Critical evaluation of p-type doping effects in Bi-Sb alloys

Relative to conventional power generators or refrigerators, thermoelectric (TE) technology provides a more stable, quiet, and environmentally-friendly method for heat-electrical energy conversion, and thus receives much research attention.¹ However, the limitation of the energy conversion efficiency of TE devices makes them less competitive in many situations. To improve the TE properties of materials and apply them to develop a device with better TE performance, one must focus on the dimensionless figure of merit zT of materials. zT is defined as:

where σ is the electrical conductivity, S is the Seebeck coefficient, and κ is the total thermal conductivity comprised of κ_e (electronic part) and κ_l (lattice part).

The Bi-Sb alloy has been known to be a promising n-type TE material system at cryogenic temperature for more than five decades.^2,3 Below 200 K, the zT of Bi₈₅Sb₁₅ alloy was reported to be ∼0.5, increasing to ∼1.2 under an applied magnetic field.³ The Bi-Sb alloy at the composition ∼Bi₉₀Sb₁₀ was also the first topological insulator discovered experimentally⁴ with a complex band structure evolution as the alloy composition changed. Between ∼7 and 22 at. % Sb, Bi-Sb alloy is a narrow-gap semiconductor with a maximum band gap of 14∼24 meV in single crystals.^5–7 Compared to the n-type Bi-Sb, the TE properties of the p-type Bi-Sb were found to be worse (zT∼0.1 below 200 K),^8–10 which limited the overall TE performance of devices made using compatible n-type and p-type Bi-Sb legs.¹¹ Therefore, it is worth studying p-type Bi-Sb alloys in order to understand the limitation and improve their TE performance.

Ideally, zT is to be as high as possible in order to have high energy conversion efficiency. However, according to the Mott relations and Wiedemann-Franz law,^12,13 because σ, S and κ mutually influences each other, it is difficult to simultaneously achieve high σ, S, and low κ. Researchers have used several techniques to resolve this problem, including doping^14,15 and nano-structuring.^16,17 It was found that, with the help of those techniques, the power factor σS² could be enhanced or the lattice thermal conductivity κ_l could be suppressed without having adverse effect on each other, thus making it possible to enhance zT.

Doping, which involves adding impurities into the matrix, can be used to tune the chemical potential and thus optimize the TE transport properties to improve the zT of materials. Bi and Sb are elements of valence 5, while Ge/Sn/Pb are elements of valence 4, which makes them p-type dopants in a Bi-Sb system. Researchers have done much work on Ge/Sn/Pb doped Bi-Sb,^8–10,18,19 but high levels of doping have not been reported due to the low or even zero solubility of those elements in Bi or Sb. Using melt-spinning (MS) and low-temperature spark plasma sintering (low-T SPS), we have successfully introduced a high concentration of dopants into Bi-Sb with higher doping efficiency. X-ray diffraction (XRD) and microstructure analyses using scanning electron microscope (SEM) have been performed to demonstrate the single-phase structure and homogeneity of the samples.

The numerical simulation based on two-band effective mass model has not been performed on the TE transport properties of p-type doped Bi-Sb alloys. To gain a deeper understanding of the doping effects, we used the two-band effective mass model to simulate the resistivity ρ and Seebeck coefficient S. By this approach, we were able to gain some insights about the effectiveness of the dopants and impact on the band structure of Bi-Sb alloys.

High purity pieces of Bi, Sb (99.999%) and Ge/Sn/Pb (99.999%) were weighted according to the composition of Bi₈₅Sb₁₅(Ge, Sn, Pb)_x (x=0.1, 0.5, 5, 10, 15 for Ge-, x=0.1, 0.5, 5 for Sn-, and x=0.1, 0.5 for Pb-doped Bi₈₅Sb₁₅). The elemental pieces were sealed in a quartz tube and placed into a furnace and heated for 2 hours up to 875 °C for Ge- doped Bi-Sb and 800 °C for Sn/Pb- doped Bi-Sb. Next, the quartz tube was quenched in liquid nitrogen. After obtaining the as-cast ingot, the ingot was broken into small pieces and placed into a quartz tube to prepare for melt-spinning (MS). Melt-spinning was performed with a linear wheel speed of 30 m/s to produce the ribbons. Melt-spinning makes it possible to quickly solidify the molten alloy and avoid phase separations. The ribbons were pulverized with a mortar and pestle and the alloy powders were compacted into a solid disk using spark plasma sintering (SPS). We have employed “low-temperature spark plasma sintering (low-T SPS)” to retain a high concentration of Ge and Pb in Bi-Sb (see supplementary material for SPS conditions). Because the solubility of Sn (∼3 at.%) is significantly higher than that of Ge (∼0 at.%) or Pb (∼0.5 at.%) in Bi-Sb, low-T SPS did not greatly affect the doping process of Sn as of Ge or Pb.

For the TE transport property measurements, we cut the sample into a rectangular bar with dimensions of 1.2×2.5×8.5mm and measured the resistivity and Seebeck coefficient on a Physical Property Measurement System (PPMS, Quantum Design). We have explored possible grain-structure orientation effect in samples compacted using pulverized ribbons as well as ribbon segments aligned in parallel. Measurements of resistivity and Seebeck coefficient were performed along two orthogonal directions. No orientational effect was detected within the experimental resolution. The Hall coefficient measurements were performed on Versalab (Quantum Design) using the Van der Pauw method.

X-ray diffraction (XRD, PANalyticalX’Pert pro MPD instrument) was employed to check if there was any second phase in the samples. Scanning electron microscope (SEM, FEI Quanta LV200) and energy dispersive spectroscopy (EDS) were used to examine the homogeneity of the samples.

Fig. 1(a) shows the XRD patterns of undoped Bi₈₅Sb₁₅, Ge-doped Bi₈₅Sb₁₅Ge_x (x=5, 10, 15), Pb-doped Bi₈₅Sb₁₅Pb_0.5, and Sn-doped Bi₈₅Sb₁₅Sn_x (x=0.5, 5) samples. We have compared the XRD patterns of those samples with the patterns of pure Ge/Sn/Pb and the binary alloys composed of Bi/Sb and Ge/Sn/Pb. Second phase is not detected in the samples except for Bi₈₅Sb₁₅Sn₅. Three extra peaks can be observed in the XRD pattern of Bi₈₅Sb₁₅Sn₅ and they are corresponding to the phase of Sn-Sb alloy.²⁰ Fig. 1(b) shows the EDS mapping of the Bi₈₅Sb₁₅Ge₁₅ sample. Ge atoms are distributed homogeneously in Bi-Sb solid solution in micro-scale based on the uniform density of bright blue dots shown in the EDS mapping and the respective atomic ratios of Bi, Sb and Ge at different spots on the sample.

To study the doping efficiency of Ge/Sn/Pb in Bi₈₅Sb₁₅, we investigated a set of samples that included Bi₈₅Sb₁₅(Ge, Sn, Pb)_0.1,0.5 and Bi₈₅Sb₁₅(Ge, Sn)₅. Considerable difficulty was encountered in producing melt-spun ribbons of Bi₈₅Sb₁₅Pb₅ due to the viscosity of the molten alloy. Hall measurements for these alloys were performed down to 50 K. The carrier concentrations of the Ge/Sn/Pb doped samples were calculated using the Hall coefficient (R_H) data between 50 K and 100 K. At these relatively low temperatures, the carrier transport is hole dominated since the electron carrier concentration is negligible. As a result, the Hall coefficient R_H (Eq. (2)) is nearly constant with positive values at 50-100 K. Fig. 2(a) shows the hole concentrations of doped samples at 50-100 K. Most of these concentrations are nearly constant, except for the Ge_0.1- and Sn_0.1- doped samples, for which the carrier concentration increases as temperature increases. It is worth noting that this rise of carrier concentration is the result of using the single band approximation as follows:

where n and p are the electron and hole concentrations, μ_n and μ_p are the electron mobility and hole mobility, respectively, and e is the unit charge. In fact, as temperature increases, excitation of electrons becomes stronger, leading to a smaller Hall coefficient R_H. This gives the false impression of a larger hole concentration in the single-band approximation.

If all the dopants occupy substitutional sites in Bi₈₅Sb₁₅, and act as acceptors, the nominal hole concentration can be calculated since Ge/Sn/Pb atoms have valence 4 while the Bi and Sb atoms have valence 5. The result is shown in Table I. However, after analyzing the Hall coefficient data, we found that not all of the Ge/Sn/Pb atoms go into substitutional sites because the number of holes in experiments was much lower than the nominal values. A proportion of the Ge/Sn/Pb atoms might go to the interstitial sites in Bi₈₅Sb₁₅, and acted as interstitial donors.²¹ 

Assuming that each Ge/Sn/Pb atom occupies a substitutional or interstitial site and acts as an acceptor or a donor in the system, we can derive the number of donor and acceptor sites and calculate the doping efficiency with the charge neutrality equation:

where $Nd+$ and $Na−$ are the concentrations of donors and acceptors.

Here the doping efficiency is defined as the ratio of the hole concentration to the acceptor concentration.²² The concentrations of donors $Nd+$ and acceptors $Na−$ and the doping efficiency of the doped samples are calculated (using the density of the samples as 9.32 g/cm³). The obtained values are shown in Table I. We have also plotted $Na−$ versus Ge/Sn/Pb doping levels in Fig. 2(b). Among Ge, Sn, and Pb, Pb has the highest doping efficiency (around 60%) in Bi-Sb. The atomic radii of Bi, Sb, Ge, Sn, and Pb are 156 pm, 140 pm, 122 pm, 140 pm, and 175 pm respectively. Because the atomic radius of Pb is larger than that of Bi and Sb, it is more likely for Pb atoms to occupy the substitutional sites rather than interstitial sites in the Bi-Sb matrix. As a result, there would be more substitutional acceptors in Pb-doped Bi-Sb than in Ge- or Sn- doped Bi-Sb given the same number of doping atoms. Ge and Sn shows comparable doping efficiencies. When the concentration of Ge/Sn is increased to Ge₅/Sn₅ into Bi₈₅Sb₁₅ matrix, however, the doping efficiency decreases. At such high dopant concentration, the local (atomic scale) clustering of dopant atoms is unavoidable, which would diminish the effectiveness of the dopants as acceptors in the Bi-Sb host.

Fig. 3 shows the variation of the band diagram of Bi-Sb alloy with respect to the alloy compositions.^23,24 Below 7% antimony, the alloy behaves as a semimetal. As the antimony content is increased to 7%∼22%, a gap opens between the L_a band and T band and the alloy becomes a semiconductor. This is a result of the inversion of L_a and L_s bands, which occurs at 4% antimony. In the semiconductor region, the L_s band becomes a direct hole band, while the T and H bands act as indirect hole bands. Once the antimony content is larger than 22%, the alloy returns to the semimetallic state.

For the composition Bi₈₅Sb₁₅, L_a, L_s and H bands are the conduction band, direct light valence band, and indirect heavy valence band, respectively. Due to the low density-of-states effective mass of the L_s band,²⁵ it makes little contribution to the TE transport properties compared to the H band. Therefore, the L_a band and H band are used as the conduction band and valence band in the simulations.

An effective mass model, together with the Boltzmann transport equations,²⁶ was used to simulate the thermoelectric transport properties (resistivity ρ and Seebeck coefficient S) of undoped and doped Bi-Sb alloys. Comparison was made with experimental results. Assuming parabolic band, the density of states D(E) is given by:

where $md*$ is the density-of-states effective mass and E is the energy of electrons.

The equations for electrical conductivity and Seebeck coefficient can be written in the following general forms:

where e is the unit charge, $mc*$ is the conductivity effective mass, f is the Fermi-Dirac distribution function, μ is the chemical potential and τ is the scattering time. Details of the simulation and comparison with measurement can be found in the supplementary material.

Previous studies of Bi₈₅Sb₁₅ alloys reported a band gap (E_g) in the range of 10∼24 meV.^5–7,23,24 Given the large variation of E_g, we used E_g as a fitting parameter in the simulation. Our result showed band gap to have a temperature dependence both for the undoped and doped samples. The gap was found to shrink in size at increasing temperature, even reaching zero at certain threshold temperatures, as shown in Fig. 2(c).

The decreasing of the band gap could be explained by the band edge movement at the L-point for Bi-Sb alloys: the L_a band is found to move down as temperature increases.²⁷ The impurity band created by adding Ge/Sn/Pb plays a definitive role here since band gap decreasing is noted in our doped samples. The effect of impurity band is manifested in the significant increase in the hole effective masses (supplementary material). Besides, it is reported that the undoped Bi-Sb alloy shows lattice deformation at around 150 K∼200 K,²⁸ which might lead to the change in its band gap. The decreasing of the band gap is clearly an obstacle when pursuing good TE properties of p-type Bi-Sb. It is known that the hole Seebeck coefficient is low in comparison with the electron thermopower in Bi-Sb alloys due to the mobility difference between electron and hole.⁹ Bipolar effect associated with the gap-decreasing would further undermine the Seebeck coefficient of the hole-doped samples. The latter would adversely affect the already low zT of the p-type Bi-Sb alloys.

Although Ge, Sn, and Pb have been studied as p-type dopants in Bi-Sb alloys, addressing doping effect was limited by the low solubility of these elements in Bi-Sb alloys, especially for Ge and Pb. We have employed rapid solidification coupled with a low-temperature spark plasma sintering method to significantly increase the doping level of Ge and doping efficiency of Pb in Bi₈₅Sb₁₅. This method can also be applied to other difficult-to-dope thermoelectric material systems. Moreover, a band-gap decreasing phenomenon was observed through simulation using the two-band effective mass model. This band-gap decreasing behavior poses an impediment to achieving higher zT in p-type Bi-Sb alloys.

See supplementary material for the spark plasma sintering (SPS) conditions and details of the simulations.

This work was supported by the Defense Advanced Research Projects Agency MATRIX Program contract HR0011-16-C-0011 (P.I.: Dr. Rama Venkatasubramanian, Johns Hopkins University Applied Physics Laboratory). The authors also thank Dr. Normand Modine (Sandia National Laboratories) for discussion.