How Do You Calculate Electrical Current Density with Complex Integrals?

[Denver Dang]

Homework Statement

Hi.

I'm having some trouble calculating the electrical current density, which in my case is given by:

$$\sigma =\frac{{{e}^{2}}}{4{{\pi }^{3}}}\int{\left( -\frac{\partial f}{\partial \varepsilon } \right)\tau \cdot \mathbf{v}\cdot \mathbf{v}}\,dk$$

Homework Equations

Lets assume that $\tau$ is constant, and that:

$$f=\frac{1}{\exp \left( \frac{\varepsilon -\mu }{{{k}_{B}}T} \right)+1},$$
the Fermi function.
And that:
$$\varepsilon =\frac{{{\hbar }^{2}}{{k}^{2}}}{2m}$$
and
$$v=\frac{\partial \varepsilon }{\partial k}$$

The Attempt at a Solution

Then, how am I supposed to do this integral ?
Exponential functions are so annoying if they are not alone.

I am supposed to end up with a value/number, but I really can't see how this is doable when I have all these exponential functions. When you differentiate the Fermi function, before putting it inside the integral, you get even more exponential functions.

So yes, I'm kinda lost here. So I was kinda hoping for someone who might be able to give a little hint :)Thanks in advance.

 

[clamtrox]

Use the chain rule. You have $v = \frac{1}{\hbar} \frac{\partial \epsilon}{\partial k}$, so $$v dk = \frac{1}{\hbar} \frac{\partial \epsilon}{\partial k} dk = \frac{1}{\hbar} d \epsilon$$ After that, I have no idea. You can try partial integration, which let's you dodge the derivative, although I didn't actually do the calculation so I don't know if it gets easier.

 

[Denver Dang]

Hmmm, I've actually narrowed it down to several integral, only one being hard, which is:

$$\int{\exp \left( \frac{\varepsilon +\mu }{{{k}_{B}}T} \right)}\,{{k}^{2}}dk,$$
where epsilon is the energy given in my first post.

Any clever way to solve this now ?