Indeed, there is a temperature dependent behavior. A diode can make a pretty good temperature sensor. Dual matched transistors on a common substrate work even better in a more complex circuit. The bandgap is affected by the substrate material and the dopants but the details are beyond my field of expertise.
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I am not an expert in solid state physics but the band gap is inherently defined as the applied voltage at which current flows in the junction. So to say that it varies with voltage doesn't really make sense. I don't know what other parameter would influence it.
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Well again my lack of expertise in solid state physics leave me to only conjecture. It may be confusion with regard to terms. From Wikipedia: "In solid state physics and related applied fields, the band gap, also called an energy gap or stop band, is a region where a particle or quasiparticle is forbidden from propagating. For insulators and semiconductors, the band gap generally refers to the energy difference between the top of the valence band and the bottom of the conduction band."
So it seems that it is an intrinsic function of the materials that comprise the junction and not applied voltage. The varicap works by varying the width of the depletion zone which is a function of the applied voltage and not a variation in the inherent band gap characteristics of the material.
In an above post other physical parameters are suggested but I could only guess as to which or how much effect they might have. I would guess maybe pressure but the only one I know of for sure is temperature and it is very well characterized.
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