Last Modified 1 May 2001

The purpose of this page is to develop and annotate the procedures we need to use to calculate the phonon spectrum for diamond structure solids using the frozen phonon approximation and an appropriate total energy calculation program. In the examples we will use the NRL static program.

Note that this is more a collection of notes than a tutorial. So you can expect that as we go on you'll find that write-up at each k-point in each structure will change. In addition, the descriptions will probably become more and more terse with time. For these reasons, I'd recommend that you go through these pages in the order in which they are written. At the moment, that means start with the diamond lattice, going through the k-points in order.

In the supercell approximation phonon frequencies can only be determined at those wave vectors which lead to reasonably sized supercells of the primitive lattice. Thus we'll derive procedures to calculate the phonons on an 85 point mesh. This is the mesh known as the Regular Order-8 mesh described in the pre-defined fcc k-point mesh page. However, there are some differences that must be noted:

• The current calculation is in the reciprocal space of lattice vibrations, not the space of electron momentum. In fact, we will see that there are two type of k-point meshes involved.
• The first, described on this page, is the k-point mesh of the phonons. At each phonon k-point there will be six phonon eigenstates, three for each atom in the primitive diamond unit cell. Each eigenstate is characterized by a phonon frequency and polarization vector. The frequencies may be degenerate, but the polarization vectors will be orthogonal in a six-dimensional space.
• The second k-point mesh is that needed for the calculation of the total energy of a given structure by summing over the eigenvalues of the occupied states in the Brillouin zone. This is a reciprocal space of electron crystal momentum variables. In tight-binding diamond carbon, where there are one s orbital and three p orbitals for each atom, there are eight electronic eigenvalues at each k-point in the primitive unit cell.
  • In all calculations shown on this we will use only the k-point mesh equivalent to the Regular Order-8 k-point mesh to sum over the Brillouin zone. This is actually overkill for the insulating diamond sturcture elements. We'll provide alternate k-point sets so that you can check convergence yourself.
• The two reciprocal lattices represent different things, the space of phonon vibrations and the space of allowed electron eigenstates. So, even though both meshes might include the point (2p/a, 2p/a, 2p/a), we need to remember that the quantities coming out of the calculations are different.

Remember that the primitive cell for all of this is the unit cell of diamond, with primitive vectors a1 = ( 0 , ½ a , ½ a ) a2 = ( ½ a , 0 , ½ a ) a3 = ( ½ a , ½ a , 0 ) and reciprocal lattice vectors b1 = (2 p/a) ( -1 , 1 , 1 ) b2 = (2 p/a) ( 1 , -1 , 1 ) b3 = (2 p/a) ( 1 , 1 , -1 )

The First Brillouin zone of a face-centered cubic lattice, with high symmetry k points marked. Note that this is also the Wigner-Seitz cell of a body-centered cubic lattice in real space.

With all of that out of the way, here is the listing of the 85 k-point mesh for the phonons. Clicking on a given k-point link will take you to a page showing how we do the calculations. You may note that the ordering is a bit unfamiliar. We've done this so that you can construct k-point meshes equivalent to the Order-0, Order-1, Order-2, Order-4 and Order-8 meshes of the fcc lattice. The ``Order'' of a mesh refers to the number of divisions made between the origin (Gamma) and the high-symmetry point X, at (0,0,2p/a).

In the table below,

• Index is the k-point number,
• Order is the smallest order k-point mesh which includes this point,
• Lattice Coordinates are the coordinates (l₁,l₂,l₃) of the k-point, i.e.,

  k = l₁ b₁ + l₂ b₂ + l₃ b₃    ,

• Cartesian Coordinates are the coordinates of the same point in the real reciprocal space,
• A Relative Weight is assigned to each point based on the number of equivalent points in the unit cell. When doing summations over the Brillouin Zone these weights should be normalized so that they sum to unity.
• Atoms indicates the number of atoms contained in the supercell.
• Notation is the designation of this point according to the scheme in the figure above. (Some points have no designation.) Until someone figures out how to get Greek and Latin characters on the same page we'll spell out the Greek letters.
• Clicking on the Index for a given k-point will take you to a page describing the calculations needed to determine the phonon frequencies at that point.
• This page is under construction, so not all k-points are hooked up at this time.

Before starting, we'll need to state a few more computational details. We will determine phonon frequencies of carbon, silicon, and germanium at their experimental lattice constants. For this we'll need to know the

We'll also need to note the location of the space group files directory.

Finally, I guess we should look at the total energy for the perfect diamond crystal for each of these elements.

Return to