Charge Density and Mobility of Charge Density Waves in the Quasi-One-Dimensional Conductor NbS₃

Three charge density waves (CDWs), two of which are formed above room temperature, are observed in the NbS₃ monoclinic phase (NbS₃-II). The charge density and mobility in the high-field limit have been determined for each of three CDWs in this work using the synchronization effect of CDWs in high-frequency fields. It has been found that the mobility of each CDW in this limit is approximately equal to the normal-state mobility of quasiparticles condensed in it. Furthermore, correlation has been observed between the temperature dependences of mobilities of CDWs and quasiparticles. The results of this work refresh problems of a mechanism of the limit conductivity of CDWs and of the distribution of CDWs between atomic chains in the unit cell.

The Frohlich charge transfer mode was proposed to explain superconductivity [1]. Although this hypothesis is invalid, it appeared that the described collective conductivity mechanism is indeed implemented in quasi-one-dimensional [2] (and even in some quasi-two-dimensional [3]) compounds with charge density waves (CDWs). However, the conductivity of CDWs is limited for at least two reasons. First, as a result of the pinning of CDWs in impurities and defects, a CDW begins motion only in a field E above the threshold value E_t. Second, an increase in the conductivity of the CDW is saturated at E ≫ E_t, and the velocity of the CDW $v(E)$ in the high field limit is proportional to E. This property allows us to formally introduce the mobility of the CDWs ${{\mu }_{{{\text{CDW}}}}} \equiv v(\infty ){\text{/}}E$.

It has been known for a long time that the conductivity of the CDW, ${{\sigma }_{{{\text{CDW}}}}}(\infty ) \equiv \sigma (\infty ) - \sigma (0)$, in various compounds in the high field limit approaches the conductivity of its constituting quasiparticles in the normal (metallic) state [4] (below, the conductivity in low fields, $\sigma (0)$, is denoted as σ and the differential conductivity is denoted as $\sigma (E)$). This means that the temperature dependence of σ has a step $\delta \sigma$ near a Peierls transition temperature T_P, which is approximately equal to the limit conductivity of the CDW below T_P. A relation between ${{\sigma }_{{{\text{CDW}}}}}(\infty )$ and $\delta \sigma$ was justified for some particular cases [5, 6]. However, a general theory relating the conductivity of electrons in single-particle and collective states is still absent. For this reason, the study of the sliding of the CDW in extremely high fields in new materials is relevant.

The NbS₃ compound is of interest in this context. Three CDWs denoted as CDW-0, CDW-1, and CDW‑2 are formed in the low-resistivity subphase of the NbS₃ monoclinic phase (NbS₃-II). The corresponding Peierls transition temperatures are T_P0 ≈ 470 K, T_P1 ≈ 360 K, and T_P2 = 150 K [7] (see Fig. 1). Such a diversity of CDWs is obviously due to a relative complexity of the unit cell of NbS₃-II, which contains four pairs of equivalent Nb chains [2, 8, 9]. The wave vectors of the CDW-0 and CDW-1 in relative units are ${\mathbf{q}_{0}} = (0,{\kern 1pt} 0.298,{\kern 1pt} 0)$ and ${\mathbf{q}_{1}} = (0,{\kern 1pt} 0.352,{\kern 1pt} 0)$, respectively [8, 9]. A superstructure related to the formation of the CDW-2 has not yet been observed; for this reason, the wave vector ${\mathbf{q}_{2}}$ is unknown.

Fig. 1.

(Color online) Temperature dependence of the conductivity $\sigma (T)$ for the (blue line, T = 100−335 K) 126 μm × 0.06 μm² and (red line, T = 298−537 K, σ/1.67) 160 μm × 0.1 μm² NbS₃-II samples with close properties. Arrows mark the temperatures of three Peierls transitions. Dashed lines are extrapolated dependences $\sigma (T)$ used to estimate δσ_i, i = 0, 1, 2; δσ₁ is shown; δσ_i and δσ₂ are determined similarly.

Each CDW can slide at E > E_t. The E_t values for three CDWs in NbS₃ are different and follow a relation known for a long time [10]: the lower the temperature T_P, the lower the threshold field E_t. This property makes it possible to observe the sliding of each CDW below the temperature of the corresponding transition in the absence of the contribution from waves formed at higher temperatures. Thus, it is possible to estimate the ${{\sigma }_{{{\text{CDW}}}}}(\infty )$ value for each of three CDWs and to compare these values with $\delta \sigma$ on the corresponding transition.

In addition to the conductivity, we can experimentally determine the charge density carried by each CDW. To this end, it is convenient to use the synchronization of the sliding of CDWs in a high-frequency (HF) field [2]. The current–voltage characteristic (CVC) of a sample in the synchronization regime contains the so-called Shapiro steps, i.e., voltage intervals where the velocity $v$ is constant or nearly constant. The frequency of the HF field f at the first Shapiro step coincides with the fundamental frequency of sliding of the CDW f_f. In this regime, each chain involved in the Frohlich conductivity carries two electrons (charge 2e) in each cycle of the HF field. Consequently, the density of chains in the cross section of the sample involved in the transfer can be easily determined in terms of the current density of the CDW ${{j}_{{{\text{CDW}}}}}$ and the frequency f:

$${{n}_{{{\text{ch}}}}} = ({{j}_{{{\text{CDW}}}}}{\text{/}}f){\text{/}}(2e).$$

(1)

Let s_c be the area of the unit cell in the plane perpendicular to chains (in the ac plane for NbS₃-II). Then, the number of chains in the cell contributing to the current of the CDW is

$${{N}_{{{\text{ch}}}}} = {{n}_{{{\text{ch}}}}}{{s}_{{\text{c}}}}.$$

(2)

The electron density in the CDW can be calculated by the formula

$$n = 2{{n}_{{{\text{ch}}}}}{\text{/}}\lambda ,$$

(3)

where λ is the period of the CDW. It is interesting to compare the charge density en carried by each CDW thus calculated with known models of the formation of the CDW on eight chains in the unit cell.

The mobility ${{\mu }_{{{\text{CDW}}}}}$ in the high-field limit can be easily estimated in terms of ${{\sigma }_{{{\text{CDW}}}}}(\infty )$ as ${{\mu }_{{{\text{CDW}}}}}$ = σ_sCDW(∞)/en. Taking into account that $v = \lambda {{f}_{{\text{f}}}}$ and $n = 2{\text{/}}(\lambda {{s}_{{\text{c}}}})$, it is easy to verify that σ_sCDW(∞)/en ≡ $v$(∞)/E. Here and below, the subscript s means the specific conductivity. In addition, the mobility of the same electrons in the normal state can be determined in terms of $\delta \sigma$: $\mu = \delta {{\sigma }_{{\text{s}}}}{\text{/}}en = \lambda \delta {{\sigma }_{{\text{s}}}}{\text{/}}({{j}_{{{\text{CDW}}}}}{\text{/}}f)$ = $L\lambda \delta \sigma {\text{/}}({{I}_{{{\text{CDW}}}}}{\text{/}}f)$ [7], where ${{I}_{{{\text{CDW}}}}}$ is the current of the CDW and L is the length of the sample. The last expression is convenient for the calculation of μ from directly measured quantities. It is interesting to compare the μ values thus obtained with the mobilities of the corresponding CDWs.

In this work, the electron density involved in the Frohlich conduction mode, the corresponding number ${{N}_{{{\text{ch}}}}}$, and the mobility of the same electrons in the CDW and in the normal state are determined for all three CDWs formed in NbS₃-II. Comparison shows that the conductivity of each CDW in the high-field limit is close to the normal-state conductivity of electrons condensed in it. The numbers ${{N}_{{{\text{ch}}}}}$ for the CDW‑0, CDW-1, and CDW-2 are related as (1.3—2.3) : 1 : (1/3–1/1000), whereas the mobilities of these CDWs (and the corresponding electrons in the normal state) are approximately 0.045, 0.6, and 3 cm²/(V s), respectively. The values obtained are in agreement with preliminary results presented in [7]. The anomalously low ${{\mu }_{{{\text{CDW}}}}}$ value for the CDW-0 indicates a hopping conductivity of quasiparticles at T > T_P0, which can explain the dielectric temperature dependence of the conductivity in this range. Furthermore, according to our preliminary data, the dielectric behavior of the conductivity (and mobility) is also observed for the CDW-0 itself, whereas the temperature dependence of these parameters for the CDW-1 is much weaker in accordance with a higher mobility.

First, we present the main characteristics of all three CDWs. For the CDW-1 in NbS₃-II, j_CDW/f = 18 А/(MHz cm²) [11]; substituting this value into Eq. (1) and then Eq. (1) and s_c = 180 Ų [2] into Eq. (2), we obtain ${{N}_{{{\text{ch1}}}}} = 1$ [11]. The period of the CDW-1 is ${{\lambda }_{1}} = b{\text{/}}{\mathbf{q}_{b}}$ = 1.12 nm, where $b = 0.335$ nm [2, 9] is the lattice constant along chains and ${\mathbf{q}_{b}} = 0.298$ is the b component of the vector q. Formula (3) gives ${{n}_{1}} = 1.0 \times {{10}^{{21}}}$ cm^–3 for the CDW-1.

Current–voltage characteristics where the nonlinear conductivity is due to the sliding of the CDW-0 are presented in [7]. Shapiro steps were observed in these characteristics under the action of an HF field. It was established that the ratio j_CDW/f for the CDW-0 is a factor of 1.3–2.3 larger than that for the CDW-1; i.e., ${{N}_{{{\text{ch0}}}}} = 1.3{-} 2.3$, and ${{n}_{0}} = ({{\lambda }_{1}}{\text{/}}{{\lambda }_{0}})(1.3{-} 2.3){{n}_{1}}$ = (1.5—2.7) × 10²¹ cm^–3, which is a factor of 1.5–2.7 larger than that for the CDW-1.

The ratio j_CDW/f for the CDW-2 varies in a wide range exceeding two orders of magnitude [12]. Correspondingly, ${{N}_{{{\text{ch2}}}}}$ is in the range of 1/3–10^–3; i.e., the number of chains per unit cell is much smaller than 1 in any case. The period ${{\lambda }_{2}}$ is still unknown, but it is most likely close to the periods of the CDW-1 and CDW-0. This conclusion follows from the calculation of the Lindhard function for NbS₃-II [13]: this function has a maximum at three vectors q; the b* components of all three vectors are close to 1/3, two vectors are close to the experimental values q₁ and q₀, and the b* component of the third vector is 0.339. It can be assumed that this vector corresponds to the CDW-2. Setting ${{\lambda }_{2}} = {{\lambda }_{1}}$,¹ we obtain ${{n}_{2}} = (1{\text{/}}3{-} {{10}^{{ - 3}}}) \times$ 10²¹ cm^–3, which can be considered as an estimate for ${{n}_{2}}$ at any reasonable ${{\lambda }_{2}}$ values. Such a low density of the CDW-0, which strongly depends on the sample, is most likely due to the formation of this CDW on structural defects in NbS₃. It was argued in [9, 14–16] that the CDW-2 appears on stacking faults, which are formed by atom-thick planes parallel to the ab plane. Cells in this layer usually contain an extra pair of chains. The recent observation of CDWs on dichalcogenide monolayers (see, e.g., [17]) corroborates this possibility. A significant concentration of stacking faults in NbS₃-II is confirmed by transmission electron and scanning tunneling microscopy data [9, 15]. Nevertheless, the formation of two-dimensional CDWs on stacking faults requires direct proof.

We now estimate the normal-state mobility of electrons condensed in the CDWs. To this end, it is necessary to determine $\delta \sigma$ on the corresponding transition. Figure 1 shows the temperature dependences of the conductivity $\sigma (T)$ for NbS₃-II. Since high temperature measurements (above 500 K) required a special mounting of samples, we present data for two whiskers with similar properties in order to represent $\sigma (T)$ in the entire temperature range. To join two curves, the conductivity of the whisker studied at high temperatures was divided by 1.67. A dielectric temperature dependence of the resistance above T_P0 is observed for all studied samples [7]: the resistance decreases with increasing temperature, although charge density waves are apparently absent at these temperatures.

The dashed lines in Fig. 1 illustrate the method for determining the conductivity jump δσ_i, i = 0, 1, 2, near three transitions. The behavior of $\sigma (T)$ below T_P1 varies strongly from sample to sample [11]. A weak temperature dependence of σ and a pronounced kink near T_P2 = 150 K mean that the sample presented in Fig. 1 is in a low-resistivity “subphase” [12].

Taking into account the δσ_s values for the CDW-1 and CDW-0, we estimate the mobilities of quasiparticles as ${{\mu }_{1}} = 0.6$ cm²/(V s) and ${{\mu }_{0}} = 0.04{-} 0.05$ cm²/(V s). Although the ratio j_CDW/f for the CDW-2 varies in a wide range, it is proportional to the jump of the specific conductivity δσ_s2 (Fig. 1) [12]. According to [12, Fig. 15], the coefficient of proportionality $\delta {{\sigma }_{{{\text{s2}}}}}{\text{/}}({{j}_{{{\text{CDW}}}}}{\text{/}}f)$ is 30 ($\Omega$ cm)^–1/(A/(MHz cm²)). The normal-state mobility of electrons condensed in the CDW-2 is the product of this coefficient and ${{\lambda }_{2}}$. Setting ${{\lambda }_{2}} = {{\lambda }_{1}}$, we estimate the mobility of quasiparticles as ${{\mu }_{2}} = 3.4$ cm²/(V s).

We now characterize the dynamics of CDWs in high fields, which is the central aim of the work. All three CDWs demonstrate a tendency to the saturation of an increase in $\sigma (E)$. The right panels of Fig. 2 present $\sigma (E)$ curves for three CDWs obtained without and with HF irradiation. The tendency to the saturation of an increase in $\sigma (E)$ is seen on all curves; it is more pronounced on curves recorded under HF irradiation. The fragments of the temperature dependences of the conductivity $\sigma (T)$ in the regions of the corresponding transitions and below are shown in the left panels of Fig. 2 together with the variation range of $\sigma (V)$ at the corresponding temperatures; i.e., the “differential CVCs” shown in the right panels are projected on the respective left panels. It is seen that ${{\sigma }_{{{\text{CDW}}}}}(\infty )$ for each CDW is close to $\delta \sigma$ in the region of the corresponding transition. Thus, the estimates obtained above for the conductivity and mobility can also be assigned to electrons in the corresponding collective states. In particular, the most mobile CDW-2 of three CDWs is formed through the condensation of the most mobile quasiparticles.

Fig. 2.

(Color online) (Top left panel) Fragment of the $\sigma (T)$ curve near T_P0 and (top right panel) the $\sigma (V)$ curve at 415 K measured (filled circles) without and (empty circles) with exposure to a 10-MHz field; all data were obtained with the 58 μm × 0.12 μm² sample. (Middle left panel) Fragment of the $\sigma (T)$ curve near T_P1 and (middle right panel) the $\sigma (V)$ curve at room temperature measured (filled circles) without and (empty circles) with exposure to a 800-MHz field; all data were obtained with the 38 μm × 0.01 μm² sample. (Bottom left panel) Fragment of the $\sigma (T)$ curve near T_P2 and (bottom right panel) the $\sigma (V)$ curve at 120 K measured (filled circles) without and (empty circles) with exposure to a 400-MHz field; all data were obtained with the 266 μm × 0.015 μm² sample. The vertical straight lines in the left panels are the projections of CVCs shown in the respective right panels.

The “superhigh-temperature” CDW-0, which is formed through the condensation of quasiparticles with a mobility of 0.04–0.05 cm²/(V s), has the lowest mobility. Such a low mobility is characteristic of a hopping conductivity mechanism. This is a natural explanation of the dielectric behavior of the conductivity in NbS₃-II above T_P0, i.e., in the normal state (Fig. 1). However, a question arises: Is the mechanism of the collective conductivity of the CDW-0 in high fields also hopping? This question cannot be answered unambiguously because a universal model explaining the similarity of scattering mechanisms for electrons condensed in CDWs and in the normal state is still absent. Nevertheless, we attempted to study the temperature dependence of σ_CDW(∞) and ${{\mu }_{{{\text{CDW}}}}}$ for the CDW-0 and CDW-1.

To study the mobility of the CDW-0, we selected one of the II phase samples where the CDW-1 is absent. As shown in [7, 15], the grown NbS₃-II batches include unconventional samples where the CDW-1 is either not observed or weakly pronounced. At the same time, the temperature T_P0, period of the CDW-0, and its mobility in high fields are the same as in conventional samples. This was confirmed by transport [7, 15] and structural studies [15]. The advantage of these samples is that the dynamics of the CDW-0 in them can be examined in a wide temperature range not only above T_P1: the contribution of the CDW-0 to the conductivity against the background of the sliding of the CDW-1 in conventional samples can hardly be determined below T_P1. Moreover, fluctuations of the CDW-1 also significantly reduce the conductivity σ above T_P1. As a result, the ${{\sigma }_{{{\text{CDW}}}}}(\infty )$ value determined as $\sigma (\infty ) - \sigma$ is overestimated. A set of CVCs for one of such samples in the temperature range from room temperature to 480 K, which is slightly above T_P0, is presented in [7]. Shapiro steps can be observed in all CVCs under 10-MHz irradiation. As seen in Fig. 2, such CVCs often allow one to more accurately estimate ${{\sigma }_{{{\text{CDW}}}}}(\infty )$ at limited voltages.

The asymptotic behavior of the conductivity in the limit $V \to \infty$ can often be described using, e.g., the formula ${{\sigma }_{{{\text{CDW}}}}} = A + B\exp ( - {{V}_{0}}{\text{/}}V)$, where A, B, and ${{V}_{0}}$ are constants, for Zener tunneling [18]. As an example, the CVC of the sample without the CDW-1 at T = 321 K with and without HF irradiation replotted as $\sigma - \sigma (0)$ versus $1{\text{/|}}V{\text{|}}$ on a semilogarithmic scale is shown in the inset of Fig. 3. Extrapolating $1{\text{/|}}V{\text{|}}$ to 0, one can obtain the estimate σ_CDW(∞) = 1.2 × 10^–6 Ω^–1 (σ_sCDW(∞) = 5.8 Ω^–1 cm^–1). Figure 3 shows σ_sCDW(∞) thus obtained versus T/T_P0. Other methods for determining σ_CDW(∞) gave close results.

Fig. 3.

(Color online) Conductivities σ_sCDW(∞) for (circles) the CDW-0 in the 58 μm × 0.12 μm² sample (see top panels in Fig. 2) versus T/T_P0 and (asterisks) the CDW-1 in the (blue) 18 μm × 0.008 μm² and (green) 38 μm × 0.01 μm² samples versus T/T_P1. The inset shows the ($\ln [\sigma - \sigma (0)]$, $1{\text{/}}V$) CVC of the 58 μm × 0.12 μm² sample (without the CDW-1) at T = 321 K.

The σ_sCDW(∞) values for the CDW-1 were determined similarly. These results obtained with two high-resistivity samples are also presented in Fig. 3. The conductivity σ_sCDW(∞) for the CDW-0 below about 0.95T_P0 decreases with decreasing temperature; i.e., σ_sCDW(∞) demonstrates a dielectric behavior. At the same time, the conductivity of the CDW-1 below 0.95T_P1 weakly depends on the temperature.

All σ_sCDW(∞) curves exhibit a sharp drop when the temperature approaches T_P1 or T_P0 from below (see Fig. 3), which is due to the decay of the CDW, i.e., to a decrease in the charge density of the CDW. Shapiro steps were observed for samples exposed to the HF field; therefore, the charge density ne of the CDW is known at each temperature. The ratio σ_sCDW(∞)/ne gives the mobility of the CDW-0, as well as of the CDW-1, for one of the samples (Fig. 3), in which the synchronization of the CDW-1 was examined (Fig. 4).

Fig. 4.

(Color online) Mobilities of (circles) the CDW-0 in the 58 μm × 0.12 μm² sample versus T/T_P0 and (asterisks) the CDW-1 in the 38 μm × 0.01 μm² sample versus T/T_P1 (see Fig. 3). Synchronization in a wide temperature range was not examined for the 18 μm × 0.008 μm² sample (Fig. 3).

As seen in Fig. 4, the mobilities are close to the previously estimated mobilities of quasiparticles near T_P0. An increase in the mobility of the CDW-0 with the temperature is observed in the entire temperature range. This means that the mobility of the CDW-0 in high fields is approximately equal to the normal-state mobility of quasiparticles condensed in it and has a similar dielectric temperature dependence.

The temperature dependence of the mobility of the CDW-1 is much weaker. This corresponds to a weak temperature dependence of the conductivity above T_P1 [7, 12] (the contribution from quasiparticles associated with the CDW-0, which has a dielectric temperature dependence, should be subtracted from this conductivity for correct comparison with σ_CDW(∞)).

To summarize, it has been shown experimentally that the mobilities of all three CDWs in NbS₃-II in high fields are approximately equal to the mobilities of their constituent quasiparticles. The temperature dependence of the conductivity (and mobility) of the CDW-0 is dielectric, repeating the behavior of the conductivity above T_P0, whereas the conductivity of the CDW-1 weakly depends on the temperature. The difference between the mobilities of different CDWs reaches almost two orders of magnitude. Because of this result, the development of a general theory relating the mobilities of electrons in CDWs and in the free state, as well as the problem of the formation of CDWs from states with hopping conductivity, becomes relevant. The Bardeen–Stephen model describing the resistance of a two-dimensional superconductor in the mode of viscous flow of Abrikosov vortices [19] can be considered as a certain analogy in the case of viscous friction.

In spite of the absence of a general theory, it is reasonable to attribute the behavior of the conductivity of the CDW-0 to the dielectric temperature dependence of the single-particle conductivity of NbS$_{3}$ above T_P0 [7]. The mobility of quasiparticles forming the CDW-0 above T_P0, 0.04–0.05 cm²/(V s), is characteristic of the hopping conductivity. The mobility of the CDW-0 is of the same order of magnitude. Thus, we arrive at a surprising conclusion that the conductivity of the CDW-0 has a hopping character. This conclusion requires further analysis. We can currently conclude that the closeness of the conductivities of charges in the collective and single-particle states for three CDWs having significantly different properties is not accidental.

It is noteworthy that the dissipation of CDWs in relatively low fields is determined by the scattering of quasiparticles excited through the Peierls gap [20, 21], whereas the mechanism of dissipation of CDWs in high fields is similar to the mechanism of normal-state scattering of electrons condensed in it. At the same time, it is not excluded that σ_CDW(∞) is also determined to a certain extent by quasiparticles coexisting with CDWs. This is indicated by the results obtained in [22], where the application of the Hall effect allowed the authors to achieve the sliding of a CDW in NbSe₃ in the absence of the codirected current of quasiparticles. It appeared that σ_CDW(∞) in this case can be much higher than that in the case of the sliding of the CDW in the external electric field and can noticeably exceed the normal conductivity of quasiparticles condensed in the CDW.

The determined charge densities of three CDWs should be discussed separately. The j_CDW/f values for the CDW-0 were obtained in the range of 23–40 А/(MHz cm²); i.e., they are 1.3–2.3 times higher than those for the CDW-1. The ratio j_CDW/f for the CDW-1 corresponds to the sliding of one chain of the CDW among eight chains per unit cell [11]. It could be assumed that the CDW-0 modulates two of eight chains (j_CDW/f = 36 А/(MHz cm²)), whereas the spread of j_CDW/f values could be attributed to the error in the determination of the area of the cross section of the nanosamples. However, the ratio ${{N}_{{{\text{ch0}}}}}{\text{/}}{{N}_{{{\text{ch1}}}}}$ for samples where Shapiro steps were observed in the case of sliding of both the CDW-0 and CDW-1 was also noninteger and depended on the sample. Thus, it can be suggested that it is impossible to “seat” CDW-0 and CDW-1 on “their” chains, and the simultaneous modulation of the same chains by different CDWs should be taken into account, as proposed in [9, 23]. This model was experimentally confirmed for NbSe₃ [24], where two CDWs are formed. However, the j_CDW/f values for both CDWs in NbSe₃ coincide with each other [25, 26] and are consistent with commonly accepted concepts on the formation of each CDW on one of three pairs of chains in the unit cell. It is not also excluded that the spread of j_CDW/f values in NbS₃ is due to the inhomogeneity of the sample.

Finally, we note that the method proposed here to estimate the mobility of quasiparticles based on the synchronization of CDWs and on the analysis of the $\sigma (T)$ curve in the transition region can be very relevant for whiskers with CDWs. In particular, NbS₃-II whiskers usually have a width of less than 1 μm, and it is very difficult to study the Hall effect in them in order to determine the mobility.