When Heat Divides in Half: AI Finds a Thermoelectric Golden Ratio

A Kyoto University-led team has found a “50–50” signature in high-performing thermoelectric materials. By mining 71,913 experimental records and screening more than 100,000 compounds, the researchers turned a 30-year-old ideal—conduct heat like glass, electrons like crystal—into a quantitative compass for recovering wasted heat.

KYOTO / OSAKA / EVANSTON — Bradley L. Bartz with AI / Japan.co.jp / July 26, 2026

Stand beside a furnace, an engine block or the warm exhaust of a data center and the energy problem becomes physical. Heat presses against the skin, proof that fuel or electricity has done useful work and then left a remainder. In industry alone, the U.S. Department of Energy estimates that 20 to 50 percent of energy input can depart as hot gases, cooling water or warm equipment. Some of that heat can be reused. Much of it is too diffuse, intermittent or awkwardly located to recover economically.

Thermoelectric materials offer a seductive answer. Put one side of a suitable solid against heat and keep the other side cooler; charge carriers diffuse across the temperature difference and generate a voltage. There is no turbine, piston or refrigerant loop. Reverse the current and the same class of material pumps heat, the basis of compact Peltier coolers. The machine is silent and solid-state. The material, however, must perform an internal contradiction.

It must let electricity travel easily while making heat travel badly. That is difficult because the electrons wanted for electrical conduction also carry heat. The crystal lattice supplies a second heat route through collective atomic vibrations called phonons. Improve one transport property and another often moves in the wrong direction.

A Japan–U.S. team now reports that decades of experiments contain a surprisingly simple marker for navigating that compromise. Yifan Sun and Ken Kurosaki at Kyoto University’s Institute for Integrated Radiation and Nuclear Science, Zhi Li and Chris Wolverton at Northwestern University, and Tetsuya Imamura and Yuji Ohishi at Osaka University analyzed 71,913 curated measurements. High thermoelectric performance clustered when lattice vibrations carried roughly half the total heat and charge carriers carried the other half.

The paper, published online July 15 in Materials Today Physics, calls the lattice share κ_L/κ, where total thermal conductivity κ equals lattice conductivity κ_L plus electronic conductivity κ_e. Near the strongest performance, that ratio approaches 0.5.

The “golden ratio” is not a promise that half of waste heat becomes electricity. It is a sign that two microscopic pathways inside an already poor heat conductor have reached a productive balance.

A golden ratio—with a crucial asterisk

The phrase “50–50” is memorable enough to mislead. The result is not a 50 percent conversion efficiency, not the mathematical golden ratio of approximately 1.618, and not a universal theorem requiring every thermoelectric to divide heat exactly in half. It is an empirical clustering in a large literature dataset: many high-zT measurements lie near κ_L/κ = 0.5, especially after researchers have deliberately optimized a material.

Nor is electronic heat conduction desirable by itself. In the standard figure of merit,

zT = S²σT / (κ_L + κ_e)

S is the Seebeck coefficient: voltage produced per unit temperature difference.
σ is electrical conductivity.
T is absolute temperature.
κ_L is heat carried by lattice vibrations; κ_e is heat carried by mobile charges.

A high numerator—the power factor S²σ multiplied by temperature—is good. A low denominator is good. Yet increasing carrier concentration often raises σ and κ_e together while reducing S. Suppressing the lattice contribution is useful, but a nearly insulating material can have low heat conduction and still generate little power. The 0.5 marker appears where a low-κ material has enough mobile carriers to perform electrical work without allowing total heat leakage to run away.

That makes the ratio a direction sign, not a destination by itself. The researchers first screen for low total thermal conductivity—generally 2 watts per meter-kelvin or less in the high-throughput stage—and then ask whether the material’s two heat channels are balanced. Low κ without strong electrical transport is not enough. A 0.5 ratio with a large total κ is not enough either.

71,913Curated experimental entries used for the final analysis and models.

≈ 0.5The lattice-to-total thermal conductivity ratio around which high zT clustered.

104,567Materials Project compositions screened by the two-model framework.

2,522Stable semiconductor candidates predicted to have ultralow thermal conductivity at 300 K.

From a deflected needle to a solid-state generator

The story began in 1821, when Baltic German physicist Thomas Johann Seebeck heated one junction in a circuit made from unlike conductors and saw a nearby magnetic needle move. Seebeck interpreted the phenomenon through magnetism; later work established that a temperature difference had produced an electromotive force and current. His name remains attached to the effect that lets a thermocouple measure temperature and a thermoelectric generator make power.

In 1834, French watchmaker and physicist Jean Charles Athanase Peltier found the reciprocal phenomenon: current passing through a junction could absorb heat on one side and release it on the other. William Thomson—later Lord Kelvin—showed in the 1850s that the Seebeck and Peltier effects belong to one thermodynamic family and predicted the third effect that bears his name, heat absorbed or evolved when current flows through a homogeneous conductor with a temperature gradient.

Those discoveries did not immediately produce efficient machines. Metals conduct electricity well, but they usually conduct heat well too and have modest Seebeck coefficients. The semiconductor era changed the possibilities. In 1954, H. Julian Goldsmid and R. W. Douglas demonstrated practical thermoelectric refrigeration with bismuth telluride, still a standard near room temperature. Abram Ioffe’s mid-century theory organized electrical conductivity, thermopower and thermal conductivity into the figure of merit that still judges the field.

Year	Milestone	What it added
1821	Seebeck effect	A temperature difference across unlike conductors produces voltage.
1834	Peltier effect	Electric current can pump heat across a junction.
1850s	Thomson relations	Thermodynamics links the reciprocal effects and predicts heat exchange in one conductor.
1954–57	Goldsmid–Douglas and Ioffe	Bismuth-telluride cooling and semiconductor theory launch modern thermoelectrics.
1993–95	Low-dimensional design and PGEC	Dresselhaus and Hicks propose quantum confinement; Slack frames the phonon-glass electron-crystal ideal.
2026	Data-driven 0.5 descriptor	Nearly 72,000 measurements make the PGEC balance quantitatively searchable.

The glass-and-crystal dream

In 1995, American physicist Glen Slack gave the field one of its most durable design metaphors: the “phonon-glass electron-crystal,” or PGEC. An ideal thermoelectric would disrupt lattice vibrations as thoroughly as a disordered glass while preserving the long-range electronic transport of a good crystal. Phonons would stumble; electrons would run.

Researchers pursued that ideal through complex structures. Skutterudites such as CoSb₃ contain cages that can be filled with loosely bound heavy atoms. Those “rattlers” scatter phonons while doping adjusts the carrier population. Clathrates place guest atoms inside framework cages. Alloying introduces mass disorder; nanostructures add interfaces that scatter heat-carrying vibrations across different wavelengths. Band engineering, resonant states and quantum confinement reshape electronic transport.

Hicks and Mildred Dresselhaus’s 1993 quantum-well analysis helped reopen a field whose best bulk figure of merit had improved slowly for decades. The subsequent nanostructuring era produced striking laboratory values, but also a lesson: a record zT at one temperature is not a generator. Contacts add electrical and thermal resistance. A module needs matched p- and n-type legs, mechanical strength, chemical stability, durable interfaces and good performance across the full temperature gradient. Scarce or toxic constituents may make a beautiful crystal commercially unattractive.

For three decades PGEC guided intuition, but it lacked one simple experimental gauge. How glass-like is the phonon channel relative to the electronic channel? How close is a doped sample to the ideal? The new work proposes that κ_L/κ answers those questions well enough to steer the next experiment.

The database made from old graphs

The discovery was possible because thousands of published figures became machine-readable. Starrydata, developed in Japan by Yukari Katsura and collaborators, is an open system in which curators digitize experimental curves from the materials literature and connect them to compositions, samples and publication metadata. Its name comes from the night sky: bright and dim stars together reveal the Milky Way’s structure; celebrated records and ordinary measurements together reveal a field.

The team began with the July 1, 2025 snapshot—202,178 curves. It retained samples reporting a complete set of Seebeck coefficient, electrical conductivity, thermal conductivity and zT. Invalid compositions, duplicate curves and physically impossible values were removed. Reported zT values above 3 were excluded because they were too rare and too vulnerable to input error to train the model reliably.

Digitized curves do not always place every property at exactly the same temperature. The researchers therefore aligned values within ±5 kelvin, recalculated zT from the component properties and rejected rows that differed from the authors’ reported value by more than 10 percent. After further restricting the practical window to 300–800 K and total thermal conductivity to at most 10 W m⁻¹ K⁻¹, 71,913 entries remained.

The electronic share was not measured independently in every paper. The team estimated κ_e from electrical conductivity using the Wiedemann–Franz relation, with a Lorenz number estimated from the Seebeck coefficient rather than held constant. In 27,140 entries that included a reported electronic thermal conductivity, the calculation had a mean absolute difference of 0.09 W m⁻¹ K⁻¹. The approximation is useful, not perfect: the authors estimate errors around 5 percent for single-parabolic-band semiconductors and as much as 20 percent for more complex systems.

An inverted U hidden in 71,913 points

When the data were plotted against the lattice fraction, performance traced an inverted U. Most ordinary thermoelectric records sat near 0.9 to 1.0, meaning that lattice vibration dominated heat flow. The highest zT values gathered nearer 0.5, especially among optimized rather than pristine samples. As total thermal conductivity fell, the high-performance region tightened toward the even split.

Two families make the pattern tangible because they approach it from opposite sides. Pristine CoSb₃ is a low-carrier semiconductor whose lattice carries too much heat. In one historical example, filling its cages with ytterbium increased electrical conductivity from 208 to 1,024 S cm⁻¹ at 623 K and cut lattice conductivity from 4.60 to 2.00 W m⁻¹ K⁻¹. Its zT rose from 0.1 to 0.9, while the lattice fraction moved from 0.95 to 0.65—toward 0.5.

Germanium telluride begins with the opposite problem. Native germanium vacancies make pristine GeTe heavily p-type: electronic transport is abundant, but carrier concentration is excessive and the Seebeck coefficient suffers. Antimony doping reduces the holes. In the cited example at 700 K, the lattice fraction rose from 0.26 to 0.44 and zT rose from 0.94 to 1.68. CoSb₃ moves down toward the line; GeTe moves up. Their convergence is stronger evidence than either trajectory alone.

Still, the inverted U is a pattern in accumulated experiments, not a controlled proof of causation. Literature favors successful compositions. Optimized materials are reported more often than failed ones. Multiple temperatures and related alloys can reflect the same research programs. The result is best understood as a semi-quantitative descriptor grounded in physical reasoning—not a new law of nature.

Two models, because heat has two carriers

Rather than ask one black box to predict zT, the researchers trained separate models for lattice and electronic thermal conductivity. This matters because the two channels arise from different microscopic processes and because the smaller electronic signal can disappear inside a model dominated by the lattice contribution.

Each composition was represented by Magpie descriptors—statistics of elemental mass, size, electronegativity, melting temperature, valence orbitals and related chemistry—plus measurement temperature. After removing redundant features, 108 remained. The dataset was split 80–20 with all records sharing a reduced chemical formula kept on one side, preventing nearly identical compositions from leaking into training and testing. Four tree-based algorithms were compared with grouped five-fold cross-validation. Random forests gave the lowest mean absolute errors: 0.35 W m⁻¹ K⁻¹ for the lattice channel and 0.23 for the electronic channel, with cross-validated R² values of 0.79 and 0.73.

Interpretability analysis found chemically sensible signals. High melting temperatures, proxies for strong bonds, tended to increase both heat channels. Late d-block elements such as copper and silver, heavy p-block elements, softer bonding and mass contrast helped frustrate phonons. More metallic, low-electronegativity-contrast environments favored delocalized carriers. The useful recipe is not simply “weaken every bond.” It is to decouple bond softness from electron mobility.

From 104,567 formulas to 2,522 leads

The trained models were applied to 104,567 inorganic compositions from the Materials Project that were absent from training and testing. The screen narrowed first to 14,830 binary and ternary compounds with a finite band gap and calculated energy within 0.1 electron volt per atom of the thermodynamic stability hull. At 300 K, 2,522 were predicted to have total thermal conductivity of 2 W m⁻¹ K⁻¹ or less.

Those 2,522 are not 2,522 new high-efficiency thermoelectrics. They are low-heat-conducting, computationally plausible semiconductors worthy of deeper work. Five reported compounds hidden from the training set—AgBiS₂, CuSbS₂, In₄SnSe₄, Sr₃AlSb₃ and TbCuTe₂—showed reasonably good agreement between predicted and published temperature trends. That is a validation of triage, not laboratory confirmation of the full list.

The framework also attempted the harder task of “polishing” a candidate. AgBiS₂ already has ultralow total thermal conductivity, but its lattice fraction exceeds 0.8. The models suggest electron-donating halogens on the sulfur site. Chlorine follows a known experimental trend; bromine and iodine are predicted to combine carrier enhancement with greater lattice suppression. Every proposed substitution still needs synthesis, phase analysis and transport measurement.

Where the model cannot see

The model knows composition and temperature, not the complete crystal. It does not know with certainty which crystallographic site a dopant occupies, whether a secondary phase formed, how grains and defects were produced, or whether a nominal formula describes the actual sample. For electronic transport, those omissions are fundamental. A dopant can donate on one site and behave differently on another.

There are other boundaries. The Wiedemann–Franz separation can break down in multiband, non-degenerate, phase-changing or otherwise complex materials. Measurement uncertainty in electrical and thermal transport is often substantial. Materials Project stability and band gaps are calculated approximations. A low thermal conductivity at 300 K says little about oxidation, thermal cycling, contact resistance, scalable synthesis or performance at an application’s operating temperature.

The result means	It does not mean
High-zT literature data cluster near κ_L/κ ≈ 0.5.	Exactly half the input heat becomes electricity.
The ratio can guide optimization after low κ is found.	Any material at 0.5 will be efficient.
2,522 candidates merit simulation and experiment.	2,522 new devices have been discovered.
Composition-only AI provides fast, interpretable triage.	The model replaces crystal structure, defect physics or synthesis.
The pattern strengthens the PGEC concept.	Correlation in the literature establishes a universal physical law.

Waste heat is not a single resource

Thermoelectrics are most valuable where reliability, compactness and the absence of moving parts matter more than peak efficiency. Radioisotope thermoelectric generators have powered spacecraft from the Viking landers to the Voyager probes; NASA’s Voyagers launched in 1977 with three units each and still draw diminishing power from plutonium decay. Terrestrial opportunities include remote sensors, exhaust streams, furnaces, pipelines, wearable electronics and localized cooling.

But “waste heat” spans a hierarchy. A red-hot exhaust has more ability to do work than lukewarm cooling water. Carnot’s limit depends on the temperature difference between hot and cold sides, and real systems fall below it. Capturing heat may add pressure drop, pumps, heat exchangers, mass and cost. A laboratory material with excellent peak zT may fail to deliver useful module power if the thermal contact is poor or the temperature gradient collapses.

That practical realism makes faster materials triage valuable. The new ratio does not repeal thermodynamics. It can reduce the number of dead ends before expensive synthesis, helping researchers match materials to temperature windows and then optimize carrier density and phonon scattering with a measurable goal.

The next loop: find, make, measure, learn

The clearest next step is structural awareness. Models that know lattice symmetry, atomic sites, defects and processing history should judge dopants more precisely than composition averages. Uncertainty estimates could rank not only predicted performance but also where a new measurement would teach the model most. Automated synthesis and rapid transport characterization could then close the loop: propose, make, measure, update.

The work also points to a quieter scientific infrastructure. Starrydata turned human labor spent tracing old plots into a common experimental memory. The Materials Project supplied a computed universe to search. Machine learning connected the two. None alone discovered a deployable material, but together they transformed published history from a shelf of conclusions into a landscape of trajectories—showing not only what worked, but how materials moved as researchers doped, filled and alloyed them.

There is an elegance to the 0.5 result. Thermoelectric design has always been a negotiation between two kinds of order: a lattice disordered enough to impede vibrations and an electronic structure ordered enough to carry charge. Slack gave that negotiation a memorable name. Thirty-one years later, tens of thousands of measurements have drawn a coordinate on the map.

The future generator will not be built from a ratio alone. It will need abundant elements, stable contacts, manufacturable modules and a heat source worth harvesting. But before any of those things can be engineered, researchers must know which way to push the crystal. The new work offers a compass needle—pointing, remarkably, toward the middle.

Primary sources and further reading

This report centers on the 2026 Materials Today Physics paper, its open preprint and Kyoto University’s release. Historical and technical context draws on original papers, official databases, reviews and U.S. government sources. “Golden ratio” is treated as the authors’ descriptive shorthand, not the mathematical constant or a conversion-efficiency claim.