Tailored Demagnetization Protocols

• Tailored demagnetization protocols are specialized methods that adjust laser and magnetic field parameters to control rapid magnetization changes in materials.
• Theoretical models use a three-temperature approach to describe energy transfer between electrons, spins, and lattice, ensuring precise control over demagnetization dynamics.
• Optimized protocols, such as those applied in HAMR, balance rapid laser-induced heating with external fields to prevent critical slowdown near the Curie temperature.

Tailored demagnetization protocols encompass methodologies designed to control and optimize the process of magnetization reduction in materials and devices—whether for fundamental studies of ultrafast spin dynamics, magnetic cooling, memory device operation, or magnetic shielding. These protocols are not generic but leverage specific knowledge of system dynamics, material properties, and relevant external stimuli (e.g., laser fluence, field geometry, pulse shaping) to achieve precise control over demagnetization behaviors such as speed, spatial selectivity, thermodynamic endpoint, and reversibility.

1. Microscopic Modeling of Laser-Induced Demagnetization

Theoretical modeling of laser-induced demagnetization, as articulated in the “three-temperature” framework, involves treating electrons, spins, and lattice as coupled thermal reservoirs with exchange dictated by specific interaction channels. After ultrafast laser irradiation, nonthermal electrons thermalize within ≲10 fs. The ensuing energy transfer to spin (magnons)—governed by electron–spin (sd exchange) interactions—and to lattice (phonons) via electron–phonon coupling, sets the timescale and extent of demagnetization.

The system Hamiltonian is formulated as the sum of constituent baths and their couplings:

• Electronic: $H_e = \sum_k \varepsilon_k c^\dagger_k c_k$
• Phonons: $$H_l = \sum_{q\lambda} \hbar\omega_{q\lambda}^{(p)} b^\dagger_{q\lambda} b_{q\lambda}$$
• Spins: $$H_s = -\sum_{\langle ij \rangle} J_{ij}\mathbf{S}_i\cdot\mathbf{S}_j - g\mu_B H_{ex}\sum_i S^z_i$$

Couplings—electron–phonon $(H_{el})$, electron–spin $(H_{es})$, and spin–lattice $(H_{sl})$ (usually weak)—are included explicitly. Using self-consistent random phase approximation transforms the spin sector to bosonic magnon representation, yielding temperature-dependent magnon energies

$$\hbar\omega_q = g\mu_B H_{ex} + 2m(T_s)\left[\sum_q (J_0 - J(q))\right],$$

where $m(T_s)$ is the magnetization at spin temperature $T_s$. The dynamic exchange equations for the subsystem temperatures are

$$\begin{aligned} C_e(T_e)\frac{dT_e}{dt} & = -\Gamma_{el}(T_e, T_l) - \Gamma_{es}(T_e, T_s) + P(t) \ C_l(T_l)\frac{dT_l}{dt} & = \Gamma_{el}(T_e, T_l) - \frac{T_l - T_{rm}}{\tau_l} \ C_s(T_s)\frac{dT_s}{dt} & = \Gamma_{es}(T_e, T_s) \ \end{aligned}$$

where $\Gamma_{el}$ and $\Gamma_{es}$ are electron–lattice and electron–spin energy transfer rates, respectively, and $P(t)$ describes laser excitation.

A key microscopic insight is that $\Gamma_{es}$ exhibits a cubic dependency on the instantaneous magnetization: $$\Gamma_{es} \propto m^3(T_s)[G_2(T_e/(DT_c)) - G_2(T_s/(DT_c))],$$ with $G_n(x) = x^{n+1}\int_0^{1/x} \frac{t^n\,dt}{e^t-1}$. As $m(T_s)\to 0$ near the Curie temperature $T_c$, the coupling collapses and “critical slowdown” arises. The steady-state or remagnetization trajectory is self-consistently built via the magnon population equation

$$m(T_s) = \frac{1}{2} - \frac{1}{N}\sum_q \frac{2m(T_s)}{e^{\beta\hbar\omega_q(T_s)}-1},$$

exhibiting the characteristic mean-field and Bloch’s law limits.

2. Control Parameters: Laser Intensity, Temperature, and Magnetic Fields

Tailoring demagnetization requires manipulating both experimental conditions and intrinsic Hamiltonian parameters:

Laser Intensity and Temperature:

• At moderate laser intensity, $T_e$, $T_s$, $T_l$ rise but remain below or near $T_c$, ensuring that $m(T_s)$ stays significant, $\Gamma_{es}$ remains strong, and ultrafast (sub-picosecond) demagnetization occurs efficiently.
• At high intensity, $T_s$ rapidly approaches $T_c$; $m(T_s)$ collapses, $\Gamma_{es}$ weakens, and a temporal plateau (critical slowdown) emerges, delaying full demagnetization/remagnetization on $\mathcal{O}$(picosecond) scales.

External Magnetic Field:

• Inclusion of $H_{ex}$ in the magnon dispersion sustains nonzero magnetization even near $T_c$.
• This restored order lifts the coupling strength $m^3(T_s)$ and thus eliminates slow plateau dynamics. Even moderate fields—experimentally accessible—restore rapid magnetization transfer.

The protocol can hence be tuned using two “knobs”: laser fluence (controlling heating and proximity to $T_c$) and field strength (controlling residual order and recovery dynamics).

3. Critical Slowdown and Its Suppression

Near $T_c$, the vanishing of instantaneous magnetization $m(T_s)$ leads to a bottleneck for spin–electron energy transfer: $\Gamma_{es} \rightarrow 0$. This critical slowing down is a generic bottleneck for thermally-driven demagnetization protocols as the system passes through a second-order phase transition.

Application of an external magnetic field increases the Zeeman energy in the magnon spectrum, shifting and rounding the order–disorder transition, and thus avoids the abrupt softening of $m(T_s)$. This enables ultrafast processes to persist even in regimes where longitudinal magnetization quenching would otherwise be limited.

Numerical solutions presented in the paper demonstrate the removal of the demagnetization plateau and restoration of high-speed response as $H_{ex}$ is increased, which is essential for device-scale rapid switching.

4. Implications for Heat-Assisted Magnetic Recording (HAMR)

In HAMR, data writing is facilitated by laser-induced transient heating of the magnetic medium close to $T_c$, at which point the magnetization easy axis can be switched by a much weaker external field. However, device speed and energy efficiency are constrained by the competition between rapid magnitude reduction and critical slowdown near $T_c$. The multiscale dynamics—fast (sub-10 ps) for longitudinal magnitude response and slower (nanoseconds) for orientational reversal—require protocols that balance heating (temperature trajectory) and external magnetic field.

By carefully coordinating laser and field settings, these protocols can:

• Expedite demagnetization while avoiding the plateau regime (via optimized field).
• Minimize energy consumption by restricting cycle time to the fast ($<10$ ps) regime whenever possible.
• Exploit sequential two-timescale dynamics for efficient write/readback in HAMR technology.

5. Protocol Optimization Framework

The theoretical results translate into concrete workflow for tailoring protocols:

• Solve the system of coupled rate equations for electrons, spins, and lattice with parameterized laser pulse shape $P(t)$ and set $H_{ex}$.
• For a given material, determine $T_c$ and magnetization curve $m(T_s)$ from ab initio or model calculations.
• Use the cubic dependency of coupling to design ramp rates and field strengths that avoid long-lived plateaus.
• For write cycles in HAMR, design the temporal sequence of heating and applied field such that the magnetization magnitude is quenched swiftly, but critical slowdown is precluded by appropriate ordering fields.

Sample equation set employed: $$\begin{aligned} C_e(T_e)\frac{dT_e}{dt} &= -\Gamma_{el}(T_e,T_l) -\Gamma_{es}(T_e,T_s)+P(t)\ C_l(T_l)\frac{dT_l}{dt} &= \Gamma_{el}(T_e,T_l) - \frac{T_l-T_{rm}}{\tau_l}\ C_s(T_s)\frac{dT_s}{dt} &= \Gamma_{es}(T_e,T_s)\ \Gamma_{es} &\propto m^3(T_s)[G_2(T_e/(D T_c))-G_2(T_s/(D T_c))]\ \end{aligned}$$

6. Applications and Future Research Directions

The theoretical framework supports targeted demagnetization protocols in various settings, including:

• Ultrafast magnetic switching for memory and logic (HAMR, spintronic devices).
• Time-domain and field-strength optimization for maximized speed and minimized power dissipation.
• Material-specific tuning, accounting for the relevant coupling strengths, $T_c$, and experimental boundary conditions.
• Multi-pulse and shaped excitation sequences, potentially further manipulating the path in ($T_e$, $T_s$, $T_l$) space to avoid critical slowdowns and optimize device throughput.

Future areas for development include:

• Incorporation of non-equilibrium magnon and electron distribution functions (beyond thermal baths).
• Stochastic protocols or feedback (real-time) adjustment to correct for inhomogeneous heating and disorder.
• Extension to antiferromagnets and systems with complex magnetic ordering, where multichannel energy flow may introduce new protocol considerations.

Table: Protocol Control Parameters and Effects

Control Parameter	Effect on Demagnetization	Notes/Dependencies
Laser fluence/intensity	Sets heating, controls proximity to $T_c$	High intensity may induce slowdown
External magnetic field	Maintains finite order near $T_c$; removes slowdown	Field strength determines efficacy
Pulse duration/shape	Adjusts timescales; sequence may avoid bottlenecks	Multi-pulse shapes may be beneficial
Material $T_c$ and $m(T)$	Sets regime transitions	Requires careful calibration

Summary

Tailored demagnetization protocols, grounded in microscopic non-equilibrium thermodynamic models, enable controlled manipulation of magnetic order using laser and field parameters. Efficient and rapid demagnetization is achieved by balancing pulse-induced heating and external ordering forces, with explicit attention to nonlinearities such as critical slowing down near the Curie temperature. The protocols inform the optimal operation of magnetic memory, switching, and recording technologies, and highlight the synergies between ultrafast laser techniques, statistical physics, and real-world device engineering (Manchon et al., 2011).