- D.45.1 Propagation speed
- D.45.2 Motion under an external force
- D.45.3 Free-electron gas with constant electric field

D.45 Motion through crystals

This note derives the semi-classical motion of noninteracting electrons in crystals. The derivations will be one-dimensional, but the generalization to three dimensions is straightforward.

D.45.1 Propagation speed

The first question is the speed with which a more or less localized electron moves. An electron in free space moves with a speed found by dividing its linear momentum by its mass. However, in a solid, the energy eigenfunctions are Bloch waves and these do not have definite momentum.

Fortunately, the analysis for the wave packet of a free particle is
virtually unchanged for a particle whose energy eigenfunctions are
Bloch waves instead of simple exponentials. In the Fourier integral
(7.64), simply add the periodic factor

As a result the group velocity is again

In the absence of external forces, the electron will keep moving with
the same velocity for all time. The large time wave function is

where

D.45.2 Motion under an external force

The acceleration due to an external force on an electrons is not that
straightforward. First of all note that you cannot just add a
constant external force. A constant force

Next there is a trick. Consider the expectation value

where

Moreover, its magnitude

The time evolution of

(D.28) |

Writing this out with the arguments of the functions explicitly shown gives:

Now assume that the external force

(D.29) |

where

It follows that the magnitude of the

It further follows that the average wave number

Since the packet remains compact, all wave numbers in the wave packet change the same way. This is Newton’s second law in terms of crystal momentum.

D.45.3 Free-electron gas with constant electric field

This book discussed the effect of an applied electric field on free electrons in a periodic box in chapter 6.20. The effect was described as a change of the velocity of the electrons. Since the velocity is proportional to the wave number for free electrons, the velocity change corresponds to a change in the wave number. In this subsection the effect of the electric field will be examined in more detail. The solution will again be taken to be one-dimensional, but the extension to three dimensions is trivial.

Assume that a constant electric field is applied, so that the
electrons experience a constant force

Assume the initial condition to be

in which a subscript 0 indicates the initial time.

The exact solution to this problem is

where the magnitude of the coefficients

Unfortunately, this solution is only periodic with period equal to the
length of the box

At intermediate times, the solution is not periodic, so it cannot be correctly described using the periodic box modes. The magnitude of the wave function is still periodic. However, the momentum has values inconsistent with the periodic box. The problem is that even though a constant force is periodic, the corresponding potential is not. Since quantum mechanics uses the potential instead of the force, the quantum solution is no longer periodic.

The problem goes away by letting the periodic box size become
infinite. But that brings back the ugly normalization problems. For
a periodic box, the periodic boundary conditions will need to be
relaxed during the application of the electric field. In particular,
a factor