**Equivalent Circuit Description**

If we pretend for a moment that we are hard-core electrical systems engineers,
we do not care at all about what a pn-junction diode consists off, how it works, or how it is made. We simply describe
it as a black box and use the two equations from
before, but now with a possible frequency dependence thrown in for the output current.
For small signal behavior we have by definition | |||||||||||||||

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i.e. we consider a frequency dependent amplitude of the current output signal and a frequency dependent phase shift I_{m} (w
)j (w). |
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The relation between and U_{0}
is simply given by the (I_{0}DC) current voltage characteristics of our black box. | |||||||||||||||

In principle, the output signal amplitude (i.e. the current amplitude) is also a function
of or U_{0}
; in other words of the I_{0}working point chosen. All
we have to do then is to tabulate (or represent graphically) the functions ) and I_{m}(w,
I_{0}
j(w, for our black box.I_{0} ) |
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This is all. We always have an output signal looking exactly like the input signal, and with
an amplitude proportional to the input amplitude because we have by definition a linear system.
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We know even more. We always will have | |||||||||||||||

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as long as the output signal can be directly obtained from the DC characteristics,
i.e. at small frequencies (leaving open at present what "small" means in numbers). |
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Looking at our diode (or the small signal (i.e. linear) behavior of about anything
else) in this way has a large advantage: | |||||||||||||||

All black boxes can now be described by a suitable network
of basic linear electronic elements - resistors, capacitors and inductors. In the most
simple case all elements have constant values; more realistically their value depends on external parameters (e.g. the voltage). |
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Calculating the behavior of such a network is relatively simple - for a computer, that is. | |||||||||||||||

However, just looking at some network or equivalent
circuit diagram as it is called, has disadvantages too: | |||||||||||||||

Many different kinds of networks will give exactly the same response; their is no unique choice - but often intelligent choices. |
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If some network does behave like the actual device, it is not necessarily easy to extract
the important physical parameters of the device from it. In other words, if some network behaves exactly like the diode
you are interested in, it does tell you how you must change resistors and capacitors
if you want to make it faster, but not which physical parameters of the device (doping, lifetime, dimensions, ...). |
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The best way therefore is to construct an equivalent circuit - keeping it as simple as possible - by looking at the physical characteristics of the real device first. We are going to do this now for our junction diode. | |||||||||||||||

Equivalent Circuit Construction for a Junction
Diode | |||||||||

If we start with the low frequency behavior of a junction diode, the equivalent circuit diagram is exceedingly simple: | |||||||||

It consists of a single resistor with a value that depends on
the voltage and is given by R_{D},
i.e. by the (inverse) slope of the R_{ D} = dU_{diode}/dI_{diode} = R_{D}(U)DC characteristic of the diode as illustrated
before. This is shown below | |||||||||

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This "network" cannot possibly give any frequency
dependence, so it can not be all there is to a diode. We now must remember
that any pn-junction has a capacitance
associated with the space charge region. It arises from the C_{SCR}ionized
dopants that provide the net charge on both sides of the junction needed for any capacitor and its value (for a
symmetric junction) is given by | |||||||||

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is obviously switched in parallel to C_{SCR}
; our equivalent circuit diagram now transforms into
this:R_{S} | |||||||||

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This network will give us a simple frequency response: High frequencies are essentially
short-circuited by the capacitor; and the current amplitude will increase with frequency (accompanied by a phase shift of
90 at the maximum).^{o} | |||||||||

The current amplitude will have increased twofold if the AC resistance of R_{C}
equals C_{SCR}, this happens forR_{D} | |||||||||

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With our formulas for and for I = I(U, doping, ...)
, we could calculate the limit frequency as a function of the prime material parameters like doping, lifetime and so
on - but we are not yet done in constructing the equivalent circuit diagram: C_{ SCR} = C_{SCR} (U, doping, ...)
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There is a second capacitance hidden in the junction
diode called "diffusion capacitance ". Lets see what it is. |
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Any
net charges to the left and right of the junction form a capacitance. Above, we considered the space charge region
capacitance stemming from ionized dopants in the SCR which are not compensated by majority carriers as we have it
in the bulk. For diodes biased in reverse direction, i.e. with only a small reverse
current flowing, this is all there is. | |||||||||

But if we look at the distribution of all
carriers for a diode in forward direction, we notice that we have an excess of minority carriers at the edge of the SCR;
a schematic picture was given before.
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These excess minorities - only present while current flows - are coming from the injection of majority carriers over the potential barrier which become minorities in the other
part of the junction. Their concentration is increased because a concentration gradient is needed to remove these carriers
by diffusion. They are not compensated by other charges and thus form a diffusion capacitance that increases with current.C_{Dif} |
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This diffusion capacitance must also be switched in parallel
to and R_{D} .C_{SCR} | |||||||||

Moreover, the ohmic resistance of the bulk Si is not zero, but has some
finite value which is constant and easily determined by the resistivity of the R_{S}Si and
all other ohmic resistors in the circuit. | |||||||||

Clearly, must be switched R_{S}in series
to everything else, and we obtain the final equivalent circuit diagram of a junction diode valid for all cases. |
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Of course, you always could lump together the two capacitors into just one, but then you loose the connection to physical reality. Note that their numerical values will depend on physical parameters in very different ways - as we will see below. | |||||||||

If we would know the value of , we could now calculate the small signal response of a junction diode. We have no ready formula, so we must derive
one.C_{Dif}
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We will do that right away; it will prove to be particular enlightening for the understanding
of the correspondence of primary material parameters
and their transformation to equivalent circuits. | |||||||||

The Diffusion Capacitance and its Relation
to Time Constants of the Device | ||||||||||||

The excess at to the left and right of the space
charge region increases with increasing (forward) voltage Q_{min} . If we have a linear relation between U
and U, the capacitance Q_{min} would be given by C_{Dif} |
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Since this is not necessarily the case, is a function
of C_{Dif} and must be defined Udifferentially as . | ||||||||||||

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Considering that the current is a function of I (given by the
basic diode equation), we may rewrite this expression as
U | ||||||||||||

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The second term is simply the derivative of the characteristics, and thus equal to the inverse diode resistance. I-U
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d is the interesting term. This differential
quotient must have the dimension of time, and thus can be seen as theQ_{min}/dI time constant
connected to . In order to compute it, we need C_{Dif}
as a function of the current flowing through the junction, i.e.
Q_{min} | ||||||||||||

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We have not encountered this functional relationship so far - but we came close. | ||||||||||||

In the "Useful Relations"
subchapter we have an equation giving the excess minority
carrier density D as a function of the distance n(x ) from
the edge of the depletion layer: x | ||||||||||||

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With D = excess density at the edge
of the depletion zone, n_{0} = diffusion length.L | ||||||||||||

In the "Junction Reconsidered"
subchapter we derived an equation relating the current
density (for one of the two current components) through the junction to the excess carrier density jD at the edge of the depletion layer:n_{0} | ||||||||||||

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So all that remains to do is to express the charge D
at the edge of the n_{0}SCR as a function of the total excess charge , substitute it in the
current equation, and do the differentiation Q_{min}d. This is easy:Q_{min}/dI | ||||||||||||

is simply the integral over Q_{ min}e · D taken from
n (x) to x = 0; i.e.x = ¥ | ||||||||||||

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Inserting D
in the current equation from above gives
n_{0} = Q_{min}/ L · e | ||||||||||||

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The differentiation (turning to current densities instead of currents j
and interpreting I as charge density) finally yieldsQ_{min} | ||||||||||||

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And this, of course, is nothing
but the minority carrier life time t! | ||||||||||||

The time constant associated with the diffusion capacitance thus has a very
clear physical meaning - it is the minority carrier life time. It is, if you like, the fundamental property that causes the diffusion capacitance. | ||||||||||||

Interpreted in other words: While separated charges always cause a (static)
capacitance Q given by C, you always will find a (dynamic) capacitance, too, if it C = Q/U = ee_{0}Q
· A/d
takes time to change
charge concentrations. The capacitance associated with some given time constant t than is more appropriately expressed as | ||||||||||||

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With being some ohmic resistor which limits current flow.R |
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Now we have all terms; the diffusion capacitance can be finally expressed as | ||||||||||||

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Finally, we look at the diffusion capacitance from yet another angle: It results from the necessity to "store" some charge on both sides of the junction for forward current flow. | ||||||||||||

The concept of stored charge and the time constant associated
with its removal will come up again later. | ||||||||||||

Consequences | |||

We now have all ingredients to analyze the small signal response of a pn-junction
- as long as it is ideal, infinitely extended, and symmetric. | |||

The first question coming to mind is the relation between the two capacitances. Electrically, they add up - but which one dominates? | |||

This is most easily seen if we consider the forward and reverse current direction separately. | |||

The reverse
current mode is easy: The diffusion capacitance is small because d is practically zero and
can be neglected. We only will find the junction capacitance I/dU. Moreover, C_{ SCR}
is very large, certainly much larger than R_{D}, and may be taken to be infinite.R_{S} |
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In
forward direction, increases with C_{SCR}
while U^{1/2} increases C_{Dif}exponentially with
(look at the formulas!). This means that in forward direction U will always win and dominate the total capacitance. C_{Dif}
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We thus can simplify the equivalent circuit diagrams for the two extremes: | |||

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This means that the small signal behavior of a pn-junction is quite different
in reverse or forward direction, and that it also depends very much on the working point (the DC current) in forward
direction. It also means that we only can transmit AC signals in reverse direction. | |||

If we now try to deduce some numbers for limiting frequencies of real diodes, we run into problems: | |||

The diffusion capacitance will short-circuit all signal with frequencies considerably larger
than 1/t. Since the minority carrier life time in good Si can be as large as 1
ms, we must expect problems in the kHz region - and even for life times of 10 µs we would not get
far beyond the MHz region. This is obviously far below limiting frequencies in actual devices.
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What went wrong? Not so easy to unravel,
lets just consider the most important point: | |||

Dimensions in real (Si) devices, especially integrated circuits, are
much, much smaller than the diffusion length . For the excess minority
carrier distribution as shown for a large
diode, this has profound consequences.L |

The excess concentration must go down to zero at the contact, i.e. after a distance that gives the length of the (almost) neutral piece of d_{Con}
Si (the distance from the edge of the space charge layer
to the contact). In a first approximation
we have to replace the diffusion length (which is in the order of magnitude of L100 µm)
by , which can be a fraction of a d_{Con}µm. | ||||||||

The minority carrier excess density now decays rapidly and reaches zero at the contact, i.e.
after a distance . For all practical purposes it is sufficient to assume a linear relationsship
as shown below. In practice, the difference between d_{Con} and L is much larger.d_{Con} |
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Accordingly, we have to replace the bulk lifetime t =
byL^{2}/D | ||||||||

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This is something we have encountered before by looking at real diodes. | ||||||||

A small device thus could be faster by several orders of magnitude. | ||||||||

And this is where, quite generally, the dimensions come
in. We always need to move carriers from here to there, and this takes some time determined by the distance and the velocity
(or mobility), respectively, as pointed out before. | ||||||||

For our case of small diodes, we may use the equation
from before in the form | ||||||||

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This tells us that the small signal frequency response of a small junction diode is ultimately controlled by the dimensions of the device and the mobility of the carriers that carry the current. | ||||||||

Of course, real small devices are even more
complicated. But all kinds of parameters neglected in this simple consideration will make the frequency response worse,
not better. The absolute limits are always determined by dimensions and mobility. | ||||||||

A few last words of caution: | ||||||||

While we replaced the (bulk) lifetime by a transit time in this simple consideration of the real device world, it would be premature to conclude that the life time is of no importance in small devices. | ||||||||

It also would be premature to conclude that we should reduce t
as much as possible, because this carries heavy penalties
in other areas - see the links 1, 2,
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However, that does not mean that "lifetime killing"
is not used on occasion. For some devices - particularly (large) power devices - some Au might be diffused into a
junction to reduce the life time as much as possible. |

© H. Föll (Semiconductors - Script)