Spatiotemporal observation of quantum crystallization of electrons

Crystallization from supercooled liquids or glasses is a fundamental nonequilibrium process typically governed by classical nucleation and growth dynamics. For Coulomb-interacting electrons, long-range interactions favor Wigner crystallization; in crystalline materials this manifests as charge order (CO) on an underlying lattice. On geometrically frustrated lattices, such as quasi-triangular networks at quarter filling, mismatch between preferred electronic ordering and the lattice can hinder long-range periodicity and instead produce a charge glass (CG). Layered molecular conductors θ-(BEDT-TTF)2X provide a platform where CO and CG compete, controlled by lattice anisotropy and frustration. Notably, θ-(BEDT-TTF)2RbZn(SCN)4 (θ-RbZn) sits near the boundary between CO and CG: slow cooling yields CO at T_CO ≈ 200 K, whereas rapid cooling produces a supercooled charge liquid (SCL) that vitrifies into CG at lower temperatures. Prior transport and NMR studies showed electron itinerancy within the CG and characterized crystallization back to CO, posing a central question: how does quantum character of electrons affect nucleation and growth during crystallization, and what aspects remain universal compared with classical systems? The present study addresses this by directly imaging, in space and time, the electronic crystallization from SCL/CG to CO using Raman microspectroscopy.

Previous work established glassy electronic states and their crystallization in frustrated organic conductors. The θ-(BEDT-TTF)2X family exhibits either CO or CG depending on triangular-lattice anisotropy, with CG showing classical glass hallmarks (slow fluctuations, medium-scale correlations, aging). Transport and NMR indicated itinerant electronic character within CG and SCL, implying a quantum-classical energetic hierarchy. In θ-RbZn, slow cooling yields CO at T_CO ≈ 200 K, while rapid cooling leads to SCL that vitrifies into CG; earlier transport and NMR measurements characterized CO evolution and suggested distinct high- and low-temperature crystallization behaviors. Raman and infrared spectroscopy have been established as sensitive probes of charge disproportionation in BEDT-TTF salts via C–C stretching modes, enabling phase identification. Theoretical advances have begun to explain CG properties via quantum simulations, including continuous charge density distributions and temperature-dependent spectral features, supporting a quantum-mechanical configuration interaction picture of charge patterns.

• Material: θ-(BEDT-TTF)2RbZn(SCN)4 single crystals with 13C enrichment at the central C=C of BEDT-TTF to isolate the charge-sensitive ν2 mode. Typical crystal size ~1 × 0.1 × 0.1 mm^3.
• Experimental setup: Raman microspectroscopy (Renishaw inVia) with 532 nm excitation, backscattering geometry, 1800 g/mm grating, CCD detection, Leica N Plan L×50 objective (NA=0.5). Polarization: excitation E∥a, scattered light collected for E∥a and E∥c. Renishaw StreamLine used for rapid imaging.
• Temperature control: Samples mounted on copper, cooled on a Linkam 10002L stage with optical access. Cooling protocols: rapid quench at 30 K/min to various quench temperatures T_q (140–200 K) to metastabilize SCL/CG; slow cooling at ≤1 K/min to realize CO. After each isothermal run, sample warmed to 210 K and sequence repeated for different T_q.
• Probed depth and spatial resolution: Observation depth ≈ min(confocal depth ~2.5 µm, optical penetration depth ~0.67 µm) ~1 µm (≈500 molecular layers). In-plane spatial resolution 6.5 µm per pixel; typical mapped areas ~65 × 130 µm^2; extended up to ~124.8 × 416 µm^2.
• Spectral analysis and phase fraction extraction: The time-dependent Raman spectrum I(ν,t) at the ν2 mode is modeled as a linear combination of reference spectra for CO and CG/SCL: I(ν,t) = A_CO(t) I_CO(ν) + A_CG(t) I_CG(ν). The CO volume fraction is defined as φ(t) = A_CO(t) / [A_CO(t) + A_CG(t)]. Spectral references and fitting details are in Supplementary Note 1.
• Time evolution protocols: After quenching to T_q, Raman spectra/imaging acquired as a function of time until φ→1. For high T_q (above ~165 K), nucleation incubation and subsequent rapid growth are captured; for low T_q (below ~165 K), gradual φ rise consistent with many micro-nucleation events.
• Kinetic modeling: At T_q below the nose temperature (~165 K), φ(t) is fit by the Avrami equation φ(t)=1−exp(−K t^n), extracting K and the Avrami exponent n to infer growth dimensionality (n≈d+1). A time–temperature–transformation (TTT) diagram constructed by contouring φ in the t–T_q plane.
• Real-space growth-rate extraction: From Raman images at T_q=195 K, the φ(x) profile across the moving SCL–CO interface is fit by φ(x) = {1 + tanh[(x−x_e)/x_0]}/2 (phenomenological interface form). The interface position x_e(t) yields a linear advance with time; the slope gives the growth rate v. Directional dependence of v tested and found negligible.
• Temperature dependence of growth: v(T) measured across 170–200 K. Comparison with classical models using v = k(T){1−exp(−Δµ/kT)}. Wilson–Frenkel framework assessed by setting k(T)∝D(T); instead, the characteristic frequency f(T) of resistivity noise (probes SCL charge fluctuations) used as kinetic factor, giving v = f(T) l {1−exp(−Δµ/kT)} with l taken as a lattice constant (~0.5 nm) and Δµ≈ΔH/T_CO (ΔH≈160 K). Alternative fits with k(T)=k0 also tested. Effective step length l back-calculated as l = v/[f(T){1−exp(−Δµ/kT)}] to assess model plausibility.

• Direct spatiotemporal imaging: Raman microspectroscopy visualizes real-space and real-time electronic crystallization from SCL/CG to CO in θ-RbZn, distinguishing the CO’s two-peak ν2 spectrum from the broad SCL/CG feature.
• Distinct regimes separated by a nose temperature: The TTT diagram shows a nose at T_n≈165 K. Above T_n, φ(t) exhibits an incubation time (≈4×10^2–10^4 s depending on T_q) with a rapid rise once nucleation occurs, often dominated by a single nucleation event that sweeps the field of view. Below T_n, φ(t) rises gradually over time, well fit by the Avrami law.
• Avrami kinetics and dimensionality: Avrami exponent n=2.6–3.5 below T_n indicates growth consistent with two-dimensional extension (n≈d+1) of numerous microcrystals within layers.
• Spatial growth profiles: At 195 K (above T_n), crystallization proceeds via nucleation of a CO microdomain followed by rapid front propagation across areas up to ~125×416 µm^2, with single nucleation dominating; the SCL–CO boundary appears graded, consistent with depth averaging and possible layer-by-layer mixing. At 155 K (below T_n), φ evolves nearly homogeneously across the sample at the 6.5 µm resolution, indicating sub-micrometer nucleation/growth scales.
• Growth rate v(T): Interface position x_e advances linearly in time, indicating interfacial reaction–limited kinetics. v increases roughly linearly with undercooling ΔT=T_CO−T near T_CO and saturates at ≈10^3 nm/s at low T. v shows little anisotropy with growth direction.
• Breakdown of classical growth models: Wilson–Frenkel-based estimates using v = f(T) l {1−exp(−Δµ/kT)} with l≈0.5 nm and ΔH≈160 K underpredict v by 3–5 orders of magnitude. Treating l as adjustable yields unrealistically large l up to ~100 µm at 170 K. A gapless kinetic factor k(T)=k0 fit requires ΔH≈4600 K (∼30× experimental) and k0≈840 nm/s, also implausible.
• Quantum-crystallization signature: The anomalously high growth kinetics point to the quantum nature of electrons in SCL/CG (finite conductivity ~10 S/cm at 150 K, extended wavefunctions) and a sharp change to localized CO upon crystallization, suggesting a quantum-to-classical transition at the interface underpinning ultrafast growth compared to classical particle systems.

The study directly addresses how quantum electronic character affects nucleation and growth during crystallization. Universality is observed in the temperature-dependent division between nucleation-dominated and growth-dominated regimes, the Avrami behavior below the nose temperature, and the TTT nose structure—features shared with classical crystallization. However, the crystal–liquid interface propagates orders of magnitude faster than predicted by classical Wilson–Frenkel-type models or by kinetics based on thermally activated diffusive steps, even when incorporating measured fluctuation frequencies and reasonable thermodynamics. The failure of classical descriptions implies a fundamentally different kinetic pathway in which electronic quantum coherence or extended wavefunctions in SCL/CG enable rapid interfacial ordering, followed by wavefunction localization in CO—a quantum-to-classical catastrophe. The minimal directional dependence and interfacial reaction–limited kinetics further point to interface physics as the rate-determining step. Lattice involvement is acknowledged (electronic configuration changes must couple to structural distortion), but phonon propagation speeds far exceed the observed growth rates, suggesting the lattice is not the bottleneck. The results thus highlight interface electronic dynamics as the key to quantum crystallization and motivate theories that explicitly include quantum electronic degrees of freedom at the moving interface.

This work achieves real-space and real-time observation of electronic crystallization in θ-(BEDT-TTF)2RbZn(SCN)4 via Raman microspectroscopy, revealing a canonical nucleation-and-growth picture with a TTT nose at ~165 K and Avrami kinetics below the nose, while uncovering extraordinarily fast interfacial growth that cannot be reconciled with classical Wilson–Frenkel-type models. The findings suggest that the quantum nature of electrons in SCL/CG underpins the ultrafast growth and a quantum-to-classical transition across the interface. These results open a pathway to study crystallization in quantum systems. Future research directions include: microscopic spectroscopy and theory of the SCL–CO interface dynamics; quantitative modeling of quantum-kinetic factors beyond diffusion; probing dimensional coherence of growth (in-plane vs out-of-plane); clarifying the role of lattice distortions and electron–phonon coupling; extending spatiotemporal studies to other frustrated electronic materials.

• Spatial resolution (6.5 µm) and depth averaging (~1 µm over ~500 layers) limit detection of sub-micrometer nucleation and preclude resolving interlayer coherence during low-temperature crystallization.
• The Avrami analysis infers growth dimensionality indirectly; direct structural probes at comparable scales are lacking.
• Growth-kinetics modeling relies on prior noise spectroscopy f(T) and thermodynamic estimates (ΔH), introducing uncertainties; effective parameters inferred from classical models become unphysical.
• Observations are focused on a single compound (θ-RbZn); generality across materials remains to be established.
• Stochastic nucleation means single-event dominance in finite fields of view; rare multiple nucleations may be under-sampled.