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My question relates to semiconductor physics. The relationship between energy E and wavenumber k for the Kronig-Penny model looks like this, where a is the period of the potential function:

enter image description here

I am referring to a statement made in the book Advanced Semiconductor Fundamentals, 2nd Edition by Robert F. Pierret, page 61. In reference to the E-k diagram of the Kronig-Penney model it says:

...the energy band slope dE/dk is zero at the k-zone boundaries.

and referring to the gradient:

This is a feature common to all E-k plots, even those characterizing real materials.

Why is the gradient zero at the k-zone boundaries (i.e. edges of Brillouin zones)? What does it signify?

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The x axis on the plot is proportional to momentum. The k-zone boundary in your diagram, is the point with zero momentum (no net motion of a carrier.); So, in the plot you show, that's why it's the minimum point as all other points have a net motion greater than it. You will notice, that the graph is symmetrical about the k=0 point in the kronig-penny model. The gradient is zero because of that symmetry. From basic calculus, when you take the derivative of a function, it is always zero at an extremium.

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