The character 眠 is a combination of 目 and 民. 眠 has the meaning of sleeping, as in “falling asleep”.
The Japanese-based pronunciation for 眠 is ねむ. The Chinese-based pronunciation of 眠 is ミン, same as 民.
As a verb,
Examples of 眠 read as ねむ
Is the Order a Rabbit? » Volume 1 » Page 63 (Left) » Panel 3
While deciding which drink everyone should order, Syaro recommends lavender because Rize had said she hasn’t been able to sleep lately.
Examples of 眠 read as ミン
Is the Order a Rabbit? » Volume 1 » Page 82 (Left) » Panel 2
After seeing strange behavior by Chino earlier in the day, Chiya e-mailed Cocoa to be sure Chino gets enough nutrition and sleep. 睡眠 combines 睡 and 眠, which both mean sleep. “To take a nap” would be 睡眠をとる, such as used in Chiya’s e-mail here.