  High pressure serves as a path finding tool towards novel structures,
including those with very high Tc.The superconductivity in sulfur hydrides with
record value (203 K) is caused by the phonon mechanism. However, the picture
differs from the conventional one in important ways. The phonon spectrum in
sulfur hydride is both broad and has a complex structure. High value of Tc is
mainly due to strong coupling to the high-frequency optical modes, although the
acoustic phonons also make a noticeable contribution. New approach is
described;it generalizes the standard treatment of the phonon mechanism and
makes it possible to obtain an analytical expression for Tc . It turns out
that, unlike in the conventional case, the value of the isotope coefficient
varies with the pressure and reflects the impact of the optical modes. The
phase diagram, that is the pressure dependence of Tc, is rather peculiar. A
crucial feature is that increasing pressure results in a series of structural
transitions, including the one, which yields the superconducting phase with the
record Tc.In a narrow region near 150GPa the critical temperature rises sharply
from 120K to 200K. The sharp structural transition, which produces the high Tc
phase, is a first-order phase transition caused by interaction between the
order parameter and lattice deformations.Remarkable feature of the electronic
spectrum in the high Tc phase is the appearance of small pockets at the Fermi
level. Their presence leads to a two-gap spectrum, which can, in principle, be
observed, with the future use of tunneling spectroscopy. This feature leads to
non-monotonic and strongly asymmetric pressure dependence of Tc. Other hydrides
can be expected to display even higher values of Tc, up to room temperature.
The fundamental challenge lays in creation a structure capable of displaying
high Tc at ambient pressure.

Superconductivity