Lattice-to-Total Thermal Conductivity Ratio

• Lattice-to-total thermal conductivity ratio is a measure that quantifies the fraction of heat transport mediated by phonons relative to the total thermal conduction in solids.
• It assists thermoelectric optimization by distinguishing between phonon and electronic contributions, guiding strategies such as doping and alloying to achieve balanced heat transport.
• Advanced methods like DFT-based calculations and machine learning enable precise evaluation of this ratio, improving the design of high-performance thermoelectric materials.

The lattice-to-total thermal conductivity ratio quantifies the fraction of total heat transport in a solid that is mediated by phonons (the lattice) as opposed to electronic charge carriers. This ratio, often written $$R(T) = \kappa_\mathrm{ph}(T)/\kappa_\mathrm{tot}(T)$$ or $L = \kappa_L/\kappa$, emerges as a critical descriptor in thermoelectric optimization, thermal metamaterials, and the interpretation of heat flow in complex materials. Its accurate evaluation requires separate determination of the lattice thermal conductivity ($\kappa_\mathrm{ph} = \kappa_L$) and electronic thermal conductivity ($\kappa_\mathrm{el} = \kappa_e$), followed by normalization with respect to the total thermal conductivity $$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$. This ratio provides direct insight into the phonon-glass electron-crystal paradigm vital for high-performance thermoelectric materials (Sun et al., 26 Nov 2025).

1. Definitions and Formalism

Total thermal conductivity can be decomposed as

$$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$

where $\kappa_\mathrm{ph}(T)$ is the lattice (phononic) component, and $\kappa_\mathrm{el}(T)$ is the electronic (or carrier) component (Chen et al., 2012, Ding et al., 2015, Sun et al., 26 Nov 2025). The lattice-to-total ratio is then

$$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$

and, in the notation of large-scale materials informatics, $L \equiv \kappa_\mathrm{L}/\kappa$ (Sun et al., 26 Nov 2025). In the context of phonon-glass electron-crystal (PGEC) design, $L = \kappa_L/\kappa$0 signals phonon-dominated (electron-poor) conduction, while $L = \kappa_L/\kappa$1 marks electron-dominated transport.

2. Extraction and Computational Approaches

Experimental and First-Principles Determination

• Electronic Component:

Often obtained via the Wiedemann–Franz law, $L = \kappa_L/\kappa$2, with Lorenz number $L = \kappa_L/\kappa$3, but this can deviate sharply, particularly for large Seebeck coefficients $L = \kappa_L/\kappa$4 (Chen et al., 2012). Corrections include $L = \kappa_L/\kappa$5 and the direct evaluation using DFT-based Boltzmann transport integrals:

$L = \kappa_L/\kappa$6

where $L = \kappa_L/\kappa$7 denotes Onsager moments (Chen et al., 2012).

• Lattice Component:

Extracted by subtracting $L = \kappa_L/\kappa$8 from measured $L = \kappa_L/\kappa$9, or directly via phonon Boltzmann transport calculations (e.g., ShengBTE with first-principles IFCs) (Ding et al., 2015).

Magnetothermal Resistance

In pure metals and semiconductors, magnetothermal suppression allows direct phonon measurement: transverse magnetic fields suppress $\kappa_\mathrm{ph} = \kappa_L$0, and extrapolation of $\kappa_\mathrm{ph} = \kappa_L$1 versus suppressed electrical conductivity gives $\kappa_\mathrm{ph} = \kappa_L$2 (Yao et al., 2017, Yao et al., 2017).

Machine Learning and Data-Driven Prediction

Separate regression models for $\kappa_\mathrm{ph} = \kappa_L$3 and $\kappa_\mathrm{ph} = \kappa_L$4 predict both components, enabling rapid high-throughput screening of $\kappa_\mathrm{ph} = \kappa_L$5 across large compound libraries (Sun et al., 26 Nov 2025). Random Forest regressors trained on 72k+ measurements achieve test MAE $\kappa_\mathrm{ph} = \kappa_L$60.34 W m$\kappa_\mathrm{ph} = \kappa_L$7 K$\kappa_\mathrm{ph} = \kappa_L$8 for $\kappa_\mathrm{ph} = \kappa_L$9 and $\kappa_\mathrm{el} = \kappa_e$00.22 for $\kappa_\mathrm{el} = \kappa_e$1 at 300–800 K. This enables practical optimization strategies targeting $\kappa_\mathrm{el} = \kappa_e$2, the empirical maximum for thermoelectric $\kappa_\mathrm{el} = \kappa_e$3.

3. Representative Numerical Results and Material Examples

Material/System	$\kappa_\mathrm{el} = \kappa_e$4 (K)	$\kappa_\mathrm{el} = \kappa_e$5 (W/m·K)	$\kappa_\mathrm{el} = \kappa_e$6 (W/m·K)	$\kappa_\mathrm{el} = \kappa_e$7 or $\kappa_\mathrm{el} = \kappa_e$8 = $\kappa_\mathrm{el} = \kappa_e$9
Ba$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$0Au$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$1Ge$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$2	300–700	0.95$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$3	1.15–0.70	0.83→0.29 (Chen et al., 2012)
TiNiSn (half-Heusler)	100–1000	16.0–2.6	16.2–3.9	0.99→0.67 (Ding et al., 2015)
Graphene (bulk)	300	2700	3000	0.90 (Kim et al., 2016)
Al single crystal	5–60	10–500	800–1100	0.01 (5K), 0.45 (40K) (Yao et al., 2017)
Bi$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$4Te$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$5 (single crystal)	5–60	30–5.7	36.5–5.8	0.82→0.98 (Yao et al., 2017)

$$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$6Values at 300 K, see text for temperature trends.

Numerical trends:

• Thermoelectrics: High-ZT compounds typically achieve $$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$7 after careful chemical tuning (doping/alloying). For pristine semiconductors, $$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$8 is typically $$\kappa_\mathrm{tot} = \kappa_\mathrm{ph} + \kappa_\mathrm{el}$$9, but maximal $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$0 follows an inverted-U profile versus $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$1 (Sun et al., 26 Nov 2025).
• Half-Heusler TiNiSn: $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$2, with $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$3 falling to 0.67 at 1000 K due to increasing electronic contribution (Ding et al., 2015). $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$4 dominates the total, implying that $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$5 enhancement requires lattice suppression (e.g., alloying).
• Graphene: $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$6 (bulk, room temperature), but in submicron crystallites, phonon boundary scattering reduces $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$7 to $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$80.7–0.8 (Kim et al., 2016).
• Metals (Al, Cu, Zn): At low $$\kappa_\mathrm{tot}(T) = \kappa_\mathrm{ph}(T) + \kappa_\mathrm{el}(T)$$9, electrons overwhelmingly dominate ($\kappa_\mathrm{ph}(T)$0 at $\kappa_\mathrm{ph}(T)$1 K), but $\kappa_\mathrm{ph}(T)$2 peaks near $\kappa_\mathrm{ph}(T)$3, with the highest value for Cu ($\kappa_\mathrm{ph}(T)$4 near $\kappa_\mathrm{ph}(T)$5 K) (Yao et al., 2017).

4. Temperature, Doping, and Microstructural Effects

The ratio $\kappa_\mathrm{ph}(T)$6 displays marked dependence on temperature and microstructure:

• Low $\kappa_\mathrm{ph}(T)$7: Electronic heat conduction is dominant in good metals; $\kappa_\mathrm{ph}(T)$8 as $\kappa_\mathrm{ph}(T)$9 (Yao et al., 2017).
• Intermediate to High $\kappa_\mathrm{el}(T)$0: Phonons gain or maintain dominance in most semiconductors and oxides; however, at extreme $\kappa_\mathrm{el}(T)$1 ($\kappa_\mathrm{el}(T)$2 K), carrier contribution can become comparable or dominant, reducing $\kappa_\mathrm{el}(T)$3 (Chen et al., 2012).
• Doping and Carrier Engineering: Increased carrier concentration raises $\kappa_\mathrm{el}(T)$4 and suppresses $\kappa_\mathrm{el}(T)$5. However, high $\kappa_\mathrm{el}(T)$6 values (thermoelectrics near band edge) reduce the “true” Lorenz number $\kappa_\mathrm{el}(T)$7, so full DFT-based calculation is necessary—naïve Wiedemann–Franz analysis may under- or overestimate $\kappa_\mathrm{el}(T)$8 by up to 40% (Chen et al., 2012).
• Microstructural Effects: In micro/nanoscale samples, boundary or grain-limited phonon scattering strongly suppresses $\kappa_\mathrm{el}(T)$9 and $$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$0, making carrier conduction relatively more important (notably in graphene) (Kim et al., 2016).

5. Role as Descriptor: Screening and Optimization in Thermoelectrics

The lattice-to-total ratio $$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$1 is established as the primary PGEC descriptor for thermoelectric optimization (Sun et al., 26 Nov 2025):

• Screening: High-throughput models identify “ultralow-$$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$2” ($$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$3 W m$$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$4 K$$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$5) materials, but true high-$$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$6 candidates cluster near $$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$7.
• Design Hierarchy:
  • If $$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$8: prioritize raising $$R(T) = \frac{\kappa_\mathrm{ph}(T)}{\kappa_\mathrm{tot}(T)} = 1 - \frac{\kappa_\mathrm{el}(T)}{\kappa_\mathrm{tot}(T)}$$9 (carrier engineering).
  • If $L \equiv \kappa_\mathrm{L}/\kappa$0: prioritize lowering $L \equiv \kappa_\mathrm{L}/\kappa$1 (phonon scattering, alloying).
• Case Studies: Doping strategies (e.g., Cl in AgBiS$L \equiv \kappa_\mathrm{L}/\kappa$2, Br in In$L \equiv \kappa_\mathrm{L}/\kappa$3SnSe$L \equiv \kappa_\mathrm{L}/\kappa$4) can quantitatively shift $L \equiv \kappa_\mathrm{L}/\kappa$5 toward $L \equiv \kappa_\mathrm{L}/\kappa$6, simultaneously lowering $L \equiv \kappa_\mathrm{L}/\kappa$7 and optimizing $L \equiv \kappa_\mathrm{L}/\kappa$8.

6. Lattice-to-Total Ratio in Lattice Metamaterials

In periodic shell-based metamaterials, the asymptotic directional conductivity (ADC) formalism enables precise analytic and numerical control over the lattice-to-total ratio. For $L \equiv \kappa_\mathrm{L}/\kappa$9 (shell thickness), the expansion

$L = \kappa_L/\kappa$00

shows that the ratio approaches unity, with the lattice (geometry-defined) conductivity accounting for nearly all thermal transport as thickness vanishes (Zhang et al., 27 Jun 2025). Maximal $L = \kappa_L/\kappa$01 is achieved for minimal surfaces (e.g., TPMS), with the average asymptotic value attaining $L = \kappa_L/\kappa$02 of the normalized maximum.

7. Experimental and Theoretical Considerations

• Measurement Artifacts: Magnetothermal techniques involve uncertainties (e.g., extrapolation at finite field), with typical errors $L = \kappa_L/\kappa$03\% in $L = \kappa_L/\kappa$04, impacting $L = \kappa_L/\kappa$05 at higher $L = \kappa_L/\kappa$06 (Yao et al., 2017).
• Modeling Limitations: Constant-$L = \kappa_L/\kappa$07 relaxation-time approximations, neglect of impurity, or boundary scattering can cause divergence from experiment at higher carrier concentrations or in nanostructured systems (Chen et al., 2012, Ding et al., 2015).
• Lorenz Number Dependence: Large deviations from $L = \kappa_L/\kappa$08 are routine in high-$L = \kappa_L/\kappa$09 or strongly correlated systems; first-principles or data-driven estimation of $L = \kappa_L/\kappa$10 is necessary for accurate $L = \kappa_L/\kappa$11 assignment (Chen et al., 2012, Sun et al., 26 Nov 2025).

• (Chen et al., 2012) Electronic thermal conductivity as derived by density functional theory
• (Ding et al., 2015) Examining the thermal conductivity of half-Heusler alloy TiNiSn by first-principles calculations
• (Kim et al., 2016) The electronic thermal conductivity of graphene
• (Yao et al., 2017) Experimental determination of phonon thermal conductivity and Lorenz ratio of single crystal metals: Al, Cu and Zn
• (Yao et al., 2017) Experimental determination of phonon thermal conductivity and Lorenz ratio of single crystal bismuth telluride
• (Zhang et al., 27 Jun 2025) Asymptotic analysis and design of shell-based thermal lattice metamaterials
• (Sun et al., 26 Nov 2025) Lattice-to-total thermal conductivity ratio: a phonon-glass electron-crystal descriptor for data-driven thermoelectric design