🔬 mesoscale physics

Electron--phonon cooling noise in monolayer and bilayer graphene

This paper presents a theoretical study of electron-phonon cooling noise in monolayer and bilayer graphene by deriving full analytical expressions for non-equilibrium heat-current fluctuations across different temperature regimes and validating them against established thermal laws and on-shell calculations.

Original author: Muhammad Waqas Haseeb, Edvin G. Idrisov

Published 2026-11-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Muhammad Waqas Haseeb, Edvin G. Idrisov

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of modern electronics, heat is not just a byproduct; it is a fundamental currency. When electrons move through a material, they carry energy, and when they slow down or change direction, they often pass that energy to the atoms of the material itself, creating vibrations known as phonons. In many tiny devices built from two-dimensional materials like graphene, this exchange of energy determines how hot the electrons get and how quickly they can cool down. This cooling process is vital for the performance of sensitive sensors and detectors that rely on measuring minute changes in temperature. For decades, scientists have understood the average rate at which this energy flows, a concept known as cooling power. However, just as the average speed of cars on a highway tells you nothing about the sudden braking or swerving of individual vehicles, the average flow of heat hides the random, microscopic jitters that occur every time an electron bumps into an atom. These tiny, random fluctuations are the focus of new research, offering a deeper look into the chaotic dance of energy at the quantum level.

Researchers at the United Arab Emirates University have now mapped out these fluctuations for the first time in both single-layer and double-layer graphene. They focused on how the heat current, the flow of energy from electrons to the vibrating atomic lattice, wobbles around its average value. To do this, they built a theoretical model that treats the electrons and the atomic vibrations as two separate systems that can have different temperatures, a common state in fast-moving electronic devices. By calculating the noise, or the statistical spread of these energy exchanges, they discovered that the story changes significantly when you look at it over very short time scales, or at high frequencies. While the average cooling power follows well-known laws, the noise reveals a more complex picture that depends heavily on the specific frequency of the measurement and the temperature difference between the electrons and the atoms.

The study confirms that at very low frequencies, the noise behaves as expected, matching the established relationship between heat flow and temperature. In this steady state, the random emission of vibrations by electrons and their absorption from the lattice balance out in a way that is predictable. However, as the researchers pushed their calculations to higher frequencies, reaching into the range of tens of gigahertz and approaching the sub-terahertz band, the behavior diverged. They found that the noise does not simply scale up from the low-frequency results. Instead, the energy exchanged in each individual event becomes weighted differently depending on the frequency of the observation. This means that a detector looking at the system at a specific high frequency sees a different pattern of fluctuations than one looking at the average over a long time. The researchers derived precise mathematical descriptions for these patterns in two distinct temperature regimes: one where the material is very cold and only the lowest energy vibrations are active, and another where the material is warmer and the vibrations behave more like a classical gas.

A key finding of this work is that the noise cannot be inferred simply by knowing the average cooling power when the system is out of equilibrium. In many other areas of physics, knowing the average flow allows you to predict the fluctuations, but here, the random nature of the energy exchange creates new information that is only visible at high frequencies. The team showed that for typical graphene parameters, these high-frequency corrections are small at ordinary radio frequencies but become measurable, reaching the level of a few percent, in the upper-gigahertz to sub-terahertz range. This is significant because it suggests that future ultra-fast thermal sensors and detectors could potentially use these noise signatures to learn more about the internal state of a device than average measurements ever could. The researchers also verified that their complex calculations correctly reproduce the known laws for average cooling when the frequency is zero, giving them confidence in their results for the more difficult high-frequency cases.

The paper distinguishes clearly between single-layer and double-layer graphene, showing that while both materials follow similar general rules, the details of their atomic structure lead to different dependencies on how many electrons are present. In the single-layer material, the noise scales with the amount of doping in a specific way, while in the double-layer version, the relationship is different due to the unique way electrons move between the two sheets. The study also explicitly rules out the idea that a simple extension of standard thermal laws can describe these high-frequency effects. Instead, the researchers demonstrate that a direct, microscopic calculation is necessary to capture the full picture. Their work provides a solid baseline for understanding the intrinsic noise caused by the interaction between electrons and sound waves in graphene, separate from other complications like impurities or interactions with the material underneath.

Ultimately, this research provides a clearer map of the thermal landscape in graphene devices. By showing exactly how heat-current noise behaves across different temperatures and frequencies, the authors offer a tool for engineers and physicists who are designing the next generation of thermal detectors. The ability to predict these fluctuations at high frequencies means that scientists can better distinguish between the signal they want to measure and the background noise of the material itself. The study does not claim to have solved all problems related to heat in graphene, as real-world devices often involve other factors like disorder or different types of vibrations, but it establishes a rigorous foundation for the intrinsic behavior of clean graphene. For those working on fast thermometry and calorimetric detection, the results indicate that the noise in the upper-gigahertz range is not just a nuisance, but a rich source of information about the fundamental energy exchange processes occurring within the material.

Technical Summary: Electron–Phonon Cooling Noise in Monolayer and Bilayer Graphene

Problem Statement
While the average electron–phonon cooling power in monolayer (MLG) and Bernal bilayer (BLG) graphene is well-established, the fluctuations (noise) surrounding this mean energy transfer rate have not been fully characterized, particularly at finite frequencies and out of equilibrium. The average cooling power represents only the first moment of the energy-transfer process. A complete microscopic description requires the second moment: the heat-current noise. Previous studies in metallic systems and mesoscopic thermal links indicate that finite-frequency heat noise contains information absent from average thermal conductance, specifically quantum contributions that cannot be inferred from a simple frequency-dependent extension of the thermal fluctuation–dissipation theorem (FDT). This work addresses the need for a direct microscopic calculation of electron–phonon heat-current noise in clean graphene coupled to longitudinal acoustic (LA) phonons, covering both equilibrium and non-equilibrium (unequal electron and phonon temperatures) regimes.

Methodology
The authors construct a microscopic theory based on the deformation-potential coupling between electrons and LA phonons.

Hamiltonian and Operator: The system is described by a total Hamiltonian comprising free electron (MLG or BLG), free phonon, and electron–phonon interaction terms. A symmetric heat-current operator, $\hat{J} = \frac{1}{2}\partial_t (\hat{H}_{ph} - \hat{H}_e)$ , is defined to ensure symmetric treatment of energy changes in both subsystems.
Noise Calculation: The unsymmetrized noise spectrum is calculated via the current–current correlator in the leading order of electron–phonon coupling. The calculation employs a full Wick expansion of the fermionic four-point function.
- The direct contraction vanishes due to momentum conservation requiring zero-momentum phonons (which carry no energy).
- The exchange contraction yields the physical noise spectrum, containing both phonon emission and absorption processes.
Regimes and Approximations: The study derives analytical expressions for two distinct temperature regimes:
- Bloch–Grüneisen (BG) Regime: Low temperatures where only long-wavelength phonons are active ( $T \ll T_{BG}$ ).
- Equipartition (EP) Regime: High temperatures where phonons are classically occupied and scattering is quasi-elastic.
Benchmarks: To validate the asymptotic limits, the authors perform a "full on-shell" intraband calculation for doped MLG without reducing to BG or EP approximations. This retains exact scattering geometry, chirality factors, and distribution functions.
Non-Equilibrium Treatment: The analysis explicitly handles cases where electron temperature ( $T_e$ ) and phonon temperature ( $T_{ph}$ ) differ, calculating the noise directly rather than inferring it from a dynamic thermal conductance.

Key Contributions and Results

Analytical Expressions: The paper provides full analytical expressions for the heat-current noise in both BG and EP regimes for MLG and BLG.
- In the BG regime, the equilibrium zero-frequency noise scales as $T^5$ for both MLG and BLG. However, the doping dependence differs: MLG noise is proportional to $|\mu|$ , while BLG noise is proportional to $|\mu|^{-1/2}$ due to the parabolic dispersion and chemical potential dependence of the Fermi velocity.
- In the EP regime, the noise behavior reflects the different electronic phase spaces. MLG noise involves a fourth-order electronic energy moment, while BLG noise involves a temperature multiplied by a logarithmic phase-space factor.
Finite-Frequency Spectra: The authors derive reduced spectra for finite frequencies. A key finding is that the heat-current vertices at finite frequency involve terms like $(\hbar\omega \mp 2E_q)^2$ $(ℏ ω \mp 2 E_{q})^{2}$ . This leads to a non-trivial frequency dependence where the noise is not simply related to the static conductance.
- At zero temperature, finite-frequency noise survives even when zero-frequency thermal noise vanishes, scaling as $|\omega|^5$ in the BG limit.
- The equilibrium zero-frequency limit recovers the standard FDT relation $S(0) = 2k_B T^2 G_{ep}(T)$ , serving as a consistency check.
Non-Equilibrium Behavior: For $T_e \neq T_{ph}$ , the noise is shown to be a rate sum governed separately by electron and phonon temperatures. It cannot be interpreted as a unique nonlinear thermal conductance. The paper derives the dependence of the noise on the temperature difference $\Delta T$ , showing that corrections become significant in the upper-gigahertz to sub-terahertz range.
Crossover Analysis: Numerical results using a full on-shell MLG calculation demonstrate a smooth crossover between the BG and EP regimes, validating the asymptotic formulas as limiting cases of a single microscopic spectrum.

Significance and Claims
The paper claims to provide the first direct microscopic derivation of finite-frequency electron–phonon heat-current noise in graphene, moving beyond the average cooling power.

Theoretical Rigor: It establishes that the standard FDT cannot be straightforwardly extended to finite frequencies for heat transport, as the noise contains quantum contributions independent of the static conductance.
Regime Distinction: The work clarifies the distinct doping and temperature dependencies of noise in MLG versus BLG, offering a theoretical basis for distinguishing the two via noise spectroscopy.
Experimental Relevance: The authors note that for representative parameters, finite-frequency corrections are small at low-gigahertz frequencies but reach the percent level in the upper-gigahertz to sub-terahertz range. This suggests the results are relevant for fast thermometry and calorimetric detection in micron-scale graphene devices.
Baseline for Future Work: The study restricts itself to clean graphene with unscreened deformation potential coupling to LA phonons. It explicitly states that effects such as screening, disorder-assisted supercollisions, flexural phonons, optical phonons, and substrate modes are outside the current scope, providing a clean intrinsic baseline against which these additional mechanisms can be identified in future studies.

The work concludes that the finite-frequency spectrum is the primary object of study for non-equilibrium energy exchange statistics, and that direct calculation is necessary rather than inferring results from conductance laws.

Technical Summary: Electron–Phonon Cooling Noise in Monolayer and Bilayer Graphene

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