Dependence of Hall coefficient on temperature in semiconductors

[Roger Dalton]

Homework Statement

By measuring the electrical conductivity and the Hall effect as a function of temperature, many characteristic parameters of semiconductors can be determined. Since charge transport in semiconductors takes place through both electrons in the conduction band and holes in the valence band, the two-band model expression for the Hall coefficient must be used.

(a) By measuring the temperature dependence of the Hall coefficient, how can the gap Eg of an n-type semiconductor be determined, as well as the distance Ed from the donor level to the edge of the conducting band? And for a p-type semiconductor, how can the distance Ea from the acceptor level to the valence band edge be determined?

(b) Can the nD density of the donors in an n-type semiconductor or the nA density of the acceptors in a p-type semiconductor be determined by measuring the Hall effect? If so, over what temperature range should the measurement be made?

Relevant Equations

The Hall coefficient for a semiconductor, in terms of the mobilities and the densities of the charge charriers (electrons and holes), is given by:

$$R_H = \frac{1}{e} \frac{p_v\mu_h^2-n_c\mu_e^2}{(p_v\mu_h+n_c\mu_e)^2}$$

Where $\mu_h$ and $\mu_e$ are the mobilities for holes and electrons respectively, and $p_v$ and $n_c$, their respective densities as well.

My first assumption is that the temperature dependence on the mobilities can be neglected, and so we would have:

$$R_H(T)= \frac{1}{e} \frac{p_v(T)\mu_h^2-n_c(T)\mu_e^2}{(p_v(T)\mu_h+n_c(T)\mu_e)^2}$$

The expression for the electron and hole densities could be derived from

$$\frac{n_c(n_c+n_A)}{n_D-n_A-n_c}=n_c^{eff}(T)e^{-E_d/k_BT} (1)$$
$$\frac{p_v(p_v+p_A)}{p_D-p_A-p_v}=p_v^{eff}(T)e^{-E_a/k_BT} (2)$$

Where $E_d$ is the ionisation energy it takes to extract an electron from the donor state to put it into the conduction band, $E_a$ is the ionisation energy it costs to extract a hole to put it in the valence band, and $n_D$, $p_D$, $n_A$, and $p_A$ are the densities of the donor and acceptor densities depending if they are electrons or holes. In this case, we can neglect both $n_A$ and $p_D$.

$$n_c^{eff}(T)=2\left(\frac{m_e^*k_BT}{2\pi\hbar^2}\right)^{3/2}$$
$$p_v^{eff}(T)=2\left(\frac{m_h^*k_BT}{2\pi\hbar^2}\right)^{3/2}$$

Solving (1) and (2) in terms of $n_c(T)$ and $p_v(T)$ and pluging the final expressions into the one for the Hall coeffcient would give us the dependence of Hall coefficient on temperature. Is this approach correct? The thing is that using this approach I do not know what the donor and acceptor densities are.

Besides, how could I calculate the energies $E_a$, $E_d$ and the energy gap of the semiconductor $E_g$ from the final expression of the Hall coeffcient?

Would it also be possible to find the donor and acceptor densities from the Hall coefficient $R_H (T)$ final expression?

Thank you.

 

[member38644]

Your approach to finding the temperature dependence of the Hall coefficient in semiconductors is generally correct. However, there are a few things to keep in mind:

1. Neglecting temperature dependence of mobilities: This assumption is valid only at low temperatures, where the temperature dependence of mobilities is small. At higher temperatures, the mobilities can vary significantly and should be taken into account in the expression for Hall coefficient.

2. Neglecting acceptor and donor densities: In some cases, the acceptor and donor densities can be neglected, but it is not always the case. These densities can have a significant impact on the overall behavior of the semiconductor and should be considered in the calculations.

3. Calculating energies and energy gap: The energies $E_a$ and $E_d$ can be calculated from the expression for Hall coefficient by making certain assumptions about the donor and acceptor densities. The energy gap $E_g$ can also be calculated from the Hall coefficient, but it is a more complex process and may require additional information about the semiconductor material.

4. Finding donor and acceptor densities: It is possible to calculate the donor and acceptor densities from the Hall coefficient, but it is a challenging task. It requires a detailed understanding of the semiconductor material and the assumptions made in the calculations. It is always better to have direct measurements of these densities rather than relying on the Hall coefficient.

In conclusion, your approach is correct, but it is important to consider the limitations and assumptions involved. It is always better to have direct measurements of the relevant parameters rather than relying on indirect calculations.