You can think of k being the momentum and E as the energy. From a mathematical point of view it is the wavevector.  The E vs k is a dispersion diagram.  

For example the probability of a photon emission indirect band gap semiconductor is much much lower than  than of the a direct band gap semiconductor. In the direct band gap the minimum of the conduction band and the maximum of the valence band are at the same k =0, so momentum is conserved and when the hole and electron recombine both energy and momentum conservation requirements are met. In the indirect semiconductor the momentum mismatch needs to be made up by a phonon which is unlikely. 

Since it is a dispersion diagram you can look at the curvature of the diagram and determine other parameters such as the electron mobility, effective mass.

Without getting into a lot detail, a crystal is a multi-body problem with forces from huge numbers of atoms. When they are organized likely as in a crystal we consider the electron as a wave with the wavevector etc. This allows one to conveniently think about the electron as some sort of billiard ball model as being simple and having a mass and if we apply a field acceleration etc. The band diagrams let us she how the effective mass etc change as the electron travels in different directions.  

You can think of k being the momentum and E as the energy. From a mathematical point of view it is the wavevector.  The E vs k is a dispersion diagram.  

For example the probability of a photon emission indirect band gap semiconductor is much much lower than  than of the a direct band gap semiconductor. In the direct band gap the minimum of the conduction band and the maximum of the valence band are at the same k =0, so momentum is conserved and when the hole and electron recombine both energy and momentum conservation requirements are met. In the indirect semiconductor the momentum mismatch needs to be made up by a phonon which is unlikely. 

Since it is a dispersion diagram you can look at the curvature of the diagram and determine other parameters such as the electron mobility, effective mass.

Without getting into a lot detail, a crystal is a multi-body problem with forces from huge numbers of atoms. When they are organized likely as in a crystal we consider the electron as a wave with the wavevector etc. This allows one to conveniently think about the electron as some sort of billiard ball model as being simple and having a mass and if we apply a field acceleration etc. The band diagrams let us she how the effective mass etc change as the electron travels in different directions.  

This wave vector what we say 'k' is nothing but one variable which states the change of the momentum of an electron in different directions/co-ordinates, while the electron traverse from VB(valance band) to C.B or C.B to V.B. If it is to be expressed thematically in scalar form that it is like, p=(kh)/2*pi ;...... Where p is momentum of electron, h is plunk constant. Now due to the variation of momentum of the electron while travelling through one non-uniform potential barrier the momentum changes instantaneously. k=(2*pi)/wavelength this is the expression illustrates the variation of wave vector, due to the generated wave length by the electron. This approach resembles with the Schrodinger wave nature approximation of objects, that reason the wave function always contains this wave vector 'k'. Actually the electron is detected as a wave while it passes through the energy barrier.

While one electron & one hole recombines then an excess energy generates from that particular position, and this energy radiates with a wave-length which equals to the visible spectrum of light wavelength, which is photon. By this method the LEDs are mostly formed. Due to indirect band the the wave vector 'k' shifts a little bit right or left side, for that reason the emitted energy gets lesser compared to direct band gap in semiconductor materials.

Caption 1: the effective mass of electron

Caption 2: wave vector visualization 

If you want to think of electrons in solids in terms of plane waves, convert the energy into a frequency $\omega=E /  \hbar$.  Then $(\omega, k)$ form the usual frequency-wave vector pair, from which you can find things like the phase and group velocity.

In terms of linear momentum, $k$ is analogous to $p /  \hbar$ where $p$ is the linear momentum of a free particle-- $E = p^2/(2m)$ in the non-relativistic case.  However, in solid state physics, $k$ actually refers to the crystal momentum (https://en.wikipedia.org/wiki/Crystal_momentum), which is similar to, but distinct from the physical momentum.  In a nut-shell, the band structure of the $E$-$k$ diagram arises from diffraction off of the periodic lattice of the crystal, which breaks the symmetry of the space the electrons move in  and causes non-conservation physical linear momentum of the electrons.

E(k) diagram is the most important entity characterising a semiconductor. E is electron (hole) energy and k is the wave vector. No theoretical study, experimentation, and technological application can take place unless the E(k) diagram of the semiconductor is first obtained by solving Schroedinger's equation. 

For practical applications, among the most important semiconductor properties are its band gap, electron (hole) mobility, electron (hole) effective mass, direct versus indirect band gap.

From the E(k) diagram, we find the band gap, i.e. the range of energy, where the density of eigenstates (allowed energy) is zero. 

As has been commented above, the effective mass is directly obtained from the curvature of the E(k) diagram, and the mobility in turn from it. 

A band gap is direct if Ec (conduction band minimum) and Ev (valence band maximum) occur at the same value k; indirect if otherwise. Silicon is the purest material obtained by man and it is the most studied among all materials; unfortunately, however, it is an indirect band gap material, so, cannot find optoelectronic applications; also, its hole mobility is very low, so, future generations of CMOSFETS will have to be made on high mobility semiconductors such as Ge, GaAlAs, etc.

I think all the answers are right, but not convincing. we need an intuition between reciprocal lattice and direct one. I think of electron wave functions as bloch waves.

we have N primitive cells in direct lattice and hence N allowed wave vectors. I think of allowed k-wave_numbers correspond to a special electron, say 3s1 , and of course it is contained within one band. but k is a state, which resembles for example all 3s1 electrons are in phase. like a 1-d chain. and i think this the reason for which we get for example for gamma point s or d .... characters. and orbitals change with varying k.

if you notice about lower bands they are flat and purely one character.

we can change the k wave vector to get the electrons not to be in phase or completely out of phase.

the dispersion of energy is indeed works for valence electrons in solids which their wave functions overlap. but for a molecule or an atoms, the dispersion is very low. and the band cannot vary too much in energy.

as a matter of fact, for strongly bonded orbitals, the dispersion of energy vs k is high.

PLEASE CORRECT ME IF I'M WRONG. THIS IS SOMETHING WHICH SHOULD BE DISCUSSED.

Adding to the colleagues, when electrons are moving in a material they move like a wave having specific wavelengths and frequencies. Since they are quantum mechanical particles, they have momentum and energy. The energy Eis proportional to the frequency f; E=hf. De Broglie set the relation between the wavelength lambda and the momentum p of the particle as p= h/lambda= (h/2pi)(2pi/ lambda)=(h/2pi) k where k is called the wave number. So for particles moving as waves k is a measure of the momentum of the particle.

When the electrons are moving in an atom , molecule or a solid material they get specific quantized values which means that not all momentum values are allowed. Continuous values of the momentum are allowed only when the particle moves in a boundless medium, that is when the particle is very.

Consequently, the energy of the particle is also quantized. Like the energy momentum relation of a free particle, there is also an energy momentum relation of electrons in any material. This relation is a characteristic of the material and is termed the dispersion relation.

Best wishes

How and why is the wave vector 'k' assumed to be analogous to Momentum P?

What would be the equation of E(k) dispersion in nanowire (one dimensional).

There is an introduction to E(k) in our recent article in the Journal of Chemical Education:

Article From Benzene to Graphene: Exploring the Electronic Structure...

Note in particular the section entitled, "Effect of Orbital Crossing on the Band Structure of Bilayer Graphene" and Figure 17 therein.

Dear Raghav,

welcome,

To answer your question:

How and why is the wave vector 'k' assumed to be analogous to Momentum P? On has to return back to the de Broglie postulate relating the wavelength lambda of the particle motion to its momentum p as:

lambda= h/p

h is the Planck consatnt.

By definition the wave vector vector k can be expressed by 2pi/lambda

Then k= 2pi p/ h,

It follows p= (h/2pi) k so k is directly a measure of the momentum of the particles.

This is borrowed from the properties of the photons

E=hc/lambda=mc^2 where c is the speed of the photon and E is the energy of the photon. The photo momentum p= mc,

substituting in the above equation then

hc/lambda= pc

from which p= h/lambda,

de Broglie took this equation and applied it to the electrons or generally any quantum particles.

Best wishes

Dear Amit, To answer your question:

What would be the equation of E(k) dispersion in nanowire (one dimensional)?

if you consider it one dimensional, then the motion of the electrons will be confined in two dimensions representing the confinement in the dimensions perpendicular to the direction of the wire. Assuming the dimensions are ax and ay,

Then ax houses nx lambdax/2, lqmbdax= 2 ax/nx

also ay houses ny lamday/2, lamday= 2ay/ny

It follows that the momentum components px and py can be expressed as:

px= h/lambdax= h nx/ 2ax

py= hny/2ay

The energy E= px^2+ py^2/2m where m is the mass of the electrons. You need only to substitute px and py in equation of E.

Then this will be the required E-p relation

Considering that

p= (h/2pi) k

one can substitute p in terms of k and get also p as a function of k.

It is assumed that the electrons are free in the conduction band and the holes in valence band where the bottom level in the conduction band is the conduction band edge Ec and the conduction band at Ev the valence band edge.

For more information how one can use the free electron model please refer to the the book: Book Electronic Devices

Best wishes