# Why does a semiconductor hole have a mass?

I have read that holes in semiconductor are nothing but vacancies created by electrons. But how can this vacancy i.e. hole has a mass?

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– Steve B Jul 10 '12 at 22:57

## 3 Answers

While the answers above are correct in essence, the inertia of the hole is due to the electrons that must flow to fill up the hole, there is a simple way to understand what the mass of a hole is in a tight-binding model.

If you have an electron in a lattice, and it can hop from one site to another with amplitude a, then the time evolution of the wavefunction is by

$${\partial_t \psi(x)} = a\sum_{\langle y,x\rangle} \psi(y)$$

where the sum is over nearest neighbors of x (not including x). If you redefine the phase of the wavefunction in a time-dependent way $\psi(x)\rightarrow e^{iNat}\psi(x)$\$ where N is the number of nearest neighbors, you subtract a term from the right hand side which leads to a standard form of the lattice Laplacian, so you have a discretized Schrodinger equation. The continuum limit of this is the normal Schrodinger equation:

$${\partial_t \psi} = a\epsilon^2 \nabla^2 \psi$$

Where $\epsilon$ is the lattice spacing. From this, you can read off the effective mass $m={1\over 2a}$.

The point is that mass of a quantum excitation is just the inverse of the hopping amplitude, so it is universal to describe any localized excitation that moves coherently with an effective mass. The harder it is to hop, the more massive the object is. The result is in principle independent of the actual mass of the fundamental particles--- if you make an optical lattice and place a metal BEC in each optical trap, the electrons have to tunnel from one trap location to the next in order to flow, and the effective mass can be as large as you like. In real materials, the effective mass is usually close to the electron mass, but sometimes can be hundreds of times bigger.

If you fill up all the electron states, and consider one hole, there is an amplitude for the hole to hop to a neighboring location. This gives an effective mass as above, proportional to the inverse hopping amplitude. Unlike in classical systems, you don't have to consider dissipation--- the effective mass description is exact. Further, if the hopping amplitude for a hole is fundamentally the same as the hopping amplitude for an electron to fill the hole from the neighbors.

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A quick answer:

Imagine an array of billiard balls with one missing in the middle of the array; there is a "hole" where the billiard ball is missing. For this hole to "move", a billiard ball must move into that position, leaving a hole at the ball's previous position. Since, in fact, the hole movement is entirely equivalent to the billiard ball movement, we can speak of the mass of the hole even though it is the billiard ball's mass that is physical.

Now, in the case of electrons moving through the unfilled valence band, the effective mass is greater than the mass of a mobile electron so hole mass is typically greater than mobile electron mass.

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For a very common practically important classical mechanical equivalent, consider a bubble of air in the water. The apparent inertia of the bubble is on order of the mass of water displaced by the bubble, as the water around the bubble has to move for the bubble to move. This is an important consideration for ships, submarines, torpedoes, fish, etc, that end up having considerably more inertia than their own mass, and end up accelerating slower with the engine on.

One time I did mathematics for realistic barrel falling into water in computer game; the apparent 'increase' of the mass, in combination with the conservation of momentum, rather accurately describes a large part of the deceleration of object when it is entering water.

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