Superconductivity in metal-hydrogen systems has been reviewed by Stritzker and Wühl.[226] Only a few highlights are mentioned here.
In order to appreciate the effect of hydrogenation on superconductivity, one must understand the role of lattice vibrations. Everyone knows from the BCS equation[16]
that increasing the density of states at the Fermi level increases Tc. This expectation is borne out by the detailed calculations of McMillan, which treat the electron-phonon coupling more exactly.[165] However, the BCS equation also gives that the impression that raising the Debye temperature by stiffening the acoustic phonons will raise Tc, which is usually false. In general, it is ``soft'' phonons rather than stiff ones which enhance Tc, the best-known example of this rule-of-thumb being the A15 superconductors.[23] In the special case of hydrogenated metals, the relationship between the electron-phonon coupling parameter Lambdaep and the lattice frequencies reduces to an especially simple form because of the large mass difference between hydrogen and the metal. This relationship is:[99]
where eta is an electron-phonon matrix element.[165] The typical phonon
frequency Omegaoptic might be thought of as
proportional to the Einstein temperature TE. This
equation, in conjunction with the McMillan formula for
Tc (Eqn. 
), shows that raising the
Debye and Einstein temperatures should in general lower
Tc.
In most of the elemental superconductors, hydrogen gas acts to depress Tc.[226] Hydrogen generally weakens the coupling of the carriers to acoustic phonons, so unless it can overcompensate with strong electron coupling to optic phonons, Tc will decrease.[98] Much of the motivation for the study of hydrogenated superconductors came from the discovery of superconductivity in PdH. Elemental palladium is not itself superconducting, so the observation of 8.8 K superconductivity in PdH aroused great interest.[226] Just as surprising is the higher Tc = 10.7 K found in the deuterated compound PdD. The Tc enhancement in PdH has been attributed primarily to the suppression of pair-breaking spin fluctuations that are seen in Pd.[226] A secondary cause (and the reason for the inverse isotope effect) is the presence of low-frequency hydrogen (or deuterium) optic modes which increase the electron-phonon coupling.[68,98] The Tc enhancement in thorium and aluminum is likewise explained as the result of low-frequency optic modes.[68,98]
In general it seems that the presence of metallic hydrogen bands seems to favor superconductivity and the presence of localized ionic hydrogen seems to suppress it. When the hydrogen bands have some metallic character, conduction electrons spend more time in the vicinity of the hydrogen ions and thus the electron-phonon coupling enhancement tends to be greater.[68] These two roles for hydrogen may be called the ``Proton Model'' and the ``Anion Model.''[233] This categorization is in reality an oversimplification: in most transition metals the hydrogen bands have some resemblance to both pictures.[233]
