IOP Institute of Physics LETTER Charge density wave soliton liquid To cite this article: T. Matsuura et al 2015 EPL 109 27005 View the article online for updates and enhancements. Related content Single-valuedness problem on charge density wave rings T. Hosokawa, T. Matsuura, M. Tsubota et al. Field-induced dielectric response saturation in o-TaS3 Yongchang Ma, Cuimin Lu, Xuewei Wang et al. High-frequency, 'quantum' and electromechanical effects in quasi-one- dimensional charge density wave conductors Vadim Ya Pokrovskii, Sergey G Zybtsev, Maksim V Nikitin et al. Recent citations Hydrodynamic model of a Bose—Einstein condensate with anisotropic short-range interaction and bright solitons in a repulsive Bose–Einstein condensate Pavel A Andreev From chiral anomaly to two-fluid hydrodynamics for electronic vortices
S. Brazovskii and N. Kirova Possible origin of nonlinear conductivity and large dielectric constant in the commensurate charge-density-wave phase of 1TTaS2 Yongchang Ma et al EPL, 109 (2015) 27005 doi: 10.1209/0295-5075/109/27005
T. M ATSUURA¹,2, J. H ARA¹, K. I NAGAKI³, M. T SUBOTA 4, T. H OSOKAWA¹ and S. T ANDA¹,2 1 Department of Applied Physics, Hokkaido University - Sapporo 060-8628, Japan ² Center of Education and Research for Topological Science and Technology, Hokkaido University Sapporo 060-8628, Japan 3 Department of Physics, Asahikawa Medical University - Asahikawa 078-8510, Japan 4 Division of Quantum Science and Engineering, Hokkaido University - Sapporo 060-8628, Japan received 10 September 2014; accepted in final form 13 January 2015 published online 2 February 2015 PACS  71.45.Lr – Charge-density-wave systems PACS  72.15.Nj – Collective modes ( e.g. in one-dimensional conductors) PACS  05.45.Yv – Nonlinear dynamics and chaos: Solitons Abstract – We investigate the charge density wave transport in a quasi–one-dimensional conduc- tor, orthorhombic tantalum trisulfide ( o-TaS 3 ), by applying a radio-frequency ac voltage. We find a new ac-dc interference spectrum in the differential conductance, which appears on both sides of the zero-bias peak. The frequency and amplitude dependences of the new spectrum do not correspond to those of any usual ac-dc interference spectrum (Shapiro steps). The results suggest that CDW phase dynamics has a hidden degree of f reedom. We propose a model in which 2 π phase solitons behave as liquid. The origin of the new spectrum is that the solitons are depinned from impurity potentials assisted by an ac field when a small dc field is applied.  Our results provide a new insight as regards our understanding of an elementary process in CDW dynamics. Copyright © EPLA, 2015 Introduction. – Phase solitons are topological defects in the phase field of a charge density wave (CDW) [1] and are similar to the vortices in superconductors and el- ementary excitation in a vacuum. The degree of freedom of the phase soliton is the key to understanding the na- ture of a CDW. Phase solitons have been expected in the one-dimensional CDW system of orthorhombic tantalum trisulfide ( o -TaS 3 ) crystals to explain the CDW transport properties [2–4], and the Aharonov-Bohm interference in o-TaS3 rings [5,6]. In addition, the coexistence of commen- surate and incommensurate CDWs in o -TaS3 crystals was observed below the Peierls transition temperature Tp = 218 K by X-ray diffraction [7]. The satellite peaks of the incommensurate CDW of Q ic ～ 0.5α* + 0.125b* + 0.255c* develop first just below T P , and those of the commensurate CDW of Q c = 0.5a* + 0.125b* + 0.25c* develop with the decrease of temperature.  Furthermore, the incommensu- rate CDW was enhanced when the CDW was sliding. The results suggest that there are unconventional degrees of freedom associated with the solitons, and that they will affect CDW dynamics. In this letter, CDW dynamics in o-TaS 3 crystals is inves- tigated using a conventional ac-dc interference measure- ment in the temperature range where the commensurate and incommensurate CDWs coexist. We find a new type of ac-dc interference spectrum, which does not originate from the usual Shapiro interference mechanism.  The re- sults suggest that an o -TaS 3 CDW system has 2π phase solitons.  The solitons are depinned from impurities and move individually as a liquid when a large ac electric field is applied. Experimental. – The ac-dc interference measurement was performed as follows. The o -TaS 3 crystals were syn- thesized using the chemical vapor transport method [8,9]. A needle-shaped o-TaS3 crystal was selected and placed on a sapphire substrate, and two silver wires were connected to the crystal with gold evaporation films and silver paste for the two-probe measurement.  The resistance of the sample present here was 7 Ω at room temperature (280 K). The cross-section area of the sample was 16 × 7 μm², the minimum distance between two terminals was 110 μ m, and the room temperature resistivity was consistent with a previously reported value [10].  The differential conduc- tance of o-TaS 3 crystals was measured as a function of dc bias current while applying a radio-frequency (RF) ac voltage.  A differential conductance measurement with a lock-in amplifier (Stanford Research SR830DSP: a low- frequency (LF) ac current I F = 1 μ A, and f LF = 13 Hz) was performed. The sample temperature was controlled by using liquid nitrogen and an electrical heater with a ±0.1 K accuracy. Results. – Figure 1(a) shows the differential conduc- tance of the o-TaS3 crystal as a function of the dc bias cur- rent at 150 K, when we applied an ac voltage of 1 MHz with various amplitudes.  The threshold current and voltage were I th  =  0.35 mA  and  V th  =  70 mV  ( E th  =  636 V/m) for V r = 0, respectively, and the zero-bias peak became sharper when V r  increased, then the unusual peaks ap- peared on both sides of the zero-bias peak for V r = 160 mV (see also the peaks indicated by red arrows in fig. 1(b)).  We note that the vertical shifts of the dI/dV curves in fig. 1(a) are not artificial, but mean that there are several CDW phase domains.  A major domain gives the Shapiro interference peaks indicated by black arrows in fig. 1, but other domains move incoherently. The thresh- old electric field of the sample was relatively larger than those reported in the previous papers. One reason is that it was a two-probe transport measurement. Generally the threshold voltage observed by two-probe transport mea- surement is observed larger than that by four-probe trans- port measurement since the charge density wave phase distortion and phase slip must be involved. The new peak structures appearing under ac and dc electric fields have never been reported explicitly before. On the other hand, the step structures in the dI/dV curve have been often observed in NbSe3 crystals [11,12], and other quasi–one-dimensional conductors.  The step struc- tures in the dI/dV curve are clearly different from these new peak structures, since there is no decrease of dI/dV in the step structures. In addition, a Joule heat effect cannot explain the decrease in the differential conductivity be- cause the conductivity of o-TaS 3 crystals should increases when the temperature increases [10] (see also fig. 3 inset). The RF amplitude and frequency dependences of the current at the new peaks are not consistent with those of the Shapiro peaks. The first Shapiro peaks can also be seen in fig. 1(b). The frequency dependence of the differ- ential conductance is shown in fig. 2(a).  The peak posi- tions and heights of the new peaks are symmetric, while the dI/dV curves are asymmetric. The asymmetry sug- gests that there is a macroscopic polarization associated with metastable states of the phase field in o -TaS 3 crys- tals [13], however, metastability is not directly related to the new peak structures. From figs. 1(a) and 2(a), the RF amplitude and fre- quency dependences of the CDW current at the new peaks and at t he first Shapiro peaks are obtained. Figure 2(b) shows the CDW current at the peaks as a function of the ac amplitude for frf = 1 MHz. The CDW current interval of the first Shapiro peaks is constant (( I CDW − I CDW )/2 = 0.065 mA at frf = 1 MHz). On the other hand, the CDW current at t he new peaks increases as the RF ampli- tude increases. As shown in fig. 2(c) the CDW current at the first Shapiro peaks is proportional to frf, while the CDW current at the new peaks decreases.  Contrary to the Shapiro peaks following the conventional Shapiro in- terference mechanism, the new peaks have clearly different RF amplitude and frequency dependences. These results suggest that the CDW dynamics has an unconventional internal degree of freedom. We also find that the new peaks disappear at low tem- perature.  Figure 3 (left panel) shows the temperature dependence of the differential conductivity when an ac voltage of V R = 200 mV and f F = 1 MHz is applied. When the temperature decreases, the current at the new peaks decreases, and the height of the zero-bias peak in- creases. Below 120 K, the new peaks are below the shoul- der of the zero-bias peak.  At t the temperature range at which the new peaks are observed, the commensurate and incommensurate CDWs coexist simultaneously [7]. The synchrotron X-ray diffraction measurement could re- veal high resolution of satellite diffraction structure of the CDW in o-TaS 3 . There were the peaks of commensurate and incommensurate CDWs while the diffraction peaks of the mother lattice did not split.  The results show that the CDW in o-TaS 3 has an internal degree of freedom in the phase structures, which would be a key to un- derstanding the origin of the new peak structures in the dI / dV curve. We note that the temperature dependence of the low- field conductivity (dI/dV for Idc = 0, and Vrf = 0) in the inset of fig. 3(a) is consistent with previous re- sults [10].  The transition temperature is 218 K and the conductivity above 110 K follows the Arrhenius curve σ(T ) = σ 0 exp(−2Δ/T) with a semiconducting energy gap of 2Δ = 828 K. The new peak structures have been observed in another o-TaS 3 whisker (sample 2).  Figure 3 (right panel) shows the temperature dependence of the differential conduc- tance under a RF electric field. In this sample the new peaks appear but no clear zero-bias peak are observed. The new peak structures are observed at 80 K to 100 K, and the peaks disappear at higher and lower temperatures. This measurement was performed with both current sweep directions. The new peak structures do not have hysteresis and dependence associated with current sweep direction. We note that all samples did not show the new peak struc- tures. We measured 10 o-TaS3 whiskers, and two samples showed the unconventional ac-dc interference peaks. The sample dependence has not been understood yet. Discussions. – The peak structures nearby the zero- bias peak were clearly observed in sample 1, and 2. The appearance of the peaks is a physical phenomenon, not a data error.  The differential conductance without RF bias has no special features. This means that there is no large crack in the crystals to make two or more thresh- old voltages. In the electrical measurement, there may be a transversal current flow.  However, in our samples, the probe to probe lengths are more than 10 times larger than the crystal width and thickness.  The conductivity anisotropy of the TaS3 is of the order of 100 at room tem- perature and increases at low temperature.  Thus only 0.1% of the conductivity is affected by the transverse cur- rent. Thus, we can neglect the effect of transverse current. The origin of the new peak structures is discussed. The inhomogeneous CDW current explains the step structures in dI/dV curves [11,12].  If the CDW current flows in- homogeneously due to impurities or inhomogeneous cur- rent injection, the step structures might appear in dI/dV curves. When there are parallel CDW chains with differ- ent CDW currents flow, total current vs. voltage curves must have many threshold voltages.  In dI/dV curves, many step structures would be observed.  Moreover, the frequency and amplitude dependences of the new peaks are completely different from those of the Shapiro inter- Thus, the new peaks cannot be produced the ference. Shapiro interference of the inhomogeneous current. Another possible internal degree of f reedom causing the new peak structures is the CDW phase solitons. Solitons have been expected to exist in the commensurate CDW of an o -TaS3 system from electrical transport and optical excitation experiments [2,3]. The existence of a fractional 2π/M  phase soliton (with a fractional charge 2e/M) is predicted in commensurate CDW systems with commen- surability M , where M = 4 for o -TaS 3 systems. Therefore, we consider the soliton dynamics and its contribution into IV characteristics with a RF field. Solitons must play an important role for the CDW dynamics, since a soliton is a topologically stable and carries a 2e/M electrical charge. Since the soliton has electrical charge, the solitons must be pinned by impurities.  Thus, the soliton pinning and depin- ning must be taken into account. When a RF electric field is applied, the solitons must be depinned and contribute dc conduction at a moment, while the bulk CDW does not contribute to dc conduction. The degree of freedom of the solitons would modify CDW dynamics. The soliton flow would be associated with the origin of the unconventional ac-dc interference peaks. Soliton liquid model. – Here we discuss the structure of o-TaS3 CDWs based on the sine-Gordon–type equation, comparing it with the structural information obtained by the X-ray diffraction experiment [7].  The high resolved X-ray diffraction experiment revealed that there are two components corresponding to the commensurate (Q c = 0.25c*) and incommensurate (Q ic = 0.255c*) CDWs, and their intensities have a temperature dependence [14]. The electron density of the CDW on an o -TaS 3 chain is expressed as ρ(z,t) = |ρ 0 |sin(Qcz + φ(z,t)), where |ρ 0 | is the amplitude of the CDW order parameter. The dynam- ical equation of the phase field φ ( z, t ) of a commensurate CDW with commensurability M is written as
where m* is the CDW effective mass, n c = Q c/π is the base one-dimensional electron density, gм is the commen- surability coupling constant, τ is the damping time, E(t) is the external electric field, and vph = √m/m*vf is the CDW phason velocity.  Fimp(z,t) = gP ∑, sin(Qcz + φ(z,t))δ(z − z i ) is the pinning potential force of impurities at z = z . The phase solitons can exist without any excitation if there is a mismatch between 2kF and 1/4c*. With o-TaS 3 systems, the true one-dimensional electron density must be Qic/2π, and the excess charges should make 2π/M If there are many phase solitons in the sys- solitons. tem, the phase φ(z) is approximately written as φ(z,t) = ∑ i φ i (z, t), where the i-th 2π phase soliton is expressed as
Here vs is the soliton velocity, γ = [1 − (vS/vph)²]−1/2 is the Lorentz factor, and d = vph/(M √9M) is the soli- ton width. Due to the experimental situation, the soliton velocity vs = λc fnbn ≪ vph, where λc = 2π/Qc is the commensurate CDW wavelengths. The X-ray diffraction [7] gave a hint about M [15]. In the experiment, the commensurate and incommensurate satellite diffraction spots were observed simultaneously at from 220 K to 50 K, while the Bragg diffraction spots of the mother lattice did not split. We found that 2π phase solitons can only reproduce the commensurate and incom- mensurate satellite diffraction spots, as follows. By calcu- lating the Fourier transformation of the electron density ρ( z, t ), we can expect the CDW satellite diffraction spec- trum.  The Fourier magnitudes have a spot at Q ic , and high-order spots at Q ic + n M ∆ Q, where n is ± 1, ± 2, · · , and ΔQ = Q ic – Q c , due to the Bessel function expansion. To reproduce the commensurate peak at Q c [7], M should be 1. The existence of the 2 π solitons is reasonable since a 2π soliton can exist when a smallest phase dislocation loop encircles a one-dimensional chain, while the chain-chain Coulomb interaction energy is minimized [5,6].  If a frac- tional soliton exists in a chain, the chain-chain Coulomb interaction energy between the neighboring chains must be increased. In other words, the fractional solitons must form a domain wall across the cross-section of the crystal to match the CDW phase with those of the neighboring chains. If the soliton-soliton interaction is strong, the soliton- soliton distance will be constant. Then the 2π phase soli- tons form a one-dimensional lattice as the curve above the dashed line in fig. 4(a) [16–18]. The Fourier magnitude of the electron density for the soliton lattice state shows not only the commensurate and incommensurate spots, but also high-order spots as the curve above the dashed line in fig. 4(b), contrary to the experimental results. The fact that only two spots at Qic and Qc = Qic – ∆Q were ob- served means that the soliton lattice has melted. When the interaction becomes weaker than the impu- rity potential or thermal fluctuation, the soliton lattice will melt and the solitons will move individually [19,20]. This behavior is analogous to that in superconducting vor- tices [21]. The soliton positions are random in the liquid state. The position of the i-th soliton zo is assumed to be distributed randomly in the M∂Q − d range. The curves below the dashed line in fig. 4(a) show dφ(z)/dz of the soliton liquid state for several soliton widths. The Fourier magnitudes for the soliton liquid state are shown below the dashed line in fig. 4(b). For the soliton liquid state of 2π phase solitons, both the commensurate and incommensurate components remain in the Fourier spectrum, and the randomness of the soliton positions causes other high-order components to become diffused. Moreover, increasing of the soliton width d reproduces the crossover from the commensurate to the incommen- When the soliton width is small, then surate  CDWs. the commensurate peak is only observed.  With the de- crease of the soliton width, the CDW is approaching to be incommensurate, then the incommensurate diffraction peak is enhanced. The results of the X-ray experiment [7] indicate that the soliton width d increases when temper- ature increases.  The temperature dependence of d could be interpreted as the development of the CDW order pa- rameter, since the commensurate coupling constant gm is an increasing function of the CDW amplitude. The existence of solitons qualitatively explains the en- hancement of the incommensurate CDW at t he sliding state [7]. The experiment found that the intensity of the incommensurate CDW diffraction spots in the sliding state is stronger than that in the pinning state. The numerical calculation of eq. (1) in overdamped situation provides a consistent result.  Figure 1(d) shows dφ/d x as a function of the applied dc electric field. The peaks of dφ/dx cor- respond to the 2π-solitons. The soliton at zero external electric field have a maximum of the peak height.  Thus, the soliton width is smallest. The soliton width for 2π- solitons is expressed by d = vph/ √ 9м, while soliton-soliton distance is larger than the soliton width. When the ex- ternal electric field exceeds the threshold field, the bulk CDW slides, and the soliton height becomes drastically small, namely, the soliton width becomes larger, as shown in fig. 1(d).  The CDW moves when the external electric field exceeds the threshold field, then the many parts of the CDW do not stay the minimum of the commensura- bility energy potential. Then the constant gм effectively decreases for the moving CDW, since an incommensurate CDW reduces the elastic energy instead of the commen- surability energy in the sliding state.  Hence, the soliton width is increased at t he CDW sliding state, and the CDW phase field becomes rather incommensurate, as s shown in fig. 4(a) and (b).  The one-dimensional soliton liquid model naturally explains the enhancement of the incommensu- rate satellite spots at the sliding state. We investigate CDW dynamics with the 2π soliton liq- uid and impurities in the presence of RF bias using numer- ical calculation of eq. (1). The dc (time-averaged) CDW current ⟨ d ⟩ d ⟩ is calculated as a function of dc electric field.  All calculations are performed with overdamped sit- uation, and the following parameters: vph = 3, gм = 0.5, gP = 5, and τ = 0.01. If there is no soliton, and there is no impurity potential, the calculation reproduces the well- known nonlinear conduction of the CDW system, as shown in fig. 5(a). When a RF field is applied, the Shapiro inter- ference occurs at ⟨ φ ⁄ dt ⟩ = nfrf, where n is integer. The steps in the IV curve correspond to the integer Shapiro steps. Secondly, if impurities are assumed to be induced ran- domly in the one-dimensional chain, the phase field φ(z) is modified since the phase is locked locally by the impu- rities. Then the threshold electric field is increased rather than that without impurities, as shown in fig. 5(b).  In the presence of the RF field, the fractional Shapiro steps appear. This must be caused by the interaction between the impurities and the CDW phase field. Thirdly, if 2 π phase solitons are introduced in the sys- tem, the current flows when a low electric field is applied, as shown in fig. 5(c). Here the impurities were neglected. The solitons move and carry charges, while the bulk CDW is pinned by the commensurability energy potential. So, the system shows an Ohmic conduction at zero voltage. Finally, fig. 5(d) shows the results when both the 2π phase solitons and the impurities are taken into account for the calculation. For ERF = 0, both the bulk CDW and the solitons are pinned by impurities when Edc < Eth, and depinned when Edc > Eth. The soliton depinning energy is much larger than the CDW depinning energy from the commensurability potential and impurity potential, then only one threshold electric field should be observed by the experiment.  We note that this is the same with our experi- mental results.  If we assumed two or more separated CDW domains, two or more threshold fields must be expected. When a RF electric field is applied, then the IV curve shows step-like structures around zero bias.  Figure 5(e) shows magnification of the step-like structure. When Edc < Es1, solitons and CDW are moved by the RF field, however, the contribution to the dc conductivity is zero, as well as at the pinning state. For ERF ≫ Eth, the solitons depinned when Edc > ES1 with the help of the RF field, and contribute to dc conduction. The soli- ton conduction is much smaller than the bulk CDW con- duction since the soliton charge is localized and sparse. Therefore, a step-like structure appears in the current- voltage characteristics. When Edc > Es2, the bulk CDW is depinned, and the sliding CDW causes the integer and fractional Shapiro steps. The differential conductivity of the calculation results indicates a peak structure beside the zero-bias peak, as shown in fig. 5(f), which is quan- titatively consistent with our experimental results. This model provides a possible explanation for both experimen- tal results of the ac-dc interference measurement and the X-ray diffraction measurement. While the strength of im- purity potential has arbitrariness, this calculation suggests that the unconventional ac-dc interference effect is associ- ated with an interaction between solitons and impurities. The sample dependence of strength of impurity potential might be the reason why a few samples show the new ac-dc interference peaks. Summary. – In summary, we found new ac-dc interfer- ence peaks in the differential conductance of o -TaS 3 crys- tals when we applied a radio-frequency (RF) ac electric field. We proposed the 2π soliton liquid model to explain our results and the coexistence of commensurate and in- commensurate CDWs. The existence of the soliton liquid may provide a consistent explanation for the low-energy excitations [2,3], and nonlocal voltages [4], and will play an important role for understanding the quantum inter- ference effect in o -TaS3 systems [5,6]. We thank K. Yamaya, K. Ichimura, J. Ishioka, T. Kuroishi, T. Satoh, and K. Nakatsugawa for fruitful discussions. This work is supported by the JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 12837528 and Iketani Science and Technology Foundation 0241040-A.