Why can Energy Bands Bend?

[thegreenlaser]

My only exposure to energy bands is coming from courses on semiconductor devices (diodes and transistors), and I only understand the very basic theories of quantum mechanics (I've only taken a course which briefly introduced it). The thing is, we keep using energy band diagrams to explain and derive the properties of diodes and transistors in my circuits courses, so I'm really trying to understand them. Band bending is one thing that gets used a lot, and I can't seem to make sense of it.

From what I understand, each discrete energy level in the band corresponds to some particular energy state that only two electrons may occupy because of the Pauli exclusion principle. This is why I don't understand why it makes sense to "bend" the energy band. Say there's only one electron occupying an energy level. When the energy level bends (e.g. due to an applied electric field), does that mean the electron's energy is now position dependent? Can it "slide" along that energy level, gaining/losing energy in a continuous fashion? That doesn't seem right to me based on what I've seen of quantum mechanics where particle energy always seem to be quantized, but again, my knowledge is very limited in that sense. I guess I just don't really understand why the energy of a given energy level would be position dependent.

 

[marvincwl]

can u clarify what you mean by "bending"

 

[M Quack]

Look at a free particle (e.g. electron). The energy "band" bends:

E=p^2/2m.

where p = hbar k. That is just the ordinary kinetic energy.

In a semiconductor or metal things become more complicated because your electrons move through a periodic potential and interact with each other.

You only get discrete, flat energy levels when the electrons cannot move freely, such as bound electrons in an atom or molecule, or a quantum well.

 

[Cthugha]

From what I understand, each discrete energy level in the band corresponds to some particular energy state that only two electrons may occupy because of the Pauli exclusion principle.

No, we are not discussing single isolated atoms or molecules here, but solids. Therefore the bands form a quasicontinuum of allowed states. While one may argue that the allowed states inside the band are technically discrete, the energy spacings are VERY tiny, so the allowed energy bands are typically treated as a continuum and the number of allowed states per energy is typically treated in terms of a density of states. Are you familiar with the basics of how energy bands form in solids?

This is why I don't understand why it makes sense to "bend" the energy band. Say there's only one electron occupying an energy level. When the energy level bends (e.g. due to an applied electric field), does that mean the electron's energy is now position dependent? Can it "slide" along that energy level, gaining/losing energy in a continuous fashion?

Band bending typically occurs at interfaces between two materials, like Schottky contacts, where they are (at least in my opinion) easiest to understand. This is basically a question of thermodynamics. If you have a metal-semiconductor junction, the Fermi level of the metal (which can be roughly pictured as the highest occupied energy state in the metal in case you are not familiar with that) must "line up" with the chemical potential in the semiconductor in thermal equilibrium. If there was a difference, some flow of particles across the junction would occur until the difference disappears. The bands then just follow up and the fact that such a junction introduces a spatial variation of the energy bands might seem somewhat more intuitive. Introducing a bias voltage is in principle not too different from having a junction as it also just introduces a spatial dependence.

edit:

Look at a free particle (e.g. electron). The energy "band" bends:

E=p^2/2m.

where p = hbar k. That is just the ordinary kinetic energy.

This is of course correct, but this intrinsic bending of the dispersion in momentum space is usually not implied with the technical term band bending.

 

[DrDu]

Let me try to give a somewhat intuitive picture:
The idealized band picture applies only to infinite periodic arrays of atoms. But e.g. a pn junction is not periodic. However if electon densities change not too rapidly near the junction you may consider the junction of being made up of homogeneous cubes which are still large on a microscopic scale and iside which the potential seen by the electrons is approximately periodic. You can calculate the band structure for each of this cubes and plot it as a function of their distance from the junction.
Then you get the picture of the bending gaps.

 

[thegreenlaser]

No, we are not discussing single isolated atoms or molecules here, but solids. Therefore the bands form a quasicontinuum of allowed states. While one may argue that the allowed states inside the band are technically discrete, the energy spacings are VERY tiny, so the allowed energy bands are typically treated as a continuum and the number of allowed states per energy is typically treated in terms of a density of states. Are you familiar with the basics of how energy bands form in solids?

I'm not entirely sure what you mean by the 'basics,' so I would guess probably not. All I really know is that bringing atoms close together causes wavefunctions interact so that the energy levels 'split.' Bringing lots of atoms close together means there are a bunch of energy levels which are very finely spaced.

I do understand that it looks like a continuum, but my impression was that it's still made of of discrete lines which run from one end of the band to the other. Kind of like a bundle of wires. The sense I'm getting from your post and DrDu's post is that this isn't really the case. So in a pn junction, a given energy level doesn't span the entire junction, it's 'localized' in a sense? Is this the correct way of thinking about it? I apologize if I'm not drawing the correct conclusion.

 

[DrDu]

In principle you could construct bands for the whole macroscopic structure. These bands would span the whole range of the "bent bands" one usually considers and you could learn much less from them (besides them being difficult to calculate). You are completely right in that the bent bands include some localization on a macroscopic scale. You may think of wavepackets (or, to use a more fashionable term, wavelets) with a very small uncertainty in energy but a macroscopically well defined localization.

 

[ZapperZ]

I think we are going in two different directions here.

I see two separate questions: (i) is the band "continuous" and (ii) how does a band bend?

The band is really continuous. This is because the band diagram that you have been using has been "momentum-averaged", so you are only mapping the energy band diagram as a function of distance of the material (inside and outside). The full band diagram is a "spaghetti" of lines that includes the crystal momentum. When you integrate out the momentum, you get your familiar band diagram.

The bending is simpler than what you think. Think of all the energy levels in the band diagram as being similar to potential energy lines. So the Fermi level, for example, can represent a reference potential for the material (that's why sometime we use the term chemical potential). And we all know from classical mechanics or classical E&M that potential energy contour can change, curve, bend, etc. depending on the topology of the potential field. So if I imbed positive ions on the surface of a material, for example, that will affect the potential field of the material and causes the field lines to change (i.e. bend).

Zz.