Synthetic conditions and their doping effect on β-K2Bi8Se13

Article abstract of DOI:10.1016/j.jallcom.2008.06.143

In this work the synthetic conditions for K₂Bi₈Se₁₃ and their effect on its thermoelectric properties were investigated. K₂Bi₈Se₁₃ was prepared as a single phase using K₂Se and B

Full text of DOI:10.1016/j.jallcom.2008.06.143

Journal of Alloys and Compounds 474 (2009) 351–357
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: www.elsevier.com/locate/jallcom
Synthetic conditions and their doping effect on ␤-K Bi Se
13
2
8
a,∗
a
b
b
b
c
Th. Kyratsi , I. Kika , E. Hatzikraniotis , K.M. Paraskevopoulos , K. Chrissafis , M.G. Kanatzidis
a
Department of Mechanical and Manufacturing Engineering, University of Cyprus, 1678 Nicosia, Cyprus
Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Department of Chemistry, Northwestern University, Evanston, IL 60208, and Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, United States
b
c
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 17 April 2008
Received in revised form 18 June 2008
Accepted 20 June 2008
Available online 28 August 2008
In this work the synthetic conditions for K Bi Se and their effect on its thermoelectric properties were
2
8
13
investigated. K2Bi8Se13 was prepared as a single phase using K2Se and Bi2Se3 as starting materials in a
furnace or via a reaction using direct flame, followed by remelting or annealing. Seebeck coefficient mea-
surements showed that the doping level in the material is sensitive to the synthetic conditions. Higher
synthesis temperatures as well as the flame reaction technique followed by annealing gave more homoge-
nous samples with higher Seebeck coefficient. IR optical spectroscopic measurements showed a wide
range of doping level achieved among the different synthetic conditions. These findings suggest that syn-
thetic conditions can act as a useful tool for the optimization of the thermoelectric properties of these
materials.
Keywords:
Thermoelectric materials
Chemical synthesis
©
2008 Elsevier B.V. All rights reserved.
1. Introduction
tures that seem to be responsible for its low thermal conductivity
(
∼1.3 W/m K) [4].
The thermoelectric figure-of-merit ZT of a material is given by
Attempts to improve the thermoelectric properties of ␤-
K Bi Se have been done based on solid solution formation as well
2
2
8
13
S ꢀ
ZT =
T
(1)
as doping experiments. Solid solutions were prepared via substi-
tution on the heavy metal sites (i.e., K Bi SbxSe13) [6,7], on the
ꢁ
2
8−x
where ꢀ is the electrical conductivity, S is the Seebeck coefficient,
is the thermal conductivity and T is the temperature. Candi-
date materials for thermoelectric applications should possess high
electrical conductivity, high Seebeck coefficient and low thermal
conductivity, so the thermoelectric figure-of-merit ZT can be max-
imized exceeding one.
alkali metal site (i.e., K2 RbxBi Se ) [8] as well as on the chalco-
−x
8
8
13
ꢁ
genide sites (i.e., K Bi Se Sx) [9]. The series were studied in
2
13−x
terms of crystallography, physical and thermoelectric properties.
Doping studies were also carried out on selected members using
varying amounts of excess Se, Pb and Sn [6]. The charge trans-
port properties of the K Bi SbxSe13 series (low Sb concentration)
2
8−x
Research on complex bismuth chalcogenide compounds has
shown that ␤-K Bi Se has many attractive features that make
as well as doped materials displayed degenerate semicon-
ductor behavior with very high free carrier concentration of
2
8
13
it promising for thermoelectric applications [1–3]. The structure
of ␤-K Bi Se [4] include two different interconnected types of
20
−3
.
∼
10 cm
2
8
13
According to band structure calculations [10] there are special
Bi/Se blocks, K ions positionally and compositionally disordered
with Bi or other K atoms, over the same crystallographic sites,
and loosely bound K atoms in tunnels. Decrease of the lattice
thermal conductivity can be accomplished based on the concept
of “phonon glass electron crystal” (PGEC) [5] through “rattling”
atoms in cages or tunnels of the structure, or by materials with
complex compositions, large unit cells, low crystal symmetry and
sites in the crystal structure that are very important for the prop-
erties of this material. The calculations indicated that, depending
on how the K and Bi atoms are arranged in the mixed occu-
pancy lattice sites, an energy gap may or may not appear at the
fermi level. Therefore, two extreme variants of the band structure
exist: from a semimetal (band overlap) to semiconductor (band
gap) based on the K/Bi arrangements. Detailed analysis [11] of
the carrier concentration as well as of the Seebeck coefficient had
site occupancy disorder provided. ␤-K Bi Se combines such fea-
2
8
13
shown that the properties of Bi-rich members of the K Bi
SbxSe13
2
8−x
series were well described with a finite-gap semiconductor model
that contained a significant concentration of shallow donors.
We believe that these shallow donors that likely originate from
defects created by local disordering between K and M (Bi, Sb),
∗ Corresponding author. Tel.: +357 22892267.
E-mail address: kyratsi@ucy.ac.cy (Th. Kyratsi).
0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jallcom.2008.06.143

352
Th. Kyratsi et al. / Journal of Alloys and Compounds 474 (2009) 351–357
Fig. 1. Crystal structure of the ␤-K2Bi8Se13 compound.
resulted in the difficulty to make materials with low carrier
concentration.
2.3. Powder X-ray diffraction
The samples were examined by X-ray powder diffraction to assess phase purity.
Powder patterns were obtained using a Rigaku Miniflex powder X-ray diffractome-
ter with Ni-filtered Cu K␣ radiation operating at 30 kV and 15 mA. The data were
In this work, we aimed to explore how the synthetic condi-
tions affect the doping level of these materials in order to find a
procedure that leads to promising and doping-sensitive materials.
We studied the K Bi Se compound with emphasis on modify-
◦
collected at a rate of 0.5 /min. The purity of phases for the solid solutions was con-
firmed by comparison of X-ray powder diffraction pattern to the calculated one from
single crystal data for ␤-K2Bi8Se13 using Cerius software [12]. The reported purity of
phases is subjected to the detection limit of the experiment which is approximately
2
8
13
2
ing the free carrier concentration in order to find more efficient
ways to control the doping level of the material through the syn-
thesis procedure. We focused mainly on using K Se and Bi Se as
4%.
2
2
3
starting materials and treat them with different heating/cooling
profiles. The products were studied with X-ray diffraction, thermal
analysis and energy dispersive spectroscopy regarding their purity
and elemental analysis while Seebeck coefficient and IR reflectivity
measurements were performed in order to study their doping level
and explore their properties.
2.4. Thermal analysis
Differential scanning calorimetry (DSC) was performed with
controlled SETARAM Setsys 16/18 TG-DTA as well as Linseis STA1600 thermal
a computer-
analyzer. The ground samples (∼30 mg total mass) were placed in alumina crucibles
◦C at 10 ◦C/min.
and heated under argon flow to 730
2
.5. Energy dispersive sectroscopy (EDS)
2. Experimental
EDS was realized using a Jeol 840A scanning microscope with an energy disper-
2
.1. Reagents
sive spectrometer attached (Oxford, model ISIS 300). The beam spot was 1 ␮m, the
accelerating voltage 20 kV, the beam current 0.4 nA, the working distance 20 mm and
the counting time 60 s real time. The matrix correction protocol was ZAF correction.
Chemicals in this work were used as obtained: (i) bismuth metal, 99.999% purity,
Alfa Aesar; (ii) Selenium metal, 99.999% purity, Alfa Aesar; (iii) potassium metal, rod,
9
9.5% purity, Aldrich Chemical Co.
2.6. Thermoelectric properties
2.2. Synthesis
Seebeck coefficient measurements were carried out at temperature range of
80–400 K with a programmable Seebeck controller SB100 from MMR Technologies,
K2Bi8Se13 was synthesized using K2Se and Bi2Se3 as starting materials and fol-
Inc., using constantan wire as reference. Heat gradient was applied along the needle
lowing different heating treatments as described below. All manipulations were
carried out under nitrogen atmosphere. The reacting materials were sealed in silica
direction (crystallographic b-axis).
−4
tubes under vacuum (10 Torr). All products however are stable and can be han-
dled and stored in air. K2Se was prepared by stoichiometric combination of K and Se
in liquid ammonia. Bi2Se3 was prepared by the stoichiometric combination of the
2.7. Optical properties
Infrared spectra were recorded at nearly normal incidence in the 70–2000 cm 1
spectral region, at room temperature, with a Bruker 113 V FT-IR spectrometer with a
−
◦
elements at 900 C.
Table 1
Synthetic conditions for ␤-K2Bi8Se13 and their minimum and maximum Seebeck coefficients
◦
Sample
Synthetic condition
In furnace
Temperature ( C)/duration (h)
Smax (␮V/K)
Smin (␮V/K)
I-1
I-2
I-3
I-4
I-5
I-6
760/0
760/3
780/0
820/0
820/1
850/1
−119.0
−142.6
−130.6
−130.7
−136.9
−143.2
−57.8
−95.6
−89.5
−116.9
−115.9
−127.7
II-1
II-2
Flame reaction + remelting
Flame reaction + annealing
760/3
780/3
−131.9
−162.5
−113.5
−151.3
III-1
III-2
III-3
500/48
500/60
500/76
−193.4
−222.7
−238.4
−184.0
−204.9
−223.2

Th. Kyratsi et al. / Journal of Alloys and Compounds 474 (2009) 351–357
353
3.2. Synthesis and characterization
Previously, ␤-K Bi Se type materials were prepared by direct
2
8
13
combination of the elements and were grown by Bridgman tech-
niques [13]. The latter gives highly oriented large ingot samples.
These samples show strong anisotropic properties and high free
carrier concentration [6]. We have also reported solid solutions of
K Bi Se with Sb [6,11] and S [9,14] prepared by stoichiometric
2
8
13
combination of the elements followed by Bridgman crystal growth.
The Seebeck coefficient of the Bi-rich members of K Bi SbxSe13
2
8−x
was below −100 ␮V/K [6] and for K Bi Se Sx it was ∼ −120 ␮V/K
2
8
13−x
for x = 4 [14]. Doping experiments were also carried out on
K Bi SbxSe13 with x = 1.6 and the Seebeck coefficient was below
2
8−x
−
90 ␮V/K using Pb and Sn as dopants. In general, all materials
prepared by stoichiometric elemental combination followed by
Bridgman crystal growth showed a relatively high doping state with
carrier concentration [6] of the order of 1020 cm . Those results
−3
suggest the ␤-K Bi Se materials are not responsive to doping
2
8
13
when they are prepared by Bridgman technique. Moreover, when
-K Bi Se was prepared using K Se, Bi and Se metals as start-
Fig. 2. Experimental XRD pattern of K2Bi8Se13 compare to the calculated from Cerius
software.
␤
2
8
13
2
ing materials [4], lower carrier concentration and higher Seebeck
coefficient (of −200 ␮V/K) were found [4]. This is probably due to
the creation of Se vacancies during heating. Doping experiments on
such materials have shown [15] that ZT can be improved, mainly
resolution of ∼2 cm−1. The reflection coefficient was determined by typical sample-
in–sample-out method with a gold mirror as the reference. Measurements were
carried out on pressed pellets (crystals were first ground using mortar and pestle
and then cold pressed).
2
by raising the power factor (S ꢀ). More specifically, in Sn doped
K Bi Se13, the power factor was increased by a factor of ∼3 compare
2
8
to the pristine material [15].
3. Results and discussion
In this work samples fo K Bi Se were prepared with three
2
8
13
different synthetic conditions (see Table 1):
3
.1. Structure of K Bi Se
2
8
13
•
Synthetic condition I: K Se and Bi Se were initially mixed in 1/4
2 2 3
The structure of ␤-K Bi Se includes two different intercon-
nected types of Bi/Se rod-shaped building blocks and K atoms
2
8
13
molar ratio. A slight excess of K Se (2–3%) was found necessary in
2
+
order to obtain pure K Bi Se13. This excess presumably compen-
2
8
(
1 1 1)
in tunnels [4], see Fig. 1. The so-called NaCl
ing blocks are cuts representing Bi Te -type structural fragments,
-type Bi/Se build-
sated for the loss of K Se due to a reaction with silica tube that
2
2
3
occurred during synthesis. For example, K Bi Se was prepared
2
8
13
(
1 1 1)
interconnected to form a step-like structure along a-axis. NaCl
type blocks are bridged, along c-axis, with the NaCl
building blocks that are separated by tunnels. The NaCl -type
-type blocks are connected on Bi(8) atom as well as at
-
by mixing 0.175 g K Se (1.1 mmol) and 2.830 g Bi Se (4.3 mmol).
2
2
3
(
1 0 0)
-type Bi/Se
Carbon coated silica tubes were used to inhibit glass attack by
(
1 1 1)
K Se. The reaction was performed in a silica tube sealed under
2
(
1 0 0)
and NaCl
◦
vacuum in a tube furnace at 750–850 C followed by cooling at a
Bi(9) metal site with Bi Se bonds. In this structure these connect-
ing sites were found to be mixed occupied by K and Bi atoms and
according to band structure calculations the electronic properties
of the material are sensitive to the K/Bi atom distribution [10].
◦
rate of ∼3 C/min.
•
Synthetic condition II: K Se and Bi Se were mixed as in synthetic
2
2
3
condition I (Table 1) but the reaction was done by melting the
constituents with a flame. This was followed by remelting the
Fig. 3. DSC graph of K2Bi8Se13 (sample from I-2).
Fig. 4. Seebeck coefficient vs. temperature at 80–400 K for selected samples.

354
Th. Kyratsi et al. / Journal of Alloys and Compounds 474 (2009) 351–357
Fig. 5. Minimum and maximum room temperature Seebeck coefficient in a given batch for various conditions of synthesis.
material in a furnace. For the so-called “flame reaction” the tube
was carefully placed under the flame of butane–propane mixture
3.3. Properties of K Bi Se
2 8 13
until the material melted. Then the tube was removed from the
flame and quenched. The flame reaction has to be carried out
by a cautious and experienced operator using protection shields
and helmet. The products of the flame reaction were a mixture of
K Bi Se and K2.5Bi8.5Se14 phases. To get pure K Bi Se phase
Seebeck coefficient measurements and IR reflectivity spectra
were carried out for samples prepared by all three types of synthetic
conditions described above.
2
8
13
2
◦
8
13
3.3.1. Seebeck coefficient
the sample was remelted in a furnace 750–850 C followed by
Seebeck coefficient was negative for all samples confirming the
n-type character of the materials. It increases with temperature at
the range of 80–400 K, see Fig. 4, and this behavior is typical of alkali
bismuth selenide compounds [1].
◦
cooling with rate of ∼3 C/min.
•
Synthetic condition III: K Se and Bi Se were mixed as described
2
2
3
above but the reaction was done by flame followed by anneal-
◦
ing treatment under vacuum at 500 C for more than 48 h. The
In order to study the effect of synthetic conditions on dop-
ing homogeneity different samples obtained from the same ingot
were measured. Fig. 5 shows the minimum and maximum val-
ues of Seebeck coefficient at room temperature that were taken
from the same ingot. The variation of the Seebeck coefficient (dif-
ference between minimum and maximum value) was found to be
decreased from I-1 to I-6. According to Table 1, the samples I-1
through I-6 were prepared at increasingly higher temperature, and
the products are expected to be more homogeneous when pre-
pared at higher temperatures. More homogeneous doping was also
observed in samples obtained from the flame reaction (conditions
II and III in Fig. 5).
products of the flame reaction were mixture of K Bi Se and
2
8
13
◦
K2.5Bi8.5Se14 phases and annealing at 400 C was necessary to get
pure K Bi Se phase.
2
8
13
All products were pure K Bi Se phase as judged by powder
2
8
13
X-ray diffraction patterns that were compared to the calculated pat-
2
tern using Cerius software, see Fig. 2. Thermal differential scanning
calorimetry (DSC) analysis showed a melting peak during heating
◦
at about 700 C, see Fig. 3.
EDS analysis was used to obtained the elemental composition
of the products. Table 2 shows the atomic percentage of the K, Bi
and Se elements in the products of the three different categories.
In order to improve limited accurancy by the EDS measurements,
several points were measured and the average values are presented
in Table 2. These results indicate an increasing trend of K percent-
age for synthetic condition III as well as an decreasing trend of Bi.
This variation is expected to affect the doping level of the sam-
Furthermore, there is a wide range of Seebeck coefficient values
with various synthetic conditions (from sample I-1 through III-4 in
+
3+
ples since K and Bi share the same crystallographic sites [4].
Excess K+ ions are expected to act as acceptors, thus a lower car-
rier concentration in samples made with this synthetic condition is
expected.
Table 2
EDS results for selected conditions of ␤-K2Bi8Se13 synthesis
Sample
K (at.%)
Se (at.%)
Bi (at.%)
I-1
I-2
I-4
7.5
7.6
8.1
55.0
55.4
54.8
37.5
37.0
37.1
II-1
II-2
8.1
7.8
55.2
54.5
36.7
37.7
III-1
III-3
8.9
8.4
55.3
54.9
35.8
36.7
Fig. 6. Reflectivity data from pressed pellets of samples prepared with various syn-
thetic conditions (I, II and III represent the sample categories shown in Table 1).

Th. Kyratsi et al. / Journal of Alloys and Compounds 474 (2009) 351–357
355
Fig. 5) suggesting a wide variation in the doping level. Regarding
the Seebeck coefficient trends for different conditions these results
show that
(
a) the higher synthesis temperature, the higher the Seebeck coef-
ficient
(
b) the flame reaction produces samples with higher Seebeck coef-
ficient especially when followed by annealing treatment.
3.3.2. Optical properties
IR reflectivity spectra in the far- and mid-infrared region on
different K Bi Se samples are shown in Fig. 6; the spectra are
2
8
13
shifted along the y-axis for clarity. The spectra were analyzed by the
Kramers–Kronig method and also fitted by classical dispersion the-
ory [16,17]. The Kramers–Kronig (K–K) algorithm was optimised for
increased accuracy in the high-end and low-end spectral regions.
Fig. 7a shows a III-type sample where two zones are characteristic
−
1
bands in the IR spectra at ∼100 and ∼150–160 cm (see arrows
in the inset). These are similar to those of other K (Bi/Sb) Se
13
Fig. 8. Dielectric loss function, Im(−1/ε), for various synthesis conditions.
2
8
compounds studied elsewhere [18]. There is a shoulder at about
2
00 cm 1 (see arrows in the inset of Fig. 7a). The lines correspond
−
to the analysis model that is described below.
Fig. 8 shows the dielectric loss function, Im(−1/ε), for the spec-
tra, as obtained by K–K analysis. In the spectra of samples belonging
to category III two clear peaks are observed (longitudinal optical
−
1
(
LO)-phonons) at about 130 and 180 cm , see arrows in Fig. 7b,
which are decreased at the expense of a broad band at higher wave
−
1
numbers (at about 1070 cm for spectrum of category I, Fig. 8).
The reflectivity R(ω) is given by [19]:
ꢀ
ꢀ
ꢁ
2
ꢀ
ε(ω) − 1ꢀ
ꢀ
ꢀ
R(ω) = ꢀꢁ
ꢀ
ꢀ
(1)
ꢀ
ε(ω) + 1
where ε(ω) is the dielectric function. For the spectra analysis, we
assume that dielectric function is given by [16]:
⎡
⎤
2
− ω2 − i · ꢂLO,j
− ω2 − i · ꢂTO,j
ꢂ ω
ω
ω
2
PL
ω
− i · (ꢂP − ꢂ0)ω
LO,j
⎣
⎦
ε(ω) = ε∞
−
(2)
2
TO,j
2
ω
ω
+ i · ꢂ0ω
j
where ε∞ is the dielectric function at high frequencies and j is the
number of modes. The first term in Eq. (2) that corresponds to the
phonon contribution, proposed by Kurosawa [17], is the extension
of the Lyddane–Sachs–Teller (LST) relation to finite frequencies,
and works well in numerous cases, including strongly damped soft
modes [20]. The second term corresponds to the free carrier con-
tribution based on extended Drude model [21]. The frequencies
ωLO,j and ωTO,j are the characteristic frequencies for longitudinal
and transverse modes. The ꢀLO,j and ꢀTO,j are the damping factors
for each mode j. Based on the Drude model ꢀ0 and ꢀP are the damp-
ing constants for frequency ω = 0 and ω = ωPL (plasma frequency),
respectively. For ꢀ0 = ꢀP we obtain the known Drude equation. The
plasma frequency (ωPL) is given by
2
n · e
2
PL
ω
=
(3)
m ∗ ε ε∞
0
where ε0 is the dielectric function of vacuum, m* the carrier effec-
tive mass and n is the free carrier concentration.
The results of the analysis are shown in Fig. 9. For the analysis we
−
1
assumed four modes with TO at 98.7, 148.4, 192.6 and 451.2 cm
−
1
Fig. 7. Example of fittingresultsfor the III-type of synthesis (a)reflectivity, (b) dielec-
tric loss function Im(−1/ε). Point are the experimental data for reflectivity and K–K
analysis for the Im(−1/ε). Solid lines represent the calculated curve from disper-
sion theory. In the inserts are shown the reflectivity and the dielectric loss function
in short wavelengths. The arrows indicate the position of the TO-phonons (a) and
LO-phonons (b).
and LO at 122.7, 171.6, 245.6 and 457.0 cm . As an example, the
results of the fit for a sample synthesized with condition III are
shown in Fig. 7. As can be seen the overall fit is good for both the
reflectivity spectrum and for the dielectric loss function Im(−1/ε).
Similar fitting results were obtained for the spectra of all measured

356
Th. Kyratsi et al. / Journal of Alloys and Compounds 474 (2009) 351–357
Fig. 10. Seebeck coefficient (average values from values in Fig. 5) vs. carrier con-
centration at room temperature. The solid lines are calculated based on Eq. (4) for
assuming scattering by acoustic phonons (r = 0) and by impurities (r = 2).
(
see Table 1) produces samples with high free carrier concentration
20
−3
of 1.2–1.3 × 10 cm . The flame reaction followed by re-melting
19
−3
decreases the free carrier concentration (5.1 × 10 cm ) by nearly
a factor of two, while the free carrier concentration is again halved
1
9
−3
(
2.3–2.6 × 10 cm ) when annealing follows the flame reaction.
Fig. 10 shows the Seebeck coefficient with the carrier concentra-
tion. As expected from theory, when the free carrier concentration
increases, the Seebeck coefficient should decrease. The Seebeck
coefficient as well as the free carrier concentration depend on fermi
level (Á = EF/kBT where EF is the fermi level, kB is the Boltzman con-
stant and T is the temperature), through the fermi integrals. Seebeck
coefficient and the free carrier concentration are given by [22]:
Fig. 9. TO and LO frequencies as a function of the free carrier concentration for
different synthesis conditions.
ꢃ
ꢄ
kB Fr+2(Á)
S = ± e
− Á
(Á)
,
n = NC · F3/2(Á)
(4)
F
r+1
samples. As shown in Fig. 9, the values of TO and LO frequencies
remain practically unchanged, as expected. The change in the spec-
tral characteristics of the dielectric loss function, Im(−1/ε), can be
understood as due to the contribution of the free carrier concen-
tration; actually, samples with higher free carrier concentration (as
samples in synthesis type-I) show a more profound band at about
where e is the electron charge and N is the density of states in the
C
conduction band. Fr(Á) are the fermi integrals, while the parameter
r depends on the scattering mechanism (r = 0 for acoustic phonon
scattering, r = 2 for impurities scattering), and are given by:
ꢅ
ꢆ
ꢇ
∞
∂f (ε, Á)
∂ε
r
Fr(Á) =
−
· ε dε
(5)
−
1
1
070 cm . This band is lowered in intensity and shifted into lower
0
wavenumbers as the free carrier concentration is decreased.
From the analysis of the IR reflectivity spectra, it is evident that
the plasmon is over-damped. The free carrier concentration was
determined with Eq. (3), taking for simplicity the effective mass
m* = 1. As can be seen in Table 3, the various conditions of synthe-
sis lead to samples with free carrier concentration ranging from
In the case of non-degeneracy, Eq. (4) is reduced to the well
known formula [22]:
kB
S = ± e [A − ln(n) + ln(N )]
(6)
C
where A is related to scattering mechanisms (A = 2 for acous-
tic phonon scattering, A = 4 for ionized impurities scattering).
As expected from theory, when the free carrier concentration
increases, Seebeck coefficient should decrease. In a linear-log graph,
in the case of a non-generate electron gas, the dependence of See-
beck coefficient to free carrier concentration should show a linear
trend, with a slope kB/e. In a more general case of arbitrary degen-
eracy, the dependence of the Seebeck coefficient on the carrier
concentration should show a sub-linear trend (curved to higher
n), as observed in Fig. 10. The solid lines in Fig. 10 are the calcu-
lated dependence based on Eq. (4) for reduced fermi level taken
from −1 to 10. Two extreme cases were considered, carriers are
scattered by acoustic phonons (r = 0) and by impurities (r = 2). As
shown in Fig. 10, the points lie well inside those two cases, indica-
tive of a mixed scattering mechanism. From Fig. 10 we conclude that
8
20
−3
9
.6 × 101 to 1.3 × 10 cm . The so-called “in furnace” reaction
Table 3
IR analysis results (plasma frequency, dumping factors and carrier concentration)
for selected conditions of ␤-K2Bi8Se13 synthesis
−
−1
−1
n (1019 cm−3
)
Sample
ωP (cm 1
)
ꢂP (cm
)
ꢂ0 (cm
)
I-1
I-2
I-4
1068.9
1110.8
1102.8
2050.9
2490.8
2408.8
2050.9
2490.8
2408.8
12.2
13.2
13.2
II-1
II-2
632.4
700.4
1271.6
1324.8
1039.8
1180.6
4.29
5.18
III-1
III-2
III-3
464.9
300.2
461.0
955.6
609.9
1254.4
955.6
609.9
1254.4
2.30
0.96
2.56

Th. Kyratsi et al. / Journal of Alloys and Compounds 474 (2009) 351–357
357
lowed by annealing. The work described here shows the response
of K Bi Se to synthetic conditions and the new information
2
8
13
reported here can be exploited in future optimization procedures.
Acknowledgments
The authors thank Dr. E. Pavlidou for her help with EDS measure-
ments. Cyprus Research Promotion Foundation is greatly acknowl-
edged for the financial support (THEMATA—TEXNO/0104/16). MGK
thanks ONR for support.
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8
13
[
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1
8
20
−3
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2
[
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8
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man technique [6] generally have high carrier concentration of
[
2
0
−3
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(
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[
[
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synthesis temperatures and (b) the flame reaction method fol-
[