Fractal dimension and size scaling of domains in thin films of multiferroic BiFeO3

Abstract: We have analyzed the morphology of ferroelectric domains in very thin films of multiferroic BiFeO3. Unlike the more common stripe domains observed in thicker films BiFeO3 or in other ferroics, the domains tend not to be straight, but irregular in shape, with significant domain wall roughening leading to a fractal dimensionality. Also contrary to what is usually observed in other ferroics, the domain size appears not to scale as the square root of the film thickness. A model is proposed in which the observed domain size as a function of film thickness can be directly linked to the fractal dimension of the domains.

• The paper reveals that ferroelectric domains in BiFeO3 films exhibit fractal, irregular patterns rather than uniform stripe configurations.
• The findings demonstrate an anomalous scaling law where domain size increases with film thickness at an exponent near 3/4, deviating from LLK predictions.
• The study employs PFM and Hausdorff measures to connect fractal dimensionality with domain wall energy, challenging conventional ferroelectric models.

Insights into the Fractal Nature of Domain Morphology in BiFeO$_3$ Thin Films

This paper presents a detailed investigation of the domain morphology and scaling in thin films of multiferroic BiFeO$_3$ (BFO), with a focus on their fractal properties. The study departs from traditional observations in ferroics by finding that the ferroelectric domains in BFO are not the common straight stripe configurations but exhibit substantial roughening and irregularities, leading to a fractal dimensionality. Additionally, the domain size does not adhere to the anticipated scaling relation with film thickness seen in other ferroic systems.

The authors propose a comprehensive model linking domain size to fractal dimensions, opposing the classical Landau-Lifshitz-Kittel (LLK) scaling which posits that domain size increases proportionally with the square root of the film thickness. Instead, an anomalous relationship is derived where the domain size scales with a thickness exponent substantially larger, specifically $\gamma \simeq 3/4$, as opposed to the expected $\gamma = 1/2$.

Methodological Approach

BiFeO$_3$ films were developed via pulsed laser deposition on SrTiO$_3$ substrates, succeeded by examination using piezoelectric atomic force microscopy (PFM) to outline the domain morphology. The analysis revealed the fractal nature of the domain walls, where the domains exhibited a rough, mosaic-like appearance, and their size expanded anomalously as the film thickness increased. The authors quantified the fractal dimension using Hausdorff measures and correlated these dimensions with the unconventional domain scaling observed.

Key Findings

1. Fractal Domain Characteristics: The study notes that the domain walls in ultra-thin BFO films display significant roughness and a fractal dimension with measured Hausdorff values around 1.45. This implies substantial deviations from a planarly uniform domain wall configuration.
2. Domain Size Scaling: An empirical scaling law derived for domain periodicity when thickness varies shows exponents of approximately 0.70 and 0.81 for out-of-plane and in-plane domains, respectively. This deviation highlights the inadequacy of LLK scaling in the presence of fractal wall roughness.
3. Implications of Fractality: The existence of fractal domains and their rough walls suggests a higher domain wall energy cost, potentially due to magnetoelectric coupling intrinsic to multiferroic systems, and necessitates reevaluation of classical scaling laws in ferroelectric domains.

Implications for Theory and Practice

Through this rigorous analysis of BFO thin films, the paper raises significant implications for both theoretical frameworks and practical applications involving multiferroic materials. The fractal characteristics of domain walls may broadly apply across various ferroic systems. This insight challenges traditional paradigms and indicates that such fractal behavior could affect polarization switching, domain dynamics, and ultimately device performance in multiferroic-based technologies.

Furthermore, the elucidation of fractal scaling exponents suggests opportunities for tuning domain behavior through altered embedding matrices or alternative deposition techniques, providing a pathway for optimizing material properties for specific applications like ferroelectric memory and spintronics.

Future Directions

Future research endeavours should consider comprehensive explorations of domain wall dynamics influenced by fractal geometries, particularly under varying environmental conditions or external fields. High-resolution studies such as time-resolved PFM may elucidate the kinetics of domain wall movement in fractal configurations, providing deeper insights into their energy landscape and pinning behavior across diverse ferroic materials. Ultimately, integrating these fractal scaling principles in theoretical models will better predict domain dynamics and improve multiferroic device engineering.