Fun with the Diode Equation

Introduction and Scope

Diodes, specifically p-n junction diodes, are a commonly used device in all manner of electronic circuits. These range from being used as very high power rectifiers dealing with hundreds (or thousands) of volts and/or amps to detecting the information signal in an AM radio that is powered only by the energy in the radio signal itself.

For the more brute-force applications, a diode is usually modeled sufficiently well by considering it to be a mostly-ideal switch that conducts no current in one direction and conducts pretty much any current (within reason) in the other direction with only a small, relatively constant, voltage drop across it. In these applications we can ignore the details of the exact nature of the voltage-current characteristic as long as we don't violate the limits at either extreme, namely putting so much reverse voltage that we exceed the PIV (peak inverse voltage) rating or the maximum forward current rating, either of which can quickly destroy the device.

Our interest here, though, are those details of how the forward current through the diode is related to the forward voltage and some of the interesting and useful aspects of it.

The Shockley Ideal Diode Equation

Let's spend a bit of time with the well-known Schockley diode equation, which models the voltage vs. current characteristic for a p-n junction diode.

\(i_d \, = \, I_s e^{\frac{v_d}{nV_T} - 1}\)

where

\(V_T \, = \, \frac{kT}{q}\)

is the "thermal voltage". The parameter k is the Boltzmann constant, 'q' is the elementary charge (the magnitude of the charge on an electron or proton) and T is absolute temperature.

\(k \, = \, 1.3806488(13) \times 10^{-23} \frac{J}{K} \\ q \, = \, 1.602176565(35) \times 10^{-19} C \\ \frac{k}{q} \, = \, 86.173324(78) \frac{\mu V}{K}\)

At "room temperature", which is often taken to be 300 K (26.85 °C or 80.33 °F), the thermal voltage is 25.85200 mV. While it would properly be rounded to 26 mV, which corresponds to 301.72 K (28.57 °C or 83.42 °F), it is often rounded to 25 mV, which corresponds to 290.12 K (16.97 °C or 62.55 °F), as this is a very convenient value for off-hand calculations and is generally "close enough".

The value of I_s, known by a variety of names but perhaps most commonly as either the "scale current" or the "reverse saturation current", is typically on the order of nA or μA and, thus, the "-1" can be safely neglected for any forward voltage much above the thermal voltage.

One of the ugly little secrets (meaning that most people that use the diode equation are unaware of it) is that I_s is pretty strongly dependent on temperature. This is actually extremely important because, if I_s were truly the constant that it appears, we would see that as the temperature increases we would need to increase the diode forward voltage in order to maintain the same diode current. However, even though v_d and V_T are in the exponential, I_s is sufficently dependent on T that as the temperature increases, the forward voltage drop across the diode at a given current actually decreases. For this reason, p-n junctions that are placed in parallel are subject to thermal runaway.

The parameter 'n' is the "quality factor" or "ideality factor" (and also known by a few other names). It is generally between the values of 1 and 2. For the most part, a value of n greater than one indicates that there are other mechanisms that result in electron-hole recombination other than thermal recombination. Probably the most dominant mechanisms are defect-induced recombination and photon-induced recombination.

Interestingly, the parameter 'n' is not truly part of the Schockley equation since he was only modeling thermal recombination in his theoretical derivation of the equation using statistical thermodynamics.

As manufacturing processes have improved, the defects have been reduced to the point that n=1 is a reasonable assumption in most cases. Having said that, in direct semiconductors, such as GaAs and InP, the ideality factor is typically close to 2, which in indirect semiconductors, such as Si and Ge, it is generally close to 1).

In photovoltaic devices, the ideality factor can be a significant factor and can vary with voltage and locally reach values of 5 or more. That's completely beyond the scope of this discussion, so it's being thrown out just to make people aware of it.

Decades ago, a rule of thumb was that n=1 for transistors (the Schockley model is also used to model the voltage-current relationship of the base-emitter junction in bipolar junction transistors) and n=2 for diodes. But now it seems that n=1 is assumed for all Si devices unless there is a compelling reason not to.

The Nature of the the Diode I-V characteristic

Getting back to our discussion,

\(i_d \, = \, I_s e^{\frac{v_d}{nV_T}}\)

is an acceptable approximation for our use since we are only interested in the forward conduction region well above the thermal voltage. We'll go ahead and continue to carry the ideality factor, but in most instances where we talk about hard numbers, we will assume that n=1 unless stated otherwise.

Since we know that I_s is a pretty unreliable number, let's see if we can make it go away. To do this, we will measure the the forward voltage drop, Vo, at some reasonable forward current level, Io, and use those as a pair of reference values. Hence

\(I_o \, = \, I_s e^{\frac{V_o}{nV_T}}\)

If we take the ratio of the the current at another other voltage to our reference current, we get

\(\frac{i_d}{I_o} \, = \, \frac{I_s e^{\frac{v_d}{nV_T}}}{I_s e^{\frac{V_o}{nV_T}}}\)

which simplifies to

\(\frac{i_d}{I_o} \, = \, e^{\frac{(v_d-V_o)}{nV_T}}\)

Notice how the saturation current simply goes away. But it does NOT mean that we can use a reference point at one temperature and an actual operating point at another temperature. The only reason the I_s parameters canceled is because we assumed they are the same value in both instances of the equation and this is, in general, only true if both instances are at the same temperature. A subtle point that is easy to overlook.

An interesting and useful result happens if we take the logarithm (to any base you like) of both sides. Since it doesn't matter which base, we'll choose the natural logarithm because it keeps the form of the results nice and clean.

\(\ln \( \frac{i_d}{I_o} \) \, = \, \frac{(v_d-V_o)}{nV_T}\)

Now let's define Δv_d as the difference between the actual diode voltage and the reference voltage.

\(\Delta v_d \, = \, (v_d-V_o) \\ \ln \( \frac{i_d}{I_o} \) \, = \, \frac{\Delta v_d}{nV_T} \\ \Delta v_d \, = \, nV_T \cdot \ln \( \frac{i_d}{I_o} \)\)

At this point, we can ask a couple of questions and develop some extremely useful rules of thumb.

1) How much does the diode forward voltage have to be increased by in order to double the diode forward current?

To answer this question, we simply solve for ΔV_d and set i_d/Io equal to two:

\(\Delta v_d \, = \, nV_T \cdot \ln \( \frac{i_d}{I_o} \) \\ i_d \, = \, 2I_o \\ \Delta v_d \, = \, nV_T \cdot \ln \( \frac{2 I_o}{I_o} \) \, = \, nV_T \cdot \ln (2) \\ \Delta v_d \, = \, 0.693 n V_T
\)

Note that the fact that we might have a specific value of Io, perhaps from a data sheet, that doesn't matter because the Io dropped out of the equation. In other words, we could have just assumed that, if needed, we would measure Io and the corresponding Vo at the appropriate current, but it turns out we don't need to.

At room temperature (and assuming n=1), this works out to

\(T = 300K; \ n=1 \ : \\ \Delta v_d \, = \, 17.92mV \)

Since a doubling in frequency is called an "octave", this term is commonly used for any doubling in a parameter, hence we can say that, for a diode (at constant temperature) that the voltage change is modeled by

\(\Delta v_d \, = \, 17.92mV/octave \\ \Delta v_d \, \approx \, 20mV/octave \)

We can pretty safely round this to 18 mV/octave and, for mental calculations and approximations, can usually use 20 mV/octave.

2) How much does the diode forward voltage have to be increased by in order to increase the diode forward current by an order of magnitude?

This is exactly the same process as above, except now we set the ratio of i_d to Io to be a factor of 10. An increase by a factor of ten is generally referred to as a "decade", so we have

\(T = 300K; \ n=1 \ ; i_d = 10I_o \ : \Delta v_d \, = \, \ln (2) V_T \\ \Delta v_d \, = \, 59.52mV/decade \\ \Delta v_d \, \approx \, 60mV/decade \)

Diode Thermometers

We use the above results to make a very useful and simple electronic thermometer, despite the strong temperature dependence of the reverse saturation current.

To do this, we replace the thermal voltage with what it is defined as and we get

\(\Delta v_d \, = \, nV_T \cdot \ln \( \frac{I_1}{I_o} \) \\ \Delta v_d \, = \, n \cdot \frac{kT}{q} \cdot \ln \( \frac{I_1}{I_o} \) \\ T \, = \, \frac{\Delta v_d}{n} \cdot \( \frac{k}{q} \cdot \ln \( \frac{I_1}{I_o} \) \)^{-1} \)

If we set up a circuit that forces a known ratio of current in two nominally identical diodes that are at the same temperature, we can measure the difference in the voltages across them and use that to back out the temperature.

For a current ratio of ten (and assuming n=1), this becomes

\(I_1 = 10I_o ; n=1 \ : \\ T \, = \, \Delta v_d \cdot \( 86.17\ln \( 10 \) \frac{\mu V}{K} \)^{-1} \\ T \, = 5040 \frac{K}{V} \cdot \Delta v_d \\ T \, \approx 5 \frac{K}{mV} \cdot \Delta v_d \)

Note that this last result is pretty obvious in hindsight, since if we have 60mV/decade at 300K, that means that we have 300K/60mV or 5K/mV when we have a decade difference in current.

Also, notice that we aren't required to use a current ratio of ten. We can choose any convenient value as long as it is reasonable to implement the necessary circuit. To get a scale factor of 3K/mV would require a current ratio of 47.9, which is very doable on an IC, while a scale factor of 2K/mV would require a current ratio of 331.2, which is still doable. However, to achieve a factor of 1K/mV would require a current ratio of nearly 110,000.

A simple way of setting up such a circuit, particularly on an IC, is to use a current mirror to produce two currents that are scaled by the desired ratio. Current mirrors can, in theory, be cascaded to achieve very high ratios, but doing so introduces additional uncertainties that can defeat the purpose.

On an IC, it is a fairly straightforward matter to manufacture diodes and transistors that are very closely matched and that have very little temperature difference between them. Doing this with discrete diodes is considerably more problematic, especially finding diodes that are sufficiently well-matched to be considered "identical". But inexpensive transistor arrays can be readily used for this purpose.

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