睡

Left eye radical, a vertical pictograph of an eye (originally drawn horizontally, then rotated). It supplies the meaning: an action involving the eyes, here their closing. 睡 is the eyelids drooping shut - sleep as an eye-event rather than a body-event. The radical anchors the seeing-and-watching family: 看, 眼, 睛, 盯, 瞧, 眠.

Right 垂 means 'to hang down, droop' - drawn as branches drooping from a tree. Combined with the eye on the left, the whole forms a compound ideograph: drooping eyelids, the visible image of falling asleep. 垂 is semantic here, not phonetic - chuí does not match shuì, though they are graphically close. Same family: 唾, 锤, 陲.