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I am currently enrolled in a solid state physics course, and have just completed a lecture on the Hall effect, though it did not go into too much detail. It was mentioned that with van der pauw geometry, material properties such as the Hall coefficient, and carrier mobility could be obtained from sample materials, but that it was most accurate for materials in which a single charge carrier type dominated. From what I understood

Applying a current through a sample material, placed within a perpendicular magnetic field, would cause the charge carriers to experience the Lorentz force, and thus their path would curve. This means the mean free path of the charge carrier increases and thus the resistivity of the material increases with increasing magnetic field (magnetoresistance). However, for sufficiently long sample materials, since the charge carriers have a curved path, they essentially build up on one wall of the material, creating a potential difference between the two sides of the material, perpendicular to both the original current and the magnetic field. This new potential essentially "straightens out" the path of the charge carriers, thereby negating the magnetoresistive effects mentioned earlier. The lecture notes then go on to say that this is why in a single-carrier model, magnetoresistive effects are negligible.

My question now is why is this the case only for a single-carrier model, and why does the presence of measurable magnetoresistance indicate the need to use a two/multi-carrier model? Surely in the case of two charge carriers (e.g. negatively charged electrons and positively charged holes), the Lorentz force causes the electrons to curve in one direction and thus build up negative charge on one side of the material, and the holes curve in the other direction and build up positive charge on the opposite side of the material? As a result, and even larger potential difference builds up across the sides of the material, which has an even stronger "straightening" effect on the charge carriers? Thus the magnetoresistive effect should be negated here as well?

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It sounds to me like there is a misunderstanding of the term single carrier. In the metal, only the electrons are mobile and holes are created by a static redistribution of the charges due to an electrical potential difference (i.e. your test body is quasi between the two plates of a capacitor).

In the present case, however, a current is applied, which causes an inflow and outflow of the electrons, whereby the following effects result because of the applied magnetic field:

  • Hall effect, i.e. the movement of the electrons sideways
  • lengthening of the path of the electrons through the material
  • Increase of the measured resistance because of the more frequent collisions of the electrons due to the longer path. And
  • Filling up the holes with electrons of the current with the movement of the electrons straight through the material.

Explain. On the left is the accumulation of electrons due to the Hall effect. On the forced way there the way of the electrons through the material is prolonged. On the right the holes which are constantly filled up. Here the motion trajectory of the electrons flattens. So on the right there will be electrons whose trajectory from hole to hole is less curved than on the left.

But maybe I misunderstood your explanation and it is clear from one comment what is meant by single carrier.

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