Crossed Fields: The Hall Effect — Question 9

Question 9

A two-carrier material can have zero Hall voltage while conducting strongly

For a weak-field electron-hole model, take the supplied relations $\sigma=e(n\mu_e+p\mu_h)$ and $R_H=(p\mu_h^2-n\mu_e^2)/[e(p\mu_h+n\mu_e)^2]$. Here n,p are positive carrier densities and both mobilities are positive.

1. Find the condition for zero Hall coefficient with nonzero conductivity.

2. For $\mu_e=2\mu_h$, find p/n at the Hall null and the corresponding conductivity. Explain why a Hall null does not establish absence of carriers.

Question 9 - Solution

A two-carrier material can have zero Hall voltage while conducting strongly

(a) Method and calculation

$$R_H=0\Longleftrightarrow p\mu_h^2=n\mu_e^2\Rightarrow p/n=(\mu_e/\mu_h)^2.$$$$\sigma=e(n\mu_e+p\mu_h)>0\quad\text{for nonzero populations}.$$

(b) Interpretation and further calculation

$$\mu_e=2\mu_h\Rightarrow p=4n.$$$$\sigma=e(2n\mu_h+4n\mu_h)=6en\mu_h.$$

Physical check

Hall contributions carry mobility-squared weights, so carrier-number majority alone does not determine the sign in this supplied model.