How Do You Sketch Energy Bands and Determine Effective Mass in a Semiconductor?

[me_master]

Hi,

I have a question regarding Energy band.

The energy band of a simple cubic semiconductor crystal with a lattice constant b can be represented as in the following function:

E(k) = E₀ + E₁[cos(k_xb) + cos(k_yb) +cos(k_zb)]

where E₀ and E₁ are independent of the wave vector k.

i) Sketch the dependence of E and k from k=0 to the edge of the Brillouin zone in the [100] direction.

For this question, please help to explain whether I need to partial differentiate in the x direction only since the question is for [100] direction?

ii) Determine the effective mass of electron at k=0 in the [100] direction.

 

[kanato]

Yeah, you only need to do the x direction.

 

[asheg]

1- Set ky=0 kz=0 plot the function (Constant+Cos(kx.b))
2- mx = Constant*1/(d^2E/dkx^2)

 

[me_master]

Thanks asheg and kanato

 

[paranormal]

Hi there,

In response to your question, let me first explain what the energy band represents. In a semiconductor crystal, the energy levels of electrons are arranged in bands, with a gap between the valence band (where electrons are bound to atoms) and the conduction band (where electrons are free to move and conduct electricity). The shape and position of these energy bands are determined by the crystal structure and the interactions between electrons and atoms.

Now, to answer your first question, yes, you are correct in assuming that the partial differentiation should be done in the x direction only since the question specifies the [100] direction. This means that you will only be considering the x component of the wave vector k in the given function, while keeping the y and z components constant at 0. This will give you a plot of E vs. k in the [100] direction, which will show the energy levels of electrons as they move from k=0 to the edge of the Brillouin zone.

Moving on to your second question, the effective mass of an electron at k=0 in the [100] direction can be determined by taking the second derivative of the energy function with respect to k. This will give you the curvature of the energy band at k=0, which is related to the effective mass of the electron. The effective mass is a measure of how easily an electron can move in the crystal lattice, and it is an important factor in determining the electrical conductivity of a semiconductor.

I hope this helps clarify your doubts. Let me know if you have any further questions. Happy studying!

Best,