Evidence for solitonic spin excitations from a charge-lattice–coupled ferroelectric order

See allHide authors and affiliations

Science Advances  30 Nov 2018:
Vol. 4, no. 11, eaau7725
DOI: 10.1126/sciadv.aau7725


Topological defects have been explored in different fields ranging from condensed matter physics and particle physics to cosmology. In condensed matter, strong coupling between charge, spin, and lattice degrees of freedom brings about emergent excitations with topological characteristics at low energies. One-dimensional (1D) systems with degenerate dimerization patterns are typical stages for the generation of topological defects, dubbed “solitons”; for instance, charged solitons are responsible for high electrical conductivity in doped trans-polyacetylene. Here, we provide evidence based on a nuclear magnetic resonance (NMR) study for mobile spin solitons deconfined from a strongly charge-lattice–coupled spin-singlet ferroelectric order in a quasi-1D organic charge-transfer complex. The NMR spectral shift and relaxation rate associated with static and dynamic spin susceptibilities indicate that the ferroelectric order is violated by dilute solitonic spin excitations, which were further demonstrated to move diffusively by the frequency dependence of the relaxation rate. The traveling solitons revealed here may promise the emergence of anomalous electrical and thermal transport.


One-dimensional (1D) electron-lattice–coupled systems exhibit various instabilities followed by lattice symmetry breaking; Peierls and spin-Peierls instabilities are well-known examples (1, 2). In these systems, topological defects, which are exotic excitations emerging in diverse physical systems (3), are expected to be generated (4, 5). The organic complex, tetrathiafulvalene-p-chloranil (TTF-CA), in which the donor molecule, TTF, and the acceptor molecule, CA, stack alternately to form 1D columns (Fig. 1A), shows charge-transfer and lattice-dimerization instabilities. At ambient pressure and temperature, TTF-CA is in a quasi-neutral (N) state (the degree of charge transfer ρ ~ 0.3), with both TTF and CA forming electronic closed shells; upon cooling, TTF-CA transitions to a quasi-ionic (I) state (ρ ~ 0.6 to 0.7), with a collective electron transferred from TTF to CA to gain electrostatic energy at 81 K (6, 7). The charge transfer is followed by simultaneous lattice dimerization (810), yielding an “electronic ferroelectricity” (11). Thus, TTF-CA is nonmagnetic in both the N phase (due to the closed-shell structure) and the dimerized I phase (due to spin-singlet pairing). As seen in Fig. 1B, dimerization can occur with one of two degenerate patterns (labeled IA and IB in Fig. 1B), leading to the emergence of solitonic spins as topological defects against long-range dimerization order. To date, these defects have been observed as frozen defects in 3D long-ranged dimerized ferroelectric order but not as topological thermal excitations (12, 13).

Fig. 1 Soliton excitation and phase diagram for TTF-CA.

(A) Molecular structures of the TTF and CA molecules. The central double-bonded carbon atoms in the TTF molecule are substituted by 13C isotopes for 13C-NMR measurements. (B) Schematic of degenerate spin-singlet dimer patterns (IA and IB) in the quasi-ionic phase. A spin soliton with spin 1/2 emerges between them. (C) A contour plot of the 1H-NMR spin-lattice relaxation rate, 1T1−1, in the pressure-temperature (P-T) plane of TTF-CA. The red circles indicate peak or kink temperatures for 1T1−1. The white squares and triangles indicate the dimerization transition temperatures as determined by previous neutron and NQR measurements, respectively (14, 16).

The simultaneous charge-transfer and dimerization transition occur at an elevated temperature as the pressure is increased (14, 15). However, a recent nuclear quadrupole resonance (NQR) study revealed that the charge transfer and dimerization are separated above ~8 kbar (1416); the charge-transfer temperature continues to increase, whereas the dimerization transition temperature turns downward with increasing pressure (Fig. 1C) (16), suggesting a paraelectric I phase without static lattice dimerization in the region between the charge-transfer and dimerization lines (labeled Ipara in Fig. 1C). Naively, the substantially charge-transferred Ipara phase should carry paramagnetic spins on uniform 1D chains free from dimerization. An infrared spectroscopy study, however, found the activation of the ag mode in CA molecules, indicative of lattice dimerization, at pressures above 8 kbar at ambient temperature (17, 18). These contradicting results suggest that the dimerization is temporally fluctuating at a characteristic frequency in between the frequency windows of the NQR (107 Hz) and infrared (1012 Hz) probes (16). Thus, the Ipara phase above 8 kbar is expected to host no homogeneous 1D spin chains but carry mobile solitonic spins that emerge at thermally activated topological boundaries separating oppositely polarized dimer domains. The purpose of the present work is to demonstrate the existence of topological magnetic excitations from a ferroelectric order for the first time, using nuclear magnetic resonance (NMR), which probes the spin states in a microscopic manner.


Magnetic excitations investigated by 1H-NMR and 13C-NMR

NMR experiments were carried out for the 1H and 13C sites in TTF molecules (see Materials and Methods). First, to reveal the profile of the magnetic excitations in the pressure-temperature (P-T) phase diagram, we measured the 1H-NMR spin-lattice relaxation rate, 1T1−1, which probes electron spin fluctuations, under various temperatures and pressures. At ambient temperature, 1T1−1 is vanishingly small in the N phase at low pressures, indicative of a nonmagnetic state, in agreement with the previous 1H-NMR results (19, 20), whereas it is rapidly enhanced with increased pressure above 8 kbar (Fig. 1B), at which point charge transfer is sharply promoted (Fig. 2B) (16, 21). Even in the high-pressure region, however, 1T1−1 rapidly decreases on cooling (Fig. 2A). The contour plot of 1T1−1 in the P-T plane (Fig. 1C) shows that the area of active spin excitations nearly coincides with the Ipara region (16), indicating that the Ipara phase contains active spins, in contrast to the N phase and ferroelectric I (Iferro) phase. The dimerization transition temperatures determined by 1T1−1 nearly reproduce the previous neutron scattering and 35Cl-NQR results (14, 16) except for 4.5 kbar, at which the pressure is possibly reduced because of the solidification of the pressure medium.

Fig. 2 1H-NMR relaxation rate and degree of charge transfer.

(A and B) Pressure dependence of the 1H-NMR spin-lattice relaxation rate, 1T1−1, under various temperatures (A) and the degree of charge transfer, ρ, evaluated by a previous NQR measurement (16) at room temperature (RT) (B). The gray line indicates the NI crossover pressure (defined as ρ = 0.5) at room temperature.

Next, we present 13C-NMR results, which reveal that the active spins are solitonic instead of emerging on every TTF site. The 13C nuclei residing on the central double-bonded carbon sites in TTF molecules (Fig. 1A) are more strongly hyperfine-coupled with electrons than 1H nuclei located on the edges of TTF so that the NMR spin shift can be well resolved. The spectral shift could not be observed in the 1H-NMR measurements (see the Supplementary Materials). Using the hyperfine coupling constant, we can quantitatively discuss the spin shift and relaxation rate involved with the static and dynamic spin susceptibilities, respectively, which provide key information for the nature of the magnetic excitations. Figure 3A shows the 13C-NMR spectra measured under several different pressures at 285 K, where the system crosses over from the N phase to the Ipara phase as pressure is increased. The spectra form two-peak structures due to dipolar coupling between adjacent 13C nuclear spins at all pressures except for 3 kbar, at which the spectrum is disturbed by the appearance of the signal from the oil pressure medium because, at low pressures, the relaxation rate of the sample signal becomes long enough to be comparable to that of the oil signal. The spectral shift defined by the midpoint of the doublet is nearly independent of pressure up to 6 kbar and gradually increases with further pressure (Fig. 3B). In general, the spectral shift has two origins: the spin shift, δs, which originates from the hyperfine interaction with electron spins that is proportional to the local spin susceptibility at the 13C site, and the chemical shift, σ, originating from the orbital motion of electrons. In the nonmagnetic N phase at low pressures, the spectral position is solely determined by the chemical shift, which yields ~82 parts per million (ppm); this value is unchanged even in the I state at high pressures, as indicated by the spectrum measured at 14 kbar and 144 K (Fig. 3A), at which the TTF-CA is in a nonmagnetic I state. The spin shift that appears at 6 kbar reaches 51 ppm at 14 kbar (Fig. 3B), and concomitantly, the 13C spin-lattice relaxation rate 13T1−1 increases with pressure before leveling off to a value of approximately 11 s−1 (Fig. 3C), indicating the emergence of paramagnetic spins in the Ipara phase, in agreement with the 1H-NMR results. There appears to be an anomaly around 12 kbar in Figs. 2A (RT) and 3C; however, it is probably not intrinsic but due to statistic errors because such an anomaly was not observed in a separate sample (for details, see the Supplementary Materials).

Fig. 3 13C-NMR spectral shift and relaxation rate.

(A) Pressure dependence of 13C-NMR spectra at 285 K (brown lines). The uppermost (gray) line shows a spectrum measured at 144 K under 14 kbar in the Iferro phase. The origin of the horizontal axis corresponds to a resonance frequency of TMS (tetramethylsilane). The broken line indicates the position of the chemical shift. The dip at approximately 30 ppm at 3 kbar is a phase-reversed signal arising from the oil pressure medium (Daphne 7373). (B) Pressure dependence of the 13C-NMR spin shift (right axis), which is the difference between the averaged value of the two peak positions and a chemical shift of 82 ppm (left axis). (C) Pressure dependence of the 13C-NMR spin-lattice relaxation rate 13T1−1 at 285 K. (D) Comparison between the spin susceptibility χ and 13T1−1 in terms of the uniform 1D Heisenberg model. The red closed circles indicate the experimental values obtained for χ and 13T1−1 for TTF-CA. The red open circles indicate the values expected for χ (13T1−1) from the uniform 1D Heisenberg model using the experimental values for 13T1−1 (χ).

We note that the previously reported 35Cl-NQR relaxation rate, 35T1−1, shows a decrease with increasing pressure above 9 kbar (16), unlike the behavior of the NMR 1T1−1 and 13T1−1. This is because NQR probes lattice fluctuations, whereas NMR probes spin excitations. The lattice fluctuations are expected to be enhanced close to the dimerization transition temperature Tc, which decreases as pressure is increased above 9 kbar. Because room temperature departs further from Tc at higher pressures, lattice fluctuations at room temperature are gradually depressed under pressurization, as observed in (16).

δs is related to the spin susceptibility, χ, through the expression (22)Embedded Image(1)where NA is the Avogadro constant, μB is the Bohr magneton, and a|| is the hyperfine coupling component parallel to an external magnetic field, H. Using the 13a|| value of 10.8 kOe/μB, the hyperfine coupling constant of the analogous material, (TMTTF)2X (X = Br and AsF6) (23, 24), because that of TTF-CA is not known, χ is estimated to be 2.6 × 10−5 emu/mol TTF at 14 kbar (Supplementary Materials), which is one or two orders of magnitude smaller than that of other organic quasi-1D spin systems [e.g., ~6 × 10−4 emu/mol spin in TTF-BA (25, 26) and (TMTTF)2AsF6 (24) at room temperature]. If the value of 2.6 × 10−5 emu/mol for TTF is the low-temperature saturating value of spin susceptibility of the uniform 1D antiferromagnetic Heisenberg model (AFHM), the exchange interaction yields an unrealistically large value, J = 5900 K, referring to the theoretical calculations by Bonner and Fisher (27) and Estes et al. (28). On the other hand, we compare the experimental 13T1−1 values with the consequence of the scaling theory for the uniform 1D AFHM, expressed as (29, 30)Embedded Image(2)where a is the hyperfine coupling component perpendicular to H, ℏ is the reduced Planck constant, and D is a nonuniversal constant giving the overall magnitude of the dynamical spin susceptibility. Using values of D = 0.15, as determined for Sr2CuO3 (30), which is an ideal 1D Heisenberg spin system, and 13a = 5.6 kOe/μB (Supplementary Materials), the uniform 1D AFHM with J = 5900 K, which is obtained from the spin shift analysis, gives an estimate of 13T1−1 = 1.1 s−1 (Fig. 3D), which is one order of magnitude smaller than the experimental value of 13T1−1 = 11 s−1 at 14 kbar. Conversely, if we estimate the spin susceptibility expected from the experimental value of 13T1−1 = 11 s−1 in the 1D AFHM scenario, Eq. 2 yields the J value of 580 K, which gives the low-temperature spin susceptibility of 3.7 × 10−4 emu/mol TTF (Fig. 3D), more than an order of magnitude larger than the experimentally determined value of 2.6 × 10−5 emu/mol TTF. Thus, the experimental results of spin susceptibility and relaxation rate are mutually incompatible in the framework of the uniform 1D AFHM.

For reference, we also examine the compatibility of the experimental values χ and 13T1−1 of the putative 1D AFHM system, (TMTTF)2AsF6, in the similar analysis; in the low-temperature paramagnetic state at 80 K (24), χ = 4.2 × 10−4 emu/mol spin and 13T1−1 = 27.5 s−1, which is the averaged value of 13T1−1 at the inner and outer carbons of the charge-rich TMTTF molecules. The 1D AFHM fitting of the χ data is reported to yield J = 410 K (31), which would result in 13T1−1 = 22 s−1 for the 1D AFHM, using the same D value of 0.15 and 13a = 13 kOe/μB, in consideration of a spin-1/2 per TMTTF dimer (Supplementary Materials). The agreement between the experimental value, 27.5 s−1, and the expected value, 22 s−1, is excellent, demonstrating that the 1D AFHM works well for (TMTTF)2AsF6. At the same time, this analysis suggests that the D value of 0.15 is widely applicable to 1D AFHM systems including inorganic and organic materials. Thus, the contradiction between the experimental values and the 1D-AFHM expectations for TTF-CA suggests that the uniform 1D AFHM is not appropriate for describing the spin state in question, invoking an alternative picture. The extraordinarily small value of χ strongly suggests the presence of solitonic spin excitations. Assuming that the spin solitons behave similarly to Curie spins, the soliton density n is estimated to be ~0.02 spins per TTF or CA molecule at 285 K under 14 kbar. Compared with n ~ 10−4 spins in the Iferro phase at ambient pressure (12), many more spin solitons are thermally activated in the Ipara phase. Both the spin shift and T1−1 are sharply enhanced around the neutral-ionic (NI) boundary, whereas the degree of charge transfer is gradually increased even in the low-pressure region (Fig. 2B). These results imply that the soliton density is not proportional to the density or volume fraction of ionic domains, but the solitonic excitations require a certain amount of ionic domains that are thermally generated, and rapidly develop under further pressure. Then, the soliton density slightly increases with pressure in the Ipara phase. The observed large value of 13T1−1 in the Ipara phase is likely due to their traveling along the 1D chains, possibly explaining why 13T1−1 shows a somewhat different pressure dependence compared to the spin shift.

Dynamics of spin solitons

To seek further evidence for mobile solitons, we investigated the frequency dependence of T1−1, which measures the spectral density of local field fluctuations, S(ω). The gyromagnetic ratio of the 1H nucleus is four times larger than that of the 13C nucleus, enabling one to reach higher frequencies in experiments; thus, we adopted 1H-NMR for this purpose and carried out measurements for frequencies over a 20-fold range (14 to 300 MHz) at 300 K under 14 kbar in the Ipara phase. If spin solitons travel in a diffusive manner, S(ω) should show a frequency dependence characteristic of the spatial dimension for soliton motion, namely, S(ω) ∝ ω−1/2 in 1D, ∝ −lnω in 2D, and independent of ω in 3D. In particular, in a 1D-3D crossover regime for quasi-1D systems, S1D-3D(ω) is expressed in a form that includes the 1D diffusion rate D|| and the 3D cutoff frequency 1/τ(32)Embedded Image(3)which yields a ω−1/2 dependence in the 1D regime of ω > 1/τ and approaches a constant, Embedded Image, in the 3D regime of ω < 1/τ. Neglecting the anisotropic part of the hyperfine coupling tensor for simplicity, the contribution of spin diffusion to the relaxation rate is given by T1−1sd = kBT(χ/NAg2μB2)aiso2Se), where aiso is the isotropic part of the hyperfine coupling tensor, ωe (= γeH) is the electron Larmor angular frequency (instead of the NMR frequency ωN), γe is the electron gyromagnetic ratio, and g is the electron g-factor (33, 34). As shown in Fig. 4, the behavior of 1T1−1 is well fitted by 1T1−1 = kBT(χ/NAg2μB2)1aiso2S1D-3De) + constant, with D|| = 5.1 × 1011 s−1 and 1/τ = 5.6 × 1010 s−1, using 1aiso = −0.39 kOe/μB of TTF-BDT(Cu) (35). This result signifies that the spin solitons move diffusively at D|| ~ 1011 s−1 along the 1D chains with weak interchain interactions of 1/τ ~ 1010 s−1. The D|| value lies in between the two frequency windows of the NQR (107 Hz) and infrared (1012 Hz) probes, consistent with the dimer-liquid picture. The constant term may derive from longitudinal local field fluctuations through the anisotropic parts of the hyperfine coupling tensor, as discussed in doped trans-polyacetylene (34). For the origin of the 3D cutoff, exchange interaction between spin solitons on adjacent 1D chains or the hopping of solitons to neighbour chains is conceivable; it is not known at present which mechanism is dominant. The 1D diffusion expressed by Embedded Image, where x(t) stands for the soliton position at a time t, leads to the consequence that the solitons travel ~5 donor-acceptor (DA) pairs in time τ, indicating that the distance between a soliton and an antisoliton running in counter directions varies by ~10 DA pairs in the time period τ. This distance is comparable with the averaged soliton-antisoliton distance of 25 DA pairs (~1/n*; n* = 2n is the soliton density per DA pair), reserving another possibility that the cutoff time, ~τ, is determined by soliton-antisoliton annihilation events. Note that the uniform 1D AF Heisenberg spin system can show diffusive behavior only for T >> J due to the contribution of uniform spin fluctuations (q ~ 0) (36). The available band parameters for TTF-CA yield a value for J of ~1800 to 3600 K (Supplementary Materials). Thus, in the present case, with T (=300 K) << J, AF fluctuations (q = π) are dominant and the diffusive behavior is not expected in the uniform 1D AFHM.

Fig. 4 Frequency dependence of the 1H-NMR relaxation rate.

Frequency dependence of 1H-NMR spin-lattice relaxation rate, 1T1−1, under 14 kbar at 300 K in the Ipara phase. The red line is a fit of the form 1T1−1 = kBT(χ/NAg2μB2)1aiso2S1D-3De) + constant to the experimental data. Note that ωe is the Larmor angular frequency of the electron spin. Inset: Plot of 1T1−1 against f−1/2. A linear extrapolation of the data to f−1/2 = 0 is indicated by the green broken line.

The present work demonstrates the emergence of solitonic spin excitations from the ferroelectric order. In the vicinity of the NI transition, another type of soliton, namely, the charge soliton, is also expected to emerge (3741). It is suggested that the charge and spin solitons carry fractional charges and generate an anomalous charge transport (37, 40, 41). Note that the anomalous charge transport requires not only charge solitons but also spin solitons; the charge solitons alone cannot sustain steady electrical current. Anomalously high electrical conductivity is reported (42). Thus, the present experiments capturing mobile solitons lend support to anomalous charge transport attributed to running solitons along the 1D chains. We expect that solitonic spin and charge excitations in organic charge-transfer complexes bring about unconventional phenomena in various research fields such as thermoelectric transport.


Sample preparation

The TTF-CA crystals with and without 13C-labeling were prepared by a co-sublimation method; a vacuum-sealed glass tube containing the powdered components in the ends separately was heated with a three-zone temperature-controlled furnace. For 13C-NMR measurements, we synthesized 13C-enriched TTF molecules, in which the central double-bonded carbon atoms were substituted by 13C isotopes (Fig. 1A), according to methods described in the literature (43, 44) with slight modifications from 13CS2 (ISOTEC, 13C = 99.9 atomic %) as a starting compound. The detail of the scheme is described in fig. S2.

1H-NMR measurements under pressure

We performed 1H-NMR measurements for a TTF-CA single crystal under pressures of up to 17 kbar; the TTF molecule contains four 1H sites. To obtain NMR signals, we used the so-called solid-echo pulse sequence [comb pulse − (π/2)x pulse − (π/2)y pulse]. The NMR spectra for a single crystal of TTF-CA were composed of three or four lines comprising two sets of 1H-1H dipolar splittings (depending on the direction of magnetic field), arising from two differently oriented 1D columns. Except for the measurements of the frequency dependence of 1T1−1, the NMR frequency used was ~156.0 MHz under an applied field of 3.66 T. The frequency dependence of 1T1−1 was investigated by varying the magnetic field in a range from 0.3 to 7.0 T. The relaxation curve of the nuclear magnetization was fitted by a stretched exponential function ~exp(−(t/T1)β), where β is the stretching exponent characterizing the distribution of relaxation rate; in TTF-CA, up to four nonequivalent 1H sites with respect to magnetic field may have different relaxation times. The typical β value was ~0.8 to 0.9. Hydrostatic pressure was applied to the sample using a BeCu clamp-type cell, with DEMNUM oil (solidification pressure of ~10 kbar) used as the hydrogen-free pressure medium. To attain hydrostatic pressures above 10 kbar, we warmed the pressure cell with a heater wound around the cell during the application of pressure for preventing solidification of the oil. We monitored the external pressure while pressurizing the sample at room temperature. The conversion coefficient used to determine the internal pressure from the applied pressure was assumed to be 0.9, which was determined separately by monitoring the resistivity of a manganin wire mounted in the pressure cell. The pressure values quoted in this paper are those determined at room temperature. The temperature variation of the internal pressure of the Daphne oil in the pressurized and clamped cell was investigated by Murata et al. (45). As seen in Figs. 1 to 3, important pressure and temperature ranges are above 5 kbar and above 200 K. According to this paper, a drop in pressure upon cooling from room temperature is ~5% at 250 K and ~10% at 200 K for pressures above 5 kbar. The dimerization transition temperatures determined by 1T1−1 in the present work (red circles in Fig. 1C) nearly coincide with the previous results (white lines in Fig. 1C), although the oil used in the variable-temperature experiments in our study is a different kind of oil, DEMUNUM.

13C-NMR measurements under pressure

13C-NMR measurements were conducted for a 13C-labeled TTF-CA single crystal in a magnetic field H of 8 T applied parallel to the a axis (corresponding to the 1D direction). In this field configuration, all TTF molecules are equivalent against H, and thus, we observed two peaks arising from dipole interactions between the two 13C nuclei on the double-bonded sites. We used a solid-echo pulse sequence to obtain the NMR signals. The spectra shown in Fig. 3A were acquired at the time tj of 5T1 to 10T1 after the saturation of nuclear magnetization except for the measurements at 3 kbar; at such a low pressure, T1 at 13C site was too long to fulfill the ideal condition of tj > 5T1, and thus, tj was inevitably set to times comparable to T1. The spin-lattice relaxation rate 13T1−1 was determined by fitting the single exponential function to the relaxation curve. To apply hydrostatic pressure to the sample, we used clamp-type pressure cells built using nonmagnetic BeCu with Daphne 7373 as a pressure medium. The conversion coefficient from applied to internal pressures is the same as in the 1H-NMR measurements.


Supplementary material for this article is available at

Section S1. Deduction of spin susceptibility from the NMR spin shift

Section S2. Hyperfine coupling constants

Section S3. Estimation of the exchange interaction energy J along the 1D axis

Section S4. Temperature dependence of 1H-NMR spectra at 13 kbar

Section S5. Temperature dependence of 1H-NMR relaxation rate

Fig. S1. Molecular frame and principal axes of TTF (X = H) and TMTTF (X = CH3) molecules.

Fig. S2. Synthetic route for a 13C-enriched TTF molecule.

Fig. S3. Pressure dependence of 13C-NMR spin-lattice relaxation rate at 285 K for a crystal different from the one used in the measurement described in the main text (Fig. 3C).

Fig. S4. Temperature dependence of 1H-NMR spectra at 13 kbar.

Fig. S5. Temperature dependence of 1H-NMR spin-lattice relaxation rate.

References (4653)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: We thank H. Fukuyama, T. Takahashi, M. Ogata, T. Hasegawa, and R. Takehara for critical discussions. Funding: This work was supported by the JSPS Grant-in-Aids for Scientific Research (S) (grant nos. JP25220709, JP18H05225, and JP16H06346) and for Scientific Research (C) (grant no. JP17K05532) and by CREST (grant no. JPMJCR1661), Japan Science and Technology Agency. Author contributions: K.S., T.N., and K.M. performed the NMR measurements. R.K. synthesized the 13C-enriched TTF molecules. S.H., T.M., and H.O. prepared the TTF-CA single crystals. K.S. wrote the manuscript with the assistance of K.M. and K.K. K.S., K.M., and K.K. designed the experiments. All authors discussed the results and commented on the manuscript. This project is headed by K.K. 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. Correspondence and requests for materials should be addressed to K.S. or K.K.
View Abstract

Navigate This Article