Question About Bloch Oscillation: Understand the Energy

[hafsa]

i have a question concerning BLOCH OSCILLTION.i studied that in the extended zone scheme, the energy of an electron with k=pi/d is same as with k=3 pi/d.
i can't understand this because by dispersion relation as wave vector increases ,energy also increases(in extended zone scheme too)
please help me

 

[cavalier3024]

well, if I'm not mistaken, bloch oscillations are based on the assumption that the dispersion relation is periodic [something like - e(k)=B+A*cos(a*k)]. when you derive the speed from this energy [by v=(1/h)*de(k)/dk] you get a sine - a periodic speed.
the reason that the dispersion relation is periodic and not e(k)=h^2*k^2/2m is that the later is true only for free electrons. when dealing with bloch electrons this formula isn't quite right. if you work with the tight binding method you get those periodic e(k) functions.

 

[hafsa]

hmm,u mean that in bloch oscillation we are actually apllying bloch theorem also.(by letting n vary,we obtain same energy for different energies of k(wave vector)and there is no role of extended zone scheme here.am i right?

 

[cavalier3024]

sure. and according to bloch's theorem:
$$\epsilon$$_n$$(k)$$=$$\epsilon$$_n$$(k+G)$$
where G is any reciprocal lattice vector.
this means that at the same band (n) the energy is periodic over k.

regarding the extended zone scheme, I am not sure but as far as i understand it will be the same as the repeated-zone scheme (that is the reduced zone repeated).