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 signalthrough a unidirectional transfer characteristic diode and then transferring the rectified signal through anintegrator 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?