Huge ultrafast spin Seebeck effect mediated by laser-excited superdiffusive magnon currents

This paper introduces an ab initio-parameterized microscopic framework based on the quantum Boltzmann equation to reveal a superdiffusive transport regime and a huge ultrafast spin Seebeck effect in laser-excited bcc Fe films, overcoming the limitations of traditional diffusive models in describing nonthermal magnon dynamics.

The Gist

The Big Picture: A Laser-Induced "Spin Storm"

Imagine a block of iron (a ferromagnetic metal) as a crowded dance floor where everyone is holding hands and spinning in the same direction. This synchronized spinning is what we call magnetism.

In 1996, scientists discovered that if you hit this dance floor with a super-fast, super-bright laser pulse, the dancers stop spinning in sync almost instantly. The magnetism disappears in less than a trillionth of a second. This is called ultrafast demagnetization.

For decades, scientists have argued about how this happens. The old theories were like trying to describe a chaotic mosh pit by assuming everyone is just slowly warming up and cooling down evenly. But this paper argues that the reality is much more violent and fast: it's a chaotic rush of energy that moves faster than simple heat diffusion can explain.

The New Theory: The "Quantum Boltzmann" Traffic Model

The authors (from Uppsala University) built a new, more detailed computer model to simulate what happens when the laser hits the iron.

1. The Old Way (The Three-Temperature Model):
Imagine a room with three groups of people: electrons (the fast movers), phonons (the vibrating floor), and magnons (the spinning dancers). The old model assumed that when the laser hits, the electrons get hot, they share heat with the floor, and the floor shares heat with the dancers. Everyone eventually reaches a "thermal equilibrium" (everyone is the same temperature). It treated the dancers as if they were just slowly warming up.

2. The New Way (The Nonlocal Model):
The authors say this is wrong for the first few picoseconds (trillionths of a second). Instead of a slow warm-up, the laser creates a shockwave.

• The Analogy: Imagine throwing a giant bucket of water onto a dry sponge. The old model says the water slowly soaks in. The new model says the water shoots out in a high-speed jet, splashing everywhere before it even has time to soak in.
• The Mechanism: The laser excites the electrons, which then violently kick the "spinners" (magnons). These spinners don't just sit there; they shoot out of the surface like bullets, carrying their spin energy deep into the material.

Key Findings: The "Superdiffusive" Rush

The paper identifies a specific regime called superdiffusive transport. Here is what that means in plain English:

• Ballistic Phase (The Bullet): Immediately after the laser hits, the excited magnons travel in a straight line, like a bullet fired from a gun. They don't bump into anything yet. They move incredibly fast (about 35–50 nanometers per picosecond).
• Diffusive Phase (The Crowd): After a few picoseconds, they start bumping into other particles, slowing down and spreading out randomly, like a crowd of people milling about in a hallway.
• The "Super" Part: The transition between the "bullet" phase and the "crowd" phase is what the authors call "superdiffusive." It's faster and more efficient than normal diffusion.

The "Spin Seebeck" Effect: A Spin Tsunami

The paper claims this process creates a massive, ultrafast Spin Seebeck Effect.

• The Analogy: Usually, the Spin Seebeck effect is like a slow river of spin flowing from a hot area to a cold area.
• The Paper's Claim: In this ultrafast scenario, it's not a river; it's a tsunami. The laser creates a sudden, massive temperature difference right at the surface. This triggers a "burst" of spin current that is 1,000 times stronger than what you would get from normal, steady heating.
• Why it matters: This burst is so strong and fast that the authors calculate it could theoretically be powerful enough to flip the magnetization of a thin layer of iron in just 10 picoseconds. This is the "holy grail" for making super-fast computer memory.

Connecting Theory to Reality: The "Kerr Angle"

How do we know this is happening? We can't see the magnons directly. Instead, scientists use a tool called the Magneto-Optical Kerr Effect (MOKE).

• The Analogy: Imagine shining a flashlight at a mirror. If the mirror is magnetic, the light bounces back with a slightly different twist (polarization).
• The Paper's Contribution: The authors used their model to predict exactly how this "twist" of light would change over time as the magnetism disappears and reappears at different depths of the iron. They found that the light signal behaves in a very specific, counter-intuitive way (sometimes the signal gets stronger even as the magnetism gets weaker) because the "tsunami" of spin is moving deep into the material.

Summary of What They Claim

1. Old models are too slow: They miss the initial "bullet-like" speed of the particles.
2. New model is accurate: By using a "Quantum Boltzmann" equation, they can track these fast-moving particles as they shoot from the surface into the deep.
3. Huge Spin Currents: The laser creates a massive, ultrafast flow of spin (magnons) that is much stronger than anything seen in steady-state experiments.
4. Two-Stage Demagnetization: First, the surface loses magnetism instantly. Then, a "wave" of demagnetization travels deeper into the material as the spin current arrives.
5. Experimental Proof: They calculated what a laser experiment would "see" (the Kerr signal) and showed that these signals contain a "fingerprint" of this super-fast, deep-traveling spin current.

In short: The paper claims that when you zap iron with a laser, you aren't just heating it up; you are launching a high-speed, super-strong wave of magnetic energy that travels deeper and faster than anyone previously thought possible.

Technical Summary

Technical Summary: Huge ultrafast spin Seebeck effect mediated by laser-excited superdiffusive magnon currents

Problem Statement
The microscopic origin of ultrafast laser-induced demagnetization in ferromagnetic metals remains a subject of debate, particularly regarding the roles of transverse versus longitudinal spin excitations and nonlocal transport mechanisms. While the widely used three-temperature model (3TM) successfully reproduces many experimental results, it assumes instantaneous thermalization of subsystems and fails to capture angular momentum transfer or the strongly nonequilibrium dynamics present on sub-picosecond timescales. Previous extensions, such as the $(N+2)$-temperature model ($(N+2)$TM), addressed nonthermal magnon populations but were limited to local (spatially homogeneous) dynamics. Consequently, theoretical descriptions of magnon transport on ultrafast timescales have largely been restricted to diffusive models, which are ill-suited for capturing the ballistic and superdiffusive regimes observed in electron spin transport. There is a need for a framework that integrates nonthermal magnon scattering with spatial transport to explain depth-resolved demagnetization and spin currents in films of arbitrary thickness.

Methodology
The authors present an ab initio-parameterized microscopic framework that extends the local $(N+2)$TM to a nonlocal version capable of describing spatial dynamics. The core of this approach is the use of the quantum Boltzmann equation (QBE) to model magnon transport, moving beyond the diffusive approximations used in earlier studies.

• Theoretical Framework: The model couples the dynamics of electrons, phonons, and magnons. Electrons and phonons are treated via heat diffusion equations (incorporating Fourier's law with source/sink terms), while magnons are governed by a drift-diffusion equation derived from the QBE. This equation includes a drift term ($v_q \partial n_q / \partial z$) representing ballistic propagation and a scattering term ($s_q$) derived from an $sp$-$d$ Hamiltonian and Fermi's golden rule.
• Scattering Mechanisms: The scattering term accounts for electron-magnon scattering events that drive magnons toward thermal equilibrium with electrons (first-order) and magnon-number-conserving scattering events that redistribute magnon populations (second-order).
• Numerical Implementation: The system is solved for body-centered cubic (bcc) Fe films using a spatially discretized domain. The electron and phonon equations are solved using an implicit Crank-Nicolson scheme, while the magnon advection equations are solved using an explicit total variation diminishing (TVD) scheme.
• Parameters: The study utilizes first-principles calculated parameters for magnon frequencies and scattering rates from Ref. [43]. Simulations are performed for Fe films of 100 nm and 1 $\mu$m thickness, excited by an 800 nm laser pulse (60 fs duration) with fluences ranging from 0.01 to 4 mJ/cm$^2$.
• Observables: To connect theory to experiment, the authors calculate depth-resolved magnetization profiles and simulate time-resolved magneto-optical Kerr effect (MOKE) signals using a generalized transfer matrix method (pyGTM).

Key Contributions and Results

1. Ultrafast Spin Seebeck Effect: The model predicts a "huge" ultrafast spin Seebeck effect driven by strong, inhomogeneous temperature gradients. Immediately following laser excitation, a transient burst of nonthermal magnons is generated near the surface. These magnons propagate into the bulk, carrying a spin current that reaches amplitudes of approximately $5 \hbar \text{ nm}^{-2} \text{ ps}^{-1}$. This current is orders of magnitude larger than those predicted by linear-response theory for steady-state temperature gradients.
2. Superdiffusive Transport Regime: The simulations reveal a crossover from initially ballistic magnon transport to a diffusive regime at later times. By analyzing the mean squared displacement (MSD), the authors identify a "superdiffusive" regime where the transport exponent $\alpha$ transitions from 2 (ballistic) to 1 (diffusive). This regime persists for up to 10 ps and distances up to 100 nm, a timescale and length scale where purely diffusive models fail.
3. Depth-Resolved Demagnetization Dynamics: The nonlocal model predicts distinct demagnetization profiles depending on film thickness and depth.
   • Surface: Exhibits rapid demagnetization followed by prompt partial remagnetization.
   • Bulk: Deeper regions show a delayed demagnetization as heat and magnons propagate from the surface.
   • Thickness Dependence: In thicker films (e.g., 1 $\mu$m), the system takes tens of picoseconds to reach a spatially uniform magnetization state, whereas ultrathin films (<20 nm) homogenize almost instantaneously, validating the local $(N+2)$TM for thin films.
4. MOKE Signal Simulation: The calculated depth-dependent magnetization profiles are used to simulate L-MOKE signals. The results indicate that Kerr rotation and ellipticity do not linearly track the average magnetization on ultrafast timescales. Notably, probing from the side opposite the pump surface reveals a delayed second demagnetization stage caused by the arrival of the ballistic magnon burst. This stage can lead to counterintuitive behaviors, such as an increase in the magnitude of Kerr components or sign changes in ellipticity, which are absent in simulations without transport.

Significance and Claims
The paper claims that its framework provides a route to describe ultrafast nonthermal magnon transport beyond diffusive models. By incorporating the quantum Boltzmann equation, the model successfully captures the transition from ballistic to diffusive transport, a feature essential for understanding dynamics in films thicker than the laser penetration depth.

The authors assert that the calculated magnonic spin currents are technologically relevant, suggesting that laser-induced magnon transport could potentially induce substantial reorientation or reversal of magnetization in thin films within 10 ps. Furthermore, the simulation of MOKE signals offers a direct method to bridge the gap between theoretical predictions and experimental observables without requiring complex depth-sensitive reconstructions. The presence of a delayed demagnetization stage in opposite-side probing is highlighted as a potential experimental signature of magnon-mediated transport, supporting the attribution of ultrafast demagnetization to transverse spin excitations. The work is presented as a tool to aid in the design and interpretation of time-resolved spin-transport experiments.