First-principles analysis of intravalley and intervalley electron-phonon scattering in thermoelectric materials

Materials that efficiently convert heat energy into electrical energy, or vice versa, display a high thermoelectric (TE) figure of merit $ZT$ = $S^{2}\sigma T/\kappa$, where $S$ is the Seebeck coefficient, $\sigma$ is the electrical conductivity, $T$ is the temperature and $\kappa$ is the sum of electronic and lattice thermal conductivities [1, 2]. One way to enhance $ZT$ is to optmize the power factor $PF$ = $S^{2}\sigma$, which can be challenging since $S$ and $\sigma$ are interrelated – increasing $|S|$ tends to decrease $\sigma$, and vice versa. Band engineering is a broad strategy to control and modify the electronic states and scattering properties of TE materials with the goal of enhancing the power factor [3, 4, 5]. Such approaches include, for example, band convergence [6], resonant states [7], energy filtering [8, 9, 10], modulation doping [11, 12, 13], warped electronic bands [14, 15, 16, 17], and optimization of scattering profiles [18, 19].

Band convergence, in particular, has been the focus of much research over the past decade, and is based on a strategy aligning multiple electronic valleys and/or bands within a narrow energy range (compared to $k_{B}T$). Multiple valleys can arise from valley degeneracy and/or secondary higher-energy bands that may themselves be degenerate. Control over band alignment can come from alloying, strain and temperature, each of which can shift the relative energy of the band edges. Aligning multiple valleys/bands can enable a higher $|S|$ with a similar $\sigma$, when assuming a fixed carrier concentration, thereby enhancing the power factor. The effect on the TE parameters depends on a number factors, including the shape of the bands, number of valleys and scattering rates.

Improved TE performance by band convergence has been illustrated in many cases. Pei et al. experimentally demonstrated $ZT$ of 1.8 in p-PbTe_1-xSe_x originating from the alignment of multiple valleys tuned through alloying of Se and temperature [20]. Liu et al. found that n-Mg₂Si_1-xSn_x exhibits a large power factor and $ZT$ reaching 1.3 when $x$ = 0.7 resulting from the convergence of light and heavy conduction bands, confirmed by density functional theory (DFT) and experiment [21]. Tang et al. explained, through a combination of measurements and DFT, how the high $ZT$ of the skutterudite n-CoSb₃ stems from the convergence of a single primary conduction band with highly degenerate secondary bands [22]. Zhang et al. combined DFT and experiment to show that the high TE performance in the Zintl phase n-Mg₃Sb_1.5Bi_0.5 arises from multi-valley conduction leading to an enhanced power factor [23]. Many other studies confirming the benefits of band convergence have been reported [24, 25, 26, 27, 27, 28, 29, 30, 31, 32].

Although having multiple valleys participate in carrier transport has been shown to benefit TE performance, the degree of improvement will depend sensitively on the scattering details – in particular, the magnitude of intervalley (or interband) scattering. In materials with a single valley, only intravalley scattering processes occur, wherein the electron (or hole) transitions between initial and final states both residing within the same valley. However, in the case of multiple valleys or bands, carriers can additionally scatter from one valley to another giving rise to intervalley scattering. Intervalley processes activate when aligning multiple valleys and add to the intravalley processes, thereby increasing the overall scattering rates and potentially reducing the conductivity. As a result, intervalley scattering will tend to offset, and in some cases potentially eliminate, the benefits of band convergence. Thus, reducing intervalley collisions can help achieve the maximum benefits of band convergence.

Theoretical studies capturing the effect of intervalley transitions have revealed that the advantage of band convergence can be significantly offset with strong intervalley scattering. DFT calculations of electron-phonon (el-ph) scattering in silicon, among its six equivalent conduction valleys, showed that the intervalley scattering rates are much larger than those from intravalley processes, which negatively impacts its TE performance [33]. A theoretical investigation of Co-based half-Heuslers by Kumarasinghe and Neophytou [34] found that in certain cases the power factor could decrease with band convergence. Using first-principles, Park et al. [35] carried out scattering and transport calculations on CaZn_2-xMg_xSb₂ Zintl alloys and the full Heusler Sr₂SbAu, and determined that intervalley scattering can render band convergence ineffective. DFT-calculated el-ph scattering and transport simulations of various two-dimensional materials showed a large reduction in conduction and TE performance due to intervalley processes [36, 37]. First-principles calculations of TE transport in p-PbTe revealed that band convergence is beneficial for $ZT$, but that intervalley scattering significantly reduces the power factor [38]. When theoretically exploring band convergence it is important that the scattering model captures the effect of intervalley processes, by using a rigorous treatment [33, 39, 35, 40] or having the scattering rates be proportional to the density of states (DOS) [41, 42] (assumes an equal probability of scattering to any valley) – a constant relaxation time is blind to changes in scattering as valleys are aligned [33, 36, 37].

These previous studies highlight the importance of understanding what controls intervalley scattering, with the goal of identifying or designing TEs with limited intervalley processes to maximize the benefits of band converge. In this work, we utilize DFT to carry out el-ph scattering and transport calculations on three lead chalcogenides (PbS, PbSe, PbTe) and three half-Heuslers (ScNiBi, ScPdSb, ZrNiSn) to separate and analyze the contributions of intravalley and intervalley scattering, and their impact on TE performance. These materials were selected since they possess multiple equivalent conduction valleys, with no secondary bands within an energy range relevant for transport. The paper is outlined as follows. Section II discusses the theoretical approach and numerical details. The results and their analysis are presented in Section III. Section IV explores how to achieve reduced intervalley scattering and proposes a computationally-efficient scheme for estimating intervalley el-ph coupling. Finally, our findings are summarized in Section V.

Decomposing the scattering rates into the phase space and average el-ph coupling helps understand the origin of the scattering characteristics. As shown above, the phase space – a measure of how many el-ph transitions are possible – is largely determined by the density of states (or equivalently in terms of effective mass), which can be easily calculated for different materials. When the average el-ph coupling is energy independent, which appears to be a reasonable starting assumption based on the calculated $\langle g^{2}_{\bf k}\rangle$ in Fig. 5, the scattering rates are simply related to the density of states. This helps explain why DFT-calculated el-ph scattering rates have been shown to agree with the DOS [33, 45].

It is also interesting to note that while the scattering rates are fairly similar among both classes of PbX and HH materials (in the range 15-25 ps^-1 for 0.1 eV above the CBM), our analysis of the phase space and average el-ph coupling indicates that this arises for different reasons; the HH materials display lower $\langle g^{2}_{\bf k}\rangle$ and higher $W_{\bf k}$ compared to the PbX materials in which this trend is reversed.

Next, we explore how one could search for or design materials with limited intervalley collisions, leading to improved thermoelectric power factor and $ZT$. Equation (4) informs us that a reduction in intervalley scattering can originate from the phase space or the average el-ph coupling. Focusing first on the phase space, Eqns. (5),(7) show that this quantity is proportional to the equilibrium occupation of the phonons participating in the scattering, $n_{\bf q}$, which for the PbX and HH materials correspond to zone-boundary phonons near $\bf q$ $\approx$ X. This suggests that materials with high-energy phonons, specifically the phonons responsible for the electronic transitions from one valley to another, can lead to reduced $W_{\bf k}^{\rm inter}$ and $1/\tau_{\bf k}^{\rm inter}$. The Bose-Einstein distribution $n_{\bf q}$ monotonically decreases with increasing energy, and for energies beyond a few $k_{B}T$ decays exponentially – thus, this approach would be expected to be most effective with relatively large phonon energies or at lower temperatures. Such a strategy has been reported in cubic boron-V compounds such as BP, BAs and BSb [78].

To explore the potential reduction in intervalley scattering, we calculate the phase space for intervalley transitions assuming that $n_{\bf q}$ = 0, which represents a hypothetical limit where the energy of the zone-edge phonons is very large, $\hbar\omega_{\bf q}$ $\gg$ $k_{B}T$. Figure 6 presents the ratio of the intervalley phase space with and without setting $n_{\bf q}$ = 0. The PbX and HH materials show a possible reduction in $W_{\bf k}^{\rm inter}$ reaching between 8-12 and 5-7.5, respectively, if the scattering-mediated phonons had negligible occupation. With the PbX materials, the effect increases going from PbS to PbSe to PbTe, since the lowest-energy phonon at $\bf q$ = X decreases in energy (increasing $n_{\bf q}$) among these materials. The same argument explains the increasing phase space ratio of the HH materials going from ZrNiSn to ScPdSb to ScNiBi. The phase space ratio has a different shape for both material classes, which is due to the location of the chemical potential; a $\mu$ located deeper in the band (PbX) allows for both absorption and emission processes around $\mu$ giving rise to a peak, while a $\mu$ below the band edge (HH) predominantly has absorption-only processes near $\mu$ giving rise to a sharp peak and rapid drop.

Figure 7 shows how the thermoelectric parameters change when $n_{\bf q}$ is set to zero (for intervalley processes only), in order to explore how this could improve the TE properties. For comparison, we also present the change in TE coefficients when eliminating intervalley transitions entirely, leaving only intravalley scattering. (The original TE properties, including all intravalley and intervalley processes, are presented in Appendix A.) The power factor increase is in the range of 45-100% among the different semiconductors. With the PbX materials this comes from a combination of enhanced electrical conductivity and Seebeck coefficient, while with the HH materials this is attributed to only the electrical conductivity. The increase in electrical conductivity, corresponding to $\sim$35% with the PbX and 50-70% with the HH, simply results from reduced el-ph scattering. This effect is most pronounced in the HH materials since intervalley collisions represent a larger fraction of the overall scattering compared to the PbX materials. By only considering intravalley scattering, the conductivity of the PbX and HH semiconductors increases further by $\sim$5% and 30-45%, respectively. The Seebeck coefficient is related to the average energy of electron flow relative to the chemical potential, $\langle E-\mu\rangle$. Suppressing intervalley scattering can alter the energy dependence of el-ph scattering profile, and in the case of the PbX materials can increase the lifetime of higher-energy electrons thereby increasing $\langle E-\mu\rangle$ and $|S|$. These results highlight the potential benefits of materials with larger-energy zone-edge phonons to reduce intervalley scattering. Moreover, Fig. 7 demonstrates that not considering intervalley scattering in theoretical calculations can significantly overestimate the conductivity and power factor.

So far we have focused on how the intervalley phase space $W_{\bf k}^{\rm inter}$ could be suppressed, however the average el-ph coupling $\langle g^{2}_{\bf k}\rangle^{\rm inter}$ is equally important. Unlike the phase space that can be reasonably approximated by the DOS, an easily computed quantity, obtaining the average el-ph coupling for intervalley processes requires a rigorous calculation that is computationally demanding. Here, we propose a simple and efficient method for estimating $\langle g^{2}_{\bf k}\rangle^{\rm inter}$ that can be used, for example, to screen through material candidates.

Inspired by the deformation potential approximation [79, 80], this approach relies on constructing a supercell in which the atoms are displaced according to the eigenvectors of the zone-edge phonons participating in the intervalley processes and calculating the change in conduction band edge energy $\Delta E_{c}$. Since intervalley transitions typically involve zone-boundary phonons with short wavelengths, the supercells can be relatively compact – for the PbX and HH materials studied in this work, the atomic displacements of the $\bf q$ = X phonons responsible for intervalley processes can be directly simulated with a 1$\times$2$\times$2 supercell (see Appendix B for details on how to construct the supercell). The normalized eigendisplacements are scaled such that the maximum atomic displacement is $d$ = 0.05 Å. Since $\Delta E_{c}$ is expected to scale with $d$, we quote $\Delta E_{c}/d$ as a measure of the intervalley el-ph coupling strength. One supercell calculation is carried out for each of the zone-edge phonon modes (six and nine in the case of PbX and HH, respectively). The atomic perturbation in the supercell alters the energy and degeneracy of the band structure, and $\Delta E_{c}$ is calculated as the average magnitude in CBM shift relative to the unperturbed supercell. Note that this supercell approach is only meant to provide a qualitative measure of the intervalley el-ph coupling, which can potentially serve to compare and assess different materials.

Figure 8 presents $\Delta E_{c}/d$ for the PbX and HH semiconductors. With the PbX materials, the largest values correspond to the phonon modes 4-5 and increase going from PbTe to PbSe to PbS. This trend is consistent with the calculated $\langle g^{2}_{\bf k}\rangle^{\rm inter}$ shown in Fig. 5. In the case of the HH materials, the largest $\Delta E_{c}/d$ values among the different phonon modes increases going from ScNiBi to ScPdSb to ZrNiSn. This is somewhat different from the $\langle g^{2}_{\bf k}\rangle^{\rm inter}$ in Fig. 5 showing that the average intervalley el-ph coupling is highest in ScPdSb and nearly identical in ScNiBi and ZrNiSn. Comparing the maximum $\Delta E_{c}/d$ among both material classes, the HH semiconductors have larger values than the PbX semiconductors. While the HH materials displayed lower overall $\langle g^{2}_{\bf k}\rangle$ in Fig. 5, which was explained in terms of non-bonding orbital hybridization of the conduction states [54], the intervalley component of $\langle g^{2}_{\bf k}\rangle$ is lower in the PbX materials for energies near the band edge.

This supercell approach can potentially identify which phonon branches control the intervalley el-ph coupling. To verify that the results of this simple method are qualitatively consistent with those from DFT, we calculate the mode-resolved el-ph coupling $g({\bf k},{\bf q})$ versus phonon wavevector $\bf q$ along the line $\Gamma$-X, as shown in Fig. 9. The initial electron state $\bf k$ is located 0.1 eV above the CBM along the $\Gamma$-X line. Note that there is a gap in $g({\bf k},{\bf q})$ for intermediate $q$ values, which separates the intravalley (small $q$) and intervalley (large $q$) processes. Since the supercell approach captures the effect of phonons at the $\bf q$ = X point, we focus on the results of $g({\bf k},{\bf q}$ = X) for comparison.

With the PbX semiconductors, Fig. 9 shows that modes 4-5 are dominant and that they increase when going from PbTe to PbSe to PbS, which is consistent with findings of the supercell approach (Fig. 8). With the HH semiconductors, Fig. 9 indicates that the two largest contributions come from modes 7 and 1 in ScNiBi, 9 and 3 in ScPdSb, and 6 and 1 in ZrNiSn – the supercell approach identified the same two dominant phonon modes for each HH (Fig. 8). The maximum $g({\bf k},{\bf q}$ = X) is found in ZrNiSn, with ScNiBi and ScPdSb having lower values that are similar, which is in qualitative agreement with the results of Fig. 8. These findings suggest that this simple and efficient supercell approach may be useful to identify materials with low intervalley el-ph coupling when searching for new TE materials.

Intervalley collisions in semiconductors comprised of multiple valleys/bands can significantly deteriorate thermoelectric performance, through enhanced scattering that lowers the electrical conductivity and power factor. In this work, first-principles electron-phonon scattering and electron transport calculations were carried out in three lead chalcogenides (PbS, PbSe, PbTe) and three half-Heuslers (ScNiBi, ScPdSb, ZrNiSn), which all possess multiple equivalent conduction valleys giving rise to intervalley scattering. The individual intravalley and intervalley el-ph scattering components were isolated and compared. The results show that intervalley collisions represent a smaller fraction of the total el-ph scattering in the lead chalcogenides (PbX) compared to the half-Heuslers (HH), and that the intervalley scattering rates scale with the valley degeneracy leading to stronger energy dependence versus intravalley processes.

To better understand what controls the scattering characteristics, the collision rates are expressed as the product of the phase space $W_{\bf k}$, a measure of how much scattering is possible given the constraints of conservation of energy and crystal momentum, and the average el-ph coupling squared $\langle g^{2}_{\bf k}\rangle$, a measure of the probability of transitions occurring. The phase space is found to follow the electronic density of states, which can easily be computed, and explains why the HH materials show larger $W_{\bf k}$ over the PbX materials. The average el-ph coupling for intravelley transitions is significantly larger than intervalley transitions with the PbX materials, due to a stronger polar interaction for the small $\bf q$ phonons participating in the intravalley processes and prohibited intervalley processes at the conduction band minimum. The HH materials displayed more similar $\langle g^{2}_{\bf k}\rangle$ for both intravalley and intervalley, however the overall el-ph coupling is weaker compared to the PbX materials as a result of their particular bonding properties. While the el-ph scattering rates are somewhat similar for both the PbX and HH semiconductors, an analysis of their phase space and average el-ph coupling reveal different underlying characteristics.

Lastly, we explored two approaches that could help guide the search for new and improved thermoelectric materials with reduced intervalley scattering. First, increasing the energy of the zone-edge phonons responsible for the intervalley processes lowers the transition rate. In the limit when $\hbar\omega$ $\gg$ $k_{B}T$, we find that the intervalley phase space can drop by an order of magnitude and the power factor can double. Second, we introduce a simple and efficient supercell approach to estimate the intervalley el-ph coupling strength arising from zone-boundary phonons. When compared to rigorous DFT-computed el-ph coupling, this supercell method is found to be in good qualitative agreement, and correctly identified trends among the different materials as well as the two dominant phonon modes in each case. The findings of this study help advance our understanding of what controls intravalley and intervalley el-ph scattering, which, coupled with the proposed strategies to identify and design materials with suppressed intervalley scattering, may lead to the discovery of new high-performance thermoelectrics and high-mobility semiconductors.

This work was supported by NSERC (Discovery Grant RGPIN-2016-04881) with computing resources provided by the Digital Research Alliance of Canada.