..and metal is needed for a capacitive surface so it knows when you hold your finger on it.
Not quite. An iPhone screen doesn't have a metal surface, does it?
It does in a way, because it's got "wires".
The capacitive touchscreen has a transparent grid of two layers (rows on one, columns on the other) of conducting material, which is used to determine where you touched the screen. A voltage is sequentially applied to each row in one layer, and then the capacitive charge at each column intersection in the other layer is measured. (The capacitance is changed by the proximity of your finger.)
The grid squares are relatively large, so extrapolation is used to determine the actual touch point.
(In the latest in-screen method that Apple uses, one plane of the grid is actually inside the display. This saves one layer of conducting material/glass.)
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A capacitive fingerprint sensor is similar, but has a far more dense grid, and is sensitive enough to read the tiny depths of fingerprint ridges (~.020 mm).
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An RF fingerprint sensor transmits a low frequency radio signal from a surrounding ring/square through your finger to a very high DPI antenna "pixel" array, where the signal intensity is measured at each column.
(The antenna array can be an area grid sensor as big as a finger, or it can be just a few rows that you swipe past. More than one row is needed to be able to recognize the swiping motion/speed, or you can just have a few offset pixels for that.)
It could also be put under the glass surface, right?
In theory your finger doesn't actually have to touch the RF transmitter ring, but I think in practice it's always been done that way.
Yes, my terminology isn't technically accurate but people seem to more immediately understand antenna after years of having them stick out of their phones and cars

Although with NFC and wireless charging coming about, hopefully inductor and/or coil come into more prominent use.
I understand and concur. There's a fine line between writing so most laymen can understand it, and being accurate enough that other engineers don't freak out. lol
