Do hall effect-based current sensors (clamps) always use a semiconductor as the material through which we apply the initial voltage difference and why?
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Yes.
Although you get the Hall effect in metals as well, the size of the Hall voltage is inversely proportional to the number of mobile charge carriers. Metals typically have an electron or two per atom in the conduction band. Semiconductors have only one mobile carrier per dopant atom, which is orders of magnitude fewer.
The Hall voltage in semiconductors is large enough to be useful, and have a decent signal to noise ratio when amplified. The Hall voltage in metals is much more difficult to measure.
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\$\begingroup\$ Note that the most common materials are even binary semiconductors because you need exceptionally high electron mobility. And then there is graphene. \$\endgroup\$– PlasmaHHFeb 8, 2017 at 15:07
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\$\begingroup\$ @PlasmaHH nocomprende. Are any of those words typos, or am I being dense? \$\endgroup\$– Neil_UKFeb 8, 2017 at 15:18
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\$\begingroup\$ The most suitable materials are GaAs, InAs, InP, and InSb, which are all binary semiconductors have electron mobilities from 8500-78000 cm²/Vs, whereas Si has ~1400. Graphene is out of that group with mobilities at the 200000 level. Metals are usually at the 50 level. And then in labs you can find two dimensional electron gasses in GaAs semiconductors at 3K with values in the many millions... \$\endgroup\$– PlasmaHHFeb 8, 2017 at 15:39
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\$\begingroup\$ @Neil_UK I can't seem to understand why is it that the size of the Hall voltage (by "size" you mean value I guess...) inversely proportional to the number of mobile charge carriers. Shouldn't it be the inverse! \$\endgroup\$ Feb 9, 2017 at 0:33
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\$\begingroup\$ Try the wikipedia article en.wikipedia.org/wiki/Hall_effect, the treatment for metals does lay out the carrier concentration ideas, equivalent to an MHDG (Magnetohydrodynamic Generator (try wikipedia again)) but the semiconductor section does rather gloss over the role of the mobilities. \$\endgroup\$– Neil_UKFeb 9, 2017 at 5:40