JOURNAL OF APPLIED PHYSICS 104, 063705 共2008兲
Transport properties of Ti-Zr-Ni quasicrystalline and glassy alloys
Y. K. Kuo,1,a兲 N. Kaurav,1 W. K. Syu,1 K. M. Sivakumar,2 U. T. Shan,3 S. T. Lin,3
Q. Wang,4 and C. Dong4
1
Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan
Department of Physics, Radharaman Group of Institutes, Ratibad, Bhopal, Madhya Pradesh 462046, India
3
Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
4
State Key Laboratory for Materials Modification, Department of Mechanical Engineering, Dalian
University of Technology, Dalian 116024, People’s Republic of China
2
共Received 16 April 2008; accepted 10 July 2008; published online 18 September 2008兲
We report on measurements of the temperature dependence of the electrical resistivity 共␳兲,
thermopower 共S兲, and thermal conductivity 共␬兲 of Ti-Zr-Ni alloys between 10 and 300 K. A series
of Ti-Zr-Ni quasicrystals 共QCs兲 Ti40Zr40Ni20, Ti45Zr35Ni20, and Ti50Zr30Ni20 and metallic glasses
Ti35Zr45Ni20 and Ti40Zr40Ni20 was prepared to systematically study the compositional and structural
dependences of their transport properties. The resistivity of all these alloys was found to be very
weakly temperature dependent with a negative temperature coefficient of resistance. Further, the
observed increase in electrical resistivity with increasing Ti/Zr ratio is most likely due to the
increase in disorder. The S / T against temperature curves exhibited a maximum between 20 and 50
K and a noticeable deviation from the expected linear behavior in S共T兲 at higher temperatures. Such
observations in the thermopower of QCs have been attributed to the electron-phonon enhancement
and phonon drag effect. The measured thermal conductivities were analyzed by separating the
electronic and phonon contributions that provide a reasonable explanation for plateau-type feature
in ␬共T兲 of QCs. Our present results suggest that the transport properties of glassy phase are
influenced by the same mechanisms as those of quasicrystalline phase. © 2008 American Institute
of Physics. 关DOI: 10.1063/1.2977721兴
I. INTRODUCTION
The quasicrystals 共QCs兲 are cluster-based structures with
long-range rotational order but lack translational symmetry.
Accordingly, they exhibit properties similar to that of amorphous alloys as they behave like a semimetal or semiconductor electronically and like a glass thermally. A fascinating
aspect of several icosahedral QCs is the combination of a
reasonably ordered atomic structure with its electronic structure markedly different from a free electronlike description
used for normal crystalline metals. Concerning the Ti-based
QC forming materials, the composition window is rather narrow and limited number of system, mainly based on Ti and
Zr transition metals, has been investigated so far. The ternary
Ti-Zr-Ni alloy presents great scientific and technological interests since a highly ordered icosahedral phase, displaying
good thermal stability, has been previously described in this
system.1–15 The corrosion resistance and high melting point
make the Ti/Zr based QCs possible candidates for high temperature protective coatings. Their potential in hydrogen
storage capability 共capable of absorbing two hydrogen per
metallic atom兲 and metal-hydride battery applications have
also been extensively studied.1–9 Recently, Lefaix et al.16
have underlined the potential interest of these alloys for biomedical application, taking advantages from the unusual
combination of surface properties. Some of their physical
properties are rather unique, leading to novel application.17
Any improvement in such application oriented properties
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-8979/2008/104共6兲/063705/7/$23.00
could be greatly facilitated if the transport properties of the
icosahedral 共i phase兲 Ti-Zr-Ni QCs would be known.
There are several competing phases known in the TiZr-Ni phase diagram.10–15,18,19 Dominant equilibrium phases
identified are the C14-like Laves phase, a Ti2Ni-like phase,
␣共Ti/ Zr兲 phase, the bcc 1/1 approximant phase 共W phase兲,
and the icosahedral phase 共i phase兲. The i phase forms over a
small compositional range from a high-temperature equilibrium phase mixture of the Laves 共C14兲 and ␤共Ti/ Zr兲 solid
solution phases.14 Lower annealing treatments have not been
found to yield the i phase, suggesting that in this alloy, the
QCs phase might be the ground state phase.15 Further, a
stable icosahedral Ti41.5Zr41.5Ni17 was prepared by lowtemperature annealing of about seven days at 570 ° C.2
Structural analysis indicates that this phase forms in an ordered icosahedral state and therefore it can be considered as
an ideal model system for the investigation of its possible
nontrivial physical properties.
In general, QCs are found to exhibit moderate electrical
resistivity, reasonably high thermopower, and low thermal
conductivity. The electrons in these QCs can transport either
by diffusion or by variable-range hopping 共VRH兲 depending
on whether they are metallic or insulating.20–22 The peculiar
behavior regarding the electrical resistivity and thermopower
has been explained by the formation of a pseudogap in the
density of states 共DOS兲 near the Fermi level 共EF兲, with localization of electrons near the EF. Several experimental investigations pointed out that the pseudogap originates most
likely due to the Hume-Rothery mechanism, whereas the
sp-d hybridization further enhances the formation of
104, 063705-1
© 2008 American Institute of Physics
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J. Appl. Phys. 104, 063705 共2008兲
Kuo et al.
II. EXPERIMENTAL PROCEDURES
Ingots of Ti35Zr45Ni20, Ti40Zr40Ni20, Ti45Zr35Ni20, and
Ti50Zr30Ni20 were obtained by arc melting of a mixture of
high purity Ti 共99.98 wt %兲, Zr 共99.9 wt %兲, and Ni
共99.99 wt %兲. Bulk QC rods with a diameter 2–3 mm were
then prepared by means of copper suction casting. Metallic
glasses 共MGs兲 Ti35Zr45Ni20 and Ti40Zr40Ni20 in ribbon forms
were obtained by pouring the melt onto a copper roller with
a diameter 15 cm at 6000 rpm. The quasicrystalline and
glassy phases of the prepared samples were assessed by examining their x-ray spectra recorded using a rotating-anode
x-ray generator 共Cu K␣, 50 kV, 120 mA兲 with a graphite
共002兲 monochromator. Figure 1 shows the XRD diffraction
patterns of MGs Ti35Zr45Ni20 and Ti40Zr40Ni20 共top panel兲
that exhibit a single broad peak, a typical feature for the
glassy materials. On the other hand, the XRD patterns for the
Ti40Zr40Ni20, Ti45Zr35Ni20, and Ti50Zr30Ni20 QCs 共lower
panel兲 can be indexed by Elser’s scheme, and the results
confirmed that the prepared alloys were of the expected
icosahedral phase. It is noted that the latter two samples contain weak but detectable impurity peaks which have been
Ti35Zr45Ni20-MG
Ti45Zr35Ni20-QC
C14 Laves phase
(423211)
(333222)
(333101)
(422211)
(332002)
Lattice constant (A)
(322101)
(322111)
Ti40Zr40Ni20-QC
(222100)
(311111)
(221001)
Ti40Zr40Ni20-MG
(211111)
pseudogap.23,24 On the other hand, at low temperatures 共well
below the Debye temperature ␪D兲, the deviation from the
linear behavior of thermopower has been attributed to the
electron-phonon mass enhancement effects.25–28 Further,
QCs exhibit low thermal conductivity ␬共T兲 because of their
phonon scattering due to both the inherent disorder in structure and the scattering on heavy ions. In general, ␬共T兲 shows
a plateau at temperature ranging from 20 K up to 70 K and
then increases monotonically at T ⬎ 100 K.29,30 It has been
emphasized that the lattice contribution dominates at temperatures T ⬍ 100 K, while the electronic contribution
gradually starts to dominate in the temperature range T
⬎ 100 K. Nevertheless, several studies have shown that the
thermal conductivity of QCs follows the Wiedemann-Franz
共WF兲 law at low as well as high temperatures.31,32
Most of the studies performed so far on present test material and its approximant structure were related to its structural characterization and in search of future application for
hydrogen storage.1–15 The available information on physical
properties includes data of electrical transport and thermoelectric properties,33,34 and mechanical properties.35–38 There
appears to be only few studies of the electrical and thermal
transport properties of the Ti-based materials by varying its
composition and comparing it with similar class of systems.
Given the importance of physical properties and the lack of
systematic studies for this class of materials due to the availability of relatively narrow composition window, we undertook transport properties on both the quasicrystalline and
glassy phase of Ti-Zr-Ni alloys. In this study we have investigated electrical and thermal properties including electrical
resistivity 共␳兲, thermopower 共S兲, and thermal conductivity
共␬兲 on Ti-Zr-Ni alloys. We have shown that the electronphonon scattering plays an important role in the temperature
variations of S and ␬ in these alloys. A detailed comparison
of transport properties of an icosahedral phase to the glassy
phase was also presented.
Intensity (arb. unit)
063705-2
5.20
5.18
5.16
5.14
40
45
50
Ti concentration (%)
Ti50Zr30Ni20-QC
C14 Laves phase
30
40
50
60
70
80
2θ (degree)
FIG. 1. XRD patterns of Ti-Zr-Ni alloys. Inset shows the variation of lattice
constants with Ti concentration.
identified to be the C14 Laves phase. As shown in the inset
of the Fig. 1, the lattice constants of these QCs were found to
decrease linearly with increasing Ti content, in accordance
with the Vegard’s law. Electrical resistivity measurements
were carried out by using standard four probe method. The
thermopower and thermal conductivity measurements were
simultaneously performed in a closed cycle refrigerator by a
heat pulse technique. The details of the measurement techniques can be found elsewhere.39
III. RESULTS AND DISCUSSION
A. Resistivity
In order to compare with the electrical transport of
glassy and icosahedral phases, the low-temperature resistivity was measured from room temperature to 4.2 K. The temperature dependent electrical resistivity ␳共T兲 of
共Ti40Zr40Ni20, Ti45Zr35Ni20, Ti50Zr30Ni20兲 QC and
共Ti35Zr45Ni20, Ti40Zr40Ni20兲 MG alloys is shown in Fig. 2.
The ␳共T兲 characteristic of all QC and MG was found to be
nearly identical, except the variations in their magnitude. In
the inset of Fig. 2, we show the normalized electrical resistivity ␳共T兲 / ␳共300 K兲 as a function of temperature to highlight the variation of ␳共T兲 for each alloy. The observed features in the electrical resistivity of MG alloys, a week
temperature variation, and a negative temperature coefficient
of resistivity 共TCR兲 with ␳ 共T ⬇ 300 K兲 ⬎ 150 ␮⍀ cm are in
accordance with the Mooij criterion, a correlation between
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063705-3
J. Appl. Phys. 104, 063705 共2008兲
Kuo et al.
Ti-Zr-Ni
600
Ti35Zr45Ni20-MG
Ti40Zr40Ni20-MG
cm
Ti40Zr40Ni20-QC
Ti45Zr35Ni20-QC
400
Ti50Zr30Ni20-QC
1.08
(T)/
(300 K)
500
1.06
1.04
1.02
1.00
0
50 100 150 200 250 300
300
200
100
0
50
100
150
200
250
300
T(K)
FIG. 2. 共Color online兲 Electrical resistivity as a function of temperature of
Ti-Zr-Ni alloys with various compositions. Inset: normalized electrical resistivity ␳共T兲 / ␳共300 K兲 as a function of temperature.
the residual resistivity ␳共0兲 and the TCR for disordered
systems.40 In particular, in the case of MG alloys, the TCR is
expected to decrease with increasing residual resistivity, then
it becomes negative for ␳共0兲 above 150 ␮⍀ cm. The increase in Ti/Zr ratio 共or an increase in Ti concentration兲 leads
to a systematic increase in resistivity, possibly due to the
enhanced disorder with increasing Ti concentration. This is
further evident from the corresponding increase in the residual resistivity ␳共0兲 for higher Ti/Zr ratio. In the case of
QC alloys, the commonly observed negative TCR is most
likely due to the quasiperiodicity rather than the chemical
disorder and is attributed to the weak localization effect.27,41
We argue that the impurity C14 Laves phase observed in
high Ti/Zr ratio 共see Fig. 1兲 does not affect the ␳共0兲 and the
temperature dependent ␳共T兲 significantly because it is well
known that in contrast with metals, the impurity and/or defect would decrease rather than enhance the resistivity of
QCs and also possibly alter the sign of TCR. Thus we believe that the large increase in ␳共0兲 about 3.2 times from
Ti/ Zr= 35/ 45 to 50/30 is mainly due to chemical effects 共see
discussions in later sections兲. Note that the ␳共0兲 for
Ti40Zr40Ni20 QC is only about 20% larger than that for
Ti40Zr40Ni20 MG, which has a structure much different from
a QC.
Actually this observed trend is in contrast to the theoretical prediction that the electrical resistivity should decrease
as a result of the charge transfer from the Ti and Zr atoms to
Ni.11 In Ti based QCs the effect of hybridization corresponds
to charge transfer from Ti and Zr to Ni. In the charge transfer
process, Ti and Zr donate 1.5 and 0.5 electrons, respectively,
while Ni gains 1.8 electrons and hence the extent of charge
transfer from Ti and Zr atoms to Ni is expected to be enhanced with increasing Ti/Zr ratio. Therefore, the origin of
the high electrical resistivity in the high Ti/Zr ratio of icosahedral phases is still a matter of debate. As pointed out previously, the increase in disorder may be one of the reasons
for the observed increase in resistivity for higher Ti/Zr ratio.
On the other hand, changes in the local atomic environment
around Ti/Zr atoms could also play an important role in increasing the resistivity through the modification of electronic
structure. Zr may occupy more volume in the given structure
because Zr 共0.162 nm兲共Ref. 42兲 is slightly larger than Ti
共0.147 nm兲.42 Consequently, the change in Ti/Zr ratio is expected to induce a corresponding modification of electronic
band structure and hence the behavior of electrical resistivity.
Our observation of increased resistivity with Ti concentration
could be understood in the band structure picture for the
following reason. Two facts seem to be generally recognized:
the existence of a pseudogap in the electron DOS near the
Fermi level 共EF兲 and the localization tendency of electrons in
states near EF.22,43
If the complicated band structure plays a key role, one
can change the behavior of carriers by changing the Fermi
level very slightly. The changing of the Fermi level can be
realized either by changing temperature or the composition
of the sample. A higher Ti/Zr concentration is expected to
broaden the band and that might lead to systematic widening
of the pseudogap and hence a rigid bandlike shifting of EF to
a deeper valley of the pseudogap may occur.43 In general,
one expects a reduction of the DOS near the Fermi level to
provide a stabilizing contribution to the electronic energy.
The pseudogap at the Fermi level is often found in QCs
where the differences in the electronic DOS determine the
relative stabilities of quasicrystalline phases.44,45
B. Thermopower
The measured thermopower S共T兲 between 10 and 300 K
for Ti40Zr40Ni20, Ti45Zr35Ni20, and Ti50Zr30Ni20 QCs and
Ti35Zr45Ni20 and Ti40Zr40Ni20 MGs is shown in Fig. 3. The
room temperature S values for studied alloys were found to
be around 7 – 9 ␮V / K for QC and 2 – 3 ␮V / K for MG, respectively, and such a typical range of value of S is generally
observed in quasicrystalline and glassy alloys.27,28 The thermopower is positive and nearly identical for all QC alloys in
the presently investigated temperature range, which reveals
that the hole-type carriers dominate the thermoelectric transport in these alloys. The thermopower of MGs is found to
exhibit marked differences as compared with QCs at low
temperatures. The value of S is initially negative for
Ti40Zr40Ni20 MG sample and then crosses zero at around 20
K. As it is well established that S is very sensitive to the
changes in the DOS near the Fermi level, such a sign reversal, in addition to the majority carriers, indicates that there
may be a substantial change in the band structure.
Further, a small hump in S共T兲 at low temperatures is
observed in all the investigated QC and MG alloys. On the
other hand, from 50 to 300 K, S increases monotonically as
the temperature rises, indicating that the diffusion thermoelectric transport prevails in the high temperature range. The
temperature dependence of S at higher temperature is linearlike in T with slopes ranging from approximately 0.007 to
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
J. Appl. Phys. 104, 063705 共2008兲
Kuo et al.
Ti-Zr-Ni
0.06
0.02
S(µ
µV/K)
8
9
S/T (
V/K)
0.04
data
fit
0.6
0.3
6
50
0.0
150 200 250 300
100
T (K)
0.00
Ti40Zr40Ni20-QC
50
-0.02
100 150 200 250 300
Ti35Zr45Ni20-MG
T(K)
Ti40Zr40Ni20-MG
0
4
0.9
data
fit
Ti40Zr40Ni20-QC
4
0.6
Ti45Zr35Ni20-QC
λ
S (
V/K)
3
6
Ti50Zr30Ni20-QC
0.3
2
50
2
0.0
150 200 250 300
100
T (K)
S(µ
µV/K)
10
12
λ
063705-4
Ti40Zr40Ni20-MG
0
0
50
100
150
200
250
300
T(K)
50
100
150
200
250
300
T(K)
FIG. 3. 共Color online兲 Thermopower as a function of temperature for TiZr-Ni QCs and MGs with various compositions. Replotting the thermopower data by S / T vs T in inset. We see more clearly the deviation from
a linear-in-T dependence of S at low temperatures.
0.03 ␮V / K2. These slopes correspond to Fermi energies EF
around 2–5 eV if we use Sb共T兲 ⬇ 关共␲2kB2 / 3eEF兲T兴 as a crude
estimation for the electron diffusion thermopower at high
temperatures. Here, Sb共T兲 is the bare diffusion thermopower,
usually taken as linear in temperature 共Sb = bT兲, which is the
usual Mott approximation for metallic diffusion
thermopower.46 It is apparent that the observed slope is
smaller for the MGs, corresponding to a larger Fermi energy,
than that of their quasicrystalline counterparts. The deviation
from a linear-in-T dependence of S is confirmed by plotting
S / T versus T, as can be seen in inset of Fig. 3 where all
curves exhibit a maximum between approximately 20 and 50
K. We may recall that the effect of electron-phonon enhancement would manifest itself as a broad hump in S at low
temperatures for highly disorder QCs and bulk metallic materials with resistivities greater than 100 ␮⍀ cm.25–28 In
present case the resistivities for all measured QC and MG
alloys are greater than 150 ␮⍀ cm 共Fig. 2兲, suggesting that
the electron-phonon enhancement effect in thermopower is
expected to be noticeable in these materials.
For QC and MG, S共T兲 is normally dominated by the
electron diffusion contribution which is proportional to the
temperature. The deviation from linearity 共increased gradient
at lower temperatures兲 and small peak at low temperatures
are undoubtedly related to electron-phonon enhancement and
phonon drag effects, respectively. S共T兲 can therefore be written as a function of temperature including the electronphonon enhancement and phonon drag effects28
S共T兲 = Sb共T兲关1 + a␭共T兲兴 + c exp关− 共T − f兲2/共2w2兲兴,
0
0
共1兲
where the second factor, within the brackets of first term,
describes the enhancement of the thermopower by the
FIG. 4. Fits of Eq. 共1兲 including electron-phonon enhancement and a lowtemperature phonon drag peak to the measured thermopower of the 共a兲
Ti40Zr40Ni30 QC and 共b兲 Ti40Zr40Ni30. The insets show the ␭共T兲 plot against
temperature for the phonon spectrum using the Debye model. Various parameters values are discussed in the text.
electron-phonon interaction, which is analogous to the wellknown electron-phonon enhancement of electronic specific
heat at low temperatures. The parameter a gives the magnitude of the enhancement factor, which is equal to unity when
only electron-phonon enhancement is considered. The temperature dependence of ␭共T兲 enhancement is given by
␭共T兲 =
2
共kB␪D兲2
冕
kB␪D
EG共E/kBT兲dE.
共2兲
0
Here, kB is Boltzmann’s constant and G共E / kBT兲 is the universal thermopower enhancement function defined earlier.25
As ␭共T兲 is sensitive at low temperature, we may further simplify the problem by using the Debye model with kB␪D as an
effective Debye cutoff energy for the electron-phonon enhancement effect. The second term in Eq. 共1兲 represents the
contribution from phonon drag thermopower at low temperatures which yields the observed plateaulike behavior. However, the magnitude of phonon drag contribution is found to
be very small 共Inset of Fig. 3兲 as compare to crystalline
metals.47 Here, the height and width of the plateau are determined by the parameters c, f, and w with the Gaussian kind
of peak characteristic resulting from the remnant phonon
drag thermopower.
We have fitted our temperature dependent thermopower
data to the Eq. 共1兲. Two representative fits of the thermopower data for Ti40Zr40Ni20 QC and Ti40Zr40Ni20 MG alloys are shown in Figs. 4共a兲 and 4共b兲, respectively. We have
also plotted the temperature dependence of electron-phonon
mass enhancement ␭共T兲 parameter in the inset of respective
figures. Particularly, Fig. 4共a兲 shows that the combination of
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063705-5
J. Appl. Phys. 104, 063705 共2008兲
Kuo et al.
TABLE I. Parameters obtained from the fitting of experimental data using Eqs. 共1兲 and 共2兲.
Ti35Zr45Ni20
Ti40Zr40Ni20
Ti40Zr40Ni20
Ti45Zr35Ni20
Ti50Zr30Ni20
MG
MG
QC
QC
QC
b
共␮V K−2兲
A
␪D
共K兲
c
f
共K兲
w
共K兲
0.008
0.007
0.026
0.029
0.031
0.16
0.14
0.13
0.13
0.12
350
357
360
365
370
0.30
0.20
0.48
0.51
0.10
45
35
48
42
30
38
25
26
15
20
electron-phonon enhancement and phonon drag effects can
give a satisfactory explanation of the observed lowtemperature behavior of thermopower for QCs. The parameter values, obtained for these fits and for similar fits for
other Ti/Zr concentration, are listed in Table I.
It is inferred from the table that the fitting parameters are
quite similar between QC and MG alloys. The increase in the
slope of diffusion thermopower 共b兲 with increasing Ti/Zr ratio indicates enhanced contribution of bare diffusion thermopower with Ti content. Further, we have found that the
Debye temperature ␪D increases while the electron-phonon
enhancement factor a decreases with Ti/Zr ratio. The decreasing trend of electron-phonon coupling ␭ with increasing
Ti/Zr ratio might be arising from the modification of the
phonon spectrum as Ti is added. With the above deduced
values of ␪D, the change in ␪D with Ti/Zr ratio implies that an
intensification of disorder occurs for higher values of Ti/Zr
ratio. Henceforth, the Ti/Zr dependence of ␪D in these alloys
suggests that increased disorder drives the system effectively
toward the weak electron-phonon coupling region. The fitting parameters and Debye temperature are consistent with
that earlier reported for QCs 共Ref. 27兲 and bulk MGs.28
However, above expression cannot account for the sign
reversal in thermopower that is seen below 20 K for
Ti40Zr40Ni20 MG alloy. The discrepancy between the data
below 20 K and the fit for MG suggests that the substantial
change in the band structure moving from quasiperiodicity to
glassy state. To get additional insight into this behavior we
have attempted the use of energy dependent diffusion thermopower rather than bare Sb共T兲 in the model calculations,
but such an effort yielded no significant improvement to the
overall fit. Since S can be positive or negative depending on
the sign of energy derivative of electrical conductivity, which
in turn depends on the position of EF across the pseudogap.48
Therefore, the sign change in S for these glassy alloys could
be qualitatively understood by this simple picture.
It is instructive to mention that the presence of additional
low-energy optical vibrational states, the boson peak, in
these alloys could also play a role as in glassy materials.49
We argue that the origin of low-temperature hump presented
here is originated from the phonon drag and not by any other
phenomenon. First, the drag and diffusion thermopower contributions to thermopower are additive.47 Figs. 3 and 4 show
that the parameters thus obtained satisfactorily describe the
experimental data. Hence, an additive low-temperature peak
term such as that in Eq. 共1兲 is appropriate due to phonon drag
contribution. Second, the factor 1 + ␣␭共T兲 with ␭共T兲 a decreasing function of temperature 共see in insets of Fig. 4兲
cannot account for the maximum 共hump兲 in S / T, phonon
drag seems to be the best explanation for the hump.
C. Thermal conductivity
Figure 5 shows the temperature variation of thermal conductivity ␬ of Ti40Zr40Ni20, Ti45Zr35Ni20, and Ti50Zr30Ni20
QCs for the whole temperature range covered in this work.
Due to the unavoidable difficulties in performing the lowtemperature thermal conductivity measurements on ribbonshaped samples with our experimental setup, no ␬共T兲 data on
MG alloys are presented here. The room temperature ␬ of all
QCs was found to be about 60– 80 mW/ cm K, which is
consistent with the values reported for other system of
QCs.27 The temperature dependent thermal conductivity of
these QCs decreases with decreasing temperature, then develops into a broad plateau at around 30 K and steeply decreases if the temperature decreases further. This overall behavior is similar to that of other quasicrystalline solids,
where generally a low-temperature plateau is observed and
ascribed to the generalized Umklapp process.50 Later, we
will put forward an argument that the monotonic decrease in
␬ with decreasing temperature above 50 K in these alloys
can be attributed to a power-law dependence of VRH
mechanism.51
Ti-Zr-Ni Quasicrystals
80
Ti45Zr35Ni20
60
κ (mW/cm K)
Alloys
Ti40Zr40Ni20
Ti50Zr30Ni20
40
Ni 20
Z r 40
i
40
T
or
κef
N i 20
Z r 35
r Ti 45
o
f
κe
i
Zr 0N 20
r Ti 5 0 3
20
κ e fo
0
0
50
100
150
200
250
300
T(K)
FIG. 5. Temperature dependence of the total thermal conductivity of TiZr-Ni QCs with various compositions. The solid lines are an estimate of the
electronic contribution ␬el as explained in the text.
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063705-6
J. Appl. Phys. 104, 063705 共2008兲
Kuo et al.
Ti-Zr-Ni Quasicrystals
50
35
30
25
ph (mW/cm K)
40
data
fit
100
30
20
ph (mW/cm K)
40
Ti40Zr40Ni20
15
10
300
T(K)
Ti40Zr40Ni20
20
Ti45Zr35Ni20
Ti50Zr30Ni20
10
10
100
300
T(K)
FIG. 6. 共Color online兲 Quasilattice contribution ␬ph to the thermal conductivity of Ti-Zr-Ni QC. In the inset, the solid line is a power-law approximation to the data between 50 and 300 K for Ti40Zr40Ni30 QC.
In most of the metals and semimetals, where WF law is
applicable, the thermal conductivity and the electrical resistivity are mutually related over certain temperature ranges
through the relationship ␬el共T兲␳共T兲 = L0T. Where ␬e共T兲 gives
the contribution to the thermal conductivity due to the charge
carriers, ␳共T兲 is the electrical resistivity and L0 = 2.45
⫻ 10−8 W ⍀ K−2 is the Lorentz number. The ␬el共T兲 was calculated using WF law and the measured electrical resistivity
presented in Fig. 2. The result of this calculation is shown in
Fig. 5 as a solid line. The thermal conductivity ␬ph共T兲 contributed by the phonons alone may be obtained by subtracting the estimated electronic contribution ␬el共T兲 from the total
thermal conductivity. It may be seen that over most of the
covered temperature range ␬el共T兲 is at least two or three
times less than that of ␬tot共T兲 and, therefore, ␬ph共T兲 may be
evaluated quite accurately.
Figure 6 shows the quasilattice thermal conductivity ␬ph
of Ti-Zr-Ni QCs on logarithmic scales, as obtained in the
way described above. The occurrence of a plateau in ␬ph is in
agreement with expectations of generalized Umklapp processes in QCs and implies that the mean free path of phonon
with a given frequency decreases with increasing temperature. We note two salient features in ␬ph versus T plot. The
first one is that the height of the low-temperature plateau
decrease gradually with increasing the substitution level, indicative of a strong enhancement in the phonon scattering by
increased Ti/Zr ratio or substitutional disorder. In general,
the impurity and defect scattering could suppress the lowtemperature peak in ␬.52 This suggests that extrinsic effects
such as impurity phase may also have strong influences on
the measured thermal conductivity at low temperatures in
these QCs. In our samples, Laves phase was clearly observed
for higher Ti/Zr concentration 共see Fig. 1兲. Also, the disorder
is expected to reduce the contribution of phonon drag effect
and indeed the same has been observed in the present thermopower results also.27
Another point in ␬共T兲 variation is the monotonic increase of ␬ with increasing T, approximately above 50 K.
This characteristic feature is also commonly observed in
quasicrystalline solids and is attributed to the activation of
localized phonon states at high T.45 The thermal activation of
localized vibrational modes opens a channel for the heat
transfer via hopping excitations at higher temperatures.
Janot51 proposed a model taking into account the highenergy and localized vibration modes with a hierarchical
VRH mechanism, indicating that the quasilattice thermal
conductivity is expected to follow a power law, i.e., ␬ph
⬀ T␭ with ␭ = 1.5. In this scenario, we have also attempted to
fit our quasilattice thermal conductivity to ␬ph = 关␣ + ␤T␭兴 in
the temperature interval between 50 and 300 K, and the result is presented as a solid line in the inset of Fig. 6 for
Ti40Zr40Ni20 QC. The deduced value of ␭ is about 1.67, close
to the theoretical prediction of ␭ = 1.5. It is noted that the ␭
values deduced from this analysis are quite similar for all
studied alloys with various Ti/Zr concentrations, indicating
that we are observing essentially the same effect for each of
the samples. Such a finding is in good agreement with earlier
reports.27
IV. CONCLUSIONS
In conclusion, we have reported a detailed investigation
of electrical and thermal transport properties of Ti-Zr-Ni
based QCs 共Ti40Zr40Ni20, Ti45Zr35Ni20, and Ti50Zr30Ni20兲 and
MGs 共Ti35Zr45Ni20 and Ti40Zr40Ni20兲. We have found weak
temperature dependence with a negative TCR in the studied
alloys. It was noticed that the electrical resistivity increases
with increasing Ti/Zr ratio, which could be due to the increased disorder. Further analysis on the microstructure may
help to understand the influence of composition and/or microstructure on the transport properties of these alloys. On
the other hand, the deviation from linearity and a small hump
at low temperatures in the measured thermopower of these
alloys are related to electron-phonon enhancement and phonon drag effects, respectively. Seemingly, the transport properties of glassy phase are influenced by the same mechanisms as those of quasicrystalline phase. The thermal
conductivity was also analyzed by separating the electronic
and phonon contributions that provided a reasonable explanation for plateau-type feature in ␬ of QCs. We have shown
that the thermal conductivity follow a power-law dependence
of VRH mechanism above the plateau region.
ACKNOWLEDGMENTS
The authors would like to thank the National Science
Council of Taiwan for financially supporting this research
under Contract No. NSC-96-2112-M-259-003 共Y.K.K.兲.
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