2.1.1.2 Velocity-Band Structure Relationship

The relation of the average electron velocity to the band structure is established by the Schrödinger equation for the amplitude of a Bloch wave:

$$-\frac{\hbar^{2}}{2m_{0}}\nabla^{2}u_{n\vec{k}}(\vec{r})-\frac{i\... ...g(\epsilon_{n}(\vec{k})-\frac{\vec{p}^{2}}{2m_{0}}\biggr)u_{n\vec{k}}(\vec{r}).$$ (2.10)

Using periodic boundary conditions and the Gauss theorem one obtains from (2.10):

$$-\frac{i\hbar}{m_{0}}(\vec{p}-\vec{p}^{'})\cdot\int u_{n^{'}\vec{... ...}}\bigr] \int u_{n^{'}\vec{k}^{'}}^{*}(\vec{r})u_{n\vec{k}}(\vec{r})\,d\vec{r}.$$ (2.11)

If $n = n^{'}$ and $\vec{p}^{'}\rightarrow\vec{p}$, so that $\epsilon_{n}(\vec{k})-\epsilon_{n}(\vec{k}^{'})\approx (\vec{p}-\vec{p}^{'})\cdot\frac{\nabla_{\vec{k}}\epsilon_{n}(\vec{k})}{\hbar}$, $\vec{p}^{2}-\vec{p}^{'2}\approx2\vec{p}\cdot(\vec{p}-\vec{p}^{'})$, it follows from (2.11):

$$-\frac{i\hbar}{m_{0}}\int u_{n\vec{k}}^{*}(\vec{r})\nabla u_{n\ve... ...r}=\frac{1}{\hbar}\nabla_{\vec{k}}\epsilon_{n}(\vec{k}) -\frac{\vec{p}}{m_{0}}.$$ (2.12)

This equation together with (2.9) gives for the average electron velocity in band $n$ with wave vector $\vec{k}$:

$$\vec{v}_{n}(\vec{k})=\frac{1}{\hbar}\nabla_{\vec{k}}\epsilon_{n}(\vec{k}).$$ (2.13)

This relationship means that the average electron velocity in the band $n$ is the gradient of the energy branch $\epsilon_{n}(\vec{k})$ in quasi-momentum space and is thus perpendicular to the surface of constant energy $\epsilon_{n}(\vec{k}) =$   const.

Figure 2.2: Momentum and quasi-momentum for different shapes of the surface of constant energy.

This also emphasizes the difference between momentum and quasi-momentum which are not parallel in general. For example, in the vicinity of a non-degenerate band extremum the surface of constant energy can be either a sphere or an ellipsoid. In the former case momentum and quasi-momentum are parallel and in the latter they are not parallel as it is shown in Fig. 2.2(a) and (b), respectively. S. Smirnov: