Not in a plasma reactor, yet. The laser Nuclear Ignition Facility at Lawrence Livermore Labs claimed "scientific breakeven" back in 2014.[1] That's the setup where they have a huge building full of pulse lasers focused on one tiny target.
This is breakeven for a very weak definition of breakeven: “thermonuclear energy out” > “energy absorbed by the fuel capsule”. Not "> energy required to run the lasers." That's for a very brief period, nanoseconds. It's taken Lawrence Livermore 45 years of zapping tiny targets with big lasers to get to this point.
This was being touted as a potential approach to fusion energy back in the 1970s. It's not, really. It's mostly a way to study bomb-type fusion without setting off H-bombs. It's now part of "stockpile stewardship", keeping some people working on fusion to prevent forgetting how to make H-bombs.
[1] http://www.hiper-laser.org/News%20and%20events/index.html
There are also just a few elements that do not produce unacceptably long lived radioisotopes under fusion neutron bombardment. This greatly limits the choice of elements from which to make the reactor structure. Right now, the best choice is RAFM steel, but it has a number of serious drawbacks.
RAFM steel is also ferromagnetic, which interferes with the design of the reactor.
https://gcep.stanford.edu/pdfs/qa4ScQIicx-kve2pX9D7Yg/baluc_...
> Brief periods of theoretical breakeven have been achieved before.
Source? I'm not aware of any reactor that has done that.
> Ahead lies "sustained theoretical breakeven" - the thing can be kept running for a while. So far, other tokomak experiments have achieved 70 seconds of plasma containment (Korea) and 120 seconds (China). That's below ignition temperature, though.
Ignition temperature is not set just by the time you run it. The larger you go the longer you need to run before you can get to ignition. ITER will need something like 1000 seconds, but SPARC will only need 10.
> Then, someday, "economic break-even" - it can pay for itself.
For ITER-like designs that's impossible because the massive manufacturing cost.
> This begins with the "first wall" problem of finding something that can survive the conditions just outside the magnetic field. Those conditions include huge numbers of neutrons, which tend to split atoms in the first wall material and cause unwanted transmutation. This causes radiation embrittlement, which is not good for materials.
Radiation embrittlement is only a problem for certain types of materials. Some material types do not absorb neutrons at fusion energies and simply pass them through. This is a complex materials problems and you can't just use steel but there's already many designs that people are experienced with for this and it's well known because of the history of fision energy research with neutrons.