s whisker would conduct
electricity in one direction but not the other.
Cat's Whisker:
http://upload.wikimedia.org/wikipedia/commons
However, the first reliable devices which acted as one-way /8/8c/CatWhisker.jpg
current carriers were vacuum tubes known as “valves,” first
developed by John A. Fleming in 1904.
In a vacuum tube valve, depicted at right, two electrodes were enclosed
in an evacuated tube, with a voltage difference applied between them.
One of the electrodes, the cathode, was heated to high temperatures by
a nearby filament (powered by a separate high-voltage circuit), while the
other, the anode (or plate), remained cool.
If a negative voltage were applied to the cathode, excess electrons would
accumulate there and boil off of the surface due to thermionic emission;
these electrons would be attracted toward the positively-charged
cathode, and current would flow. If a positive voltage were applied to
the cathode, however, there would be no excess electrons to boil off,
and no current would flow.
Vacuum Tube Diode (Valve)
Both the cat’s whisker and the diode valve, despite their long and storied histories in the development
of radio and other electronics developments of the early 20th century, have long since been supplanted
by the semiconductor diode (specifically, a type known as the “junction diode”), about which the rest of
this module is written.
III.
Characterizing Diodes: The I-V Plot
One very simple way to characterize electronic devices is with an “I-V plot,” which depicts an applied
voltage along the x-axis, and the current drawn along the y-axis.
An ideal conductor appears on this plot as a vertical line coincident with the y-axis (red trace in left plot
below): any amount of current can flow through an ideal conductor with zero voltage drop across it.
An ideal insulator, however, appears as a horizontal line overlying the x-axis (blue trace in left plot): no
matter how much or how little voltage is applied across an ideal insulator, zero current flows through it.
An ideal resistor (that is, one which obeys Ohm’s Law, green trace on left plot) is characterized by a line
with a y-intercept of zero (no current flows when zero voltage is applied) and a positive slope of G = 1/R
(since I = VG = V/R). A non-Ohmic device is characterized by an I-V curve departing from this ideal linear
trend (such as the orange trace on the left
plot).
The I-V curve for an ideal diode (purple trace,
right plot) may be deduced from its bipolar
behavior: an ideal conductor for V > 0, and an
ideal insulator for V < 0. Therefore, a diode’s IV curve follows the x-axis for V < 0 (i.e. I = 0),
then sharply heads up the y-axis (i.e. any
I-V characteristics for various devices. red: ideal conductor. blue: ideal
insulator. green: ideal (ohmic) resistor. orange: non-ohmic resistor. purple:
ideal diode.
positive value of I corresponds to V = 0).
Capacitors and inductors (under DC) do not have a representation in the I-V plane, since their
fundamental equations involve derivatives: IC = C dV/dt and VL = L dI/dt. Under AC, however, XC and XL
have Ohm-like I-V curves: a linear relationship between I and V with intercept zero and slope 1/X L,C = BL,C
(susceptance).
Finally, real diodes have I-V curves similar to that of the ideal diode model, but with some added
complications; we will return to “real diodes” shortly.
IV.
Some basic materials physics: what do electrons do in matter?
Note: you don’t need the material in the next three sections, in order to understand the practical aspects
of how diodes work. Therefore, I will not be testing you on this material. However, it’s nice to see how
diodes work “on the inside,” and you will probably have to learn this material eventually anyway (if you
take solid state physics), so I present it here.
In order to understand semiconductor diodes, we will have to learn some materials physics.
We developed an understanding of the ideal conductor by reasoning that good metals have free
(unbound) electrons which are capable of traveling through the material (i.e. current) with negligible
loss of energy (i.e. voltage drop). As long as there is an abundance of mobile electrons which may travel
without loss of energy, a material will behave as an ideal conductor.
Our model for the ideal insulator centered around the absence of free electrons. Since all the electrons
are tightly bound to immobile atomic cores, no current can flow, even if a comparatively high voltage
were applied.
Of course, no material actually follows either extreme (that is, aside from superconductors!), so it makes
sense to consider what happens for real materials which lie somewhere in between.
While Ohm’s Law for resistors was discovered empirically by Georg Ohm in the 1830’s, Paul Drude
showed decades later that a linear relationship between V and I was to be expected for a material with
an abundance of free electrons, which could travel through the material relatively freely, with the
“resistivity” arising from the fact that the electrons suffered frequent collisions with the stationary cores
(nuclei plus tightly bound inner electrons) of atoms which make up a rigid crystalline lattice. The linear
relationship between V and I formed the basis for our “ideal resistor” model.
While the Drude model was one of the crowning jewels of classical materials science, with Ohm’s Law
and several other significant experimental successes to its credit, we have since found out that this
picture, like our pictures for the ideal conductor and ideal insulator, is highly limited, and only applies to
certain materials under certain conditions.
Well, let’s consider the limitations of these models.
Well, in real materials – even those that are very good conductors – some loss of energy associated with
current will exist. As it turns out for most metals, the dominant phenomena leading to resistivity are
impurities (the moving electrons scatter off of impurity atoms or defects) and increased temperature
(as temperature increases the atoms “jiggle” more and the mean time between collisions  decreases,
leading to an increase in ). This led to our model of “metallic” conductivity, in which  increases with
temperature, and is a function of the metal’s purity.
What about our ideal insulator? Well, it always turns out that there are some conduction electrons
available, since it’s at a finite temperature, which gives enough energy to those electrons that a small
number of them can be free at any given time. This corresponds to an astoundingly high resistivity, but
one which results in a nonzero (albeit very small) current flow as a function of applied voltage; as
temperature increases the number of such free electrons increases, so insulators typically have a
resistivity which decreases with increasing temperature, giving rise to our model of “insulator”
conductivity.
However, even the insulator’s bound electrons are able to “wiggle” a little bit; even if they don’t actually
conduct current from one end of the material to the other, by systematically shifting a bit they can
induce an internal electric field. This is the origin of the “dielectric effect” that makes certain insulators
very useful to increase the effective capacitance of a capacitor – the positive and negative charges shift
a small amount, resulting in an induced electric field which partially cancels the applied field. Finally,
even those tightly bound electrons are only so tightly bound. Apply a high enough voltage, and some of
those electrons will be freed from their “prisons,” and the material will “break down” and begin to
conduct wholesale. Typically this results in real, physical damage to the insulator, and when it occurs
there is often a spark or discharge (think: lightning). This is why capacitors and many other devices have
maximum voltage ratings!
The ideal resistor is subject to a combination of the effects that plague our insulators and conductors.
While materials and impurity levels may be carefully selected to yield materials with an appropriate
value of , the value of  is itself a function of temperature. In some materials (“metals”) the limiting
factor on the resistivity is the frequency of collisions; as temperature increases the resistivity increases.
In others (“insulators”) the availability of conduction electrons is the limiting factor; as temperature
increases the number of conduction electrons increases so the resistivity decreases. This “temperature
dependence of resistivity” is to blame for a great chunk of the non-Ohmic behavior out there, although
there are some curious materials which exhibit non-Ohmic behavior for other reasons.
Is there a way to reconcile these two contradictory pictures with each other?
developments in physics have revealed that the answer is an emphatic Yes!
More modern
First, we recognize that electrons have both a particle and a wave nature, with a de Broglie wavelength
 = h/p. These electrons also have a kinetic energy given by p2/2m. Electrons, being fermions, obey the
Pauli exclusion principle, which means that no two electrons can occupy the same quantum state.
Although all of the electrons in a material would “like” to have the lowest energy possible, they can only
occupy the lowest-energy available states. This is much like a parking lot – the first to arrive can take
the “best” spots, while the later arrivals are left to take the “closest available” spots. This continues
until all the cars get in, and the spots are filled roughly in order from the “closest” ones to the “farthest”
ones up to a certain level. In materials this threshold “level” is known as the “Fermi level” – states with
energies below the Fermi level are generally filled, whereas states above the Fermi level are generally
empty.
Since the atomic cores are arranged in a regularly-spaced periodic lattice, it is possible for electrons –
and their concomitant de Broglie wavelength, which is a function of the electrons’ energies – to interact
with this lattice. Felix Bloch showed that for certain wavelengths, electrons interact very strongly, with
the lattice, and this drives their energy upward or downward. The net effect is that certain energies are
“forbidden,” and there are no available states at those energy levels. All of the electronic states,
therefore, are concentrated into certain energy “bands,” each of which are separated from each other
by “gaps.”
V.
Band Theory: Metals, Insulators and Semiconductors (continued, optional)
Here is where it gets interesting.
If the Fermi level falls in the middle of a band, that band is “half filled,” meaning that there are both
occupied and empty states in the band. Therefore electrons can move about freely since they can hop
from one state in the band to another, and this material acts as a metal (a term which has much more
meaning to solid state physicists than it does in everyday life!), and has high conductivity.
If the Fermi level falls in a gap, then the highest band below the gap (the “valence band”) is fully
occupied, and the lowest band above the gap (the “conduction band”) is completely empty. It is very
difficult for an electron to move from one state to another within the valence band, for much the same
reason that it is difficult to move around in a very crowded bar – if you want to move from where you
are, there are no “empty spaces” between people you can move into - and in general very little
conduction occurs within the valence band. And since it takes a fairly large amount of energy (the “band
gap energy,” Eg) for an electron to jump from the valence to the empty conduction band, this material
acts as an insulator, with very high resistivity.
Because in a metal the highest occupied band is half-filled, the resultant picture (as far as we’re
concerned right now) is not all that much different from that of the Drude model, with lots of free
electrons able to roam about. However, our insulator picture is vastly different! But the actual physics
is a bit more subtle than this.
First, it’s worth pointing out one significant assumption we made in developing this model. We stated
that all states below the Fermi energy are filled, whereas all states above it are empty. This is only true
at a temperature of absolute zero. At finite temperatures, thermal fluctuations (which have an energy
of the order of kBT, where kB is Boltzmann’s constant) cause some electrons to populate states above
the Fermi energy, resulting in unoccupied states (known as “holes”) below the Fermi energy. As
temperature increases (going from the blue curves
toward the red in the plot at right), the number of
such electrons, and the range of energy above and
below Ef with “abnormally” filled or unfilled states,
increases. For energies differing from Ef on the
order of the band gap Eg, in the case of an insulator,
increasing temperature results in an increasing
number of “abnormally” occupied states in the
conduction band, and an increasing number of
unoccupied states (holes) in the valence band.
Both these conduction electrons (in an otherwise empty band) and valence holes (in an otherwise full
band) are capable of conducting electricity. (An interesting fact which may perplex you for now, but
hopefully will become clearer when you take quantum mechanics and solid state physics, is that the
effective conductivity for conduction electrons and for valence holes may differ significantly. This only
makes sense in light of band theory, since in our “classical” picture there should be no difference
between an electron moving one way, and the lack of an electron (a hole) moving the other way.)
Next, we need to realize that the gap between the conduction and valence bands is finite in size. In the
limit that this gap size is enormous (i.e. Eg >> kBT), at room temperature (about 300K) the number of
conduction electrons and valence holes is still going to be tiny, and the resistivity is going to be huge.
This is why materials like quartz and Teflon have such enormous resistivities at room temperature –
their band gaps are so large that very, very few electrons (or holes) manage to “jump” across the gap!
But if the gap is smaller (typically Eg is a few eV or less), thermal fluctuations at room temperature are
capable of significantly populating electrons into the conduction band (and thus holes into the valence
band), and in fact resistivities can be relatively small (although typically not as small as highly-conducting
metals), but with the telltale difference that, as in insulators, resistivity decreases with increasing
temperature. This happens in materials known as semiconductors, which are simply insulators with
relatively small gaps.
Metals typically hail from the “left side” of the periodic table, corresponding to elements and alloys with
a relatively small number of loosely-bound electrons, which generally result in half-filled bands at the
Fermi surface. Good insulators typically include elements from the “right side” of the periodic table, and
highly ionic compounds in which the metal’s loosely-bound outer electrons are tightly bound to a highly
electronegative anion (think solid NaCl, oxides of metals), and typically result in the Fermi energy falling
inside a very large gap.
Semiconductors typically hail from somewhere in between, generally from Group IV (or 14) of the
periodic table, although compounds with Group III – V (13-15) elements, and Group II-VI (12-16)
elements (so called III-V and II-VI semiconductors) also have substantial applications. You may also
recall that as you travel downwards in the periodic table within a group, materials gradually become
“more metallic” as the outermost electrons become less tightly bound to the nucleus.
Let’s take a stroll down Group IV (14). The first element in Group IV, carbon, exhibits semiconducting
behavior, albeit with a relatively large gap, so it is typically used in resistors. Silicon and germanium
exhibit very typical semiconducting behavior, with germanium “more metallic” than silicon (meaning
that its conductivity at a given temperature is typically several orders of magnitude higher). Continuing
down Group IV, we get to Sn and Pb. Sn and Pb are actually quite metallic in behavior, although their
resistivities are much higher than good metals like Cu or Al. But they have a relatively low melting point,
which is why Sn and Pb are used in solder.
Among III-V compounds, GaAs is quite widely used, although there are tons of others, including
(Al,Ga)As (in which Al atoms take the place of some fraction of Ga), InAs, GaSb and InSb. II-VI
compounds such as CdTe and CdSe find use as well, often in photocells, since visible-light photons have
about the right energy to cause excitations from the valence band to the conduction band, with a
concomitant reduction in resistivity.
However, Si is used for the bulk of semiconductor devices, and we will focus on Si devices for now.
VI.
Doping and the pn Junction: What Makes It All Possible (optional)
Now that we have a successful model which explains the transport (conductivity/resistivity) properties
of most materials, let’s think about how we actually use them in electronics.
A chunk of silicon or germanium isn’t all that interesting. It acts as a resistor, since it’s a chunk of
material with a resistivity . Since Si has a much lower resistivity than carbon (which is typically used in
resistors) but higher than most metals (which are typically used to make wires), the resistance of a
chunk of silicon is typically on the order of ohms to tens of ohms, somewhere in between that of wires
(tenths of ohms) and of typical resistors (hundreds to thousands of ohms).
Now, what if we take a chunk of silicon (each atom of which has four unpaired outer electrons), and
replace some of those silicon atoms with elements from Group V such as P, As or Sb (each of which has
five unpaired electrons)? The net effect is that there is an “extra” electron added to the Fermi sea for
each Group V atom doped into the compound. This means that there are extra negatively charged
electrons in the material, so this is referred to as n-type doping. If we replace some of the Si atoms with
a Group III atom such as Al, Ga or In (each of which has only three electrons), the material ends up with
a paucity of electrons relative to the undoped material, and is referred to as p-type doping.
What does this do to the band structure? Since there are additional
electrons in the n-type material, its Fermi energy must be higher, relative to
the bands. In the p-type material, there are fewer electrons, pushing the
Fermi energy down with respect to the bands. If we imagine piling
electrons into the semiconductor one by one, they will, like water, occupy
the lowest available energy levels, and progressively occupy higher and
higher levels until they “level off” at the Fermi energy when all electrons
have been added. Because of the foregoing, this means that the Fermi
energy in the n-type material has shifted upwards relative to the bands, and
the Fermi energy in the p-type material has shifted downwards relative to the bands.
An isolated piece of p-doped or n-doped Si isn’t all that interesting, either. The resistivity may be shifted
up or down relative to that of undoped Si, because one of the bands may have been pushed closer to or
further away from the Fermi level. Thus, at a given temperature there will be more electrons in the
conduction band, or more holes in the valence band. Whichever carrier type predominates is known as
the “majority carrier,” and the other is known as the “minority carrier;” the total conductivity is the sum
of the conductivities of the two carrier types. In n-doped materials, conduction electrons will be the
majority carrier, and in p-doped materials, valence holes are the majority carrier.
But what happens if I abut a piece of n-type semiconductor to a piece of p-type semiconductor? This is
where things get interesting. Electrons will fill the material up to the Fermi energy, so in diagrams we
conventionally draw the Fermi energy at one level, and adjust the bands up or down to match the Fermi
energies of both materials.
Since the n-type material has lower bands
(relative to Ef) than the p-type, there is a
discontinuity where the two materials meet (the
“pn junction,” depicted at right). Since there is a
much smaller energy gap between the top of the
p-type valence band, and the bottom of the ntype conduction band, there is a tendency for
some of the electrons in the conduction band of
the n-type material (and there will be more of
them in the n-type material due to the extra electrons from the Group V dopants!) to jump across the
“barrier” to occupy a lower-energy unoccupied state in the valence band on the p-doped side This
results in an accumulation of excess negative charges (electrons) on the p-doped side of the junction,
and an accumulation of excess positive charges (holes) on the n-doped side.
This results (below left) in a region around the junction with nonzero overall electric charge, and a sharp
reduction in the availability of carriers (either conduction band electrons or valence band holes) on
either side, known as the depletion region The energy (and thus
population) difference between the p-type and n-type band
levels, and thus the width of the resultant depletion region, may
be controlled with great precision by the level of doping in the ptype and n-type materials.
As a result of this strong carrier depletion, one may expect that the pn junction has a very high effective
resistance. This is true, although we have to be a little more careful before making any statements.
VII.
Biasing the pn Junction: The Diode in all its Glory (back to required stuff!)
(If you skipped the last three sections, the upshot is that a semiconductor diode is made of a material
such as silicon, where there is a junction between two chunks of silicon containing two different types of
doping – p-type and n-type. It’s this junction that makes “diode action” possible.)
The heart of the semiconductor diode is the pn junction, as a diode is composed of a piece of p-type
semiconductor (usually silicon, but other materials are sometimes used) abutted to a piece of n-type
semiconductor. The semiconductor chunks themselves – intrinsic, p-doped, or n-doped - are rather
boring; they’re just “resistors” with a rather piddly resistance of 10  or so. But the pn junction
between them is where all the action happens.
Just sitting there in a drawer unconnected to anything, the diode’s pn junction accumulates a charge
imbalance due to the difference in Fermi energies on the n-type and the p-type side. Electrons from the
n-type side jump across and “down” to the attractive (and relatively nearby) lower-energy states in the
p-type material, resulting in a net positive charge on the n-doped side, and a net negative charge on the
p-doped side. This happens until there are very few available carriers left over, so that there is a strong
depletion of carriers in the vicinity of the pn junction; this is known as a depletion region.
Ok, that’s what the diode does when it’s sitting in a drawer. Let’s consider what happens when I apply
a voltage difference across this junction.
If I apply a positive voltage to the n-type side, and a negative voltage to the p-type side, electrons will
begin flowing toward the p-type side of the junction (which already has an excess of negative charges),
and electrons will be drawn away from (and, equivalently, holes will be drawn toward) the n-type side of
the junction, which already has an excess of positive charge.
This tends to increase the size of the depletion zone (more
unbalanced charge on either side). Since there are so few
available carriers, very little current can flow.
This
configuration - (+) terminal connected to n-type, (-) terminal
connected to p-type - corresponds to reverse bias in our ideal
diode model.
But if I apply a negative voltage to the n-type material, and a
positive voltage to the p-type material, those excess charges in
the depletion zone begin getting neutralized by charges flowing
from the source. As I increase the applied voltage, the
depletion region decreases in thickness. As the depletion layer
eventually disappears, the available carrier population starts
increasing, and conductivity rapidly increases! This is the basis
of forward bias in our ideal diode model.
Therefore, in analogy with the labeling of the electrodes in our vacuum tube diode (cathode for the
negative terminal, anode for the positive terminal), we call the p-facing side of the diode the anode, and
the n-facing side the cathode.
As long as the cathode (p-side, “downstream” of the arrow) is at a negative voltage relative to the anode
(n-side, “upstream” of the arrow), the diode is in forward bias. If the cathode’s voltage is more positive
than the anode, then the diode is in reverse bias.
VIII.
Beyond the Ideal Diode Model: More precise modeling of diode behavior
There is a phenomenological model (like Ohm’s Law) due to William Shockley, that is a more accurate
model of real diodes than our simple ideal diode model.
Under an applied voltage V (with V>0 corresponding to forward bias and V<0 to reverse bias), the
current drawn I increases exponentially with V, as described by the Shockley equation:
I = IS (exp(V/(Vt)) – 1)
where IS is known as the reverse bias saturation current (which is a very small temperature-dependent
current, typically on the order of A, drawn when the diode is in reverse bias),  is an “ideality factor”
which is typically between 1 and 2 (usually assumed to be 1) and depends on particular details of how
the diode was made, and Vt = kBT/e is the “thermal voltage,” which represents, roughly, the “average
voltage” of electrons due to thermal energy: thermal energy goes as kBT, and the electron’s charge is e,
so the Vt = kBT/e is the thermal energy per unit charge). T in the above must be expressed in absolute
temperature (i.e. Kelvin, where room temperature is approximately 300 K).
Since Vt is usually very small (~ 25-30 mV at room temperature) the term in the exponential is >> 1, if V
>> Vt the Shockley equation reduces to
I = IS exp(V/(Vt))
With this, we’ll develop a more realistic model of the diode we’ll term the “complete diode model,”
which also accounts for some of the limitations of the Shockley model. We’ll also develop a “practical
diode model” which is not quite as accurate, but is easier to deal with.
As the Shockley equation suggests, the current drawn in reverse bias is not exactly zero, but a small
value IS. So the “horizontal” branch of our ideal diode I-V curve is shifted from zero to a slight negative
value –IS. And the “vertical” branch is no longer a vertical line coincident with the y-axis, but an
exponential curve that rapidly swings upward as V increases beyond zero.
But it is important to note that under reverse bias, the depletion region becomes larger and larger,
making the diode a good insulator. But, like any insulator, a sufficiently high voltage will result in a
breakdown once the potential difference across the depletion region is large enough to cause electrons
to break free of their bonds. This is known as breakdown, and occurs when the reverse-bias voltage V >
Vbd. In typical off-the-shelf diodes, Vbd (often referred to as the peak inverse voltage, or PIV, which is
usually given on the package, just as maximum voltage and maximum power ratings for capacitors and
resistors are given) is at least 50 V, though it has some temperature dependence, and diodes can be
obtained with PIV ratings of thousands of volts. This breakdown voltage is not accounted for in the
Shockley model, but we will do so in our complete diode model.
In most diodes, damage will result if the breakdown voltage is exceeded, so the advice is to never allow
diodes to operate in this region! Our ideal diode model assumed Vbd  ∞, so breakdown never occurs.
(However, there is a special type of diode – the Zener diode – which is designed to operate in breakdown
mode. We will talk about the Zener diode later, as it has important applications in voltage regulation.)
Another limitation of the Shockley model is that it does not account for plain old ordinary resistance.
The really interesting “diode” stuff is all happening at the pn junction. But the bulk p-type and n-type
materials have ordinary resistivity as well, so a real diode has an internal resistance, typically on the
order of tens of ohms (although this is often negligibly small, since diode circuits almost always include
resistors on the order of k or larger). The effect of this extra resistance is to depress the I-V curve a
bit, meaning that a higher voltage is required to pull a given amount of current, as part of the voltage is
being dropped by the resistance of the bulk diode. Our complete diode model accounts for this.
Of course, real diodes have finite current- and power-handling capabilities. Diodes are typically rated by
their maximum forward current, usually specified as the “average forward current” I0, and by their
maximum power dissipation, usually specified as the “forward power dissipation” PD. (The power
dissipated by a diode is given by P = VI, which is always zero for an ideal diode, but the actual values of V
and I can be read off of an I-V curve for a real diode, or measured with a multimeter.)
Lastly, as you may have already figured out, if a diode in depletion mode has an accumulation of positive
and negative charges on either side of the junction, it also must have some capacitance. This comes in
two parts, a “junction capacitance” and a “diffusion capacitance. They are typically rather small (~ 100
pF total), although in some cases (such as high-frequency applications) it can become a significant
problem, as reverse-biased current can “leak” through the effective XC resulting from the capacitance!
With all of these points taken into account, we may develop our complete diode model, which takes into
account the reverse leakage current IS, the exponentially increasing current with forward voltage, the
failure of the current to reach the levels predicted by the Shockley equation due to ordinary resistance,
and the dramatic failure of the diode at large reverse voltages. This complete model turns out to be a
very accurate description of the behavior of most diodes, as long as we use parameters that accurately
fit the diode’s true I-V curve (chiefly, IS, n, VT and PIV). However, you can’t straightforwardly deal with
capacitance in I-V curves, so that effect is something that we will have to leave out of our complete
diode model. (But it can be significant!)
IX.
Dealing with Real Diodes in Circuits
Now that we’ve developed our complete diode model and its corresponding I-V curve, let’s consider
how to analyze circuits containing diodes.
The simplest diode circuit (the “RD” circuit) is a battery connected to a diode in
series with a resistor. In this case, we can apply Kirchhoff’s Voltage Law to find:
 - VR – VD = 0
where VR = IR is the familiar resistor voltage, and VD is the diode voltage. Unlike
resistors, which are very simply modeled by Ohm’s law so it’s more or less a
mindless task to determine VR, determining VD is a bit more complicated and depends on which model
we assume (and how closely our particular real diode conforms to the model chosen).
According to our ideal (but somewhat unphysical) diode model, if the diode is in forward bias, the diode
acts as an ideal conductor, so VD = 0, and I = /R. If the ideal diode is in reverse bias, it acts as an ideal
insulator (open circuit), so VD = , and I = 0.
If we wish to apply our complete diode model, the first question to ask is still whether the diode is
operating in forward or reverse bias.
If the diode is in reverse bias and |V| < Vbd, then we know that the current drawn will be extremely
small (on the order of IS), and roughly independent of both the applied voltage  and the resistance R.
We could be a little more careful, but for most intents and purposes we’re done: the diode acts like an
enormous resistance, almost to the point of being an “open circuit.”
If the reverse-bias voltage |V| > Vbd, then we’re in the breakdown region, and if this is an ordinary diode
for which breakdown is a bad thing, we’re in trouble! Something’s going to get cooked…
If the diode is in forward bias, then there is a relation between VD and I given by our I-V characteristic. It
is approximated by the Shockley equation, although this does not account for the finite resistance of the
diode. Unfortunately, these equations are highly nonlinear, so it is not straightforward to determine I.
One can use an analytical approximation for the I-V curve (such as the Shockley equation, perhaps with
an extra term for the internal resistance, or a piecewise linear approximation to the device’s real I-V
curve), and go through the hairy algebra to solve for I in closed form.
We can also use an iterative technique: guess a value of I, compute VR = IR and determine VD for that I by
picking off the value of I from our I-V curve, and plug our VR and VD into our loop equation. If  - IR –
VD(I) = 0, we’re done. If not, guess a new value of I and try again.
This can get very unwieldy, however, if there are multiple diodes, especially because some of them may
be in reverse bias (and this can only be found by determining that no I > 0 solves the loop equations).
We could also solve for I numerically (which often amounts to performing an iterative technique, but
letting the computer do all the dirty work).
Finally, I can attempt to solve for I graphically. The I-V characteristic curve tells me I as a function of the
diode voltage VD. If I can use Kirchhoff’s Laws to determine what VD should be in terms of I, based on
other constraints in the circuit, I can draw a line corresponding to that constraint; this is known as the
DC load line. The intersection of the diode’s I-V curve and the DC load line represent the values of VD
and I that will exist in the circuit.
As an example, consider our “RD” circuit. We know from KVL that  - VD – VR = 0, which implies that
VD =  - VR =  - IR

I = (-VD)/R
On a plot of I versus VD, this appears as a line with slope -1/R, with a y-intercept (that is, where the line
strikes the y-axis, corresponding to VD=0) at /R, and an x-intercept (where the line strikes the x-axis,
corresponding to I=0) at VD = .
This DC load line intersects the diode’s I-V characteristic at a certain point, from which the values of I
and VD can be read off. Then VR =  - VD. On the following page is a sample I-V characteristic (in black)
for a diode. I have plotted five possible DC load lines corresponding to combinations of  and R:
(red):  = 1.0 V, R = 200 

x-intercept =  = 1.0 V, y-intercept = /R = 5.0 mA
 intersection at I = 1.95 mA, VD = 0.605 V. VR = -VD = 0.395 V
(blue):  = 1.25 V, R = 200 

x-intercept =  = 1.25 V, y-intercept = /R = 6.25 mA
 intersection at I = 3.00 mA, VD = 0.655 V. VR = -VD = 0.595 V
(orange):  = 5.0 V, R = 1 k

x-intercept =  = 5.0 V, y-intercept = /R = 5.0 mA
 intersection at I = 4.40 mA, VD = 0.69 V. VR = -VD = 4.31 V
(violet):  = 0.7 V, R = 100 

x-intercept =  = 0.7 V, y-intercept = /R = 7.0 mA
 intersection at I = 1.40 mA, VD = 0.56 V. VR = -VD = 0.14 V
(green):  = 1.0 V, R = 650 

x-intercept =  = 1.0 V, y-intercept = /R = 1.54 mA
 intersection at I = 0.8 mA, VD = 0.46 V. VR = -VD = 0.54 V
You can verify using Ohm’s Law that the computed VR’s and I’s for each of the five circuits are consistent
with each other (at least as closely as you can expect from a graphical solution.
Power in Diodes
The power dissipated by a diode is, as usual, given by the relation P = VI.
In the case of an ideal diode, either V = 0 (forward bias), or I = 0 (reverse bias), so an ideal diode never
dissipates any power!
Real diodes, however, do actually dissipate power. This power is dissipated both by the bulk resistance
of the diode, and in the voltage drop that occurs when carriers pass through the pn junction. Both of
these voltage drops contribute to the VD in the diode’s I-V characteristic curve.
If subjected to a reverse bias voltage Vr, the power dissipated is P  IrVr, where Ir is the reverse current
that flows through the diode, and Vr is the reverse bias voltage (which is dropped entirely across the
diode). However, since Ir is typically very small (~ A), and the maximum allowable Vr is limited by the
peak inverse voltage (typically hundreds to thousands of volts), the power dissipated by a reverse-biased
diode is typically very small. (Unless, that is, the peak inverse voltage is exceeded and the diode breaks
down… but we’ve already decided never to do that, right?)
If the diode is placed into a circuit where it is subjected to forward bias, then the diode will conduct
some amount of forward current If which can be analytically determined (as we did in the above
examples for our simple “RD circuit.” The voltage drop across the diode will be V D, and the power
dissipated will be P = IfVD. Although VD does not typically get very large (and is rarely more than a volt
or so), If can become quite large, so some care must be taken to ensure that a diode is adequately rated
for the expected load; I discuss this in some more detail in the final section of this module.
We now know how to analyze diode circuits using our “complete diode model, but discovered that it can
be rather time-consuming.
Is there a simpler diode model which still captures most of the essential characteristics of the diode’s
behavior?
X.
A More Practical Diode Model
With the computational limitations of our “complete diode model” in mind, we may develop a more
“practical” diode model which is more realistic than the ideal diode model, but is analytically more
tractable than the complete diode model.
You will notice for the diode in the above example that over a wide range of conditions, VD varies rather
little, from about 0.5 to 0.7 V. This is due to the fact that under forward bias, the current drawn
increases exponentially with increasing VD.
With this in mind, we may approximate the diode’s behavior by saying that the diode conducts very little
(i.e. zero) current for small forward voltages, and suddenly “turns on” when the voltage reaches a
“knee” voltage VK. In our ideal diode model we assumed VK = 0, but in real diodes VK > 0 (for silicon VK ~
0.7 V and for germanium VK ~ 0.3 V, for example). The effect of this is to take the “vertical” branch of
the ideal diode’s I-V curve and displace it slightly to the right, shooting rapidly upwards when V = VK.
Heuristically, we can think of this knee voltage in terms of the depletion zone being, well, depleted. In
order to do so, a minimum forward voltage VK must be applied. If a forward voltage V < VK is applied,
the depletion zone will still exist, and conductivity will be low. When we reach V ~ VK, the depletion
region has become small enough that the conductivity increases very rapidly.
To apply the practical diode model in circuits, the simplest technique is to calculate the expected diode
voltage VD0 assuming zero current flow.
This is because there are three “regions” of the practical diode I-V curve:
1.
If VD0 < -Vbd (that is, large and negative!), the diode is in breakdown mode. Then the current is
found by Kirchhoff and Ohm, treating the diode as having a fixed voltage VD = -Vbd, and the diode
is in trouble…
2. If –Vbd < VD0 < VK, the diode is in its depletion mode. Then the current is negligibly small (zero in
the practical diode model, but in reality slightly nonzero, on the order of IS), and the diode
voltage is equal to VD0.
3. If VD0 > VK, then the diode is in its forward-biased mode. In this case, the actual diode voltage VD
= VK, and the currents are found by Ohm’s Law by treating the diode as a fixed voltage VD = VK.
If there are multiple diodes, we may need to iterate through a few times to determine which diodes are
in which limit until we get a self-consistent result.
In summary, assuming that none of our diodes are in breakdown (which they should never be), the
practical diode has only two states, an “off” state corresponding to reverse bias or insufficient forward
bias (VD < VK), and an “on” state in which the diode drops a fixed voltage VD = VK for any finite amount of
current, and the current in our simple series “RD” circuit can be found via Ohm’s Law as
I = (-VK)/R
The power dissipated by a forward-biased diode is given as P = VKI, which sets an effective limit on how
much current the diode can handle (or, equivalently, tells you how large a resistance must be placed in
series with the resistor for a given supply voltage).
This practical diode model is much easier to work with, and in most cases gives fairly good results.
Specifically, this is where IS is negligibly small in comparison to relevant currents in the circuit, the
diode’s internal resistance is quite small in comparison to the external resistances, and the actual “zero
current voltages” VD0 are never very close to VK, so the only parts of the “exponential” behavior in the
complete diode model we see are the extreme upper and lower limits for which the practical diode
model correctly approximates) gives fairly good results.
One point to keep in mind is that this knee voltage VK, being an approximation of that temperaturedependent exponential in the Shockley equation, is dependent on temperature. This means that your
circuit’s behavior can change in subtle ways as the voltages VD of your forward-biased diodes drifts up
and down.
Comparison of our three diode models: ideal diode model (left, red), complete diode model (center,
blue), practical diode model (right, green). Note the gap on the negative V-axis, as Vbd is typically orders
of magnitude larger than the threshold forward voltage!
XI.
Practical Issues: Identifying, Rating and Testing Diodes
At right are three views of a diode. The top picture shows the slab of
semiconductor, with p- and n-type pieces portrayed. The middle picture
shows the corresponding schematic diagram (mnemonic: the arrow points
toward n). The bottom picture shows what a typical diode physically looks
like (generally cylindrical, like a resistor), the cathode (negative, or n-type
terminal) is identified by a band (which also corresponds with the bar
connecting across the tip of the arrow in the schematic symbol).
In all three pictures, the anode is the terminal at left, and the cathode is the
terminal on the right side.
Classifying diodes
In this section we will deal only with “ordinary” diodes. Specialty diodes (such as Zener, Schottky and
light-emitting diodes) will be dealt with in the next module.
Diodes are generally rated in terms of a number of parameters. Notation varies quite a bit, so I will not
use many acronyms or symbols, but only the proper names.
First, there is the reverse breakdown voltage, also known as the peak inverse voltage (PIV) or peak
reverse voltage (PRV). If a diode in reverse bias is subjected to a voltage in excess of the PIV, its currentlimiting capability ends (i.e. it shorts out), and the diode can be damaged. For most diodes, this is on the
order of hundreds of volts, which is usually enough for most purposes. But high-voltage diodes do exist
if you need them.
Second, there is a maximum forward current, which is the largest amount of current a diode in forward
bias can carry. This limitation means that diodes are almost always used in series with resistors (or in
circuits that already contain resistors), in order to limit current flow. That said, “power diodes,” which
can handle very large forward currents, exist as well.
Third, there is a maximum power dissipation (where P = VDI, as usual), based on how much heat the
diode can dissipate. Typical small diodes will have a maximum power dissipation on the order of 1/8 W
or 1/16 W, although “power diodes” exist as well. While the maximum power dissipation and the
maximum forward current are related to each other, it’s not a strict 1:1 dependence since VD can vary.
So both of these limitations must be taken into account, and the more stringent one in a particular
scenario used to design the circuit.
Fourth, there are the “imperfections,” – the deviations in diode behavior from the ideal diode model.
These include:
-
-
the saturation current IS (which represents the amount of current, typically ~A, which leaks
through the diode in reverse bias, and sometimes represented as “reverse leakage current”),
the diode’s bulk resistance RB (which results in an additional voltage drop for large current
draws).
the finite “recovery time” (or “turn-on/turn-off time, ” which the diode takes in order to build
up or dissipate the depletion region, which limits how rapidly it can respond to changing
voltages), and the diode’s junction and diffusion capacitances; these mainly lead to losses at
high frequencies).
in the practical diode model, the finite voltage required to drain the depletion region and start
wholesale forward conduction is parametrized by a “knee” voltage VK.
Finally, diodes are highly temperature-dependent – the knee voltage and the saturation current, for
example, are sensitive to temperature changes, so the derivatives of these quantities with temperature
(or values given at a number of temperatures) are needed for some high-precision work. And at a high
enough temperature, the diode itself can be damaged.
A diode spec sheet will typically include all or most of these values, and others as well. Sometimes
separate values are given for continuous and surge (instantaneous) loads, since components can often
handle much higher loads if they are short in duration, than if they are continuously applied.
Testing Diodes
Given that diodes have a “bipolar” dependence, with diametrically opposite properties depending on
polarity, diode testing generally is done by measuring the diode connected in both directions.
When an ohmmeter is connected with its voltage in forward bias, the resistance is expected to be rather
small. Since the voltage of the ohmmeter is relatively small (often smaller than VK), the diode is not
fully in forward-bias. But resistances on the order of 1 k or less are typical.
When an ohmmeter is connected in reverse bias, the effective resistance should be enormous (typically,
it will overload the meter’s range, just as we saw for capacitors).
Many meters also have a diode-testing function. This function measures the voltage across the diode:
-
In forward bias, the diode should have a voltage of the order of VK. For silicon diodes, VK ~ 0.7 V,
although other types can have forward voltages from 0.2V up to a little over 1V.
In reverse bias, the diode should “max out” the multimeter’s voltage source (typically a few
volts).
As a rule of thumb, if a diode in reverse bias reads a value less than about 1 V, or significantly more than
1 V in forward bias, it is probably faulty. If the polarity of a diode is unknown, a multimeter in either
ohmmeter or diode testing mode can be used to determine which is which.