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Electronic crystal growth

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Science  29 Sep 2017:
Vol. 357, Issue 6358, pp. 1378-1381
DOI: 10.1126/science.aal2426

Ordering and disordering electrons

When a liquid is cooled rapidly, it can form a glass, a state stuck between liquid and solid. Two groups looked in detail into analogous dynamics in electronic systems. Sato et al. and Sasaki et al. studied layered organic materials with a triangular in-plane lattice. These materials can assume a state in which their charge distribution has a regular pattern—an electronic or charge crystal. When the materials were cooled rapidly, a charge glass was formed instead and then allowed to crystallize. The dynamics of crystallization showed similarities to the analogous processes in conventional glasses.

Science, this issue p. 1378, p. 1381

Abstract

Interacting atoms or molecules condense into liquid, and, when cooled further, they form a crystal. The time evolution of the atomic or molecular ordering has been widely studied as a nonequilibrium emergence of order from a supercooled liquid or a glass. Interacting electrons in a variety of correlated electron systems also form crystals, but observing the time evolution of electronic crystallization has been experimentally challenging. Here, working with an organic conductor exhibiting a supercooled charge liquid or charge glass as a metastable state, we observed electronic crystal growth through resistivity and nuclear magnetic resonance measurements. The temperature profile of the crystal growth is similar to those observed in classical systems and reveals two distinct regimes for the mechanism of electronic crystallization.

A crystal is the most fundamental form of order in interacting many-body systems. The process of crystallization in atomic or molecular systems has been a major subject in the science and technology of condensed matter (1, 2). Experimentally, the crystallization of atoms or molecules is observable as a time-dependent phenomenon that occurs from a supercooled liquid or glass (3). In a system of electrons, the electrons interact repulsively and can form an electronic crystal called the Wigner crystal (4). In solids, electronic crystals with a periodicity distinct from that of the underlying lattice have been widely observed in the form of charge order (CO) in strongly correlated electronic systems (5). An open question in this field is whether crystallization in a quantum system proceeds in the same manner as in conventional classical systems. To address this question, we observe and study the process of electronic crystallization in the layered organic material θ-(BEDT-TTF)2RbZn(SCN)4 (hereafter θ-RbZn) (Fig. 1A) (6).

In the conducting layers of θ-RbZn, BEDT-TTF molecules form a geometrically frustrated quasitriangular lattice (Fig. 1, B and C) and accommodate one hole per two molecules (a quarter-filled band). In this band filling condition, Coulomb repulsion among electrons generally forces them to form an insulating electronic crystal (called charge order), as has been observed in many materials (7). In triangular lattices, however, geometrical frustration works against long-range electronic crystallization (810) and is suggested to lead to glassy electronic states, even in the absence of quenched disorder (11, 12). Consequently, θ-RbZn exhibits an electronic crystal with the in-plane stripy order (upper inset in Fig. 1D) stacked out-of-plane below the ordering temperature, TCO = 198 K, only when cooled slowly (<1 K/min) (Fig. 1D) (1315); when cooled rapidly (>5 K/min) (Fig. 1D), the transition is bypassed to produce a supercooled state (16, 17), followed by a charge glass (CG) transition at about Tg= 160 to 170 K (8). Our previous study on a series of θ-(BEDT-TTF)2X with various anions X revealed that the critical cooling speed required for glass formation reduces as the lattice frustration becomes stronger; for example, θ-(BEDT-TTF)2CsZn(SCN)4, with a nearly equilateral triangular lattice, shows CG but not CO even when cooled slowly on a laboratory time scale (9, 10). θ-RbZn is suitable for investigating electronic crystallization on an experimentally accessible time scale. We note that the CG in the present system is distinct from the previously observed Coulomb glass (18, 19), which is a state associated with quenched disorder that does not undergo the type of crystallization exhibited by θ-RbZn. Unlike atomic or molecular systems, an electronic crystal and glass (or supercooled liquid) can be probed by charge- and spin-sensitive measurements. The resistivity is much lower in the supercooled electronic liquid/glass state than in the electronic crystal state and thus characterizes the electronic state macroscopically. Moreover, to examine the crystallization process on a molecular scale, we exploit 13C nuclear magnetic resonance (NMR), which is capable of distinguishing the electronic liquid/glass and crystal states through NMR spectra reflecting local charge/spin disproportionation characteristic of the liquid, glass, and crystal states [section III of (20)]. In the present work, θ-RbZn was quenched (4 to 6 K/min) to the target temperature, Tq, from above TCO to prepare a supercooled or glass state. Next, the time evolution of the electronic state during electronic crystallization was investigated by monitoring the resistivity and NMR spectrum, while keeping the temperature at Tq to unveil the process of formation of the electronic crystal.

Fig. 1 Structure and transport properties of θ-(BEDT-TTF)2RbZn(SCN)4.

(A) A large-scale view of a crystal structure of θ-(BEDT-TTF)2RbZn(SCN)4. BEDT-TTF denotes the molecule of bis(ethylenedithio)-tetrathiafulvalene. (B) A side view of the layered structure. The two-dimensional conducting layers of BEDT-TTF molecules are separated by insulating anion layers of RbZn(SCN)4. (C) Horizontal cross section showing the structure of the conducting layer, in which BEDT-TTF molecules form an anisotropic triangular lattice, providing geometrical frustration against charge ordering. (D) The temperature dependence of resistivity at different cooling rates. Slow cooling (<1 K/min, red plot) results in a CO state at TCO = 198 K. The CO pattern of the horizontal cross section is illustrated in the upper inset. The colors (dark red, red, and light red) in the insets indicate the magnitude of the charge density. When the system is cooled rapidly (>5 K/min, blue points), the first-order CO transition is avoided and gives way to a supercooled state, followed by a CG transition around Tg ~ 165 K. The lower left inset illustrates the CG state.

Figure 2A displays the time dependence of resistivity at several temperatures. At Tq= 191 K (just below TCO), it takes more than 104 sec for the crystallization to start; however, once it begins, it is completed rapidly. When Tq is lowered to 188 K and subsequently to 180 K, the onset time becomes shortened. When Tq is lowered further, however, the onset time becomes longer, and the profile of the time evolution of the resistivity changes such that the crystallization proceeds gradually after it sets in. We define the crystallization time (tcry) as the time at which resistivity reaches a fixed fraction of the saturated values. The plot of tcry versus Tq for several fixed fractions (Fig. 2B) shows dome-like structures. Curves of this shape have been widely observed for the crystallization of structural glasses, such as ionic glasses, chalcogenide glasses, metallic alloys, and so on (2123); they are referred to as time-temperature-transformation (TTT) curves. The present results provide evidence that the TTT curve also applies to electronic crystallization. The characteristic TTT curve was reproduced for all the samples studied, although the onset time for the resistivity increase was slightly sample-dependent, probably owing to disorder in the sample, which generally affects nucleation [section V of (20)].

Fig. 2 Time evolution of electronic crystallization from a supercooled liquid or a glass.

(A) The time dependence of resistivity measured at different Tq after rapid cooling across TCO. The plot for each temperature is normalized by the initial value, ρ(Embedded Image), and the saturated value, ρ(Embedded Image), which is determined by the resistivity in a CO state under slow cooling. (B) The time-temperature-transformation (TTT) diagram of θ-RbZn determined from (A). The color scale represents the percentage of resistivity recovery. The circles indicate the tcry for particular percentages (1, 10, 20, 30, 40, and 50%) in resistance recovery. (C) Schematics of the crystallization rate as a function of temperature. The time of crystallization (red curve) is determined by the combination of nucleation (blue dashed curve) and growth (yellow dashed curve) processes.

The TTT curve for the structural crystallization is explained by classical nucleation theory, which is based on the two processes governing crystallization: nucleation and growth (24, 25). In the first step, small nuclei are generated in a supercooled liquid or a glass through structural fluctuations at a rate given byEmbedded Image(1)where I0 is a constant, Embedded Image is a kinetic factor, and Embedded Image is the free-energy barrier for the formation of a critical nucleus (26). The key competing factors that determine Embedded Image are the crystal/liquid interface energy cost, Embedded Image, which acts against nucleation, and the free-energy difference between crystal and liquid, Embedded Image, which promotes nucleation as a thermodynamic driving force and increases as T is lowered. Embedded Image is known to be expressed in the form of Embedded Image (27). The kinetic factor Embedded Image, which measures the diffusivity in the configurational space in the glass state, decreases in magnitude toward low temperatures. In the second process, the formed nuclei increase in size at a rate given byEmbedded Image(2)where Embedded Image is a constant, and Embedded Image is the volume per particle (28). The general temperature profiles of I and V expected from these equations are such that the growth rate reaches a maximum at a higher temperature than that of the nucleation rate (29, 30) (Fig. 2C), resulting in the total crystallization rate that reaches a maximum in between the two domes (Fig. 2C). Thus, the maximum crystallization rate at the “nose temperature” (Tn), which is approximately 170 to 180 K in the TTT curves for the tens of percent resistance recovery that is indicative of sizable crystallization (Fig. 2B), signifies a crossover between nucleation- and growth-dominated regimes. This explains the distinctive time evolutions of resistivity above and below the Tn (Fig. 2A); the abrupt resistivity increase after a long quiescent time for T > Tn indicates that the nucleation (as a rare event) is immediately followed by a rapid growth of the crystal seeds, whereas the gradual resistivity increase for T < Tn suggests that the growth rate (which is suppressed under the nucleation rate at low T) regulates the speed of crystallization. In the former regime, macroscopic inhomogeneity likely occurs; indeed, we often encountered irregular time evolutions of resistivity [sections V and VI of (20)] suggestive of the macroscopic inhomogeneity above Tn, whereas the time evolution of resistivity was always smooth below Tn, as expected for finely distributed microcrystals, which are expected to behave like an effectively homogeneous medium.

The two regimes of crystal growth are clearly distinguished in the resistivity-fraction dependence of the TTT curves in Fig. 2B. Noticeably, the Tn nominally shifts from 170 to 180 K in the TTT curve of the 50% resistance recovery to 160 K for that of the 1% resistance recovery. The initial resistivity increase is caused by nucleation. Given that it is measured by the 1% TTT curve, the nucleation rate is suggested to peak at ~160 K, a value lower than Tn for the total crystallization, in accordance with the general behavior of structural glasses (Fig. 2C).

Next, we examine the crystallization process through investigation of the 13C-NMR spectra, which serve as a probe of the charge density profile [section III of (20)]. The magnetic field of 6 T was applied in a particular direction within the ab plane to form the so-called “magic angle” with the 13C=13C vector in the 13C-enriched BEDT-TTF molecule to avoid the unwanted complications of the NMR spectra arising from nuclear dipolar splitting and molecular inequivalence against the field direction [section III of (20)]. At room temperature, the spectrum has a two-line structure (Fig. 3A) originating from the two adjacent 13C sites in the center of the molecule (Fig. 1A). As T is lowered, the lines become broadened because of the slowing down of charge fluctuations (Fig. 3B). When T slowly passes through TCO, the spectrum exhibits the structural characteristics of a CO state (Fig. 3C). The complicated structure is experimentally separated into two components arising from the charge-rich and -poor molecular sites by taking advantage of the ~40-fold difference in the relaxation times at the two sites (the relaxation time is inversely proportional to the square of charge density in the paramagnetic state) (14, 31) [section III of (20)]. When the system is rapidly cooled to Tq through TCO, however, the spectrum exhibits a broad feature indicative of a CG state with spatially inhomogeneous charge density. Next, the spectrum was traced as a function of time at each Tq. The case of Tq = 140 K is shown in Fig. 3D; the broad spectrum observed just after cooling to 140 K gradually changes its structure with time and eventually takes the shape characteristic of the CO state. This is a microscopic observation of the electronic crystal growth from a glass state.

Fig. 3 Evolution of 13C NMR spectra during electronic crystallization.

(A and B) 13C-NMR spectra under a magnetic field of 6 T applied in the “magic angle” direction within the ab plane [section III of (20)] at room temperature (A) and at 210 K (B). At room temperature, the charge density is homogeneous, resulting in a sharp two-line structure originating from the two nonequivalent 13C nuclei in a BEDT-TTF molecule [section III of (20)]. Upon cooling, the charge dynamics slows down, resulting in the spectral broadening to the degree that the two-line structure in the spectrum is unresolved at 210 K. (C) 13C-NMR spectra of a CO state obtained at 190 K after slow cooling from above TCO (~200 K). The sharp and broad components (shaded with light red and dark red colors, respectively) originate from charge-poor and charge-rich sites, respectively. (D) The time evolution of 13C-NMR spectra during the crystallization process at 140 K. The blue and red parts indicate the CG and CO components fitting the spectra, respectively [section IV of (20)]. The sum of the two components in the fitting (dashed lines) reproduce the observed spectra.

The time dependence of the volume fraction of the CO domains is determined by decomposing the spectrum at each time into the CG and CO components with the relative intensity as a fitting parameter [section IV of (20)]. Figure 4, A and C, respectively, show the time evolution of the CO-domain fraction at 191 and 150 K. Remarkably, the two curves are distinctive in terms of their relation to the time evolution of conductivity. At 191 K, the CO-domain fraction rapidly increases in parallel with a steep decrease in conductivity; the correspondence between the two evolution curves appears reasonable, considering that conductivity should show an appreciable change when the CO domains develop to a sizable fraction of the sample volume. At 150 K, however, the CO domains remain undeveloped, whereas the conductivity shows a considerable decrease during the period of 400 to 2000 s (highlighted by yellow shading in Fig. 4C). This feature was reproduced at 160 K as well [section VII of (20)]. This puzzling behavior suggests that, although a considerable fraction of the sample becomes highly resistive, this fraction is not the CO domains. A clue for resolving this puzzle is found in the recent study of colloidal hard-sphere systems, which revealed that metastable intermediate domains different from both glass and crystalline phases are formed before the formation of true crystallite (32, 33) (Fig. 4D). Below Tg, where charge fluctuations are nearly frozen in the present case, the formation of crystal domains, which are formed at the cost of the interface energy, is less promoted than at higher temperatures. In this situation, an intermediate structure between true crystal and glass likely can reduce the interface energy cost Embedded Image, thereby decreasing the free-energy barrier for nucleation, Embedded Image. In the present system, the metastable intermediate domains are highly resistive but show no clear difference in the NMR spectra from the glass state. Because the NMR spectra reflect the charge disproportionation (14), the results indicate that the disproportionate charge density is not appreciably developed in the intermediate state. Instead, the spatial configuration of charge density is probably changed from that of the initial glass state. Subsequently, the CO domains emerge after the intermediate domains grow considerably (Fig. 4, C and D). The present results suggest that this distinct two-step nucleation mechanism occurs at low temperatures below Tn. The profile of the time evolution of the crystal fraction evaluated by NMR appears different at 191 and 150 K. We examined the profile in terms of the so-called Avrami formula, which contains the information on the dimensionality of the crystal growth. The analysis suggests that the crystal growth occurs in a three-dimensional manner at 191 K but in one or two dimensions at 150 K, which is supportive of the differentiation in the crystal-growth mechanism in the high- and low-temperature regimes [section VIII of (20)].

Fig. 4 Comparison of the time evolution of conductivity and NMR spectra during electronic crystallization.

(A and C) Time dependences of the CO-domain fraction (red), which is deduced from the NMR spectra, and conductivity (blue) at (A) 191 K and (C) 150 K. (Insets) Conductivity for 191 and 150 K are plotted on a logarithmic scale. (B and D) Schematics of the CO-domain evolution in (B) the high-T regime and (D) the low-T regime.

We note that the lattice degrees of freedom are also involved in the charge crystallization and vitrification, which are accompanied by a long-range two-fold lattice modulation and a short-range three-fold lattice deformation, respectively. The clear correlation between the charge frustration and the glass-forming ability (10) and a theoretical indication of the crystallization and vitrification without the lattice degrees of freedom (11) suggest that the frustrated Coulomb interactions play a vital role in the present phenomena. Notwithstanding, how the lattice degrees of freedom affect the mechanism of the crystallization is an interesting and open issue.

Supplementary Materials

www.sciencemag.org/content/357/6358/1378/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

References (3436)

References and Notes

  1. Materials and methods are available as supplementary materials.
  2. Acknowledgments: This work was supported in part by Japan Society for the Promotion of Science (JSPS) KAKENHI under grant no. 25220709. T.S. was supported by a JSPS Research Fellowship (no. 26-7870). Requests for data should be addressed to the corresponding authors.
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