Enhanced Figure of Merit in Bismuth-Antimony Fine-Grained Alloys at Cryogenic Temperatures

Thermoelectric (TE) materials research plays a vital role in heat-to-electrical energy conversion and refrigeration applications. Bismuth-antimony (Bi-Sb) alloy is a promising material for thermoelectric cooling. Herein, a high figure of merit, ZT, near 0.6 at cryogenic temperatures (100–150 K) has been achieved in melt-spun n-type Bi85Sb15 bulk samples consisting of micron-size grains. The achieved ZT is nearly 50% higher than polycrystalline averaged single crystal ZT of ~0.4, and it is also significantly higher than ZT of less than ~0.3 measured below 150 K in Bi-Te alloys commonly used for cryogenic cooling applications. The improved thermoelectric properties can be attributed to the fine-grained microstructure achieved from rapid solidification, which not only significantly reduced the thermal conductivity but also mitigated a segregation effect. A record low thermal conductivity of ~1.5 W m−1 K−1 near 100 K was measured using the hot disk method. The thermoelectric properties for this intriguing semimetal-semiconductor alloy system were analyzed within a two-band effective mass model. The study revealed a gradual narrowing of the band gap at increasing temperature in Bi-Sb alloy for the first time. Magneto-thermoelectric effects of this Bi-Sb alloy further improved the TE properties, leading to ZT of about 0.7. The magneto-TE effect was further demonstrated in a combined NdFeB/BiSb/NdFeB system. The compactness of the BiSb-magnet system with high ZT enables the utilization of magneto-TE effect in thermoelectric cooling applications.


S2. Simulation method and parameters for Bi85Sb15 alloy
In order to calculate the carrier concentrations (electron concentration n and hole concentration p), the charge neutrality equation was applied: where + , − are the ionized donor and acceptor concentration, respectively.
For Bi85Sb15, the Hall coefficient data at 50 K showed | − | = | + − − | ≈ 1.6 * 10 23 −3 . It was based on the assumption that the donors and acceptors were fully ionized above 50 K, and + and − would stay constant as the temperature changed.
The equations for the carrier concentrations were given by: where is the Fermi-Dirac distribution function, ( ) ( ) is the density of states of the conduction (valence) band, CBM and VBM are the conduction band minimum and valence band maximum, respectively.
The Matthiessen's rule was used to calculate the scattering rate 1 in the undoped Bi85Sb15.
The first term 1 represents the acoustic deformation potential scattering (ADP), which is related to the electron-lattice interaction. The trend of the mobility caused by the ADP scattering is proportional to −1.5 . The equation for 1 was given by 1 : where is the acoustic deformation potential, is the longitudinal elastic constant, ( ) = 6.6 × 10 10 6.6 × 10 10 Table S1. The parameters in the simulation of undoped Bi85Sb15. The acoustic deformation potential is around 20 eV.
The second term 1 was coming from the ionized impurity scattering in this n-type material. It has the form = 0 ( ⁄ ) 3/2 , where 0 is constant. 1 The mobility trend will be proportional to 1.5 if the ionized impurity scattering dominates, resulting in an increase in resistivity at low temperatures.
The following equation is usually used to describe the relationship between resistivity and energy gap in a semiconductor: where 0 is a constant and is the band gap.
In view of the impurity scattering term, instead of directly using Eq. 6 to determine the band gap at low temperature (<100K), we have used a bandgap value of 14 meV based on the result from single crystal Bi-Sb alloy. 2 The plots of the carrier concentration and mobility are shown in Fig. S2. It was found that the rising of the resistivity of our SPSed Bi85Sb15 was not entirely caused by the carrier concentration related to the band gap; it was also due to the ionized impurity scattering which significantly changed the mobility trend below 100K. Based on the electronic structure of bismuth, the quasi-ellipsoid centered at the L-point of Brillouin zone where the electrons are located are strongly elongated along a direction tilted by an angle φe out of the binary-bisectrix plane (φe=6 ± 0.2° at 4.2 K) 3 . This highly anisotropic shape led to unusually small effective masses along with two directions that resulted in a very high electron mobility 4 , especially when compared with the hole mobility due to the much larger effective hole mass. Therefore, the undoped Bi85Sb15 system behaved as a strong n-type semiconductor dominated by electrons. By analyzing the transport properties using a two-band effective mass model, we were able to quantitatively decouple the contributions from electron and hole channels. Results are shown in Fig. S2. The electrical conductivity and Seebeck coefficient components can be calculated using Eq. 7 and Eq. 8. and are the electrical conductivity component and Seebeck coefficient component of the electron channel, and and are the electrical conductivity component and Seebeck coefficient component of the hole channel. * is the conductivity effective mass, the values can be found in Table S1. is the chemical potential and is the band gap. The same scattering mechanisms were used for electrons and holes to calculate the TE transport properties.
At low temperature, the alloy is extrinsic with electrons dominating the mobility, and ionized impurity scattering (~1 .5 ) is the main scattering mechanism. While above 110K, acoustic deformation potential scattering (~− 1.5 ) becomes the primary scattering mechanism, as shown in Fig. S2. There was a turning point for at 70 K. This could be approximately explained using Mott's equation 5 for a single band: where * is the density of state effective mass, n is the carrier concentration.
At low temperature, the system is n-type extrinsic, the number of electrons does not change dramatically with temperature. So, | | increases nearly linearly as the temperature increased. However, at higher temperatures, intrinsic excitation must be considered. The bipolar effect becomes increasingly important and the magnitude of the total Seebeck coefficient It is worthy to note that was found to have a relatively larger value compared with | | due to two main factors for hole carriers: (1) larger density of states effective mass than electrons; (2) lower carrier concentrations than electrons.

S4. Magnetic fields simulation and correction of TE measurements with magnet plates
The magnetic field generated by the magnet plates was simulated using standard textbook formulae. The two uniform magnet plates were divided into infinitely small magnetic dipoles dm ⃗⃗⃗ with magnetization M ⃗⃗⃗ along the x-direction, the coordinates were set as shown in Fig Therefore, from the basic electromagnetic field law, the magnetic field generated by this infinitely small dipole at point (x, y, z) will be: where = ( − ′ , − ′ , − ′) , and 0 is the vacuum permeability.
By Integrating over the magnet plates space, each component of the magnetic field at point (x, y, z) can be calculated as: where i=x, y, z, and V' are the magnet plates. Therefore, the magnitude and direction of the magnetic field produced by the magnet plates can be determined.
As shown in Fig. 9 in the main text, to calculate the TE properties for the segment LA of the sample between the magnet plates from the measurement that also included the field-free segment LB, we assume the two segments LA and LB were connected in series. Therefore, the measured total resistivity and Seebeck coefficient, denoted by subscript t, can be expressed as: where the symbol R represents electrical resistance Here we assume the thermal gradient was constant through the sample approximately. Therefore, the TE properties between the magnet plates could be calculated as: and the are the corrected resistivity and Seebeck coefficient shown in Fig. 10 in the main text.