The Tascam DR-680 is a very convenient device for recording from a soundfield microphone such as the TetraMic, Brahma, or SPS-200. However, it has a serious defect which affects some microphones (though not the TetraMic or the AKG C404ULS, the only mics I've used with it). This is that the phantom power is not adequately smoothed; the smoothing provided, per channel, is two 1uF surface-mount capacitors - in series (therefore effectively 0.5uF - oops!). The effect of this is that some microphones generate a lot of LF noise which could be avoided with adequate capacitative smoothing of the phantom power.
The Swiss branch of Teac acknowledged this, and modified the recorder of the man who reported it to them and pressed them to find a solution. They simply wired a 47uF electrlytic across the phantom power feed to each of the recorder's six channels, and in some cases got a noise improvement of 12dB!
So long as you are happy to dismantle the recorder and solder onto a circuit board, it is easy to fit the capacitors to the four XLR inputs. Further dismantling is required to reach the two jack socket inputs, and I didn't bother with this on my unit.
It's worth noting that the DR-680 MkII has been improved, and appears not to have this issue. In my experience the inputs are a couple of dB quieter as well, but I haven't yet done a proper comparison comparing the gain as well.