On current density calculation Current density in semiconductors is often expressed as
$$
\mathbf{\vec j}=\mathbf{\vec j}_n+\mathbf{\vec j}_h
$$
where $\mathbf{\vec j}_n$ is the current density due to mobile electrons, and $\mathbf{\vec j}_h$ is the current density due to mobile holes. But why?
The current in an electric circuit is not given by the sum of current density going one direction, plus the current due to positive charges going the opposite direction: it's just the same thing, you'd calculate twice the expected value.
Why should this case be different? Isn't a hole created whenever an electron is promoted from valence to conduction band? And why just for semiconductors?
 A: 
The current in an electric circuit is not given by the sum of current density going one direction, plus the current due to positive charges going the opposite direction

The current in a conductor most definitely is given by the sum of the current densities of each charge carrier. The overall charge density is $$J=\sum_{i\in S} J_i$$ where $S$ is the set of all species of charge carriers in the conductor. In a metal the set of charge carriers is just $S=\{e^-\}$ so it is particularly simple, but in the electrolyte in a lead acid battery $S=\{ H^+, OH^-, HSO_4^-, Pb^{2+}\}$
So the general rule is indeed to add the current due to all species of charge carrier. Metals are just very simple since they have only one charge carrier. But other conductors or semiconductors are not as simple. In semiconductors the holes are a little weird, but they are legitimate charge carriers, distinct from the electrons. So they need to be accounted for separately.
A: $\vec j_n$ is the current density due to electrons
$\vec j_p$ is the current density due to holes
You have two different charge carriers contributing to current. The total current is, well, the total current. Add up all the parts to get the whole.
Contrary to what you wrote, the current in an electric circuit is given by the sum of current density due to negative charges going one direction, plus the current due to positive charges going the opposite direction. You just don't have positive charge carriers in most circuits you are thinking about.

Isn't a hole created whenever an electron is promoted from valence to conduction band?

That depends on how the electrons or holes are created. In a (non-degenerate) semiconductor in equilibrium you have $pn=n_i^2$. In all semiconductors except intrinsic semiconductors the electron and hole concentrations are different.

And why just for semiconductors?

You typically only have both positive and negative charge carriers in a semiconductor, so it doesn't typically come up outside semiconductors where you typically only have electrons.
Electrons and holes are not the same thing. A lot of people will try to tell you that holes are just missing electrons. That's a vastly oversimplified model that leads to confusion like this. Ignore those people. Electrons and holes are each their own separate individual things.
