How much bigger does an aluminum grounding conductor need to be?
One to two AWG sizes larger than copper for the same overcurrent device rating. At a 100 A device it is 8 AWG copper or 6 AWG aluminum. At 400 A it is 3 AWG copper or 1 AWG aluminum. NEC Table 250.122 lists both columns side by side, so this is a lookup, not a conversion you do in your head.
Why does aluminum have to be bigger at all?
Aluminum carries less current than copper for the same cross-section, so it takes more metal to do the same work. That is the entire reason the table carries two columns. The code already did the comparison for you at every device rating it lists, which is why nobody needs a rule of thumb here.
And a rule of thumb would be wrong anyway, because the step between the two columns is not constant across the table. It runs one to two AWG sizes depending on where you are. Read the row for your device rating instead of applying a factor you remember from a different job.
What are the actual pairs?
Down low, a 15 A device takes 14 AWG copper or 12 AWG aluminum, and a 20 A device takes 12 AWG copper or 10 AWG aluminum. A 60 A device takes 10 AWG copper or 8 AWG aluminum. At 100 A it is 8 AWG copper or 6 AWG aluminum, and at 200 A it is 6 AWG copper or 4 AWG aluminum.
Above that the sizes climb fast. A 300 A device takes 4 AWG copper or 2 AWG aluminum, 400 A takes 3 AWG copper or 1 AWG aluminum, and 600 A takes 1 AWG copper or 2/0 AWG aluminum. At 1000 A it is 2/0 AWG copper or 4/0 AWG aluminum, and at 1200 A the aluminum column has left AWG behind entirely: 3/0 AWG copper or 250 kcmil aluminum. Every one of those pairs comes out of NEC Table 250.122, and the Cable Tray Bonding Jumper Size Calculator returns them one at a time so you are reading a single answer instead of scanning a grid.
When does the aluminum column pay for itself?
On weight and cost, mostly on the larger sizes and the longer runs. A 4/0 aluminum conductor is a very different thing to pull through a long raceway than the copper it replaces, and on a big job the material difference is real money.
What you give back is space and care at the terminations. The bigger conductor needs somewhere to go, so check the raceway and the wireway before you commit. Check that the lug you are landing in accepts the conductor you actually bought, in that size and that material, because a termination that was fine for the copper number is not automatically fine for the aluminum one. And whoever makes up that connection should know how aluminum wants to be handled. None of that is a reason to avoid it. It is a reason to decide before the material is on the truck rather than after.
Does the 250.122(A) cap still apply?
Yes, and it does not care which material you picked. The bonding jumper or grounding conductor is not required to be larger than the circuit conductors it serves. Check it after you read the table, not instead of reading the table. More on how that fits the method in whether you size the grounding conductor from the breaker or the wire.
Does any of this change for cable tray?
No. A conductor running in or bonded to a metallic tray is sized from the overcurrent device the same as anywhere else, in whichever material you are running. What the tray can change is whether a separate conductor is required in the first place, which is a question for NEC 392.60(A) and is covered in whether cable tray can be used as the equipment grounding conductor. The calculator itself lives in the tool catalog, and it will hand you both columns for whatever device rating you give it.
Values follow NEC Table 250.122. Codes are adopted and amended locally and change between editions. Confirm against the NEC edition your jurisdiction enforces and the project specifications before you build. This is a reference, not a stamped engineering decision.