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So, one can ask what is the smallest size of matrix which can attempt to model the band
structure of diamond-structure semiconductors. The answer is 8x8, since there are two
independent atoms per unit cell (at [0,0,0] and [1/4,1/4,1/4] in the f.c.c arrangement),
and we need a minimal basis set of four states per atom to model the valence band. If we
consider these as two s- and two p-electrons, we will not stabilise the diamond structure;
but, as explained in the last lecture, if we consider one s-
and three p- electrons, then we do. These electrons can be thought of (notionally, maybe)
as forming sp^{3} hybrids; the key point is that we need to get back more energy,
from overlap (of the hybrids), than we have to give up by *promoting* one s-electron
to a p-state. In any case we need at least an 8x8 matrix to solve this problem by matrix
methods. Note that this does not include any d-states, which have been shown to be
important in getting the calculated band structure to agree with experiment. Silicon has
an indirect gap (see problem 7.4.1), and the position
of the minimum in the conduction band turns out to be a sensitive test of the involvement
of d-states. These points are all too detailed for our present purposes.

There are two elements to the calculation of matrix elements involving only nearest neighbour overlap. The first is the size of the on-site and overlap integrals, which in principle are exactly the same as in the diatomic molecules discussed in lecture 1. However, now we have to take the 3D nature of the wavefunctions and the direction of bonding into account. This is summarised by Sutton on page 122, and by Pettifor on page 200-201. The largest contribution to the overlap is between hybrids pointing towards each other along the bond, and this is given by

or h =
1/4(sss - 2Ö
3sps - 3pps)
(8.1) |

Here we need to recognise s-bonds, as in
sss or sometimes
V_{sss}, and also
p-bonds. There are actually 4 of these, the other three
being V_{sps}, V_{pps}
and V_{ppp}, in the notation used by Yu & Cardona (1996).
The notation can be confusing here: Pettifor uses f for
the hybrid and h for the bond integral,
whereas Sutton uses h for the hybrid and b for the bond
integral as in equation 8.1 above. In addition the promotion energy is called
D, with

E_{p} - E_{s})
(8.2) |

If hybrids are used as the basis set, then D is the on-site matrix element between different hybrids on the same atom, and b is the matrix element between the main overlapping hybrids on the two different atoms. A gross approximation, which nevetheless retains some of the essential features, is to neglect all other overlaps; this is known as the Weaire-Thorpe model, which depends on just these two parameters D and b.

The classic tight binding calculation, using the minimal basis set needed to describe silicon, was developed in a series of papers by Chadi and Cohen, and is described in detail by Yu and Cardona (1996, section 2.7, pages 78-91).

The first decision to make is whether to use the sp^{3} basis set or the s,
p_{x}, p_{y}, p_{z} set. It turns out that unless one is going
to make further approximations, such as neglecting small overlaps, there is no advantage
in the hybrid set, and the latter set is just fine. Then the matrix elements are
proportional to V_{ss}, V_{sp}, V_{xx} and V_{yy},
in the notation used by Yu & Cardona (1996, equation 2.80, page 84). Each of these are
linear combinations of the 4 V's given previously, as shown below in equation 8.3.
These details can be explored further via
problem 7.4.2, or equivalently by reverse
engineering Edward's computer program for the silicon band structure. In addition, Jony
Hudson did a project on this topic in 2000, and produced a mathematica program which can
be seen here, complete with animations. In the next
project of this type Fridrik Magnus has produced a MatLab program to explore the
differences between C(diamond), Si and Ge: why are the band structures of these
supposedly similar materials different?

_{ss} = 4V_{sss};
V_{sp} = -(4/Ö
3)V_{sps};3V _{xx} = 4V_{pps} +
8V_{ppp} and3V _{yy} = 4V_{pps} -
4V_{ppp}
(8.3) |

The sums over phase factors, called g_{1} to g_{4}, are given by Yu &
Cardona (1996, equations 2.81-82, page 85). They have a structure very like the hybrids
themselves, reflecting the four neighbours and whether the phases are positive or negative,
as in equation 7.2. But, as a warning, take care to get
the signs right if you actually use these equations, don't take my word for it.

Now, we can begin to construct the 8x8 matrix itself, in terms of 8 columns labelled
S1, S2, X1, Y1, Z1, X2, Y2, Z2 and equivalently 8 rows, as in Table 8.1 below. Here 1,2 represent
the two non-equivalent atoms, S is an s-state, and X,Y,Z are p-states oriented along the
corresponding axes. Since we are using *atomic* s- and p-orbitals for the basis,
you should be able to fill in which matrix elements are zero by orthogonality. The diagonal
elements should also be clear. The matrix, as all Hamiltonian matrices, is
Hermitian, so that gives a
further clue. The rest, however, is in the details given below the table, which you can
find in Yu and Cardona (1996, Table 2.25, page 85). In Edward's program, which was
implemented from this reference, specific directions for the **k**-values have been
chosen so as to produce the equivalent of Yu and Cardona's figure 2.24. This is not a trivial
exercise- Jony Hudson spent a good deal of time on this during his project in 2000.

Here A = *E*_{s} - *E*_{k};
B = *E*_{p} - *E*_{k};
C = V_{ss}g_{1}; D = V_{sp}g_{2};
E = V_{sp}g_{3}; F = V_{sp}g_{4};
G = V_{xx}g_{1}; H = V_{xy}g_{2};
J = V_{xy}g_{3} and K = V_{xy}g_{4}.
From Yu and Cardona (1996, Table 2.25).

The V_{ss} etc., values to be used in this program were given by Qian and Chadi (1987),
as discussed by Sutton, pages 124-131. These values (in eV) are D =
1.61, (sss) = -1.9375, (sps) = 1.745,
(pps) = 3.050, and (ppp) = -1.075.
(Don't be overawed by the apparent accuracy here, the .025's must result from dividing by 4).
Thus such a calculation represents in practice a five-parameter fit to the experimental band
structure of silicon. Different values can be used to model the band structures of diamond or
germanium, with appropriate values given by Yu and Cardona (1996, Table 2.26, page 86), based
on Chadi and Cohen's work. The figures and discussion which follow indicate, both
qualitatively and quantitatively, some of the differences between these solids. Here, we
have set out on a walk through the foothills of this topic; there are a lot more details
to explore if we had the time or inclination to put on our climbing boots, and the stamina
to hold on tight and keep going.

When computers had small memories, it was essential to use symmetry arguments to the maximum extent to reduce the size of the matrix, and hence the time of the calculation. It still is a good idea, and at the limit of what is possible, this will always be important. However, as speeds and memory have increased dramatically, symmetry (or equivalently group theory) arguments have become less important, and small clusters can now be routinely solved very quickly. The development of these cluster codes was the reason why John Pople received his share of the 1998 Nobel prize for Chemistry - the other half going to Walter Kohn for density functional methods. Several cluster and DFT codes are available locally on our bfg computer, and the Theoretical Chemistry group is linked into various consortia which use and develop such codes, e.g. AIMPRO at Exeter and Newcastle.

As one can appreciate from this outline description, actually doing a real calculation is
computationally very intensive, and it also has to be repeated for different ionic positions
**R**, and relaxed to the minimum energy configuration in the most efficient way.
Individual orbitals are often expanded in 'Gaussian orbitals', or in plane waves with a large
number M of independent coefficients c_{k} of the various **k**-vectors which
have to be computed as the calculation proceeds. For N electrons, the number of computing
operations to diagonalise a matrix scales as (MN)^{3}. However, many of the
operations required for each **k** are identical, so the code can be written for
implemention on parallel (super)computers. Now, systems with N ~several 100’s and
typical M~1000 can be tackled. The virtue of pseudo-potential calculations is that they
reduce the number of electrons per atomic site, at the cost of increased complexity of the
ionic (external) potential. There is a strong impetus to reduce the cubic power law to
something lower, which one can summarise by exclaiming: *O(N) methods are in!* This is
a major area for development, with even small advances being highly prized.

However, no actual methods are as good as O(N), the best perhaps scaling as Nln(N), and one
also needs to look critically at the multiplying constants. Many careers have
been spent trying to crack these highly technical conceptual and computational problems.
A feel for how this is going can be gauged by consulting Goringe *et al.* (1997),
Turchi *et al.* (1998), or Gödecker (1999), and doubtless yet more in the period 2000-05.
If you do that, you will know more than we do: so we're done for *this* year.
See you sometime in the next.