How could a science fiction book explain the Earth's half dark/half light state?

Actually, there are planets for which this is the case, but it's not because the planet has stopped. If the earth were not rotating (but still revolving around the sun) a day would be equal to a year. If the earth were rotating slowly at the perfect rate, it could be as if the sun weren't moving in the sky at all. This is the same as why we always see one side of the moon even though the moon is rotating and revolving. Imagine that the moon is the earth and the earth is the sun: the so-called dark side of the moon would truly be in eternal night. This is all totally plausible and is known to happen in the real world.