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The velocity of the Bloch electrons is given by $${\vec v}({\vec k})=\frac{1}{\hbar}\nabla_{\vec k}\varepsilon({\vec k}).$$ By we know that the gradient $\nabla\phi$ to a surface $\phi(x,y,x)={\rm constant}$ is normal to the surface. Therefore, the velocity of Bloch electrons should be perpendicular to the surfaces of constant energy.

However, the diagram that is given in Ashcroft and Mermin (Fig. 12.6, page 229) draws the velocity vector tangential to the constant energy surface. See a snapshot from the book below.

enter image description here

How can the velocity of the electron orbits in presence of a magnetic field lies tangential to the constant energy surfaces in the $k-$space if the equation above has to hold?

Solidification
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2 Answers2

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Yes, $\dot{\vec{r}}=\vec{v}(\vec{k})$ is perpendicular to a surface of constant energy. But the figure you posted shows a $k$-space orbit, not a real-space trajectory. In other words, at each point along the line the tangent points along $ \dot{\vec{k}} \propto \vec{v}(\vec{k}) \times \vec{H}, $ which is perpendicular to $\vec{v}(\vec{k})$. If you keep reading a bit, you'll see that Ashcroft & Mermin do discuss real-space trajectories too, see e.g. Fig. 12.7 and footnote 26.

Anyon
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The equation given in the question is the generalization from the two-dimensional case for free particle, where $$ \epsilon(\mathbf{k}) = \frac{\hbar^2\mathbf{k}^2}{2m}. $$ Another way to justify this equation is by using the operator equation of motion: $$ \mathbf{v} = \frac{d}{dt}\mathbf{x}=\frac{1}{i\hbar}[\mathbf{x}, H(\mathbf{p}]_-=\nabla_\mathbf{p}H(\mathbf{p}). $$ In other wors, it should not be put in doubt (beyond the more subtle caveat that the true momentum here is replaced by the Bloch momentum).

What could be put in doubt is the correctness of the illustration in the Ashkroft&Mermin or, more precisely, whether they had intention at all to show the direction of velocity in respect to the energy surfaces, given that velocity is the direction of the electron movement in real space, whereas the energy is a more abstract concept. Mixing of real and energy spaces in figures in books on semiconductor theory is quite common - just think of a typical illustration of a p-n junction.

Roger V.
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