How can I calculate the chemical potential and the effective masses in an intrinsic semiconductor? [closed]

Question

I want to calculate the chemical potential and the effective masses of the germanium charge carriers considering that the relaxation time is the same for the two carriers: electrons and holes. Germanium has an energy gap $E_{\text{gap}}=0,68\, \mathrm{eV}$, its electron mobility being $\mu_{e}=3900\, \mathrm{cm^2/V\cdot s}$ and the mobility of the holes $\mu_{h}=1900\, \mathrm{cm^2/V\cdot s}$ to 300 K. In this case, $\tau_e=\tau_h$. I've been searching the Internet and discovered this expression for the chemical potential: \begin{equation} \mu(T)=\frac{1}{2}(E_c+E_v)+(K_BT)\frac{3}{4}\left(\frac{m_{h}*}{m_e*}\right) \end{equation}

I have already calculated the chemical potential according to the data that I have and obtained a relation between the effective masses. I want to know the effective masses of holes and electrons as being one digit times the mass of the electron. How can I find the masses effective? I'm missing some detail or some equation that I can not remember.

\begin{equation} \mu_e=\frac{e\tau}{m_e^{*}} \end{equation}

\begin{equation} \mu_h=\frac{e\tau}{m_h^{*}} \end{equation}

Putting together the two equations I get a relation between the effective masses:

\begin{equation} \frac{\mu_e}{\mu_h}=\frac{3900}{1900}=\frac{m_h^{*}}{m_e^{*}} \end{equation}

Substituting the relation in the chemical potential I obtain: \begin{equation} \mu(T)=\frac{1}{2}\times 0.68 + (8.617\times 10^{-5} \times 300)\times \frac{3}{4} ln\bigg(\frac{3900}{1900}\bigg)=4.74\times 10^{-3}\,eV \end{equation}

Answer 1

The effective mass can be calculated as

$$m^* = \frac{\hbar^2}{\left(\frac{\partial^2E(k)}{\partial k^2}\right)}$$

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Edited:

Normally, an energy level is identified by some quantum numbers: ${n, \mathcal{l}, s, j, m}$. For example: $n=1; l=0; m=0; s=½$.

When the levels "expand" into bands, those quantum numbers refer to a band. Not a level anymore. Now it is a band.

But, there are many energies inside one band. How to differentiate them?

Well, there is a new quantum number, called "quasi-momentum", $\vec{k}$. This $k$ identifies a concrete energy inside the band (the band is an interval, so we need something else to pick a concrete energy).

So, we would need to know the function $E(k)$. Then, apply the first formula, and you've got it.

BUT. The problem is that we need to solve the Schrödinger's equation for the whole solid if we want $E(k)$. And that is impossible. Solving for one ataom requires approximation. Imagine $1 mol$ of atoms. That's absolutely crazy.

So, we just cannot know $E(k)$ anallytically. However, we can know some things:

1. $E(k)$ is approximately parabollic near the ends of the band.
2. Consequently, the second derivative is constant in the borders of the bands.
3. That means that the effective mass is constant in the band borders. The effective mass varies withing the band, but it is constant at the borders.
4. And, that constant depends on the material. $Si$ has its values, $Ge$ has its values, $GaAs$ has its values.

In short: We cannot know $E(k)$, so we cannot know the effective masses; but we can find them experimentally. There's nothing wrong with working experimental data. You do it all the time!

When you use $G=6.67\cdot10^-11 Nm^2/kg^2$, you are using experimental data, and it works fine. Just substitute when you cannot avoid it more, and just that.