If you apply an externtal force on an object, its acceleration is inversely proportional to its mass (Newton’s low). Electrons in a lattice; however, accelerate at a different rate compared to electrons in free space. Hence, an electron may show a different “effective mass” in different environments and under different conditions. To explore this concept, we will first start with the relationship between kinetic energy and momentum in classical physics and then look at the quantum mechanical picture for an electron inside a one dimensional lattice.
In Classical world, the kinetic energy and momentum have a simple relationship.
Hit the balls with different intensities to give them different momenta. Can you find the relationship?
Can you use the equations for kinetic energy and momentum to write the relationship between the two?
See what happens to the curvature of the plot when you increase the mass.
Within a lattice, electrons experience a periodic potential due to the positive charges in the nucelli of atoms. For simplicity, let’s consider a one dimensional array of atoms shown on the right which creates a periodic potential energy shown below.
Distance(nm)
Remember that electrons have wave properties and its momentum actually is a function of its wavelength. One of the amazing results of this is that the relationship between the kinetic energy and momentum is dictated by the periodic potential that electrons experience.
Change the distance between atoms to change the periodic potential and see how the kinetic energy versus momentum relationship changes.
Distance(nm)
Unlike the classical model, the relationship is no longer quadratic. However, for low energy values, you can still approximate the curve with a parabola.
Play with the slider to match the dashed line (parabola) with the actual relationship.
Effective mass:
You can drag atoms in the lattice to change the effective mass.
\[F = Eq = {m_{eff}a}\]
Why is effectice mass so important? Let’s say you apply a constant electric field. What happens if you have different effective masses (adjust the slider)?
Effective mass:
What would happen if the effective mass is negative?
When the effective mass is negative, the KE vs. p graph is flipped. And velocity of the electron become reversed as well. Momemtum, which is the product of mass and velocity, would remain positive.
Effective mass:
Within the lattice, the valence band diagram assigns different values to effective masses of electrons. Thus, electrons move in opposite directions depending on where they locate on the valence band diagram.