Due to its good thermoelectric figure of merit at room temperature, bismuth
telluride (Bi
Te
) as well as some related ternary alloys are often used
for cooling applications in commercial Peltier elements. Commonly applied
ternary alloys consist of bismuth telluride with either bismuth selenide
(Bi
Se
) or antimony telluride (Sb
Te
) [152]. Their
common crystal structure is hexagonal [153], although some authors
also describe the unit cell as rhombohedral [154], which is not a
discrepancy. The hexagonal description outlines the layered structure of the
material, and its unit cell has the lattice constants
and
at
[155]. Furthermore, the
corresponding linear thermal expansion coefficients are
and
[156,157]. According to
[158], Bi
Te
has a mass density of
and
a melting point of
which limits the temperature range for
thermoelectric applications.
A change of the free carrier concentration can, similarly to lead telluride,
either be performed by changing the material composition or with extra dopants.
In contrast to lead telluride, stoichiometric bismuth telluride is of p-type
with a free carrier concentration of approximately
. A
shift to excess tellurium leads to an n-type material.
Bismuth telluride is a narrow gap semiconductor with an indirect band gap of
at
. As most semiconductors, the temperature
dependence of its band-gap is negative with a value of
[159,160,161]. According to
pseudopotential band structure calculations [162], both the highest
valence band and lowest conduction band have six valleys. Beside these two
bands, each a second conduction and valence band with energy separations of
and
, respectively are proposed in
[163,164]. Due to the low density of states, the population
of higher energy levels is relatively high. Thus, the large non-parabolicity
of the band structure becomes important [165]. Recently,
experimental work has been accomplished with first principle calculations
[166,167], which serves as a basis for further performance
optimization, such as the introduction of low-dimensional structures
[168].
Reduction of the thermal conductivity is one important possibility to increase
the figure of merit. Within ternary alloys, the lattice thermal conductivity
depends on the additional phonon scattering introduced by alloy disordering.
The lowest values are achieved at the highest lattice disorder, for bismuth
antimony telluride, this is achieved in (Bi
Sb
)
Te
[169]. However, the according maximum figure of merit is
obtained at higher antimony content due to the contra-productive evolution of
the electrical conductivity and the carrier contribution to the total thermal
conductivity [170,171]. In sintered samples, the lattice
thermal conductivity is reduced by additional grain boundary scattering
[172]. The influence of several dopants on the thermal
conductivity is examined in [173]. Specific heat as well as the
influence of dopants has been studied in [163,174].
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While pioneering work has focused on pure bismuth telluride [176], the electrical properties for many ternary alloys have been investigated extensively later on [177,178,179,180,181]. Additional doping of bismuth antimony telluride with lead telluride causes a more favorable ratio of electrical and thermal conductivity and thus results in an elevated figure of merit [182].
Since the figure of merit reaches its maximum in a narrow temperature range of
about
, the overall device performance of a thermoelectric
generator is lower than the theoretical maximum. An approach to overcome
this fact is the introduction of graded or segmented materials along the
temperature gradient in order to match the optimum material properties to the
given thermal conditions [183].
Several mechanical, optical, and transport parameters show a strong anisotropy.
While anisotropy ratios of 4-6 and 2-2.5 are reported for the electrical
resistivity and the thermal conductivity, respectively
[169,184,185], the Seebeck coefficient is rather
isotropic with a deviation of about
between the according extrema.
Both p-type and n-type samples have Seebeck coefficients between
and
which depend on the material
composition [175,186]. The maximum figure of merit can be
observed parallel to the cleavage plains and outperforms the normal direction
by a factor of 2.
Figures 4.5 and 4.6 depict
thermoelectrically relevant data of bismuth telluride alloys with respect to
the material composition at room temperature. Transport properties have been
measured parallel to the cleavage plains [175] since this direction
is the most favorable for thermoelectric applications.
Thermal conductivity values shown in Fig. 4.6 change due to
the influence of the material composition on phonon scattering as well as an
additional carrier contribution at elevated free carrier concentrations.
Electrical resistivity is mainly influenced by the free carrier concentration
and the rate of ionized impurity scattering. The resulting figure of merit has
a reported maximum for n-type materials at a tellurium content of
.
Due to the high sensitivity of the figure of merit to the material composition,
an exact stoichiometric control during fabrication is necessary. For p-type
material, the maximum figure of merit is lower than that of n-type material.
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M. Wagner: Simulation of Thermoelectric Devices