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?

• In a metal you have free electrons moving around without the need for holes. There's no band gap in a metal.
– hft
Commented Jun 3, 2022 at 18:10
• I learned that in metals the Fermi level is located at half the band (in alkali metals for instance), so... why can't the free part be considered as a bunch of holes? Commented Jun 3, 2022 at 18:13
• When a metal wire carries a current, you are basically putting a bunch of physical electrons on the surface of the wire and moving them around. You can't do the same thing with holes. Electrons are real physical elementary particles and you can plop a bunch of them down on a piece of metal and move them around. Holes are a construct that only makes sense in the context of many-particle condensed systems. You can't exactly manipulate them in the same way. Like, how would you put a bunch of holes down onto a piece of metal? I don't know how.
– hft
Commented Jun 3, 2022 at 18:29
• But, I see what you are wondering about. It seem that you are wondering, why can't I just lift one electron out of the middle of the band and then I created one hole in the middle of the band. It just doesn't work that way, partly because band theory is a very rough approximation that only works because of some miracle. (Why would a non-interacting theory of electrons and atoms ever work in the first place?) In a semiconductor you have a band gap and can use light to pop one electron out and create one hole and the hole isn't immediately filled. It just doesn't work like that for a metal.
– hft
Commented Jun 3, 2022 at 18:32

$$\vec j_n$$ 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.

• Ok, but... what are holes then? What I'm trying to understand is why $\mathbf{{\vec j}}_n\neq \mathbf{{\vec j}}_h$. Commented Jun 3, 2022 at 18:49
• @ric.san In semiconductors the things we call "electrons" and the things we call "holes" are Bloch states of the crystal lattice. They arent "normal" electrons. They are solutions to the Schrodinger equation for electrons in a periodic potential. Because of where they are in the electronic band structure they have significantly different behavior from each other and especially for holes leads to some very unintuative behavior.
– Matt
Commented Jun 4, 2022 at 22:22

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.