🔬 applied physics

Reverse heat flow with Peltier-induced thermoinductive effect

This paper reports the experimental demonstration and theoretical derivation of a Peltier-induced "thermoinductive" effect that enables controllable, temporary reverse heat flow from cold to hot regions in solid-state materials, thereby realizing the missing inductive component for thermal circuit design.

Original authors: Kenjiro Okawa, Yasutaka Amagai, Hiroyuki Fujiki, Nobu-Hisa Kaneko

Published 2026-05-08

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CC BY 4.0

Original authors: Kenjiro Okawa, Yasutaka Amagai, Hiroyuki Fujiki, Nobu-Hisa Kaneko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long, thin metal rod. In the normal world, if you heat one end, the heat naturally flows to the cold end, like water flowing downhill. It never flows "uphill" on its own. This is a fundamental rule of nature (the second law of thermodynamics).

However, a team of researchers at Japan's National Institute of Advanced Industrial Science and Technology (AIST) has found a clever way to make heat flow "uphill" for a split second, using a material that acts like a thermal version of an electrical coil.

Here is the simple breakdown of what they did and what they found:

The Missing Piece: The "Thermal Inductor"

In electrical circuits, we have resistors, capacitors, and inductors. An inductor is a component that resists changes in current; it stores energy in a magnetic field and can make current keep flowing even after the power source is turned off or reversed.

In the world of heat (thermal circuits), we have "resistors" (insulators) and "capacitors" (materials that store heat), but we have been missing a true "inductor." Scientists have wanted one because it would allow for complex thermal designs, like thermal logic gates or memory, just like we have in computers. The problem is that heat usually just flows from hot to cold; it doesn't have that "inertia" to keep flowing backward.

The Trick: The Peltier Effect as a "Thermal Inductor"

The researchers used a special material called (Bi,Sb)₂Te₃ (a type of thermoelectric material). They applied an alternating current (AC) to it.

Think of the AC current like a person rapidly pushing and pulling a heavy swing.

The Push and Pull: When the electricity flows one way, the Peltier effect makes one end of the rod get hot and the other cold. When the electricity reverses direction, the ends swap: the hot end becomes cold, and the cold end becomes hot.
The Lag (Thermal Inertia): Heat doesn't change temperature instantly. It takes time to move through the material. This is called "thermal inertia."
The Collision: Because the electricity is switching directions faster than the heat can fully catch up, the "heat waves" traveling from the ends crash into each other in the middle of the rod.
The Reverse Flow: At a very specific speed (frequency) of switching, this collision creates a momentary situation where heat actually flows from the cold side back toward the hot side. It's like a wave crashing back up the beach before it settles.

The researchers call this the "thermoinductive effect." It's not magic; it's just using the delay in heat movement combined with rapid switching to create a temporary "uphill" flow.

The Experiment: Seeing the Invisible

You can't easily see heat flowing backward, so how did they prove it happened?

They used the material's electrical resistance as a thermometer. When the material gets slightly cooler, its electrical resistance changes.

They set up a four-wire measurement system (like a very precise scale) on the (Bi,Sb)₂Te₃ rod.
They ran the AC current at different speeds.
The Result: At a specific "sweet spot" frequency, they detected a tiny dip in electrical resistance. This dip proved that the material had cooled down locally by about 25 millikelvin (a tiny fraction of a degree) in the middle of the rod, even though the ends were being heated and cooled.

Why This Material Matters

They tried this with a standard copper wire too. While the physics says it should happen in copper, the effect was so tiny (about 10,000 times smaller) that it was practically impossible to measure. The (Bi,Sb)₂Te₃ material is special because it has a high "Seebeck coefficient" (it's very good at converting temperature differences into electricity and vice versa), which amplifies this reverse flow effect, making it detectable.

The Bottom Line

The paper claims to have successfully:

Theoretically modeled how to create this reverse heat flow using exact math.
Experimentally observed a local, temporary cooling effect caused by this reverse flow in a single material.
Demonstrated that this acts as a "thermoinductor," a component that was previously missing from thermal circuit design.

They did not claim this can cool your house or power a refrigerator yet. They simply proved that, under very specific conditions, you can trick heat into flowing backward for a brief moment, opening the door to designing more complex "thermal circuits" in the future.

Technical Summary: Reverse Heat Flow with Peltier-Induced Thermoinductive Effect

Problem Statement
Thermal circuit design aims to manipulate phonon transport to create components analogous to electronic circuits, such as thermal diodes, transistors, and logic gates. While thermal analogs for resistors and capacitors are well-established, a thermal "inductor" remains a missing component. The concept of thermal inductance requires oscillatory behavior where heat flow reverses direction (flowing from cold to hot), a phenomenon typically considered a violation of the second law of thermodynamics in steady-state systems. Previous attempts to demonstrate a "thermoinductive" effect relied on cumbersome external interventions, such as natural convection or cooling electrical coils to liquid helium temperatures, which hindered practical application and theoretical clarity.

Methodology
The authors propose and demonstrate a method to generate a locally and temporally reversed heat flow within a single thermoelectric (TE) material using the Peltier effect under periodic modulation.

Theoretical Model: The study derives an exact solution to the one-dimensional unsteady-state heat transfer equation for a rectangular parallelepiped TE sample subjected to an alternating current (AC). The model accounts for Peltier heating and cooling at the sample-electrode interfaces, treating the resulting temperature distribution as a thermal wave.
Key Parameter: A characteristic dimensionless parameter, $F \equiv (l/2)^2 f / 2\alpha$ , is introduced, representing the product of the sample's thermal time constant and the current frequency ( $f$ ), where $l$ is the sample length and $\alpha$ is thermal diffusivity.
Experimental Setup: The researchers utilized a polycrystalline p-type $(Bi,Sb)_2Te_3$ sample (15 mm length) in a vacuum chamber to minimize convection and radiation losses. They employed a four-terminal pair impedance measurement configuration to detect minute temperature changes reflected as electrical resistance variations via the Seebeck effect. Measurements were taken across a range of frequencies and voltage probe positions ( $l_V/l$ ).

Key Contributions

Exact Solution for Thermoinductance: The paper provides an exact analytical solution describing the temperature distribution and heat flow in a TE material under AC excitation. This solution reveals that reverse heat flow is a universal phenomenon in solid-state systems but is significantly enhanced by thermoelectric properties.
Identification of the "Thermoinductive" Regime: The study classifies thermal behavior into three regimes based on $F$ $F$ :
- DC-like ( $F \ll 0.05$ ): Linear temperature gradient.
- Thermoinductive ( $F \sim 1$ ): A regime where the thermal inertia causes a phase delay, resulting in a local, temporary negative temperature gradient (heat flowing from cold to hot) in the center of the sample.
- Cancellation ( $F \gtrsim 15$ ): The TE effect is canceled, resulting in a flat temperature distribution.
Electrical Detection Mechanism: The authors demonstrate that the reverse heat flow can be detected electrically as a "negative resistance change" (a dip in impedance) when using a four-probe configuration. This avoids the difficulty of direct calorimetric measurement of minute heat flows.

Results

Material Comparison: Calculations show that while reverse heat flow occurs in all solids, it is orders of magnitude more prominent in TE materials compared to metals. For $(Bi,Sb)_2Te_3$ at $F=1$ , a local cooling of approximately 25 mK was predicted and observed. In contrast, copper (Cu) under similar conditions showed a temperature decrease of only 3.5 µK (roughly 10,000 times smaller), attributed to Cu's low Seebeck coefficient and high thermal conductivity.
Experimental Verification: Experiments on $(Bi,Sb)_2Te_3$ confirmed the theoretical predictions. A distinct dip in electrical resistance was observed around 40 mHz (corresponding to $F \approx 1$ ). The magnitude of this dip varied with the voltage probe position ( $l_V/l$ ), matching the exact solution's prediction that the reverse heat flow is spatially dependent.
Magnitude of Effect: The study achieved a partial temperature inversion of approximately 20% of the Peltier heating effect at room temperature. The resistance change was measurable in the milliohm range for the TE material, whereas it would be in the nanohm range for copper, making the effect experimentally accessible only in high-performance TE materials.

Significance and Claims
The paper claims to have successfully formulated and experimentally observed a Peltier-induced reverse heat flow, which serves as a functional equivalent to the "thermoinductive" effect.

Controllability: Unlike previous methods requiring complex external hardware, this effect is electrically controllable by simply tuning the AC current frequency.
Thermal Circuit Design: The results suggest that TE materials can act as "thermoinductors," expanding the toolkit for thermal circuit design. This could enable the creation of thermal resonant circuits and other advanced thermal management architectures.
Mechanism: The study clarifies that the effect arises from the interplay between thermal inertia (RC component) and the phase delay induced by current reversal, rather than a violation of thermodynamic laws. The authors note that defining a "thermal self-inductance" in a lumped-parameter model is difficult for this time-dependent phenomenon, emphasizing the necessity of their exact solution derived from the heat transfer equation.

The work concludes that utilizing TE materials is a promising strategy for manipulating heat flow, offering a pathway to realize novel thermal components previously considered missing from the thermal circuit paradigm.