Analytical formulas for far-field radiated energy and angular momentum of metallic thin films

This paper employs the non-equilibrium Green's function framework to derive analytical formulas for the far-field radiated energy, linear momentum, and angular momentum from metallic thin films, demonstrating how an external magnetic field induces gyrotropy to enable torque radiation and establishing a unified connection between these radiative quantities and generalized Fresnel coefficients.

The Gist

Imagine a piece of metal so thin it's almost like a sheet of paper, but instead of being made of wood or plastic, it's made of a sea of tiny, jittering electrons. This is a metallic thin film.

Usually, when we think of heat or light, we just think of energy moving from a hot object to a cold one. But this paper explores a more complex story: what happens when that heat doesn't just move, but also pushes and twists?

Here is the breakdown of the paper's discovery, translated into everyday language.

1. The Setup: A Jittery Crowd in a Magnetic Field

Think of the electrons in this metal film as a massive, chaotic crowd of people dancing in a dark room.

• The Heat: The "dancing" is caused by heat. The hotter the room, the wilder the dance.
• The Radiation: As they dance, they shake the air around them, creating invisible waves (light/radiation) that fly out into the distance. This is how the room loses heat.
• The Twist (The Magnetic Field): Now, imagine you turn on a giant magnet pointing straight up through the floor. This magnetic field acts like a strict dance instructor. It forces the crowd to dance in a specific, swirling pattern instead of randomly.

2. The Big Discovery: Light that Pushes and Twists

In the old days, scientists thought light only carried energy (heat). Later, they realized light also carries momentum (a gentle push, like wind blowing a sail).

This paper asks: Can light also carry "twist" (angular momentum)?

The answer is yes, but only if you use that magnetic field.

• Without the magnet: The electrons dance randomly. The light they emit is like a straight beam of wind. It pushes things, but it doesn't spin them.
• With the magnet: The electrons are forced to swirl. The light they emit becomes like a helicopter blade or a corkscrew. It doesn't just push; it twists.

3. The "Recipe" for the Math

The authors didn't just guess this; they wrote a perfect mathematical recipe (analytical formulas) to predict exactly how much energy, push, and twist the metal will emit.

• The "Fresnel Coefficients": Imagine you are looking at a window. Some light bounces off, some goes through. The "Fresnel coefficients" are just numbers that tell you how much light bounces and how much goes through.
• The Connection: The authors found a beautiful link between the light the metal emits and the light it absorbs. It's like a mirror: if the metal is good at absorbing a certain type of "twisted" light, it's also good at emitting it. This is a modern version of an old rule called Kirchhoff's Law.

4. The "Ghost" Problem and the Solution

Calculating how a spinning crowd of electrons creates a twisting force is incredibly hard. It's like trying to calculate the exact wind force on a spinning fan blade while the fan is moving.

To solve this, the authors used a clever mathematical trick called the Wigner Transform.

• The Analogy: Imagine trying to measure the speed of a car. If you just look at the car at one spot, you don't know where it's going. If you look at the road, you don't know the car's position. The Wigner Transform is like a special camera that lets you see both the car's position and its speed at the exact same time, allowing the authors to calculate the "twist" without getting lost in the math.

5. The Real-World Test: Bismuth

To prove their math works, they tested it on a real metal called Bismuth (a shiny, brittle metal often used in cosmetics or medicine).

• They simulated a thin film of Bismuth at room temperature.
• They turned up the magnetic field.
• The Result:
  • Energy: The total heat emitted went up a little bit.
  • Push (Force): The light pushed harder.
  • Twist (Torque): This was the surprise! The light started to spin. The stronger the magnet, the more the light spun—but only up to a point. After a certain magnetic strength (around 3 to 4 Tesla), the spinning actually started to slow down again. It's like pushing a swing: push too hard at the wrong time, and you stop the motion instead of helping it.

Why Does This Matter?

This isn't just about math for math's sake. It opens the door to controlling light in new ways.

• Quantum Computers: We might use this "twisting" light to spin tiny quantum bits (qubits) to store information.
• Nano-Machines: Imagine tiny machines powered by light that don't just move forward, but actually rotate to do work, like a microscopic windmill.
• Better Heat Management: We could design materials that radiate heat in specific directions or spins, helping to cool down tiny computer chips more efficiently.

In a Nutshell

This paper is like discovering that a hot piece of metal, when placed in a magnetic field, doesn't just glow; it screws the air around it. The authors wrote the perfect instruction manual for predicting exactly how hard that metal will push and how fast it will spin the light it emits. It's a bridge between the chaotic dance of electrons and the precise, controllable forces of the future.

Technical Summary

1. Problem Statement

While thermal radiation and energy transport are well-understood phenomena governed by classical laws (Planck, Kirchhoff, Stefan-Boltzmann), the radiative transport of angular momentum (AM) remains less explored, particularly in analytical frameworks.

• The Gap: Existing theoretical treatments often rely on approximations (e.g., Born approximation) that neglect the full interaction between the electromagnetic field and the material, or they focus on semi-infinite slabs rather than thin films.
• The Challenge: Deriving exact analytical expressions for radiative power, linear momentum, and angular momentum from a metallic thin film that accounts for non-equilibrium effects, material dispersion, and the breaking of reciprocity (necessary for net AM radiation) without resorting to purely numerical simulations.

2. Methodology

The authors employ a rigorous non-equilibrium Green's function (NEGF) framework using the Keldysh formalism.

• System Model: A two-dimensional (2D) metallic thin film described by the Drude conductivity model. To enable angular momentum radiation, an out-of-plane magnetic field ($B = B\hat{z}$) is applied to break reciprocity, introducing gyrotropic terms into the permittivity tensor.
• Green's Functions:
  • The photon Green's function (GF) is defined on the Keldysh contour.
  • The Dyson equation is solved exactly for the retarded GF ($D^R$) by treating the thin film as an infinitesimally thin layer where electron fluctuations in the out-of-plane direction are neglected.
  • The lesser GF ($D^<$), which governs field correlations and thermal fluctuations, is derived using the fluctuation-dissipation theorem.
• Coordinate Transformation: To simplify the matrix algebra, the authors rotate the coordinate system to align with the in-plane wavevector ($\hat{k}_{\parallel}, \hat{s}, \hat{z}$), where $\hat{s}$ is the transverse direction.
• Wigner Transform: A critical methodological step involves using the Wigner transform to handle the calculation of radiative torque. This allows the conversion of integrals involving position coordinates (required for torque, $M = \int \mathbf{r} \times \mathbf{T} \, dA$) into a form dependent on the flux density and area, avoiding the vanishing integral problem associated with translational symmetry.

3. Key Contributions

A. Exact Analytical Solutions

The paper derives exact analytical expressions for the photon Green's functions in terms of generalized Fresnel coefficients. Unlike previous works that approximated the photon GF with the vacuum version, this work fully accounts for the material's presence via the Dyson equation.

B. Unified Framework for Energy, Momentum, and Torque

The authors establish a unified set of formulas where the radiative quantities are expressed as integrals over frequency ($\omega$) and in-plane wavevector ($k_{\parallel}$):

• Radiated Power ($\langle I \rangle$): Proportional to $\hbar \omega$.
• Radiated Force ($\langle N_z \rangle$): Proportional to $\hbar \gamma$ (momentum component).
• Radiated Torque ($\langle M_z \rangle$): Proportional to $\hbar$.

Crucially, the integrands ($A_I$ and $A_M$) are expressed entirely in terms of Fresnel transmission coefficients ($t_{ss}, t_{pp}, t_{sp}, t_{ps}$).

C. Connection to Kirchhoff's Law

• Zero Magnetic Field ($B=0$): The derived power formula recovers Kirchhoff's law of thermal radiation, demonstrating that emission equals absorption (detailed balance) via the relation $1 - |r|^2 - |t|^2$.
• Non-Zero Magnetic Field ($B \neq 0$): The reciprocity is broken. The paper shows that while standard Kirchhoff's law does not strictly hold due to non-reciprocal terms, the emission is still governed by generalized Fresnel coefficients, including cross-polarization terms ($t_{sp}$) arising from the magneto-optical (Kerr) effect.

D. Mechanism for Angular Momentum Radiation

The study identifies that off-diagonal terms in the lesser Green's function ($D^<_{xy}$ and $D^<_{yx}$) are responsible for angular momentum transport. These terms vanish without a magnetic field. The presence of $B$ induces polarization rotation (gyrotropy), allowing the emission of photons carrying net angular momentum.

4. Results

Numerical Validation (Bismuth)

The theory was applied to a Bismuth (Bi) thin film ($L_z = 1$ nm) at room temperature (300 K) using Drude parameters.

• Power Spectrum: The application of a magnetic field (0 to 1 T) slightly enhances the total radiated power and shifts the spectral peak to higher frequencies, effectively mimicking a higher temperature.
• Torque Spectrum:
  • Magnitude: The radiated torque is non-zero only when $B \neq 0$. The magnitude scales approximately linearly with $B$ at low fields.
  • Direction: The torque is negative, indicating it opposes the direction of the external magnetic field.
  • Non-Monotonic Behavior: A key finding is that the total radiated torque does not increase monotonically with the magnetic field. Instead, it reaches a maximum around 3 T to 4 T before decreasing. This suggests a complex interplay between the cyclotron frequency and the plasma frequency.
• Spectral Distribution: Both power and torque spectra are concentrated in the infrared range ($10^{12}$ to $10^{15}$ s$^{-1}$), consistent with thermal emission at 300 K.

5. Significance

• Theoretical Advancement: This work provides the first exact analytical framework connecting non-equilibrium Green's functions, generalized Fresnel coefficients, and the radiative transport of angular momentum in thin films.
• Experimental Relevance: The derived formulas offer a predictive tool for designing nanoscale photonic devices that manipulate thermal radiation, such as:
  • Thermal Photovoltaics: Enhancing efficiency by controlling spectral emission.
  • Optical Manipulation: Using thermal radiation to exert torque on nearby objects (optical tweezers for thermal photons).
  • Quantum Communication: Utilizing the angular momentum degree of freedom for information encoding.
• Fundamental Physics: It clarifies the role of magneto-optical effects in breaking reciprocity to generate net angular momentum flux, bridging the gap between classical thermodynamics and quantum electrodynamics in low-dimensional systems.

In conclusion, Zhang et al. successfully unify the description of energy, momentum, and angular momentum radiation in gyrotropic thin films, providing a robust analytical toolset for future research in thermal nanophotonics.