The discussion in the previous section primarily focused on strategies to reduce the lattice thermal conductivity through the creation of multiscale phonon scattering centers. In all these cases, however, the power factor is actually reduced relative to that of the single-phase host material because of the additions of these multiscale centers that often have the potential to scatter the carriers. Therefore, care has to be taken in implementing the strategy to avoid this scenario to achieve further ZT improvements. A strategy targeting the control and minimization of the band offsets between two phases can be realized in bulk systems. If the conduction band minimum of the matrix is close in energy to that of the second phase, then electron transmission through the system should be more facile. Similarly, hole transport should also be facile if the two valence band maxima are close. Therefore, another generic operating concept for high-ZT thermoelectrics other than the creation of multiscale phonon scattering centers has emerged: a small energy difference in the relevant valence or conduction bands between the host and second phases ensures relatively fast electron or hole mobilities.

Calculation methods

Bulk calculations alone are insufficient to provide band alignments because they contain no absolute reference for the electrostatic potential.50 To calculate band alignments, the band structures of the two materials must be aligned on a common energy scale. The common energy can be the universal hydrogen transition energy, the vacuum energy level, or the electrostatic potential across the interface between the two materials.51 The universal hydrogen transition level approach has been reported to be able to predict the electrical activity of hydrogen in any host material once some basic information about the band structure of that host is known.52 Briefly, DFT calculations within the general gradient approximation are used to determine the key quantities of the formation energy of interstitial H in the host and the electronic transition level. The formation energy of interstitial H in charge state q (where q = + 1, 0, and −1) can be determined by placing H in a volume of the host material, \(E^f\left( {{\mathrm {H}}^q} \right) = E_{{\mathrm {tot}}}\left( {{\mathrm {H}}^q} \right) - E_{{\mathrm {tot}}}\left( {{\mathrm {bulk}}} \right) - \frac{1}{2}E_{{\mathrm {tot}}}\left( {{\mathrm {H}}_2} \right) + qE_{\mathrm {F}}\), where E tot (Hq) is the total energy of H with q charge in this structure, E tot (bulk) is the total energy of the pure host material, E tot (H 2 ) is given by an H 2 molecule at T = 0, and E F is the Fermi level or the electron chemical potential. The Fermi-level position labeled as ε(+/−), where the positive and negative charge states are equal in energy, can be identified. Using this approach, the band energies of the materials are thus aligned with the universal hydrogen transition level.

The vacuum level or the ionization potential is another common energy that can be used to align the valence and conduction bands of two materials. For this process, two separate calculations are needed: (i) a bulk calculation to determine the bulk band structure relative to the average electrostatic potential and (ii) a slab calculation to determine the difference between the average of the electrostatic potential in the bulk and in vacuum. Ideally, the slabs should be thick enough to allow the electron density in the center of the slab to be identical to the bulk electron density. Generally, the two methods should give the same band alignment results for a pair of materials. We have used both the H transition level and vacuum level methods to align PbS and CdS band structures. The valence band maximum difference between PbS and CdS is 0.13 eV using the universal H method, which is in good agreement with the difference of 0.21 eV obtained using the vacuum level method with eight-layer-thick (001) slab calculations.

Simple band alignment

The generality of the concept of band alignment between the host and second phase is evident in the example of PbS system with addition of the second phases of MS (M = Cd, Zn, Ca, Sr).28,53 By adding different NaCl-type metal sulfides MS (M = Cd, Zn, Ca, Sr) as second phases, the carrier mobility in PbS host are adjusted significantly, due to the differences of alignment between the energy bands of the host and second phases. Carrier mobility is typically reduced by host/precipitate interfaces, but as the (valence or conduction) bands become aligned across this interface, the smaller the degradation of mobility. The energy difference between the host PbS valence band maximum and those of the metal sulfides are 0.13, 0.16, 0.53, and 0.63 eV for CdS, ZnS, CaS, and SrS, respectively. These energy differences suggest that the hole mobility in PbS with the above four separated second-phase additions should decrease. This point has been proven by experimental observations, where the hole mobilities at 923 K were approximately 37, 28, 25, and 22 cm2/Vs, respectively.28,53 As a potential thermoelectric material, PbS was abandoned in the past as not promising, but the second-phase band alignment approach gave the highest ZT value of 1.3 at 923 K for p-type Pb 0.975 Na 0.025 S–3%CdS over a 160% improvement.28

Band alignment using compositionally alloyed nanostructures

The PbSe system is another interesting example for which the enhanced thermoelectric performance can be partly attributed to the large carrier mobility induced by the host/precipitate band alignment. In this case, the band energy offsets between the PbSe host and second phases can be reduced using compositionally alloyed CdS 1 − x Se x /ZnS 1 − x Se x nanostructures.29 A series of experiments have been performed in the PbSe host for the second-phase addition of CdSe/ZnSe and CdS/ZnS. A very small difference in the carrier mobility in PbSe was observed with the addition of 1% CdSe, ZnSe, CdS, or ZnS. However, upon increasing the second-phase fraction to 4%, the carrier mobility at 300 K remained almost constant for the CdS and ZnS additions but decreased for the CdSe and ZnSe additions. As previously mentioned, the variations in carrier mobility are thought to be associated with valence band offsets between the host PbSe matrix and the nanostructured second-phase precipitates. Therefore, the DFT calculations of band alignment relative to PbSe were applied to try to explain the above experimental carrier mobility observations. Surprisingly, the valence band energy differences relative to PbSe were 0.06, 0.26, 0.13, and 0.30 eV for CdSe, CdS, ZnSe, and ZnS,29 respectively, as shown in Fig. 4. The band offsets between perfectly ordered, stoichiometric phases were clearly insufficient to explain the small changes in the carrier mobilities for PbSe–(CdS/ZnS).

Fig. 4 Band alignment of PbSe with rock salt type CdS, CdSe, CdSe 0.75 S 0.25 , CdS 0.9 Se 0.1 , ZnS, and ZnSe Full size image

To obtain further insight into the carrier mobilities, we used DFT together with cluster expansion and Monte Carlo methods to investigate the quaternary phase diagram of (Pb,Cd)(S,Se) and the corresponding phase diagrams of (Pb,Cd)S, (Pb,Cd)Se, Cd(S,Se), and Pb(S,Se). Figure 5 clearly reveals that in the 0 K quaternary phase diagram, the tie line connects PbSe–CdS rather than PbS–CdSe, implying that the formation energy of PbSe + CdS is more favorable than that of PbS + CdSe. For the 200 K quaternary phase diagram, the single-phase points in the 0 K case grew into small single-phase regions because of the small amount of off-stoichiometry solid solutions. The tie lines in the 0 K case all grew into two-phase regions, which are shaded in yellow. For example, the diagonal two-phase region PbSe 1 − x S x + CdS 1 − x Se x grew from PbSe + CdS at 0 K. In addition, the white triangle areas are the three-phase regions, which are surrounded by two-phase regions. It is interesting to see the quaternary phase diagram at 300 K, where the two-phase region connecting PbS 1 − x Se x and PbSe 1 − x S x grew into a large single phase of solid-solution Pb(Se,S). Based on the quaternary phase diagram, with the addition of up to 4% CdS into PbSe, the CdS in the host PbSe will induce the coexistence of the two phases PbSe 1 − x S x + CdS 1 − x Se x at 300 K, which can be expressed as a chemical reaction of PbSe + CdS → PbSe 1 − x S x + CdS 1 − x Se x . These calculation results support the high-resolution energy-dispersive X-ray spectroscopy observation of these nanoscale precipitates.29 In addition, the band alignments of CdS 0.9 Se 0.1 and CdSe 0.75 S 0.25 are much closer to PbSe than CdS and CdSe. We call these compositionally alloyed nanostructures. The valence-band energy levels for CdS 1 − x Se x are thus intermediate between those of CdS and CdSe. Therefore, the very small experimental carrier mobility difference with 1% addition of CdSe or CdS can be understood based on the 0.06-eV band offset of CdSe and the intermediate valence band energy of CdS 1 − x Se x , which is very close to that of PbSe. Upon increasing the addition of CdSe and CdS to 4%, strong scattering of holes across the CdSe/PbSe interfaces is expected, whereas CdS is expected to remain in the form of CdS 1 − x Se x with weak hole scattering.

Fig. 5 Quaternary phase diagrams of (Pb,Cd)(S,Se) at a 0 K, b 200 K, and e 300 K. Binary phase diagram of c Cd(S,Se), d (Pb,Cd)S, f (Pb,Cd)Se, and g Pb(S,Se). All the two-phase regions are shaded in yellow Full size image

In summary, in this section we have shown that by tailoring the alignment of electronic bands of the compositionally alloyed nanostructures relative to the host matrix, we can use this as design tool to tune desirable thermoelectric properties, specifically electronic mobility. Computational screening of band alignment can be used to select second-phase nanostructures with the host matrix with aligned band energies. Together with the large reductions of the lattice thermal conductivity induced by the embedded nanostructures (Strategy A), the band-offset engineering (Strategy B) provides a powerful design strategy for advanced thermoelectric systems.