Photodiodes in general tend to be very fast, much faster than phototransistors. You need to use them with a transimpedance amplifier though and if you're using a high sensitivity photodiode they tend to have large areas and correspondingly high capacitance so you normally run them with a negative bias which effectively reduces the capacitance.
You might find this useful. https://www.analog.com/designtools/en/photodiode/
Photodiodes are NORMALLY run with negative bias (it's not special to high sensitivity ones as your comment implied). In positive bias, they just conduct like a wire. A diode's special ability comes into play, ONLY when in reverse bias. Normal diodes block in the reverse direction. Photodiodes become photosensitive in the reverse direction (more current flows for more light hitting it).
As for transimpedance amplifier (current to voltage converter), you'll see my circuit does just that. The light sensor is wired in series with a resistor, and the voltage is measured across the resistor. As for the "amplifier" aspect of "transimpedance amplifier", my voltage measuring device (my Picoscope 2204A) has one builtin. The device itself is basically just a voltage amplifier who's output goes to the input of an analog to digital converter, and the output of the ADC then goes to a USB interface to send the data to the computer.
If you have any questions about my circuit, check the circuit diagram in my opening post in this thread. For the values of the components, read the last sentence of paragraph 4 in the opening post.
From your original post, it sounds like the device you have has around 200pF effective capacitance, so that it achieves around 15kHz bandwidth with a load resistance of 47 kohm. To get a faster response, you could use a photodiode with a smaller capacitance - but to achieve this, it would have a smaller active area, and so give a lower photocurrent for the same luminous flux. In some cases, such as your optical data link, this can be mitigated by focussing the received light into a smaller area. Think of a telescope pointed at your LED, focussed to generate a real image in front of the eyepiece: place your photodiode here. It needs to be only just big enough so that the whole of the image falls on its active area. Higher magnification gives a smaller image (until diffraction becomes significant) but makes alignment more critical.
Using a focussing telescope has the additional advantage that (if correctly-designed baffles are used) the amount of stray light reaching the photodiode can be greatly reduced. As @tggzz points out, this gives excess photocurrent, which even if it doesn't saturate your amplifier, will give increased shot noise.
Also, of course, the telescope has to be correctly aligned & focussed, and kept that way. This may not be trivial.
As most other posters have suggested, you can make things a lot easier for yourself by using a transimpedance amplifier,
with a low input impedance. This means that instead of being loaded by 47 kohm your photodiode may see only an ohm or so. This raises the bandwidth dramatically. You do this by moving your load resistor to the feedback path of an OPAMP, with the photodiode connected directly to the non-inverting input. The Analog Devices tool posted by @james_s shows this configuration. Using this tool, and assuming a 200 pF photodiode capacitance, I was able to achieve a 3MHz bandwidth and a nicely-damped pulse rise time of 170ns, using the two stage configuration. (You should AC-couple between the two stages to remove the effect of background light)
The AD tool allows you to set the photodiode bias voltage, but this makes little difference. Often, you use a DC 'bias' voltage of 0V. This gives the maximum photodiode capacitance, but also gives zero leakage current - important if you want to measure absolute light levels. Your application may benefit from a reverse bias of a few volts (check the photodiode datasheet) but beware - noise on the bias supply plays
directly into the amplifier input through the 200pF photodiode capacitance & gets the full gain. As you appreciate, you don't want to operate in photovoltaic mode, where the photodiode is forward-biased - stored charge will massively slow its response.
Almost none of the above depends on whether the device is a photodiode or a phototransistor, except for the speed of response. For that, you want a photodiode and preferably a
p-i-n photodiode. But at your modest bandwidth I don't think that is essential. The actual photoelectric effect within the device is very fast (picoseconds). What slows it down is the transit time of the photocarriers to the device terminals, and the self-capacitance of the device, if working into a finite load.
Bob Pease wrote some about this here:
https://www.electronicdesign.com/technologies/analog/article/21801223/whats-all-this-transimpedance-amplifier-stuff-anyhow-part-1, and the book by Jerald Graeme he mentions is also a good read.
Graeme, Jerald G.
Photodiode Amplifiers: Op Amp Solutions. New York: McGraw Hill, 1996, ISBN 978-0-07-024247-0
Phil Hobbs also covers the development of transimpedance amplifiers for photodiodes, for example in:
Hobbs, Philip C. D.
Photodiode Front Ends: The Real Story. Optics and Photonics News 12, no. 4 (1 April 2001): 44–47.
https://doi.org/10.1364/OPN.12.4.000044.And of course, the 'Bible":
Hobbs, Philip C. D.
Building Electro-Optical Systems: Making It All Work. Wiley Series in Pure and Applied Optics. New York: Wiley, 2nd Ed. 2009.
I believe there is a PDF of the Graeme book lurking on the web somewhere