Understanding of isoelectronic alloying induced energy gap variation for a large enhancement of thermoelectric power factor

Band-structure engineering is an important strategy that can improve the properties of functional materials or even bring new features to existing systems. Band gap (or energy gap, Eg) plays a crucial role in deciding the electronic or optical properties of one material. Isoelectronic and isostructural alloys usually exhibit similar electronic band structures, but the related effect of Eg variation was found to be distinct and sometimes controversial. Herein, we provided a deep understanding of the origin of band tuning in isoelectronic alloys based on experimental characterizations and theoretical calculations. The prerequisites of an isoelectronic alloy system with an Eg that is sensitive to composition are thoroughly disclosed by studying ${\left(\mathrm{Sb},\mathrm{Bi}\right)}_2{\mathrm{Te}}_3$ and Pb(Se,Te). In the promising ${\mathrm{Mg}}_3{\left(\mathrm{Sb},\mathrm{Bi}\right)}_2$ thermoelectric materials, it is found that Bi isoelectronic substitution can significantly decrease the Eg, due to the combination of the higher energy of Bi $6p$ orbit than that of Sb $5p$ orbit and the varied bond strength induced by lattice expanding. A high room-temperature power factor (PF) value of $38.5\mu \mathrm{W}\mathrm{c}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$ is obtained in nearly zero-gap ${\mathrm{Mg}}_{3.2}{\mathrm{Sb}}_{0.3}{\mathrm{Bi}}_{1.7}$ samples, which rationally guides the design of thermoelectric materials in the aspect of electronic transport.

Figure 1

The relationship between the electronegativity difference and Eg difference of 13 well-studied semiconductor systems [28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. The cases of ${\mathrm{Mg}}_2\mathrm{Si}$, GeTe, and their isoelectronic counterparts are discussed in detail in the SM (Fig. S1) [8].

Figure 2

Critical factors induce the band-gap variation process. (a) The ratios of the energy gap change of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ and PbSe after and before 0.5 and 1% strains. (b), (c). The schematic illustration of SOC-driven band narrowing (b) and band inversion (c). (d), (e) Orbital projected band structure of PbSe and (f), (g) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, Fermi level is set at 0 eV.

Figure 3

Calculation results of DOS and pCOHP of ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$. (a) Partial density of states of selected atoms in ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$. (b)–(e) pCOHP of Sb-Mg1, Sb-Mg2, Mg1-Mg1, and Mg1-Mg2 interaction. (f) The structure schematic of ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$: one Sb atom and its ligand in the perspective of ligand-field theory.

Figure 4

Eg calculation and chemical bonds analyzing for strained ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$. For (a) and (c), ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$ Eg as a function of uniaxial tensile strain along the $a$ axis and $c$ axis, respectively. For (b) and (d) pCOHP and −ICOHP of Sb-Mg1 and Sb-Mg2, the former is plotted in solid lines and the latter dashed lines. The black line, the orange line, and the blue line are the results of the original cell, the $a$-axis strained cell, and the $c$-axis strained cell, respectively. Fermi level is set at 0 eV. The SOC effect is not considered herein.

Figure 5

mBJ band structures for (a) ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$ and (b) ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$ of (a) 1-3. orbital projected band structure of ${\mathrm{Mg}}_3{\mathrm{Sb}}_2$ for Mg1, Mg2, and Sb, respectively, and (b) 1-3. orbital projected band structure of ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$ for Mg1, Mg2, and Bi respectively. Fermi level is set at 0 eV.

Figure 6

Orbital projected band structure of ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$ without SOC. The $6s$ orbit and $6p$ orbit of Bi are projected on the band structure.

Figure 7

Electric properties and structural properties simulation and examination of ${\mathrm{Mg}}_3$(Sb,Bi) and the ${\mathrm{Mg}}_3{\mathrm{Sb}}_{0.3}{\mathrm{Bi}}_{1.7}$ of interest: (a) The composition-dependent band gap Eg by first-principles calculation. (b) Effective band structure and DOS for ${\mathrm{Mg}}_3{\mathrm{Sb}}_{0.25}{\mathrm{Bi}}_{1.75}$; Fermi level is set at 0 eV. (c) The Rietveld refinement results of the ${\mathrm{Mg}}_3{\mathrm{Sb}}_{0.3}{\mathrm{Bi}}_{1.7}$ sample. The lattice constants ($a=4.666\AA$, $c=7.401\AA$) of the ${\mathrm{Mg}}_{3.2}{\mathrm{Sb}}_{0.3}{\mathrm{Bi}}_{1.7}$ sample are obtained based on the Rietveld refinement. An expansion of ∼1.6% and an expansion of ∼1.7% in the a- and $c$ directions are obtained, respectively. (d) Comparing the lattice constants of ${\mathrm{Mg}}_3{\mathrm{Sb}}_{0.3}{\mathrm{Bi}}_{1.7}$ with former experimental and calculation results [60].

Figure 8

Electric properties measurement and simulation of ${\mathrm{Mg}}_3{\left(\mathrm{Sb},\mathrm{Bi}\right)}_2$ systems and the composition of interests ${\mathrm{Mg}}_{3.2}\left({\mathrm{Sb}}_{0.3–0.5y}{\mathrm{Bi}}_{1.7–0.5y}\right){\mathrm{Te}}_y$ ($y=0$, 0.002, 0.005, 0.0075, and 0.01): (a) electronic conductivity, (b) Seebeck coefficient, (c) power factor, (d) Hall carrier concentration and Hall carrier mobility, (e) The Hall mobility for $n$-type ${\mathrm{Mg}}_{3+\delta }{\mathrm{Sb}}_{2-x}{\mathrm{Bi}}_x$-based samples [22, 67, 76, 77] (whose carrier concentration ranges from 3 to 5 × ${10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$) depends on $x$ in ${\mathrm{Mg}}_{3+}{\mathrm{Sb}}_{2-x}{\mathrm{Bi}}_x$-based $n$-type TE materials, as simulated by the SKB model. The blue scatters represent experimental results from early works, while the black solid line represents the simulated result by the SKB model. The black dashed line indicates the extrapolated result due to the failure of the SKB model when the Eg approaches zero or becomes negative Additionally, the gray dashed line represents the simulated result where only the alloy scattering effect is considered, and the band sharpening effect is absent. (f) PF at room temperature, compared with other ${\mathrm{Mg}}_3{\left(\mathrm{Sb},\mathrm{Bi}\right)}_2$ materials, Refs. [22, 23, 70, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86]. (g) The simulated cooling power ($T_{\mathrm{C}}=300$ K, ΔT = 5 K) of the unicouple devices where commercial ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is selected as the $p$-type counterpart (see Fig. S10 for the TE performance of the commercial ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ [24]) of the advanced ${\mathrm{Mg}}_3{\left(\mathrm{Sb},\mathrm{Bi}\right)}_2$ materials for cooling application [23, 24, 58, 59]. The blue scatters represent the cooling-power density ($Q_C$) under the optimized condition (see Table S3 for the optimized current and the optimized An/Ap values for each $n$-type material). (h) Thermal conductivity and (i) ZT value of each sample with different Te doping concentrations.