Energy Levels in a Crystalline Solid — Question 4

Question 4

Band curvature gives opposite effective-mass signs at the two edges

For $E(k)=E_0-2t\cos(ka)$ define $1/m^*=(1/\hbar^2)d^2E/dk^2$. A weak force changes crystal momentum according to $\hbar\dot k=F$.

1. Find the effective mass at the bottom and top of the band.

2. Derive the acceleration and interpret its sign near the top.

Question 4 - Solution

Band curvature gives opposite effective-mass signs at the two edges

(a) Method and calculation

$$\frac{d^2E}{dk^2}=2ta^2\cos(ka)\Rightarrow m^*(k)=\frac{\hbar^2}{2ta^2\cos(ka)}.$$$$\boxed{m^*(0)=+\hbar^2/(2ta^2),\quad m^*(\pi/a)=-\hbar^2/(2ta^2)}.$$

(b) Interpretation and further calculation

$$\dot v=\frac1\hbar\frac{d^2E}{dk^2}\dot k=\frac{F}{\hbar^2}\frac{d^2E}{dk^2}.$$$$\boxed{\dot v=F/m^*};\quad\text{near the top, acceleration opposes }F.$$

Physical check

Negative band curvature is often described more simply using positive-mass holes. The effective mass is a response parameter, not a change in the fundamental electron mass.