Phonon-Glass Electron-Crystal Design

• PGEC is a design paradigm that targets low lattice thermal conductivity while maintaining high electronic conductivity to maximize the thermoelectric figure of merit.
• It employs strategies like nanoscale engineering, inclusion of rattler atoms in caged structures, and controlled disorder to disrupt phonon transport.
• Data-driven screening and tailored chemical doping provide actionable guidelines for fine-tuning the κL/κ ratio toward an optimal balance around 0.5.

The phonon-glass electron-crystal (PGEC) paradigm describes the design of materials for thermoelectric applications in which the lattice component of thermal conductivity is suppressed to the level of an amorphous (glassy) state, while the electronic properties remain close to those of a well-ordered crystal. This duality enables the maximization of the thermoelectric figure of merit, $ZT = \frac{S^2\sigma T}{\kappa_L + \kappa_e}$, where $S$ is the Seebeck coefficient, $\sigma$ is electrical conductivity, $T$ is absolute temperature, and the total thermal conductivity $\kappa = \kappa_L + \kappa_e$ combines the lattice (phonon, $\kappa_L$) and electronic ($\kappa_e$) contributions. Recent advances demonstrate that high $ZT$ materials cluster near a lattice-to-total thermal conductivity ratio $L = \kappa_L/\kappa \approx 0.5$, indicating optimal PGEC behavior (Sun et al., 26 Nov 2025). The concept has been operationalized both via nanoscale engineering and through the exploitation of intrinsic structural and bonding motifs, as summarized below.

1. Core PGEC Mechanisms and Descriptor

The PGEC model, following Slack and subsequent refinements, seeks materials where phonons are strongly scattered (yielding low $\kappa_L$) while electrons conduct efficiently (yielding high $S$0) (Takabatake et al., 2014). Detailed data-driven analysis based on a curated dataset of 71,913 thermoelectric measurements establishes that peak $S$1 values concentrate near $S$2—not at the lowest possible $S$3, but where electronic and lattice contributions to heat transport are balanced (Sun et al., 26 Nov 2025). This semi-quantitative target arises because $S$4 strikes the optimal trade-off between suppressing the lattice heat channel and preserving the electronic heat channel, thereby maximizing $S$5. The PGEC descriptor $S$6 thus serves as a practical guide for screening and optimization.

2. Structural Motifs and Rattling: Clathrates, Caged, and Disorder-Driven PGECs

Type-I clathrates and related caged structures epitomize the PGEC principle. These materials consist of a rigid, electronically conductive framework encaging loosely bound guest atoms, termed “rattlers,” which scatter phonons via low-frequency localized modes without perturbing the electronic band edges (Takabatake et al., 2014, Roy et al., 2023, Feng et al., 2020). The glass-like $S$7 emerges from the hybridization of rattler modes with acoustic phonons, formation of avoided crossings (flattened dispersion), and resonant or tunneling scattering. Clathrate Ba$S$8Ga$S$9Ge$\sigma$0 demonstrates how occupational disorder (Ga/Ge exchange) broadens rattler-band energies, proliferates anticrossings, and strongly reduces phonon group velocity, thereby halving $\sigma$1 compared to the ordered phase while leaving electronic conduction largely intact (Roy et al., 2023).

Descriptive metrics such as the site-projected rattling frequency ratio $\sigma$2 and spring-constant ratio $\sigma$3 quantitatively link low-frequency “rattler” dynamics to reduced $\sigma$4. In half-Heusler and similar frameworks, compounds with $\sigma$5 or $\sigma$6 consistently show $\sigma$7 below $\sigma$8 (Feng et al., 2020).

3. Nanoscale Engineering: Inclusion, Porosity, and Defect-Patterned PGECs

A distinct route to PGEC behavior employs controlled nanostructuring in bulk crystalline hosts. Ge:Mn thin films synthesized by molecular beam epitaxy followed by annealing exhibit embedded Ge$\sigma$9Mn$T$0 inclusions, 20–50 nm in diameter, spaced at $T$1–$T$2 nm—sufficient to scatter phonons yet too large to degrade the mobility of charge carriers (whose mean free path is $T$3 nm). This engineering enables a $T$430-fold reduction in $T$5 (from $T$6 to $T$7) while maintaining bulk-like $T$8 and $T$9, yielding $\kappa = \kappa_L + \kappa_e$0 (Tainoff et al., 2013).

In single-crystal silicon, nanoperforation (pore diameter $\kappa = \kappa_L + \kappa_e$1 nm, pitch $\kappa = \kappa_L + \kappa_e$2 nm) or patterned ion-beam–induced defects (cylinder diameter $\kappa = \kappa_L + \kappa_e$3–$\kappa = \kappa_L + \kappa_e$4 nm, spacing $\kappa = \kappa_L + \kappa_e$5–$\kappa = \kappa_L + \kappa_e$6 nm) suppresses lattice thermal conductivity by one or two orders of magnitude, yet electrons with much smaller mean free paths traverse the structure with minimal scattering. This approach yields $\kappa = \kappa_L + \kappa_e$7 enhancements by up to a factor of 18 over bulk Si, with experimental and simulation results indicating that optimal spacing is bounded by $\kappa = \kappa_L + \kappa_e$8 (electron and phonon mean free paths, respectively) (Bah et al., 2022, Zhu et al., 2017).

4. Chemical and Electronic Strategies: Low-Dimensional and Procrystalline PGECs

The decoupling of thermal and electronic transport may also be engineered chemically. In $\kappa = \kappa_L + \kappa_e$9–$\kappa_L$0 stacked molecular crystals, exemplified by bis-dithienothiophene (BDTMC), one-dimensional (1D) conjugated stacking facilitates highly dispersive electronic bands (yielding high $\kappa_L$1 and $\kappa_L$2), while weak van der Waals inter-stack interactions result in “glassy” lattice heat transport ($\kappa_L$3, $\kappa_L$4 at 300 K) (Mi et al., 2015). Similarly, procrystalline solids, i.e., dense packings of orientationally disordered but locally ordered motifs on periodic lattices, manifest strongly correlated topological disorder. This selectively broadens phonon linewidths at specific Brillouin zone wave vectors, producing “waterfall” effects in the phonon spectrum and substantial $\kappa_L$5 suppression, while leaving high-symmetry electronic dispersion largely unaffected (Overy et al., 2015).

5. Machine Learning and Data-Driven PGEC Screening

High-throughput screening of PGEC materials is enabled by machine learning models trained on large datasets of thermoelectric measurements with explicit separation of $\kappa_L$6 and $\kappa_L$7 (Sun et al., 26 Nov 2025). Models (Random Forest, XGBoost, etc.) predict $\kappa_L$8 and $\kappa_L$9 from compositional Magpie features and measurement temperature, achieving mean absolute errors of 0.34 and 0.20~W m$\kappa_e$0 K$\kappa_e$1 for lattice and electronic thermal conductivity, respectively. Experimental validation confirms that the models generalize to out-of-dataset compounds. Coupling with the $\kappa_e$2 descriptor enables workflow optimization: (1) identify low-κ materials; (2) evaluate if $\kappa_e$3 (call for boosting $\kappa_e$4, e.g., via doping) or $\kappa_e$5 (suppress $\kappa_e$6 further, e.g., via disorder or inclusion); (3) propose dopants or alloy schemes to target $\kappa_e$7.

Case studies in materials such as AgBiS$\kappa_e$8, In$\kappa_e$9SnSe$ZT$0, and TbCuTe$ZT$1 illustrate how suggested dopants quantitatively steer $ZT$2 toward the PGEC regime, indicating both routes of $ZT$3 suppression and $ZT$4 enhancement can be optimized iteratively (Sun et al., 26 Nov 2025).

6. Quantitative Models, Structure–Property Guidelines, and Future Directions

First-principles anharmonic lattice dynamics (e.g., Simoncelli–Marzari–Mauri framework) combine particle-like (Peierls) and wave-like (coherent, tunneling) terms in the calculation of total $ZT$5 (Semwal et al., 26 Aug 2025). For materials such as TlAgTe, glass-like lattice heat transport arises via strongly localized “rattler” modes of heavy atoms in oversized cavities, enhanced quartic anharmonicity, and four-phonon Umklapp scattering, as well as coherent tunneling. Structure–property design rules emerging from these studies include: (1) use of rigid, electronically conductive frameworks with loosely bound heavy “rattler” atoms (preferably with stereoactive lone pairs); (2) maximization of inclusion or pore sizes within the phonon wavelength regime but not detrimentally affecting carrier mobility; (3) controlled structural disorder or occupation randomness to flatten phonon dispersion; (4) chemical modulation to induce Q1D electronic bands combined with glassy vibrational networks.

A summary of structural, chemical, and physical design attributes for realizing PGEC behavior is provided below:

Design Attribute	PGEC Function	Example Implementation
Large cages/voids	Rattling, low-frequency modes	Type-I clathrates, TlAgTe, half-Heusler
Nanoinclusions/pores	Phonon scattering, preserve $ZT$6	Ge:Mn thin films, perforated Si
Low-dimensional bands	Enhance $ZT$7 (Q1D/2D)	$ZT$8–$ZT$9 stacked organics
Topological disorder	Selective phonon broadening	Procrystalline/aperiodic frameworks
Doping/alloying	Tune $L = \kappa_L/\kappa \approx 0.5$0 or $L = \kappa_L/\kappa \approx 0.5$1	Halide/metal substitutions in TEs

This table summarizes the mapping of specific design motifs to their functional role within the PGEC paradigm as documented in (Sun et al., 26 Nov 2025, Takabatake et al., 2014, Tainoff et al., 2013, Bah et al., 2022, Roy et al., 2023, Semwal et al., 26 Aug 2025, Feng et al., 2020, Mi et al., 2015, Zhu et al., 2017, Overy et al., 2015).

7. Broader Impacts and Outlook

The PGEC design paradigm has led to a reorientation of thermoelectric materials development from the pursuit of ultralow lattice thermal conductivity alone to the simultaneous and balanced optimization of electronic and phononic channels, as captured quantitatively by the $L = \kappa_L/\kappa \approx 0.5$2 criterion (Sun et al., 26 Nov 2025). Multi-target strategies that decouple and separately optimize $L = \kappa_L/\kappa \approx 0.5$3 and $L = \kappa_L/\kappa \approx 0.5$4 via machine learning, nanoscale engineering, and chemical design rulebooks are now the focus of the field. Future advances are expected from the integration of atomic-scale structural data, high-throughput defect/dopant extraction using LLMs, and direct structure-by-design methods for realizing new PGEC candidates across chemical families. The convergence of computational, data-driven, and experimental approaches has clarified practical, material-agnostic guidelines for pushing $L = \kappa_L/\kappa \approx 0.5$5 toward—and beyond—the intrinsic PGEC regime.