  NaxCoO2 has emerged as a material of exceptional scientific interest due to
the potential for thermoelectric applications, and because the strong interplay
between the magnetic and superconducting properties has led to close
comparisons with the physics of the high-Tc cuprates. The density, x, of the
sodium in the intercalation layers can be altered electrochemically, directly
changing the number of conduction electrons on triangular Co layers. Recent
electron diffraction measurements reveal a kaleidoscope of Na+ ion patterns as
a function of concentration. Here we use single-crystal neutron diffraction
supported by numerical simulations to determine the long-range
three-dimensional superstructures of these ions. We show that the sodium
ordering and its associated distortion field are governed by pure
electrostatics, and the organizational principle is the stabilization of charge
droplets that order long range at some simple fractional fillings. Our results
provide a good starting point to understand the electronic properties in terms
of a Hubbard Hamiltonian taking into account the electrostatic potential from
the Na superstructures. The resulting depth of potential wells in the Co layer
is greater than the single-particle hopping kinetic energy. As a consequence,
holes occupy preferentially the lowest potential regions and, therefore, the
Na+ ion patterning plays a decisive role in the transport and magnetic
properties.
