Crossed Fields: The Hall Effect — Question 10

Question 10

A heating limit changes the thickness scaling of Hall sensitivity

A one-carrier strip has resistivity rho, length L, width w and thickness t. Its longitudinal resistance is rho L/(wt). Maximum allowed Joule heating is Pmax. The field B, density n and charge magnitude e are fixed; contacts are ideal.

1. Derive the maximum permitted current and the largest Hall-voltage magnitude under the heating limit.

2. If t is halved with all other quantities fixed, compare the new maximum current and maximum Hall voltage with the old values.

Question 10 - Solution

A heating limit changes the thickness scaling of Hall sensitivity

(a) Method and calculation

$$I_{\max}=\sqrt{\frac{P_{\max}}R}=\sqrt{\frac{P_{\max}wt}{\rho L}}.$$$$|V_H|_{\max}=\frac{BI_{\max}}{net}=\frac B{ne}\sqrt{\frac{P_{\max}w}{\rho Lt}}.$$

(b) Interpretation and further calculation

$$t\prime=t/2\Rightarrow I_{\max}\prime=I_{\max}/\sqrt2.$$$$|V_H|_{\max}\prime=\sqrt2|V_H|_{\max}.$$

Physical check

At fixed current, halving thickness doubles the voltage; under fixed allowable heating, current must also decrease.