# Origin of square law behaviour in Schottky diode microwave power detectors

The instructional schematic for a Schottky diode power detector is usually given as something like this (see eg page 2 of this pdf, or this page), where the detector consists of the diode and capacitor: simulate this circuit – Schematic created using CircuitLab

As stated in the infineon document:

The principle of diode detection is rectifying the AC signal through a unidirectional transfer characteristic diode and then transferring the rectified signal through an integrator to obtain the DC component.

Which is easily understood by looking at the circuit. However (assuming ideal diode behaviour), this gives an output voltage which is proportional to the rf or microwave voltage, not the power. At lower input powers, this is not true: the slope of the output voltage changes such that it is proportional to the input power. This is the region where these detectors are usually used, but application notes tend to simply state this as a fact with absolutely no justification. What is the origin of this effect? Some property of low-voltage non-ideal diode behaviour presumably, but what specifically?

$$I_D = I_S ( e^{\frac{V_D}{nV_T}}-1)$$

If your calculus is up to it (mine isn't), write out the expression for the expected mean output voltage across the load with various AC drive levels input as V1. Hint, this is somewhat easier if the load is a constant current sink, rather than a resistor, because then you're solving at a fixed current, but the overall behaviour is the same.

If your calculus isn't up to it, write a simple simulation in the language of your choice, even a spreadsheet would do, calculating the currents and voltages over a full cycle of input, and taking the mean.

When you plot the relevant curves, the behaviour emerges as clear as day, output voltage proportional to incident power for low powers, and at a linear slope, with an offset from the input voltage at high powers. If it happens, it must be true.

I did this exercise myself a couple of decades ago. Partly because like you I didn't trust what I was being told, wondering whether there was anything qualitatively different about microwave 'detector diodes' from say a 1N4148, and partly to examine the detail of the curve where the diode changes between the square law and the linear law, to improve my power detector. Hint, the deviation from the asymptotes is not symmetrical.

You may have issues with this answer. The Shockley equation is only a model, not 'truth', but it is a well established model that describes diodes pretty well. It has a 'fiddle factor' n that is needed to adjust to particular diodes. However the behaviour is much the same for any n over reasonable values. I also haven't explained how the square/linear break actually emerges from the equation, only observe that it does. The diode equation is all that's needed to demonstrate the behaviour1. Try it yourself, or better still, measure a few real diodes.

If you want to try to get some 'how' through intuition, notice that at very high currents and voltages, the diode tends to act like a rectifier, with a forward voltage drop that is somewhat insensitive to the current flowing. At very low currents and voltages, especially where that '-1' term in the diode equation is starting to get significant, the voltage current graph is gently curving, and you might not expect get any real rectification at all, just a tendency for more conduction one way than the other. Note that according to this equation (and in practice if you can measure small enough currents) diodes conduct at all voltages, including in reverse. I measured some diodes for low leakage clamping/protection duty a while ago, and it was nice to observe almost constant resistance within a few mV either side of the zero bias point.

1 - I wouldn't be surprised if conduction laws of a different detailed shape, but following a reasonable smooth curve with small response at low voltage, big at high voltage, with zero at zero, also gave rise to the same type of emergent behaviour, but I'll leave testing that to somebody else.

Large signals deteriorate diode mixer's performance: first, nonlinearity results in conversion gain degradation and causes the third order harmonic distortion; second, when two signals are present at the RF input, intermodulation distortion occurs.

You are correct when indicating that at lower input powers the slope of the output voltage is proportional to the input power and that this is the region where these detectors are usually used. It may seem that application notes tend to simply state this as a fact. But I cannot agree that application notes state this as a fact with absolutely no justification: although the context may seem to be somewhat understated, it is certainly present in sections covering gain compression (1dB point), the third-order intercept point, and intermodulation distortion.

Actually, the document considers so-called "well-behaved" scenarios, in which the small-signal approximation is valid. It clearly (although maybe not quite persistently) states that it is not a property of low-voltage non-ideal diode behavior that is responsible for the focus of this document, but requirements of applications (RF mixers, power detectors, etc.) and use scenarios considered.

UPDATE

If you agree with this explanation of apparent omission of discussion about power/voltage proportionality by application notes (not only Infineon's, but others too), I believe I can indicate the property of low-voltage non-ideal diode behavior which makes possible sufficiently wide square law detection range in Schottky diode microwave power detectors. It is the low potential barrier across Schottky diode junction.

Infineon boasts about their low-barrier Schottky diode products: best-in-class RF performance, advanced package miniaturization technology, highest part-to-part uniformity. Notice the characterization of Schottky diode products in their own words: Infineon has more than 30 years development expertise for the RF small signal discretes etc. In the context of your question, the most pertinent part is 'RF small signal discretes'.

More detailed explanation of how the diode detection characteristic ranges from square law through a transition region to linear detection is given in a classic application note by Agilent Technologies Fundamentals of RF and Microwave Power Measurements, AN 64-1B. It considers a power series approximating a diode equation:

It is the second and other even-order terms of this series which provide the rectification. For small signals, only the second-order term is significant so the diode is said to be operating in the square-law region. In that region, output i (and output v) is proportional to RF input voltage squared. When v is so high that the fourth and higher order terms become significant the diode response is no longer in the square law region. It then rectifies according to a quasi-square-law i-v region which is sometimes called the transition region. Above that range it moves into the linear detection region (output v proportional to input v).

A detailed derivation of detection characteristic curves can be found elsewhere.

It seems the paragraphs of this document cited below directly address your question "What is the origin of this effect? Some property of low-voltage non-ideal diode behaviour presumably, but what specifically?"

Agilent's AN 64-1B compares 'an ordinary silicon p-n junction diode' and 'a low-barrier Schottky diode' as candidates for an RF detector device.

First, it shows how little can be achieved with ordinary silicon p-n junction diodes used to convert high frequency energy to DC by way of their rectification properties

... It might seem that an ordinary silicon p-n junction diode would, when suitably packaged, be a sensitive RF detector. However, p-n junctions have limited bandwidth. In addition, the silicon p-n junction, without bias, has an extremely high impedance and will supply little detected power to a load. An RF signal would have to be quite large to drive the junction voltage up to 0.7 volts where significant current begins to flow. One alternative is to bias the diode to 0.7 volts, at which point it only takes a small RF signal to cause significant rectified current. This effort turns out to be fruitless, however, because the forward current bias gives rise to large amounts of noise and thermal drift. There is little, if any, improvement in the minimum power that can be metered.

Fortunately, there exist devices with much lower potential barrier across their junction:

Metal-semiconductor junctions, exemplified by point-contact technology, exhibit a low potential barrier across their junction, with a forward voltage of about 0.3 volts. They have superior RF and microwave performance, and were popular in earlier decades. Low-barrier Schottky diodes, which are metal-semiconductor junctions, succeeded point-contacts, and vastly improved the repeatability and reliability.

And later the document gives figures characterizing the usefulness of low-barrier Schottky diodes for practical implementation of RF diode detectors:

For a typical packaged diode, the square-law detection region exists from the noise level up to approximately −20 dBm. The transition region ranges from approximately −20 to 0 dBm input power, while the linear detection region extends above approximately 0 dBm. Zero dBm RF input voltage is equivalent to approximately 220 mV (rms) in a 50 Ω system. For wide-dynamic-range power sensors, it is crucial to have a well-characterized expression of the transition and linear detection range.

Figure 5-2 shows a typical detection curve, starting near the noise level of −70 dBm and extending up to +20 dBm. It is divided up into the square law, transition and linear regions. (Noise is assumed to be zero to show the square-law curve extends theoretically to infinitely small power.) Detection diodes can now be fabricated which exhibit transfer characteristics that are highly stable with time and temperature. Building on those features, data correction and compensation techniques can take advantage of the entire 90 dB of power detection range.

[ starting near the noise level of −70 dBm: It appears AN 64-1B calculates the noise level (-70dBm) at room temperature given a 24GHz bandwidth: Johnson-Nyquist noise power is P_dBm=10·log((kT·f_bw)/1mW) = 10·log((k·300)/1mJ) + 10·log(24GHz/1Hz) = 173.8 + 10·10.38 = -70dBm; 1mW = 1mJ·1Hz ]

Low-barrier Schottky diodes does have the transition and linear detection characteristic ranges at large signal levels, but this wide square law detection range make them especially fit for use in RF detection circuits. The wide square-law power detection range of 50dB (from -70dB to -20dB) may explain why presently application notes omit the discussion of linear detection range appearing at larger input power levels (above 0dB). Notice that an RF detector can operate even in a linear detection range, using 'data correction and compensation techniques', which enable us to 'take advantage of the entire 90 dB of power detection range.'