# by Oldenbourg Wissenschaftsverlag, München
Electronic structures of Al-based transition metal compounds
and effective valence of transition elements
Kaoru Watanabe and Yasushi Ishii*
Department of Physics, Chuo University, Kasuga, Tokyo, 112-8551, Japan
Received June 10, 2008; accepted July 21, 2008
Electronic structure / Band structure / Quasicrystal /
Intermetallic compound
Abstract. Electronic structures of Al-TM (transition metal) compounds with rather simple B2, L12 and A15 structures are studied. Focusing on the energy bands, which are
not hybridized strongly with the TM-d bands, we estimate
the number of electrons occupying the states below the
gap induced by periodic potentials and hence obtain the
average number of valence electrons per atom e=a.
Although the values of e=a obtained for the different
structures are scattered in spite of combination of the
same chemical species, these values are obtained if one
assumes similar negative values for effective valence of
TM. Roles of the interference effect opening the gap in
the nearly-free-electron bands are discussed.
1. Introduction
Icosahedral quasicrystals (QCs) are formed in many Albased transition-metal (TM) alloys and stable ones are
usually obtained by controlling the average number of valence electrons par atom, which is denoted as e=a [1]. It is
known that the electronic density of states (DOS) for the
QCs and approximant crystals has universally a minimum
near the Fermi energy and it is believed that this pseudogap in the electronic states contributes to stabilization of
these compounds [2]. If the Fermi energy is adjusted to
the pseudogap, the band energy is reduced. So the valence
electron density e=a has been considered as a key parameter in predicting new QCs.
Aluminum has three valence electrons and TM has a
partially occupied d band. In Al-TM compounds, the valence electrons of Al, which is less electro-negative than
TM, fill the TM-d band and the TM elements behave as
acceptors of electrons. Raynor [3] expressed such effects
in terms of the effective negative valence for the TM elements. It is known that the valence electron concentration
for the stable Al-TM QCs falls about e=a ¼ 1:7, which is
obtained by assuming the negative valence for TM proposed by Raynor. Physics of the effective negative valence
* Correspondence author (e-mail: [email protected])
has been discussed repeatedly [4, 5, 6] but is still not fully
clarified.
In this paper, we investigate electronic structures of AlTM compounds with rather simple structures. By studying
the systems without structural complexity, we try to understand an essence of an interplay between nearly-free valence electrons and TM-d electrons. We shall estimate the
number of electrons occupying the nearly-free-electron
bands below the gap induced by periodic potentials and
discuss roles of interference effects opening the gap in the
nearly-free-electron bands and the effective valence of TM
in Al-TM compounds.
2. Models and calculations
We calculate electronic structures of Al-based TM alloys
with B2, L12 and A15 structures. Atomic arrangements in
a unit cell are shown in Fig. 1. Space group of these structures is Pm3m and atomic compositions in a cubic unit
cell are AlTM, Al3 TM and Al6 TM2 , respectively, for the
B2, L12 and A15 structures. Except for a few systems, the
Al-TM compounds calculated here are hypothetical and a
lattice constant is determined to minimize the total energies. In this paper we shall report the results for Al––Mn
compounds. The lattice constants of a cubic cell are 2.92,
3.82 and 4.86 [
A] for the B2, L12 and A15 structures,
respectively.
Calculations have been done with a program package,
WIEN2k [7]. We use energy-independent augmented plane
waves with local orbitals (APW+lo) for valence electrons
and localized orbitals for the semicore states of Al-2p and
Fig. 1. Crystal structures of binary Al-TM compounds: B2, L12 and
A15 structures.
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Z. Kristallogr. 223 (2008) 739–741 / DOI 10.1524/zkri.2008.1045
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TM-3p as basis functions. Electron-electron interaction is
treated within the density functional theory with the generalized gradient approximation (GGA) proposed by Perdew,
Burke and Erzerhof [8]. Muffin-tin radius for Al is taken
as 2.2 [au] whereas that for TM is chosen so that neighbouring muffin-tin spheres are touching with each other.
Typical kinetic energy cut-off for APW is about 230 [eV].
3. Results and discussions
In Fig. 2, we show band structures of Al-Mn with different structures. The TM-3d character of the energy bands is
indicated by circles. One can see that narrow bands near
the Fermi energy have strong TM-3d character whereas
wide bands with less TM-3d character show free-electron
like dispersion. The TM-3d bands are located above
3.0 eV and the number of branches of the wide bands
below them increases with increasing a size of the unit
cell. In the energy range where the TM-3d and wide
bands coexists, most of the bands have considerable
amount of the TM-3d components but there are some
branches which are not hybridized strongly with the TM3d bands as seen in the A15 case. We focus on such
branches with the free-electron like dispersion.
In Fig. 3, we show dispersion of free electrons on an
empty lattice. Degenerate states at the symmetric zone
boundary split into several levels by onset of periodic potentials. Splitting of the lowest states at X, M and R
points and the second lowest state at G point are summarized in Table 1 where K and eðKÞ denote, respectively,
wavevector in unit of p=a and kinetic energy in unit of
ðp=aÞ2 =2 with a being the lattice constant. Vlmn is a Fourier
component
of
the
periodic
potential
at
G ¼ 2p=a ðl; m; nÞ. For the A15 structure, where V111
and V100 vanish, the lowest states at R and M points split
into two levels whereas the second lowest states at G point
split into three levels.
Considering the level splittings at the zone boundary
for the free-electron bands, we can assign the free-electron
bands to the wide bands in the Al-TM compound. For
example, for the A15 compound, six branches near R
point below 4 eV are assigned, from bottom to top, as
R1, R2, R2, R3, R3 and R4 bands for the free-electron
model shown in Fig. 3. We have band crossings not only
at the zone boundary but also at points inside the Brillouin zone (BZ). For example, the band R3 crosses with
K. Watanabe and Y. Ishii
the band R5 at a point L where the corresponding wave
vector K is (7=3, 1=3, 1=3) and (5=3, 5=3, 1=3) in unit of
p=a.
Band splittings at the crossing point L are seen for the
energy bands in the A15 compound but the states near the
point L are strongly hybridized with the TM-3d band for
the L12 and B2 compounds. In the nearly-free-electron
(NFE) model, interference of electronic waves with the
wavevectors k and k þ G, where G is a reciprocal lattice
vector, induces splitting of the energy bands near the band
crossing where eðkÞ eðk þ GÞ. Let us assume that the
valence electrons are sp electrons occupying the NFE
bands below the gap induced by the interference effect
and other sp electrons participate in the sp-d hybridization
to yield a significant part of cohesion. To analyze the
number of valence electrons, we assign zone planes responsible for the energy gap in the NFE bands. In the
assignment, if the free-electron branch is hybridized
strongly with the TM-3d band near the energy gap, we
exclude it.
For the B2 structure, we assign the zone plane as a
plane passing the point K ¼ ð1; 0; 0Þ and perpendicular to
K because the states near the M point (K ¼ ð1; 1; 0Þ) have
significant amounts of the TM-3d character. For the L12
structure, the states at the R point (K ¼ ð1; 1; 1Þ) are still
free-electron like but those near the second lowest state at
the G point (K ¼ ð2; 0; 0Þ) are hybridized with the TM-3d
band. So the zone plane responsible for the gap induced
by periodic potentials is assumed to be a plane passing the
point (1, 1, 1). For the A15 structure, the free-electron
band extends to much higher energies. We assign the zone
planes responsible for the gap in the free-electron band as
planes passing the points (7=3, 1=3, 1=3) and (5=3, 5=3, 1=3)
corresponding the band crossing at the point L, and those
passing the points (7=3, 1=3, 0) and (5=3, 5=3, 0) corresponding that at the point S in Fig. 3.
The number of electrons below the gap is estimated
from a volume of a zonohedron (a pseudo-BZ) constructed
from the zone planes assumed above. The results are 1, 4
and 7.68, in unit of ð2p=aÞ3 , for the B2, L12 and A15
structures, respectively. As the number of electrons occupying a single band in the first BZ, whose volume is
ð2p=aÞ3 , is 2, the number of electrons is given by 2 a
ratio of a volume of the pseudo-BZ to that of the 1st BZ.
The results are 2, 8 and 15.4 and thus the valence electron
density e=a is estimated as 1.0, 2.0 and 1.92, respectively,
for the B2, L12 and A15 structures. Although the values
Fig. 2. Dispersion curves for the energy
bands of Al-Mn with B2, L12 and A15
structures. The TM-3d character of the
energy bands is indicated by circles.
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Fig. 3. Dispersion curves for free electrons on an empty lattice.
Table 1. Splitting of degenerate free-electron states.
Point
K
eðKÞ
Splitting (degeneracy)
X
ð1; 0; 0Þ
1
V100
M
ð1; 1; 0Þ
2
V110 2V100
R
ð1; 1; 1Þ
3
V110 (2 fold)
3V110 ð3V100 þ V111 Þ
G
ð2; 0; 0Þ
ð0; 2; 0Þ
4
V110 ðV100 V111 Þ (3 fold)
ð0; 0; 2Þ
V200 (3 fold)
V200 2V110 (2 fold)
V200 þ 4V110
low the Fermi energy, not the total number of electrons in
the occupied states.
The valence electron density e=a less than 2 is interpreted as that only a part of Al and TM-4s electrons occupies the NFE band. The rest of the valence electrons, including those occupying the states above the gap in the
NFE bands, are hybridized with the TM-3d states to yield
significant stability. The pseudogap coming from the interference effect of the nearly-free electronic waves is located
at more-than 1 eV below the Fermi level and that near the
Fermi level comes from the hybridization effect. So we
conclude that the most important role of the interference
effect in Al-TM compounds is not to gain the band energy
of the NFE bands but to control the number of sp electrons participating in the hybridized states with the TM-3d
states.
We have not presented in this report the results for
other TM’s. If one changes TM from early transition elements to late ones, the bottoms of the wide sp band and
TM-3d band shift to lower energies and the TM-3d band
width decreases. Within the present analysis, however, the
zone plane, at which the gap in the NFE bands is open, is
not changed significantly for different TM’s, To elucidate
the individual effective valence for TM elements, more
quantitative and systematic assignment of the electronic
states as NFE ones might be desired.
Acknowledgments. This work is partly supported by Chuo University
Grant for Special Research.
References
of e=a are scattered in spite of combination of the same
chemical species, it is interesting to note that these values
are obtained if one assumes similar negative values about
1 for effective valence of Mn.
4. Conclusion
We have studied the electronic structures of Al-TM compounds with the B2, L12 and A15 structures. We focused
on the energy bands, which are not hybridized strongly
with the TM-3d bands. Assuming the zone plane, at
which the energy gap in the NFE bands is open, we estimated the number of valence electrons and hence obtained
the effective valence electron density e=a. Although the
values of e=a are scattered in spite of combination of the
same chemical species, these values are obtained if one
assumes similar negative values for effective valence of
TM. Here we note that a negative value for the effective
valence of TM is obtained by counting the number of
electrons only in the NFE bands up to the energy far be-
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Electronic structures of Al-based transition metal compounds