Introduction

Electronic band structure plays a critical role in thermoelectric (TE) materials, where their energy conversion efficiency of electricity from heat is given by the materials’ figure of merit \({zT}=T\bullet {\rm{PF}}/\kappa\), where \(\kappa\) is the total thermal conductivity and \({\rm{PF}}={S}^{2}\sigma\) is the power factor given by Seebeck coefficient \(S\) and the electrical conductivity \(\sigma\). The thermoelectric (electronic) quality factor \(B\) is inversely proportional to the inertial effective mass \({m}_{{\rm{I}}}^{* }\) and increases with increasing electronic density of state (DOS) effective mass \({m}_{{\rm{D}}}^{* }\), and has served as an indicator of the best possible TE performance a material can achieve1,2,3,4,5,6. Since the quality factor \(B\) is determined in terms of materials’ electronic structure, different ways have been devised to achieve higher \(B\) factors through band structure engineering in recent years. For example, convergence of multiple conduction valleys can be used to increase the DOS effective mass7,8,9. Band convergence can be achieved either intrinsically in crystal structures with high crystal symmetry10 or by doping, as illustrated in some recent works8,11,12,13. A complex and anisotropic Fermi surface is another way to increase the thermoelectric quality factor \(B\), as highlighted by Gibbs et al.14. Indeed, anisotropic Fermi surface can lead to unusual transport properties15 and high power factors16,17. In the extreme case, the electronic bands are flat along certain high-symmetry directions of the Brillouin zone (BZ) but dispersive in others. Known as pudding-mold bands, they give rise to high thermoelectric power factors18,19,20,21. Finally, improving carrier mobility for higher power factors can be achieved by reducing the inertial effective mass \({m}_{{\rm{I}}}^{* }\) and/or improving the electronic lifetime, which depends on electron-phonon scattering and in general sample quality22,23,24.

Obtaining a desirable band structure for better TE performance is largely a trial-and-error process that requires large experiment efforts25,26. While recent high-throughput calculations have helped researchers to identify promising compounds with desirable electronic structure at a faster speed27,28,29,30, they can provide little guidelines for doping or electronic structure engineering, which can often improve the TE performance of a specific compound dramatically7,8,9,11,31,32,33. On the other hand, chemical bonding description of the electronic structure, pioneered by Hoffmann34, promises to give clear insights to the origins of various electronic/phononic properties35,36,37,38 and guidelines for understanding, as well as tunning, compounds’ electronic structure39.

The commonly used density-functional theory (DFT) orbital projection method, although providing knowledge of the wavefunctions of bands that participate in transport properties, generally provides only partial information about orbital interactions and their quantitative effects40. The tight-binding method, on the other hand, can establish straightforward relationships between a set of orbital interaction parameters and the band energies, providing a qualitative description of the electronic structure41,42. However, due to the diagonalization process, the relationships between band eigen-energy \({\varepsilon }_{i{\bf{k}}}\) and orbital interaction parameters \({H}_{\mu \nu }({\bf{R}})\), where \(\mu\) and \(\nu\) index atomic orbitals, and \({\bf{R}}\) is the translation vector, are still difficult to apprehend. Furthermore, a direct relationship between transport properties and orbital interactions have not been examined yet.

In this work, we propose to use a sensitivity analysis method, based on the tight-binding parameterization, to pinpoint the role of orbital interactions in band structure and thermoelectric transport properties. The sensitivity analysis assumes that the orbital interaction parameters can be independently perturbed, but in practice, the orbital interactions are related to each other by crystal symmetry. Therefore, we also introduce a computational procedure to automate the setup of a symmetry-adapted tight-binding problem to facilitate the orbital interaction sensitivity analysis. A tight-binding pipeline can be created directly from the structure that takes the value of the symmetry reduced set of independent interaction parameters as input and calculate electronic properties, such as the thermoelectric power factor.

As related works, Brod et al.43,44 used tight-binding methods to study the electronic structure in the thermoelectric PbTe systems using derived expressions for valence band edges at a few \(k\) points as a function of orbital interaction parameters. Based on their analytic expressions, they studied the band convergence behavior in PbTe. However, they did not study the relationship between the interaction parameters with transport property directly and their method cannot be extended to other material systems easily. The generation of symmetry-adapted tight-binding problem was studied by Varjas et al.45 and Zhang et al.46. However, their methods symmetrize the Hamiltonian in the reciprocal \(k\) space and require additional manual inputs such as symmetry operations, which is cumbersome to use. In this work, the process of constructing a symmetry-adapted tight-binding problem is fully automated starting from the crystal structure, and the computational procedure of the calculation of various electronic properties, such as density of states (DOS) and transport properties directly from the set of independent tight-binding orbital interaction parameters, are also included.

Results

Basic workflow

Our quantitative analysis of the effect of the orbital interactions on electronic properties is illustrated by the workflow shown in Fig. 1. The analysis workflow requires the crystal structure and desired set of atomic orbitals on each atom as input to construct a tight-binding problem, which gives the set of independent interaction parameters required and is able to output calculated electronic properties once these parameter values are provided. To calculate sensitivities, the orbital interaction parameters are perturbed around their initial values in a specified range and corresponding perturbed properties are calculated. Finally, sensitivity analysis (see Methods section below) is performed using the perturbed orbital interaction parameter values and the resulting electronic properties.

Fig. 1: Flowchart for the orbital interaction sensitivity analysis.
figure 1

From the input crystal structure, a tight-binding problem can be constructed. To compute the sensitivity indices, the tight-binding orbital interaction parameters are perturbed, and electronic properties are then calculated based on the corresponding tight-binding electronic structures.

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Application of band sensitivity analysis

As described in the introduction, a systematic approach to band structure engineering through orbital interactions would not only require knowledge of the wavefunction components but also how the band energies \({\varepsilon }_{{\bf{k}}}\) are determined by various orbital interaction terms. However, due to the diagonalization process, expressions of \({\varepsilon }_{{\bf{k}}}=f(\{{H}_{\mu \nu }\left({\bf{R}}\right)\})\), where {\({H}_{\mu \nu }\left({\bf{R}}\right)\)} is the set of tight-binding interaction parameters, are not readily available. This difficulty motivates us to use a sensitivity analysis method which aims to determine how target variable are affected by changes in the input variables47. (Recently, sensitivity analysis methods are also used to interpret input–output relationships in deep neural network models48) Here, we first use a toy model to illustrate how a sensitivity analysis of the electronic band structure can help us understand the role of orbital interaction parameters.

In Fig. 2, the result of orbital interaction sensitivity analysis for a 2D toy lattice is compared with the conventional band coefficient plot. The toy lattice is shown in Fig. 2a where two atomic orbitals (\({{\rm{p}}}_{x}\) and \({{\rm{p}}}_{y}\)) at the origin of the unit cell are used as the tight-binding basis. The unit cell is rectangular with \(a\ne b\) to differentiate the interaction between the \(p\) orbitals along the \(x\) or \(y\) directions. Apart from the onsite energy of the orbitals \({E}_{{\rm{p}}}=0\), the relevant nearest neighbor interactions considered are \({V}_{{\rm{pp}}{\rm{\sigma }}}^{y}\), \({V}_{{\rm{pp}}{\rm{\pi }}}^{y}\), \({V}_{{\rm{pp}}{\rm{\sigma }}}^{x}\) and \({V}_{{\rm{pp}}{\rm{\pi }}}^{x}\) with values set with reference to Froyen et al.49. With these parameters, a tight-binding band diagram can be obtained, as shown in Fig. 2b. Since in this toy model, the \({{\rm{p}}}_{x}\) orbital does not interact with the \({{\rm{p}}}_{y}\) orbital, the component of each band is therefore either pure \({{\rm{p}}}_{x}\) or pure \({{\rm{p}}}_{y}\) in character. However, Fig. 2b only shows that the two bands are composed of \({{\rm{p}}}_{x}\) and \({{\rm{p}}}_{y}\) contribution, respectively. How the bands run up and down from \(\varGamma\) to \(X\) or \(Y\) is unexplained.

$${E}_{{{\rm{p}}}_{x}}\left(X\right)={E}_{{\rm{p}}}+2\,{V}_{{\rm{pp}}{\rm{\pi }}}^{y}-2\,{V}_{{\rm{pp}}{\rm{\sigma }}}^{x}$$
(1)
$${E}_{{{\rm{p}}}_{x}}\left(Y\right)={E}_{{\rm{p}}}-2\,{V}_{{\rm{pp}}{\rm{\pi }}}^{y}+2\,{V}_{{\rm{pp}}{\rm{\sigma }}}^{x}$$
(2)
Fig. 2: Orbital interaction sensitivity analysis of a toy model.
figure 2

a The 2D lattice with \({{\rm{p}}}_{x}\) and \({{\rm{p}}}_{y}\) orbitals as the basis, b The fat-band plot showing the two bands, \({{\rm{p}}}_{x}\) and \({{\rm{p}}}_{y}\), and c The sensitivity plot showing how the band energies, relative to that in \(\varGamma\) point correspondingly, are related to the interaction parameters.

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For each band, the energy at \(\varGamma\) is equal to \({E}_{{\rm{p}}}+2\,{V}_{{\rm{pp}}{\rm{\pi }}}+2\,{V}_{{\rm{pp}}{\rm{\sigma }}}\). Equations (1) and (2) then determines how the bands run along the \(\varGamma\)–\(X\) or \(\varGamma\)–\(Y\) directions: the change from bonding to anti-bonding character and vice versa increases or decreases the band energies. In Fig. 2c, the calculated orbital interaction sensitivity for the relative energies of each band with respect to its energy at \(\varGamma\) (i.e., \({E}_{{{\rm{p}}}_{x}}\left({\bf{k}}\right)-{E}_{{{\rm{p}}}_{x}}\left(\varGamma \right)\) and \({E}_{{{\rm{p}}}_{y}}\left({\bf{k}}\right)-{E}_{{{\rm{p}}}_{y}}\left(\varGamma \right)\)) is shown. It indicates how each interaction parameter plays a role in determining how the bands run up and down from the \(\varGamma\) point. We can see that the plot immediately reveals information corresponding to the derived result in Eqs. (1) and (2). For example, the figure shows that the difference in energy between \({E}_{{{\rm{p}}}_{x}}\left(\varGamma \right)\) and \({E}_{{{\rm{p}}}_{x}}\left(X\right)\) comes mainly from the term \({V}_{{\rm{pp}}{\rm{\sigma }}}^{x}\). Therefore, if we want to modify the band energy \({E}_{{{\rm{p}}}_{x}}\left(X\right)\), we need to play with the term \({V}_{{\rm{pp}}{\rm{\sigma }}}^{x}\).

Comparing Fig. 2b, 2c, we see that the benefit of the proposed orbital interaction sensitivity is that it reveals the importance of the orbital interaction terms (Fig. 2c) in determining the energy of the band at each \(k\) point in the band structure, instead of the wavefunction components (Fig. 2b). Indeed, if the goal of band structure engineering is to change the energy of the bands at the specific \(k\) points, then the wavefunction components do not provide enough (in this case, none) information on what should be done to change the band energies. On the other hand, we showed that such an information can be obtained using the sensitivity-based approach with a tight-binding model. In Supplementary Note 2, we further illustrate the use of the orbital sensitivity analysis on metal halide perovskites35,50, which reveals clearly the two mirror-like symmetries in their electronic structure that originate from orbital interactions, which cannot be observed easily using wavefunction projection method.

Application to the thermoelectric PbTe compound

Let us apply this band sensitivity analysis method to the thermoelectric compound PbTe to study the relationship between orbital interactions, band structure and its thermoelectric transport properties32,51,52. This compound is chosen as an example here because of its simple crystal structure and that its electronic structure–orbital interaction relationship has been partly studied before by analytic methods44, which allow us to verify the current numerical approach using band sensitivity. Here, the tight-binding problem is automatically constructed considering the orbital interactions of both the \({\rm{s}}\) and the \({\rm{p}}\) orbitals of Pb and Te up to secondary neighbors. In PbTe, crystallizing in the cubic \({Fm}\bar{3}m\) space group, each atom in the unit cell is surrounded by six nearest neighbor atoms at 3.23 Å and twelve secondary neighbors at 4.57 Å. Using the point group symmetry of each site, we can obtain the symmetry relationship of the orbital interactions between each atom and its neighbors (SALCs used to find the symmetry relationships between orbital interactions are listed in Supplementary Table 1, the \({\rm{p}}\) orbitals are oriented along the \(x\), \(y\) and \(z\) Cartesian directions).

Using symmetry relationship, a total of 19 independent orbital interaction parameters can be found for PbTe from a total of 576 orbital interaction terms. The obtained set of independent interaction parameters is similar to that used by Brod et al. in their study of PbTe43, except for the difference in definition (for example, \({V}_{{{\rm{p}}}_{x}{{\rm{p}}}_{x}},\,{V}_{{{\rm{p}}}_{x}{{\rm{p}}}_{y}}\) instead of \({V}_{{\rm{pp}}{\rm{\pi }}}\) and \({V}_{{\rm{pp}}{\rm{\sigma }}}\)) and two additional terms (Te and Pb respectively) describing secondary neighbor \({\rm{p}}\) orbital interactions, which were ignored previously. The initial values of these tight-binding interaction parameters are set to the ones derived by Brod et al. for the current study, since they give a good description of the band structure near the valence band maximum (the tight-binding parameters and their values are listed in Supplementary Table 2).

Band structure and orbital interactions

Using the same sensitivity analysis method introduced previously, we show here the importance of orbital interaction parameters for states located in the entire valence band structure of PbTe (see Fig. 3a for the energy \({\varepsilon }_{n{\bf{k}}}\) and Fig. 3b for the energies of each electronic state relative to the valence band maximum at \(L\)). From the valence band edge regions shown in Fig. 3a, it can be seen that the energies near the valence band maximum depend broadly on the energy of \({E}_{{\rm{p}}}^{{\rm{Te}}}\) terms (deep blue), corresponding to the fact that the Te \({\rm{p}}\) orbitals are the main components of the upper part of the valence band. Furthermore, the figure shows straightforwardly that the band energies at point \(W\) and valence maximum along \(\Delta\) (\({\Delta }_{{\rm{max }}}\)) are considerably affected by the \({V}_{{{\rm{p}}}_{y},{{\rm{p}}}_{y}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) \({\rm{pp}}{\rm{\pi }}\) orbital interaction term (purple), while at valence band maximum at point \(L\) and along \(\Sigma\) (\({\Sigma }_{{\rm{max }}}\)), the nearest neighbor interaction term \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\) (red) is important.

Fig. 3: Orbital interaction sensitivity of band dispersion in PbTe.
figure 3

a Sensitivity of band energy to the selected orbital interaction parameter and b sensitivity of relative band energy, with respect to the VBM \(L\) point, with respect to selected orbital interaction parameters.

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The total orbital interaction sensitivity indices \({S}_{{\rm{T}}}\) for the energy of four important valence band maximums, as well as the conduction minimum at point \(L\), are listed in Table 1. These indices indicate the importance of orbital interaction parameters in determining these band edge energies (in range of 0 to 1, with 0 indicating no functional relationship between results and parameters). The results shown here are consistent with the analytic expression derived by Brod et al.44 for these states, but are obtained directly from the tight-binding sensitivity analysis approach.

Table 1 Orbital interaction sensitivity indexes \({{\boldsymbol{S}}}_{{\bf{T}}}\) with respect to band energies and power factor in PbTe.
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Figure 3b shows the sensitivity analysis of the relative energy for the valence band states with respect to the energy at the valence band maximum at point \(L\), revealing the important interaction terms that should be modified or controlled if we want to modify the relative energy difference between band edges by doping or alloying. It can be observed that the Te–Te secondary neighbor interactions (shown in purple and light green) are important interaction terms to determine the relative position of the valence band edge energies with respect to \({E}_{L}^{{\rm{vbm}}}\). The nearest neighbor Pb–Te \({\rm{sp}}{\rm{\sigma }}\) interaction also plays a role in determining the relative energy \({E}_{L}^{{\rm{vbm}}}-{E}_{{\Sigma }_{{\rm{max }}}}\). However, playing with other orbital interaction terms, such as the on-site terms \({E}_{{\rm{p}}}^{{\rm{Te}}}\) or \({E}_{{\rm{s}}}^{{\rm{Pb}}}\), is less effective in changing the shape of the upper valence band region. Using the relevant orbital interaction parameters determined by sensitivity analysis (Table 1) and linear regression, it is possible to obtain linear expressions for the energy differences between band edges (Eqs. (3) and (4)) based on a training set of perturbed tight-binding parameters and the resulting band energies, which gives the trend for band convergence and divergence with respect to the value of orbital interactions (the root mean squared error (RMSE) is 0.04 eV and 0.02 eV, respectively, and is plotted in Supplementary Fig. 1).

$${E}_{L}-{E}_{{\Sigma }_{{\rm{max }}}}=9.53\,{V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)+2.64\,{V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)-0.16\left({E}_{{\rm{p}}}^{{\rm{Te}}}-{E}_{{\rm{s}}}^{{\rm{Pb}}}\right)-0.24$$
(3)
$${E}_{L}-{E}_{W}=6.07{\,V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)+3.54{\,V}_{{{\rm{p}}}_{y},{{\rm{p}}}_{y}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)+1.5{3\,V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)-0.10\left({E}_{{\rm{p}}}^{{\rm{Te}}}-{E}_{{\rm{s}}}^{{\rm{Pb}}}\right)-0.04$$
(4)

Although we showed here how orbital interaction sensitivity can be applied to band convergence at different \(k\) points in the Brillouin zone of PbTe, it can also be used to study the convergence behavior of multiply bands at the same \(k\) points, which have simpler chemical interpretations and are perhaps easier to achieve. For example, alloying or crystal strain can lead to the convergence of valence band in the thermoelectric Mg3Sb2 compound53,54. We note that the latter (crystal strain) is recently realized in experiments by Xie et al.54.

Transport properties and orbital interactions

Apart from the band energies, our sensitivity analysis also revealed the important orbital interaction terms, the value of which can greatly affect transport properties. Numerical solution of the Boltzmann transport equations was used to simulate electronic transport properties of PbTe, from the perturbed tight-binding interaction parameters. The carrier concentration \({N}_{{\rm{c}}}\), The Seebeck coefficient tensor \({\boldsymbol{S}}\) and the electronic conductivity tensor \({\boldsymbol{\sigma }}\) were calculated from the eigen-energies \({\varepsilon }_{n{\bf{k}}}\) and the band velocity \({{\boldsymbol{v}}}_{n{\bf{k}}}\) obtained from the tight-binding model55,56. Since the tight-binding model itself cannot provide a description of carrier scattering, we approximated the energy dependence of the electronic relaxation time using the relationship \({\tau }_{n{\bf{k}}}\propto {g\left({\varepsilon }_{n{\bf{k}}}\right)}^{-1}\), where \(g\left({\varepsilon }_{n{\bf{k}}}\right)\) is the calculated total density of states extracted from the tight-binding electronic structure using the tetrahedral summation57. This form of carrier relaxation time, although simple, should capture the effect of the electronic structure on the scattering of carriers within the acoustic phonon scattering assumption2,15,44,58. A series of chemical potentials, from \(-0.1\) eV to \(0.04\) eV relative to the valance band maximum energy, was used for varying carrier concentrations in the Boltzmann transport calculation.

The orbital interaction sensitivity indices are listed in Table 1 for power factors (obtained at the optimal carrier concentration) at 300 K and 600 K. All the interaction terms that affect the band edge energies are shown to have a role in determining the resulting power factor. It can be understood that the \({E}_{{\rm{p}}}^{{\rm{Pb}}}\), \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Pb}}}\left(10\bar{1}\right)\) and \({E}_{{\rm{p}}}^{{\rm{Te}}}\) terms affect the direct band gap \({E}_{{\rm{g}}}\) at point \(L\), which could diminish the Seebeck coefficient significantly if \({E}_{{\rm{g}}}\) becomes too small. The orbital interaction terms \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\), \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) and the onsite energy term \({E}_{{\rm{s}}}^{{\rm{Pb}}}\) control the band convergence or divergence behavior, as will be discussed in the following section. According to the sensitivity analysis, apart from these orbital interaction terms, other interactions play little role in the electronic transport properties. It is also worth noticing that with increasing temperature, the importance of the orbital interaction term \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) increases while other terms become less important in determining the power factor, stressing the importance of the anion–anion interaction \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) in transport properties in PbTe. This intriguing observation hints that the importance of orbital interaction parameters in determining thermoelectric power factor can change with temperature, a phenomenon that is worth future investigation.

We studied the effect of the nearest neighbor anion—cation interaction term \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\), the anion–anion interaction term \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) and the onsite energy term \({E}_{{\rm{s}}}^{{\rm{Pb}}}\) on transport properties in more details. Fig. 4a, d and g shows the effect of changing these three interaction terms on the valence band structure. The change in relative energies at the valence band maximum points due to different interactions are as predicted from that given by Eqs. (3) and (4). The valence band maximum at point \(L\) converges/diverges with respect to other valence maximum energies with increasing/decreasing values of the \({E}_{{\rm{s}}}^{{\rm{Pb}}}\), \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) or \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\) terms (a clear illustration of band convergence/divergence can be found in Supplementary Fig. 2, where all bands in Fig. 4a, d and g are aligned at the valence band maximum point at L). The most effective term to converge the bands are the anion–anion interaction \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\), since the band edge energy at point \(W\) is not affected by it.

Fig. 4: Band structure, power factor enhancement and effective mass as functions of different orbital interaction values.
figure 4

a The electronic band structure of PbTe calculated with different values of the orbital interaction parameters \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\). Legend values indicate the range of values of \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\), b power factor enhancement, and c the Seebeck effective mass \({m}_{{\rm{S}}}^{* }\) as a function of the interaction parameter \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) at different temperatures. d–i electronic band structure, power factor enhancement and Seebeck effective mass variation with orbital interaction parameter \({V}_{{{\rm{s}},{\rm{p}}}_{{\rm{z}}}}^{{\rm{Pb}},{\rm{Te}}}(001)\) and \({E}_{{\rm{s}}}^{{\rm{Pb}}}\), respectively.

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Figure 4b, e and h shows the improvement of the thermoelectric electronic power factor as a function of these orbital interaction parameters, at different temperatures. Surprisingly, the result shows that band convergence, corresponding to decreasing the values of the \({E}_{{\rm{s}}}^{{\rm{Pb}}}\), \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) or \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\) interaction terms, are not always beneficial. To understand the orbital interaction dependence of power factor, we recall that the thermoelectric power factor is closely related to the number of degenerate transport valleys \({N}_{{\rm{v}}}\) and the inertial effective mass \({m}_{{\rm{I}}}^{* }\) via the relationship \({PF}\propto {N}_{{\rm{v}}}/{m}_{{\rm{I}}}^{* }\). When the temperature is not high enough to excite carriers in nearly converged bands, the reduction of effective mass \({m}_{{\rm{I}}}^{* }\) dominates the change in \({PF}\), so that the band divergence (sharpening) leads to a higher power factor and the modification of \({E}_{{\rm{s}}}^{{\rm{Pb}}}\) or \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\) energy terms for band convergence can be seen to have a detrimental effect, unless the band convergence is achieved within an energy difference of \({k}_{{\rm{B}}}T\) so that the carriers from the converging bands can be thermally excited. We calculated the Seebeck effective mass59 (note that \({m}_{{\rm{S}}}^{* }\,\ne \,{m}_{{\rm{DOS}}}^{* }\)) as function of the orbital interaction parameters shown in Fig. 4c, f and i. Indeed, the large effective mass due to band convergence occurs at higher temperature (Fig. 4f), or when the bands are well converged (Fig. 4c). Only then, the larger effective mass can lead to a greatly improved power factor.

At lower temperature, band divergence improves the thermoelectric power factor. Increasing the valence band maximum energy at point \(L\) (i.e., band divergence) will lead to a lighter band, as shown in Fig. 4a. Not only does it increase electron’s group velocity, but it also leads to a smaller electronic density of states, resulting in a smaller electron-phonon scattering phase space and a longer carrier lifetime, under the assumption \(\tau \propto 1/g\). The same phenomenon is observed experimentally in Mg3Sb2-xBix alloy system in which Bi alloying reduce the electron effective mass, leading to higher mobility22. Our analysis also echoes the observation by Park et al.53, that band convergence could have a negative impact on the power factor by severely reducing carrier mobility. Our study importantly demonstrated that, when tuning band convergence via modification of orbital interaction terms, a not-well-converged band cannot benefit the electronic power factor, especially when the low temperature region is of concern. Finally, comparing the effect of different orbital interaction terms, it can be observed that the power factor is mostly sensitively determined by the anion–anion interaction term \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\), explaining the result given in Table 1. It is also worth mentioning that the increase of the anion energy term \({E}_{{\rm{p}}}^{{\rm{Te}}}\), although effective in converging the bands, can rapidly close the electronic band gap and harm thermoelectric transport, as shown in Supplementary Fig. 3.

In Fig. 4, the changes in electronic structure and power factor are studied with orbital interaction parameters \({E}_{{\rm{s}}}^{{\rm{Pb}}}\), \({V}_{{{\rm{p}}}_{x},{{\rm{p}}}_{z}}^{{\rm{Te}},{\rm{Te}}}\left(10\bar{1}\right)\) or \({V}_{{\rm{s}},{{\rm{p}}}_{z}}^{{\rm{Pb}},{\rm{Te}}}\left(001\right)\) varied one at a time. However, further enhancement of the power factor can be achieved if different orbital interaction parameters are allowed to change at the same time. In Fig. 5, the power factor, scaled so that the maximum power factor for the initial orbital interaction parameters equal to one, as function of carrier concentration, are plotted for \(\sim\)1400 calculations of transport properties with systematically perturbed tight-binding orbital interaction parameters (estimated \({zT}\) is plotted in Supplementary Fig. 4 for T = 300 K). The figure thus presents the best possible power factors, within the PbTe system, when both carrier concentration and orbital interactions can be perfectly controlled. The two cases, Case 1 and Case 2, highlighted in Fig. 5a, b, correspond to band divergence (lighter band at point \(L\), Fig. 5c) and band convergence (Fig. 5d). It can be observed that at room temperature, pursuing band divergence by tuning the related orbital interactions is as effective to improve power factor as pursuing band convergence. However, at higher temperature, band convergence has a much larger beneficial effect. The possible power factor enhancement also depends on the achievable carrier concentration. The best power factor enhancement is achieved in Case 2 with perfectly converged bands, as seen in Fig. 5d, giving rise to a one-dimensional electronic density of state60 (its Fermi surface is shown in Supplementary Fig. 5).

Fig. 5: Power factor enhancement with perturbed tight-binding orbital interaction parameters.
figure 5

a and b Scaled power factor as a function of carrier concentration calculated with perturbed orbital interaction parameters at 300 K and 600 K, the empty points correspond to the obtained result for the PbTe tight-binding band structure without perturbation, c Band structure corresponding to maximum performance of Case 1. The band structure obtained without change in interaction parameters is shown in gray and the energy derivative of the Fermi-Dirac distribution function\(-\partial f/\partial E\) is shown on the left for T = 300 K. d Band structure corresponding to the maximum performance of Case 2 with converged bands.

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Discussion

Several limitations in this work should be pointed out. Firstly, it should be reminded that the tight-binding method with a small number of interaction parameters provide only a reasonably good approximation of the electronic structure with a trade-off between accuracy and number of interaction parameters. It will be generally less efficient to determine sensitivity for a large parameter set, which will be the case when crystal symmetry or orbital interactions are long-ranged, due to the large sampling required. We also ignored the effect of the spin orbital coupling (SOC) in constructing the tight-binding model of PbTe for simplicity. SOC reduces the energy at valence band edge at point \(W\) and along Δ61, but the sensitivity analysis of the band energy and the trend in band convergence/divergence and the roles of orbital interaction parameters obtained by the current analysis are not expected to change much44. Regarding the calculation of transport properties, the use of a relaxation time inversely proportional to the total density of states could introduce error since it ignores other source of scattering (microstructure related, such as scattering by grain boundaries and secondary phase or defect related, such as impurity scattering) and ignores as well the difference between intra- and inter-band electron-phonon scattering. As a result, the carrier lifetime in the band convergence case (Case 2) may be underestimated due to overestimation of inter-band scattering. A better treatment would include additional parameters such as the phonon energy58,62,63 and the \(k\) vector dependence, which is beyond the scope of this article. However, more realistic scatterings, including deformation potential, ionized impurity and polar optical phonon scattering can be easily incorporated with a suitable set of material parameters58. Finally, although our method can construct a symmetry-adapted tight-binding problem with minimum human effort, a set of tight-binding parameters should be provided that describes the specific electronic structure of the compound reasonably well. Finding such a set of parameters, typically by fitting the tight-binding model with DFT-calculated electronic structure, could be a difficult task. But we note that the direct formulation of a tight-binding model from DFT calculations, or from Wannierization is possible64,65,66. Recently, a tight-binding database that can provide a good starting point for the analysis described here, was published67.

Our study in this work presents a theoretical analysis on how to obtain the desired electronic structure features or properties by tuning the value of the orbital interaction parameters. We assumed that some small perturbation of the value of orbital interactions can be introduced and then studied how the band structure and thermoelectric power factor vary with respect to the introduced perturbation using sensitivity analysis and Boltzmann transport theory. However, whether such a tuning of orbital interactions can be achieved or possible at all is an open question, especially in terms of the neighboring interactions. Doping or alloying can be expected to modify the onsite energy terms or the nearest neighbor interactions by averaging the atomic potentials (according to the virtual crystal approximation68) and the relationship \({H}_{\mu \nu }=0.5{K}\left({H}_{\mu \mu }+{H}_{\nu \nu }\right){S}_{\mu \nu }\) in the extended Hückel theory (where the \({H}_{\mu \mu }\) and \({H}_{\nu \nu }\) elements are the onsite energies of orbitals \(\mu\) and \(\nu\), and \({S}_{\mu \nu }\) is the overlap integral between the two orbitals)41,69. However, it should be noted that the virtual crystal approximation ignores the structure relaxation near the defect sites, which could influence the properties70. Furthermore, when the orbital energy difference between the dopant and the substituted atoms are significant, the local chemical bonding around the dopants can be different from that of the substituted atoms39. If impurity bands are formed, the perturbative picture assumed in this work will break down and is not applicable. The overlap integrals \({S}_{\mu \nu }\) are related to the geometry of the atomic arrangement in the solid49,71. Consequently, it is also possible to tune the interatomic orbital interactions by lattice strain, which can also be introduced by doping. Finally, we also would like to mention that although there are reports that band convergence was achieved via substitutional doping8,11,12,13,32, it still remains to be seen how well they correspond to tuning of the orbital interactions as outlined in here. Such a verification is important for future works aiming at improving thermoelectric transport properties by modifying orbital interactions.

In summary, we used a sensitivity analysis method based on the symmetry-adapted tight-binding approximation to numerically investigate the importance of orbital interaction parameters in the electronic structure and transport properties. The method was applied to the PbTe thermoelectric compound. The purposed band sensitivity analysis clearly indicates the role of orbital interactions in the entire band structure of PbTe. The possible variation of the thermoelectric power factor of PbTe was studied by varying orbital interaction parameters using Boltzmann transport equation. We surprisingly find that the variation of the power factor can mostly be attributed to only a few orbital interaction terms. Modifying other orbital interaction parameters will not affect the power factor much. It was also discovered that when converging electronic bands by modifying the related orbital interaction parameters, a less well-converged band (not converged within \({k}_{{\rm{B}}}T\)) can have a detrimental effect on transport properties due to a larger mobility effective mass. On the other hand, tuning the orbital interaction parameters for a lighter valence band maximum can readily improve the power factor at low temperature. The sensitivity analysis with the help of an automatic construction of a symmetry-adapted tight-binding problem described in this work aims at understanding the role of orbital interactions in electronic properties and is not restricted to the compound studied here but is also applicable to any general compounds (preferably with high crystal symmetry) and electronic properties. We hope that the method introduced here will open venues for orbital interaction engineering by providing clearer guidelines about ‘recipes’ to achieve desired electronic properties.

Methods

Construction of the symmetry-adapted tight-binding model

The tight-binding approximation constructs the reciprocal space Hamiltonian \({H}_{\mu \nu }\left({\bf{k}}\right)\) matrix elements from the real space Hamiltonian that describes orbital interactions as \({H}_{\mu \nu }\left({\bf{k}}\right)={\sum }_{{\bf{R}}}{e}^{i{\bf{kR}}}{H}_{\mu \nu }\left({\bf{R}}\right)\)42. The electronic structure can then be solved by the secular equation:

$$\sum _{\nu }\left({H}_{\mu \nu }\left({\bf{k}}\right)-{\varepsilon }_{i{\bf{k}}}{S}_{\mu \nu }\left({\bf{k}}\right)\right){c}_{i{\bf{k}}}=0$$
(5)

where \({\varepsilon }_{i{\bf{k}}}\) are the eigen-state energies and \({c}_{i{\bf{k}}}\) are the coefficients which determine the crystal wavefunction as a combination of atomic orbitals \(\mu\) and \(\nu\). \({S}_{\mu \nu }\left({\bf{k}}\right)\) is the overlap matrix. The symmetry relationship between orbital interaction parameters can reduce the number of independent tight-binding parameters. For example, the interactions between orbitals of atom \(i\) with orbitals of its neighbors show symmetry relationship given by its site symmetry point group and can be approached by considering the formation of symmetry-adapted linear combinations of atomic orbitals (SALCs). SALCs with different symmetry properties (indicated by different representations \({\varGamma }_{i}\) and \({\varGamma }_{j}\) of the symmetry group72) are not allowed to interact with each other41,72,73. In other words,

$${\big\langle }{\phi }_{{\rm{SALC}}}^{{\varGamma }_{i}}\left|H\right|{\phi }_{{\rm{SALC}}}^{{\varGamma }_{j}}{\big\rangle }=0$$
(6)

The set of these symmetry-forbidden interactions between SALCs leads to a set of equations specifying the symmetry constraints between the orbital interaction terms \({H}_{\mu \nu }\left({\bf{R}}\right)\), as illustrated in Fig. 6. Furthermore, using the symmetry relationship between equivalent sites and that \({H}_{\mu \nu }^{\dagger }({\bf{R}})={H}_{\nu \mu }(-{\bf{R}})\), the set of symmetry independent orbital interaction parameters for the whole set of orbital interactions in the unit cell can be obtained. The symmetry constraints between orbital interactions can be written as a homogeneous matrix. To obtain the independent orbital interaction parameters, the homogeneous matrix is brought to a row-echelon form. The non-leading terms can be identified to be independent parameters74. The same homogeneous matrix can be used to recover all tight-binding parameters from only the independent ones. This processed is illustrated through an example in Supplementary Note 1. Calculation of various electronic properties can be done after solving the eigen-equation.

Fig. 6: Finding the independent terms for the symmetry-adapted tight-binding model.
figure 6

Symmetry-adapted linear combination (SALC) of atomic orbitals can be formed using the site-symmetry operations. SALCs of different symmetry properties are not allowed to interaction with each other.

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Sensitivity analysis

We used the Sobol’s sensitivity analysis method to calculate the orbital interaction sensitivity in this work. For a given function \(y=f\left({\bf{x}}\right)\), the Sobol’s variance-based method decomposes the variance of \(y\) into fractions that can be attributed to each of the inputs \({x}_{i}\in {\bf{x}}\) 75,76,77. For a subset of parameters \({\bf{u}}\subset {\bf{x}}\), the variance of \(y\) that can be attributed to the set of parameters \({\bf{u}}\), can be written as \({{\mathbb{V}}}_{{\bf{u}}}\left[{\mathbb{E}}\left(y{\rm{| }}{\bf{u}}\right)\right]\) (where \({\mathbb{E}}\) is the expectation value and \({\mathbb{V}}\) is the variance), whose value suggests whether the different values of \({\bf{u}}\) lead to different expected values of the output \(y\). The Sobol sensitivity coefficient is then defined to be \({S}_{{\bf{u}}}={{\mathbb{V}}}_{{\bf{u}}}\left[{\mathbb{E}}\left(y{\rm{| }}{\bf{u}}\right)\right]{\mathbb{/}}{\mathbb{V}}\left[y\right]\). In this work, we use the total effect Sobol index \({S}_{{{\rm{T}}}_{i}}\) of a single parameter \({x}_{i}\) defined as

$${S}_{{{\rm{T}}}_{i}}={\sum }_{{\bf{u}}{\rm{;}}{x}_{i}\in {\bf{u}}}{S}_{{\bf{u}}}$$
(7)

where the sum is over all the subsets of parameters \({\bf{u}}\) that include \({x}_{i}\). The total effect coefficient thus includes not only the direct contribution of change in the single variable \({x}_{i}\) to the change of \(y\), but also includes any interaction effect between \({x}_{i}\) and other variables, when the effect of \({x}_{i}\) depends on the value of other parameters. A large value of \({S}_{{{\rm{T}}}_{i}}\) of one variable (close to \(1\)) implies that changing its value will lead to large change (response) in the resulting properties. On the other hand, a small \({S}_{{{\rm{T}}}_{i}}\) (close to \(0\)) means that changing these terms will not have noticeable impact on the output. To calculate the orbital interaction sensitivity using the Sobol method and the tight-binding model, the independent orbital interaction parameters are perturbed around their initial values in a small range. Then, the corresponding electronic properties, such as band eigen-energies and transport properties, are calculated using the tight-binding model. Finally, Sobol indices are calculated using the Python library SALib based on the perturbation and the resulting properties. In this article, we found it convenient to use the product \({S}_{{{\rm{T}}}_{i}}{\mathbb{\cdot }}{\mathbb{V}}\left[y\right]\), i.e., without scaling with the total variance of \(y\) when plotting orbital sensitivities on band structure plot. This gives information of the absolute variance when we compare different \(k\) points across band structures.

Transport properties

Thermoelectric transport properties based on the tight-binding model were calculated by the following equations:56

$${N}_{c}\left(\mu ,T\right)=\frac{1}{V}{\sum }_{n{\bf{k}}}f\left({\varepsilon }_{n{\bf{k}}},\mu ,T\right)$$
(8)
$${{\boldsymbol{\sigma }}}_{\alpha \beta }\left(\mu ,T\right)=\frac{{e}^{2}}{V}{\sum }_{n{\bf{k}}}{v}_{n{\bf{k}}}^{\alpha }\,{v}_{n{\bf{k}}}^{\beta }{\tau }_{n{\bf{k}}}\left({\left.-\frac{\partial f\left(\mu ,T\right)}{\partial E}\right|}_{{\varepsilon }_{n{\bf{k}}}}\right)$$
(9)
$${\left[{\boldsymbol{\sigma }}{\boldsymbol{S}}\right]}_{\alpha \beta }\left(\mu ,T\right)=\frac{e}{VT}{\sum }_{n{\bf{k}}}{v}_{n{\bf{k}}}^{\alpha }{v}_{n{\bf{k}}}^{\beta }{\tau }_{n{\bf{k}}}\left({\varepsilon }_{n{\bf{k}}}-\mu \right)\left({\left.-\frac{\partial f\left(\mu ,T\right)}{\partial E}\right|}_{{\varepsilon }_{n{\bf{k}}}}\right)$$
(10)

where \({N}_{c}\left(\mu ,T\right)\), \({\boldsymbol{\sigma }}\left(\mu ,T\right)\) and \({\boldsymbol{S}}\left(\mu ,T\right)\) are the carrier concentration, electrical conductivity and Seebeck coefficient tensor, respectively, at temperature \(T\) and chemical potential \(\mu\). In these equations, \(f\left({\varepsilon }_{n{\bf{k}}},\mu ,T\right)\) is the Fermi-Dirac function for the energy of electronic states of the \(n\)-th band at point \({\bf{k}}\). \(\mu\) is the chemical potential and \(T\) is the temperature. \({v}_{n{\bf{k}}}^{\alpha /\beta }\) is the electronic group velocity in directions \(\alpha\) or \(\beta\). \({\tau }_{n{\bf{k}}}\) is the electronic relaxation time. Density of states were calculated using an accurate tetrahedron summation.