Multilevel Exchange Bias

• Multilevel exchange bias is defined as the discrete shift of a ferromagnetic hysteresis loop induced by engineered coupling with antiferromagnetic or ferrimagnetic phases.
• Key methodologies include lateral interfacial engineering in 2D magnets, bulk sublattice coupling via field-cooling, and spacer-controlled RKKY exchange in multilayers.
• Its applications span multilevel magnetic memory, spin-valves, and nanoscale logic devices, offering enhanced data storage density and precise magnetic control.

Multilevel exchange bias denotes the phenomenon where the exchange-bias field $H_{EB}$—the shift of the ferromagnetic hysteresis loop induced by exchange coupling with an adjacent antiferromagnetic or ferrimagnetic or otherwise "pinning" phase—can be precisely controlled to yield discrete, switchable levels rather than a single fixed value. This tunability is realized through specific device architectures, interfacial engineering, or sublattice coupling, enabling functionalities such as multi-state magnetic memory and nanoscale logic. Recent research establishes multilevel exchange bias as a central motif in low-dimensional magnets, magnetic multilayers, and single-phase materials with multiple magnetic sublattices (Wang et al., 3 Dec 2025, Sun et al., 2013, Polishchuk et al., 2022).

1. Physical Origins and Classification

Exchange bias traditionally refers to the shift of the ferromagnet’s ($FM$) hysteresis loop due to exchange coupling across the $FM/AFM$ (ferromagnet/antiferromagnet) interface—the interfacial pinning imparted by uncompensated spins or domain walls in the antiferromagnet ($AFM$). Multilevel exchange bias generalizes this effect through mechanisms that couple multiple discrete magnetic states to the $FM$ layer and allow external control over the exchange field amplitude and sign.

Three distinct physical mechanisms underpin multilevel exchange bias:

• Lateral interfacial bias in 2D magnets: In materials like atomically thin CrPS$_4$, lateral $AFM/FM$ interfaces formed by parity alternation between even and odd layers yield a set of controlled exchange-bias levels associated with the domain configuration of adjacent $AFM$ blocks (Wang et al., 3 Dec 2025).
• Bulk sublattice coupling: Single-phase magnets with two distinct magnetic sublattices (e.g., YbFe$_2$O$_4$) exhibit volume-wide coupling rather than interfacial pinning. Manipulation of sublattice alignment via field cooling can produce multiple bias levels intrinsic to the bulk material, not limited by thin-film geometry (Sun et al., 2013).
• Superposed interfacial (direct) and indirect (RKKY) exchange: Multilayers such as Py*/FeMn/Cr/Py enable multilevel $H_{EB}$ by controlling the superposition of direct exchange at one interface and an oscillatory Ruderman–Kittel–Kasuya–Yosida ($RKKY$) exchange field across a normal metal ($N$) spacer. Varying $N$ thickness or toggling free-layer magnetization accesses multiple stable $H_{EB}$ states (Polishchuk et al., 2022).

2. Theoretical Frameworks and Energy Functionals

The energy landscapes supporting multilevel exchange bias are governed by competition between anisotropy, Zeeman, and various exchange terms. Key models include:

• Stoner–Wohlfarth single-domain model: In laterally-coupled 2D systems (CrPS$_4$), the total energy of an odd-layer region interfaced with $N$ even-layer domains is:

$$E(\theta) = -K S t_{\text{FM}} \cos^2\theta - \mu_0 M_s S t_{\text{FM}} H \cos(\theta-\phi) + \epsilon_{\text{DW}} C t_e (1-\cos\theta)/2$$

Here, the interfacial domain-wall cost leads to a bias field proportional to perimeter/area ($H_{EB} \propto C/S$), and the discrete antiphase states of each $AFM$ neighbor yield $2^N$ possible $H_{EB}$ values (Wang et al., 3 Dec 2025).

• Bulk sublattice-coupling energy functional: For YbFe$_2$O$_4$,

$$E = -\mu_0 H M_1 \cos\alpha - \mu_0 H M_2 \cos\beta + K_1 \sin^2\alpha + K_2 \sin^2\beta - J_{\text{int}} \cos(\alpha - \beta)$$

In the field-cooled regime, one sublattice acts as a rigid bias on the switching layer, yielding a bias field $H_{\text{EB}} = J_{\text{int}} / (\mu_0 M_1)$ with multilevel states governed by cooling field magnitude and orientation (Sun et al., 2013).

• Superposition model in multilayers: The effective field acting on the AFM is $H_{\text{eff}} = H^* + H_{\text{RKKY}}(d_N)$, where $H^*$ is direct exchange and $H_{\text{RKKY}}(d_N)$ oscillates with spacer thickness. In the Landau expansion:

$$H_{\text{EB}}(d_N) \propto L(d_N) \simeq C \left[ H^* + H_{\text{RKKY}}(d_N) \right]$$

Multiple exchange-bias states are accessed by varying $d_N$ and magnetic configuration (Polishchuk et al., 2022).

3. Experimental Realizations and Detection

Key experimental techniques for probing multilevel exchange bias include:

• Scanning NV magnetometry: Utilized in CrPS$_4$ structures to directly image uncompensated surface magnetization, switching fields of each domain, and polarity of antiphase boundaries. Real-space mapping reconstructs $\sigma(x,y)$ from stray fields $B(x,y,z)$ using Fourier propagation, quantifying discrete bias levels tied to domain configurations (Wang et al., 3 Dec 2025).
• Hysteresis loop measurements (VSM/FMR): In multilayers, precision control of the spacer thickness ($d_N$) yields oscillatory $H_{\text{EB}}$ signatures; minor-loop experiments demonstrate distinct bias states corresponding to free-layer reversal (Polishchuk et al., 2022).
• Bulk magnetometry: In single-phase magnets, coercive field shifts post-field-cooling are measured; variation with $H_{\text{cool}}$ and the training effect (decay law $H_{\text{EB}}(n) = H_{\text{EB}}(\infty) + K/n$) are quantitative indicators of bulk multilevel coupling (Sun et al., 2013).

Mechanism	Key Experimental Probe	Bias Levels Achieved
Lateral 2D (CrPS$_4$)	Scanning NV magnetometry	$2^N$ (for $N$ domains)
Bulk sublattice (YbFe$_2$O$_4$)	Field-cooling, hysteresis loop	Up to two decades (0–2 T)
Indirect/direct superposition	VSM, FMR, minor-loops	Oscillatory (up to $\,\pm400\%$ swing)

4. Tunability and Multilevel Register Encoding

Multilevel exchange bias emerges when the energy landscape supports discrete, stable minima associated to switchable magnetic states in the coupled phase. Specific tunability pathways include:

• AFM domain engineering in CrPS$_4$: Each even-layer domain (e.g., 2L, 4L regions) adjacent to an FM-like odd layer can be toggled independently, yielding $2^N$ possible bias levels, with each neighboring domain’s critical field $B_c^i$ influencing the net $H_{EB}$ by perimeter and thickness ($$H_{EB} \simeq \sum_i [\epsilon_{\text{DW}} C_i t_{e,i}/(2 M_s S t_{\text{FM}})]$$). Experimentally, discrete coercivity steps ($\approx 70$ mT to $27$ mT) are observed as domain patterns are switched (Wang et al., 3 Dec 2025).
• Sublattice programming in YbFe$_2$O$_4$: By adjusting the magnitude and temperature profile of the cooling field, different inter-sublattice alignments are frozen in, producing well-separated bias states from zero up to $\sim$2 T (Sun et al., 2013).
• Spacer-controlled RKKY superposition and polarity switching: In multilayer valves, varying the Cr spacer thickness tunes both the amplitude and sign of $H_{RKKY}$; toggling the free-layer magnetization allows in situ reversal of coupling, enabling up to four stable $H_{EB}$ states within a single device element (Polishchuk et al., 2022).

5. Applications in Magnetoelectronics and Data Storage

Multilevel exchange bias supports advanced device concepts:

• Multilevel magnetic memory: Each bias state encodes a distinct magnetic bit, surpassing binary storage. For example, CrPS$_4$ devices can encode $2^N$ states per bit cell, while bulk sublattice-coupled materials achieve robust multibit registers resilient to scaling (Wang et al., 3 Dec 2025, Sun et al., 2013).
• Lateral spin-valves and nanoscale AFM logic: FM-like regions with variable $H_{EB}$ function as “free” and “pinned” electrodes. Lateral control of AFM domains enables programmable logic gates and nonvolatile operation, with switching speeds set by AFM domain dynamics (Wang et al., 3 Dec 2025).
• Polarity-tunable sensors and memory elements: In multilayer devices, exchange bias can be engineered during fabrication (via spacer thickness) or reversibly toggled by field-driven magnetization reversal, suitable for precision sensors and multi-bit memory cells (Polishchuk et al., 2022).

6. Comparative Analysis and Limitations

Multilevel exchange bias offers several advantages over conventional single-level bias, including expanded data storage density, discrete register control, and device scalability. However, certain limitations persist:

• Training effect: The progressive decay of $H_{EB}$ under repeated cycling, observed as a $1/n$ law in YbFe$_2$O$_4$, necessitates refresh strategies to maintain stability (Sun et al., 2013).
• Geometric scaling: Interfacial bias is sensitive to the ratio $C/S$ (perimeter/area), as in lateral 2D systems. As device dimensions shrink, careful engineering is required to avoid dilution of exchange coupling (Wang et al., 3 Dec 2025).
• Oscillatory RKKY contributions: Multilevel states accessible via RKKY oscillations are contingent on precise fabrication control of spacer thickness; interface roughness or interdiffusion can degrade switching fidelity (Polishchuk et al., 2022). This suggests ongoing needs for atomic-layer deposition and interface characterization.

7. Outlook and Future Directions

Multilevel exchange bias is central to emerging AFM/FM hybrid technologies, offering new paradigms for high-density storage, spin logic, and reconfigurable magnetoelectronics. Systematic exploration of 2D lateral heterostructures, bulk sublattice-coupled magnets, and indirect exchange-mediated multilayers continues to reveal new mechanisms and device architectures. A plausible implication is the integration of multilevel bias schemes into ultra-scaled, nonvolatile memory arrays and spintronic logic co-processors, where programmable exchange springs and domain engineering provide both functional diversity and operational speed. Further advances in atomic-scale imaging, micromagnetic simulation, and interfacial engineering are expected to extend the reach and tunability of multilevel exchange bias systems.