What does k mean in E-k diagram?

[msaleh87]

In the E-k diagrams for a semiconductor material,what exactly is k?

-is it just the quantized quantity in the bloch theorem :ψ(x+a)=exp(i.k.a)ψ(x)

-is it the wavevector of the electron wave

-does it determines the direction of the motion, i.e a given k corresponds to motion in a certain direction in the crystal and only certain energy levels E are accessible to an electron moving in this direction. Correspondingly, electrons with a certain energy E can only move in certain directions.

-is it related to the momentum of the electron. And if so, shouldn't it be related to the energy with simple relations, not those complex diagrams

please give me the exact meaning of k or tell me where to find such an precise meaning

 

[M Quack]

It is a wave vector, i.e. it contains both a direction and a spatial frequency or wavelength lambda = 2pi/|k|. The momentum is $p = \hbar k$.

Electrons in a crystal are not free particles, due to the presence of nuclei and plenty of other electrons. Therefore the free particle dispersion relation E=p^2/2m is not valid in crystals. Instead you get the "spaghetti diagrams" of band structure. In crystals, the energy can depend on the direction of p or k.

This is the fundamental foundation of solid state physics. Read any textbook on solid state physics, for example Ashcroft and Mermin.