SUMMARY Ch. 14 "CELL METHODS"
Until now, all bandstructure methods discussed (PW, OPW, Pseudopotentials)
are based on a Bloch ansatz, i.e., the calculated basic functions and
wavefunctions are defined for the whole volume of the crystal.
14.1 Basic idea of cell methods
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Wigner and Seitz, 1933:
(*) The crystal volume is subdivided in unit cells which reflect the
symmetry of the lattice as good as possible:
---> Wigner-Seitz cell (WSC).
(*) The Schroedinger equation is solved only inside and on the
surface of the WSC, but including boundary conditions that
guarantee the fulfillment of Bloch's theorem.
It is a property of the WSC that its surface consists of pieces
of area that are pairwise parallel with a primitive translation vector
R between them (see the figure on p. 174; NOTE THE WRONG DESCRIPTION).
Using r_B = r_A + R , the Bloch condition for a wavefunction
reads
psi_k(r_B) = psi_k(r_A+R) = exp{i vec(k).vec(R)} psi_k(r_A) (14.1)
That means: we are looking for a solution of the Schroedinger equation
H psi_k(r) = E_k psi_k(r) for vec(r) in WSC with
H = -hbar^2/(2m) nabla^2 + V(r) ,
where V(r) is lattice-periodic. Additionally, the relatively complicated
boundary conditions are to be fulfilled.
This problem can be significantly simplified by neglecting the spatial
anisotropy of the potential:
V(vec(r)) approx V(r) .
Remember: if we had an atomic problem, we would proceed as follows:
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Separation ansatz: psi_{nlm}(vec(r)) = R_{nl} Y_{lm}(theta,phi) .
For the radial part of the wavefunction, one has the differential
equation (14.2) with the conditions
R_{nl}(r -> 0) must not be infinite, and R_{nl}(r -> infinity) = 0 .
These functions define an eigenvalue problem which gives non-trivial
solutions only for (descrete) energies E_{nl}.
However, in our "crystal problem", the situation is quite different!
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The condition for r -> infinity has to be reset by the k-dependent
condition (14.1) on the surface of the WSC.
If the fulfillment of this condition is - for the moment - neglected,
one can find a regular solution of Eq. (14.2) for ANY value of E;
in that case, the principal quantumnumber "disappears", and only the
quantumnumber l for the angular momentum is further valid [see Eq. (14.3)].
The corresponding solution of the Schroedinger equation reads
psi_{l,m}(r;E) = R_{l}(r;E) Y_{l,m}(theta,phi)
where E is no more an eigenvalue but a parameter of the solution!
It can be shown that no one of these partial solutions for any E is able
to fulfill the boundary condition (14.1) on the surface of the WSC.
In principle, however, it should be possible to succeed by a linear
expansion of such partial solutions [see Eq. (14.4)].
It should be possible to determine the coefficients b_{lm} and the
quantity E such the boundary conditions (14.1) are completely fulfilled.
Unfortunately, due to the complexity of such a problem, the practical
use of this method is very small.
14.2 The muffin-tin concept
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The definition of a MUFFIN-TIN potential is demonstrated in the figure
on page 176 of the lecture notes: the crystal potential is approximated
by a spherically symmetrical function inside so-called "muffin-tin spheres"
with the "muffin-tin radius" r_{MT}; between these spheres (which are not
allowed to overlap!), in the "interstitial region", the potential is
approximated by a constant [see Eq. (14.5)].
Is such a muffin-tin approximation is a good one for a crystal potential?
This question is to be answered in Sec. 15.3.1 of the next chapter about
the APW method.
(***) The muffin-tin concept makes the cell method to a useful procedure!
WHAT IS THE PROGRESS ?
In the intestitial region where the potential is considered to be constant,
the corresponding solution of the Schroedinger equation is - of course -
a plane wave.
Due to the fact that the surface of the WSC lies always in the interstitial
region, the boundary condition (14.1) has to be applied on plane waves
which IS NO PROBLEM because these functions "automatically" fulfill the Bloch
condition (14.1).
Therefore, in the muffin-tin case, the following ansatz for the solution
phi_k(vec(r)) of the Schroedinger equation within and on the surface of the
WSC is possible:
(*) Inside the muffin-tin sphere, phi_k(vec(r)) is given by Eq. (14.6);
(*) Outside the muffin-tin sphere (in the interstitial region), phi_k(vec(r))
is given by a plane wave [Eq. (14.7)].
Now to an IMPORTANT QUESTION:
Is the function defined by Eqs. (14.6) and (14.7) a quantummechanical
wavefunction?
In that case, phi_k(vec(r)) has to fulfill the following conditions:
(1) phi_k(vec(r)) has to be continuous function within the whole WSC.
This condition is fulfilled for the whole WSC EXCEPT on the surface
of the muffin-tin sphere. The latter problem can be solved by choosing
the parameters A_{lm}(k) in Eq. 14.6 such that the functions (14.6)
and (14.7) have the same values on the muffin-tin surface.
This is done by expanding the plane wave of Eq. (14.7) with respect
of spherical Bessel functions and spherical harmonics
("Rayleigh expansion"); the corresponding calculation, shown on
page 177, leads to the result Eq. (14.8).
By this procedure, the ansatz function phi_k(vec(r)) becomes a
so-called AUGMENTED PLANE WAVE (APW), given in Eq. (14.9).
(2) the first derivative of phi_k(vec(r)) has to be a continuous function
within the whole WSC.
It is not possible to fulfill this condition by a single APW.
However, it can be shown that it is possible to describe the desired
wavefunction of a Bloch electron by a linear expansion of APWs like
Eq. (14.10).
This expansion represents the basis of a very successful bandstructure method,
which is called
THE AUGMENTED PLANE WAVE METHOD (APW METHOD).