Ê by itself, I think, occurs only as an interjection (where you often have - across all languages - sounds that don't otherwise occur in words; compare the "tsk tsk" sound in English, which doesn't otherwise have clicks), so it's pretty marginal as a phoneme.
This is totally off-topic now, but I'm now wondering how these interjections are written in Chinese... for example, if you're doing a verbatim transcription of speech in English, you might have something like: "So then she did this, um, thing." "Tsk tsk, what was she thinking?" — is there a Chinese character for "um"? Or "tsk"? How are newly-minted sounds written, for example if I was trying to transcribe someone howling in a particular way? Like in English, I might write something like "aaaaaaooooooooohhhh".
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This is totally off-topic now, but I'm now wondering how these interjections are written in Chinese... for example, if you're doing a verbatim transcription of speech in English, you might have something like: "So then she did this, um, thing." "Tsk tsk, what was she thinking?" — is there a Chinese character for "um"? Or "tsk"? How are newly-minted sounds written, for example if I was trying to transcribe someone howling in a particular way? Like in English, I might write something like "aaaaaaooooooooohhhh".