Nonequilibrium Ferrimagnetic Phase

• Nonequilibrium ferrimagnetic phase is a state where external stimuli like oscillatory magnetic fields disrupt thermal equilibrium, leading to dynamic magnetization and novel phase transitions.
• It is modeled using mixed-spin Ising systems with dynamic order parameters that reveal compensation points, spin-glass behavior, and distinct hysteresis effects in materials such as Heusler alloys and spinels.
• Understanding these phases aids the development of spintronic devices by enabling controlled magnetization reversals, tunable compensation, and differentiation between nonthermal and electrothermal effects.

The nonequilibrium ferrimagnetic phase is a state of a ferrimagnet where the system is driven out of thermal equilibrium by external stimuli such as time-dependent magnetic fields, electrical currents, or thermal protocols, resulting in magnetization dynamics, phase transitions, and magnetic ordering distinct from those in equilibrium. Ferrimagnets consist of at least two crystallographically or electronically distinct magnetic sublattices that couple antiparallel and possess unequal moments, yielding a net magnetization. Nonequilibrium variants display fluctuations, compensation phenomena, phase coexistence, glassiness, hysteresis, and critical phenomena inaccessible in equilibrium. The subject spans both theoretical studies (e.g., dynamic Monte Carlo modeling under oscillatory fields) and experimental investigations of real materials (e.g., Heusler alloys, spinels) under non-static conditions.

1. Model Systems and Hamiltonian Formulation

Nonequilibrium ferrimagnetic phases are prototypically studied in mixed-spin Ising models on square or cubic lattices subjected to oscillating magnetic fields. The model Hamiltonian comprises sublattice couplings, anisotropy, and time-dependent external fields. For a mixed spin-½/spin-3/2 Ising system, the Hamiltonian is: $$\mathcal{H} = -J_1\!\sum_{\langle i\in A,\;j\in B\rangle} \sigma_i S_j - J_2\!\sum_{\langle\langle i,k\rangle\rangle_A} \sigma_i\sigma_k - D\!\sum_{j\in B} S_j^2 - H(t)\!\left[\sum_{i\in A}\sigma_i + \sum_{j\in B}S_j\right]$$ where $J_1 < 0$ is nearest-neighbor antiferromagnetic (AFM) coupling (A–B), $J_2 \geq 0$ is next-nearest-neighbor ferromagnetic (FM) coupling within A, $D$ is the single-ion anisotropy on B, and $H(t)=H_0\cos(\omega t)$ is the time-dependent field source (Vatansever et al., 2013).

In real materials, such as hexagonal Mn₂CuGe and spinel Co₂.₇₅Fe₀.₂₅O₄, the nonequilibrium phases arise from competing exchange interactions, cation disorder, and experimental manipulations such as field-cycled thermal protocols, pulsed currents, or temperature ramps (Khorwal et al., 20 Jul 2024, Bhowmik et al., 2020).

2. Dynamic Order Parameters and Measurement Protocols

Instantaneous sublattice magnetizations $m_{A,B}(t)$ are measured during Monte Carlo or experimental time evolution, and dynamic order parameters $Q_A, Q_B, Q_T$ are defined via time averages over one field period: $$Q_A = \frac{1}{\tau}\int_{t_0}^{t_0+\tau} m_A(t)\,dt,\quad Q_B = \frac{1}{\tau}\int_{t_0}^{t_0+\tau} m_B(t)\,dt,\quad Q_T = \frac{1}{2}(Q_A + Q_B)$$ These capture the stationary component of magnetization under periodic drive and serve to locate dynamic phase transitions. In experimental systems, protocols include zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves, thermoremanent magnetization, magnetic memory protocols, and AC susceptibility; the manifestation of bifurcation, slow relaxation, and memory plateaus below characteristic temperatures signals nonequilibrium behavior (Khorwal et al., 20 Jul 2024).

3. Phase Transitions, Compensation, and Nonequilibrium Dynamics

Ferrimagnets often exhibit a magnetization compensation point $T_{\text{comp}}$ where sublattice moments cancel $(|Q_A|=|Q_B|,\mathrm{sgn}(Q_A)=-\mathrm{sgn}(Q_B))$ below a critical temperature $T_c$. The dynamic compensation may occur only if the next-nearest-neighbor coupling $J_2$ exceeds a threshold $J_2^c(D,H_0)$ (Vatansever et al., 2013). The compensation temperature is sensitive to field amplitude $H_0$ and anisotropy $D$, and is largely independent of $J_2$ once above $J_2^c$. Increased field amplitude lowers both $T_{\text{comp}}$ and $T_c$; for sufficiently large $H_0$ compensation vanishes.

Phase transitions in these nonequilibrium systems are typically second order, as determined by peaks in dynamic heat capacity: $$C_{\mathrm{coop}}(T) = \frac{dE_{\mathrm{coop}}}{dT}$$ No evidence for dynamically first-order transitions or tricritical points is found, contrary to some molecular-field predictions (Vatansever et al., 2013).

4. Spin-Glass Behavior and Magnetic Memory

At low temperatures, ferrimagnets with frustrated or competing exchange interactions may enter a spin-glass–like nonequilibrium phase, as seen in Mn₂CuGe and Co₂.₇₅Fe₀.₂₅O₄. Experimentally, this is signaled by ZFC/FC bifurcation, slow non-exponential relaxation of $M(t)$, and magnetic memory effects—e.g., stepwise recovery in magnetization after temperature cycling. The time dependence is well-described by the stretched exponential (Kohlrausch-Williams-Watts) law: $M(t) = M_0 + M_g\,\exp\left[-(t/\tau)^\beta\right]$ with $\beta \approx 0.5$, $\tau \sim 10^2-10^3$ s (Khorwal et al., 20 Jul 2024). C_p(T) data fit with a glassy $T^{3/2}$ term and large electronic contribution $\gamma T$: $C_p(T) = \gamma T + \beta T^3 + \delta T^{3/2}$ A lack of frequency shift in AC susceptibility peaks and absence of Vogel–Fulcher or canonical Arrhenius behavior distinguishes this glassiness from canonical spin glasses (Khorwal et al., 20 Jul 2024, Bhowmik et al., 2020).

5. Field and Current-Driven Nonequilibrium Phenomena

External magnetic fields tune the balance between FIM and AFM phases. In Co₂.₇₅Fe₀.₂₅O₄, increasing $H$ quenches high-T FIM blocking peaks and enhances AFM order, with critical fields $H \gtrsim 10$ kOe needed to suppress the FIM signature. The peak positions $T_{m,i}(H)$ shift as power laws in $H$: $T_{m,i}(H) = a_i - b_i H^{n_i}$ for $i=1,2$, with $n_1 \sim 0.14-0.19$, $n_2 \sim 0.24-0.34$ (Bhowmik et al., 2020). Exchange bias and nonlinear hysteresis form when AFM and FIM clusters coexist at interfaces with pinned domains.

For current-induced nonequilibrium in Mn₃Si₂Te₆, dynamic phase transitions mimic first-order switching but are fundamentally electrothermal: pulsed current drive yields resistance dynamics and magnetic domain collapse consistent with Joule heating above $T_c$, not true non-thermal quantum states. Genuine nonequilibrium phases would require rapid, nonthermal electronic reorganization, not achieved under present conditions; all observed hysteresis, NDR, and nonreciprocity are direct consequences of electrothermal physics (Fang et al., 16 Feb 2025).

6. Material-Specific Realizations and Parameter Regimes

Real material systems exhibit a rich spectrum of nonequilibrium ferrimagnetic phases:

Material	Nonequilibrium Features	Key Parameters
Mn₂CuGe (Heusler) (Khorwal et al., 20 Jul 2024)	Spin-glass at low $T$, compensation at $T_{\text{comp}}$	$T_c\sim683$ K, $T_{\text{comp}}\sim250$ K, $\beta=0.56$, $H_{EB}=255$ Oe
Co₂.₇₅Fe₀.₂₅O₄ (spinel) (Bhowmik et al., 2020)	Coexistence of FIM and AFM phases, field-tunable compensation	$T_{m1}\sim 30\!-\!50$ K, $T_{m2}\sim 150\!-\!330$ K, $H_c=790\!-\!7930$ Oe
Mn₃Si₂Te₆ (Fang et al., 16 Feb 2025)	Abrupt current-driven transitions, domain collapse	$T_c\sim83$ K, $I_c\sim17$ mA, $\tau_p\sim500\,\mu$s

In all systems, molecular-field constants and sublattice Curie parameters govern phase boundaries and compensation, as shown for Mn₂CuGe using Néel theory, where $N_{AB}$ far exceeds $|N_{AA}|,|N_{BB}|$, ensuring a compensation point in the phase diagram.

7. Implications for Spintronic Devices and Future Directions

The ability to manipulate compensation points, magnetization reversals, and nonequilibrium relaxation in ferrimagnetic materials has direct relevance for spintronic technologies, particularly for low-stray-field, high-frequency devices. Tunable compensation and engineered glassy or domain memory effects enable tailored switching, enhanced thermal stability, and complex magnetic-history-dependent functionalities. However, achieving genuine nonthermal nonequilibrium ferrimagnetic phases—distinct from electrothermal or metastable glassy phenomena—requires ultrafast control and characterization protocols that go beyond current time and energy scales.

A plausible implication is that stringent identification of nonequilibrium quantum states in ferrimagnets must rely on dynamical probes (ultrafast Kerr, THz-pump), control of heating, and direct order-parameter characterization, distinguishing them sharply from glassy, thermal, or multicluster states governed mostly by frustration, domain pinning, and conventional compensation mechanisms.