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For example, in this paper (also arXiv), it uses a.u. as the unit in multiple graphs for the quantiy $\frac{dI}{dV}$, for example in Figure 3. I am suspecting a.u. means "appropriate unit," but I am not entirely sure because in the same figure it also marks an extra $10^{-12}$ for that $\frac{dI}{dV}$ quantity. If it were meaning "appropriate unit", why would that extra bit of $10^{-12}$ be useful?

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It is short for arbitrary unit. When I use it I prefer “arb. unit” to “a.u.” to avoid confusion with astronomical unit or atomic unit.

https://en.m.wikipedia.org/wiki/Arbitrary_unit

Edit: the “$\times 10^{-12}$” are indeed superfluous. In my opinion they should be removed since arb. units are used. They probably arise because the data which are shown were collected using some instrument that was recording some uncalibrated voltage or current and the software the collects the data does some conversion and the final result is at the $10^{-12}$ level and the authors just didn’t bother to rescale the data before plotting.

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    $\begingroup$ The scale $10^{-12}$ is useful to compare their different graphs with eachother. $\endgroup$
    – user137289
    Mar 8 at 9:02
  • $\begingroup$ I even request to change a.u. into arb. un. Many times I ask for removal as usually people like to assign arbitrary units to adimensional quantities. $\endgroup$
    – Alchimista
    Mar 8 at 12:21
  • $\begingroup$ @Pieter I haven't read closely, but are there also data shown at the unity level in their $dI/dV$ arb. units? I see data as small as the $10^{-13}$ level and as large as $10^{-10}$. Without knowing anything else, my preference would be scale all of the data up by either $10^{10}$ or $10^{13}$ so that the data range from 0.001 to 1 or 1 to 1000. I would be surprised if they're measurement has 13 orders of magnitude of dynamic range.. $\endgroup$
    – Jagerber48
    Mar 8 at 14:52
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    $\begingroup$ And this point of course gets to the whole point of arb. unit. The absolute value of the vertical axis is understood to be pretty much 100% meaningless when you use arb. units (if this is against the wishes of the author they should not use arb. unit in my opinion). Only relative scaling are important, so the author is free to scale all of the data by whatever factor they wish to make relative comparisons as easy as possible. $\endgroup$
    – Jagerber48
    Mar 8 at 14:54
  • $\begingroup$ The currents are given in pA, and I would guess that the modulation $\Delta I$ is on the order of $10^{-13}$ A. The referees of these STM papers have no objections. $\endgroup$
    – user137289
    Mar 8 at 16:06
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With regards to Atomic/Nuclear physics a.u. could mean Hartree atomic units.

In general, it can also mean some normalized unit so that the unit values are relevant to the scales used in that particular field. These are called Arbitrary units as already mentioned by @jgerber.

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