KINETIC SELECTION AND NON-EQUILIBRIUM STRUCTURE FORMATION IN RARE-EARTH–DIFFUSED SILICON PROBED BY IMPEDANCE SPECTROSCOPY

ABSTRACT

High-temperature diffusion of rare-earth elements into silicon often produces a structurally and electronically heterogeneous near-surface region while the bulk largely retains crystallinity. This paper proposes a unified physical scheme linking thermodynamic constraints, non-equilibrium diffusion pathways and “kinetic selection” to the formation of metastable interfacial states. The model is supported by impedance spectroscopy, which reveals characteristic relaxations associated with interfacial polarization and defect-controlled charge transport. We discuss how finite-source conditions and quenching can freeze non-equilibrium configurations and shift the dominant contribution from bulk conduction to near-surface response. The proposed framework provides practical criteria for interpreting impedance spectra of rare-earth-modified Si and for distinguishing near-surface processes from bulk-related contributions.

АННОТАЦИЯ

Высокотемпературная диффузия редкоземельных элементов в кремний приводит к формированию структурно и электронно неоднородной приповерхностной области при сохранении кристалличности объёма. В статье предложена единая физическая схема, связывающая термодинамические ограничения, неравновесные пути диффузии и механизм «кинетического отбора» с образованием метастабильных межфазных состояний. Подход обоснован данными импедансной спектроскопии: в спектрах выделяются релаксации, обусловленные межфазной поляризацией и дефектно-контролируемым переносом заряда. Показано, что конечный запас источника и закалка могут «замораживать» неравновесные конфигурации и смещать вклад от объёмной проводимости к приповерхностному отклику. Предложены практические критерии интерпретации спектров импеданса модифицированного кремния и разделения приповерхностных и объёмных процессов.

Keywords: silicon; rare-earth diffusion; impedance spectroscopy; interfacial polarization.

Ключевые слова: кремний; диффузия редкоземельных элементов; импедансная спектроскопия; межфазная поляризация.

Introduction

Rare-earth (RE) modification of silicon has attracted sustained attention over several decades due to its potential impact on the electronic, optical, and functional properties of semiconductor materials [1–3]. High-temperature diffusion of rare-earth elements has been extensively explored as an alternative route for tailoring silicon properties beyond the limitations of conventional substitutional doping [4,5]. Despite the substantial volume of experimental studies, the physical interpretation of rare-earth-diffused silicon remains fragmented, with many reported observations discussed in isolation rather than within a unified causal framework linking diffusion conditions, structure formation, and resulting electrical behavior.

A fundamental difficulty arises from the fact that diffusion of rare-earth elements in silicon differs qualitatively from classical dopant diffusion. Owing to their large atomic radii, extremely low equilibrium solubility, and high chemical reactivity, rare-earth atoms do not readily occupy substitutional lattice sites. Instead, they interact strongly with intrinsic point defects and residual impurities, most notably oxygen [6–8]. As a result, rare-earth diffusion activates a complex interplay of defect generation, strain accumulation, clustering, phase competition, and chemical stabilization, occurring simultaneously across atomic, nanoscale, and mesoscale regimes.

In this context, oxygen should not be regarded as a secondary or uncontrolled impurity. Due to its strong chemical affinity to rare-earth elements, oxygen introduces deep local minima into the free-energy landscape, stabilizing rare-earth-related complexes and clusters already at early   stages of diffusion. Consequently, oxygen effectively transforms thermodynamically permissible configurations into kinetically stabilized ones, suppressing long-range redistribution and promoting the kinetic fixation of metastable structural states during cooling.

From a thermodynamic standpoint, rare-earth–silicon systems are commonly discussed in terms of equilibrium phase diagrams and bulk phase stability [9]. However, numerous experimental studies demonstrate the formation of nanoscale clusters, metastable silicides, oxides, and mixed RE–Si–O configurations that cannot be adequately rationalized within equilibrium considerations alone [10–12]. Thermodynamics delineates the set of energetically accessible local configurations but does not uniquely determine which of these configurations will be realized under diffusion conditions characterized by limited atomic mobility and finite processing times. The final structural state is therefore governed not by global free-energy minimization, but by kinetic selection among multiple locally stable configurations, controlled by diffusion pathways, defect activity, chemical stabilization, and cooling history.

Kinetic aspects of rare-earth diffusion—such as defect-assisted transport, strain-driven aggregation, competition between RE–Si and RE–O bonding, and the decisive role of cooling— have often been addressed in a fragmented manner [13–15]. In particular, oxygen is frequently treated as a passive background impurity, while its active role in stabilizing clusters and metastable configurations remains insufficiently emphasized. Similarly, weak or suppressed diffraction signatures of rare-earth-related phases are commonly interpreted as evidence of their absence, rather than as manifestations of nanoscale size, low volume fraction, and kinetic stabilization.

The structural manifestations of this non-equilibrium evolution include lattice distortions, diffuse scattering contributions in X-ray diffraction patterns, broadened or weak reflections of rare-earth-related phases, and pronounced surface and interfacial heterogeneity [16–18]. These features are not parasitic artifacts of diffusion processing but intrinsic fingerprints of the diffusion scenario and its kinetic history, reflecting the competition between diffusion-driven aggregation, chemical stabilization, and arrested phase evolution.

A similar conceptual gap emerges in the interpretation of electrical and electrodynamic properties. While DC transport measurements probe long-range percolation pathways and average conductivity, they are inherently insensitive to localized charge dynamics and interfacial polarization phenomena. In contrast, AC excitation probes spatially confined charge displacement on time scales shorter than those required for macroscopic transport, rendering impedance spectroscopy intrinsically sensitive to non-equilibrium structural heterogeneity and extended interfacial regions. Impedance studies of rare-earth-modified silicon consistently reveal non-Debye relaxation behavior, constant phase element responses, and pronounced interfacial polarization effects [19–21]. These features are often analyzed phenomenologically, without explicit linkage to the underlying non-equilibrium structure formation.

In the present review, impedance spectroscopy is therefore not treated as a complementary characterization technique, but is explicitly repositioned as a physical probe of non-equilibrium structure formation. By accessing collective charge dynamics and interfacial polarization, impedance measurements provide direct insight into kinetically stabilized heterogeneity that remains largely inaccessible to equilibrium-based structural techniques and DC transport analysis. This review establishes a unified methodological framework linking thermodynamic constraints, kinetic selection, non-equilibrium structure formation, and electrodynamic response in rare-earth- modified silicon. Europium-diffused silicon is employed as a physically instructive representative system due to its strong oxygen affinity and pronounced tendency toward kinetic clustering, which accentuate non-equilibrium effects common to a broad class of rare-earth–silicon systems.

1. Thermodynamic constraints versus kinetic selection

Thermodynamic descriptions of rare-earth–silicon systems are traditionally based on equilibrium phase diagrams and bulk phase stability considerations [1,2]. Within this framework, the structural state of the system is assumed to be determined by global free-energy minimization under conditions of sufficient atomic mobility and extended relaxation times. While such an approach is appropriate for substitutional dopants with moderate solubility and weak chemical activity, it becomes fundamentally limited when applied to rare-earth diffusion in silicon.

In rare-earth-modified silicon, thermodynamics should therefore be regarded not as a predictive tool for the final structural state, but as a constraining framework that defines the boundaries of energetically accessible configurations. In this sense, equilibrium phase diagrams are necessary but fundamentally insufficient descriptors of the structural outcomes of rare-earth diffusion in silicon. Equilibrium phase diagrams specify which bonding states, clusters, and phases are thermodynamically permissible, but they do not determine which of these configurations will form under diffusion conditions characterized by finite processing times, defect-mediated transport, and rapid cooling.

Rare-earth elements are distinguished by large atomic radii, extremely low equilibrium solubility in silicon, and strong chemical affinity to impurities, particularly oxygen [3–6]. As a consequence, thermodynamics alone does not uniquely define the structural outcome of rare-earth diffusion. Instead, it delineates a complex free-energy landscape containing multiple local minima corresponding to rare-earth–silicon bonds, rare-earth–oxygen complexes, strain-stabilized clusters, and incipient phase nuclei. These configurations represent competing locally stable states rather than a single equilibrium phase assemblage.

The mere existence of multiple energetically accessible configurations does not imply their equal physical realizability. While thermodynamics defines the landscape of admissible states, kinetic factors govern how this landscape is explored. Under high-temperature diffusion conditions, defect activation in the silicon lattice, defect-assisted migration of rare-earth atoms, elastic strain accumulation, and chemical competition between RE–Si and RE–O bonding processes occur simultaneously and on overlapping spatial and temporal scales [7–9]. As a result, the system evolves along kinetically accessible pathways rather than relaxing toward an equilibrium configuration.

Kinetic selection therefore plays a decisive role in determining the final structural state. Kinetic factors control which thermodynamically allowed configurations are realized, how clusters nucleate and evolve, and how chemical stabilization—particularly via oxygen—arrests further structural reorganization. Cooling represents a critical kinetic stage, effectively freezing the system into metastable configurations and preventing subsequent equilibration, even when such equilibration would be thermodynamically favorable [10–12]. From a physical standpoint, cooling acts as an irreversible kinetic barrier that prevents subsequent relaxation toward thermodynamic equilibrium. Consequently, the final structure reflects the history of diffusion and cooling rather than equilibrium phase stability.

Within this framework, thermodynamics and kinetics should not be viewed as competing descriptions, but as complementary levels of physical constraint. Thermodynamics defines the boundaries of the accessible energy landscape, whereas kinetic selection governs the actual pathways through this landscape and determines which configurations are stabilized and retained. The structural state formed during rare-earth diffusion in silicon can thus be regarded as the result of constrained exploration of the thermodynamic energy landscape followed by kinetic fixation of selected configurations.

This conceptual distinction provides the methodological basis for analyzing specific kinetic pathways of diffusion, clustering, chemical stabilization, and phase evolution discussed in the following sections (Fig. 1).

Figure 1. Schematic representation of thermodynamic constraints and kinetic selection during rare-earth diffusion in silicon. Thermodynamics defines the accessible energy landscape, while kinetic pathways determine the selection and stabilization of metastable configurations during diffusion and cooling

2. Kinetic pathways of rare-earth diffusion

High-temperature diffusion of rare-earth elements in silicon proceeds through a set of competing kinetic pathways that operate simultaneously across different spatial and temporal scales [3,7,10]. Unlike classical dopant diffusion, which can often be described by a single effective diffusion coefficient, rare-earth diffusion involves defect-mediated transport, elastic strain effects, chemical interactions, and phase competition. The coexistence of multiple diffusion pathways renders any effective “averaged” description inadequate and necessitates a kinetic framework capable of capturing pathway-dependent structural evolution.

At the atomic scale, diffusion of rare-earth atoms is enabled by the activation of intrinsic point defects in the silicon lattice, including vacancies and interstitials [4,11]. Owing to their large atomic size and extremely low substitutional solubility, rare-earth atoms preferentially migrate via defect-assisted mechanisms rather than occupying regular lattice sites. These early atomic-scale processes determine the initial spatial localization of rare-earth species and generate localized lattice distortions that subsequently influence higher-level structural organization.

As diffusion progresses, elastic strain accumulates due to lattice mismatch between rare-earth- related species and the silicon matrix. This strain provides a driving force for aggregation, promoting the formation of nanoscale clusters as a means of reducing local elastic energy [12–14]. Importantly, strain-driven clustering is not an isolated phenomenon but emerges as a direct consequence of defect-assisted localization, linking atomic-scale diffusion processes to mesoscopic structural evolution.

A defining feature of rare-earth diffusion in silicon is the strong chemical affinity of rare-earth elements to oxygen. Even trace amounts of oxygen profoundly modify diffusion pathways by stabilizing rare-earth–oxygen complexes and mixed RE–Si–O configurations [6,15]. Oxygen- mediated stabilization suppresses long-range redistribution of rare-earth atoms, effectivelyconverting diffusive transport into an anchoring growth regime in which clusters become chemically and elastically pinned. This chemically induced anchoring suppresses alternative diffusion pathways that would otherwise promote long-range homogenization of the rare-earth distribution. This transition from diffusion-controlled redistribution to kinetically stabilized localization represents a critical mechanism underlying the formation of structurally heterogeneous regions embedded within the silicon matrix.

At elevated temperatures, kinetic competition between rare-earth–silicon bonding, rare-earth– oxygen complex formation, and the possible nucleation of silicide or oxide phases governs the evolution of the system [8,16]. However, owing to limited diffusion times and restricted atomic mobility, phase nucleation remains kinetically constrained. Consequently, the absence or weakness of diffraction signatures associated with rare-earth-related phases does not indicate the absence of phase-forming processes, but rather reflects their nanoscale character and kinetic stabilization. Phase evolution therefore proceeds without reaching equilibrium phase assemblages.

Cooling constitutes a critical kinetic step that fixes the entire diffusion history into a metastable structural state. Rapid or moderate cooling arrests atomic mobility and freezes the system into configurations selected during earlier diffusion stages, preventing subsequent equilibration even when thermodynamically favorable [9,17]. From a physical standpoint, cooling acts as an irreversible kinetic barrier that prevents subsequent relaxation toward thermodynamic equilibrium. In this sense, cooling does not merely terminate diffusion but acts as a physical mechanism of kinetic fixation, imparting irreversibility and structural memory to rare-earth-modified silicon.

Taken together, the kinetic pathways described above form a hierarchically organized cascade. Early atomic-scale processes associated with defect activation and chemical bonding define localization tendencies; intermediate processes involving elastic strain and cluster formation determine mesoscopic structure; and late-stage cooling fixes these configurations into metastable states. This hierarchy explains why rare-earth-diffused silicon cannot be characterized by a single diffusion parameter and why its structural state must be interpreted as a materialized record of its kinetic history rather than as an equilibrium configuration.

The dominant kinetic processes, their characteristic spatial and temporal scales, and their experimental manifestations are summarized in Table A.

Table A.

Hierarchy of thermodynamic constraints and kinetically selected processes governing non-equilibrium rare-earth diffusion in silicon

3. Structural and morphological manifestations

The kinetic pathways discussed in the previous section manifest structurally as a heterogeneous modification layer rather than as a uniformly doped region. This heterogeneity is a direct consequence of defect-assisted diffusion, strain-driven aggregation, oxygen-mediated chemical stabilization, and kinetic phase selection acting concurrently during diffusion and cooling. As a result, rare-earth-modified silicon develops a complex multiscale structural organization that cannot be reduced to equilibrium phase assemblages.

X-ray diffraction studies of rare-earth-diffused silicon consistently reveal several characteristic features, including broadening of silicon reflections, diffuse scattering contributions, and weak or suppressed diffraction peaks associated with rare-earth-related phases [14–16]. Peak broadening and asymmetry reflect elastic strain fields and local lattice distortions induced by embedded clusters and defect complexes. Diffuse background contributions indicate the presence of disordered or partially amorphous regions formed through non-equilibrium incorporation of rare-earth and oxygen species. Importantly, the absence of intense diffraction peaks corresponding to rare-earth phases does not imply their absence; rather, it reflects their nanoscale dimensions, low volume fraction, and kinetic stabilization.

At the nanoscale, rare-earth diffusion leads to the formation of chemically and elastically distinct regions embedded within the silicon matrix. These regions may consist of rare-earth– silicon, rare-earth–oxygen, or mixed RE–Si–O configurations, depending on the local chemical environment and diffusion history [6,15]. Their characteristic size, spatial distribution, and compositional variability are governed by kinetic constraints rather than equilibrium phase stability, resulting in a broad dispersion of structural motifs.

Morphological investigations using atomic force microscopy further corroborate the heterogeneous nature of rare-earth-modified silicon layers. Increased surface roughness, multiscale topographical features, and spatially varying contrast are commonly observed, reflecting subsurface clustering, strain accumulation, and interfacial organization [17,18]. These morphological signatures should not be interpreted as surface artifacts; instead, they represent the surface expression of underlying non-equilibrium structural heterogeneity within the diffusion- modified layer.

A central outcome of this non-equilibrium structural evolution is the formation of extended interfacial regions. Interfaces between clusters and the silicon matrix, as well as between chemically distinct domains, occupy a significant fraction of the modified layer volume. These interfacial regions act as sites of charge localization, defect accumulation, and polarization, thereby playing a decisive role in governing the electrodynamic behavior of the system.

The structural and morphological manifestations described above therefore constitute intrinsic fingerprints of the diffusion scenario and its kinetic history. Recognizing their non-equilibrium origin is essential for establishing causal links between diffusion conditions, structural organization, and functional properties. In particular, the dominance of interfacial regions and nanoscale heterogeneity provides the structural basis for the collective electrodynamic effects discussed in the following section. The principal structural and morphological manifestations of non-equilibrium rare-earth diffusion in silicon are summarized in Table B.

Table B.

Structural and morphological manifestations of non-equilibrium rare-earth diffusion in silicon

4. Impedance response and collective electrodynamic effects

The non-equilibrium structural heterogeneity formed during rare-earth diffusion has direct and unavoidable consequences for the electrodynamic response of silicon. In contrast to DC transport

measurements, which average over long-range percolation pathways, AC excitation probes localized charge displacement and interfacial polarization processes occurring on time scales shorter than those required for macroscopic charge transport. As a result, impedance spectroscopy is inherently sensitive to heterogeneous and kinetically stabilized structures.

Impedance spectra of rare-earth-modified silicon consistently exhibit non-Debye relaxation behavior, manifested by depressed semicircles in Nyquist representations and broad distributions of relaxation times [19–21]. Such behavior naturally arises in systems where charge carriers experience a wide spectrum of local environments associated with clusters, interfaces, and structurally disordered regions. Importantly, any spatially heterogeneous distribution of clusters and interfaces necessarily leads to a dispersion of local relaxation times, making non-Debye impedance response an intrinsic consequence of non-equilibrium structural organization rather than an experimental artifact.

Within this framework, constant phase element (CPE) behavior acquires a clear physical interpretation. Rather than representing a purely phenomenological fitting parameter, the CPE response reflects a continuous distribution of interfacial capacitances and resistances associated with rough, spatially extended, and often fractal-like networks of clusters and interfaces. Such networks emerge as a natural outcome of non-equilibrium diffusion, strain-driven aggregation, and kinetic phase stabilization.

At low and intermediate frequencies, Maxwell–Wagner–Sillars (MWS) polarization plays a dominant role in the dielectric response. Charge accumulation at interfaces between clusters and the silicon matrix, as well as between chemically distinct regions such as RE–Si and RE–O domains, leads to enhanced interfacial polarization and pronounced frequency dependence of the impedance response. The contribution of MWS polarization is therefore confined to frequency regimes where interfacial charge storage can develop and relax, rather than constituting a universal mechanism across the entire spectral range.

The causal relationship between non-equilibrium structural heterogeneity and the resulting impedance response is schematically illustrated in Fig. 2, which emphasizes the hierarchical connection between kinetic structure formation, interfacial organization, and collective electrodynamic effects.

Figure 2. Causal relationship between non-equilibrium structure formation and impedance response in rare-earth–diffused silicon

Kinetically stabilized clustering, strain-induced heterogeneity, and extended interfacial regions formed during high-temperature diffusion and subsequent cooling give rise to distributed relaxation times, interfacial charge accumulation, and collective electrodynamic effects, manifested as non-Debye impedance behavior under AC excitation.

A critical distinction must be drawn between AC and DC transport properties. While DC conductivity reflects the connectivity and continuity of percolation pathways across the heterogeneous layer, AC measurements probe localized and collective charge dynamics that do not require long-range charge transport. Consequently, many electrodynamic features associated with non-equilibrium structure formation remain inaccessible to DC measurements but emerge clearly in impedance spectra.

No specific equivalent circuit model is imposed in the present analysis. This choice is deliberate, as the primary objective of this review is to elucidate the physical mechanisms linking non-equilibrium structure formation to electrodynamic response, rather than to provide phenomenological fitting schemes. This choice is methodological rather than experimental and reflects the intent to preserve a direct physical linkage between impedance signatures and kinetically stabilized structural heterogeneity.

The electrodynamic manifestations discussed above therefore represent collective effects arising from the interplay between nanoscale clustering, extended interfacial regions, and kinetic phase stabilization. The characteristic impedance signatures associated with non-equilibrium rare-earth–silicon systems are summarized in Table C.

Table C.

Impedance signatures of non-equilibrium rare-earth–silicon systems.

This table summarizes characteristic impedance features in a phenomenological–structural mapping form, complementing the physical interpretation and causal analysis provided in the main text rather than duplicating it

5. Conclusions

This review has demonstrated that silicon modified by high-temperature rare-earth diffusion should be regarded as a kinetically stabilized, structurally heterogeneous system rather than as an equilibrium-doped semiconductor. The central novelty of the present work lies in repositioning impedance spectroscopy as a structure-sensitive probe of kinetic stabilization, capable of accessing non-equilibrium structural features that remain largely invisible to equilibrium-based structural techniques and DC transport measurements.

Thermodynamic considerations define the boundaries of the accessible energy landscape by constraining possible bonding configurations, cluster stability, and phase formation. However, the final structural state is not determined by thermodynamic equilibrium. Instead, it is governed by kinetic factors, including defect-assisted transport, strain-driven aggregation, oxygen-mediated chemical stabilization, and cooling-induced freeze-in. These processes selectively stabilize metastable configurations and impart kinetic irreversibility, rendering the structural state explicitly dependent on diffusion and cooling history. The resulting non-equilibrium structures manifest as lattice distortions, diffuse diffraction features, nanoscale clusters, and extended interfacial regions. These features should be interpreted as intrinsic fingerprints of the diffusion scenario rather than as secondary artifacts or indicators of poor crystallinity. Their collective presence reflects the constrained exploration of the thermodynamic landscape followed by kinetic fixation of selected configurations. A key consequence of this structural heterogeneity is its direct impact on electrodynamic behavior. Distributed relaxation times, non-Debye impedance response, constant phase element behavior, and Maxwell–Wagner–Sillars polarization emerge naturally from the dominance of interfaces and kinetically stabilized clusters. These effects are inherently collective and manifest primarily under AC excitation, explaining why they are largely inaccessible through DC conductivity measurements.

Although europium-diffused silicon was employed as a physically instructive representative system, the methodological framework developed in this review is broadly applicable to rare-earth–silicon systems in general. The explicit linkage between thermodynamic constraints, kinetic selection, non-equilibrium structure formation, and impedance response provides a transferable basis for interpreting experimental data beyond equilibrium-based approaches. Beyond interpretation, the proposed framework offers practical guidance for the design of diffusion processes and characterization strategies aimed at controlling structural heterogeneity and electrodynamic properties in rare-earth-modified silicon. By emphasizing kinetic pathways and interfacial organization, this work establishes a coherent physical methodology for future studies of non-equilibrium semiconductor systems.

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