# Can the Schrödinger equation only give eigenvalues greater than the Fermi level in a semiconductor potential well?

I'm an undergrad with no direct coursework dealing with solid state physics so please forgive misunderstandings.

Say you have a well of semiconducting material surrounded by an infinite potential barrier.

If there is no voltage applied, the valence band is full and the conduction band is empty. Solving the Schrödinger equation using $$E_c = 0$$ as a reference point and using the appropriate effective mass of an electron in the material produces available wavefunctions that are in the conduction band, but have essentially 0 probability of being occupied since the eigenvalues are above the Fermi level.

In some instances, I gather that you should be able to apply a voltage $$V$$ to the semiconductor to raise the Fermi level into the conduction band. In that case, that should also create a potential energy landscape $$U = -V$$, so shouldn't the solutions to the Schrodinger equation also be offset by that potential $$U$$, meaning that there are still no aailable wavefunctions which are thermally accessible?

Edit: let me try to rephrase because I don't think I was being clear. When you apply a gate bias voltage $$V_{\text{gate}}$$ to a semiconductor, you increase the potential inside the semiconductor by approximately $$V_{\text{gate}}$$. I've been assuming that this increases the Fermi level by $$V_{\text{gate}}$$ based on a paper I'm working with.

Essentially, the Schrodinger equation says that all states in a potential well have energy greater than the minimum potential in the well, but Fermi-Dirac says that those states are all unlikely to be occupied since the Fermi level is equal to the potential in the well. What am I missing?

• A pure semiconductor at T=0 (and often even at room temperature) is an isolator. Heating, irradiation with light and ionizing radiation, doping, surface defects and probably a few more "external effects" that I can't remember right now can move charge carriers into the conduction band. Is that what you are asking? The point of doping is to control the gap by making significantly smaller than the natural gap. Apr 14 at 0:12
• Your question is kinda weirdly written. Schrödinger's equation can give you the energy eigenvalues of the filled states too, i.e. all the negative energy levels below the Fermi level too. Also, valence band is full and conduction band is empty (you wrote wrongly). Apr 14 at 7:50
• replying to FlatterMan, does that mean that electric potential alone can't move charges into the conduction band, and in a simple model that is only possible through thermal excitation or external effects?
– fhh
Apr 14 at 15:40
• replying to naturallyinconsistent, I understand that you can get solutions for valence electrons below the Fermi level, but how can you get filled conducting states with the Schrodinger equation? If you model conducting electrons as particles in a potential well, eigenvalues are always greater than the minimum potential in the well, so they are higher than the Fermi level. Is it just the case that conducting states are only filled through thermal excitation or other effects?
– fhh
Apr 14 at 15:51
• @fhh didnt get a notif of you replying that way. Fermi level is far higher than the bottom of the potential well. Have you also considered thermal excitation so that conduction states are filled (a little) and valence states a little unoccupied? Apr 15 at 3:09

Also, there is no reason to suppose that the Fermi level is the bottom of the potential well. That is given by the conduction band energy $$E_c$$.