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In high-pressure superconductors like hydrogen sulfide…

In high-pressure superconductors like hydrogen sulfide, what is the pressure physically doing to make superconductivity possible at higher temperatures?

I understand the basics of BCS theory, electrons pairing via lattice vibrations. What I don't understand is the mechanism by which extreme pressure changes that picture. Is it stiffening the lattice to raise phonon frequencies, changing the electronic density of states, or something else entirely? And is there a theoretical reason this seems to require hydrogen-rich compounds specifically?
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