The Hall Effect can be used to measure the velocity of charge carriers (moving electrons or protons) in a material, if enough other information is known (magnetic field, width of the material, and Hall voltage)
Added by Donald L.
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Let current flow along the x direction, the magnetic field B be applied along the z direction (perpendicular to the current), and the sample width across which the Hall voltage is measured be w (distance along y). The measured Hall voltage is V_H (voltage Show more…
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The Hall effect is to be used to find the sign of charge carriers in a semiconductor sample. The probe is placed between the poles of a magnet so that magnetic field is pointed up. A current is passed through a rectangular sample placed horizontally. As current is passed through the sample in the east direction, the north side of the sample is found to be at a higher potential than the south side. Decide if the number density of charge carriers is positively or negatively charged.
-?. The Hall effect and the sign of charge carriers. Consider a rectangular conductor of width $w$ and thickness $t$ carrying a current $I$ in a uniform magnetic field $\mathbf{B}$ perpendicular to the plane of the conductor, as shown in Fig. $14.45 .$ The charge carriers, with charge $q$ and drift velocity $\mathbf{v}$, experience a Lorentz force $\mathbf{F}=q \mathbf{v} \times \mathbf{B}$, which tends to push them to one side of the conductor. The resulting steady-state charge buildup causes a voltage difference $V_{\mathrm{H}}$, called the Hall voltage, between the sides of the conductor. (a) Show that the Hall voltage is given by $$ V_{\mathrm{H}}=\frac{B I}{n q t} $$ where $n$ is the concentration of charge carriers. [Hint: In steady state, there is no net force on the charge carriers transverse to the current direction, so the force due to the transverse $E$ field must be canceled by the force from the $B$ field.] (b) Show, with an appropriate diagram, that the sign of the Hall voltage gives the sign of the charge carriers. Thus, the Hall voltage allows one to determine whether the carriers in a doped semiconductor are electrons or holes. (c) Compute the magnitude of the Hall voltage in an $n$ -type semiconductor sample at room temperature with donor concentration $$ \begin{aligned} &N_{\mathrm{d}}=10^{22} \mathrm{~m}^{-3} \text { and the following parameters: } t=1 \mathrm{~mm} \text { , }\\ &I=0.01 \mathrm{~A}, B=1.0 \mathrm{~T} \end{aligned} $$
Solids-Applications
Semiconductors
The Hall effect finds important application in the electronics industry. It is used to find the sign and density of the carriers of electric current in semiconductor chips. The arrangement is shown in Figure $\mathrm{P} 22.66 .$ A semiconducting block of thickness $t$ and width $d$ carries a current $I$ in the $x$ direction. A uniform magnetic field $B$ is applied in the $y$ direction. If the charge carriers are positive, the magnetic force deflects them in the $z$ direction. Positive charge accumulates on the top surface of the sample and negative charge on the bottom surface, creating a downward electric field. In equilibrium, the downward electric force on the charge carriers balances the upward magnetic force and the carriers move through the sample without deflection. The Hall voltage $\Delta V_{\mathrm{H}}=V_{c}-V_{a}$ between the top and bottom surfaces is measured, and the density of the charge carriers can be calculated from it. (a) Demonstrate that if the charge carriers are negative the Hall voltage will be negative. Hence, the Hall effect reveals the sign of the charge carriers, so the sample can be classified as $p$ -type (with positive majority charge carriers) or $n$ -type (with negative). (b) Determine the number of charge carriers per unit volume $n$ in terms of $I, t, B$ $\Delta V_{\mathrm{H}},$ and the magnitude $q$ of the carrier charge.
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University Physics with Modern Physics
Physics: Principles with Applications
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