Research ArticleCONDENSED MATTER PHYSICS

Slow dynamics of electrons at a metal–Mott insulator boundary in an organic system with disorder

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Science Advances  11 Aug 2017:
Vol. 3, no. 8, e1601594
DOI: 10.1126/sciadv.1601594

Abstract

The Mott transition—a metal-insulator transition caused by repulsive Coulomb interactions between electrons—is a central issue in condensed matter physics because it is the mother earth of various attractive phenomena. Outstanding examples are high–Tc (critical temperature) cuprates and manganites exhibiting colossal magnetoresistance. Furthermore, spin liquid states, which are quantum-fluctuation–driven disordered ground states in antiferromagnets, have recently been found in magnetic systems very near the Mott transition. To date, intensive studies on the Mott transition have been conducted and appear to have established a nearly complete framework for understanding the Mott transition. We found an unknown type of Mott transition in an organic spin liquid material with a slightly disordered lattice. Around the Mott transition region of this material under pressure, nuclear magnetic resonance experiments capture the emergence of slow electronic fluctuations of the order of kilohertz or lower, which is not expected in the conventional Mott transition that appears as a clear first-order transition at low temperatures. We suggest that they are due to the unconventional metal-insulator fluctuations emerging around the disordered Mott transition in analogy to the slowly fluctuating spin phase, or Griffiths phase, realized in Ising spin systems with disordered lattices.

INTRODUCTION

The Mott transition is realized when the U/W ratio of the electron correlation energy U to the typical kinetic energy (or bandwidth) W exceeds a critical value. In several real materials, pressure can experimentally control W, causing pressure-induced Mott transitions. Extensive research on the Mott transition has been conducted over the last few decades and has established a consensus that the Mott transition is of the first order at low temperatures and a crossover at high temperatures in the pressure-temperature phase diagram (16), although the nature of the criticality is under debate; unconventional critical exponents are indicated by conductivity (7) and nuclear magnetic resonance (NMR) relaxation rate (8) and by the theories of marginal quantum criticality (9) and dynamical mean-field theory (DMFT) (10), whereas the mean-field or Ising values are indicated by expansivity (11) and conductivity (12, 13) as well as theoretical studies (1416).

The experimental studies on the Mott transition have been conducted in both organic and inorganic materials. One studied type of organic Mott insulator is the X[Pd(dmit)2]2 family, where dmit denotes an organic ligand, 1,3-dithiole-2-thione-4,5-dithiolate, and X is a monovalent closed-shell cation (17). In particular, X[Pd(dmit)2]2 salts with X = EtMe3Sb and EtMe3P have attracted considerable attention from the magnetism perspective because they have quasi–two-dimensional (2D) nearly regular triangular lattices. The EtMe3P salt shows a valence bond–solid ground state with a spin gap (18), whereas the EtMe3Sb salt shows a spin liquid ground state (19), the nature of which remains highly controversial (2024). The important difference between the crystal structures of these two salts is that the EtMe3Sb salt (space group C2/c) has a quenched random orientation of the ethyl groups in the cation layer (17), as shown in Fig. 1A, whereas the EtMe3P salt (space group P21/m) has an ordered orientation of the ethyl groups with no randomness.

Fig. 1 Structure and transport properties of EtMe3Sb[Pd(dmit)2]2.

(A) Side view of the layered structure of EtMe3Sb[Pd(dmit)2]2 (41). Mott-insulating Pd(dmit)2 layers are separated by closed-shell cation layers of EtMe3Sb. The cation layers have random orientations of the ethyl groups. (B) Temperature dependence of the in-plane resistivity of EtMe3Sb[Pd(dmit)2]2 under several pressures measured using the standard four-electrode method.

The EtMe3P salt undergoes a conventional Mott transition under pressure, as expected; transport studies have revealed that the EtMe3P salt shows a first-order Mott transition with a critical end point of 30 K (13, 25). However, the EtMe3Sb salt, which has inevitable randomness in the cation layer, does not show any resistivity jump indicating a first-order transition, as shown in Fig. 1B. Note that, even in the highly conductive state stabilized under high pressure, the resistivity increases slightly during cooling at low temperatures. This behavior is likely the manifestation of the randomness, and the randomness could be responsible for the peculiarity in the Mott transition in the EtMe3Sb salt. Stimulated by this observation, we further examined the nature of the Mott transition in the EtMe3Sb salt by 13C NMR and found anomalously slow dynamics of electrons never seen before.

RESULTS

Figure 2 shows the temperature dependence of the spin-lattice relaxation rate (T1−1; see Materials and Methods) of the EtMe3Sb salt, which reflects the magnitude of the fluctuations of the internal magnetic field on the 13C nuclei on a megahertz time scale. In the Mott-insulating state at ambient pressure, T1−1 shows a large drop below 1 K. This drop appears to suggest a marginal spin-gapped ground state, as discussed by the works of Itou and co-workers (21, 23), whereas there are also reports claiming the existence of fully gapless fermion-like excitations (20, 22, 24). For pressures below 5 kbar, the temperature dependence of T1−1 is almost the same, indicating that the nature of the Mott-insulating phase is maintained in this pressure region. However, at higher pressures, T1−1 is depressed and comes to follow the Korringa relation, namely, T1−1 decreases in proportion to the temperature. This behavior indicates that the Mott boundary is situated at approximately 6 kbar, consistent with the resistivity result shown in Fig. 1B.

Fig. 2 Temperature dependence of the 13C nuclear spin-lattice relaxation rate (T1−1) of EtMe3Sb[Pd(dmit)2]2 under several pressures, reflecting the amplitude of the fluctuations of the internal magnetic field on a megahertz time scale.

The dashed line shows a fit of the data at 15 kbar to the Korringa relation T1−1T.

Figure 3A shows the temperature dependence of the spin-spin relaxation rate (T2−1; see Materials and Methods) of X[Pd(dmit)2]2. The spin-spin relaxation rate T2−1 is generally expressed as the sum of two different components—the Lorentzian relaxation rate T2l−1 and the Gaussian relaxation rate T2g−1 (T2−1 = T2l−1 + T2g−1)—which measure the electron fluctuations on a kilohertz time scale and nuclear-nuclear magnetic coupling, respectively. In X[Pd(dmit)2]2 systems, T2g−1 is determined by the nuclear dipolar interaction and is, thus, invariant with respect to temperature. A constant value of 640 s−1 is observed in the EtMe3P salt, the conventional Mott insulator. In contrast, the EtMe3Sb salt shows appreciable enhancement relative to this value. This enhancement indicates the significant contribution of T2l−1, which can be estimated by subtracting T2g−1 (=640 s−1) from the observed value of T2−1, and is shown on the right axis of Fig. 3A (for details, see the Supplementary Materials). Note that, although the cation has the rotational motion of the methyl groups (23), this lattice motion does not contribute to the relaxations (for details, see the Supplementary Materials).

Fig. 3 Profiles of the 13C nuclear spin-spin relaxation rate (T2−1) of EtMe3Sb[Pd(dmit)2]2 under several pressures.

(A) Temperature dependence of T2−1 of EtMe3Sb[Pd(dmit)2]2 under several pressures. For comparison, data for EtMe3P[Pd(dmit)2]2 at ambient pressure are also plotted as open black circles. The left axis shows the observed raw value of the whole spin-spin relaxation rate. The value of T2l−1 (the right axis) is estimated by subtracting 640 s−1 from T2−1 (see Materials and Methods and the Supplementary Materials). (B) The two components of T2l−1 of EtMe3Sb[Pd(dmit)2]2. The amplitude of the blue component shows a peak at 10 to 30 K. The amplitude of the blue component at low temperatures (where both components exist) is estimated by extrapolating the trend at higher temperatures. The amplitude of the red component is enhanced at low temperatures. This enhancement is particularly prominent at 7 kbar, which is near the Mott boundary.

In typical electron systems, no significant difference in the intensity of the fluctuations between the kilohertz and megahertz regions should exist, namely, T2l−1 = T1−1, because the electron fluctuations are dominated by the Hamiltonian of the system and, thus, generally white well below the energy scales of the transfer integrals, exchange interactions, and Coulomb repulsions, all of which reside in the terahertz region. In contrast to this common understanding, in the EtMe3Sb salt, T2l−1 increases to ~103 s−1 at low temperatures, which is several orders of magnitude greater than T1−1, as seen in Figs. 2 and 3A. This difference signifies the emergence of an unusual electronic phase that has extraordinarily slow fluctuations, where the kilohertz fluctuations are several orders of magnitude stronger than the megahertz ones.

The anomalous electronic fluctuations should occur in either the spin or the charge channel: (i) direct spin fluctuations, which cause NMR relaxations through hyperfine interactions, and (ii) charge fluctuations, which also cause NMR relaxations through appendant temporal modulations of hyperfine interactions accompanying the charge fluctuations. Note that the temperature dependence of the enhancement of T2l−1 is not monotonous [this can be distinctly observed around the Mott boundary pressure (7 kbar)], as shown in Fig. 3A. At ambient pressure, T2l−1 increases appreciably below 30 K, peaks at approximately 12 K, and then monotonously decreases down to 1 K. At 4 kbar, the peak is somewhat suppressed; however, a separate increase in T2l−1 appears below 2.5 K. When the pressure is increased to 7 kbar, the low-temperature increase in T2l−1 is substantially enhanced, whereas the high-temperature peak is not. At 15 kbar, both of the features are suppressed. Two structures are thus evident in the temperature profile of T2l−1, as shown in Fig. 3B, implying two different types of slow fluctuations. One type is a component that grows at low temperatures, whereas the other is a component that shows a peak at 10 to 30 K. The former component (the red component in Fig. 3B) is markedly prominent at 7 kbar, which is near the Mott boundary. Therefore, the former component is most likely attributable to charge fluctuations related to the Mott transition, that is, slow fluctuations between the metallic state and the Mott-insulating state develop strongly around the Mott boundary. Because such a feature has never been detected in previous research on the Mott transition and we focus most of the Discussion on this anomaly, the latter component (the blue component in Fig. 3B) increases its prominence at low pressure, where the system is insulating. Because the charge degree of freedom is gapped out in the Mott-insulating phase, the latter component likely stems from spin fluctuations. The overall schematic pressure-temperature phase diagram is summarized in Fig. 4, where red and blue represent the magnitudes of the slow fluctuations in the charge and spin sectors, respectively. Note that a recent theoretical work (26) has suggested that spinon fluctuations present deep on the Mott-insulating side are strongly suppressed by scattering from charge fluctuations as soon as the Mott gap closes.

Fig. 4 Pressure-temperature phase diagram of the slow fluctuations in EtMe3Sb[Pd(dmit)2]2.

(A) Schematic pressure-temperature phase diagram of the contour plot of the slow fluctuations in EtMe3Sb[Pd(dmit)2]2. The magnitudes of the charge and spin fluctuations are represented in red and blue, respectively. The intensities are determined by the amplitudes of the blue and red components of T2l−1 in Fig. 3B. (B) Pressure-temperature phase diagram of the contour plot of the conductivity (the inverse of the resistivity shown in Fig. 1B), for comparison. Note that the horizontal axes in the two panels stand for the value of the pressure applied at room temperature, and the actual pressure in the temperature region discussed in these phase diagrams is 1.5 to 2 kbar lower than the value on the horizontal axes.

DISCUSSION

The anomalous slow charge fluctuations observed in the Mott boundary contradict the conventional picture that the Mott transition is of the first order at low temperatures. To explain this anomaly, we exploit the concept of the Griffiths phase, which was originally developed in the area of spin physics, and invoke the notion of the “electronic Griffiths phase” as follows. It has been well recognized that the Mott transition of repulsively interacting electrons is mapped onto the ferromagnetic transition of exchange-interacting Ising spins, and the pressure-temperature phase diagram of the Mott transition system corresponds to the field-temperature phase diagram of the Ising system (1, 12, 13, 27), as shown in Fig. 5 (A and B). This is because both the transitions have a scalar-nature order parameter, although the critical exponents of the Mott transition are controversial. When randomness is introduced into the Ising system, the randomness suppresses the first-order transition separating the ferromagnetic up-spin and down-spin states, pushing the critical end point of the first-order transition to absolute zero; in the case of strong randomness, the first-order transition completely vanishes. According to the celebrated concept introduced by Griffiths (28), the first-order transition line is replaced by a widely spread critical region—the Griffiths phase—in which spins are slowly fluctuating between the ferromagnetic up-spin and down-spin states (29), as shown in Fig. 5C.

Fig. 5 Schematic phase diagrams of Ising spin systems and Mott transition systems.

(A) Field-temperature phase diagram of a clean Ising spin system. (B) Pressure-temperature phase diagram of a clean Mott transition system. (C) Field-temperature phase diagram of an Ising spin system with randomness. (D) Our proposed pressure-temperature phase diagram of a Mott transition system with randomness.

We propose that this concept can be applied to the Mott transition because of the equivalency of the two systems, namely, introducing randomness replaces the original first-order Mott transition line with an electronic Griffiths phase, in which electrons slowly fluctuate between the metallic and Mott-insulating states, as shown in Fig. 5D. This picture assumes the first-order transition in the pristine system and is fully consistent with the DMFT, which predicts that the first-order Mott transition line ending at a finite temperature divides metallic and insulating phases (2). On the transition line, the two phases would coexist in a macroscopic scale in the pristine system; however, disorder would make the coexistence finely divided into microscopic scales in the present system, possibly causing slow fluctuations between the insulating and metallic states. The DMFT also suggests the emergence of quantum critical metal-insulator fluctuations at high temperatures well above the critical end point (30, 31), which was later demonstrated by experiment (32). It is intriguing to know whether this quantum critical regime spreads to lower energies and, if so, how the Griffiths phase is affected by its quantum nature. On the other hand, a continuous Mott transition in a spin liquid is also theoretically proposed (33, 34). The above scenario based on the first-order transition is not straightforwardly applicable to the case of the continuous transition; however, we reserve the possibility that the slow dynamics can be a ubiquitous feature irrespective of the order of the metal-insulator transition in the pristine system. In actuality, a disordered 2D electron system in Si is reported to exhibit slow fluctuations in conductance (3537).

The electronic Griffiths phase concept has also been proposed in a recent theoretical work (38), which predicts that the electronic Griffiths phase around the Mott boundary tends to spread toward the metallic side. Figure 4 shows that the present electronic Griffiths phase starts appearing at 4 kbar, attains prominence at 7 kbar, and then widely spreads toward the higher-pressure side. This behavior is consistent with the theoretical prediction. In addition, the theoretical work discusses the electronic Griffiths phase from the perspective of the “infinite randomness fixed point (IRFP),” around which the Griffiths singularities are much stronger than those in the usual Griffiths phase. The IRFP scenario is suggested to be generally realized in quantum phase transition systems with a lower critical dimension dc = 1 (39). For the Mott transition, dc = 1, as is the case with the Ising transition. Although whether the present Mott transition would be a quantum phase transition in the clean limit is unclear, it is expected to have the quantum transition nature in that this material fulfils the quantum critical scaling of resistivity (32). Note again that a theoretical framework of the Mott instability from spin liquids assumes a continuous Mott transition (33, 34). Therefore, the IRFP scenario may be applicable to the present material and explain why the slow dynamics, which are one form of Griffiths singularity, are observed so strongly.

The peculiarity of the Mott transition in the present material may be related to the origin of the spin liquid ground state. Note that, in addition to the charge fluctuations, the present Mott insulator also yields slow spin fluctuations (the blue-colored region in Fig. 4). These are not observed in the EtMe3P salt, where the Mott transition is conventional and the ground state of the Mott-insulating phase is a clear spin-gapped state with valence bond–solid formation. The slow spin fluctuations observed in the EtMe3Sb salt are not conventional critical spin fluctuations toward the antiferromagnetic ordering because the fluctuations are weak and observed only for T2l−1. A slight amount of inhomogeneous spin freezing in the spin liquid background has been reported in organic spin liquid materials (19, 40). The slow spin fluctuations observed here likely capture the dynamic aspect of the inhomogeneous spin freezing and provide additional insight into the spin liquid nature in organics, which has been an unresolved problem.

MATERIALS AND METHODS

Sample preparation

We prepared fine high-quality single crystals of EtMe3Sb[Pd(dmit)2]2 with typical sizes of 0.3 mm × 0.3 mm × 0.01 mm using an aerial oxidation method (41). For the 13C NMR measurements, we enriched the inner carbon sites of the Pd(dmit)2 molecule, which have large hyperfine coupling with electrons (23), as shown in Fig. 1A.

Resistivity measurements

A sample to which four electrical leads were attached was encased in a Teflon capsule filled with pressure medium. Resistivity measurements were carried out using a conventional dc method with the electrical current along the 2D plane.

Overall NMR measurements

We packed a number of fine single crystals into a Teflon tube with no particular orientation. The Teflon tube was encased in a Teflon capsule filled with pressure medium. The NMR measurements were performed at a field of 7.65 T (corresponding to the 13C NMR frequency of 80.4 MHz). The NMR signals were obtained by the spin-echo method with a π/2-π pulse sequence. The typical pulse widths of the π/2 and π pulses were 2.5 and 5 μs, respectively. These values were sufficiently smaller than the inverse of the spectral widths, and thus, the pulses could cover the whole NMR spectra.

Applying pressure

The Teflon capsules for the resistivity and NMR measurements mentioned above were set in clamp-type pressure cells, and hydrostatic pressure was applied at room temperature. The pressure medium used was Idemitsu Daphne 7373 oil. Note that the applied pressures decrease by 1.5 to 2 kbar from the room temperature values around the oil solidification temperature (200 to 300 K) upon cooling. The pressure values shown in this paper are those at room temperature.

Spin-lattice relaxation rate

The spin-lattice relaxation rates (T1−1) were obtained from the recovery of the spin-echo intensity as a function of t, where t is the time interval between the saturation comb pulses and the π/2-π pulses to form echoes. The spin-lattice decay curves, defined by 1 − M(t)/M(∞), are not single exponential functions because of the distribution of the angle between the sample direction and the applied magnetic field and the existence of four different crystallographical 13C sites. Therefore, we define T1−1 as the time when the recovery curves reach 1/e.

Spin-spin relaxation rate

The spin-spin relaxation rates (T2−1) were obtained from the decay of the spin-echo intensity as a function of 2τ, where τ is the time interval between the π/2 and π pulses. We define T2 as the time when the spin-spin decay curves [M(2τ)] reach 1/e, as for T1 (for details, see the Supplementary Materials).

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/8/e1601594/DC1

section S1. Spin-spin relaxation

section S2. Cation molecular motion

fig. S1. 13C spin-spin relaxation curves as functions of 2τ (left) and (2τ)2 (right) for EtMe3Sb[Pd(dmit)2]2 and EtMe3P[Pd(dmit)2]2.

fig. S2. Temperature dependence of the spin-lattice relaxation rate (T1−1) of the protons in the cation molecule of EtMe3Sb[Pd(dmit)2]2 at several pressures.

Reference (42)

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REFERENCES AND NOTES

Acknowledgments: We thank K. Miyagawa for supporting the NMR experiments and H. Oike and T. Furukawa for the stimulating discussions. Funding: This work was supported in part by the Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (grant nos. 15H02108, 16H06346, 23684025, 24340082, 25287089, 25287082, and 25220709), the NSF (grant no. PHYS-1066293), and the hospitality of the Aspen Center for Physics. Author contributions: T.I. designed the experiments. T.I. and E.W. conducted the NMR measurements and analyses. S.M. provided experimental support and suggestions for the NMR measurements. A.T. and N.T. performed the resistivity measurements. K. Kubo and R.K. synthesized the 13C-enriched dmit ligand and grew the single crystals used for the study. T.I. and K. Kanoda interpreted the data. T.I. wrote the paper with significant assistance from K. Kanoda. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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