What is band bending and how does it relate to Fermi Energy?

[HunterDX77M]

Homework Statement

Consider a pn junction in Si at 300K (other parameters given), with doping N_A = 10²¹/m³ and N_D = 10²³/m³. Assume all impurities are ionized. On this basis find the Fermi level on each side. From this find the band bending V_B and make a sketch of the pn junction.

Homework Equations

$N_e = N_C e^{\frac{-(E_G - E_F)}{k_B T}}$

The Attempt at a Solution

The Fermi energy calculation was fairly straightforward to solve for, since I just used the formula above for both sides and solved for E_F. My question is about band bending. What is it and how do I calculate it? I looked through the relevant chapter in my text-book, but I couldn't find any reference to it. Can someone show me how it relates to the Fermi energy that I have already calculated?

 

[Simon Bridge]

My question is about band bending. What is it and how do I calculate it?

Is usually covered in your course textbook

I looked through the relevant chapter in my text-book, but I couldn't find any reference to it.

Then the chapter you looked in was not relevant to "band bending" ... look back to where it talks about how energy bands form in the first place - conduction and valence bands etc. Then read forward until you see diagrams of these bands being bent - usually where it starts talking about P-N junctions.

Basically - different materials will have energy bands at different energies.
The bands want to be continuous. The only way this happens for two different materials close enough together for electrical contact is if the bands bend in some way. This usually means that charge carriers get trapped close to the junction or something like that.

See also:
https://www.physicsforums.com/showthread.php?t=626885
https://www.physicsforums.com/showthread.php?t=639129

 

[HunterDX77M]

I found the following two equations in my lecture slides. Due to notational differences between my textbook and the lecture slides, I'm not sure if the variable E_C represents the band gap energy (which is known in this problem). I am assuming that E_FN is the Fermi level energy.

$E_{FN} = E_C - k_B T \times ln(N_C/N_e) \\ E_C - k_B T [ln(N_C/N_e) + ln(N_V/N_h)] = eV_B = E_{FN} - k_B T \times ln(N_V/N_h)$

If E_C, is the band gap energy does this look like the correct relationship between Fermi Energy and the band bending V_B?