Introduction

Owing to the growing energy crisis, thermoelectric conversion as a clean and sustainable solution has attracted increasing attention because it can directly convert waste heat to electricity.1 However, the greatest challenge that prevents its widespread application is its relatively low conversion efficiency, which is evaluated by the TE figure of merit, zT=(S2T)/(ρ(κE+κL)), where S, T, ρ, κE and κL are the Seebeck coefficient, absolute temperature, electrical resistivity, and electronic and lattice contributions to the thermal conductivity, respectively.

It is commonly recognized that the Seebeck coefficient, resistivity and electronic thermal conductivity are strongly coupled with each other. This led to most of the research efforts in TE being focused on minimizing the lattice thermal conductivity, which is the only independent material parameter. Effective approaches include nanostructuring,2, 3, 4, 5, 6, 7 lattice anharmonicity,8, 9 liquid phonons,10, 11 vacancy12, 13, 14 or interstitial15 point defects and a low sound velocity.16 Band engineering concepts including a large number of degenerated bands,17, 18, 19, 20, 21, 22, 23, 24, 25 a low band effective mass26 and weak carrier scattering27 have also proven to be successful in enhancing the TE performance. The knowledge of the band structure is thus critical for band engineering and optimizing the electrical transport of TE materials.

P-type PbTe and SnTe have long been studied as TE materials, but GeTe, another member of the group IV monotellurides, has been less studied. PbTe28, 29, 30 and SnTe31, 32, 33, 34 both have a higher-energy L and a lower-energy Σ valence band. In pristine materials with optimized carrier concentrations, the lower-energy offset between these two bands in PbTe (0.17 eV29, 30, 35 at 300 K) compared with that in SnTe (0.3–0.4 eV32, 34 at 300 K) leads to a much higher-power factor (30 μW cm−1 K−236) for PbTe than that for SnTe (20 μW cm−1 K−237). For GeTe, the available literature shows a power factor of ∼40 μW cm−1 K−2,38, 39, 40, 41 which is the highest among the three compounds. The underlying origin of this phenomenon, however, is ambiguous, to our best knowledge. The lack of a detailed investigation of GeTe has also hindered the understanding of some GeTe-based alloys with decent TE properties, such as GeTe-AgSbTe (TAGS),42, 43, 44, 45 (GeTe)nSb2Te3 (GST),46, 47, 48 (Bi2Te3)x(GeTe)1−x49, 50, 51, 52 and Ge1−xPbxTe.53, 54, 55, 56, 57, 58 Therefore, it is of great importance to reveal the essence of the superior electronic performance in GeTe, particularly from the perspective of the electronic structure.

In addition to the reported high TE performance of GeTe-based materials, its phase transition deforms the lattice of GeTe from a NaCl-type face-centered cubic (β-GeTe) structure to a rhombohedral (α-GeTe) structure59, 60 at ∼700 K or below.61 The subtle changes of the Seebeck coefficient and resistivity in the vicinity of the phase transition temperature have been barely discussed. This phase transition, limited only in GeTe among these three group IV monotellurides, in principle enables one more degree of freedom to tune the transport properties.

Since band engineering approaches such as band convergence have recently proven to enhance effectively the TE performance of both PbTe17, 21, 23, 62 and SnTe63, 64 in their p-type forms, it is believed that a detailed investigation of the band structure and the inherent transport properties of GeTe in both its low- and high-temperature phases would be helpful not only for guiding the band/microstructure engineering approaches for further improvements but also for evaluating their net contributions.

In this work, we synthesized and measured ∼50 GeTe samples with a broad range of carrier concentrations and performed first-principles band structure calculations. The transport properties can be well understood based on a rigid band approximation and are well consistent with our calculations. When fully optimized, GeTe surprisingly shows the highest zT (a peak value of 1.8) among the three group IV monotellurides in the entire temperature range. As it is well demonstrated that a high band degeneracy is beneficial for a high power factor,17, 62 the high zT of GeTe in the low-temperature rhombohedral phase (α) electronically originates from the unique dominance of the conduction by its high-degeneracy Σ band (In the Brillouin zone of the rhombohedral phase, the original 8 half valleys of L valence band in the Brillouin zone of the cubic phase is split into 6 half valleys at L point and the rest 2 at Z point, whereas the original 12 full valleys of Σ valence band is split into 6 full valleys along Σ line and the rest 6 along η line.), whereas the conduction of the other two group IV monotellurides is dominated by the low-degeneracy L band. In the high-temperature cubic phase (β), the smallest effective masses for both the L and Σ bands are responsible for the high power factor in GeTe. This work not only demonstrates that the band structure guarantees a high TE performance inherent to GeTe in both the low- and high-temperature phases but it also establishes a reference substance for evaluating the net effects of additional approaches such as band engineering and nanostructuring for possible further improvements of this material.

Materials and methods

Because of the stoichiometric deviation in GeTe, Ge vacancies65, 66 cause a very high hole concentration on the order of 1021 cm−3, which is significantly higher than the optimal value for this material. Polycrystalline Ge1−xBixTe (x⩽0.1) and Ge1+δTe (−0.04⩽δ<0.06) samples with controlled carrier concentrations were synthesized by melting stoichiometric amounts of high-purity elements (>99.99%) in vacuum quartz ampoules at 1073 K for 6 h, quenching in cold water and annealing at 873 K for 3 days. The ingots were hand ground into fine powders for X-ray diffraction, and for hot-pressing by induction heating67 at 853 K for 40 min under a uniaxial pressure of ∼80 MPa. The obtained dense pellets (>98% of the theoretical density) were ∼12 mm in diameter and 1.5 mm in thickness for the transport property measurements.

The resistivity, Seebeck coefficient and Hall coefficient were simultaneously measured on the pellet samples, where the hot and cold side temperatures were measured by two K-type thermocouples attached to the opposite edges of the pellet sample and the thermopower was measured by two Nb wires welded to the thermocouple tips. This technique is quite important because GeTe undergoes a phase transition. The resistivity and Hall coefficient were measured using the Van der Pauw technique under a reversible magnetic field of 1.5 T. The Seebeck coefficient was obtained from the slope of the thermopower vs temperature gradient from 0 to 5 K.68 The thermal diffusivity (D) was measured by a laser flash technique (model: LFA-457, Netzsch, Selb, Germany). A Dulong–Petit limit of the heat capacity (Cp) is used, which is consistent with the measurements40 for GeTe alloys and with those in many recent reports47, 69, 70 The thermal conductivity is determined via κ=dCpD, where d is the density. All of the transport property measurements were carried out from 300 to 800 K, and the reproducibility is confirmed by multiple measurements on the same sample and in multiple samples with similar carrier concentrations.

The first-principles calculations on the band structures were performed with the Vienna ab initio simulation package. The generalized gradient approximation functional and the projected augmented wave method were used. The energy cutoff was 300 eV, and the spin–orbit coupling was included. For simplicity, the band structure for α-GeTe is shown without spin–orbit coupling because the valence band in the vicinity of the Fermi level show no difference other than a weak split for the Σ band.

Results and discussion

The X-ray diffraction patterns of Ge1−xBixTe (0⩽x⩽0.1) are shown in Supplementary Figure 1. All of the diffraction peaks can be indexed to the rhombohedral structure when x⩽0.05. With a further increase in x, trace Ge impurity peaks are observable, which is consistent with our scanning electron microscope observations, as shown in Supplementary Figure 2.

The room-temperature carrier concentration-dependent power factor (PF=S2σ) for GeTe is consistent with available literature results, as shown in Figure 1. The optimal carrier concentration for GeTe is found to be 1–3 × 1020 cm−3. The similarity among these three compounds is that they all have a two-valence band structure. The difference can be seen in the calculated band structures for α-GeTe, β-GeTe, PbTe and SnTe, as shown in Figure 2, with the detailed band parameters listed in Table 1. The calculated band structures for GeTe are consistent with those in the available literature.71, 72, 73, 74, 75, 76, 77, 78, 79 It is seen that face-centered cubic-structured β-GeTe, PbTe and SnTe are all direct-gap semiconductors with the first valence band edge located at the L point, whereas the second one is along the Σ line, as shown in Figures 2b–d, respectively. A similar two-valence band configuration can be seen in the low-temperature rhombohedral phase (α). The most distinguishing feature for α-GeTe is the switch in energy levels between the L and Σ bands, as displayed in Figure 2a, meaning a higher energy of the Σ band than that of the L band. The much heavier effective mass of the Σ band nicely explains the much higher carrier concentration80 required in GeTe for its first peak in the power factor, as shown in Figure 1. The unique Σ band dominant carrier transport in rhombohedral GeTe, along with the smallest effective mass for an individual valence band (Table 1), should lead to its TE properties being significantly different from those of cubic SnTe and PbTe, as will be discussed in detail.

Figure 1
figure 1

Room-temperature Hall carrier concentration-dependent power factor (PF=S2σ) for SnTe1−xIx,37 NaxPb1−xTe36 and GeTe in comparison with literature data for p-GeTe.38, 39, 40, 88 The optimal carrier concentration for GeTe is found to be 1–3 × 1020 cm−3 (shadow area).

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Figure 2
figure 2

Calculated band structures for GeTe in both rhombohedral (a) and cubic structures (b), PbTe (c) and SnTe (d). A few Fermi levels with their corresponding carrier concentrations are given in (a) for rhombohedral GeTe.

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Table 1 Calculated band parameters for GeTe, PbTe and SnTe
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We first achieved a broad range of carrier concentrations by both intrinsic Ge doping (Ge1+δTe, −0.04⩽δ<0.06) and extrinsic Bi doping at Ge sites (Ge1−xBixTe, x⩽0.1), showing a linear decrease with x increasing up to 0.07 and saturation afterwards (Figure 3a). In the case of intrinsic Ge doping (δ>0 for Ge excess and δ<0 for Ge deficiency), the carrier concentration saturates at 7 × 1020 cm−3 on the Ge excess side (δ=0.02) and at 2.3 × 1021 cm−3 on the Ge deficiency side (δ=−0.04). The composition and the carrier concentration for the samples are listed in Supplementary Table 1. The obtained broad range of the carrier concentration not only enables a meaningful discussion of the transport properties but also the estimation of the optimal range for the TE properties. The comparison of the temperature-dependent Hall mobility (μH; Figure 3b) for these three compounds shows an unchanged dominant carrier scattering by acoustic phonons, as μH rapidly decreases with the increasing T. This validates the exclusion of the effects on the transport properties due to carrier scattering. It should be noted that the Hall mobility decreases much faster at high temperatures (T>~650 K) than what acoustic scattering predicts. This is presumably due to the bipolar conduction, as well as the redistribution of carriers between the two valence bands with significantly different mobilities.

Figure 3
figure 3

Linear increase of room-temperature Hall carrier concentration with Bi-doping concentration (a) for BixGe1−xTe with x⩽0.07 and a saturation with x>0.07. Temperature-dependent Hall mobility (b), Hall carrier concentration-dependent Seebeck coefficient (c) and Hall mobility (d) for SnTe1−xIx,37 NaxPb1−xTe,36 BixGe1−xTe and Ge1+δTe in comparison with literature data on p-GeTe.38, 39, 40, 51, 79, 88, 89, 90

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It is then straightforward that the unique dominant heavy Σ band leads to the highest Seebeck coefficient in rhombohedral GeTe (α phase) compared with the other two compounds at the carrier concentrations (<1021 cm−3) experimentally achievable so far. According to the two-valence band structure with acoustic scattering, the band parameters listed in Table 1 enable a nice prediction (red curve in Figure 3c) of the carrier concentration-dependent Seebeck coefficient at room temperature, indicating the validity of a rigid band approximation in this work. The model prediction assumes deformation potential coefficients of 9.5 and 23.5 eV for the Σ and L bands, respectively. It should be noted that in the optimal carrier concentration range of 1–3 × 1020 cm−3, the transport of carriers in α-GeTe at low temperatures has negligible contributions from the lower L band owing to the fact that most of the carriers are within the Σ band. This is reflected by the calculated relationship between the Fermi level and the carrier concentration in Figure 2a.

Owing to the heavy effective mass (of the Σ band) and the high concentration of dopants, α-GeTe has the lowest mobility (Figure 3d) at room temperature. In addition to carrier scattering by dopant impurities,81 the further reduction in the Hall mobility at high Bi-doping levels (low hole concentrations, <~2 × 1020 cm−3) in GeTe is presumably due to the redistribution of carriers from the high-mobility L band to the low-mobility Σ band. Such a redistribution is normally seen in similar materials including PbTe,19, 23, 82 SnTe83 and PbSe.27

When the temperature increases, GeTe undergoes a phase transition from a rhombohedral structure (α phase) to one that is cubic (β phase), leading to a switch in energy between the L and Σ bands. This indicates that a dominant heavy Σ band conduction at low temperatures transits to a conduction contributed by both the L and Σ bands at high temperatures (Figure 2). This band switch results in the redistribution of large population carriers from the heavy Σ to the light L band in the high-temperature phase. Both the resistivity and the Seebeck coefficient (or, more specifically, the slope of S vs T) are expected to be reduced because of the much higher mobility and the much smaller density of state effective mass of the light L band. All of these expectations are well supported by the experimental measurements, and an example is given in Figure 4 showing the resistivity and the Seebeck coefficient in the vicinity of the phase transition temperature for high-zT samples with a room temperature nH=1.37 × 1020 cm−3. The evolution of the Fermi surface during the phase transition is also shown. This phenomenon has rarely been reported, but it is very helpful for understanding the mechanism and therefore for guiding the further improvement of the TE performance in this material.

Figure 4
figure 4

Temperature-dependent Seebeck coefficient and resistivity for BixGe1−xTe with x=0.075 (nH=1.37 × 1020 cm−3) showing a transition from conduction by the Σ band in α-GeTe at T<~550 K to that by both the L and Σ bands in β-GeTe at T> ~625 K due to the phase transition. The insets show that the Fermi surface evolution is related to the transition. The much larger effective mass of the Σ band than that of the L band leads to a decrease in the Seebeck coefficient and resistivity, which is consistent with the band structure calculation showing a switch of the principal valence band from Σ to L due to the phase transition.

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The temperature-dependent resistivity, Seebeck coefficient, thermal conductivity and figure of merit zT for BixGe1−xTe are shown in Figures 5a–d. The majority of the samples studied here show a degenerated semiconducting behavior with an observable change in the phase transition region. The decrease in both the Seebeck coefficient Figure 5a and resistivity Figure 5b at high temperatures (T>~700 K) for samples with nH⩽3.5 × 1020 cm−3 is a good indication of mixed conduction by both the majority holes and the thermally excited minority electrons. The lattice thermal conductivity (inset of Figure 5c) is obtained by subtracting the electronic contribution (κe=LT/ρ) from the total thermal conductivity Figure 5c, where the Lorenz factor (L; Supplementary Figure 3) is derived based on a single Kane band model approximation. The fact that the lower carrier concentration leads to a more obvious change in the transport properties can be well understood by the resulting less degenerated Fermi level that better probes the character of the first valence band edge in different phases.

Figure 5
figure 5

Temperature-dependent resistivity (a), Seebeck coefficient (b), thermal conductivity and figure of merit (d) for BixGe1−xTe with room-temperature Hall carrier concentrations ranging from ∼1020 to 1021 cm−3. The inset in c shows the lattice thermal conductivity and an average value for a few optimally doped (nH~2±0.6 × 1020 cm−3) samples.

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The figure of merit, zT, shown in Figure 5d is high in both the low- and high-temperature phases, showing a peak value of up to ∼1.8. This is actually the highest among the three group IV monotellurides, as evidenced from Figure 6, where detailed temperature-dependent properties including the Seebeck coefficient (Figure 6a), power factor (Figure 6b), thermal conductivity (Figure 6c) and zT (Figure 6d) are included for a comparison among the three pristine compounds with respective optimal carrier concentrations.

Figure 6
figure 6

Comparison on the temperature-dependent Seebeck coefficient (a), power factor (b), total and lattice thermal conductivity (c) and figure of merit (d) for SnTe:I,37 PbTe:Na36 and GeTe:Bi, showing the highest zT by hole concentration optimization only.

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To reveal the electronic origins of the observed superior TE performance of GeTe, it is known that the maximal power factor is proportional to the number of degenerated band valleys (Nv) and inversely proportional to the inertial mass (mI*) of each valley (PF∝μm*3/2∝Nv/mI*) when the scattering of carriers is dominated by acoustic phonons,26, 82, 84, 85 as is the case in the group IV monotellurides studied here. This indicates that a large Nv and a small mI* are beneficial for TE s.

Comparing the band effective masses of PbTe and GeTe in the cubic phase, it is seen that the L and Σ bands are ∼20% and 40% lighter, respectively, in the latter (Table 1). This difference explains the higher power factor of GeTe by ∼30% at T>~650 K (Figure 6b) without involving the complexity of the temperature-dependent energy offset between these two bands in these compounds.82, 86, 87 Therefore, the lower effective mass can be considered the main reason for the superior TE performance in cubic GeTe (at high temperatures). For low temperatures, because of the overall high valley degeneracy (Nv) with a sufficiently low effective mass (Table 1) for the dominant Σ valence band in rhombohedral GeTe (Figure 2), the power factor remains slightly higher. It should be noted that the lower lattice thermal conductivity (Figure 6c) contributes to the high zT in GeTe at low temperatures (T<~600 K) as well. As a whole, both low- and high-temperature phase GeTe obtain electronic advantages from their favorable band structures for superior TE performance.

Conclusion

By comparing GeTe with PbTe and SnTe, it is concluded that the superior TE performance of GeTe mainly stems from its favorable energy band structure. The lower effective mass of both the L band and Σ bands at high temperatures and the dominance of the high-degeneracy Σ band conduction at low temperatures, as the main electronic origin, guarantee a high power factor in GeTe. The large number of samples synthesized here with a broad range of hole concentrations enables a meaningful understanding of the transport properties inherent to this material, setting up a reference substance for both guiding and evaluating further improvements by other proven strategies. The obtained high zT inherent to this compound in the entire temperature range demonstrates that GeTe itself is an excellent matrix material for realizing extraordinary TE performance.