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Holes are electrons, but with negative mass. That's said, so by applying electric field, electrons (n) move in the opposite direction of the field, while holes (other electrons) move in the same direction. Shouldn't they cancel each other?

We consider holes to be positive charged particles with positive mass, but that's only because they are negative charged particles with negative mass. We can still think of them as "normal" electrons cancelling each other if one's motion opposes another's, right?

Qmechanic
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Samuel Shokry
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    Where did you read that holes are electrons with negative mass? Please read https://en.wikipedia.org/wiki/Electron_hole and then reconsider your question. – my2cts Mar 05 '19 at 21:23
  • I guess I was misleaded by my prof's explanation somehow, I get it better now, Thanks. – Samuel Shokry Mar 05 '19 at 22:40
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    There is an analogy with bubbles falling upward in a gravitational field. So one could describe an air bubble in water as having negative mass. –  Jan 31 '20 at 20:04
  • You can describe the buble as having negative mass but not as water with negative mass. Water moves in the direction opposite to the upward motion of the buble, as do the electrons that move opposite the motion of the hole. In this case the driving force is gravity. The hole does not move opposite to the field due to negative mass but to effective positive charge. The efefctiv mass of the holes is a positive number for semiconductors. – nasu Jan 30 '21 at 03:43

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There are a few non-intuitive effects that lead to holes.

An entirely filled band does not contribute to current. This is because for every charged particle moving in one direction, there is another moving in the opposite direction that cancels it out. The conduction band is easy because it is nearly empty and we can treat those few electrons that are in it as "normal" electrons with a certain effective mass based on the curvature of the band near the bottom.

Current in a Mostly Filled Band
The valence band on the other hand is nearly completely filled. Mathematically we can model current in a mostly filled band as the current of an entirely filled band, which is equal to zero, minus whatever current would have been contributed by the unfilled states had they been filled.

States at the Top of a Band Have Negative Effective Mass
But holes don't act like electrons. You can tell holes and electrons apart using a Hall measurement, which wouldn't happen if holes were "just" electrons. So there has to be something else going on. Earlier I mentioned effective mass, electrons at the top of a band also have an effective mass driven by the curvature of the band in that region. At the top of a band the effective mass is negative. This applies equally to both filled and unfilled states. An electron at the top of a band has a negative effective mass and also an unfilled state at the top of a band has a negative effective mass. (Note: I didn't call them holes, holes are different)

Empty States Move In the Same Direction as Filled States
Okay, but now as you stated, you might think these should cancel out current with the conduction band electrons. But the final part that I think a lot of people easily miss (at least I did when I was first introduced to this) is how to think about the "unfilled" states. A lot of people like to describe holes using the bubble model, where you have a tube of liquid, the liquid moves one way and the bubble moves the other. This is NOT what happens in semiconductors. In a semiconductor filled and unfilled states move in the SAME direction. Its like a conveyor belt with boxes that are either filled or empty. No matter what is on the belt it moves in the same direction.

Holes
So we have negative effective mass unfilled states that move in the same direction as negative effective mass electrons. Put all of this together and you find that its exactly the same mathematically as positively charged and positive effective mass charge carriers called holes.

Matt
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  • How can a negative effective mass unfilled state, which is positive, move in the same direction as a negative effective mass filled state (electron)? – Shashank V M Jul 03 '21 at 14:00
  • A general comment on mass. I think it will be confusing if one says "X has negative effective mass" without some further explanation of how the term "mass" is being used. For example, it might be a way of talking about the dispersion relation, or the relation between energy and momentum, or it might be to do with force and acceleration, or all of the above. But it is better to spell out what you mean rather than just saying "X has negative effective mass". – Andrew Steane Jul 03 '21 at 14:12
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I would not definitely say holes are electrons with negative mass. Holes are called for the points in atom where there is a lack of electron. One can say that they are due to the absence of electrons. So as electrons have negative charge and move opposite direction as compared to the applied electric field, it's quite straightforward to understand the motion of the holes in a material.

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    But it is not the lack of electron that contributes to the current, it is the corresponding electron at the top of the valence band, which has negative mass. I guess I can imagine the motion of a hole, what confounds me is during current calculations we consider the net current to be equal to the sum of both hole and electron currents, though they are all electrons moving in two opposite direction. – Samuel Shokry Mar 05 '19 at 19:46
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    Yes, it is the lack of an electron that allows current. A full band does not conduct. Further, for semiconductors you’d better wrap your head around holes or device physics will make no sense. – Jon Custer Mar 05 '19 at 19:57
  • If the band is full, the motion of one electron is canceled by another's. But, when there is a lack of an electron, this means that there exists an electron which motion is not canceled by an electron from the same valence band, I understand that we call that electron a hole. Nevertheless, if there is an electron in the conduction band moving in the opposite direction, they should cancel each other again, but for some reason we don't assume that. Sorry if my explanation is confusing. – Samuel Shokry Mar 05 '19 at 20:14
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    @SamuelShokry Your explanation doesn't make sense. Have you read standard solid state physics books? Perhaps you could recast your question in light of the usual treatments; I have no idea where your "cancelation" is coming from, for example. – Raghu Parthasarathy Oct 01 '20 at 15:22
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In a classic picture, holes are conceptualized as the “absence of an electron” in an otherwise full electron system. The movement of holes is effectively achieved through electron motion as shown in Figure 3.17. This hole can be perceived as moving when another electron fills it. When an electric field is applied, electrons and holes move in opposite directions. Electrons, which are negatively charged, move in the opposite direction of the electric field. Conversely, holes are viewed as positive charge carriers (absence of electron) and move in the same direction as the electric field. enter image description here

from Neamen, D. A. (2012). Semiconductor physics and devices: Basic principles (4th ed.). McGraw-Hill.

However, in the context of energy band theory, holes and electrons actually move in the same direction under an electric field. This might seem counterintuitive given their opposite charges, but it can be explained through the concept of effective mass. In the energy band theory, the effective mass of a carrier is inversely related to the curvature of the energy band. For electrons, the effective mass is negative in the top of valence band (positive in the bottem of conductive band), which means their response to an electric field is as if they had positive charge. Thus, both holes (viewed as positive charge carriers) and electrons in the top of valence band move in the same direction when an electric field is applied.

An explanation provided by Wiki enter image description here

(from https://en.wikipedia.org/wiki/Electron_hole)

The net current in a semiconductor is the total effect of the electron near the bottom of the conduction band and the hole near the top of the valence band.

One perspective to understand this involves imagining a picture where only holes exist near the top of a valence band. This can be likened to the 'empty seat' analogy explained in Wiki. Here, the electrons responsible for the movement of holes do not follow the parabolic approximation of the energy band model (therefore do not move in the same direction as the electric field). This is similar to visualizing a crowded theater where people's (electrons) movement to occupy empty seats (holes) results in the apparent movement of the empty seat. The empty seat appears to move against the crowd, mirroring how a hole moves in the same direction as an electric field, which is opposite to the natural movement of an electron.

Sihang Li
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