Empty lattice energy bands (Kittel, problem 7.2)

[Franz101010]

"empty lattice" energy bands (Kittel, problem 7.2)

Homework Statement

This is problem 2 of chapter 7 in Kittel's "Introduction to Solid State Physics".
The student is required to sketch the free electron energy bands in the
empty lattice approximation and in the reduced zone scheme, for a fcc lattice,
in the [111] direction.

Homework Equations

It is the equation at page 189 of Kittel's book. Basically what one has to do is
to express the free electron energy as

e(kx,ky,kz) = \hbar / 2 m (k+G)²

where G is a reciprocal vector and k lies in the first Brillouin Zone.

The Attempt at a Solution

A similar problem is treated explicitly for a simple cubic crystal in the [100] direction at
pages 189-190. The resulting bands are plotted in fig. 8.

The problem does not look very hard. Since the desired direction is [111] one gets
kx= ky = kz = k. The range of k is the interval [-\pi/a, pi/a] as in fig 8, since the boundary
of the first BZ in the direction [111] is [pi/a pi/a pi/a]

I have attempted a solution, and ended up with the same as in this "[URL
handout[/URL] I have found on the web.

So far so good... I guess...

What really puzzles me is the "fcc" ingredient. Is this solution really specific to a fcc lattice??
It seems to me that one would get exactly the same result for a simple cubic lattice.
Perhaps it is not very surprising that an "empty lattice approximation" does not depend on the lattice geometry, but I find Kittel's choice quite confusing. Even more so because just below the table at page 189 Kittel suggests to carry out the calculation for a simple cubic lattice...
Of all the possible direction kittel chooses the one where the bands look the same for sc and fcc lattices (and probably for bcc as well). And it seems to me that he does not emphasize this...

Am I missing something? Is my idea (that the band structure is lattice independent in the [111] direction) wrong? It seems to me that the band wrapping would be different for fcc and sc lattices in the [100] direction...

Thanks a lot for any insight

Franz

 

[Franz101010]

I gave some more thinking about this problem, and I realized that there's a further
difference between sc and fcc lattices, other than the BZs.
The difference is in the allowed reciprocal lattice vectors (of course...).
The reciprocal lattice of a sc lattice is another sc lattice.
A possible basis is (in units of 2 \pi/a, where a is the lattice constant)
(1) [1 0 0], [0 1 0], [0 0 1]

The reciprocal lattice for a fcc lattice is a bcc lattice, and a possible basis is
(2) [1 1 -1], [1 -1 1], [1 1 -1]

Now the G's in the above equation must be combinations of vectors
in the basis (with integer coefficients). Of course (2) can be obtained from (1),
but the converse is not true.
Hence, in my opinion, the bands corresponding to the vectors in (1) appear
only in the sc band structure.
If I'm not taking a blunder, the band structure in this http://www.colorado.edu/physics/phys7440/phys7440_sp03/HOMEWORK/Homework/S6.htm" is not for a fcc lattice, but for a sc lattice.
The band structure for the fcc lattice is obtained by removing the bands from 2 to 6
(I count them from bottom up, on the left side of the last figure in the handout).

Could someone comment on this? Thanks a lot

Franz

 

[Franz101010]

I think my previous comment can be made clearer easily.
In doing this I take advantage of the very cool tex feature I've just discovered :!), and
correct some typos I've made.

So the main equation is

$$\epsilon = \frac{\hbar^2}{2 m} ({\mathbf k}+{\mathbf G})^2$$

where G is an allowed reciprocal lattice vector and k is in the first BZ. According to the above discussion this means that

$${\mathbf k} = \frac{2 \pi}{a} x [1,1,1],\qquad\qquad |x| \leq \frac{1}{2}$$

The reciprocal lattice basis is formed (e.g.) by the three vectors

$${\mathbf g}_1 = \frac{2 \pi}{a} [1,1,-1],\qquad {\mathbf g}_2 = \frac{2 \pi}{a} [1,-1,1],\qquad {\mathbf g}_3 = \frac{2 \pi}{a} [-1,1,1]$$

so that a generic reciprocal lattice vector is

$${\mathbf G} = p\, {\mathbf g}_1+q\, {\mathbf g}_2 +r\, {\mathbf g}_3 = \frac{2 \pi}{a} [p+q-r,p-q+r,-p+q+r],\qquad\qquad p,q,r \in {\mathbb Z}\qquad (*)$$

whence

$$\epsilon= \frac{\hbar^2 \pi}{m a} \left[ (x+p+q-r)^2+(x+p-q+r)^2+(x-p+q+r)^2 \right] ,\qquad\qquad p,q,r \in {\mathbb Z} ,\qquad\qquad |x| \leq \frac{1}{2}$$

Using the last equation for drawing the bands gives the attached result Kittel_7_2.jpg, where the numbers in the legend are not the normalized components of G , but the above p, q and r. The components of G in units of $$2 \pi/a$$ are easily found by using Eq. (*). For instance

[p q r] ---> [G_x G_y G_z]
[0 0 0] ---> [0 0 0]
[1 0 0] ---> [1 1 -1]
[1 1 0] ---> [2 0 0]
[1 -1 0] ---> [0 2 -2]

If I'm not forgetting anything, this is in agreement with what I have argued in my previous post: reciprocal vectors such as

$${\mathbf G} = \frac{2 \pi}{a} [1 0 0]$$

are not allowed in a fcc lattice, but only in a sc lattice. Hence some bands which are present in the latter are missing from the former.

 

[nathan12343]

It problably doesn't matter since this thread is several months old, but anyway, the plot posted above is incorrect, all values should be multiplied by 4/3. Franz101010 did not normalize the energies correctly. See the attached mathematica notebook.

 

[FranzDiCoccio]

It problably doesn't matter since this thread is several months old, but anyway, the plot posted above is incorrect, all values should be multiplied by 4/3. Franz101010 did not normalize the energies correctly. See the attached mathematica notebook.

Nathan12343,

thanks for replying to my old post. I have not had time to look at your mathematica notebook yet, but I guess you're right about normalization. I'll peruse it asap.
Anyway if I get it right it is an overall factor, so it does not really matter for the original question. I do not recall what I was meaning by "normalized" in the plot I've attached, but probably you're right and I've got some factor wrong.

Thanks again

Franz

 

[nathan12343]

Kittel wants everything in units of the maximum energy of the lowest energy band, which happens to be 3/4 modulo a bunch of constants.

 

[FranzDiCoccio]

Kittel wants everything in units of the maximum energy of the lowest energy band, which happens to be 3/4 modulo a bunch of constants.

You're definitely right about this, I've read the text of problem 2 again. Kittel suggests the normalization convention you mention, I've adopted a different one.
But, again, that seems to me a definitely minor point, since the normalization factor is to some extent arbitrary.

My question was of a different nature. Today I wanted to take a look at your notebook, but it's still pending approvation. I'm curious to see whether you ended up with the same band structure as me (modulo a 4/3 coefficient, of course ).

Cheers

F