SUMMARY Ch. 11 "BANDSTRUCTURE METHODS -
DEVELOPMENT OF THE PLANE-WAVE (PW) METHOD"
11.1 Introduction
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We start from the stationary one-particle Schroedinger equation (11.1) including
an effective crystal potential which can be expanded into a Fourier series,
according to Eq. (11.2) for a Bravais lattice or to Eq. (11.3) for a lattice
with more than one ion per unit cell ("lattice with basis").
Eq. (11.4) shows the expansion of the wave function psi by using a basis of
Bloch functions phi_{k}(r). Inserting this ansatz into the Schroedinger
equation, the coefficients a_{t}(k) in Eq. (11.4) can be obtained by evaluating
a system of homogenous, linear equations given by Eqs. (11.5-11.8).
11.2 The plane-wave (PW) basis
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is given by Eq. (11.9) with k as a Bloch vector and K_{s} as a vector of the
reciprocal lattice.
This orthonormal [Eq. (11.10)] basis has Bloch character, and the Hamilton
matrix is given by Eq. (11.11).
For a crystal lattice with and without a basis, the corresponding system of
linear equations for the expansion coefficients reads as Eq. (11.12) and
Eq. (11.13), respectively. As ut can be seen, these equations contain the
crystal potential only by its Fourier coefficients V(K) with the following
properties:
(*) V(K_{t} - K_{s}) = V*(K_{s} - K_{t}) holds because the operator of the
potential energy is self-adjoint,
(*) if the crystal structure is inversion-symmetric in real space, V(K) has
the property V(K) = V(/K/).
It can be easily seen that - in case of V(r)=0 - one gets Sommerfelds's
dispersion rule E_{s}(k) prop /k + K_{s}/^2.
11.3 Advantages and disadvantages of the PW method
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Advantages: Eqs. (11.12) and (11.13) are extremely simple equations, needing
only the Fourier coefficients of the crystal potential and the
positions of the ions within the unit cell. The only approximation
included is the "one-particle approximation".
Disadvanteges: the convergence both of the energy bands and the Bloch functions
with respect to the number of the reciprocal vectors is
very poor!
Consequence: the order of the Hamilton matrix [Eq. (11.11)] is
used to be extremely large!
These statements are - in the following - demonstrated by the (chemically)
simplest metal, lithium: Z=3 electronic structure per atom = 1s2 2s1 .
In order to judge the quality of the PW results, the following test values are
compared to one of the state-of-the-art methods, the
"augmented plane-wave method" APW (see. Ch. 15).
11.3.1 The convergence of the Li energy bands
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Fig. 11.1: Electron bandstructure of Li along the Gamma H direction, calculated
by the APW method.
One sees they ver flat "core band" due to the 1s electrons, and the
2s "valence band" with a band width of 0.64309 Ry.
Fig. 11.2: Electron bandstructure of Li along the Gamma H direction, calculated
by APW (black dots) and by PW using a basis of 87 reciprocal-lattice
vectors (PW-87) (red dots).
The agreement is very poor: see the quite different positions of the
1s core band (PE-87 = -1.3 Ry, APW = -3 Ry);
see also the valence bands (the 2s band of PW is too narrow by
33 percent).
Of course, this situation will be improved by an increase of the PW basis =
an increase of the number of reciprocal-lattice vectors: this is documented by
Figs. 11.3 and 11.4 where the APW results are compared with PW-2123 results.
It's disappointing: even such high number of reciprocal-lattice vectors used
for the PW calculation does not lead to a perfect agreement
between PW and APW. The 1s core band obtained by the PW
method is still significantly too high (see Fig. 11.3),
and there are still significant differences concerning
the valence bands (see Fig. 11.4):
Fig. 11.3 and 11.4: Electron bandstructure of Li along the Gamma H direction,
calculated by APW (black dots) and by PW using a basis of
2123 reciprocal-lattice vectors (PW-2123) (red dots).
11.3.2 The PW convergence of Bloch functions
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After having solved the eigenvalue problem (11.12) or (11.13), the obtained
coefficients a_{s}(k) can be used to calculate the eigenfunctions of Eq. (11.1)
by inserting the PW basis (11.9) into the expansion (11.4): see Eq. (11.14).
If this equation is inserted into the normalization condition (11.15), one gets
the simple rule [Eq. (11.16)]:
the sum over all absolute squares of the (generally complex) PW coefficients
a_{s}(n,k) must be 1.0 !! [n is the band index, k is a Bloch vector of the
first BZ].
The following PW test is based on three different Bloch functions of electrons
in metallic copper. These electron states are indicated by (A), (B), and (C)
in the APW bandstructure diagram Fig. 11.5.
Fig. 11.5: The bandstructure of valence electrons in Cu along the Gamma X
direction.
The indicated states (A), (B), and (C) are discussed in Sec. 11.3.2,
and the numbers are discussed in Sec. 11.3.3.
(A) is a significantly localized state of one of the 3d bands of copper, lying
about 0.3 Ry below the Fermi energy. Such localized Bloch states generally
show a poor convergence of the PW expansion.
(B) means an electron Bloch state of the lowest valence band (4s) near Gamma,
the centre of the BZ.
(C) means an electron Bloch state of the lowest valence band (4s) near X on the
surface of the BZ.
The result of the correspondent "normalization study" is presented in the
Fig. 11.6: Dependence of the normalization sum [see Eq. (11.16)] of the Bloch
states (A), (B), and (C) in copper as a function of the numer of
plane waves used for the expansion (11.14).
As to be expected, the localized state (A) needs a very large number of plane
waves (>> 900) to fulfill the normalization condition (11.16).
Also not surprising: the good convergence of the strongly delocalized 4s
valence state (B).
At first sight - a surprise: the rather poor convergence of the state (C) which
also belongs to the 4s valence band.
This last effect is very important: therefore, the last section of Ch. 11 is
dedicated to this subject!
11.3. The notation of Bloch electron bands
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All physicists who deal with electron Bloch states in crystals should be
irridated by the fact that - even in high-level papers and textbooks in
solid-state physics - one can frequently read the terms "s-bands", "p-bands",
"d-bands" etc.
You know that - in ATOMIC physics - these letters s, p, d etc. are connected
to quantum numbers of eigenstates of the Hamiltonian including spherically
symmetric potentials. However, in crystalline materials, the potentials are
lattice-periodical and by no means spherically symmetric.
Therefore, for Bloch electrons, these quantum numbers of the angular momentum
(l=0, 1, 2, ... for s-, p-, d-states etc.) are not appropriate: they are no
"good quantum numbers" for Bloch states.
That's the theory. In practice, however, one observes that especially the
localized Bloch states in the core bands often have a marked "atomic-like"
character:
in such cases, it might be justified to call them 1s-, 2s-, 2p- etc.
"core bands".
But what's about the valence bands? Here one should be careful to use this
notation. To demonstrate this, the Fig. 11.5 with the valence bands of Cu
includes some numbers which are now to be discussed in detail:
Although it is wrong to qualify a valence Bloch state by a quantum number of the
angular momentum, it is always possible to expand such a Bloch state with
respect to the eigenfunctions of the operators L^2 and L_{z}, the well known
"spherical harmonics".
By doing so, each Bloch state can be described as a mixture of different states
of the angular momentum: I did this calculation for some Bloch states included
into Fig. 11.5, and the "boxes" in this figure give the percentages of the
different quantum numbers of the angular momentum.
Let's start with the the lowest valence band: for small values of k (point B),
this band is "almost s-like". Nevertheless, it is wrong to denote this band as
an 4s-band, because the s-character of this band significantly changes with
increasing k: the s-character decreases in favour of p and d, and close to the
surface of the BZ (point C), the band has become a (nearly) perfect d-band!
Now consider the next-higher, flat band in Fig. 11.5: here the notation
"3d-band" is clearly justified within the whole BZ.
In contrast to that, the valence band that intersects the Fermi energy line is
again a mixture of different angular momenta: it begins as an almost pure
d-band, and after a "mixed state" of s+p+d, it end as a band with pure
p-character.
Resume: be careful if you use the words "s-, p-, d-bands": such a notation is
sometimes ok, but sometimes completely wrong !!!