Antennas used at VLF are most commonly just an e-field probe (there are other options). Although often built using antenna-type hardware, such as a huge perhaps-extendable whip element, they're not really operating as a normal current fed element (way too short in terms of wavelength).
Somewhat counterintuitively if your mind is thinking in terms of "short" antennas, but making much more sense if you look past the antenna hardware and realize that it's just an e-field probe, the preamplifier needs to have as high an input Z as humanly possible.
The common design approaches for e-field probe antennas used in VLF reception are as follows:
Treat the antenna element as ultra-high impedance; this requirement impacts mounting techniques
Don't use any feed line at the base of the element
The preamp needs to be at the base of the probe
The element must feed directly into the preamp
It's common to use FETs; feeding the element directly into the gate with a short high Z wire
The preamp output can be 50 ohms (or audio Z such as 1k ohm); depending where it's headed - radio or audio
The preamp power can be fed through the same coax as the signal returns using normal splitting techniques
The probe needs to be well away from local noise sources
Think about lightning and take precautions
Use of batteries is universal; trying to use AC adapters is reportedly difficult due to carrying in common mode noise from the power lines
9v batteries are easier to deal with than 12v
There's next to no current involved
The preamp itself is not difficult; it's more or less audio frequency range. If it oscillates, then you're doing it wrong (normal build rules).
The noise floor should be set externally by the environment. The noise (signal) you hear should be external, with nothing more than a faint hiss generated by your equipment.
My collection of VLF compatible antennas includes two Sony AN-1 and an indoor MFJ desktop type that violates some of the rules about antenna placement. These are SW antennas that include some VLF.
