4.5 Bismuth Telluride and its Alloys

Due to its good thermoelectric figure of merit at room temperature, bismuth telluride (Bi$_2$ Te$_3$ ) 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$_2$ Se$_3$ ) or antimony telluride (Sb$_2$ Te$_3$ ) [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 $a=4.38\,\ensuremath{\mathrm{\AA}}$ and $c=30.36\,\ensuremath{\mathrm{\AA}}$ at $77\,\ensuremath{\mathrm{K}}$ [155]. Furthermore, the corresponding linear thermal expansion coefficients are $14.4\times10^{-6}\,\ensuremath{\mathrm{K^{-1}}}$ and $21.3\times10^{-6}\,\ensuremath{\mathrm{K^{-1}}}$ [156,157]. According to [158], Bi$_2$ Te$_3$ has a mass density of $7.86\,\ensuremath{\mathrm{g/cm^{-3}}}$ and a melting point of $858\,\ensuremath{\mathrm{K}}$ 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 $10^{19}\,\ensuremath{\mathrm{cm^{-3}}}$ . A shift to excess tellurium leads to an n-type material.

Bismuth telluride is a narrow gap semiconductor with an indirect band gap of $160\,\ensuremath{\mathrm{meV}}$ at $300\,\ensuremath{\mathrm{K}}$ . As most semiconductors, the temperature dependence of its band-gap is negative with a value of $-1.5\times10^{-4}\,\ensuremath{\mathrm{eV/K}}$ [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 $30\,\ensuremath{\mathrm{meV}}$ and $20\,\ensuremath{\mathrm{meV}}$ , 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$_{0.5}$ Sb$_{0.5}$ )$_2$ Te$_3$ [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].

Figure 4.5: Free carrier concentration as well as figure of merit with respect to material composition for Bi$_2$ Te$_3$ , after [175].

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 $50\,\ensuremath{\mathrm{K}}$ , 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 $10\,\%$ between the according extrema. Both p-type and n-type samples have Seebeck coefficients between $100\,\ensuremath{\mathrm{\mu V/K}}$ and $250\,\ensuremath{\mathrm{\mu V/K}}$ 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 $64.5\,\%$ . 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.

Figure 4.6: Resistivity as well as thermal conductivity with respect to material composition for Bi$_2$ Te$_3$ , after [175].

M. Wagner: Simulation of Thermoelectric Devices