Debye-Callaway model simulator: an interactive slider-based program for fitting theoretical and experimental lattice thermal conductivity

Abstract

Reducing lattice thermal conductivity (κ_latt) is essential for advancing thermoelectric materials. Achieving this requires deeper insight into how microstructural defects influence phonon scattering and the ability to model these interactions effectively via the Debye–Callaway model. However, the mathematical complexity of its nonlinear integral form has historically limited its use to specialists with advanced coding skills. In this study, we present a comprehensive review of the Debye–Callaway model, emphasizing the physical parameters governing nine key phonon scattering mechanisms. Building on this, we introduce a novel, standalone simulation program with an intuitive, slider-based graphical interface that enables real-time visualization of how variations in microstructural parameters affect κ_latt. The tool features editable inputs for experimental datasets, temperature ranges, and material-specific parameters, with instant graphical feedback. Through three case studies, we demonstrate its capabilities in deconvoluting defect contributions, identifying modelling errors, and predicting defect impacts, providing a significant advance in phonon transport analysis.

Introduction

Thermoelectric (TE) technology is emerging as a promising solution for addressing climate change by harnessing solar energy and converting waste heat into electricity via the Seebeck effect, contributing to global efforts to meet the 1.5 °C global warming target. However, its commercialization remains hindered by uncompetitive performance, necessitating strategies to enhance TE efficiency^1,2. TE efficiency is enhanced through approaches that foster the realization of the phonon-glass electron-crystal (PGEC) paradigm, which optimally balances semiconductor-like properties (high Seebeck coefficient, S), metallic characteristics (high electrical conductivity, σ), and glass-like behavior (low electronic (κ_ele) and lattice (κ_latt) thermal conductivity)^3,4. The synergy of these properties is summarized in the definition of the fundamental metric of TE performance, commonly referred to as figure of merit (zT) in Eq. 1, where T, PF, and κ denote the absolute temperature, power factor, and total thermal conductivity, respectively^5,6.

$$z{\rm{T}}=\frac{{{\rm{S}}}^{2}\sigma }{{\kappa }_{{ele}}+{\kappa }_{{latt}}}T=\frac{\mathrm{PF}}{\kappa }T$$

(1)

So far, efforts to maximize PF while maintaining low κ through conventional doping have reached a deadlock due to the intricate interdependence among S, σ, and κ_ele on the carrier concentration (n)⁷. To circumvent this limitation, recent strategies have focused on optimizing the thermoelectric quality factor (B). This approach avoids direct manipulation of the n, and instead focuses on enhancing μ and m*, while simultaneously striving to suppress κ_latt, as described in Eq. 2^6,8.

$${\rm{B}}=\frac{{{\rm{\mu }}{m}^{* }}^{1.5}{T}^{2.5}}{{\kappa }_{{latt}}}$$

(2)

Accordingly, maximizing the effectiveness of phonon-retarding mechanisms is essential for unlocking the commercial potential of thermoelectric materials. This can be accelerated through rational defect engineering and fostering a deeper understanding of the individual contributions of various defects in scattering phonons across a broad frequency spectrum. However, both qualitative and quantitative insights into the interplay between lattice structure modifications and the roles of different phonon scattering centers in reducing κ_latt remain limited, hindering efforts to achieve systematic and progressive thermal conductivity reduction.

Over the years, extensive experimental studies have demonstrated the collective impact of primary phonon scattering mechanisms, including point defects such as substitutional and interstitial atoms, vacancies, nano-inclusions, grain boundaries, and dislocations. For instance, Kihoi et al. reported a significant reduction in κ_latt in Sn_0.67Ge_0.2Bi_0.03Sb_0.1Te due to the combined effects of Ge and Sb point defects and Ge-rich precipitates⁹. Similarly, Tan et al. achieved a κ_latt reduction to 0.7 W/mK in SnTe through AgBiTe₂ alloying that simultaneously introduced point defect and Ag-rich nanostructures¹⁰. More recently, double half-Heusler alloys have exhibited over a 70% drop in κ_latt compared to their parent half-Heusler compounds, an improvement attributed to the synergy between point defects and lattice anharmonicity, induced by aliovalent substitution of a single element with two elements of opposite valence^11,12,13. Despite these significant advancements in κ_latt reduction through the synergy of multiple scattering mechanisms, further enhancing thermal conductivity suppression via targeted defect engineering requires a deeper understanding of the individual phonon scattering strengths of these defects and their contributions to the overall reduction in κ_latt.

Numerous experimental studies have successfully isolated specific phonon scattering mechanisms, which can aid in enabling a reasonable quantification of their phonon scattering strength. For example, grain boundary scattering was experimentally shown to yield up to an 18% reduction in κ_latt when the average grain size decreased from 1320 nm in Nb_0.95Ti_0.05FeSb to 200 nm, while the addition of Ti solute in Nb_0.8Ti_0.2FeSb yielded a drastic drop in κ_latt¹⁴. In PbTe, κ_latt dropped from 2–2.5 W/mK to 0.5–0.8 W/mK in Ag(Pb₁₋ₓSnₓ)_mSbTe₂₊_m (commonly referred to as LAST) due to phonon scattering from endotaxially embedded nanostructures¹⁵. Similarly, Sootsman et al. demonstrated a ~ 27% κ_latt reduction by incorporating 4% Sb precipitates¹⁶. These quantified contributions of individual defects to κ_latt reduction serve as valuable references to the constitution of a database of individual defect phonon-scattering strength, a critical requisite in guiding and expediting targeted defect engineering for further suppression of thermal conductivity. However, while these reductions are often attributed to the sole contribution of the dominant defects, secondary scattering mechanisms may also play a significant role, a factor that is often overlooked. For instance, the presence of point defects can alter the bonding structure due to the introduction of foreign atoms, intensifying phonon-phonon (Umklapp) scattering and reducing the phonon group velocity. On the contrary, stiffer bonds may weaken Umklapp scattering, fostering high κ_latt in the presence of dopants in line with previous results¹⁷. Furthermore, aliovalent dopants can induce charge density variations or annihilate innate vacancies, altering the interactions of phonons with electrons and vacancies, which may significantly affect κ_latt suppression^18,19,20.

The influence of nanoinclusions on κ_latt has been experimentally shown to have a nonlinear dependence on multiple parameters, including the mass differences between the inclusion and matrix, as well as size and volume fraction. The mass difference effect was exemplified in the PbTe system, where Sb and InSb species demonstrated strong phonon scattering and reduced κ_latt, while Bi nanostructures had adverse effects¹⁶. Inclusion size and volume fraction dependence were also revealed in the same study, where 2% Sb yielded the lowest κ_latt, but a higher κ_latt was recorded at >4% Sb, corresponding with observed precipitate size growth. The thermal conductivity of the inclusions relative to that of the matrix is another crucial factor influencing their phonon scattering efficacy, consistent with previous reports of adverse effects of inclusions in different TE materials where κ_inclusions was proven or presumed larger than κ_matrix^17,21,22,23. These complex and sometimes contradictory findings emphasize the importance of integrating theoretical modelling with experimental data to accurately deconvolute and quantify the individual contributions of various defects. Such integration is essential for achieving systematic and progressive reductions in κ_latt.

Phonon dispersion simulations have been extensively applied to gain a theoretical understanding of the underlying changes in phonon transport. They reveal variations in critical phonon parameters, such as phonon group velocity, phonon density of states, phonon lifetime, and the dominant phonon types, where acoustic phonons correspond to low frequencies and optical phonons to high frequencies²⁴. These simulations are built from advanced computational methods such as density functional theorem (DFT), while Boltzmann transport Equations and electron-phonon coupling calculations offer an advanced mechanism to estimate theoretical lattice thermal conductivity from the phonon dispersion²⁵. In addition, the green function methods are increasingly gaining traction in phonon transport across interfaces²⁶. While these computational methods provide valuable insights into phonon transport, they remain computationally intensive and often inaccessible to many experimental groups. Similarly, experimental measurements, such as sound velocity and Raman spectroscopy, have been widely employed to demonstrate changes in lattice structure, including softening or hardening^27,28. However, while these results indicate structural modifications, they do not quantify the extent to which these changes contribute to the overall κ_latt reduction. These research gaps underscore the critical need for an integrated approach combining both experimental and theoretical frameworks to establish a comprehensive database of individual defect scattering strengths. Such insights will accelerate targeted defect engineering efforts, bringing thermoelectric materials closer to achieving glass-like thermal conductivity for enhanced performance. To expedite such endeavors, simpler relaxation-time-based models, like the Debye–Callaway framework, should be continually improved.

The Debye–Callaway model remains a powerful predictive tool, capable of providing detailed and accurate assessments of the individual contributions of various phonon scattering mechanisms to κ_latt²⁹. Despite its potential, the model has been historically underutilized due to the mathematical complexity of its nonlinear integral equations, which include dynamic upper limits and dependence on multiple relaxation times. Additionally, a centralized and accessible reference that systematically outlines the theoretical relationships between phonon scattering mechanisms, their controlling parameters, and the corresponding relaxation times is still lacking. Matthiessen’s rule is commonly cited, but typically in the context of only a subset of scattering processes, leaving a fragmented understanding of the full scattering landscape³⁰.

This work presents a comprehensive review of the Debye–Callaway model, offering a detailed analysis of nine common phonon scattering mechanisms in TE materials, along with their associated experimentally measurable physical parameters and fitting parameters that ideally represent their scattering strengths. Secondly, a user-friendly standalone program is introduced to simplify the model’s implementation and facilitate theoretical κ_latt simulations through an interactive, slider-based interface. The program includes features that enhance its flexibility, enabling real-time modelling and fitting of theoretical κ_latt to experimental data for virtually any thermoelectric material. Finally, the program’s capabilities are demonstrated through a series of examples, showcasing its effectiveness in fitting theoretical κ_latt, deconvoluting the contributions of individual phonon-scattering mechanisms, and estimating the scattering strengths of constituent defects.

Results

The Debye-Callaway model and Matthiessen’s rule

The Debye-Callaway model and Matthiessen rule outline a theoretical account of the individual contribution of each mechanism in κ_latt reduction³¹. Based on the Debye-Callaway model, temperature-dependent κ_latt is given by Eq. 3^29,32,33,34.

$${\kappa }_{{latt}}\left(T\right)=\frac{{k}^{4}{T}^{3}}{2{\pi }^{2}{\hslash }^{3}{v}_{s}}{\int }_{0}^{{\varTheta }_{D}/T}\frac{{x}^{4}{e}^{x}}{{\left({e}^{x}-1\right)}^{2}}{\tau }_{{total}}{dx}$$

(3)

Here, ℏ is the reduced Planck constant, k is the Boltzmann constant, $x=\hslash \omega /{kT}$ where $\omega$ is the phonon frequency, T is the absolute temperature, τ_total is the total relaxation time, and ${\varTheta }_{D}$ is the Debye temperature given by Eq. 4.

$${\varTheta }_{D}=\frac{\hslash }{k}{\left(6{\pi }^{2}\frac{N}{V}\right)}^{\frac{1}{3}}{v}_{s}$$

(4)

Where $N$ is the number of atoms per unit cell, $V$ is the unit cell volume (m³). v_s is the mean sound velocity (m/s) calculated using Eq. 5.

$${v}_{s}={\left[\frac{1}{3}\left(\frac{1}{{v}_{L}^{3}}+\frac{2}{{v}_{T}^{3}}\right)\right]}^{-1/3}$$

(5)

Here, v_L and v_T are sound velocities in the longitudinal and transverse/shear directions, respectively, often measured using the pulse-echo method or ultrasound methods.

The total phonon relaxation time, τ_total, is evaluated using Matthiessen’s rule, which involves the reciprocal summation of the individual phonon relaxation times relevant to the system. In thermoelectric materials, the most dominant phonon scattering mechanisms include normal process scattering (τ_N), Umklapp scattering (τ_U), boundary scattering (τ_B), point defect scattering (τ_PD), electron–phonon interaction scattering (τ_EP), nanoinclusion scattering (τ_ni), phonon–vacancy scattering (τ_vac), and dislocation scattering, which comprises both dislocation core (τ_DC) and dislocation strain (τ_DS) contributions. Accordingly, τ_total can be calculated as shown in Eq. 6.

$$\frac{1}{{\tau }_{{total}}}=\frac{1}{{\tau }_{N}}+\frac{1}{{\tau }_{U}}+\frac{1}{{\tau }_{B}}+\frac{1}{{\tau }_{{PD}}}+\frac{1}{{\tau }_{{EP}}}+\frac{1}{{\tau }_{{ni}}}+\frac{1}{{\tau }_{{vac}}}+\frac{1}{{\tau }_{{DC}}}+\frac{1}{{\tau }_{{DS}}}$$

(6)

The following section expands this foundation by examining each scattering mechanism individually.

Phonon relaxation times

Relaxation time due to normal processes (τ_N) accounts for scattering resulting from phonon-phonon interactions in which momentum is conserved within the first Brillouin zone. Normal processes, or N-processes, play a crucial role in redistributing phonon momentum, particularly at intermediate temperatures. τ_N is expressed by Eq. 7, where NP denotes the normal process scattering coefficient, and a and c are empirical constants representing the frequency and temperature exponents, respectively²⁹.

$$\frac{1}{{\tau }_{N}}={NP}.{\omega }^{a}.{T}^{c}$$

(7)

In some cases, ${\tau }_{N}$ has been empirically given as a fraction of the Umklapp scattering relaxation time (${\tau }_{U}$)³⁴.

The relaxation time due to phonon-phonon interaction, alias Umklapp or U process (τ_U), represents scattering due to phonon-phonon interactions where the resulting momentum is not conserved within the first Brillouin zone. τ_U is typically evaluated using Eq. 8, with the Umklapp scattering constant (U_m) and the Umklapp scattering parameter (b) serving as the primary input parameters²⁹. U_m characterizes the phonon scattering strength of U-processes and is usually determined through fitting, while b is an empirical constant that reflects the vibrational spectrum of the material. τ_U is particularly influenced by microstructural defects that alter local bond anharmonicity, such as point defects and vacancies. Additionally, grain boundaries and nanoinclusions may induce local strain fields, disrupt the bonding environment, or introduce irregular phonon propagation paths, further promoting Umklapp scattering, though typically to a lesser degree.

$$\frac{1}{{\tau }_{U}}={U}_{m}.{\omega }^{2}.T{.\,e}^{{-\varTheta }_{D}/b.T}$$

(8)

Relaxation time due to grain boundary scattering, τ_B evaluates the average time between phonon scatterings at grain boundaries in polycrystalline materials. Assuming that phonons are scattered upon each encounter with a grain boundary, τ_B is typically estimated using Eq. 9, where L is the grain size in meters, often determined through microscopic analysis^29,35. L is typically determined from microscopic analysis of fracture or etched surfaces, either by direct grain size measurement or by averaging using the linear intercept method^36,37,38. The use of advanced image analysis tools, such as ImageJ or EBSD-based grain size mapping, can significantly enhance the accuracy and reliability of this measurement^14,39.

$$\frac{1}{{\tau }_{B}}=\frac{{v}_{s}}{L}$$

(9)

Relaxation time due to point defect scattering, τ_PD, represents the average time between phonon scatterings caused by point defects such as interstitial or substitutional atoms. It is typically evaluated using Eq. 10, where P_d denotes the point defect scattering constant²⁹.

$$\frac{1}{{\tau }_{{PD}}}={P}_{d}.{\omega }^{4}$$

(10)

Numerous studies have presented P_d as the primary descriptor of phonon scattering strength due to atomic defects⁴⁰. However, from the definition of ${P}_{d}$ in Eq. 11, it is important to emphasize that V_atom, the average volume per atom (V/N), and v_s are independent, experimentally measurable physical quantities.

$${P}_{d}=\frac{{V}_{{atom}}.\varGamma }{4\pi {v}_{s}^{3}}$$

(11)

Therefore, these parameters should not be treated as fitting variables during theoretical modelling, and only Γ, the point defect scattering parameter, should be adjusted to reflect changes in point defect-induced scattering. Consequently, ${\tau }_{{PD}}$ can be calculated using Eq. 12⁴¹.

$$\frac{1}{{\tau }_{{PD}}}=\frac{{V}_{{atom}}.{\omega }^{4}}{4\pi {v}_{s}^{3}}\varGamma$$

(12)

As shown in Eq. 13, Γ is the summation of scattering parameters due to mass (Γ_m, (Eq. 14)) and size (Γ_s, (Eq. 15)) contrast contributions induced by a certain fractional concentration of defects^42,43,44. Here, $\bar{M}$ denotes the average atomic mass of the alloy, $\bar{m}$ and $\bar{r}$ denote the average atomic mass and radius of the species occupying the substituted site, ${x}_{i}$, ${m}_{i}$, and ${r}_{i}$ are the fractional concentration, atomic mass, and atomic radius of the atom at the ith sublattice, respectively⁴⁵.

$$\varGamma =\frac{1}{N}{\left(\frac{\bar{m}}{\bar{M}}\right)}^{2}\left[{{\sum }_{i}{x}_{i}\left(1-\frac{{m}_{i}}{\bar{m}}\right)}^{2}+{\varepsilon {\sum }_{i}{x}_{i}\left(1-\frac{{r}_{i}}{\bar{r}}\right)}^{2}\right]$$

(13)

$${\varGamma }_{m}=\frac{1}{N}{\left(\frac{\bar{m}}{\bar{M}}\right)}^{2}\left[{{\sum }_{i}{x}_{i}\left(1-\frac{{m}_{i}}{\bar{m}}\right)}^{2}\right]$$

(14)

$${\varGamma }_{s}=\frac{1}{N}{\left(\frac{\bar{m}}{\bar{M}}\right)}^{2}\left[{\varepsilon {\sum }_{i}{x}_{i}\left(1-\frac{{r}_{i}}{\bar{r}}\right)}^{2}\right]$$

(15)

The strain field fitting parameter (ε) is a phenomenological strain parameter for scaling Γ_s to account for elastic field disturbances originating from the size mismatch. Due to the complexity of calculating ε, it is often treated as a fitting parameter, with reported values spanning a wide range, from 1 to 125^9,42,43. Some studies have suggested that ε can be estimated from the Grüneisen parameter (γ) using Eq. 16^46,47,48.

$$\varepsilon =\frac{2}{9}{\left[\frac{6.4\gamma \left(1+{v}_{s}\right)}{1-{v}_{s}}\right]}^{2}$$

(16)

However, as shown later in the Case Studies sections, this approach tends to overestimate ε, which in turn inflates Γ_s and consequently the total point-defect scattering strength. This suggests that a fitting-based determination of ε may provide more physically realistic and reliable values.

Point defects can also contribute to Umklapp scattering by altering local bond characteristics and increasing anharmonicity. In such cases, the calculated Γ, fails to fit the theoretical κ_latt to the experimental plot unless the Umklapp fitting parameter or its exponent are tuned accordingly. Such coupled fitting adjustments are therefore essential in deconvoluting the respective contributions to κ_latt reduction due to point defect-induced scattering and anharmonicity.

Interactions between electrons and phonons also play a significant role in suppressing κ_latt, particularly in degenerate semiconductors. The electron-phonon scattering relaxation time, τ_EP, quantifies the average time between phonon scattering events resulting from interactions with charge carriers. It is typically evaluated using Eq. 17, where C represents the electron-phonon scattering constant^29,49.

$$\frac{1}{{\tau }_{{EP}}}=C.{\omega }^{2}$$

(17)

C inherently includes the contribution of n, which numerous experimental and theoretical investigations have shown to be inversely proportional to τ_EP^50,51. Since $n$ is a physical property, typically measured via the Hall effect⁵², it is reasonable to exclude it from C. Accordingly, incorporating $n$ explicitly, τ_EP can be formulated as shown in Eq. 18, where C_eph denotes the electron-phonon scattering strength, encompassing electron-phonon interaction energies and other material-specific properties.

$$\frac{1}{{\tau }_{{EP}}}={C}_{{eph}}.n.{\omega }^{2}$$

(18)

Inclusions play a critical role in phonon scattering through two mechanisms: as phonon scattering centers, particularly at the micro-to-nanoscale through the Rayleigh-type mechanisms, and as composite structures via the effective medium theory at the micro-to-bulk scale. As scattering centers, the extent of phonon scattering induced by these inclusions is quantified by the nanoinclusion relaxation time, τ_ni, which represents the average time between phonon scattering events as they interact with inclusions dispersed within the matrix phase White and Klemens demonstrated that the phonon-scattering efficiency of precipitates scales with the v_s, as well as with the size and density of the inclusions⁵³. Accordingly, τ_ni can be expressed by Eq. 19, where f denotes the volume fraction of the inclusions, r is their average radius, and ρ_c represents the density contrast between the matrix and the inclusion phases⁵⁴.

$$\frac{1}{{\tau }_{{ni}}}={A}_{{nano}}.f.\left(\frac{{v}_{s}}{r}\right).{\rho }_{c}^{2}.{\omega }^{4}$$

(19)

The density contrast is calculated using Eq. 20, where ${\rho }_{m}$ and ${\rho }_{i}$ correspond to the matrix and inclusion’s density, respectively. These structural parameters are complemented by the nanoinclusion scattering constant (A_nano), which indicates the inclusion-phonon scattering strength, and can be determined via model fitting to experimental data.

$${\rho }_{c=\,\frac{{\rho }_{m}-{\rho }_{i}}{{\rho }_{m}}}$$

(20)

Given the inverse correlation between τ_ni and r in Eq. 19, as the inclusion size approaches the micro-to-bulk scale, their Rayleigh scattering strength weakens, and the influence of τ_ni diminishes. In this intermediate regime, phonon–inclusion interactions are governed by a combination of Rayleigh and geometrical scattering mechanisms. Near the upper limit of Rayleigh scattering validity, a recent study has provided practical guidelines for identifying the transition between the Rayleigh and geometrical regimes, highlighting the size scales at which each mechanism prevails⁵⁵. Beyond this transition, a variety of effective-medium theories describe the thermal coupling between the matrix and bulk inclusions. These models generally assume that the matrix and inclusion phases are thermally connected in series and predict the resulting effective thermal conductivity (κ_eff) of the composite system. In this framework, the effective thermal conductivity becomes a function of the volume fraction and the intrinsic thermal conductivities of the matrix and inclusion phases. Maxwell developed the simplest model under the effective medium theory to predict κ_eff for heterogeneous systems, as expressed by Eq. 21^56,57.

$${\kappa }_{{eff}}={\kappa }_{m}\frac{{\kappa }_{i}+2{\kappa }_{m}+2f\left({\kappa }_{i}-{\kappa }_{m}\right)}{{\kappa }_{i}+2{\kappa }_{m}-f\left({\kappa }_{i}-{\kappa }_{m}\right)}\,$$

(21)

Where κᵢ and κ_m are the thermal conductivities of the inclusions and the matrix, respectively. This implies that when κᵢ < κ_m, the inclusions function as thermal barriers, thereby reducing κ_eff. Conversely, when κᵢ > κ_m, the inclusions act as thermal conduits, enhancing κ_eff. Other models, including the Rayleigh and Hasselman–Johnson models, which account for the influence of interface thermal resistance, have been developed^57,58 However, incorporating the influence of τ_ni from such bulk inclusions into τ_total becomes challenging, as these models typically address thermal conductivity at the macroscopic scale without explicitly separating electronic and lattice contributions. Furthermore, if f exceeds the percolation threshold, the critical volume fraction at which inclusions begin to form continuous paths, heat transfer through interconnected thermal channels becomes more efficient⁵⁹. In this case, κ_eff can increase sharply. At this point, the applicability of effective medium theory becomes limited, necessitating more sophisticated numerical simulation approaches to accurately capture the evolving heat conduction behavior.

Nevertheless, it is important to note that although Eq. 21 and the associated discussion help illustrate the physical limits of inclusion-driven scattering, this equation is not implemented in the Debye–Callaway interface, because it does not correspond to a relaxation-time expression and therefore cannot be incorporated within Matthiessen’s rule or the Boltzmann transport formulation used in this work. Instead, it is presented solely to delineate the regime where the relaxation-time approximation becomes invalid and where effective-medium models provide a more appropriate description of heat transport in micro- and bulk-scale composite systems.

Vacancies are typically classified as point defects. However, unlike substitutional or interstitial atomic point defects, the formation of a vacancy results in a massless void and the elimination of bonding interactions with neighbouring atoms. ⁶¹ These distinctions cause fundamentally different lattice perturbations, thereby undermining the applicability of Γ and its associated expression in Eq. 13 for accurately describing phonon scattering due to vacancies. Consequently, the phonon-vacancy scattering relaxation time (τ_vac) is more appropriately characterized by the vacancy concentration per atom (f_vac), the phonon-vacancy scattering strength (s²), and the v_s, as expressed in Eq. 22^32,45,60.

$$\frac{1}{{\tau }_{{vac}}}={f}_{{vac}}.{\frac{3{\omega }^{4}}{\pi {{v}_{s}}^{3}}s}^{2}$$

(22)

s² represents the phonon-vacancy scattering strength arising from the missing mass and bonding associated with an isolated vacancy. It can be experimentally determined by analyzing samples with varying f_vac, ideally using single crystals to minimize extrinsic scattering contributions. For example, the s² value for Sn vacancies in SnTe has been experimentally estimated to be approximately 0.89, based on fitting phonon scattering models to thermal conductivity data from SnTe single crystals with systematically varied f_vac⁴⁵. Since the number of vacancies is typically proportional to the number of charge carriers they generate, f_vac can be estimated from the simplified charge neutrality equation under the assumption of thermal equilibrium⁶⁰. This relation is expressed in Eq. 23, where q denotes the effective charge state of the vacancy (e.g., q = 2 for V²⁺), and both n and V_atom are expressed in consistent units.

$${f}_{{vac}}=\frac{n.{V}_{{atom}}}{N.q}$$

(23)

This proportionality holds true for stoichiometrically imbalanced alloys, where vacancies serve as the dominant source of charge carriers. However, its validity diminishes in heavily doped systems, where a significant fraction of carriers originates from extrinsic aliovalent dopants. In such cases, the reliability of this approach is limited, and alternative models become necessary for accurately estimating vacancy concentrations. These alternative approaches can be developed based on the established frameworks for vacancy–defect interactions, as exemplified by the existing model for Se-doped Bi₂Te₃ systems⁶¹. While vacancies are widely recognized as the dominant defects in chalcogenide thermoelectric materials^18,62,63,64, stoichiometric imbalances have also been observed in other systems, such as half-Heusler alloys^{65,66,67,68,69}. This underscores the need for intensified investigation into the role of vacancies in phonon transport across a broader range of thermoelectric materials.

All semiconducting materials exhibit vestiges of dislocations, which contribute to phonon scattering through two distinct mechanisms: dislocation core scattering (τ_DC) and dislocation strain scattering (τ_DS). τ_DC accounts for the direct interaction between phonons and the atomic-scale distortion surrounding the dislocation core. It is primarily governed by the dislocation density (N_D) per cm² and the v_s, as described in Eq. 24⁴⁴.

$$\frac{1}{{\tau }_{{DC}}}={N}_{D}.{\frac{{{V}_{{atom}}}^{4/3}}{{{v}_{s}}^{2}}\omega }^{3}$$

(24)

τ_DS accounts for phonon scattering due to the long-range elastic strain fields generated around dislocations. It scales with the dislocation density (N_D), the magnitude of the Burgers vector (B_D), and key material properties such as the Grüneisen parameter (γ), Poisson’s ratio (ν), and the v_s, as described in Eq. 25. A′ represents the pre-factor for dislocation strain scattering, encapsulating constants and geometric factors relevant to the interaction.

$$\frac{1}{{\tau }_{{DS}}}=A{\prime} .{B}_{D}^{2}{N}_{D}{\omega }^{2}{\gamma }^{2}\left[\frac{1}{2}+\frac{1}{24}{\left(\frac{1-2v}{1-v}\right)}^{2}{\left(1+\sqrt{2}{\left(\frac{{v}_{L}}{{v}_{T}}\right)}^{2}\right)}^{2}\right]$$

(25)

For simplicity in modelling, the velocity ratio terms can be replaced by the v_s in Eq. 26, based on the established relationship that the dislocation strain scattering rate scales inversely with the square of v_s.

$$\frac{1}{{\tau }_{{DS}}}=\frac{A.{B}_{D}^{2}{N}_{D}{\gamma }^{2}{\omega }^{2}}{{v}_{s}^{2}}$$

(26)

Here, A encapsulates the constant and approximate contributions from the bracketed term in Eq. 25. The N_D and B_D are physical parameters typically obtained through advanced microstructural characterization techniques, such as transmission electron microscopy (TEM)⁷⁰. With all other quantities determined independently, A and γ remain as the fitting parameters and can therefore be extracted by fitting theoretical κ_latt to experimental data from samples with well-characterized values of N_D and B_D.

In essence, modelling κ_latt using the Debye–Callaway framework, with all nine phonon-scattering mechanisms in Eq. 9, requires 23 distinct input parameters, as listed in Table 1. Of these, 13 are experimentally measurable physical quantities, 3 are empirical constants, and 7 serve as fitting parameters. The large number of required inputs, combined with the mathematical complexity of evaluating the nonlinear integral form of the Debye–Callaway equation, complicates implementation of this model. Consequently, many researchers have defaulted to the more limited spectral κ_latt approach^9,71,72. To promote the effective utilization of this model, we present a user-friendly standalone program that simplifies its implementation.

Table 1 Measurable microstructural parameters, empirical constants, and fitting parameters, along with associated phonon scattering relaxation times for modelling lattice thermal conductivity (κ_latt) using the Debye-Callaway model, when nine distinct defect scattering mechanisms are considered

Developed Debye-Callaway model simulator’s structure and features

The simulator enables users to explore the contributions of various phonon scattering mechanisms interactively through the intuitive user-friendly interface shown in Fig. 1. Specifically, the program allows users to input up to two experimental datasets for pristine and doped samples through editable textboxes that accept comma-separated strings of the experimental κ_latt and corresponding temperature values. The textboxes accept copy-and-paste (Ctrl + V) functionality to improve convenience and reduce errors when entering long data strings. Pressing the keyboard “enter” button updates the scatter plots corresponding to these samples in the software’s interface, allowing visual interaction between the experimental and theoretical κ_latt line plots displayed in the same interface. This facilitates seamless comparison and fine-tuning of the predictive model for improved accuracy.

Fig. 1: Screenshot of the developed Debye-Callaway Simulator’s interface.

This figure shows the complete graphical interface of the Debye–Callaway simulator developed in this work. The central plot displays the theoretical conductivity as a solid blue line, alongside experimental data for the pristine sample (black filled rectangles) and the doped sample (red filled circles). The left panel contains the Experimental Data section, where users input temperature and conductivity values, which update the plot immediately. The Temperature Range section allows selection of minimum and maximum temperatures and the number of calculation points. The Physical and Empirical Parameters section provides textboxes and sliders for adjusting thirteen physical parameters and three empirical constants. The Chemical Parameters section accepts atomic radii, masses, and fractional occupancies for host and dopant species, used to compute mass-fluctuation and strain-field contributions. The Fitting Parameters section allows users to deactivate irrelevant scattering mechanisms via the checkboxes (dims all associated parameters), enable tuning of scattering strengths via the sliders, display the chi-square value, and includes an Auto-Fit χ² button that activates differential-evolution optimization. Radio buttons toggle datasets and select κ_min models, and an export button saves computed curves.

The theoretical κ_latt line plot is calculated from the implementation of the Debye–Callaway equation (Eq. 3) within the editable temperature range specified in the corresponding textboxes and using the default-set physical, empirical, and fitting parameters displayed in their respective textboxes and slider values. The textboxes allow users to modify all 13 physical parameters and the 3 empirical constants to match values obtained from microstructural analysis. Upon pressing the keyboard “enter” button, κ_latt is promptly recalculated using the updated values, and the theoretical κ_latt plot is immediately refreshed.

Since advanced material and defect characterizations are not universally available, which sometimes forces researchers to treat several physical parameters as fitting variables, the program features mini sliders next to each physical parameter textbox. This gives the user fine-grained control over these parameters. To ensure physically meaningful exploration, the slider limits automatically rescale to 0.1–10 times the current textbox value. Updating the slider updates the textbox, and updating the textbox resets the slider limits accordingly. This dynamic scaling effectively makes the allowable adjustment range unbounded, thereby maximizing flexibility while avoiding artificial restrictions on parameter tuning.

At the same time, to prevent the misuse of this freedom or the selection of physically nonsensical parameter values, an inbuilt comprehensive input-validation system is implemented. Every user entry is checked for text-format validity (to ensure the input is numerical), numerical validity (e.g., no NaN or undefined values), and physical plausibility (e.g., grain size > 0, defect fraction between 0 and 1, densities > 0, N as a positive integer, etc.). If an invalid value is detected, the interface automatically generates an appropriate error message and reverts the textbox to the last valid input. This prevents the solver or sliders from attempting to process unphysical or undefined parameter combinations. A dedicated information/warning/error display panel has also been added at the bottom of the interface, providing green (info), yellow (warning), or red (error) messages for clear user guidance.

The software also includes a dedicated Chemical Parameters section in which users can directly input the element fraction, molar mass (g/mol), and atomic radius (pm) for each atomic species. Host and dopant data are entered using a colon-separated format within the doped-site textbox, and the corresponding undoped-site atomic masses are provided in a separate textbox. Accordingly, the program computes Γ and Γ_s internally using the input chemical data and the strain-field fluctuation parameter in the fitting parameters section. The computed Γ and Γ_s values are displayed dynamically beneath the chemistry panel for user transparency.

The simulator enables users to explicitly minimize the number of active scattering channels during fitting through a dedicated set of activation checkboxes placed next to the associated fitting parameter. These checkboxes allow users to instantly deactivate any mechanism that is not physically justified for the material under investigation, ensuring that the optimization space remains minimal, transparent, and physically meaningful. Unchecking a fitting-parameter checkbox not only deactivates the corresponding relaxation-time expression but also visually dims its associated slider and textbox, as well as any linked physical parameters, substantially improving clarity regarding which mechanisms are active. This is exemplified by the checkboxes for nanoinclusion, vacancy, and dislocation scattering, which are set to inactive by default, as these mechanisms are not universally present across all materials. Unchecking the checkboxes returns an infinitely small relaxation time (10-1000) of the associated defect, thereby rendering its contribution to the combined relaxation time negligible. This functionality enables users to selectively model κ_latt using only the relevant phonon scattering mechanisms of interest.

To fit the theoretical κ_latt to experimental data, the program features a dedicated chi-square (χ²) calculator that quantitatively indicates the goodness of fit relative to the selected experimental dataset. Users can toggle between the pristine and doped datasets using the corresponding radio buttons, and the χ² value updates in real time to reflect the current model configuration. In addition, a dedicated “Auto-Fit χ²” button is provided, which initiates an automated χ² minimization routine based on a differential-evolution global optimization algorithm. When activated, the algorithm numerically optimizes only the active scattering parameters, ensuring that the search remains physically meaningful. Most relaxation-time expressions in the Debye–Callaway framework exhibit unique functional forms, which constrain the optimizer and naturally guide parameter tuning. The optimization proceeds until the χ² value falls below a predefined threshold of 1.4, at which point the process terminates, and the fitting parameter values κ_latt plot are updated accordingly. This feature provides a robust starting point from which researchers may further refine the fit using the sliders.

The intuitive sliders are provided to independently fine-tune all the scattering fitting parameters to predetermined or best-fit values, if the users desire to refine the fitting further after χ² auto-optimization fitting. Adjusting these sliders instantly updates the calculated κ_latt plot, providing a clear and dynamic visualization of how each parameter influences κ_latt and its temperature dependence. All sliders, except for s² and A, are implemented in logarithmic scale to facilitate efficient adjustment across a broad range, while the real value (slider output) is displayed in scientific notation beside each slider for quick reference. On the left edge of the sliders, independent editable manual input textboxes are provided, allowing users to directly input any desired fitting parameter value. Upon pressing enter after changing any slider’s manual input value, the program not only updates the corresponding fitting parameter and theoretical κ_latt, but also automatically shifts the minimum and maximum slider limits by three orders of magnitude below and above the new value for logarithmic sliders, and by three units for the s² and A sliders.

More interestingly, the y- and x-axes dynamically adjust in response to variations in either the experimental or theoretical κ_latt and temperature range. These features enable the program to surpass the predefined slider limits and ensure optimal display of both extremely low and high temperature and κ_latt values. This enhances the model’s applicability across a broad spectrum of thermoelectric materials and establishes it as a versatile and powerful tool for materials research and optimization.

It is important to recognize that the lattice thermal conductivity cannot be reduced to zero. Even under extremely strong scattering conditions, a finite amount of heat is still transported through the diffusive motion of vibrational modes. This concept originates from the Cahill–Pohl model, which assumes that, at sufficiently high disorder, phonons lose their coherent wave character and instead propagate through a localized hopping mechanism analogous to heat transport in amorphous solids. This behavior defines a fundamental lower bound to lattice thermal conductivity, κ_min. In this framework, the Cahill model yields the so-called glass-limit conductivity, κ_glass, given by Eq. 27, where n_atom denotes the number density of atoms per unit cell⁷³.

$${\kappa }_{{glass}}\approx \frac{1.21}{3}{k}_{B}{{n}_{{atom}}}^{2/3}(2{v}_{T}+{v}_{L})$$

(27)

More recent theoretical developments extend this idea by incorporating the contribution of diffusons, which are non-propagating, strongly scattered vibrational modes, providing a more comprehensive description of thermal transport in complex and highly disordered systems. The corresponding diffuson-based minimum conductivity, κ_diff, expressed in Eq. 28, generally predicts lower values than the glass limit and aligns more closely with some experimental observations in such regimes⁷⁴.

$${\kappa }_{{diff}}\approx \frac{0.76}{3}{k}_{B}{{n}_{{atom}}}^{2/3}(2{v}_{T}+{v}_{L})$$

(28)

In recognition of the importance of these lower bounds, the software includes an internal estimation of κ_min. Users can select between the glass-limit model and the diffusion-based model via dedicated radio buttons (κ_glass and κ_diff). The κ_min value updates immediately when toggled between κ_glass and κ_diff or when its underlying parameters are modified. Importantly, the simulated theoretical κ_latt value automatically switches to κ_min whenever κ_latt would otherwise fall below this limit, thereby enforcing the physically meaningful lower bound dictated by the selected κ_min model. This ensures that the interface reflects modern theoretical understanding of phonon, diffuson, and coherent vibrational transport in strongly scattered systems.

Finally, dedicated export buttons allow users to seamlessly export the theoretical κ_latt plot into Excel spreadsheets, facilitating quick and efficient data retrieval with a single click. This feature significantly enhances usability by streamlining data analysis and enabling smooth integration into broader research workflows. The underlying workflow of the program is illustrated in Fig. 2, while the standalone executable file, licensed under the Korean Copyright Commission, is openly accessible at https://doi.org/10.5281/zenodo.17662313. To demonstrate the widespread applicability of this program in fitting the theoretical to experimental κ_latt, experimental κ_latt, and microstructural parameters from three prototypical systems: SnTe, GeTe, and NbFeSb, are used.

Fig. 2: Workflow diagram of the developed Debye-Callaway simulation program.

This diagram summarizes the operational flow of the Debye–Callaway simulator. User inputs, physical parameters, chemical information, empirical constants, and experimental data enter the computational engine, where relaxation times for activated phonon-scattering mechanisms are calculated. The simulator applies Matthiessen’s rule and the Debye–Callaway equation to generate the theoretical lattice thermal conductivity. A real-time feedback loop updates results whenever sliders, textboxes, or activation checkboxes are modified until the best experimental-theoretical fit is achieved. An optional differential-evolution routine performs automated chi-square fitting. The workflow ends with a graphical output of theoretical and experimental data and an export option for saving computed results.

Case study 1: GeTe materials

To validate the effectiveness of the developed software in advancing theoretical κ_latt modelling for GeTe-based TE materials, experimental data and microstructural parameters of pristine GeTe and doped Ge_0.85Pb_0.05Bi_0.06Ga_0.04Te samples were extracted from Parashchuk et al.'s work⁴⁸. The multicomponent doping introduced solute atoms that altered the local bonding characteristics and annihilated intrinsic vacancies, as indicated by the decrease in sound velocity and reduction in n. Additionally, amorphous Ge precipitates were observed, although their characteristics were not quantitatively analyzed. Nevertheless, a substantial drop in κ_latt was observed, from ~3.06 to 0.87 W/m K at room temperature, and from ~0.65 to 0.51 W/m K at 600 K, representing an average reduction of about 53% across the measured temperature range. While this represents a significant advancement, a deeper understanding of the individual contributions from each phonon scattering mechanism, including introduced solutes, annihilated vacancies, induced U-processes, and nanoprecipitates, would provide further critical insight for the rational design of GeTe-based TE materials.

To address this research gap, the Debye–Callaway model simulator developed in this study was employed. The experimental κ_latt data and corresponding microstructural parameters, as listed in Table 2, were input into the simulator, as demonstrated in the accompanying screengrab video (Video S1). The parameters Γ and f_vac were calculated using Eqs. 14 and 21, respectively, where ε = 1 was adopted from fitting. To isolate the contributions of specific scattering mechanisms, physical parameters associated with absent mechanisms were set to zero, effectively omitting their corresponding relaxation times from τ_total. Theoretical κ_latt was then fitted to the pristine sample’s experimental data by iteratively adjusting the fitting parameters using the slider interface until optimal alignment was achieved. The best-fit values of these parameters are also reported in Table 2. A similar fitting procedure was subsequently applied to the doped sample. To deconvolute the contributions of dominant scattering mechanisms, the best-fit theoretical κ_latt curve for the pristine sample was incrementally adjusted by independently reducing the values of v_s, C_eph, f_vac, Γ, and U_m to those obtained for the doped sample, and the resulting changes in κ_latt were exported.

Table 2 Microstructural and fitting parameters used for best-fit modelling of experimental and theoretical κ_latt for GeTe (pristine) and Ge_0.85Pd_0.05Bi_0.06Ga_0.04Te (doped) samples, Sn_0.9Ge_0.1Te (pristine) and Sn_0.67Ge_0.2Bi_0.03Sb_0.1Te (doped) samples, and Nb_0.95Ti_0.05FeSb sintered at 1273 K (pristine) and 1123 K (nanostructured), as well as for the Ti-rich Nb_0.8Ti_0.2FeSb (doped) samples

The corresponding theoretical curves are presented in Fig. 3. The results revealed that changes in sound velocity and n had a negligible influence on κ_latt, as indicated by their curves closely overlapping the pristine data. This suggests minimal electron-phonon interactions and insignificant modifications in phonon group velocity. In contrast, removal of intrinsic Ge vacancies had a profound effect, eliminating a key phonon scattering mechanism. This was reflected by a marked increase in κ_latt, approximately 70% at room temperature and 41% on average, when an order-of-magnitude reduction in f_vac was simulated. Fortunately, the phonon scattering induced by the substituted atoms overcompensated for the loss in vacancy scattering, yielding a 35% reduction in κ_latt relative to the pristine case, and a notable 62% reduction compared to the simulated scenario with removed vacancies alone. Further tuning of the U_m resulted in an additional κ_latt reduction, culminating in a total decrease of 56% compared to the experimental pristine sample and 33% when vacancy absence was considered. However, the theoretical fit could not fully replicate the experimental κ_latt trend of the doped sample, particularly at low temperatures, possibly implying a significant contribution from the stated Ge inclusion scattering. This contribution could not be captured due to the lack of key microstructural inputs, such as volume fraction. Nevertheless, the quantified contributions of the modeled scattering mechanisms provide valuable insights into the interplay between engineered defects and phonon transport, offering a pathway for further refinement of thermoelectric materials through defect-driven thermal conductivity tuning. Furthermore, the fitted defect scattering strengths, such as s² for Ge vacancies, establish a foundational basis for building a comprehensive database of defect scattering strengths. This process is greatly facilitated by the simulator, enabling efficient and systematic evaluation of a wide range of defect types.

Fig. 3: Experimental and simulated κ_latt data for GeTe (pristine) and Ge_0.85Pd_0.05Bi_0.06Ga_0.04Te (doped) samples.

This figure compares the experimental and simulated lattice thermal conductivity for GeTe and Ge_0.85Pb_0.05Bi_0.06Ga_0.04Te samples, demonstrating the ability of the Debye–Callaway simulator to deconvolute individual phonon scattering mechanisms. Experimental data for the pristine sample are plotted as black filled rectangles, while those for the doped sample are plotted as red filled circles. The best-fit theoretical curves for each sample are shown as solid lines, with the pristine curve in black and the doped curve in orange. Additional simulated curves illustrate the independent effects of modifying specific parameters that change from the pristine to the doped state. These include separate curves corresponding to changes in sound velocity(v_s), electron–phonon coupling (C_eph), vacancy concentration (f_vac), point-defect scattering strength (Γ), and the Umklapp scattering constant (U_m). Each of these curves is shown in a distinct color to indicate its isolated effect on lattice thermal conductivity. Analysis of these theoretical data plots leads to the quantified contributions of individual defects to the overall lattice thermal conductivity drop.

Case study 2: SnTe TE materials

For SnTe, data for Sn_0.9Ge_0.1Te and Sn_0.67Ge_0.2Bi_0.03Sb_0.1Te from the work of Kihoi et al. were employed. ⁹ In this study, the introduction of Sb and Bi into the Ge-doped SnTe system led to the annihilation of vacancies and the formation of point defects as well as amorphous Ge precipitates within the quinary alloy. These modifications resulted in notable enhancements in thermoelectric performance, driven in part by a significant reduction in κ_latt. Leveraging the intuitive capabilities of the developed Debye–Callaway model simulator, this work provides further insight into the individual contributions of the various mechanisms responsible for the observed κ_latt suppression.

The Γ for Sn_0.9Ge_0.1Te was initially calculated using Eq. 13 with ε = 83, as adopted in the study by Kihoi et al., yielding a value of Γ = 7.17 × 10⁻². Surprisingly, at this value, the theoretical κ_latt consistently fell below the experimental data, even after excluding contributions from vacancies and electrons, as shown in Fig. 4a, and minimizing the U_m to a point where further reduction no longer influenced the theoretical curve (Video S2). This indicates that ε = 83 significantly overestimates Γ_s, thereby leading to an exaggerated theoretical κ_latt. This interpretation is supported by the best-fit Γ value of 9.7 × 10⁻³, obtained by fine-tuning Γ via the program’s slider interface. This corresponds to ε = 4, which is substantially lower than both the value adopted by Kihoi et al. and those reported in previous studies based on Eq. 14⁴⁶. This suggests a potential systematic bias in earlier analyses that may have led to the overestimation of Γ_s, which accounts for nearly a third of Γ (Table 2) in SnTe, underscoring the importance of empirical validation in modelling ε. It also indicates that calculating ε using Eq. 14 may not be suitable for accurately estimating Γ. Similarly, the literature-reported value of s² = 0.89 resulted in a considerably lower κ_latt, prompting the adoption of a revised value (s² = 0.065) based on the GeTe system, which yielded a more reasonable fit. This adjustment suggests that vacancy scattering strength in bulk materials is substantially weaker, nearly an order of magnitude lower than values derived from fitting κ_latt data in single-crystal samples.

Fig. 4: Experimental and simulated κ_latt data for Sn_0.9Ge_0.1Te (pristine) and Sn_0.67Ge_0.2Bi_0.03Sb_0.1Te (doped) samples.

a A screenshot of the Debye–Callaway simulator showing the theoretical lattice thermal conductivity curve computed using a point-defect scattering strength derived from a strain-field parameter of 83. The theoretical curve, plotted as a solid blue line, falls significantly below the experimental data (green and red circles), even after minimizing other scattering contributions, revealing that the originally adopted strain-field parameter strongly overestimates the mass-fluctuation and strain-field scattering terms. This is further demonstrated in Video S2. b shows the experimental and theoretical lattice thermal conductivity fitting and defect contribution deconvolution results. Best-fit theoretical curves are plotted as solid lines in black (overlapped by the red line) for the pristine sample and cyan for the doped sample. Additional dashed curves represent isolated changes to carrier (C_eph) and vacancy concentration (f_vac), point-defect scattering (Γ), and inclusions (inc), clarifying their individual influence. An orange dashed horizontal line indicates the glass-limit thermal conductivity by the Cahill model, while an overlaid pink dashed curve shows the simulated lattice thermal conductivity when the grain size (L) is reduced to 400 nm. The deconvoluted percentage contributions of each defect mechanism are also listed.

The parameters and experimental κ_latt data from Kihoi et al.’s work, as listed in Table 2, were input into the program to fit the theoretical κ_latt to the experimental data, following the procedure described in the previous case study. The slider values corresponding to the best fit, along with the relevant microstructural parameters for both Sn_0.9Ge_0.1Te and Sn_0.67Ge_0.2Bi_0.03Sb_0.1Te, are also presented in Table 2, while the fitted curves are shown in Fig. 4b. The negligible deviation between the best-fit and C_eph-adjusted plots suggests weak electron–phonon interaction. In contrast, the strong influence of the eliminated vacancies is evident from the pronounced κ_latt increase, averaging ~42% across the analyzed temperature range, when f_vac was reduced. Interestingly, the point defects that replace the annihilated vacancies appear to only serve in compensating the loss in phonon scattering strength from the vacancies, as only a 6% average κ_latt decrease is observed when Γ is adjusted to the value of the doped sample. This reduction, however, translates to ~47% when the effect of the removed vacancies is considered with the remaining κ_latt reduction tracked to Ge nanoinclusions, captured by a best-fit nanoinclusion scattering strength (A_nano) of 1.03 × 10⁻⁴⁸. This means that much of the observed κ_latt reduction (94%) came from nanoinclusions scattering as the introduced point defects compensated for the impact of the eliminated vacancies. Furthermore, the program’s predictive capability is highlighted by simulating a reduction in grain size from 5.4 µm to 400 nm, resulting in a further κ_latt decrease that approaches the phonon-glass limit (dashed line) typical of SnTe systems (~0.4 W/mK)⁷⁵. These findings demonstrate not only the simulator’s utility in estimating individual defect scattering strengths but also its power to predict κ_latt reductions and identify systematic modelling errors, serving as a valuable tool for guiding future defect-engineering strategies aimed at zT enhancement.

Case study 3: NbFeSb half-Heusler TE alloys

For half-Heusler thermoelectric materials, Ti-doped NbFeSb data from Villoro et al.‘s work were utilized. ¹⁴ Following the procedures described in the previous sections, theoretical κ_latt plots were fitted to Nb_0.95Ti_0.05FeSb samples sintered at 1273 K (pristine) and 1123 K (nanostructured), as shown in Fig. 5a. The corresponding microstructural and scattering parameters are listed in Table 2. The best-fit Γ value for the pristine sample was 2.51 × 10⁻³, corresponding to ε = 10 when calculated using Eq. 13. By varying the grain size only, from 1.32 µm in the sample sintered at 1273 K–0.2 µm in the same composition sintered at 1123 K, the theoretical κ_latt closely tracked the experimental κ_latt of the nanostructured sample as shown in Fig. 5a. This emphasizes the critical role of grain boundary scattering in κ_latt reduction and affirms the validity of this theoretical modelling.

Fig. 5: Experimental and modelled lattice thermoconductivity of NbFeSb-based case study samples.

a Experimental and theoretical κ_latt for Nb_0.95Ti_0.05FeSb sintered at 1273 K (pristine) and 1123 K (nanostructured). Experimental data are shown as black filled blocks for the pristine sample and red filled circles for the nanostructured sample. Corresponding best-fit theoretical curves are plotted as solid black and cyan lines. The reasonable fit between experimental and theoretic plots when grain size is reduced from 1.2 to 0.2 µm demonstrates that grain boundary scattering substantially accounts for the thermal conductivity reduction in the nanostructured sample. b Experimental and theoretical κ_latt data for Nb_0.95Ti_0.05FeSb (pristine) and Nb_0.8Ti_0.2FeSb (doped). The solid lines show the deconvoluted contributions of grain boundary (red), electron-phonon (green), the point-defect (blue), and the Umklapp contribution (red). The height and temperature dependence of each curve quantify its respective impact on total lattice thermal conductivity.

With higher Ti concentration in the Nb_0.8Ti_0.2FeSb sample, additional point defects were introduced, the grain size decreased significantly, and the n rose substantially compared to the Nb_0.95Ti_0.05FeSb sample sintered at the same temperature (1273 K). Thus, the observed κ_latt reduction arises from the combined influence of these three defect mechanisms. The microstructural and fitting parameters used to achieve the best-fit theoretical and experimental κ_latt curves for both pristine (Nb_0.95Ti_0.05FeSb) and doped (Nb_0.8Ti_0.2FeSb) samples are listed in Table 2. The deconvoluted contributions of the defects are presented in Fig. 5b. Point defect scattering clearly dominates, accounting for ~57% of the total reduction, followed by Umklapp processes (~24%), grain boundary scattering (~19%), and electron-phonon scattering (~1%). This finding offers critical insight into the defect-specific contribution to the observed κ_latt reduction. Furthermore, the cumulative data on defect scattering strengths (i.e., the fitting parameters) serve as valuable references for future research. They provide foundational insights for progressive κ_latt reduction via targeted microstructural engineering and, more importantly, foster a deeper and accelerated understanding of how individual defects modulate phonon transport in thermoelectric materials.

Discussion

This study presents a comprehensive review of the Debye–Callaway model, incorporating nine phonon scattering relaxation mechanisms associated with dominant defect types found in thermoelectric materials. The review outlines the corresponding physical and fitting parameters, offering a valuable quick-reference resource for researchers aiming to enhance TE performance through targeted defect engineering. The review shows that modelling lattice thermal conductivity using the nine phonon-scattering mechanisms requires more than 23 independent input parameters, as summarized in Table 1. The sheer volume of these inputs, coupled with the mathematical complexity of solving the nonlinear integral form of the Debye–Callaway equation, necessitates use of programming capabilities to implement. Additionally, the manual optimization of multiple fitting parameters to achieve the best fit with experimental data imposes substantial cognitive effort, often making the process laborious and prone to error. This barrier has significantly constrained the model’s broader adoption and its ability to provide a comprehensive and quantitative assessment of the individual contributions of each scattering mechanism in materials featuring a diverse range of coexisting defects. This challenge, however, is evidently suppressed significantly with the introduction of a user-friendly computational tool and the integration of an intuitive interface incorporated in the software developed in this work.

The developed software is a novel, interactive Debye–Callaway model simulator, which is an intuitive, standalone program that simplifies the traditionally complex implementation of the model. This is the first user-friendly tool developed specifically for systematically deconvoluting individual phonon scattering contributions. The program features editable textboxes for inputting experimental κ_latt data, microstructural physical and chemical parameters, and empirical constants, enabling real-time calculation and fitting of theoretical κ_latt. Intuitive adjustments of fitting parameters through sliders with editable limits, along with dynamic x- and y-axis scaling, ensure optimal visualization of κ_latt trends over wide temperature ranges and defect densities. These features make the simulator a powerful and flexible platform for studying a wide variety of thermoelectric materials. By replacing the traditional mathematical complexity with a dynamic slider-based interface, the simulator allows real-time fitting of theoretical predictions to experimental data, thereby significantly expanding the model’s accessibility and usability within the research community. Finally, the simulator’s capability to model κ_latt, deconvolute the effects of individual phonon-scattering mechanisms, identify systematic errors around κ_latt modelling, and predict the impact of specific defect concentrations on thermal conductivity has been demonstrated through three representative case studies.

This program represents a significant advancement over the previous AICON Python program implementation⁷⁶, which only considered Umklapp processes, normal processes, and isotopic scattering and lacked the ability to dynamically vary parameters. The versatile features of this program effectively bridge the longstanding gap, proliferating the accessibility of the Debye–Callaway model and making it practical for researchers focused on optimizing TE materials through κ_latt reduction. Furthermore, as demonstrated in the case studies presented in the Results section, the program enables detailed deconvolution of the individual contributions of each phonon scattering mechanism and can also be utilized to predict the achievable κ_latt reductions resulting from the introduction of predetermined quantities of specific defects into a material. Its application, in tandem with recently developed theoretical frameworks for first-principles lattice dynamics⁷⁷, is expected to significantly accelerate the discovery and optimization of functional materials for advanced thermal management applications. It holds strong potential to accelerate defect-driven thermal conductivity engineering and facilitate more systematic, data-informed optimization of thermoelectric materials.

Methods

Program architecture and implementation

The Debye–Callaway model simulator created in this work is a stand-alone Python application intended for lattice thermal conductivity modelling and fitting in real time. The program makes use of common scientific computing libraries, such as Matplotlib to generate the interactive graphical user interface, NumPy for numerical operations, and SciPy for integral evaluation and global optimization. The Debye–Callaway equation, distinct phonon relaxation-time expressions, user interface components, and optimization procedures are all implemented as separate functional blocks in the modular program architecture.

Graphical interface and optimisation framework

The Matplotlib widget framework, which offers interactive slider bars, editable textboxes, check buttons, and radio buttons, was used to build the graphical user interface. These interface components enable users to dynamically modify fitting parameters, empirical constants, and physical parameters; all modifications instantly cause the Debye-Callaway equation to recalculate κ_latt. Callback functions provide real-time updates by automatically updating the plotted experimental and theoretical κ_latt curves in response to changes in parameters or user input. The Differential Evolution global optimization algorithm included in the SciPy optimize module was used to implement the “Auto-Fit χ²” function.

Deconvolution of scattering mechanism contributions

The individual contributions of each phonon-scattering mechanism were quantified using a sequential parameter perturbation analysis, in which the values of defect-related parameters were varied sequentially from the pristine best-fit value set to its doped best-fit value while all other parameters were held constant. The resulting change in lattice thermal conductivity was expressed as a percentage of the total reduction between the pristine and doped samples. Averaging these normalized changes across the entire temperature range yielded the deconvoluted contribution of each defect mechanism to the overall κ_latt suppression.