Excercise 1
Expain and Plot Free electron bands in 2D square lattice in First Brillouin zone –
Plot along k directions (see figure below for notation), G-K, G-M and M-G
See link:
http://lampx.tugraz.at/~hadley/ss2/fermisurface/2d_fermisurface/2dsquare.php
M
G
Your plot should roughly look like below fig. You can follow the
same units as shown in the example of 1D in the next page
X
Below is an example for a 1D lattice
1D example
a
Real Space
2p/a
Reciprocal Space
-2p/a
p/a
-p/a
BZ
I used lattice constant a = p Å
2p/a
1 D example: Repeated Zone Scheme: I used Mathematica to Plot
1 D example: In First Brillouin Zone: I used Mathematica to Plot
Graphene
Given the real space (Direct space) primitive vectors for Graphene
a) Calculate the reciprocal lattice vectors (b1 and b2). For the purpose of calculation, you
can use a3={0,0,1}- unit vector along the z direction.
b) Draw the first Brillouin zone (the shaded region in the figure and find the K and M points
( note G= {0,0}. ). You should get G– K distance to be 1.705 inv. angs and G–M distance to
be1.477 inv. angs
c) From the tight binding model one gets the energy versus momentum dispersion for the
pi and pi* band to be
Assume t=3 and plot the electronic band structure of the pi and pi* bands along various symmetry lines.
You should obtain (see next page)
Let us orient the sample again same as above.
What is the minimum photon energy required to
see the pi band at the fermi level. Note the
fermilevel has zero binding energy. The important
thing to think about here is we not only need to
binding energy but also see if you can get to the
right parallel momentum.
Let us say that by some means, we create a flat
band at the fermilevel (blue line)- By flat band i
mean, the energy of the band does not vary with
k. (it resembles a core level). What is the
minimum photon energy required to see the flat
band at the fermi level. Use as before the work
function to tbe 4.5eV
Excercise 4:
Write atleast 2 question, that you would like to know from what you have learned.
These can be quite general questions or very specific questions.
Some examples :
1. How can I understand from the band structure various material properties - you can
even be quite specific- – why different materials have different electrical properties,
optical properties, or …... Do I need more information than just band structure.
2. Ponders over the Bloch’s theorem and ask yourself some questions
3. What is for example difference between various waves, electron waves, lattice
vibrations, electromagnetic waves-all these are waves and do they have respective
band structre and what they tell. That is try to connect between various courses you
have learned.
Apart from whatever questions you ask-try to make a guess of the answer.