High thermoelectric performance enabled by convergence of nested conduction bands in Pb7Bi4Se13 with low thermal conductivity

Thermoelectrics enable waste heat recovery, holding promises in relieving energy and environmental crisis. Lillianite materials have been long-term ignored due to low thermoelectric efficiency. Herein we report the discovery of superior thermoelectric performance in Pb7Bi4Se13 based lillianites, with a peak figure of merit, zT of 1.35 at 800 K and a high average zT of 0.92 (450–800 K). A unique quality factor is established to predict and evaluate thermoelectric performances. It considers both band nonparabolicity and band gaps, commonly negligible in conventional quality factors. Such appealing performance is attributed to the convergence of effectively nested conduction bands, providing a high number of valley degeneracy, and a low thermal conductivity, stemming from large lattice anharmonicity, low-frequency localized Einstein modes and the coexistence of high-density moiré fringes and nanoscale defects. This work rekindles the vision that Pb7Bi4Se13 based lillianites are promising candidates for highly efficient thermoelectric energy conversion.

To clarify the anisotropy of thermoelectric properties of Pb7Bi4Se13 with nH = 1.2×10 20 cm -3 , we present the Seebeck coefficient, electrical conductivities, thermal conductivities and zT values measured parallel and perpendicular to the SPS direction.
The measurement geometry is presented in Fig. S3a. Clearly, there exist a certain degree of anisotropy of thermoelectric properties as shown in Fig. S4a-4c. As shown in Fig.   S4d, the thermoelectric performance parallel to the SPS direction is slightly better than that perpendicular to the SPS direction. For convenience of discussion, thermoelectric properties of doped Pb7Bi4Se13 in the main text are results parallel to the SPS direction.
In this paragraph, a detailed discussion on the anisotropy of thermoelectric properties is performed. Thermoelectric materials with non-cubic structures demonstrate different degrees of anisotropy of charge and phonon transport. The anisotropy is considerably large in some layered structures, such as Bi2Se3, BiSe and their derivates. 21,22,23 Pb5Bi6Se14, a member of cannizzarite-type structure, also exhibits different electrical and thermal conductivities in parallel and perpendicular directions. 17 It mainly originates from its structural features, in which PbSe and Bi2Se3 subunit stack alternatively along the c axis. This layered structure is easily to form preferred orientation in the pressing direction. In contrast, other compounds, such as Pb3Bi2S6, 1 SnPb2Bi2S6 (members of lillianite homologous series), 4 and PbBi2S4 (a member of S9 galenobismuthite homologous series), 1 demonstrate nearly isotropic electrical and thermal transport properties. Taking Pb3Bi2S6 as an instance, it consists of NaCl-type (Pb/Bi)S layers with a mirror as twinning operation. These layers form the NaCl-type strips and avoid the formation of single layered structure preferred stacking to any crystal axis. Under pressing, crystal grains tend to distribute randomly, which gives rise to the near isotropy of electrical and thermal properties of parallel and perpendicular directions. For Pb7Bi4Se13, its crystal structure includes the NaCl-type (Pb/Bi)Se strips without separate PbSe and BiSe layer stacking alternatively. This might give rise to this anisotropy of thermoelectric properties. In total, the anisotropy of electrical and thermal transport properties is structural dependence. Nearly isotropic electrical and thermal transport properties are also available in Pb3Bi2S6, SnPb2Bi2S6, and PbBi2S4. Pb7Bi4Se13 indeed demonstrates a certain degree of anisotropy in electrical and thermal conductivities.   Where kB is the Boltzmann constant, is the reduced Planck constant, S is the Seebeck coefficient, η is the reduced Fermi energy, Fj is the Fermi integral, L is the Lorenz number, Nv is band degeneracy, is the speed of sound, d is the sample density, and mI * is the inert effective mass, and is the transport coefficient. S16

Supplementary Note 3: The derivation of unique quality factor
For n-type Pb7Bi4Se13, the electron is the major carrier. We use the two-band model to derive this unique quality factor.
Firstly, several physical parameters, such as reduced Fermi level for conduction, valence bands and reduced band gap, are introduced as follows.
, To consider the band nonparabolicity and bipolar effect, we take both conduction band and valence band into consideration, as well as the bipolar thermal conductivity, kb. The zT is expressed as follows: S17 , (S9) Ⅰ. For the first term in the denominator, , it is rewritten as follows.
, (S10) This first term could be expressed as . And the unique quality factor is derived as , among which is the transport coefficient, .
Ⅱ. For the second term in the denominator , , (S11) Ⅲ . For the final term , it could be described as below.
, (S12) By combining the above equations, the zT could be written as follows S18 , (S13) And finally, zT based on two-band Kane model could be expressed as the following equation.

Supplementary Note 4: Two Kane band model for a weak bipolar effect
Since there exists a weak bipolar effect, these contributions of major and minor carriers to thermoelectric parameters, including Seebeck coefficient, electrical conductivity and thermal conductivity are intertwined. It is necessary to decouple the major carrier contribution from that of the minor carrier by using a well-developed work by Pan, et al. 21 , (S15) , (S16) , (S17) , (S18) , (S19) , (S20) , (S23) The above expressions include a series of fundamental parameters. The critical parameters are effective mass, band degeneracy, mobility and deformational potential, which could be summarized into a significant parameter, weighted mobility, μW.

, (S24)
In the Pb7Bi4Se13 based compounds, the electron is the major carrier and the hole is the minor carrier. The chemical potential of electron could be defined by . S20 And the chemical potential of hole is denoted as . All charge transport parameters could be obtained after the determination of chemical potential and weighted mobilities. At first, the charge conduction at room temperature could be ascribed to the electron due to the enough band gap and almost unchanged carrier concentration shown in the inset of Fig. 3c of the main text. Based on this, the chemical potential and electron weighted mobility could be obtained. Secondly, by assuming the temperature dependence of chemical potential and weighted mobility, the remaining charge transport properties at higher temperatures could be obtained. It should be noted that the temperature dependence exponent could be changed slightly due to the existing diverse crystal defects, which introduce multiple charge scattering mechanisms. The temperature dependent parameters are tabulated in Table S3.  Here by using the two-band model demonstrated in the Section 4, the physical parameters could be decoupled from major and minor carriers. And the experimental data could be fitted reasonably as shown in Fig. S11. Considering the band gap, the contribution from the minor carrier is not so strong at high temperatures. The bipolar and carrier thermal conductivity are be estimated to be 0.05 Wm -1 K -1 and 0.14 Wm -1 K -1 . Consequently, the lattice thermal conductivity is estimated to be 0.14 Wm -1 K -1 , in a good agreement with the value estimated in the main text. These parameters obtained by this method are reliable and acceptable. It should be noted that the fit is not unique.

TableS3. Physical parameters for two-band modeling
Other sets of parameters might also offer reasonable simulations. For the calculation of quality factor B * Kane, the physical parameters of the major carrier are estimated from the two-band model and tabulated in TableS5.

TableS4. Physical parameters for the estimation of B * Kane
Sample

Ga2
-238 -0.5 110 0.14 4.6 6  factor, B * Kane is calculated to be 6 by using physical parameters from Pan et al. 21 The quality factor derivation has been described in the section 4. In Fig. S13b, the experiment data is revealed by the white point with the measured zT of ~ 0.76. The highest zT is predicted to be 1.54, shown by the red point. The red arrow indicates the optimization of η and ξ to achieve the highest zT. Not only can the unique quality factor S27 be applied into the typical thermoelectric material, it also can be adopted into the recently reported van der Waals crystal Ta4SiTe4, which exhibits a narrow band gap of ~ 0.08eV. 28 At low temperature, such as 50 K, it is reasonable to assume there is only one kind of carrier in Ta4SiTe4. Consequently, the weighted mobility of electron and chemical potential can be derived. By assuming the linear temperature dependence of chemical potential, the remaining parameters could be obtained. The B * Kane is calculated to be 0.24 for Ta4SiTe4 at 300 K as shown in Fig. S13c. The experiment zT is 0.18, which could be optimized to the highest zT of 0.27 shown by the red arrow.
The following table shows the physical parameters to derivate the B * Kane.