Structural instability and photoacoustic study of AlSb prepared by mechanical alloying

Article abstract of DOI:10.1016/j.jallcom.2010.06.137

High-purity elemental Al and Sb powders were blended with equiatomic composition and submitted to mechanical alloying. For all milling times, the milled powders showed a mixture of AlSb and elemental Sb. The largest amount of AlSb was reached for milling times between 7 and 10 h. For milling times larger than 10 h, decomposition of AlSb was observed. The volume fractions of the crystalline and interfacial components were estimated using the X-ray diffraction pattern of a sample milled for 10 h. Photoacoustic absorption spectroscopy (PAS) was used to determine the thermal diffusivity and other heat transport parameters in the same sample. A combination of XRD and PAS data was used to estimate the thermal diffusivity of the interfacial component, which has a significant contribution to the thermal diffusivity of the sample.

Full text of DOI:10.1016/j.jallcom.2010.06.137

Journal of Alloys and Compounds 505 (2010) 762–767
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: www.elsevier.com/locate/jallcom
Structural instability and photoacoustic study of AlSb prepared by mechanical
alloying
D.M. Trichêsa, S.M. Souzaa, C.M. Poffoa, J.C. de Limab,∗, T.A. Grandib, R.S. de Biasic
a Departamento de Engenharia Mecânica, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil
b Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil
c Sec¸ ão de Engenharia Mecânica e de Materiais, Instituto Militar de Engenharia, 22290-270 Rio de Janeiro, RJ, Brazil
a r t i c l e i n f o
a b s t r a c t
Article history:
High-purity elemental Al and Sb powders were blended with equiatomic composition and submitted to
mechanical alloying. For all milling times, the milled powders showed a mixture of AlSb and elemental
Sb. The largest amount of AlSb was reached for milling times between 7 and 10 h. For milling times larger
than 10 h, decomposition of AlSb was observed. The volume fractions of the crystalline and interfacial
components were estimated using the X-ray diffraction pattern of a sample milled for 10 h. Photoacoustic
absorption spectroscopy (PAS) was used to determine the thermal diffusivity and other heat transport
parameters in the same sample. A combination of XRD and PAS data was used to estimate the thermal
diffusivity of the interfacial component, which has a significant contribution to the thermal diffusivity of
the sample.
Received 3 March 2010
Received in revised form 21 June 2010
Accepted 23 June 2010
Available online 1 July 2010
PACS:
61.05.cp
64.70.kg
66.30.Xj
66.10.cd
© 2010 Elsevier B.V. All rights reserved.
Keywords:
Semiconductor
Thermal diffusivity
X-ray diffraction
Photoacoustic absorption
Mechanical alloying
1. Introduction
control, inexpensive equipment, and the possibility of scaling up.
Its disadvantage is contamination by the milling media and possibly
Aluminum antimonide (AlSb) is a semiconductor of consider-
able importance. Potentially it can be highly resistive and is closely
lattice matched to GaSb and therefore has been used as semi-
insulating substrate or buffer layer for the epitaxial growth of GaSb
[1,2]. AlSb has an indirect band gap with an energy of 1.62 eV [3]
and thus is a good candidate for applications such as high-energy
photon detectors and as a barrier material to confine electrons in
antimonide heterostructure devices [4], in InAs-channel high elec-
tron mobility transistors [5,6] and magnetoelectronic hybrid Hall
effect devices [7].
AlSb can be produced using several techniques [8–10], includ-
ing mechanical alloying (MA). MA is an efficient technique to
synthesize many unique materials such as nanostructured alloys,
amorphous alloys, and metastable solid solutions [11–13]. It has
also been used to produce commercially important alloys whose
components have a high melting point [14]. MA has many advan-
tages, including processing at low temperatures, easy composition
by the milling atmosphere.
Nanostructured materials are metastable and have two com-
ponents: crystallites with nanometer dimensions, <100 nm, which
have same characteristics as the bulk crystal, and an interfacial
phase formed by different kinds of defects (grain boundaries, inter-
phase boundaries, dislocations, etc.) [15]. The number of atoms is
similar in both components, making the properties of the nanos-
tructured materials to dependent on the atomic arrangements of
the interfacial component. From the technological point of view,
manipulation of the interfacial component offers the possibility
of designing new materials with physical properties for specific
applications [15,16].
The Al–Sb phase diagram [17] exhibits only the equiatomic
AlSb compound, which has a cubic structure. Honda et al. [18]
prepared AlSb by MA using a high-energy planetary ball mill
(Itoh, LP-4/2) at 300 rpm and at room temperature. The compound
was formed after 5 h of milling and no changes on the mea-
sured X-ray diffraction (XRD) patterns were observed for milling
times up to 60 h. In another study, Park and Sohn [19] produced
mechanically alloyed AlSb powder using a HEMM vibratory mill for
24 h.
∗
Corresponding author. Tel.: +55 48 3331 6832; fax: +55 48 3331 9758.
E-mail address: fsc1jcd@fsc.ufsc.br (J.C. de Lima).
0925-8388/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jallcom.2010.06.137

D.M. Trichês et al. / Journal of Alloys and Compounds 505 (2010) 762–767
763
Due to the technological importance of the Al–Sb system and
nanostructured materials, we focused our research on producing
AlSb in nanometric form by MA. Since the MA technique yields
materials containing a high concentration of defects, we inves-
tigated the influence of these defects on the thermal diffusivity
measured by photoacoustic absorption spectroscopy (PAS). This
paper reports the results of XRD and PAS measurements on AlSb
produced by MA.
achieved through carrier–phonon interactions (phonon emis-
sion through electron and hole scattering), in which the excess
kinetic energy of the carriers is transferred to phonons. Phonon
emission results in cooling of the carriers and heating of the
lattice until the carrier and lattice temperatures become equal.
This process is called carrier cooling [22]. González de la Cruz
and Gurevich [23,24] reported the calculation of electron and
phonon temperature distribution functions in semiconductors
and they also carried out a deeper analysis of these temperature
distribution functions in PAS measurements.The contribution
of this process to the PAS signal decreases exponentially with
the modulation frequency according to the equation
2. Theoretical background
Thermal diffusivity is defined as ˛ = k/ꢀc, where k is the ther-
mal conductivity, ꢀ the mass density and c the specific heat. It is an
important physical parameter due not only to its intrinsic physical
interest but also to its use in the modeling and designing of tech-
nological devices based on semiconductor materials. Physically, the
inverse of ˛ is a measure of the time required to establish thermal
equilibrium in a given material. As in the case of the optical absorp-
tion coefficient, its value is different for each material. Furthermore,
the thermal diffusivity is known to be strongly dependent on the
effects of compositional and microstructural variables [20], as well
as processing conditions [21]. Thus, an appropriate strategy to mon-
itor the structural changes in mixtures submitted to the MA process
is to measure the ˛ parameter.
When a modulated light beam impinges on the material inside
a photoacoustic gas cell, the absorbed light is converted into peri-
odic heat and a PAS signal is created. The dependence of the PAS
signal on the optical absorption coefficient and the light-to-heat
conversion efficiency may be used to determine the nonradiative
de-excitation efficiency, the photoinduced energy conversion, etc.
The PAS signal is directly proportional to the light-to-heat con-
version efficiency due to nonradiative processes in the material
[20]. For a thermally thick semiconductor sample, there are four
processes that may contribute to the PAS signal:
ꢀ
A
S = exp(−a f )
(1)
f
√
ꢁ
where a = ls
⁄
˛, f is the modulation frequency, ls is the sample
thickness, and ˛ its thermal diffusivity. The PAS signal phase
shows a modulation frequency dependence of the type
ꢀ
ꢁ
˚
=
− a
f
(2)
ph
2
when this process is present, it occurs in the low frequency
range;
(2) Nonradiative bulk recombination: a complete relaxation of the
system is reached after electron and hole recombine, either
radiatively or nonradiatively. When nonradiative recombina-
tion of excess electron–hole pairs occurs after diffusion, it takes
√
place over a distance Dꢂ. The contribution of this process to
the PAS signal shows a modulation frequency dependence of
the type f−1.5. The thermal diffusivity ˛, carrier diffusion coef-
ficient D, surface recombination velocity v, and recombination
time ꢂ can be determined by fitting the PAS signal phase to the
expression given by Pinto Neto et al. [25],
ꢁ
ꢂ
ꢁ
(bD/v)(ωꢂeff + 1)
˚
=
+ tan−1
,
(3)
ph
2
2
(bD/v)(1 − ωꢂeff) − 1 − (ωꢂeff
)
√
where ꢂeff = ꢂ(D⁄
˛
s − 1), b =
⁄˛, and ω = 2ꢁf . When present,
ꢁf
(1) Intraband nonradiative thermalization (thermal diffusion):
when the absorbed photon energy is greater than the band gap,
an electron is created in the conduction band and a hole is left
in the valence band. The excess energies of the electron (the
energy difference between the lower edge of the conduction
band and the initial energy of the photogenerated electron)
and hole (the energy difference between the upper edge of
the valence band and the initial energy of the photogener-
ated hole) appear in the respective carriers as kinetic energy.
The initial electron and hole distributions are not Boltzmann-
like, and the first step toward establishing equilibrium is for
the electrons and holes to interact among themselves, through
carrier-carrier collisions and intervalley scattering, to estab-
lish separate Boltzmann distributions of electrons and holes.
The Boltzmann distributions of electrons and holes can then be
separately assigned electron and hole temperatures that reflect
the distributions of kinetic energy in the respective charge car-
rier populations. If photon absorption produces electrons and
holes, each with initial excess kinetic energy at least kT above
the conduction and valence bands, both initial carrier temper-
atures are always above the lattice temperature; these carriers
are called hot carriers (i.e., hot electrons and hot holes). This
first stage of relaxation or equilibration occurs very rapidly
(<100 fs), and this process is often referred to as carrier thermal-
ization (i.e., establishment of a thermal distribution described
by Boltzmann statistics). The next step of equilibration is for
the hot electrons and hot holes to reach equilibrium with the
lattice. The initial lattice temperature is the room temperature,
which is lower than the initial hot-electron and hot-hole tem-
peratures. Equilibrium of the hot carriers with the lattice is
this process occurs in the high frequency range after intraband
nonradiative thermalizations;
(3) Nonradiative surface recombination: when nonradiative sur-
face recombination of excess electron–hole pairs occurs, it takes
place at the sample surface. The contribution of this process to
the PAS signal shows a modulation frequency dependence of
the type f−1.0. Similarly to process (2), the ˛, D, v, and ␶ param-
eters can be determined by fitting the PAS signal phase to the
expression given by Pinto Neto et al. [25]. When present, this
process occurs in the high frequency range after the nonradia-
tive bulk recombination process; and
(4) Thermoelastic bending: when a temperature gradient is gener-
ated within the sample, across its thickness, the thermoelastic
bending process contributes to the PAS signal. This contribution
shows a modulation frequency dependence of the type f−1.0
.
The thermal diffusivity ˛ can be determined by fitting the PAS
signal phase to the expression
ꢃ
ꢄ
1
˚
= ꢃ0 + tan−1
,
(4)
ꢀ
ph
a
f − 1
where the constant a is the same defined for process (1). The PAS
signal for processes (3) and (4) shows the same dependence on
the modulation frequency. However, the PAS signal phase for
each process has a different dependence on the modulation fre-
quency. Thus, analysis of the PAS signal phase can be used to
distinguish these processes and determine the thermal diffusiv-
ity ˛. When present, this process occurs in the high frequency
range after the nonradiative surface recombination process.In

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D.M. Trichês et al. / Journal of Alloys and Compounds 505 (2010) 762–767
practice, the contribution of each process to the pressure vari-
ation in the photoacoustic cell can easily be found through the
ꢀ
following procedure: (i) in plots of ln S versus
f and ˚ph (in
ꢀ
radians) versus
f , a is the slope of the straight line fitting
the data in the low modulation frequency range and the val-
ues calculated from both plots must be equal; (ii) in a plot of
log S versus log f, the slope of the straight line fitting the data
in the high modulation frequency range highlights the contri-
bution of nonradiative bulk recombination and nonradiative
surface recombination. In this case, the ˛, D, v and ꢂ param-
eters are obtained by fitting the plot ˚ph (in radians) versus f,
in the appropriate modulation frequency range, to Eq. (3), and
(iii) a similar analysis of that described in (ii) is performed to
find the contribution of thermoelastic bending. The ˛ param-
eter is obtained by fitting the plot ˚ph (in radians) versus f, in
the appropriate modulation frequency range, to the Eq. (4). This
procedure can be executed using the Origin software [26].
3. Experimental procedure
Fig. 1. XRD patterns of an AlSb mixture for several milling times.
Two equiatomic batches of elemental aluminum (Alfa Aesar, 99.999%) and anti-
mony (Alfa Aesar, 99.999%) powders were prepared. They were sealed together with
several steel balls with average diameter 11.5 × 10−3 m in a cylindrical steel con-
tainer under argon atmosphere. The ball-to-powder weight ratio was 5:1. The vial
was mounted on a SPEX mixer/mill model 8000 and the milling was performed at
room temperature. A cooling system was used to keep the container temperature
close to room temperature. Structural changes with milling times were followed by
recording XRD patterns, using a Miniflex Rigaku powder diffractometer, equipped
with CuK␣ radiation (ꢄ = 0.15418 nm). The first batch was milled for 10 and 17 h.
The XRD pattern of as-milled powder was well indexed to AlSb and elemental Sb. A
decrease of AlSb phase and increase of Sb with milling time was observed. In order
to confirm this result, a second batch was milled for 1, 2, 3, 4, 5, 7, 16, 26 and 36 h.
The results were similar.
The PAS measurements were performed on a homebuilt open photoacoustic cell
(OPC) that includes a 250 W quartz-tungsten-halogen lamp and a Bentham 605 cur-
rent power supply. After being heat filtered by a water lens, the light is mechanically
chopped by a PerkinElmer chopper (model 197) and focused onto the sample. The
sample is mounted directly onto the front sound inlet of an electret microphone
[27]. The output voltage from the microphone is connected to a computer through
a lock-in amplifier in order to record the amplitude and phase of the PAS signal as
functions of the modulation frequency. The samples for PAS measurements were
prepared by compressing the powder at the same pressure (6 tons or ≈749 MPa) to
form tiny circular pellets of 10−2 m in diameter. The as-milled sample has thickness
of 450 ␮m. The PAS measurement was taken in the modulation frequency range of
10–270 Hz in order to achieve the thermally thick regime.
sity: −179.949 kJ mol−1, a = 0.60959 nm and 5612 kg/m3 for GaSb;
−69.723 kJ mol−1, a = 0.61355 nm and 4276 kg/m3 for AlSb; and
−56.201 kJ mol−1, a = 0.64791 nm and5775 kg/m3 forInSb. The AlSb
compound has the smallest density, indicating that it has the least
stable structure. For milling times larger than 10 h, a high concen-
tration of defects and strains is introduced into the lattice, causing
lattice expansion and, consequently, increasing the internal energy.
The combination of these factors may promote the dissociation of
AlSb. In another paper [31], we suggested an approach to estimate
the volume fractions of the crystalline and interfacial components
in nanostructured materials that consists in estimating the con-
tribution of nanometric crystallites to the XRD pattern after the
measured intensity is corrected for polarization, reabsorption and
inelastic scattering and converted to electron units using the mean
square scattering factor ꢀf2ꢁ [32]. An evaluation of the background
contribution to the normalized XRD pattern and its subtraction
yields the contribution of nanometric crystallites. The background
was estimated using the OriginLab software [26]. Due to the fact
that the contribution of the interfacial component to the XRD pat-
tern is diffuse, its evaluation and interpretation require care. This
approach was applied in this study to the AlSb compound obtained
after 10 h of milling because this sample has the largest amount of
AlSb; the estimated volume fractions of the crystalline and interfa-
cial components were ≈68% and 32%, respectively.
4. Results and discussion
4.1. XRD measurements
Fig. 1 shows representative XRD patterns of AlSb for several
milling times. The XRD patterns were compared with those given in
the ICSD database [28] for the Al–Sb system and with those of ele-
mental Al and Sb. Up to 7 h of milling, the patterns are well indexed
to the patterns of cubic AlSb (ICSD code No. 44325) and elemen-
tal Sb (ICSD code No. 9859) phases. For 10 h of milling, the XRD
pattern is well indexed to cubic AlSb with a small contribution of
elemental Sb. For larger milling times, the XRD patterns are well
indexed to elemental Sb phase with a small contribution of cubic
AlSb. It is interesting to note that elemental Al diffraction lines are
not present in the XRD patterns, indicating that a solid solution of
Al atoms in the Sb lattice could also be present.
It is interesting to try to understand the partial dissocia-
tion of the cubic AlSb phase for milling times larger than 10 h.
The ZnS structure is the prototype structure for AlSb, InSb and
GaSb. In previous studies, we produced InSb and GaSb by MA
and no dissociation of these phases was observed for milling
times up 10 h [29,30]. Thus, a comparison with these compounds
may be helpful. The TAPP software [17] gives the following
values for the enthalpy of formation, lattice parameter and den-
All XRD patterns were simulated using the Rietveld structural
refinement procedure [33] and the DBWS-9807 code. For this,
the structural models given in the ICSD codes above were used.
The best simulation of the XRD pattern for 10 h of milling was
achieved assuming a lattice parameter a = 0.61151 nm, which is
slightly smaller than that given in Ref. [17]. The experimental and
simulated XRD patterns are shown in Fig. 2, where one can see an
excellent agreement between the experimental and simulated pat-
terns. The relative amounts of the phases were also obtained from
the Rietveld analysis and the values for AlSb and elemental Sb were
73 (50)% and 27 (18)%, respectively. The numbers between paren-
theses are the relative amounts of the phases taking into account
the estimated volume fraction of the interfacial component.
When the cell volume of AlSb is plotted as a function of milling
time, one can see a minimum at about 7 h of milling as shown in
Fig. 3. According to Fig. 1, the maximum volume fraction of AlSb
is reached for about 10 h of milling. Thus, one can conclude that
between 7 and 10 h of milling the maximum volume fraction of
AlSb phase is obtained and that there is a lattice expansion of the
AlSb for times larger than 7 h that promotes its dissociation.

D.M. Trichês et al. / Journal of Alloys and Compounds 505 (2010) 762–767
765
Fig. 4. PAS signal amplitude versus modulation frequency for a sample milled for
10 h.
Fig. 2. Measured and simulated XRD patterns of a sample milled for 10 h.
All peaks of the XRD pattern for 10 h of milling shown in Fig. 1
are broadened, indicating a nanometric AlSb phase. The mean size
of the crystallites can be estimated from the simulated XRD pattern
taking into account the line broadening caused by both crystallite
size and strain through the equation [34]
4.2. PAS measurements
The characteristic frequency fc = ˛/ꢁl2 is the modulation fre-
quency for the transition from the thermally thin regime (f < fc)
to the thermally thick regime (f > fc). In this expression, ˛ is the
thermal diffusivity and l is the sample thickness. Since the thermal
diffusivities of AlSb phase was not found in the literature, the value
was estimated using the equation K = ꢀCp˛, where K is the ther-
mal conductivity, ꢀ is the density, Cp is the specific heat, and ˛ is
the thermal diffusivity. The TAPP software [17] gives the values of
ꢀ = 4276 kg/m3 and Cp = 333 J/kg K for AlSb and ꢀ = 6741 kg/m3 and
Cp = 207 J/kg K for elemental Sb. Borca-Tasciuc et al. [35] reported a
value of 57 W/mK for the thermal conductivity of bulk AlSb, while
the PTOE software [36] gives a value of 24.3 W/mK for elemen-
tal Sb. These values were used in the equation above to calculate
the thermal diffusivity and the results were ˛ = 0.40 × 10−4 m2/s for
bulk AlSb and 0.17 × 10−4 m2/s for elemental Sb. These values were
used to calculate the characteristic frequency fc. In this study only
the sample milled for 10 h was investigated by PAS because it has
the largest volume fraction of AlSb. Since its thickness is 450 ␮m, a
characteristic frequency of 62 Hz was calculated. The PAS data were
acquired between 10 and 270 Hz in order to achieve a thermally
thick regime.
ꢅ
ꢆ
ꢅ
ꢆ
2
2
ˇt cos Â
1
d2
sin Â
Kꢄ
2
p
=
+
,
(5)
Kꢄ
where  is the diffraction angle, ꢄ is the X-ray wavelength, ˇt is the
total peak broadening measured at the full-width at half-maximum
(FWHM) in radians, d is the crystallite size, p is the strain, and K is a
constant which depends on the measurement conditions and on the
definitions of ˇt and d. Here, K was taken as 0.91 as is generally the
case in the Scherrer formula. Graphical linearization of the above
relationship, i.e., plotting ˇt2 cos2 Â/ꢄ2 versus sin2 Â/ꢄ2, yields the
mean crystallite size free from strain effects from the values of the
intercept of the straight line, as well as the strain from the slope. The
DBWS 9807 code generates the ˇt and 2Â positions for all simulated
peaks. Considering these values in Eq. (5), the values found were
d ≈ 33 nm and p ≈ 1.7%. These results confirm the nanometric form
of AlSb.
Figs. 4 and 5 show the PAS signal amplitude and phase for the
sample milled for 10 h. The procedure described in Section 2 to
find the contribution of each process to PAS signal amplitude was
applied. From Fig. 4, one can see that between 70 and 270 Hz, the
PAS signal amplitude changes with modulation frequency as f −1.25
,
which is near f −1.0, characteristic of nonradiative surface recom-
bination, thermoelastic bending, or thermal dilation [37]. Thermal
dilation produces a signal whose phase is independent of the modu-
lation frequency and equal to −90◦. As shown in Fig. 4, this behavior
is not observed, allowing us to disregard this heat transfer mecha-
nism. The absence of thermoelastic bending was verified by it not
being possible to fit the phase data to the Eq. (4). On the other hand,
the expression for the phase corresponding to the nonradiative sur-
face recombination mechanism, reproduced as Eq. (3), was fitted to
the ˚ph (radians) versus f plot in the modulation frequency range
of 90–270 Hz. The observed deviation of frequency dependence of
the PAS signal from the theoretical expectation f −1.0 to f −1.25 can
be associated with a partial overlap of nonradiative bulk and sur-
face recombination mechanisms because both processes have the
same expression for the signal phases.
Fig. 3. Volume fraction of the AlSb phase as a function of milling time. The solid line
is a guide to the eye.

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D.M. Trichês et al. / Journal of Alloys and Compounds 505 (2010) 762–767
46 W/mK is obtained. This value is ≈19% smaller than that reported
in Ref. [35]. This result seems to corroborate the suggestion, found
in the literature, that the thermal conductivity can be smaller in
materials having nanometric crystallites [39].
In a previous study [40], we measured the thermal diffusivity
of nanostructured single phase ZnSe produced by MA. The value
for as-milled ZnSe was 51% larger than after annealing, which was
similar to the value of bulk ZnSe (˛ = 0.101 × 10−4 m2/s). Based
on these results, it is interesting to estimate the contribution of
the interfacial component to thermal diffusivity value. For this, we
will assume that the measured thermal diffusivity is equal to the
weighted sum of thermal diffusivity of the crystalline and inter-
facial components and elemental Sb. Thermal diffusivity values of
bulk AlSb and elemental Sb and the volume fractions obtained by
taking into account the interfacial component will be considered.
The weighted value of the thermal diffusivity is written as
˛
= x˛AlSb + y˛int + z˛Sb
(8)
sample
Fig. 5. PAS signal phase versus modulation frequency for a sample milled for 10 h.
The solid line corresponds to the best fit of experimental data to Eq. (3).
where x = 0.50, y = 0.32, z = 0.18, ˛AlSb = 0.40 × 10−4 m2/s, and
˛
˛
= 0.17 × 10−4 m2/s. Using these values in Eq. (8) a value of
Sb
= 0.282 × 10−4 m2/s is calculated. This value represents ≈88%
int
In order to fit the theoretical result to the experimental curve,
we estimated an initial value for ˛ assuming that the thermal dif-
fusivity of sample could be described as a random binary mixture
of the phases present in the sample. Landauerˇıs paper [38] reports
an expression that describes quite well the electrical properties of
binary metallic mixtures. Since electrons drive heat and the elec-
tricity (see Wiedemann–Franz law), this expression has also been
used in this study to evaluate the thermal properties (thermal dif-
fusivity and thermal conductivity) of a random binary mixture of
AlSb and elemental Sb, and is written as
of the obtained from the PAS data. This result shows that for milled
materials having a large interfacial component, the contribution of
this component to thermal diffusivity of sample is very significant.
Annealing the sample can significantly reduce this contribution, as
it was shown in Ref. [40].
5. Conclusions
Several conclusions are obtained from this study. The main ones
are the following:
1
4
˛
=
(3x2 − 1)˛2 + (3x1 − 1)˛1
mixture
Milling mixture of Al and Sb with nominal AlSb composition, leads
the formation of nanostructured AlSb and elemental Sb.
The maximum volume fraction of AlSb is obtained for milling times
between 7 and 10 h.
For milling times larger than 10 h, the AlSb phase starts to disso-
ciate, leading to an increase in elemental Sb.
The volume fractions of the crystalline and interfacial components
were evaluated through an approach in which the XRD pattern is
converted to electron units by using the mean square scattering
factor ꢀf2ꢁ. Its combination with the Rietveld structural refinement
procedure is used to evaluate the volume fractions of AlSb and
elemental Sb.
The thermal diffusivity and other heat transport parameters for
nanostructured AlSb were obtained from the PAS amplitude and
phase.
ꢉ
1
ꢇ
ꢈ
⁄
2
+ ((3x2 − 1)˛2 + (3x1 − 1)˛1)2 + 8˛1˛2
(6)
where xi (i = 1,2) is the relative amount of AlSb and elemental Sb
phases obtained from the Rietveld analysis and ˛1 and ˛2 are
the thermal diffusivities of bulk AlSb and elemental Sb, respec-
tively. A value of ˛mixture = 0.326 × 10−4 m2/s was obtained. Bennett
et al. [10] reported values for the mobility of the excess carri-
ers at room temperature between 61 and 300 × 10−4 m2/s. Under
these conditions, the best fit was reached considering−t4he follow-
ingvalues: ˛sample = 0.321 × 10−4 m2/s, D = 62.6 × 10 m2/s, v =
205 × 10−2 m/s and ꢂ = 0.41 ␮s. The fit is shown in Fig. 5. The
good agreement of the thermal diffusivity values corroborates our
assumption.
Based on the previous expression (6), it is interesting to com-
pare the thermal conductivity reported in Ref. [35] with the one
calculated for a random binary mixture of the phases present in
the sample, using the expression [38].
Thermal behavior (thermal diffusivity and thermal conductivity)
of AlSb after 10 h of milling is similar to that of a random binary
mixture of the AlSb and elemental Sb.
The contribution of the interfacial component to the thermal dif-
fusivity of milled materials can be very significant. Annealing the
sample can reduce this contribution.
˛
meas
Kmixture
=
(3x2 − 1)A + (3x1 − 1)B
4
ꢉ
1
ꢇ
ꢈ
⁄2
+ ((3x2 − 1)A + (3x1 − 1)B)2 + 8AB
Acknowledgments
The authors thank Conselho Nacional de Desenvolvimento Cien-
tífico e Tecnológico (CNPq), FINEP, CAPES, and FAPESC, Brazil, for
financial support.
A = (ꢀCp)Sb and B = (ꢀCp)AlSb
,
(7)
where ꢀ is density obtained from the Rietveld analysis, Cp is the
specific heat listed above, and xi (i = 1,2) are AlSb and elemental
Sb concentrations, respectively. The term between brackets is the
volumetric heat capacity of a random binary mixture. Considering
the values of ꢀ = 4320 kg/m3 and 6757 kg/m3 for as-milled AlSb and
Sb, respectively, and the values of Cp given in Ref. [17], a value of
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