According to George F. Black in his ‘Surnames of Scotland’, in the Highlands, Carmichael is a ‘translation’ or equivalent of MacGillmichael (MacMichael) meaning ‘son  of the servant of  Saint Michael’. The surname was at one time common on the Island of Lismore in Loch Linnhe, the McIlmichells of Lismore and Appin having changed their name to Carmichael.

Account of ‘Clann ‘Ic Illemhicheil’ [Carmichaels] 14 December 1864. (download)
From the Field and transcription notebook of Alexander Carmichael, dated 1864-67. The Carmichael Watson Project / Pròiseact MhicGilleMhìcheil MhicBhatair, University of Edinburgh. Proofread, and corrected to Gnàthachas Litreachaidh na Gàidhlig 2009/Gaelic Orthographic Conventions 2009, translated into English, and annotated by Mìcheal Bauer, 7 July 2019.