Thermoelectric properties of quasicrystals

THERMOELECTRIC
PROPERTIES OF
QUASICRYSTALS
Ante Bilušić
University of Split, Croatia

Outline
General introduction to quasicrystals
Transport in
icosahedral quasicrystal
decagonal quasicrystal
approximant
Figure of merit
Conclusions

A general intro to quasicrystals
Daniel Shechtman, April 8, 1984:
„There can be no such creature.”

A general intro to quasicrystals
„Hard”, intermetallic quasicrystals
binary and ternary intermetallics

A general intro to quasicrystals
Mathematical QC structures
Penrose tiling
Arab artists (13th – 15th century)

A general intro to quasicrystals
Fibbonacci series

A general intro to quasicrystals
Fibbonacci chain

Crystallography of quasicrystals
Two ways of QC structure description
tiling („quasilattice”) approach
e.g., decagonal d-Al-Co-Ni

Crystallography of quasicrystals
Two ways of QC structure description
higher-dimensional approach
• Fibonacci chain (1D QC)
• decagonal QC:
projection of 5D hypercube
on 3D real space

Crystallography of quasicrystals
Two ways of QC structure description
higher-dimensional approach
• Fibonacci chain (1D QC)
• 1D approximant

Motivation
What makes a main influence on transport in a
quasicrystal: quasiperiodicity or complex local
atomic structure?
The role of electronic structure of quasicrystals
to electronic transport?

Anisotropy in transport:
icosahedral quasicrystals
Icosahedron: 20 triangular faces, 30 edges, 12 vertices
C 2 axes: perpendicular to twofold planes and mutually
Cartesian axes
C 3 and C 5 : three- and five-fold axes
E.g.: electrical conductivity tensor (rank two)
C 2 axes in x, y, and z directions
off-diagonal elements are zero
x,y, and z axes are related by C 3 symmetry ops.
diagonal elements are equal
No anisotropy

Anisotropy in transport:
decagonal quasicrystals (d-Al-Co-
Ni)
Quasiperiodic plane is perpendicular to
periodic [00001] 10-fold axes
Two sets of ten 2-fold axes (2 and 2’)
mutually rotated by 18°
two emphasized ones are perpendicular
Axes:
x: axis 2
y: axis 2’
z: axis [00001]
Anisotropy between quasiperiodic and periodic axes.

Icosahedral quasicrystals:
example of i-Ag 42 In 42 Yb 16
Electrical resistivity
Four-probe, dc method
r 2 , r 3 , and r 5 : measured along
2-, 3-, and 5-fold axes
When normalized, no anisotropy is
observed, unles for 5-fold axis direction,
probably due to enhanced concentration
of defects
Bobnar et al., PRB 84 134205

Icosahedral quasicrystals:
example of i-Ag 42 In 42 Yb 16
Thermoelectric power
Steady-state method
S 2 , S 3 , S 5 coincide.
Thermal conductivity
Steady-state method
Anisotropy is small and comparable
to measuremnts uncertainity
Bobnar et al., PRB 84 134205

Icosahedral quasicrystals:
example of i-Al 64 Cu 23 Fe 13
Kubo-Greenwood formalism:
Dolinšek et al., PRB 76 054201

Icosahedral quasicrystals:
example of i-Al 64 Cu 23 Fe 13
Kubo-Greenwood formalism:
H. Solbrig, C.V. Landauro
Dolinšek et al., PRB 76 054201

Icosahedral quasicrystals:
example of i-Al 64 Cu 23 Fe 13
Kubo-Greenwood formalism:
H. Solbrig, C.V. Landauro
Dolinšek et al., PRB 76 054201
Tunneling spectra (differential conductivity)
of quasicrystals and approximant phases.
Davydov et al., PRL 77 3173

Icosahedral quasicrystals:
example of i-Al 64 Cu 23 Fe 13
Thermal conductivity: electronic + quasilattice
electron thermal conductivity???
Drude theory (Wiedemann-Franz law)
Kubo-Greenwood formalism
Dolinšek et al., PRB 76 054201

Icosahedral quasicrystals:
example of i-Al 64 Cu 23 Fe 13
Thermal conductivity: electronic + quasilattice
quasilattice thermal conductivity
Debye contribution (long-wavelength phonons only)
(Matthiesen’s rule)
Vibration spectrum
of Fibonacci chain
Kalugin et al., PRB 536 14145
Dolinšek et al., PRB 76 054201

Icosahedral quasicrystals:
example of i-Al 64 Cu 23 Fe 13
Thermal conductivity: electronic + quasilattice
quasilattice thermal conductivity
Contribution of inter-cluster hopping
of localized vibrations:
Total thermal conductivity
Dolinšek et al., PRB 76 054201

Bobnar et al, PRB 85 02420
Decagonal quasicrystals:
example of d-Al 69.7 Co 10.0 Ni 20.3

Bobnar et al, PRB 85 02420
Decagonal quasicrystals:
example of d-Al 69.7 Co 10.0 Ni 20.3

Approximant phase:
Dolinšek et al., PRB 76 17420
o-Al 80 Cr 15 Fe 5
a-direction corresponds to
periodic one in d-QC while
b and c to quasiperiodic directions

Thermoelectric figure of merit:
example of i-AlPdRe
An ideal thermoelectric material: phonon glass
– electron crystal
Seebeck effect
cold
warm
Peltier effect
Thermoelectric device
heating
cooling

Thermoelectric figure of merit:
example of i-AlPdRe
An ideal thermoelectric material: phonon glass
– electron crystal
has to create as less as possible Joule heat
(small resistance)
has to transfer electricity to heat as better as
possible
(large thermoelectric power)
has to be as poorer as possible heat conductor
thermoelectric figure of merit
(the good one: ZT > 3 or 4):

Thermoelectric figure of merit:
example of i-AlPdRe
3
2
1
0

Conclusions
Quasiperidicity does not have a key-role in
transport; the complex atomic order defines it.
Pseudogap in vicinity of Fermi level is important
for electronic transport in i-QC.
No transport anisotropy for i-QC; anisotropy
between periodic and quasiperiodic directions in
d-QC.