Field-induced dielectric response saturation in o-TaS₃

The charge density wave (CDW) is a periodic distribution of conduction electrons in real space, accompanied by a crystal lattice distortion. It is usually observed in a wide class of quasi-one-dimensional (Q1D) solids below the Peierls transition temperature (T_P) [1–3]. There has been considerable progress in the understanding of the macroscopic carrier condensate in connection to CDW phase solitons [4–11].

Solitons allow for hopping between different ground states. Since the energy levels of these states are very similar, the excitation energy of the charged particles would be lower than the bare electronic gap. CDW phase solitons are the excitations of topological phase defects whose presence has long been expected in Q1D systems [12–14]. More than three decades ago, Su et al showed that excitations (solitons) with fractional charges may exist in half-filled or one-third-filled Peierls systems [13, 14]. In [9], x-ray data indicated possible internal degrees of freedom of the CDW solitons. Moreover, the experimental results from optical methods obtained by Zaitsev-Zotov et al convincingly supported the existence of solitons in orthorhombic TaS₃ (o-TaS₃) [5]. Further, in very recent experiments, the charged phase solitons have been directly observed in NbSe₃ by Brun et al using scanning tunnelling microscopy [15]. In an idealized CDW system the spacial arrangement of the density of conduction electrons would have a perfect period, but in a real crystal there are defects distributed randomly in phase field (CDW phase solitons). As the CDW phase is the manifestation of the charge density distribution, the excitations of phase solitons should respond to the external electric field.

In [10], two thresholds in field dependent conductivity ($\sigma -E$) were identified below 60 K in TaS₃. The lower $E_{\text{T}}^{\prime}$ was proposed to be related to solitons, whereas above 100 K there was only one threshold due to the classical CDW depinning. The polarization process activated below about 70 K, was also attributed to solitons (localized pinned CDW excitations) [16]. A new phase diagram was suggested where the Coulomb blockade threshold lower than the classical CDW depinning threshold field ${{E}_{\text{T}-\text{cl}}}$ emerges for charge soliton nucleation [17, 18]. Furthermore, a new ac–dc interference spectrum in the differential conductivities of o-TaS₃ were observed, attributing the driving force to the depinning of solitons [6]. Solitons contribution therefore could be rather important for the charge dynamics of CDW.

o-TaS₃ is a typical quasi-one-dimensional semiconducting CDW material with the ${{T}_{\text{P}}}\simeq 220$ K, below which a temperature dependence of the linear conductivity obeys the activation law with an activation energy $\Delta =850$ K. At $T\lesssim$ ${{T}_{\text{P}}}/2$, the linear conductivity begins to deviate from the activation law with a smaller activation energy, indicating possible excitations of charged solitons [10]. Considering that the lower threshold $E_{\text{T}}^{\prime}$ is possibly related to the pinned or bounded solitons, in this work we investigate charge transport and dielectric properties in external electric field $E\gtrsim E_{\text{T}}^{\prime}$.

Temperature dependent complex dielectric or conductivity spectrum is a powerful probe for studying charge dynamics [4, 19–22]. At a fixed temperature T₀, the complex dielectric spectrum of a material $\epsilon \left(\omega,{{T}_{0}}\right)$ is ultimately defined by the polarizations of the corresponding charged microscopic particles. In CDW systems exceptionally large dielectric constants and strong dispersion effects below 10⁵ Hz were observed [16, 23–25]. Further, under various E established by dc bias, dielectric spectroscopy would be of great interest since the collective transport or the solitons excitations in CDW systems could only occur above the threshold. In order to study CDW dynamics and to eliminate the screening effect of the normal electrons excited through the Peierls gap, experiments should be carried out at a much lower temperature. To the best of our knowledge the bias dependent dielectric response below 50 K has been inadequately reported for TaS₃. Zhilinskii et al observed that the dielectric constant did not change in o-TaS₃ even up to 100 V cm⁻¹ at 4.3 K though the conductivity increased [11]. However, the evolution of the dielectric behavior under external field covering $E_{\text{T}}^{\prime}$ below T  =  50 K has not been studied and is of interest.

Here we studied the temperature and electric field dependent charge dynamics along the chain direction of o-TaS₃ below 50 K. With increasing the external electric field, two threshold-like features could be identified. For fields larger than $E_{\text{T}}^{\prime}$, the dielectric constant starts to decrease while the conductivity increases due to the tunnelling of solitons. As E continues to increase, there is a saturation of dielectric response. With decreasing temperature, the effect of the external field on the dielectric responses of the system weakens gradually, and at 13 K it diminishes.

TaS₃ crystals were successfully grown by the conventional chemical vapor transport method. The stoichiometric tantalum and sulfur pieces were put into an alumina crucible before sealing in a quartz tube and reacting at 700 °C for 50 h. Single crystals were grown by vapor transport method in a gradient furnace. The temperature of hot (cold) end was 700 °C (500 °C). Consistent with the earlier reports [26], the as-grown single crystals showed thin needle-like shape with dark black color, typical sizes with diameter about 25 $\mu$m and length up to 10.0 mm. In the experiments, we focused on the axis direction of the sample, which is corresponding to the chain direction of o-TaS₃ [2].

Temperature dependent resistivity was measured with current I  =  10 $\mu$A before probing the dielectric spectrum by four-terminal-pair approach. The I–V curves showed linear behavior, confirming good ohmic contacts by Indium compression⁶. We also compared the contact resistance of the 4-contact with 2-contact probe and found the difference was negligible from 300 K to 12 K. In addition, we confirmed that the impedance spectra were unaffected by the difference between substrates by comparing data obtained with the LaAlO₃ and sapphire. The complex impedance properties under sweeping dc bias $z\left(\omega,E\right)=|z\left(\omega,E\right)|\cos \theta \left(\omega,E\right)+\text{i}|z\left(\omega,E\right)|\sin \theta \left(\omega,E\right)$ from 0 to 50 V cm⁻¹ were probed on Agilent 4294A precision impedance analyzer through four-coaxial cables in two pairs (corresponding to two electrodes on the sample), where $\theta \left(\omega,E\right)$ was the phase difference between the ac potential and current at frequency ω. The sample was mounted on a cryostat with the experimental temperature range 13–300 K. All the impedance spectra were measured with driving voltage V_p−p  =  30 mV(rms), far below the electric threshold field obtained in I–V measurements. In analysis, the sample was a leakage capacitor, both dielectric constant $\epsilon \left(\omega,E\right)={{\epsilon}_{1}}\left(\omega,E\right)+\text{i}{{\epsilon}_{2}}\left(\omega,E\right)$ and the complex conductivity spectrum $\sigma \left(\omega,E\right)={{\sigma}_{1}}\left(\omega,E\right)+\text{i}{{\sigma}_{2}}\left(\omega,E\right)$ could be deduced from the raw data $z\left(\omega,E\right)$ based on the universal relations: $\text{i}\omega \epsilon \left(\omega,E\right)=\sigma \left(\omega,E\right)=1/z\left(\omega,E\right)\cdot (d/S)$, where ${{\epsilon}_{i}}\left(\omega,E\right)={{\epsilon}_{0}}{{\epsilon}_{ri}}\left(\omega,E\right)$ (i  =  1,2), d and S were the length and the cross-sectional area of the sample, respectively.

The temperature dependent resistivity ${{\rho}_{\text{dc}}}$(T), the Peierls transition temperature ${{T}_{\text{P}}}=218$ K, I–V relations and ${{\epsilon}_{r1}}\left(\omega,E=0\right)$ from 300 K to 13 K along the chain direction of o-TaS₃ crystal were all consistent with previous results [9, 10]. Here we showed the $\sigma -E$ curves below 60 K since it is the interesting temperature range, see figure 1. The lower threshold $E_{\text{T}}^{\prime}$ could be clearly seen below 50 K, whereas ${{E}_{\text{T}-\text{cl}}}$ would be up to a much larger value⁷. In our experiment, ${{E}_{\text{T}-\text{cl}}}=100$ V cm⁻¹ at T  =  20 K, whereas ${{E}_{\text{T}-\text{cl}}}\sim 350$ V cm⁻¹ at 4.3 K in [10].

Zoom In Zoom Out Reset image size

Before studying the dielectric properties under external dc electric field E, we probed the temperature dependent dc conductivity shown in figure 2(a). Below T  =  60 K, the log${{\sigma}_{\text{dc}}}(1/T)$ reveals a linear behavior and the activation energy obtained is about $\Delta {{E}_{\text{LT}}}=215$ K, whereas above T  =  100 K, the activation energy is about $\Delta {{E}_{\text{HT}}}=780$ K, in agreement with the earlier reports [10]. The dielectric spectra under zero external field are shown in figures 2(b) and (c), from which it could be seen that the evolution of the polarization characteristics with temperature is very similar to [16]. At T  =  185 K, with the establishment of the CDW, one relaxation process emerges at higher frequencies. The peak of the imaginary part of the dielectric spectrum shifts towards low frequencies with decreasing temperature, indicating the gradually freezing of this process, as shown by the arrows marking in figure 2(c). At T  =  77 K and below, another process appears at higher frequencies as denoted by the double arrow. To be consistent with [16], we call the first process α and the second one β. Clearly, at T  =  50 K, the α process due to the elastic pinned CDW screened by uncondensed carriers shifts out of the measurable frequency window and the β process which was attributed to solitons (localized pinned CDW excitations) dominates the charge dynamics [16]⁸. These two processes under dc field E  =  0 were systematically investigated in [16].

Zoom In Zoom Out Reset image size

In order to investigate the charge dynamics in current carrying states, we probed the dielectric properties by sweeping dc bias below T  =  50 K. Corresponding to the lower threshold $E_{\text{T}}^{\prime}$ in conductivity, for example, at T  =  35 K as shown in figures 3(a) and (b), the ${{\epsilon}_{r1}}-E$ curve starts to decrease. Above this threshold, at about E_s  =  7.0 V cm⁻¹, the relation of ${{\epsilon}_{r,1}}-E$ tends to saturate. Clearly, E_s is larger than $E_{\text{T}}^{\prime}$ but much smaller than ${{E}_{\text{T}-\text{cl}}}\simeq 90$ V cm⁻¹ at T  =  35 K. To the best of our knowledge, such a decrease and then saturation in field dependent dielectric constant has not been reported yet.

Zoom In Zoom Out Reset image size

Generally, the structural defects can be polarized under dc field with a smooth dielectric response, hence the experimental data excludes this possibility. Considering that the β process (dynamics of solitons pinned at impurities) would dominate the charge dynamics below the freezing temperature of the α process and that the activation energy $\Delta {{E}_{\text{LT}}}=215$ K obtained above is consistent with the existence of a band within CDW gap, we propose that the main feature in electronic polarization properties of the $o$-TaS₃ crystal may originate from trapped or pinned CDW phase solitons [27–29].

The presence of CDW phase solitons has been convincingly shown in $o$-TaS₃ [5, 6, 9, 15]. As the phase of CDW presents the distribution of the charge density in the real space, it is plausible that the phase defects would response to the external electric field. In zero field limit, all pinned solitons could be polarized and contribute to the dielectric constant. In figure 4(a), the electronic polarization of charged solitons presents as a small displacement (solid bell shaped curve shifts somewhat from the dashed curve) in external dc field.

Zoom In Zoom Out Reset image size

With increasing E the increasing conductance corresponds to a decreasing dielectric constant. We suggest that only the pinned solitons could be locally polarized and the others could tunnel through the energy barrier with the help of external field. Hence the real part of the dielectric constant decreased with increasing external electric field, see figure 4. Having in mind that the CDW phase solitons are the phase defects, then electronic polarization could occur locally around the equilibrium positions of the solitons in the potential well under external field. As expected, if the reduction of the real part of the dielectric constant is caused by the tunneling of the phase solitons, the imaginary part of dielectric constant ${{\epsilon}_{r,2}}$ must increase. Indeed, we observed such behavior, see figure 5(a). When E is increased further, it is plausible that more and more solitons would be depinned and eventually the contribution to polarization from pinned solitons would diminish and the dielectric constant saturate. Corresponding to this saturation in ${{\epsilon}_{r,1}}$ or the delocalization of the whole trapped solitons, ${{\epsilon}_{r,2}}$ turns downward above E_s, revealing a change in the mechanisms of energy loss, see figure 5(a). In very recent reports, Matsuura et al proposed soliton liquid model to interpret the observed ac–dc interference spectrum [6]. Here we suggest that the main contribution for the observed features in the dielectric properties may come from the charged phase solitons in o-TaS₃.

Zoom In Zoom Out Reset image size

With decreasing temperature, the kink feature in ${{\epsilon}_{1}}(E)$ weakens. For example, at 30 K, see figure 5(a), the dielectric constant starts to decrease to E  =  8.0 V cm⁻¹ and then it saturates at about $2.2\times {{10}^{6}}$ from E  =  8.0 V cm⁻¹ to 40.0 V cm⁻¹. The difference between the saturation field and the zero-limit field $\Delta {{\epsilon}_{r1}}$ decreases when compared to T  =35 K. In figure 5(b), the frequency dependent dielectric spectra under various dc bias are shown. The obvious feature in figure 5(b) is that the β relaxation process becomes suppressed as most pinned solitons tunnel through barriers under larger electric fields. For example, the difference of the dielectric response at E  =  10.0 V cm⁻¹ and E  =  20.0 V cm⁻¹ can not be identified within the experiment error. With further cooling, at 13 K the kink is nearly negligible, as shown in figure 6(a), the dielectric constant only increases slightly in lower frequency region below several kHz at E  =  50 V cm⁻¹. The spectra remains nearly unchanged up to 1.0 MHz below 40 V cm⁻¹, see figure 6(b), indicating bias independent dielectric properties in moderate electric field.

Zoom In Zoom Out Reset image size

The temperature dependent $\Delta {{\epsilon}_{r1}}-T$ has been shown in figure 7. From this tendency, it is natural that the dielectric response would remain constant with increasing electric field at liquid helium temperatuare [11]. The diminished $\Delta {{\epsilon}_{r1}}$ may originate from the freezing of β relaxation process. Meanwhile this is consistent with the model proposed by Matsuura et al, that the number of solitons are determined by the electron density [6, 9]. Below 50 K the lower threshold field $E_{\text{T}}^{\prime}$ decreases with decreasing temperature. This can be explained based on the relation that the dielectric permittivity multiplied by the threshold field should be a constant, regardless of the details of the pinning mechanism of the CDW [16].

Zoom In Zoom Out Reset image size

As shown in figure 7, $\Delta {{\epsilon}_{r1}}$ decreases with decreasing temperature, possibly indicating that the density of solitons would decrease with cooling. Such a phenomenon might be related with the reports from [9], in which the commensurate CDW develops with a decrease of temperature. Nevertheless, research work about the dynamics of solitons in TaS₃ is not sufficient up to date, especially at temperatures below T  =  50 K. The qualitatively similar behavior between $\Delta {{\epsilon}_{r1}}-T$ and $E_{\text{T}}^{\prime}-T$ reveals that the ease for solitons tunnelling diminishes the dielectric constant of the system. In this tendency, the β process would be replaced by the ${{\beta}_{0}}$ process [16]. Based on our experimental data at T  =  13 K, it seems that the ${{\beta}_{0}}$ process is not related to the locally pinned solitons since its independence to the external electric field. Then one may ask the reason for the essential increase of conductivity with increasing electric field at T  =  13 K. In previous report [11], Zhilinskii et al also reported that the similar results: the real part of the dielectric constant $\epsilon _{r}^{\prime}$ remained unchanged at dc fields of about 100 V cm⁻¹ (T  =  4.3 K), while the conductivity increased by several times. Here we suggest that the freezing of one process means that the electronic polarization can not respond timely to the ac external field at a finite frequency, while for the dc field, the hopping mobility of the excited solitons may increase nontrivially and hence the conductivity increases.

In summary, we measured the dependence of the dielectric properties of TaS₃ crystals on temperature and electric field. Below T  =  50 K, the electrical threshold $E_{\text{T}}^{\prime}$ in I–V curve could be clearly identified. Above $E_{\text{T}}^{\prime}$, with increasing E, the dielectric constant starts to decrease while the dc conductivity increases due to tunnelling of solitons. The saturation of the field dependent dielectric constant was suggested to occur when the solitons could no longer be polarized locally under a field E  >  E_s. With further cooling, the threshold $E_{\text{T}}^{\prime}$ decreases and the difference of the dielectric constant between the saturation and the zero-limit field value $\Delta {{\epsilon}_{r1}}$ also decreases due to soliton freezing.

The authors are very grateful for instructive discussions with Toru Matsuura. The research work was supported by the National Science Foundation of China (Grant No. 10704054). Work at Brookhaven National Laboratory is supported by the US DOE, Contract No. DE-SC00112704.