Publication in the honor of Prof. T. Fujiwara.
Editors: Y. Hatsugai, M. Arai, S. Yamamoto, 2007, p. 128-144
Effect of transition-metal elements on the
electronic properties of quasicrystals and
complex aluminides
Guy TRAMBLY de LAISSARDIÈRE 1, Didier MAYOU 2
1 Laboratoire de Physique Théorique et Modélisation, CNRS et
Université de Cergy–Pontoise, France.
guy.trambly@u-cergy.fr
2 Institut Néel, CNRS et Université Joseph Fourier, Grenoble, France.
didier.mayou@grenoble.cnrs.fr
1 Introduction
It is with great pleasure that we contribute to this book in honor of Prof.
Takeo Fujiwara. GTL enjoyed eighteen months of Prof. Fujiwara’s hospi-
tality at the University of Tokyo during the early 1990’s. At that time the
work of Prof. Fujiwara in the field of electronic structure of quasicrystals
had already made a major contribution to the literature (see for instance
[1]). Since that time our research owes much to his work.
Prof. Fujiwara was the first who performed realistic calculations of the
electronic structure in quasicrystalline materials without adjustable param-
eters (ab-initio calculations) [2]. Indeed these complex alloys [3] have very
exotic physical properties (see Refs. [4, 5] and Refs therein), and it rapidly
appeared that realistic calculations on the actual quasicrystalline materials
are necessary to understand the physical mechanism that govern this prop-
erties. In particular, these calculations allow to analyze numerically the role
http://arxiv.org/abs/0704.0532v1
2 ELECTRONIC STRUCTURE 2
of transition-metal elements which is essential in those materials.
In this paper, we briefly present our work on the role of transition-metal
element in electronic structure and transport properties of quasicrystals and
related complex phases. Several Parts of these works have been done or
initiated in collaboration with Prof. T. Fujiwara.
2 Electronic structure
2.1 Ab-initio determination of the density of states
A way to study the electronic structure of quasicrystal is to consider the case
of approximants. Approximants are crystallines phases, with very large unit
cell, which reproduce the atomic order of quasicrystals locally. Experiments
indicate that approximant phases, like α-AlMnSi, α-AlCuFeSi, R-AlCuFe,
etc., have transport properties similar to those of quasicrystals [4, 6]. In 1989
and 1991, Prof. Fujiwara performed the first numerical calculations of the
electronic structure in realistic approximants of quasicrystals [2, 7, 8]. He
showed that their density of states (DOS, see figure 1) is characterized by a
depletion near the Fermi energy EF, called “pseudo-gap”, in agreement with
experimental results (for review see Ref. [4, 9, 18]) and a Hume-Rothery
stabilization [10, 11]. The electronic structure of simpler crystals such as
orthorhombic Al6Mn, cubic Al12Mn, present also a pseudo-gap near EF
which is less pronounced than in complex approximants phases (figure 1)
[11].
2.2 Models to analyze the role of transition-metal element
sp–d hybridization model
The role of the transition-metal (TM, TM= Ti, Cr, Mn, Fe, Co, Ni) elements
in the pseudo-gap formation has been shown from experiments, ab-initio
calculations and model analysis [4,13–19,11]. Indeed the formation of the
pseudo-gap results from a strong sp–d coupling associated to an ordered
2 ELECTRONIC STRUCTURE 3
-12 -10 -8 -6 -4 -2 0 2 4
-12 -10 -8 -6 -4 -2 0 2 4
Energy (eV)
Fα-Al
Figure 1: Ab-initio total DOS of Al6Mn (simple crystal) and α-
Al69.6Si13.0Mn17.4 (approximant of icosahedral quasicrystals) [11, 12].
sub-lattice of TM atoms [19, 11]. Consequently, the electronic structure, the
magnetic properties and the stability, depend strongly on the TM positions,
as was shown from ab-initio calculations [28–33,20,21].
How an effective TM–TM interaction induces stability?
Just as for Hume-Rothery phases a description of the band energy can be
made in terms of pair interactions (figure 2) [17, 19]. Indeed, it has been
shown that an effective medium-range Mn–Mn interaction mediated by the
sp(Al)–d(Mn) hybridization plays a determinant role in the occurrence of
the pseudo-gap [19]. We have shown that this interaction, up to distances
10–20 Å, is essential in stabilizing these phases, since it can create a Hume-
Rothery pseudo-gap close to EF. The band energy is then minimized as
shown on figure 3 [20, 11].
2 ELECTRONIC STRUCTURE 4
2 3 4 5 6 7 8 9 10 11 12 13
Mn−Mn distance r (A)
-0.04
-0.02
with repulsive term
without repulsive term
repulsive term : b e
− a r 
r = 4.8 A
r = 6.7 A
Figure 2: Effective medium-range Mn–Mn interaction between two non-
magnetic manganese atoms in a free electron matrix which models aluminum
atoms. [11]
0 4 8 12 16 20 24 28 32 36 40
L (A)
α-AlMnSi
β-AlMnSi
Figure 3: Variation of the band energy due to the effective Mn–Mn interac-
tion in o-Al6Mn, α-AlMnSi and β-Al9Mn3Si. [20]
The effect of these effective Mn–Mn interactions has been also studied
by several groups [17, 20, 21] (see also Refs in [11]). It has also explained the
origin of large vacancies in the hexagonal β-Al9Mn3Si and ϕ-Al10Mn3 phases
on some sites, whereas equivalent sites are occupied by Mn in µ-Al4.12Mn and
λ-Al4Mn, and by Co in Al5Co2 [20]. On the other hand, an spin-polarized
2 ELECTRONIC STRUCTURE 5
effective Mn–Mn interaction is also determinant for the existence (or not)
of magnetic moments in AlMn quasicrystals and approximants [21, 22, 32].
The analysis can be applied to any Al(rich)-Mn phases, where a small
number of Mn atoms are embedded in the free electron like Al matrix. The
studied effects are not specific to quasicrystals and their approximants, but
they are more important for those alloys. Such a Hume-Rothery stabiliza-
tion, governed by the effective medium-range Mn–Mn interaction, might
therefore be intrinsically linked to the emergence of quasi-periodicity in
Al(rich)-Mn system.
Cluster Virtual Bound states
One of the main results of the ab-initio calculations performed by Prof.
Fujiwara for realistic approximant phases, is the small energy dispersion
of electrons in the reciprocal space. Consequently, the density of states of
approximants is characterized by “spiky” peaks [2, 7, 8, 28]. In order to
analyze the origin of this spiky structure of the DOS, we developed a model
that show a new kind of localization by atomic cluster [23].
As for the local atomic order, one of the characteristics of the quasicrys-
tals and approximants is the occurrence of atomic clusters on a scale of 10–30
Å [25]. The role of clusters has been much debated in particular by C. Janot
[24] and G. Trambly de Laissardière [23]. Our model is based on a standard
description of inter-metallic alloys. Considering the cluster embedded in a
metallic medium, the variation ∆n(E) of the DOS due to the cluster is cal-
culated. For electrons, which have energy in the vicinity of the Fermi level,
transition atoms (such as Mn and Fe) are strong scatters whereas Al atoms
are weak scatters. In the figure 4 the variation, ∆n(E), of the density of
states due to different clusters are shown. The Mn icosahedron is the actual
Mn icosahedron of the α-AlMnSi approximant. As an example of a larger
cluster, we consider one icosahedron of Mn icosahedra.
∆n(E) of clusters exhibits strong deviations from the Virtual Bound
2 ELECTRONIC STRUCTURE 6
6 7 8 9 10 11 12 13 14 15
Energy E (eV)
)) 1 Mn atom
     1 Mn 
 icosahedron
 1 icosahedron 
    of 12 Mn 
    icosahedra
Figure 4: Variation ∆n(E) of the DOS due to Mn atoms. Mn atoms are
embedded in a metallic medium (Al matrix). From [23].
States (1 Mn atom) [26]. Indeed several peaks and shoulders appear. The
width of the most narrow peaks (50 − 100meV) are comparable to the fine
peaks of the calculated DOS in the approximants (figure 1). Each peak
indicates a resonance due to the scattering by the cluster. These peaks
correspond to states “localized” by the icosahedron or the icosahedron of
icosahedra. They are not eigenstate, they have finite lifetime of the order of
~/δE, where δE is the width of the peak. Therefore, the stronger the effect
of the localization by cluster is, the narrower is the peak. A large lifetime
is the proof of a localization, but in the real space these states have a quite
large extension on length scale of the cluster.
The physical origin of these states can be understood as follows. Elec-
trons are scattered by the Mn atoms of a cluster. By an effect similar to
that of a Faraday cage, electrons can by confined by the cluster provided
that their wavelength λ satisfies λ & l, where l is the distance between two
Mn spheres. Consequently, we expect to observe such a confinement by the
3 TRANSPORT PROPERTIES 7
cluster. This effect is a multiple scattering effect, and it is not due to an
overlap between d-orbitals because Mn atoms are not first neighbor.
3 Transport properties
Quasicrystals have many fascinating electronic properties, and in particular
quasicrystals with high structural quality, such as the icosahedral AlCuFe
and AlPdMn alloys, have unconventional conduction properties when com-
pared with standard inter-metallic alloys. Their conductivities can be as low
as 150–200 (Ω cm)−1 (see Refs. [4, 5, 27] and Refs. therein). Furthermore
the conductivity increases with disorder and with temperature, a behavior
just at the opposite of that of standard metal. In a sense the most striking
property is the so-called “inverse Mathiessen rule” according to which the
increases of conductivity due to different sources of disorder seems to be ad-
ditive. This is just the opposite that happens with normal metals where the
increases of resistivity due to several sources of scattering are additive. An
important result is also that many approximants of these quasicrystalline
phases have similar conduction properties. For example the crystalline α-
AlMnSi phase with a unit cell size of about 12 Å and 138 atoms in the unit
cell has a conductivity of about 300 (Ω cm)−1 at low temperature [4].
3.1 Small Boltzmann velocity
Prof. Fujiwara et al. was the first to show that the electronic structure
of AlTM approximants and related phases is characterized by two energy
scales [2, 7, 8, 28, 29] (see previous section). The largest energy scale, of
about 0.5−1 eV, is the width of the pseudogap near the Fermi energy EF. It
is related to the Hume–Rothery stabilization via the scattering of electrons
by the TM sub-lattice because of a strong sp–d hybridization. The smallest
energy scale, less than 0.1 eV, is characteristic of the small dispersion of the
band energy E(k). This energy scale seems more specific to phases related to
3 TRANSPORT PROPERTIES 8
Temperature
Metallic alloys
"Perfect" stable quasicrystals
Doped semi-conductors
4 K 300 K
Metastable quasicrystals (i-AlMn),
(i-AlCuFe and i- AlPdMn)
"Imperfect" stable quasicrystals
(i-AlLiCu)
Amorphous alloys
ρ Mott
Figure 5: Schematic temperature dependencies of the experimental resistiv-
ity of quasicrystals, amorphous and metallic crystals.
1e+14 1e+15
1/τ     (s−1)
Al (f.c.c.)
Temperature
Figure 6: Ab-initio elec-
trical resistivity versus
inverse scattering time,
in cubic approximant α-
Al69.6Si13.0Mn17.4, pure Al
(f.c.c.), and cubic Al12Mn.
the quasi-periodicity. The first consequence on transport is a small velocity
at Fermi energy, Boltzmann velocity, VB = (∂E/∂k)E=EF . From numerical
calculations, Prof. Fujiwara et al. evaluated the Bloch–Boltzmann dc con-
ductivity σB in the relaxation time approximation. With a realistic value
3 TRANSPORT PROPERTIES 9
of scattering time, τ ∼ 10−14 s [27], one obtains σB ∼ 10 − 150 (Ωcm)
−1 for
a α-AlMn model [8] and 1/1-AlFeCu model [28]. This corresponds to the
measured values [4, 6], which are anomalously low for metallic alloys. For
decagonal approximant the anisotropy found experimentally in the conduc-
tivity is also reproduced correctly [29].
3.2 Quantum transport in Quasicrystals and approximants
The semi-classical Bloch–Boltzmann description of transport gives inter-
esting results for the intra-band conductivity in crystalline approximants,
but it is insufficient to take into account many aspects due to the spe-
cial localization of electrons by the quasi-periodicity (see Refs. [34–43] and
Refs. therein). Some specific transport mechanisms like the temperature
dependence of the conductivity (inverse Mathiessen rule, the defects influ-
ence, the proximity of a metal / insulator transition), require to go beyond
a Bloch–Boltzmann analysis. Thus, it appears that in quasicrystals and re-
lated complex metallic alloys a new type of breakdown of the semi-classical
Bloch-Boltzmann theory operates. In the literature, two different unconven-
tional transport mechanisms have been proposed for these materials. Trans-
port could be dominated, for short relaxation time τ by hopping between
“critical localized states”, whereas for long time τ the regime could be dom-
inated by non-ballistic propagation of wave packets between two scattering
events.
We develop a theory of quantum transport that applies to a normal bal-
listic law but also to these specific diffusion laws. As we show phenomenolog-
ical models based on this theory describe correctly the experimental trans-
port properties [41, 42, 43] (compare figures 5 and 6).
3.3 Ab-initio calculations of quantum transport
According to the Einstein relation the conductivity σ depends on the diffu-
sivity D(E) of electrons of energy E and the density of states n(E) (summing
3 TRANSPORT PROPERTIES 10
the spin up and spin down contribution). We assume that n(E) and D(E)
vary weakly on the thermal energy scale kT , which is justified here. In that
case, the Einstein formula writes
σ = e2n(EF)D(EF) (1)
where EF is the chemical potential and e is the electronic charge. The tem-
perature dependence of σ is due to the variation of the diffusivity D(EF )
with temperature. The central quantity is thus the diffusivity which is re-
lated to quantum diffusion. Within the relaxation time approximation, the
diffusivity is written [41]
D(E) =
C0(E, t) e
−|t|/τ dt (2)
where C0(E, t) =
Vx(t)Vx(0) + Vx(0)Vx(t)
it the velocity correlation
functions without disorder, and τ is the relaxation time. Here, the effect
of defects and temperature (scattering by phonons ...) is taken into account
through the relaxation time τ . τ decreases as disorder increases. In the
case of crystals phases (such as approximants of quasicrystals), one obtains
[42, 43]:
σ = σB + σNB (3)
σB = e
2n(EF)V
B τ and σNB = e
2n(EF)
L2(τ)
where σB is actual the Bolzmann contribution to the conductivity and σNB
a non-Boltzmann contribution. L2(τ) is smaller than the square of the unit
cell size L0. L
2(τ) can be calculated numerically for the ab-initio electronic
structure [42]. From (3) and (4), it is clear that the Bolzmann term domi-
nates when L0 ≪ VBτ : The diffusion of electrons is then ballistic, which is
the case in normal metallic crystals. But, when L0 ≃ VBτ , i.e. when the
Bolzmann velocity VB is very low, the non-Bolzmann term is essential. In
the case of α-Al69.6Si13.0Mn17.4 approximant (figure 7) [42], with realistic
value of τ (τ equals a few 10−14 s [27]), σNB dominates and σ increases when
3 TRANSPORT PROPERTIES 11
1e+14 2e+14 3e+14 4e+14
1/τ     (s−1)
Figure 7: Ab-initio dc-conductivity σ in cubic approximant α-
Al69.6Si13.0Mn17.4 versus inverse scattering time. [42]
1e+14 2e+14 3e+14 4e+14
1/τ     (s−1)
1e+05
2e+05
3e+05
4e+05
Figure 8: Ab-initio dc-conductivity σ in an hypothetical cubic approximant
α-Al69.6Si13.0Cu17.4 versus inverse scattering time. [43]
1/τ increases, i.e. when defects or temperature increases, in agreement with
experimental measurement (compare figures 5 and 6).
To evaluate the effect of TM elements on the conductivity, we have
considered an hypothetical α-Al69.6Si13.0Cu17.4 constructed by putting Cu
atoms in place of Mn atoms in the actual α-Al69.6Si13.0Mn17.4 structure.
Cu atoms have almost the same number of sp electrons as Mn atoms, but
their d DOS is very small at EF. Therefore in α-Al69.6Si13.0Cu17.4, the effect
of sp(Al)–d(TM) hybridization on electronic states with energy near EF is
4 CONCLUSION 12
very small. As a result, the pseudogap disappears in total DOS, and the
conductivity is now ballistic (metallic), σ ≃ σB, as shown on figure 8.
4 Conclusion
In this article we present the effect of transition-metal atoms on the physical
properties of quasicrystals and related complex phases. These studies lead
to consider these aluminides as spd electron phases [11], where a specific
electronic structure governs stability, magnetism and quantum transport
properties. The principal aspects of this new physics are now understood
particularly thanks to seminal work of Prof. T. Fujiwara and subsequent
developpements of his ideas.
References
[1] Fujiwara T, Tsunetsugu H. In: Di Vincenxo DP, Steinhart PJ, editors,
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	Introduction
	Electronic structure
	Ab-initio determination of the density of states
	Models to analyze the role of transition-metal element
	Transport properties
	Small Boltzmann velocity
	Quantum transport in Quasicrystals and approximants
	Ab-initio calculations of quantum transport
	Conclusion

  In this paper, we briefly present our work on the role of transition-metal
element in electronic structure and transport properties of quasicrystals and
related complex phases. Several Parts of these works have been done or
initiated in collaboration with Prof. T. Fujiwara.

Introduction
It is with great pleasure that we contribute to this book in honor of Prof.
Takeo Fujiwara. GTL enjoyed eighteen months of Prof. Fujiwara’s hospi-
tality at the University of Tokyo during the early 1990’s. At that time the
work of Prof. Fujiwara in the field of electronic structure of quasicrystals
had already made a major contribution to the literature (see for instance
[1]). Since that time our research owes much to his work.
Prof. Fujiwara was the first who performed realistic calculations of the
electronic structure in quasicrystalline materials without adjustable param-
eters (ab-initio calculations) [2]. Indeed these complex alloys [3] have very
exotic physical properties (see Refs. [4, 5] and Refs therein), and it rapidly
appeared that realistic calculations on the actual quasicrystalline materials
are necessary to understand the physical mechanism that govern this prop-
erties. In particular, these calculations allow to analyze numerically the role
http://arxiv.org/abs/0704.0532v1
2 ELECTRONIC STRUCTURE 2
of transition-metal elements which is essential in those materials.
In this paper, we briefly present our work on the role of transition-metal
element in electronic structure and transport properties of quasicrystals and
related complex phases. Several Parts of these works have been done or
initiated in collaboration with Prof. T. Fujiwara.
2 Electronic structure
2.1 Ab-initio determination of the density of states
A way to study the electronic structure of quasicrystal is to consider the case
of approximants. Approximants are crystallines phases, with very large unit
cell, which reproduce the atomic order of quasicrystals locally. Experiments
indicate that approximant phases, like α-AlMnSi, α-AlCuFeSi, R-AlCuFe,
etc., have transport properties similar to those of quasicrystals [4, 6]. In 1989
and 1991, Prof. Fujiwara performed the first numerical calculations of the
electronic structure in realistic approximants of quasicrystals [2, 7, 8]. He
showed that their density of states (DOS, see figure 1) is characterized by a
depletion near the Fermi energy EF, called “pseudo-gap”, in agreement with
experimental results (for review see Ref. [4, 9, 18]) and a Hume-Rothery
stabilization [10, 11]. The electronic structure of simpler crystals such as
orthorhombic Al6Mn, cubic Al12Mn, present also a pseudo-gap near EF
which is less pronounced than in complex approximants phases (figure 1)
[11].
2.2 Models to analyze the role of transition-metal element
sp–d hybridization model
The role of the transition-metal (TM, TM= Ti, Cr, Mn, Fe, Co, Ni) elements
in the pseudo-gap formation has been shown from experiments, ab-initio
calculations and model analysis [4,13–19,11]. Indeed the formation of the
pseudo-gap results from a strong sp–d coupling associated to an ordered
2 ELECTRONIC STRUCTURE 3
-12 -10 -8 -6 -4 -2 0 2 4
-12 -10 -8 -6 -4 -2 0 2 4
Energy (eV)
Fα-Al
Figure 1: Ab-initio total DOS of Al6Mn (simple crystal) and α-
Al69.6Si13.0Mn17.4 (approximant of icosahedral quasicrystals) [11, 12].
sub-lattice of TM atoms [19, 11]. Consequently, the electronic structure, the
magnetic properties and the stability, depend strongly on the TM positions,
as was shown from ab-initio calculations [28–33,20,21].
How an effective TM–TM interaction induces stability?
Just as for Hume-Rothery phases a description of the band energy can be
made in terms of pair interactions (figure 2) [17, 19]. Indeed, it has been
shown that an effective medium-range Mn–Mn interaction mediated by the
sp(Al)–d(Mn) hybridization plays a determinant role in the occurrence of
the pseudo-gap [19]. We have shown that this interaction, up to distances
10–20 Å, is essential in stabilizing these phases, since it can create a Hume-
Rothery pseudo-gap close to EF. The band energy is then minimized as
shown on figure 3 [20, 11].
2 ELECTRONIC STRUCTURE 4
2 3 4 5 6 7 8 9 10 11 12 13
Mn−Mn distance r (A)
-0.04
-0.02
with repulsive term
without repulsive term
repulsive term : b e
− a r 
r = 4.8 A
r = 6.7 A
Figure 2: Effective medium-range Mn–Mn interaction between two non-
magnetic manganese atoms in a free electron matrix which models aluminum
atoms. [11]
0 4 8 12 16 20 24 28 32 36 40
L (A)
α-AlMnSi
β-AlMnSi
Figure 3: Variation of the band energy due to the effective Mn–Mn interac-
tion in o-Al6Mn, α-AlMnSi and β-Al9Mn3Si. [20]
The effect of these effective Mn–Mn interactions has been also studied
by several groups [17, 20, 21] (see also Refs in [11]). It has also explained the
origin of large vacancies in the hexagonal β-Al9Mn3Si and ϕ-Al10Mn3 phases
on some sites, whereas equivalent sites are occupied by Mn in µ-Al4.12Mn and
λ-Al4Mn, and by Co in Al5Co2 [20]. On the other hand, an spin-polarized
2 ELECTRONIC STRUCTURE 5
effective Mn–Mn interaction is also determinant for the existence (or not)
of magnetic moments in AlMn quasicrystals and approximants [21, 22, 32].
The analysis can be applied to any Al(rich)-Mn phases, where a small
number of Mn atoms are embedded in the free electron like Al matrix. The
studied effects are not specific to quasicrystals and their approximants, but
they are more important for those alloys. Such a Hume-Rothery stabiliza-
tion, governed by the effective medium-range Mn–Mn interaction, might
therefore be intrinsically linked to the emergence of quasi-periodicity in
Al(rich)-Mn system.
Cluster Virtual Bound states
One of the main results of the ab-initio calculations performed by Prof.
Fujiwara for realistic approximant phases, is the small energy dispersion
of electrons in the reciprocal space. Consequently, the density of states of
approximants is characterized by “spiky” peaks [2, 7, 8, 28]. In order to
analyze the origin of this spiky structure of the DOS, we developed a model
that show a new kind of localization by atomic cluster [23].
As for the local atomic order, one of the characteristics of the quasicrys-
tals and approximants is the occurrence of atomic clusters on a scale of 10–30
Å [25]. The role of clusters has been much debated in particular by C. Janot
[24] and G. Trambly de Laissardière [23]. Our model is based on a standard
description of inter-metallic alloys. Considering the cluster embedded in a
metallic medium, the variation ∆n(E) of the DOS due to the cluster is cal-
culated. For electrons, which have energy in the vicinity of the Fermi level,
transition atoms (such as Mn and Fe) are strong scatters whereas Al atoms
are weak scatters. In the figure 4 the variation, ∆n(E), of the density of
states due to different clusters are shown. The Mn icosahedron is the actual
Mn icosahedron of the α-AlMnSi approximant. As an example of a larger
cluster, we consider one icosahedron of Mn icosahedra.
∆n(E) of clusters exhibits strong deviations from the Virtual Bound
2 ELECTRONIC STRUCTURE 6
6 7 8 9 10 11 12 13 14 15
Energy E (eV)
)) 1 Mn atom
     1 Mn 
 icosahedron
 1 icosahedron 
    of 12 Mn 
    icosahedra
Figure 4: Variation ∆n(E) of the DOS due to Mn atoms. Mn atoms are
embedded in a metallic medium (Al matrix). From [23].
States (1 Mn atom) [26]. Indeed several peaks and shoulders appear. The
width of the most narrow peaks (50 − 100meV) are comparable to the fine
peaks of the calculated DOS in the approximants (figure 1). Each peak
indicates a resonance due to the scattering by the cluster. These peaks
correspond to states “localized” by the icosahedron or the icosahedron of
icosahedra. They are not eigenstate, they have finite lifetime of the order of
~/δE, where δE is the width of the peak. Therefore, the stronger the effect
of the localization by cluster is, the narrower is the peak. A large lifetime
is the proof of a localization, but in the real space these states have a quite
large extension on length scale of the cluster.
The physical origin of these states can be understood as follows. Elec-
trons are scattered by the Mn atoms of a cluster. By an effect similar to
that of a Faraday cage, electrons can by confined by the cluster provided
that their wavelength λ satisfies λ & l, where l is the distance between two
Mn spheres. Consequently, we expect to observe such a confinement by the
3 TRANSPORT PROPERTIES 7
cluster. This effect is a multiple scattering effect, and it is not due to an
overlap between d-orbitals because Mn atoms are not first neighbor.
3 Transport properties
Quasicrystals have many fascinating electronic properties, and in particular
quasicrystals with high structural quality, such as the icosahedral AlCuFe
and AlPdMn alloys, have unconventional conduction properties when com-
pared with standard inter-metallic alloys. Their conductivities can be as low
as 150–200 (Ω cm)−1 (see Refs. [4, 5, 27] and Refs. therein). Furthermore
the conductivity increases with disorder and with temperature, a behavior
just at the opposite of that of standard metal. In a sense the most striking
property is the so-called “inverse Mathiessen rule” according to which the
increases of conductivity due to different sources of disorder seems to be ad-
ditive. This is just the opposite that happens with normal metals where the
increases of resistivity due to several sources of scattering are additive. An
important result is also that many approximants of these quasicrystalline
phases have similar conduction properties. For example the crystalline α-
AlMnSi phase with a unit cell size of about 12 Å and 138 atoms in the unit
cell has a conductivity of about 300 (Ω cm)−1 at low temperature [4].
3.1 Small Boltzmann velocity
Prof. Fujiwara et al. was the first to show that the electronic structure
of AlTM approximants and related phases is characterized by two energy
scales [2, 7, 8, 28, 29] (see previous section). The largest energy scale, of
about 0.5−1 eV, is the width of the pseudogap near the Fermi energy EF. It
is related to the Hume–Rothery stabilization via the scattering of electrons
by the TM sub-lattice because of a strong sp–d hybridization. The smallest
energy scale, less than 0.1 eV, is characteristic of the small dispersion of the
band energy E(k). This energy scale seems more specific to phases related to
3 TRANSPORT PROPERTIES 8
Temperature
Metallic alloys
"Perfect" stable quasicrystals
Doped semi-conductors
4 K 300 K
Metastable quasicrystals (i-AlMn),
(i-AlCuFe and i- AlPdMn)
"Imperfect" stable quasicrystals
(i-AlLiCu)
Amorphous alloys
ρ Mott
Figure 5: Schematic temperature dependencies of the experimental resistiv-
ity of quasicrystals, amorphous and metallic crystals.
1e+14 1e+15
1/τ     (s−1)
Al (f.c.c.)
Temperature
Figure 6: Ab-initio elec-
trical resistivity versus
inverse scattering time,
in cubic approximant α-
Al69.6Si13.0Mn17.4, pure Al
(f.c.c.), and cubic Al12Mn.
the quasi-periodicity. The first consequence on transport is a small velocity
at Fermi energy, Boltzmann velocity, VB = (∂E/∂k)E=EF . From numerical
calculations, Prof. Fujiwara et al. evaluated the Bloch–Boltzmann dc con-
ductivity σB in the relaxation time approximation. With a realistic value
3 TRANSPORT PROPERTIES 9
of scattering time, τ ∼ 10−14 s [27], one obtains σB ∼ 10 − 150 (Ωcm)
−1 for
a α-AlMn model [8] and 1/1-AlFeCu model [28]. This corresponds to the
measured values [4, 6], which are anomalously low for metallic alloys. For
decagonal approximant the anisotropy found experimentally in the conduc-
tivity is also reproduced correctly [29].
3.2 Quantum transport in Quasicrystals and approximants
The semi-classical Bloch–Boltzmann description of transport gives inter-
esting results for the intra-band conductivity in crystalline approximants,
but it is insufficient to take into account many aspects due to the spe-
cial localization of electrons by the quasi-periodicity (see Refs. [34–43] and
Refs. therein). Some specific transport mechanisms like the temperature
dependence of the conductivity (inverse Mathiessen rule, the defects influ-
ence, the proximity of a metal / insulator transition), require to go beyond
a Bloch–Boltzmann analysis. Thus, it appears that in quasicrystals and re-
lated complex metallic alloys a new type of breakdown of the semi-classical
Bloch-Boltzmann theory operates. In the literature, two different unconven-
tional transport mechanisms have been proposed for these materials. Trans-
port could be dominated, for short relaxation time τ by hopping between
“critical localized states”, whereas for long time τ the regime could be dom-
inated by non-ballistic propagation of wave packets between two scattering
events.
We develop a theory of quantum transport that applies to a normal bal-
listic law but also to these specific diffusion laws. As we show phenomenolog-
ical models based on this theory describe correctly the experimental trans-
port properties [41, 42, 43] (compare figures 5 and 6).
3.3 Ab-initio calculations of quantum transport
According to the Einstein relation the conductivity σ depends on the diffu-
sivity D(E) of electrons of energy E and the density of states n(E) (summing
3 TRANSPORT PROPERTIES 10
the spin up and spin down contribution). We assume that n(E) and D(E)
vary weakly on the thermal energy scale kT , which is justified here. In that
case, the Einstein formula writes
σ = e2n(EF)D(EF) (1)
where EF is the chemical potential and e is the electronic charge. The tem-
perature dependence of σ is due to the variation of the diffusivity D(EF )
with temperature. The central quantity is thus the diffusivity which is re-
lated to quantum diffusion. Within the relaxation time approximation, the
diffusivity is written [41]
D(E) =
C0(E, t) e
−|t|/τ dt (2)
where C0(E, t) =
Vx(t)Vx(0) + Vx(0)Vx(t)
it the velocity correlation
functions without disorder, and τ is the relaxation time. Here, the effect
of defects and temperature (scattering by phonons ...) is taken into account
through the relaxation time τ . τ decreases as disorder increases. In the
case of crystals phases (such as approximants of quasicrystals), one obtains
[42, 43]:
σ = σB + σNB (3)
σB = e
2n(EF)V
B τ and σNB = e
2n(EF)
L2(τ)
where σB is actual the Bolzmann contribution to the conductivity and σNB
a non-Boltzmann contribution. L2(τ) is smaller than the square of the unit
cell size L0. L
2(τ) can be calculated numerically for the ab-initio electronic
structure [42]. From (3) and (4), it is clear that the Bolzmann term domi-
nates when L0 ≪ VBτ : The diffusion of electrons is then ballistic, which is
the case in normal metallic crystals. But, when L0 ≃ VBτ , i.e. when the
Bolzmann velocity VB is very low, the non-Bolzmann term is essential. In
the case of α-Al69.6Si13.0Mn17.4 approximant (figure 7) [42], with realistic
value of τ (τ equals a few 10−14 s [27]), σNB dominates and σ increases when
3 TRANSPORT PROPERTIES 11
1e+14 2e+14 3e+14 4e+14
1/τ     (s−1)
Figure 7: Ab-initio dc-conductivity σ in cubic approximant α-
Al69.6Si13.0Mn17.4 versus inverse scattering time. [42]
1e+14 2e+14 3e+14 4e+14
1/τ     (s−1)
1e+05
2e+05
3e+05
4e+05
Figure 8: Ab-initio dc-conductivity σ in an hypothetical cubic approximant
α-Al69.6Si13.0Cu17.4 versus inverse scattering time. [43]
1/τ increases, i.e. when defects or temperature increases, in agreement with
experimental measurement (compare figures 5 and 6).
To evaluate the effect of TM elements on the conductivity, we have
considered an hypothetical α-Al69.6Si13.0Cu17.4 constructed by putting Cu
atoms in place of Mn atoms in the actual α-Al69.6Si13.0Mn17.4 structure.
Cu atoms have almost the same number of sp electrons as Mn atoms, but
their d DOS is very small at EF. Therefore in α-Al69.6Si13.0Cu17.4, the effect
of sp(Al)–d(TM) hybridization on electronic states with energy near EF is
4 CONCLUSION 12
very small. As a result, the pseudogap disappears in total DOS, and the
conductivity is now ballistic (metallic), σ ≃ σB, as shown on figure 8.
4 Conclusion
In this article we present the effect of transition-metal atoms on the physical
properties of quasicrystals and related complex phases. These studies lead
to consider these aluminides as spd electron phases [11], where a specific
electronic structure governs stability, magnetism and quantum transport
properties. The principal aspects of this new physics are now understood
particularly thanks to seminal work of Prof. T. Fujiwara and subsequent
developpements of his ideas.
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	Introduction
	Electronic structure
	Ab-initio determination of the density of states
	Models to analyze the role of transition-metal element
	Transport properties
	Small Boltzmann velocity
	Quantum transport in Quasicrystals and approximants
	Ab-initio calculations of quantum transport
	Conclusion
