Q2Electrical Materials
Question
Q.2. What is Hall effect? Derive the relation between hall coefficient and carrier density.
Answer
The Hall effect is the development of a transverse voltage across a current-carrying conductor placed in a perpendicular magnetic field; balancing the magnetic and electric forces on carriers yields the Hall coefficient RH = 1/(nq), inversely proportional to carrier density.
The Hall effect is the appearance of a transverse (sideways) voltage — the Hall voltage — across a conductor or semiconductor carrying current I in a magnetic field B applied perpendicular to the current direction. Moving charge carriers experience the Lorentz force F = qv×B, which deflects them toward one side face of the specimen; charge accumulates on that face until the resulting transverse electric field EH exerts an equal and opposite force on subsequent carriers, establishing a steady state.
Derivation of the Hall coefficient: consider a slab of thickness t and width w carrying current I along x, with B along z. In steady state, the magnetic force on a carrier equals the electric force from the Hall field:
The drift velocity relates to current density by J = nqv_d, so v_d = J/(nq). Substituting:
The Hall coefficient is defined as RH = EH/(JB), giving:
Thus the Hall coefficient is inversely proportional to the carrier density n, and its sign follows the sign of the charge carriers (negative for electrons, positive for holes). In terms of measurable quantities, the Hall voltage is VH = EH·w = IB/(nqt). Measuring VH therefore directly yields both the carrier type (from polarity) and the carrier concentration n = IB/(qtVH), which is why the Hall effect is the standard laboratory method for characterizing semiconductors — determining whether a sample is N-type or P-type, its carrier density, and (combined with a resistivity measurement) the carrier mobility.