  The physics of an electron solid, held on a cryogenic liquid surface by a
pressing electric field, is examined in a low-density regime that has not been
explored before. We consider the effect of the pressing field in distorting the
surface at the position of each electron and hence inducing an attractive force
between the electrons. The system behavior is described in terms of an
interplay between the repulsive Coulomb interaction and the attractive
surface-induced interaction between individual electrons. For small densities
and large enough pressing fields, we find a parameter regime where a square
lattice is more favorable than the usual triangular lattice; we map out the
first-order transition curve separating the two lattice geometries at zero
temperature. In addition, our description allows an alternate static
perspective on the charge-density wave instability of the system, corresponding
to the formation of multi-electron dimples.
