# Why absence of electron is called hole?

I am having hard time in understanding the concept of holes:

1. If there is no electron than how can it be a hole?

2. For a moment lets assume absence of electron is termed as hole but how can this absent particle have mass?

3. Under equilibrium no electrons are present in conduction band. Why cant we termed this absence as holes in conduction band?

1. The absence of a material in a volume embedded into a larger volume that is fully filled by the material is always known as the hole. For example, look at this hole in soil. In semiconductors, the "soil" is replaced by the semiconductor material itself, with the right number of electrons per nucleus to make it neutral. So if there is an electron missing relatively to the expectations, it is a positively charged hole. The positive charge "microscopically" comes from the nuclear protons that are not "cancelled" but we are assuming that the nuclei together with all the electrons behave like a simple environment, like soil or the vacuum, so we may measure charges relatively to that.

2. Holes have (positive) mass and many other features analogous to the electrons themselves because there exists (at least qualitatively) a symmetry that replaces all occupied electron states by unoccupied, or vice versa. For fermions, the occupation numbers are only $N=1$ or $N=0$ for a given state, and $N\leftrightarrow 1-N$ maps this set of possibilities onto itself. That's why the exchange of electrons with holes must keep the formalism pretty much unchanged. A careful scrutiny of signs implies that the holes have a positive inertial mass, the $m$ from the kinetic energy term $p^2/2m$. That's effectively because holes like to exist with momenta $p$ near the maxima of a function of $p$ (this momentum parameterizes states) – the opposite than electrons – so one would get a negative mass but this sign is changed once again because we're talking about holes i.e. absence of electrons.

3. The states in the empty conduction band aren't called "holes" because the "fully filled state" (recall the soil analogy) isn't allowed. By definition, the conduction band is empty at rest. If there's no soil, there can't be a hole in the soil, either.

• I got your first and third point but still have difficulty in understanding second point. Could you please elaborate this point in simple language? – johndaniel Feb 9 '14 at 9:09
• Dear @johndaniel, I am afraid that the answer is No. One may only fully understand why holes behave as particles if he understands and applies some maths of quantum mechanics. If you want a classical picture, it just doesn't work well. You may still think of bubbles in water - they also move similarly to other objects even though they are holes, missing water - but for things like whether the mass is positive or negative etc., the classical picture is useless. – Luboš Motl Feb 9 '14 at 9:59
• @LubošMotl : Can you please explain your answer in terms of field theory, if at all it is involved (I was planning to ask a separate question for that). – user35952 Feb 9 '14 at 13:52
• My question: are holes in this context localized? I thought that "holes" existed "below" the conduction band - meaning those locations don't move because they exist w.r.t. a specific impurity, so a hole is like an unoccupied orbital. This is a good answer, I'm sure I'm wrong on some account but don't understand why. On @johndaniel's point, might this be similar to a helium balloon in a train car that moves toward the back of the car when it brakes? – Alan Rominger Feb 13 '14 at 0:56

In an idealized semiconductor at absolute zero, all the valence states are occupied by electrons, and all the conduction states are empty. When you take one electron an place it into the conduction band, you leave behind a state that is no longer occupied in the valence band. Now, lets say you have a sample with a billion electons, all in the valence band, and you pop one into the conduction band. Figuring out how that one electron can move about in the conduction band is easy - there isn't anything else there to worry about. For the valence band, you could now worry about how the 999,999,999 electrons move, or you could instead say, wow, there is one empty state in a sea of full states - wouldn't it be easier to figure out how the empty state moves and focus on just that one? This is a common technique in physics, be it called renormalization, or quasi-particles, or whatnot - reframe the problem to make it simpler. So, the "hole" is what we call the empty state - there is an electron "missing" in the valence band, and we watch it move around. To avoid saying 'the state not occupied by an electron' lots of times, it got shortened to 'hole'.

• Thanks for your comment. I understand the concept of holes but my main concern is how these absent particles have mass? – johndaniel Feb 13 '14 at 2:23
• If that is your concern, than no, you don't quite yet get what the hole is. The empty state (hole) can only move around because the full states (valence electrons) are, and they definitely do have mass, so the apparent motion of the hole will display the characteristics of mass. – Jon Custer Feb 13 '14 at 2:40
• I think the questioner agrees he doesn't quite yet get the concept of a hole - hence the question. – Sridhar Oct 16 '14 at 2:28