A quieter state of charge and ultra-low-noise of the collective current in quasi-1D charge-density-wave nanowires

Domain: physics
Published: 2025-12-31
Editorial outcome: major_revision

Abstract
Electronic flicker noise limits phase stability in communication systems, reduces the sensitivity and selectivity of sensors, and degrades coherence in quantum devices. There is a strong need for unconventional materials and strategies for achieving ultra-low-noise performance in nanoscale and quantum electronics. Here, we demonstrate that in nanowires of the quasi-one-dimensional, fully gapped charge-density-wave material (TaSe4)2I, low-frequency electronic noise is suppressed below the limit of thermalized charge carriers in passive resistors. When the current is dominated by the sliding Frohlich condensate, the normalized noise spectral density, S=I2, decreases linearly with current, I — a striking departure from the constant value of S=I2, observed in conventional conductors. No residual minimum noise level is reached for the current of the electron-lattice condensate in (TaSe4)2I nanowires. Repeating the measurements for another charge-density wave conductor, NbS3-II, we found a similar reduction below the normal electron limit at room temperature. Our findings signal intrinsically lower current fluctuations within a correlated electron transport regime.

Introduction
Recent years have witnessed renewed interest in charge density waves (CDW) driven by the exciting physics of strongly correlated phenomena, the growing number of quasi-one-dimensional (1D) and quasi-two-dimensional (2D) van der Waals materials with CDW phases, and possibilities for practical applications. In an ideal 1D system, a collective transport mode of the CDW condensate can slide without dissipation as the Frohlich current; in real materials, however, due to pinning by defects and surface imperfections, a finite electric field exceeding the threshold field, Et, is required to depin the CDW and initiate sliding. The CDW sliding results in the collective, electron-lattice condensate contribution to the total current, evident from the appearance of an AC component under DC bias, Shapiro-like steps in the current with applied RF signal, and an overall nonlinear increase in current. While achieving the dissipation-less Frohlich current of the sliding electron–lattice condensate appears to be impossible in real materials, one can imagine an important related target, namely reaching the electron transport regime where electronic noise is inhibited due to the collective, strongly-correlated nature of the electron-lattice condensate current.

We address here the fundamental questions in the physics of electron transport: is the collective current of the sliding, strongly-correlated CDW condensate less noisy than that of the normal current? Can we inhibit the low-frequency electronic noise in CDW conductors below the noise limit of normal electrons in conventional conductors?

The low-frequency noise in electronic materials, also known as excess or flicker noise, has the current spectral density, S, scaling as 1=f, and, in some cases, it reveals Lorentzian bulges superimposed on the 1=f background (f is the frequency). In the CDW context, it has been termed as “broadband noise”. The importance of this type of noise is well-known. It up-converts via the device’s non-linearities and appears in high-frequency signals as phase noise, thus degrading the communication systems. It sets the fundamental limit to the sensitivity and selectivity of any sensor because its power increases with the measurement time, owing to the 1=f spectral density. Continuous downscaling of electronic technologies leads to an increasing relative level of 1=f noise with decreasing device area and thus numerous problems with noise control. Reducing this noise type is crucial in quantum computing technologies because it degrades qubit coherence times and limits quantum sensors.

We hypothesized that the current carried by a sliding CDW strongly correlated electron-lattice condensate is inherently less noisy than that of thermalized single electrons. Accordingly, when transport is dominated by the condensate, the noise is suppressed relative to the noise level of the current of drifting, uncorrelated, thermally excited electrons. To test this, we focused on CDW systems in the incommensurate phase (IC-CDW), where the density of normal carriers is low and the current is dominated by the collective CDW contribution. Noise suppression is expected to be most pronounced at bias voltages above the depinning threshold, since at the depinning point, CDW creep and nonuniform sliding generate excess fluctuations. Upon considering various quasi-1D CDW van der Waals materials, we selected two candidates: a Weyl CDW semimetal (TaSe4)2I and the monoclinic CDW polymorph, NbS3-II. The first material, (TaSe4)2I, undergoes a Peierls-type transition to the IC-CDW phase, accompanied by an opening of a relatively large energy gap, at a temperature TP in the range of 245 K – 260 K. The second material, NbS3-II, reveals three CDW phases at Peierls temperatures TP0 = 460 K, TP1 = 330 K to 370 K, and TP2 = 150 K. The fact that NbS3-II exhibits a CDW electron-lattice condensate phase above room temperature (RT) makes this material promising for practical applications. Using these two quasi-1D materials, we fabricated nanowire-geometry test structures to promote coherent CDW sliding.

In this work, we demonstrate that the low-frequency noise in CDW nanowire conductors can drop below the noise limit of normal electrons. In both (TaSe4)2I and NbS3-II, the total normalized noise is lower than that of individual electrons. Our experiments with (TaSe4)2I found no residual minimum noise level at higher biases, suggesting minimal or no intrinsic noise from the sliding electron-lattice condensate. Therefore, the intrinsic noise of the sliding condensate is negligible compared to the noise of the normal electrons or the fluctuations in the threshold field. The noise due to fluctuations at the depinning threshold is extrinsic and caused by lattice imperfections that locally pin the condensate. Once the bias voltage is well past threshold and the sliding mode is established, the total normalized noise drops below the noise of normal electrons. In the case of NbS3-II, the noise of the CDW condensate appears to emerge at higher biases, saturating the overall noise at a level lower than that of the normal electron noise. The total normalized noise spectral density in CDW conductors decreases with current, a striking difference from conventional metals or semiconductors, where the normalized noise level remains constant with the current. The noise reduction at the higher current densities gives an advantage to CDW conductors in signal-to-noise ratios. These insights into the fundamental physics of CDW nanowires could be transformative for quantum technologies and suggest a way for future noiseless electronics.

Results
First, we confirmed that our (TaSe4)2I device structures are of high quality and support a fully-gapped CDW transport regime. Figure 1a shows scanning electron microscopy (SEM) images and energy-dispersive spectroscopy (EDS) maps of the synthesized material, revealing the ribbon-like structure of the quasi-1D van der Waals crystal and uniform elemental distribution. Figure 1b is an optical microscopy image of one of the four-contact nanowire structures with a channel thickness of ~83 nm. Figure 1c shows the channel resistance normalized by the RT resistance as a function of inverse temperature, 1000/T, measured in the linear regime at a small bias. The peak in the resistance slope is associated with the material’s transition from the normal metal to IC-CDW phase at TP ≈ 246 K. Below TP, an opening of the Peierls bandgap in the Fermi surface is accompanied by a change in the resistance, described as σ=σ0=exp(−Δ=KBT), where σ is the conductivity, σ0 is the conductivity at RT, KB is the Boltzmann constant, and Δ is the activation energy. For the device in Fig. 1b-f, we extracted the bandgap, Eg = 2Δ ≈ 300 meV, in agreement with the literature. To further confirm the IC-CDW phase, Fig. 1d shows the differential resistance as a function of bias voltage for temperatures between 150 K and 220 K. This dependence was obtained by numerical differentiation of measured I–Vs. The nearly constant differential resistance at low bias corresponds to the Ohmic resistance of the normal charge carriers. At the threshold bias voltage, Vt, the resistance decreases due to the onset of CDW sliding. The Vt value shows a non-monotonic temperature dependence consistent with trends reported for other quasi-1D CDW systems. Slight deviations from the perfectly flat derivative characteristics at low bias and kinks are attributed to CDW creep before the depinning and not completely coherent CDW depinning and sliding.

The I–Vs and differential conductance, G, for the (TaSe4)2I nanowire are presented in Fig. 1e, f, respectively. These characteristics are measured for the second time during the noise measurements. They are further used in the noise data analysis. The differential conductance includes the normal and collective current components, G=Gn+Gc. The onset of CDW sliding is seen at Vt ≈ 0.08 V, when the total current becomes super-linear, I/Vα (α=1.5–2.0). The nonlinear increase in total current is due to the contribution of the CDW condensate. The collective component of the current, Ic, and conductance, Gc, are shown by the blue circles in panels (e) and (f), respectively. The data in Fig. 1a–f prove that we have high-quality CDW nanowire conductors with the current dominated by sliding electron-lattice condensate at the bias of ~1 V.

To probe noise behavior in (TaSe4)2I devices in the CDW regime, we performed spectral measurements below the Peierls transition temperature TP, ensuring the system remained in the IC-CDW phase. Figure 2a–d show the normalized current spectral density, S=I2, as a function of frequency, f, for four different bias regions: (I) low-bias–purely normal carrier transport; (II) near the CDW depinning threshold; (III) at the onset of CDW sliding; and (IV) high bias, where the sliding condensate dominates. Across all regions, the noise exhibits 1=f spectral behavior. The Lorentzian bulges appearing near the CDW depinning threshold in regions II and III are a signature of phase transitions or depinning. Figure 2e shows the noise, S=I2, at a fixed frequency f= 10 Hz, as a function of bias voltage, across all four regions. The noise peaks near the depinning voltage, Vt, where the CDW starts to slide. As the bias voltage increases above Vt, the noise decreases, approaching the level of normal electrons. However, at the bias voltage of ~0.7 V, where the current is dominated by the sliding CDW condensate, we observe a striking feature: instead of saturating at the noise level of electrons in a passive conductor (region I), the total noise level decreases below this limit (yellow shade part of region IV). The noise in this fully-gapped quasi-1D CDW nanowire is, therefore, inhibited below the normal metal limit when the current is dominated by the sliding electron-lattice condensate. This behavior was consistently observed across multiple devices.

Figure 3a–f illustrates the evolution of the noise spectra in the bias region II, near the CDW depinning threshold, indicating the association of the Lorentzian features with the depinning. The Lorentzian noise component diminishes as the bias voltage surpasses the depinning threshold and enters the CDW sliding regime. The observed Lorentzian bulges originate from threshold fluctuations; they are associated with the system fluctuations between the pinned and depinned states. In mathematical description, the transitions between two phases in the material are similar to those in the two-level systems, which result in the Lorentzian features in the noise spectra.

The noise reduction in the (TaSe4)2I nanowires is intriguing and counterintuitive, thus requiring theoretical explanation. In a simple approach, the fluctuations of the conductance of the normal and collective currents can be described by the noise spectral density, S=I2:

S=I2 = S Gn (Gn/(Gn+Gc))2 + S Gc (Gc/(Gn+Gc))2

Here, the conventional assumption is that the fluctuations in the two conductivities are independent and the cross-correlation can be neglected. At low bias, V<Vt, the contribution of the CDW to the current is zero, Gc=0, and one can determine the normal conductance noise, SGn=Gn2, from the experimental data. The relative contributions of the normal, (Gn/(Gn+Gc))2, and the collective, (Gc/(Gn+Gc))2, components of electrical conductivity are also taken from the experiment (Fig. 4a). The nonlinear CDW conductance scales as Gc/Iβ, where β ≈ 0.5 (Fig. 4b), and no saturation is observed. Figure 4c presents the weighted noise contributions of the normal and CDW currents to the overall noise. At low bias, all noise is due to the normal current of individual electrons, SGn=Gn2, independent of voltage (horizontal line, gray symbols). At high current, all noise is due to the CDW current, S=I2 ≈ SGc(Gc/(Gn+Gc))2 ≈ SGc=Gc2 (red). The noise from fluctuations of the normal conductance remain the same, but its contribution to the total noise, S=Gn2(Gn/(Gn+Gc))2 (blue) decreases because it is shunted by increasing CDW conductance. The shunting of the noise of normal electrons occurs because of the dominance of the CDW conductance (Fig. 4a) and the intrinsically lower noise of sliding CDWs. A simple Eq. (1) can be extended to include the fluctuations in the threshold voltage. We have accomplished this task using a conventional linear model approximation for the current. The noise model in the linear approximation confirms the principal possibility of reducing the total normalized noise below the normal electron limit. However, this model does not describe all noise features accurately since the CDW current is strongly nonlinear. The accurate noise description requires a comprehensive phenomenological theory.

The noise reduction below the normal resistor limit is not the only surprise feature observed in experimental data. Figure 4d shows the total normalized noise spectral density, S=I2, as a function of current, I, in the CDW sliding regime. The noise reduces with increasing current as S=I2/1=I. Such inverse scaling with current contrasts drastically with the noise dependence on current for linear passive resistors, where the noise level does not depend on current: S=I2/constant (or S/I2). The fact that we observed a different dependence on the normalized noise level allows for the introduction of the phenomenological Hooge formula, S=I2(αH/fN), where αH is the Hooge parameter, and N is the number of charge carriers in the resistor. There are known cases of deviation from S=I2/constant dependence in specific electronic devices, like diodes. In this work, we report the observation of a S=I2/1=I scaling of noise in a generic resistor with two metal contacts. This means that increasing the current density will simultaneously decrease the noise level, which can become a game-changing advantage for interconnect applications.

To further understand the unusual noise reduction in fully-gapped CDW nanowires, we developed a comprehensive phenomenological noise theory in nonlinear approximation. It has been accomplished by extending the “two-fluid” approach to include explicitly the nonlinear voltage dependence of the collective current in (TaSe4)2I. The total current, I=In+Ic, consists of a normal current In=GnV, and the sliding CDW current modeled as:

Ic = Γc1(V2-Vt2)a + Γc2(V-Vt)2a θ(V-Vt)

Here, θ(V-Vt) is the unit step function that ensures the CDW contribution begins only after depinning. Equation (2) provides a phenomenological description of the CDW contribution to the current in our nanowire devices. The two power-law terms originate from the standard CDW current expressions quoted by Grüner, but the exponents a and 2a are employed as effective exponents that capture the voltage dependence of the sliding condensate over the full experimental voltage range. We further constrain the ratio of the exponents of the two nonlinear terms to share a single generalized conductance Γc with fixed dimensionality, which keeps the parametrization minimal and avoids introducing additional prefactors. This yields an accurate minimal model that reproduces the full nonlinear I–V characteristics and forms the basis for the quantitative noise analysis below.

Assuming uncorrelated, zero-mean fluctuations in the physical parameters, Gn, Γc, and Vt, the total first-order fluctuation in current can be expressed as:

δI = (∂I/∂Gn)δGn + (∂I/∂Γc)δΓc + (∂I/∂Vt)δVt

Dividing by the total current, squaring, and taking the ensemble average, we obtain the normalized noise power expressed as:

(δI2/I2) = (δGn2/Gn2)(In2/I2) + (δΓc2/Γc2)(Ic2/I2) + (δVt2/Vt2) [ ... ] θ(V-Vt)

The first term is the noise of the normal current, the second term is the noise due to fluctuations of the generalized CDW conductance, Γc, and the third term is the noise due to fluctuations in the threshold voltage, Vt. At large voltages (V≫Vt), the terms proportional to hδVt2i/Vt2 are suppressed. At V=Vt+, the normalized noise due to fluctuations in Vt is singular for a<1, but in the limit of large V≫Vt, it falls off rapidly.

Let us look closer at Fig. 4f, which shows the noise calculated from Eq. (4) superimposed on the experimental data. The fitting to the experimental I–Vs (see Fig. 4e), allows one to extract the parameters, which are then used as inputs to the noise model. The first observation is that our theory accurately describes the noise peak due to fluctuations in Vt and how it rolls off as 1=I. The bias dependence of the weighted normal electron noise and the CDW condensate noise are described as [(hδGn2i/Gn2)(In2/I2)] and [(hδΓc2i/Γc2)(Ic2/I2)], respectively. The noise of the normal electrons dominates below the threshold voltage, Vt. Both noise components are constant at lower and higher bias regimes, when I ≈ In and I ≈ Ic, respectively. In the high-bias regime where CDW noise emerges, it can saturate the total noise if its amplitudes grow large enough. The bias dependence of the threshold field fluctuations, hδVt2i/Vt2, becomes pronounced near Vt. The total contribution from the threshold fluctuations to the noise contains the three distinct components within the square brackets of Eq. (4), each with a characteristic bias dependence. The first one, S=I2 ≈ (hδVt2i/Vt2) dominates at V ≈ Vt, resulting in the noise peak. The second one, S=I2 ≈ (hδVt2i/Vt2) dominates at V≫Vt, resulting in noise scaling as 1=I. The third one drops approximately as 1=I1.5 in the intermediate regime. Together, these components capture accurately the evolution of the noise across the depinning transition and into the CDW sliding regime.

Significantly, our general phenomenological theory and Eq. (4) resolve the half-century-old argument about the role of fluctuations in the threshold field on the overall “broadband noise” level of CDW materials. The “threshold noise” is the dominant mechanism for typical voltages in the CDW sliding regime, studied previously for different materials. It is extrinsic, defect-related, rather than intrinsic to CDWs. The same lattice imperfections and defects that pin the CDW also give rise to stochastic variations in the depinning threshold, establishing a direct connection between material disorder and low-frequency noise. However, in our (TaSe4)2I nanowires, the total noise reduces rapidly beyond Vt, not only returning to the noise limit before the CDW depinning but falling below it. This raises the most important question: how noisy is the sliding CDW condensate itself compared to the current of normal electrons?

Our theory, via Eq. (4), establishes the relevant noise mechanisms and the noise scaling with the bias voltage. The noise amplitudes, hδGn2i/Gn2, hδΓc2i/Γc2, and hδVt2i/Vt2 cannot be calculated from the first principles, but rather determined by comparing the experimental data with the model. Up to the maximum measured bias we only observe the noise due to fluctuations in Vt, and we do not observe any indication of a lower limit set by fluctuations in Γc. The normal and threshold noise amplitudes hδGn2i/Gn2 and hδVt2i/Vt2 can be determined from the fitting to experimental data as 4.9×10−9 and 3.5×10−7, respectively. The CDW noise amplitude, hδΓc2i/Γc2, which can saturate the total noise level at high currents, is never revealed in our experiments, as seen clearly from the total noise, which continues falling below the normal electron limit (Fig. 2e, f). The upper bound on hδΓc2i/Γc2 has to be taken as orders-of-magnitude smaller than that of hδGn2i/Gn2 and hδVt2i/Vt2 to obtain an accurate fit to the experimental data. In the low-bias linear regime, the noise is constant; it originates from the normal carrier fluctuations described by [(hδGn2i/Gn2)(In2/I2)]. The prominent noise peak at the threshold voltage, Vt, arises from the S=I2 component, while the gradual noise decay at higher biases follows a 1=I dependence and is governed by the S=I2. The CDW noise, [(hδΓc2i/Γc2)(Ic2/I2)], is found to be insignificant across the entire measured voltage range and does not contribute to the observed noise. The total modeled noise, with all the noise components superimposed on the experimental data, is shown in the Supplemental Materials. To summarize, we found that the intrinsic noise of the sliding condensate in our (TaSe4)2I nanowires is negligible compared to the noise of normal electrons and the extrinsic noise of the threshold-field fluctuations. This is consistent with the picture of the ideal sliding Fröhlich condensate as a dissipation-less, noise-free process, with all observed noise originating from defects and disorder.

Given the fundamental nature of the noise reduction effect, we verified its reproducibility across multiple test structures. In Fig. 5a–d, we present the transport and noise characteristics of a second representative (TaSe4)2I nanowire device. In all cases, we observed a consistent nonlinear conduction scaling as, Gc/Iβ (Fig. 5b) and noise reduction in the high-bias sliding regime following the S=I2/1=I law (Fig. 5c), which indicates that the noise reduction below the normal electron limit with no observed noise saturation is a robust and reproducible noise feature intrinsic to this CDW material.

In the data analysis, it is also important to exclude local heating effects. No Joule self-heating effects are expected in our test structures at the small bias voltages, about ~1 V, and low current range, about ~1 μA. One can readily estimate the temperature rise, ΔT, from an analytical formula derived from Fourier’s law: ΔT=P×RT=I×V×(Tox/Kox×L×W). Here, P is the dissipated power in the nanowire, RT is the thermal resistance of the SiO2 layer, Tox is the thickness of the SiO2 layer, Kox is the thermal conductivity of SiO2, L is the length of the channel, and W is the width of the channel. The formula assumes that the bulk Si substrate is the thermal sink, and the SiO2 layer with its low thermal conductivity is the thermal barrier. Using the typical values, i.e., I=1 μA, V=1 V, Tox=300 nm, Kox=1 W/mK, L=5 μm, and W=300 nm, we obtain for the thermal resistance RT ~0.2 MK/W and, correspondingly, for the temperature rise ΔT~0.2 K. We can elaborate the estimate and consider the thermal boundary resistance, RC, between the nanowire and SiO2 layer. It is well known that it is primarily defined by the interface quality rather than specific channel material. Assuming the worst-case scenario of low interface conductance, Gc=15 MW/m2K, we obtained an extra thermal resistance of ~0.2 MK/W, which doubles the total resistance and temperature rise. On the other side, conduction to metal contacts along the channel may reduce the temperature rise. These simple but reliable estimates show that in our devices, in the considered bias and current ranges, the local heating is negligible. One has to pass currents in the mA range, a factor of ×1000 larger, to induce significant Joule heating. The temperature rise was also assessed using COMSOL software tools for a given device structure and bias voltages. The maximum temperature increase at the hotspot (center of the channel) is approximately 1 K at a voltage drop of 0.9 V. The simulation results confirmed negligible self-heating effects. For an independent experimental proof of the absence of local heating effects, we took Raman spectra in situ of the representative devices under the same bias as used in the noise experiments. No characteristic phonon peak shifts, expected for local heating, have been observed. We have previously detected local heating effects with Raman spectroscopy in a range of materials and different device structures, and are confident in the accuracy of such a method.

We now address another fundamental question: Is the ultra-low noise of sliding condensate unique to the IC-CDW phase of (TaSe4)2I, or is it a property of many quasi-1D CDW conductors? The early studies of the “broadband” noise in bulk CDW crystals focused on the threshold noise and never considered the possibility of noise reduction below the normal electron limit. A few recent studies for TaS3 and NbSe3 found that beyond Vt, the noise returns to the normal electron limit, in contrast to our observation of continuous noise reduction in the sliding regime of (TaSe4)2I nanowire devices. Other potential candidates for ultra-low noise transport should include CDW materials where the collective, strongly-correlated current of the sliding electron-lattice condensate is dominant. We experimented with nanowires of NbS3-II, which exists in the CDW phase at RT. The NbS3-II nanowire device fabrication and measurements followed the same protocols as with (TaSe4)2I nanowire devices. Figure 6a–f shows transport and noise characteristics for a representative NbS3-II nanowire device. The most important observation is that the normalized spectral density in the NbS3-II device reduces below the normal electron limit, similar to that in the (TaSe4)2I devices. The noise spectral density scaling with current is also consistent, i.e., noise reduces with increasing current. It reveals a similar, S=I2/1=I, dependence for the current above 10 μA. We note that the noise reduction in NbS3-II is achieved at temperatures above RT. The latter is important for practical applications, i.e., in interconnects. There is one difference in the noise behavior in the two materials: it appears that the noise spectral density in NbS3-II nanowires shows saturation behavior, even though the noise level is below the normal electron limit. This suggests that we may be reaching the intrinsic electron-lattice condensate noise in NbS3-II nanowires but not in (TaSe4)2I nanowire devices. The origin of the ultra-low-noise collective current in (TaSe4)2I nanowire, and the lack of noise saturation, may be related to the fact that the CDW transport in this fully-gapped material is described by the under-damped oscillator model. Compared to other CDW materials, (TaSe4)2I has the largest energy gap and heaviest CDW effective mass, m*=m ≈ 104, yielding the lowest CDW damping. The nonlinear CDW conductivity in this material continues to increase rather than saturate due to damping, in the examined bias voltages. The current in NbS3-II shows signs of saturation at higher bias voltages (see Fig. 6b), indicating the onset of stronger CDW current damping, which likely translates to higher intrinsic CDW noise and eventual flattening of the total noise level. Based on the recent realization that (TaSe4)2I is a Weyl semimetal, one may also argue that topological scattering suppression has relevance to the noiseless current of the sliding condensate in this material. Surveying any other possible material systems where normalized noise can be suppressed, we noted systems with weak localization effects.

It is interesting to compare the noise dependence on bias in (TaSe4)2I and NbS3-II nanowires with that in NbSe3 nanowires. The noise in (TaSe4)2I and NbS3-II nanowires decreases below the normal electron limit when the CDW sliding current component starts to dominate. The exact nature of multiple polymorphs of NbS3 and the origin of several CDW transitions in NbS3-II is still under investigation. However, both (TaSe4)2I and NbS3-II are known to have a large contribution of electron-lattice condensate to the total current after depinning. The electrical conduction is different in NbSe3 nanowires. This partially gapped CDW material always remains metallic, with the significant normal electron contribution to conduction, even at low temperatures. After CDW depinning, the current in NbSe3 nanowires is dominated by the normal electron contribution. Correspondingly, the noise level does not decrease below the normal electron limit. These considerations add to the importance of the present finding of noise reduction and unusual scaling with the current in nanowires made of CDW materials such as (TaSe4)2I and NbS3-II.

Finally, to benchmark the noise performance of CDW nanowires, we compared their noise level with that of advanced Si metal-oxide-semiconductor field-effect transistors (MOSFETs), calculated using the standard McWhorter model for a transistor channel with the same area as the nanowires. At the conductivity G=10−5 (S), midway through our measurement range, the noise in a (TaSe4)2I nanowire in the CDW sliding regime is S=I2 ≈ 1.7×10−8 (Hz−1) at a reference f=10 Hz. This value is lower than the noise in MOSFETs, S=I2 ≈ 3.2×10−8 (Hz−1), with the realistic trap density of Nt = 1020 cm−3 eV−1. Given that our nanowires and devices, fabricated at university facilities, are less perfect compared to commercial MOSFETs, this comparison underscores the promise of CDW-based devices for ultra-low-noise applications.

Methods
Chemical vapor transport synthesis and growth of (TaSe4)2I
On the benchtop, 0.6468 g (3.575 mmol) of Ta powder (Strem, 99.98%) and 1.2511 g (15.85 mmol) of Se powder (Strem, 99.99%) were gently mixed. I2 crystals (0.3270 g, 2.577 mmol, JT Baker, 99.9%) were placed at the bottom of a pre-cleaned and dried fused quartz ampule (~18×1 cm length, 10 mm inner diameter, 14 mm outer diameter, ~13 cm3 volume). The mixed Ta and Se powders were then transferred into the ampule, using a glass funnel and anti-static brush to ensure a clean and complete transfer. After loading, the ampule was capped with an adapter and valve, then submerged in an acetonitrile/dry ice bath and subjected to four evacuation/backfilling cycles with Ar(g) using a Schlenk line. The ampule was subsequently sealed under vacuum. The sealed ampule was placed in a horizontal tube furnace, where the temperature was ramped over 6 h to establish a thermal gradient of 600°C (source zone) to 550°C (growth zone). This gradient was maintained for 240 h. Then the ampule was cooled to RT over 12.25 h. Upon completion, 0.5929 g of lustrous, shard-like crystals were recovered from the growth zone and left to sit in a fume hood for 1 h (29.60% isolated yield). These crystals were stored in an Ar-filled glovebox.

Chemical vapor transport synthesis and growth of NbS3
On the benchtop, 1.4806 g (15.94 mmol) of Nb powder (Strem, 99.99%) and 1.6921 g (52.77 mmol) of S powder (J.T. Baker, ≥99.5%) were gently mixed. This mixture was placed at the bottom of a pre-cleaned and dried fused quartz ampule (~9 cm×2.2 cm length, 1.9 cm inner diameter, 2.2 cm outer diameter, volume ~33 cm3). A glass funnel and anti-static brush were used to ensure a clean and complete transfer. After loading, the ampule was capped with an adapter and valve, then submerged in an acetonitrile/dry ice bath and subjected to four evacuation/backfilling cycles with Ar(g) using a Schlenk line. The ampule was subsequently sealed under vacuum. The sealed ampule was placed in a horizontal tube furnace, where the temperature was ramped over 3.7 h to establish a thermal gradient of 700°C (source zone) to 670°C (growth zone). This gradient was maintained for 240 h. Then the ampule was cooled to RT over 110 h. Upon completion, 3.0247 g of dense, dark silver-gray, wirelike crystals were recovered from throughout the ampule (95.43% isolated yield). These crystals were stored in an Ar-filled glovebox.

Material characterization of (TaSe4)2I samples
(TaSe4)2I crystallizes in the tetragonal space group I422 with RT lattice parameters a=b=9.5310 Å, c=12.8240 Å, V=1164.932 Å. Covalently bonded Ta and Se atoms form chiral, helical chains, with I atoms weakly bonded to them along the c axis (Fig. S1a), allowing exfoliation to nanometer-scale thicknesses. SEM imaging was performed using an FEI Teneo FE-SEM at 10 keV with a spot size of 10. Energy-dispersive X-ray spectroscopy (EDS) was performed using an Aztec Oxford Instruments X-MAXN detector operated at 10 keV with a spot size of 10. For SEM and EDS analysis, the sample was prepared by mounting the as-grown crystals onto a stub using carbon tape and mechanically exfoliating the crystal surfaces with Scotch tape. EDS maps demonstrate homogeneity of the constituent elements (Fig. 1a). In addition, EDS spectra and measured atomic percentages are consistent with stoichiometric Ta2Se8I (Fig. S1c). Raman spectroscopy of (TaSe4)2I crystals was conducted at RT using a Thermo Fisher DXR Raman Microscope with ×50 magnification and λ=532 nm laser excitation at 2 mW. The characteristic peaks of (TaSe4)2I, being observed at 70, 101, 147, 160, 182, and 271 cm−1, are consistent with previous observations and are sharp, further indicating the high quality of the sample (Fig. S1d). Transmission electron microscopy (TEM) imaging was performed with a JEOL JEM-2100PLUS microscope at 200 kV. Sample preparation: 13.3 mg of (TaSe4)2I crystals were probed and bath sonicated in 20 mL ethanol, and the resulting dispersion was drop cast onto a lacey carbon copper-supported grid. High-resolution TEM data align well with the crystal structure of (TaSe4)2I (note overlay in Fig. S1e) and Fast-Fourier Transforms (FFTs) match theoretical patterns, verifying that the atomic structure of (TaSe4)2I is maintained upon exfoliation. Powder X-ray diffraction data were collected using a Bruker D2 Phaser diffractometer equipped with a LYNXEYE XE-T linear position-sensitive detector and Cu Kα (λ=1.5418 Å) radiation. Sample preparation: 20.7 mg of (TaSe4)2I crystals were finely ground with 11.5 mg of amorphous quartz glass using a mortar/pestle. Rietveld refinement (Fig. S1f) was performed using TOPAS; maintaining the tetragonal I422 space group of (TaSe4)2I, the a parameter was refined to 9.5551 Å and c to 12.7966 Å, for a final cell volume of 1168.331 Å. These results closely match the parameters of ICDD reference pattern 04-011-3118 (cell parameters listed above), with no evidence of additional phases observed. Goodness of fit was 2.79 due to preferred orientation effects still being present in the ground sample.

Material characterizations of NbS3-II samples
NbS3-II crystallizes in the monoclinic space group P2/m with RT lattice parameters of a=9.6509(8) Å, b=3.3459(2) Å, and c=19.850(1) Å. The structure of NbS3-II is composed of bilayers of sulfur-bridged [NbS6] chains, connected weakly through vdW interactions (Fig. S11a). SEM imaging was performed using an FEI Teneo FE-SEM at 10 keV with a spot size of 10, showing facile cleavage into nanowires (Fig. S11b). EDS was performed using an Aztec Oxford Instruments X-MAXN detector operated at 10 keV with a spot size of 10. For SEM and EDS analysis, the sample was prepared by mounting the as-grown crystals onto a stub using carbon tape and mechanically exfoliating the crystal surfaces with Scotch tape. EDS maps demonstrate the homogeneity of sulfur and niobium (Fig. S11c), and measured atomic percentages are consistent with a slightly S-deficient sample (Fig. S11d). TEM imaging of exfoliated NbS3 nanowires was conducted with a Thermo Fisher Titan 80–300 probe aberration-corrected scanning transmission electron microscope (STEM) at RT with 200 kV accelerating voltage. This sample was prepared by bath sonication of NbS3 crystals in ethanol, and the resulting dispersion was drop cast onto a lacey carbon copper-supported grid. High-resolution TEM data align well with the crystal structure of NbS3 (note overlay in Fig. S11e) and FFTs match theoretical patterns, verifying that the atomic structure of NbS3 is maintained upon exfoliation. Raman measurement was performed using a Renishaw inVia system in the backscattering geometry with a 633 nm excitation laser.

Fabrication of the CDW nanowire test structures
The (TaSe4)2I and NbS3-II nanowire test structures were prepared by exfoliating high-quality bulk needle-like crystals into thinner nanowire samples of high aspect ratio on top of Si/SiO2 substrates. The nanowire test structures were fabricated in the cleanroom environment with electron beam lithography and electron beam metal deposition. The thickness of the nanowires was on the order of ~50–100 nm, whereas the channel length, L, was in the range from 2 μm to 12 μm. The preparation of (TaSe4)2I nanowire multi-channel test structures began with cutting small substrates (~0.5 cm×0.5 cm) from a large SiO2/Si wafer (University Wafer, p-type, Si/SiO2, <100>) and cleaning them multiple times using acetone and isopropyl alcohol (IPA), followed by rinsing with deionized (DI) water. Next, the bulk (TaSe4)2I crystal was exfoliated into thin nanowires using the conventional mechanical exfoliation technique on clean substrates. Using an optical microscope, nanowires with varying thicknesses and cross-sectional areas were selected as channels for fabrication. The test structures were fabricated in a class-100 cleanroom. The process began by loading the substrates containing the exfoliated nanowires into a spin coater (Headway Research), spin-coating them twice with a PPMA A4 (Kayaku Advanced Materials, 495 PPMA) solution, and then baking them at 150°C for 5 min each. Next, the spin-coated samples were placed inside an electron-beam lithography (EBL) system (JEOL JSM 6610), where predesigned patterns for electrodes and contact pads were written at specific accelerating voltages, area doses, and beam currents. Afterward, the samples were developed in a solution (MIBK:IPA, 1:3) for 1 min and rinsed with DI water for 30 s. Immediately after, the samples were transferred to an etcher system (Oxford 80+), where they underwent an Ar-gas plasma-based cleaning process for 30 s to remove any chemical residue from the exposed patterns after the development solution. Next, the samples were placed into an electron-beam evaporation (EBE) system (CHA Mark 40) for metal deposition, where a 10 nm Ti adhesive layer and 90 nm Au metal layers were deposited at rates of 0.5 Å/sec and 1.0 Å/sec, respectively. Finally, the substrates with the test structures and deposited Ti/Au contacts were submerged in acetone for the lift-off process. That completed the process of fabricating (TaSe4)2I and NbS3-II nanowire test structures with varying nanowire channel lengths defined by the metal electrodes. The (TaSe4)2I device 1, described in most detail in the main text, had a channel width of 350 nm and a thickness of 83 nm.

Electronic transport and noise measurements in the CDW nanowires
The temperature-dependent transport measurements of the nanowire test structures were conducted inside a cryogenic probe station (Lakeshore TTPX) under high vacuum (~1×10−5 torr). A semiconductor analyzer (Agilent B1500A) was used for the two-terminal and four-terminal I–V measurements. In the case of the four-terminal measurements of the middle channel, a constant current was applied to the outer two contacts while measuring the voltage drop between the inner two contacts. The contact resistance of the channel was calculated using the difference between the resistances from the two-terminal and four-terminal configurations. For temperature-dependent resistance measurements, the channel current at different temperatures was measured at a small bias. The noise measurements were performed using a voltage divider circuit with the DUT and a load resistor connected in series with a low-noise DC biasing battery. A potentiometer controlled the voltage drop across the circuit. During the noise measurements, the voltage across the circuit, VT, and the grounded DUT, VD, were measured using digital multimeters (DMM). The current across the DUT, ID, was calculated from the voltage, VL=VT−VD, on the load resistor with known resistance, RL. The resistance of the DUT was determined using the voltage reading of the DUT, VD, from the DMM and the calculated current across the load resistor, ID=VL/RL. The voltage fluctuations across the DUT were transferred to a dynamic signal analyzer (Photon+), which converted the time-domain signal to the corresponding voltage spectral density, SV, using the FFT. The noise spectral density was converted into its equivalent short-circuit current spectral density, S=I2.

The theory of 1=f noise in CDW systems in the linear approximation
To gain insight into the physics of current fluctuations in CDW conductors, we developed a theory of 1=f noise in linear approximation. It includes contributions of three main components of noise and shows the principal possibility of the total noise reduction below the limit of normal electrons. Let us consider the expression for the CDW current based on J=ρv where ρ=qn and v=μc(E−Et), where n is the charge density of the CDW, μc is its mobility, E is the electric field, and Et is the threshold field. We note that E=VDS/L and Et=Vt/L where L is the channel length. Therefore, for CDW conductors, we obtain:

I = (Wd/L) qn μc (VDS−Vt) θ(VDS−Vt)

Here W is the channel width and d is the channel depth. Note that this is the modified Ohm’s law for a conductor with a threshold voltage, σc = qn μc. Starting with Eq. (M1), and adding the normal current, we have:

I = GnV + Gc(V−Vt)θ(V−Vt)

where Gn is the normal current conductivity and Gc = (Wd/L) qn μc is the CDW conductivity. Here, θ(V−Vt) is the unit step function that ensures the CDW contribution begins only after depinning. We can write the current fluctuation as:

δI = δGnV + δGc(V−Vt) − GcδVt θ(V−Vt)

Dividing by the total current, we obtain:

δI/I = (δGn/Gn) (GnV/(GnV+Gc(V−Vt))) + (δGc/Gc) (Gc(V−Vt)/(GnV+Gc(V−Vt))) − (δVt/Vt) (GcVt/(GnV+Gc(V−Vt)))

As the next step, we square, take the ensemble average, and assume no cross correlation to get to the final expression for noise in the linear model:

(δI2/I2) = (δGn2/Gn2) (GnV/(GnV+Gc(V−Vt)))2 + (δGc2/Gc2) (Gc(V−Vt)/(GnV+Gc(V−Vt)))2 + (δVt2/Vt2) (GcVt/(GnV+Gc(V−Vt)))2 θ(V−Vt)

(δI2/I2) = (δGn2/Gn2) (In2/I2) + (δGc2/Gc2) (Ic2/I2) + (δVt2/Vt2) (Ic2/I2) θ(V−Vt)

At V=Vt, the noise jumps up by (δVt2/Vt2) (Gc2/Gn2).

At V=Vt+, the noise jumps up by (δVt2/Vt2) (Gc2/Gn2).

In the limit of large V≫Vt,

(δI2/I2) = (δGn2/Gn2) (Gn2/(Gn+Gc)2) + (δGc2/Gc2) (Gc2/(Gn+Gc)2) + (δVt2/Vt2) (Gc2/(Gn+Gc)2) (Vt2/V2)

So that we reach a limit set by the normal current noise in parallel with a noiseless channel, i.e., for V→∞,

(δI2/I2) = (δGn2/Gn2) (Gn2/(Gn+Gc)2)

Compared to Eq. (1) of the main text, the derived phenomenological theory of 1=f noise of CDW conductors has a third term, related to the fluctuations in the threshold voltage. This is an extrinsic noise component since the threshold is related to the defects pinning the CDW condensate. The linear theory does not give an accurate description of the observed noise, since the conductance is nonlinear, but provides an insight into the physics of current fluctuations.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Ghosh et al., presents a detailed investigation of noise behavior in a nanoscale charge density wave system. The main claim of the manuscript is that, in the sliding mode of CDW, the normalized noise drops below that of normal electrons. The study and the results are of interest to researchers working in CDW materials in general and to researchers exploring low-noise electronic systems.\nOverall, the manuscript presents high quality data, a systematic study and the results are interesting. However, I have a couple of concerns and suggestions for improvements:\n(1) Well above V_t, say V~1.0V, it is possible that Joule heating plays an important role given the dimensions of the nanowires and the current generated. It will be good to estimate the effect of this and include a discussion. This could be relevant since the contribution from normal electrons (Figure 1 e, for example) is assumed to be unchanged at all voltages.\n(2) Figure 2 needs improvement: a Lorentzian bulge is shown in 2(b) in the depinning region whereas 2(a) shows only a smaller frequency range below where the bulge was observed in (b). It would be important to show data from comparable ranges in (a) and (b).\n(3) It is not clear if the Lorentzian bulge persists at other voltages than shown in 2(b). Additional data at various voltages in 2(b) can be helpful as well as a discussion from the authors on the evolution of the bulge with voltage.\n(4) Four different bias regions (Figure 2) were mentioned with handwaving assignments based on the behavior. It will be helpful to mark the regions in the figures and to state the voltage ranges to avoid confusion.\n(5) It will be helpful to state clearly how the normal and collective components of I and G are calculated, the explanation surrounding Figure 3 is confusing and could be improved.\n(6) A major concern is about the author's earlier work on noise behavior in another CDW system (reference 36). The observation of the noise magnitude trending significantly lower in the sliding CDW regime has been reported in 36. It will be important for the authors to clearly provide the additional understanding and/or newer result in the present manuscript.", "reviewer_2": "As known for decades, for some quasi -1D materials with a charge density wave (CDW) ground sate, above a threshold value, the pinned CDW can move and contribute to a collective current - static and modulated at a given frequency associated to 1/f low frequency noise (unfortunately called narrow and broad band noise).\nThe manuscript reports studies of non-linear transport properties, specifically noise, in (TaSe4)2I nanostructures, the Peierls transition of which occurs at TP = 245K. Data are presented for two different devices at a single temperature below TP. The authors show that far away from the threshold (10 times its value at the given temperature), the noise is suppressed below the value of normal charge carriers, and that the noise of the CDW current is negligible. Less elaborated a similar conclusion was suggested (see Ref.36) in noise measurements in NbSe3 which at the difference of (TaSe4)2I remains metallic below the Peierls transitions.\nThese conclusions are questionable as discussed in the following:\n-in Fig.1d the variation of the differential resistance in a semi-log plot is shown versus the applied voltage at different temperatures: clearly the resistance at fixed T is not constant indicating a heating effect. At T=120K the drop of resistance at the threshold (see the arrow) is even not visible. No estimation of thermodynamics appears in the paper.\n- While data in Fig.1d concern the small values of V (V  0,2V) those in Fig.1e are extended to 1V. How are the dV/dI caracteristics till 1V in order to see possible jumps or inhomogeneities masked in the log-log I-V curves in fig.1e ?.\n-Data in Fig1e show that the CDW current (evaluated from the extended (dashed line) Ohmic law (without taking care of any thermal effect) follows a power law without saturation till more than 10 times the threshold value. How to reconciliate this result with theories which evaluate the high field limit of the CDW conductance?.\n-Eq. 2 is the combination of Eq.(5.6) and 5.14) from Ref.1\nThe first term results from the “single particle” model – classical particle moving in a sinusoidal washboard potential -in the case where the voltage is regulated yields j a (E2-E2t)1/2 , (however in the part but” Electronic transport and noise measurement in the (TaSe4)2I nanowires” it seems that measurements were performed with a regulated current. In that case the threshold is manifested by an infinite negative differential resistance that is not the case in Fig.1d).\nThe second term provides from a different model due to dynamical critical phenomenon with the CDW velocity or the CDW current is a power law (E-Et) with an exponent 3/2 to be compared with ½ in the single particle model (however see   and 2 in eq.2). No real justification is made for the addition of two different threshold current with non-linear voltages with different exponents, even a third one - see figure caption of Extended data Fig.6).\nFrom their Eq.2 the authors derive the noise spectral density formed of 3 terms: noise of the normal current, noise of the fluctuations of the threshold voltage (near Vt) and fluctuations of the CDW conductance. They found that the noise spectra can be accurently fitted with only the noise amplitude of the normal current and that of threshold fluctuations but consequently without noise for the CDW conductance. However the values of different parameters such these three values of Vt are not indicated.\nI have taken consideration of this work but I consider that the conclusion of a CDW condensate moving without noise is not justified based on the presented data. There are two many uncertainties either in the experimental measurements and in the theoretical and fitting process which raise some, in my sense, raisonable doubts on the conclusions.\nThus I will not recommend publication of this work in Nature Communications", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The manuscript NCOMMS-25-46228-T by Subhajit Ghosh et al. describes a study of low-frequency electronic noise suppression in quasi-one-dimensional charge-density-wave (CDW) systems, specifically in (TaSe ) I nanowires, under conditions of sliding Frohlich condensate current.\nThe reported suppression of low-frequency 1/f noise in the sliding CDW regime is highly unusual and underlines a significant difference from conventional transport behavior in resistive materials. In this respect, the linear decrease of normalized noise spectral density, as opposed to the typical constant value in normal conductors, suggests a fundamentally different noise mechanism linked to collective transport. This is an indicator of new physics in the correlated transport regime. Moreover, the connection to real-world technological relevance, emphasizing technological relevance to quantum devices, low-noise communication, and sensing, strengthens the case for broad impact.\nIn the context of existing literature, the submitted work appears scientifically novel in its experimental observation of sub-thermal electronic noise in the sliding CDW regime, a behavior not established in previous CDW literature where broadband 1/f noise has been usually interpreted as a consequence of CDW dynamics near depinning. Moreover, recent studies, exploring CDW systems in reduced dimensions (nanowires, thin films), are concerned with transport phenomena but not noise suppression, as the case of the article here considered.\nAlthough the work appears sufficiently novel and innovative for publication in Nature Communications, two important points deserve more clarification and should be strongly taken into account before a resubmission.\n(I) The first point concerns the fact that the notion of using strong correlations to suppress noise, especially leveraging a collective mode like Frohlich conduction, is still underexplored and seems to be largely speculative. The authors need to better clarify this aspect, demonstrating that the theory developed is based on solid principles and the results obtained are not just material-specific observations. In this respect, further experimental investigations showing an analogous noise behavior as the one here reported, performed on different systems undergoing a similar CDW regime, could be very useful to address such a criticism.\n(II) The second point, which is someway connected to the first one, regards an improvement of the experimental robustness and generality of the work, in order to elevate its impact from a specialized CDW study to a wider platform as, for example, for quantum/nanoelectronics.\nAs minor comment, due to the high visibility and high impact of Nature Communications journal, my suggestion to the authors is to mention other systems and materials where an evident reduction of the 1/f noise level has been observed. This is the case, for example, of low-dimensional systems and topological insulators. A detailed description of key similarities and key differences in noise suppression mechanisms could also be given.\nFor all the reasons specified above, in my opinion, the paper is not suitable for publication in Nature Communications in its present form. A major revision, by addressing all the points indicated, is needed before the manuscript can be reconsidered."}, "author_response": "RESPONSE TO THE REVIEWERS’ COMMENTS\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\nThe manuscript by Ghosh et al., presents a detailed investigation of noise behavior in a\nnanoscale charge density wave system. The main claim of the manuscript is that, in the\nsliding mode of CDW, the normalized noise drops below that of normal electrons. The\nstudy and the results are of interest to researchers working in CDW materials in general\nand to researchers exploring low-noise electronic systems.\nOverall, the manuscript presents high quality data, a systematic study and the results\nare interesting.\nRESPONSE: We thank the Reviewer for stating that our manuscript “presents high-\nquality data, a systematic study, and the results are interesting.”\nHowever, I have a couple of concerns and suggestions for improvements:\n(1) Well above V_t, say V~1.0V, it is possible that Joule heating plays an important role\ngiven the dimensions of the nanowires and the current generated. It will be good to\nestimate the effect of this and include a discussion. This could be relevant since the\ncontribution from normal electrons (Figure 1 e, for example) is assumed to be\nunchanged at all voltages.\nRESPONSE: The reviewer correctly notes that any significant Joule heating during\ntransport measurements would affect the results. However, we can state with high\nconfidence that Joule heating does not play any role in our measurements. We\n1 | P age\nused bias voltages below or around 1 V with the current levels always remaining in the\nmicro-Ampere range, resulting in minimal power dissipation.\nIn the revised manuscript, we included analytical estimates of the thermal resistance of\nthe structure and the resulting temperature rise, which was below 1 K at 0.7 V bias,\nwhen the noise drops below the normal electron level. We also conducted COMSOL\nmodeling to verify this estimate. We included noise data for a new material system,\nNbS at room temperature. Additionally, in situ bias-dependent Raman measurements\n3,\nwere conducted on both (TaSe ) I and NbS nanowire devices, where we monitored\n4 2 3\nphonon peak positions in situ. The local heating would reveal itself via softening of\nphonon peaks. None was observed at the highest bias voltages, presenting a direct\nproof of the absence of Joule heating. We discussed the temperature rise estimation\nand the in-situ Raman behavior in the revised text and included the relevant figures in\nthe Supplemental file. In the revised manuscript, we added the following section.\n“In the data analysis, it is also important to exclude local heating effects. No Joule self-heating\neffects are expected in our test structures at the small bias voltages, about ~1 V, and low current\nrange, about ~ 1 A. One can readily estimate the temperature rise, Δ𝑇, from an analytical formula\nderived from Fourier’s law: Δ𝑇 = 𝑃×𝑅 = 𝐼 ×𝑉 ×(𝑇 /𝐾×𝐿×𝑊). Here, P is the dissipated\n𝑇 𝑜𝑥\npower in the nanowire, R is the thermal resistance of the SiO layer, T is the thickness of the\nT 2 ox\nSiO layer, K is the thermal conductivity of SiO , L is the length of the channel, and W is the\n2 ox 2\nwidth of the channel. The formula assumes that the bulk Si substrate is the thermal sink, and the\nSiO layer with its low thermal conductivity is the thermal barrier. Using the typical values, i.e.,\n2\n𝐼 =1 A, 𝑉 =1 V, 𝑇 = 300 nm, 𝐾 = 1 W/mK, L= 5 m, and 𝑊=300 nm, we obtain for the\n𝑜𝑥 𝑜𝑥\nthermal resistance R ~0.2 MK/W and, correspondingly, for the temperature rise Δ𝑇~0.2 K. We\nT\ncan elaborate the estimate and consider the thermal boundary resistance, 𝑅 , between the nanowire\n𝐶\nand SiO layer. It is well known that it is primarily defined by the interface quality rather than\n2\nspecific channel material. Assuming the worst-case scenario of low interface conductance, 𝐺 =15\n𝑇\nMW/m2K44,45, we obtained an extra thermal resistance of ~0.2 MK/W, which doubles the total\nresistance and temperature rise. On the other side, conduction to metal contacts along the channel\n2 | P age\nmay reduce the temperature rise. These simple but reliable estimates show that in our devices, in\nthe considered bias and current ranges, the local heating is negligible. One has to pass currents in\nthe mA range, a factor of ×1000 larger, to induce significant Joule heating. The temperature rise\nwas also assessed using COMSOL software tools for a given device structure and bias voltages.\nThe maximum temperature increase at the hot spot (center of the channel) is approximately 1 K at\na voltage drop of 0.9 V. The simulation results confirmed negligible self-heating effects (see\nSupplemental Materials).\nFor an independent experimental proof of the absence of local heating effects, we took Raman\nspectra in situ of the representative devices under the same bias as used in the noise experiments.\nNo characteristic phonon peak shifts, expected for local heating, have been observed (see\nexperimental data in the Supplemental Information). We have previously detected local heating\neffects with Raman spectroscopy in a range of materials and different device structures, and are\nconfident in the accuracy of such a method 46,47.”\nRegarding the second part of the question, we note that the constant noise level of the\nnormal electrons is not an assumption. It is an established experimental fact (in our\nmeasurements, and well before us), which comes from the linear response theory and\nrelates to the fact that current does not drive the fluctuations but only makes them\nvisible via Ohm’s law (see page 499 of P. Dutta and P. M. Horn: Low-frequency\nfluctuations: 1/f noise, Reviews of Modern Physics, Vol. 53, No. 3, July 1981).\nMathematically, it means that for normal current S/I2=S /V2=constant. This fact also\nI V\nemphasizes the importance of our results and their fundamental nature. We observed\nthat the scaling of 1/f noise of the sliding CDW condensate with the voltage is different.\nIn the revised manuscript, we added an extra explanation to clarify this point.\n“The fact that we observed a different dependence on the normalized noise spectral density in a\nCDW conducting channel is of great importance. It is well-known that the noise level, i.e.,\nnormalized noise spectral density, in passive resistors does not depend on current: 𝑆 ⁄𝐼2\n𝐼\nconstant (or 𝑆  𝐼2). This fundamental property originates in the linear response theory38 . In\n𝐼\nterms of physics, it means that the current does not drive the resistance fluctuations in the sample\n3 | P age\nbut rather makes them visible via Ohm’s law. The fact that 𝑆  𝐼2 allowed for the introduction\n𝐼\nof the phenomenological Hooge formula, 𝑆 /𝐼2~𝛼 /𝑓𝑁, where 𝛼 is the Hooge parameter, and\n𝐼 𝐻 𝐻\n𝑁 is the number of charge carriers in the resistor. There are known cases of deviation from\n𝑆 ⁄𝐼2 constant dependence in specific electronic devices, like diodes39 . In this work, we report\n𝐼\nthe first observation of as 𝑆 ⁄𝐼21/𝐼 scaling of noise in a generic resistor with two metal\n𝐼\ncontacts.”\n(2) Figure 2 needs improvement: a Lorentzian bulge is shown in 2(b) in the depinning\nregion whereas 2(a) shows only a smaller frequency range below where the bulge was\nobserved in (b). It would be important to show data from comparable ranges in (a) and\n(b).\nRESPONSE: At higher frequencies, the 1/f noise level reaches the background noise\nfloor of the thermal noise or shot noise (at low temperature in some devices). The noise\nspectrum can be closer to the noise floor when measured at very low biases.\nConventionally, the 1/f noise is shown for the relevant frequency range where it is above\nthe thermal floor at different bias ranges (this explains the name “excess noise” for 1/f\nnoise). It is common to report the 1/f noise level at 1 Hz or 10 Hz. In most cases, we\nreported noise over the 1 kHz range. Since our primary focus is on low-frequency noise\nbehavior, specifically around 10 Hz, which is most relevant to our analysis, we have\nemphasized that region accordingly. We added more data to the Supplemental\nMaterials regarding the noise spectral behavior of other devices to highlight\nreproducibility.\n(3) It is not clear if the Lorentzian bulge persists at other voltages than shown in 2(b).\nAdditional data at various voltages in 2(b) can be helpful as well as a discussion from\nthe authors on the evolution of the bulge with voltage.\nRESPONSE: We have included an additional figure in the revised manuscript to\nhighlight the noise behavior across the depinning transition and added an extra\n4 | P age\ndiscussion of the evolution of Lorentzian bulges with bias voltage to the revised\nmanuscript. As shown in the revised Figure 3, the noise spectrum shifts from a purely\n1/f pattern at lower bias to one showing Lorentzian bulges superimposed on the 1/f\nbackground at intermediate bias levels, indicating the depinning transition. At higher\nbiases, as the device approaches the sliding regime, the spectrum returns to a 1/f noise\ntype, with the Lorentzian features fading and eventually vanishing. This pattern confirms\nthat the origin of the Lorentzian component is related to threshold fluctuations.\nAdditional noise measurements on NbS also confirmed Lorentzian bulges near\n3\ndepinning, which disappear in the sliding regime.\n“Figure 3 (a-f) illustrates the evolution of the noise spectra in the bias region II, near the CDW\ndepinning threshold, indicating the association of the Lorentzian features with the depinning.\nThe Lorentzian noise component diminishes as the bias voltage surpasses the depinning threshold and\nenters the CDW sliding regime. The observed Lorentzian bulges originate from threshold\nfluctuations; they are associated with the system fluctuations between the pinned and depinned\nstates. In mathematical description, the transitions between two phases in the material are similar\nto those in the two-level systems, which results in the Lorentzian features in the noise spectra.\nAdditional data is available in the Supplementary Materials.”\n(4) Four different bias regions (Figure 2) were mentioned with handwaving assignments\nbased on the behavior. It will be helpful to mark the regions in the figures and to state\nthe voltage ranges to avoid confusion.\nRESPONSE: The four regions are introduced in Figure 2 e) based on noise\ndependence on bias voltage. It is a conventional approach in noise data analysis (see\nfor example, G. Liu, et al., “Low-frequency current fluctuations and sliding of the charge\ndensity waves in two-dimensional materials,” Nano Letters, 18, 3630 (2018)). The\nvoltage range may change from device to device due to dimensional differences, which\naffect the depinning. The main feature of S (V) remains the same. We marked the\nV\nregions more clearly in the revised manuscript.\n5 | P age\n(5) It will be helpful to state clearly how the normal and collective components of I and G\nare calculated, the explanation surrounding Figure 3 is confusing and could be\nimproved.\nRESPONSE: In the revised manuscript, we significantly expanded the description and\nprovided the details in the main text rather than in the Extended Data (as it was in the\noriginal submission). Additionally, a comprehensive explanation of the transport\nparameter extraction and noise modeling procedures has been included in the\nSupplemental Materials.\n(6) A major concern is about the author's earlier work on noise behavior in another\nCDW system (reference 36). The observation of the noise magnitude trending\nsignificantly lower in the sliding CDW regime has been reported in 36. It will be\nimportant for the authors to clearly provide the additional understanding and/or newer\nresult in the present manuscript.\nRESPONSE: We appreciate that the Reviewer found and read our paper S. Ghosh et\nal., “The noise of the charge density waves in quasi-1D NbSe nanowires —\n3\ncontributions of electrons and quantum condensate,” Appl. Phys. Rev. 11, 21405\n(2024). In that paper, we analyzed the noise behavior in NbSe . This is a different CDW\n3\nmaterial, which is not fully gapped. For this reason, the current is always dominated by\nnormal electrons with some contribution from CDWs. In that paper, we focused on noise\nspecifically near the depinning. We have attempted to study the noise at higher bias\nlevels. However, we have not managed to demonstrate the total noise reduction\nbelow the noise level of normal electrons. Please see Figure 7: the red and gray\ndata points at the far-right merge! We stated this in the text as “It is important to note\nthat at high currents, the noise of the normal electron current and total noise\nlevels coincide.” The physical reason for this is that NbSe material is not fully gapped.\nanalyze the noise accurately. In extracting the noise of the sliding CDW in that past\n6 | P age\nwork, we simply subtracted the noise at low bias (before CDW depinning) from the total\nnoise. This is a rough approximation. It served as an indication that the CDW noise is\nreducing and gave us an idea to study the fully gapped CDW materials and\ndevelop a theoretical framework to analyze noise. However, again, we never\nreached the total noise level below the normal electrons, the major task. We never\nobserved noise scaling with the current as S/I2~ 1/I. These two major tasks required\nI\ninvestigation of different material systems, more than a year of hard work. The\nfundamental discovery of the total noise below the normal electron level and its\nunusual scaling with the current is reported in the present manuscript for the first\ntime.\nOur new results present a major step forward and novelty: (1) clear demonstration of\nthe noise below normal electron noise (fundamental science discovery); (2) scaling of\nthe CDW condensate noise with bias voltage, which is completely different from scaling\nof the noise of normal electrons; (3) comprehensive phenomenological theory of 1/f\nnoise in CDW materials (something that has not been done before after 50 years of\ninvestigations). To add on top, we included data for another quasi-1D material, NbS ,\n3\nwhich shows CDW sliding and noise reduction at room temperature. The additional\nnoise analysis of NbS also suggested that the sliding CDW condensate is inherently\n3\nless noisy than that of thermalized single electrons. We feel that our manuscript has\nsufficient novelty for a couple of Nature family publications.\n7 | P age\nReviewer #2 (Remarks to the Author):\nAs known for decades, for some quasi -1D materials with a charge density wave (CDW)\nground sate, above a threshold value, the pinned CDW can move and contribute to a\ncollective current - static and modulated at a given frequency associated to 1/f low\nfrequency noise (unfortunately called narrow and broad band noise).\nThe manuscript reports studies of non-linear transport properties, specifically noise, in\n(TaSe4)2I nanostructures, the Peierls transition of which occurs at TP = 245K. Data are\npresented for two different devices at a single temperature below TP.\nRESPONSE: For the original submission, we tested five devices, and the results were\nconsistent. We included two different devices at two different temperatures (180 K and\nwe included data for the third (Ta Se ) I device and data for a different material system\n2 4 2\nNbS , where we observed a similar effect at room temperature. We included several\n3\ndifferent temperatures. We are confident in the reproducibility of the data.\nThe authors show that far away from the threshold (10 times its value at the given\ntemperature), the noise is suppressed below the value of normal charge carriers, and\nthat the noise of the CDW current is negligible. Less elaborated a similar conclusion\nwas suggested (see Ref.36) in noise measurements in NbSe3 which at the difference of\n(TaSe4)2I remains metallic below the Peierls transitions.\nRESPONSE: Regarding the Ref. 36 (original manuscript numbering), we repeat the\nresponse given to Reviewer #1. We appreciate that the Reviewer found and read our\npaper S. Ghosh et al., “The noise of the charge density waves in quasi-1D NbSe\n3\nnanowires — contributions of electrons and quantum condensate,” Appl. Phys. Rev. 11,\n21405 (2024). In that paper, we analyzed the noise behavior in NbSe . This is a\n3\ndifferent CDW material, which is not fully gapped. For this reason, the current is\n8 | P age\nalways dominated by normal electrons with some contribution from CDWs. In that\npaper, we focused on noise specifically near the depinning. We have attempted to study\nthe noise at higher bias levels. However, we have not managed to demonstrate the\ntotal noise reduction below the noise level of normal electrons. Please see Figure\n7: the red and gray data points at the far-right merge! We stated this in the text as “It is\nimportant to note that at high currents, the noise of the normal electron current\nand total noise levels coincide.” The physical reason for this is that NbSe material is\n3\nnot fully gapped. The current is dominated by the normal electrons. We also did not\nhave a theory to analyze the noise accurately. In extracting the noise of the sliding CDW\nin that past work, we simply subtracted the noise at low bias (before CDW depinning)\nfrom the total noise. This is a rough approximation. It served as an indication that the\nCDW noise is reducing and gave us an idea to study the fully gapped CDW\nmaterials and develop a theoretical framework to analyze noise. However, again,\nwe never reached the total noise level below the normal electrons, the major task. We\nnever observed noise scaling with the current as S/I2~ 1/I. These two major tasks\nI\nrequired investigation of different material systems, more than a year of hard work.\nThe fundamental discovery of the total noise below the normal electron level and\nits unusual scaling with the current is reported in the present manuscript for the\nfirst time.\nOur new results present a major step forward and novelty: (1) clear demonstration of\nthe noise below normal electron noise (fundamental science discovery); (2) scaling of\nthe CDW condensate noise with bias voltage, which is completely different from scaling\nof the noise of normal electrons; (3) comprehensive phenomenological theory of 1/f\nnoise in CDW materials (something that has not been done before after 50 years of\ninvestigations). To add on top, we included data for another quasi-1D material, NbS ,\n3\nwhich shows CDW sliding and noise reduction at room temperature. The additional\nnoise analysis of NbS also suggested that the sliding CDW condensate is inherently\n3\nless noisy than that of thermalized single electrons. We feel that our manuscript has\nsufficient novelty for a couple of Nature family publications.\n9 | P age\nThese conclusions are questionable as discussed in the following:\nRESPONSE: We hope that our response and additional data provided in the revised\nmanuscript will remove the questions.\n-in Fig.1d the variation of the differential resistance in a semi-log plot is shown versus\nthe applied voltage at different temperatures: clearly the resistance at fixed T is not\nconstant indicating a heating effect. At T=120K the drop of resistance at the threshold\n(see the arrow) is even not visible. No estimation of thermodynamics appears in the\npaper.\nRESPONSE: We would like to state clearly that there are NO heating effects in our\nmeasurements. Below, we repeat the response given to Reviewer 1.\nThe reviewer correctly notes that any significant Joule heating during transport\nmeasurements would affect the results. However, we can state with high confidence\nthat Joule heating does not play any role in our measurements. We used bias\nvoltages below or around 1 V with the current levels always remaining in the micro-\nAmpere range, resulting in minimal power dissipation.\nIn the revised manuscript, we included analytical estimates of the thermal resistance of\nthe structure and the resulting temperature rise, which was below 1 K at 0.7 V bias,\nwhen the noise drops below the normal electron level. We also conducted COMSOL\nmodeling to verify this estimate. We included noise data for a new material system,\nNbS at room temperature. Additionally, in situ bias-dependent Raman measurements\n3,\nwere conducted on both (TaSe ) I and NbS nanowire devices, where we monitored\n4 2 3\nphonon peak positions in situ. The local heating would reveal itself via softening of\nphonon peaks. None was observed at the highest bias voltages, presenting a direct\nproof of the absence of Joule heating. We discussed the temperature rise estimation\n10 | P age\nand the in-situ Raman behavior in the revised text and included the relevant figures in\nthe Supplemental file. In the revised manuscript, we added the following section.\n“In the data analysis, it is also important to exclude local heating effects. No Joule self-heating\neffects are expected in our test structures at the small bias voltages, about ~1 V, and low current\nrange, about ~ 1 A. One can readily estimate the temperature rise, Δ𝑇, from an analytical formula\nderived from Fourier’s law: Δ𝑇 = 𝑃×𝑅 = 𝐼 ×𝑉 ×(𝑇 /𝐾×𝐿×𝑊). Here, P is the dissipated\n𝑇 𝑜𝑥\npower in the nanowire, R is the thermal resistance of the SiO layer, T is the thickness of the\nT 2 ox\nSiO layer, K is the thermal conductivity of SiO , L is the length of the channel, and W is the\n2 ox 2\nwidth of the channel. The formula assumes that the bulk Si substrate is the thermal sink, and the\nSiO layer with its low thermal conductivity is the thermal barrier. Using the typical values, i.e.,\n2\n𝐼 =1 A, 𝑉 =1 V, 𝑇 = 300 nm, 𝐾 = 1 W/mK, L= 5 m, and 𝑊=300 nm, we obtain for the\n𝑜𝑥 𝑜𝑥\nthermal resistance R ~0.2 MK/W and, correspondingly, for the temperature rise Δ𝑇~0.2 K. We\nT\ncan elaborate the estimate and consider the thermal boundary resistance, 𝑅 , between the nanowire\n𝐶\nand SiO layer. It is well known that it is primarily defined by the interface quality rather than\n2\nspecific channel material. Assuming the worst-case scenario of low interface conductance, 𝐺 =15\n𝑇\nMW/m2K44,45, we obtained an extra thermal resistance of ~0.2 MK/W, which doubles the total\nresistance and temperature rise. On the other side, conduction to metal contacts along the channel\nmay reduce the temperature rise. These simple but reliable estimates show that in our devices, in\nthe considered bias and current ranges, the local heating is negligible. One has to pass currents in\nthe mA range, a factor of ×1000 larger, to induce significant Joule heating. The temperature rise\nwas also assessed using COMSOL software tools for a given device structure and bias voltages.\nThe maximum temperature increase at the hot spot (center of the channel) is approximately 1 K at\na voltage drop of 0.9 V. The simulation results confirmed negligible self-heating effects (see\nSupplemental Materials).\nFor an independent experimental proof of the absence of local heating effects, we took Raman\nspectra in situ of the representative devices under the same bias as used in the noise experiments.\nNo characteristic phonon peak shifts, expected for local heating, have been observed (see\nexperimental data in the Supplemental Information). We have previously detected local heating\n11 | P age\neffects with Raman spectroscopy in a range of materials and different device structures, and are\nconfident in the accuracy of such a method 46,47.”\nFigure 1 d) is a standard plot for the CDW data, which illustrates the resistance change\ndue to the depinning of CDW. It does not show any heating effect at the used\nvoltages (below 1 V) and current (micro-Amperes) levels. The drop in the resistance is\nvisible (it is not abrupt) and consistent with I-Vs. Moreover, the nonlinearity in the\ndifferential resistance vanishes as the temperature rises above the CDW transition\ntemperature (see Figure S3c), further confirming that the observed nonlinear transport\nis intrinsic to CDW behavior. Additionally, the temperature-dependent shift in the\nthreshold field, V, is a well-established characteristic of CDW systems and has been\nt\nreported in various quasi-one-dimensional materials. We observe similar behavior in our\nNbS₃ devices (Fig. 6c), supporting the CDW origin of the threshold phenomena. In the\nprior studies with NbS , we also confirmed it with Shapiro step measurements (M.\ncharge-density-wave quantum condensate in h-BN/NbS quasi-2D/1D heterostructure\n3\ndevices.” Appl. Phys. Lett., 123, 233101 (2023)).\n- While data in Fig.1d concern the small values of V (V< 0,2V) those in Fig.1e are\nextended to 1V. How are the dV/dI characteristics till 1V in order to see possible jumps\nor inhomogeneities masked in the log-log I-V curves in fig.1e ?.\nRESPONSE: The differential resistance vs. voltage plots in Figure 1d are presented on\na semi-logarithmic scale to clearly illustrate the resistance behavior under both positive\nand negative bias polarities. This data presentation style is commonly used in the CDW\nresearch community to highlight symmetric or homogeneous changes in resistance\nacross both bias directions. Regarding the reviewer’s question about the limited voltage\nrange in Figure 1d, we intentionally restricted the bias to lower voltages because the\nprimary objective of these measurements was to investigate the behavior of the\nthreshold field, V. In contrast, during the noise measurements, we simultaneously\nt\n12 | P age\nrecorded the transport characteristics over an extended bias range (~1 V). We have\nprovided the full transport behavior plots in the Supplemental file for the three devices\npresented in the manuscript. Additionally, we updated Fig 1d with a new figure showing\ndata at more temperatures and higher bias ranges.\n-Data in Fig1e show that the CDW current (evaluated from the extended (dashed line)\nOhmic law (without taking care of any thermal effect) follows a power law without\nsaturation till more than 10 times the threshold value. How to reconciliate this result with\ntheories which evaluate the high field limit of the CDW conductance?.\nRESPONSE: There were no self-heating effects at the examined bias ranges\n(please see the response above, information added to the main text and\nSupplementary). Indeed, the CDW current continues to increase in (TaSe ) I nanowires\n4 2\nwithout saturation in the examined bias voltage range. In contrast, in the NbS\n3\nnanowires, the current shows the signs of saturation. In (TaSe ) I nanowires, the current\n4 2\nwill likely saturate at biases higher than what we examined, but we do not want to bias\nour devices too high to avoid possible self-heating (mentioned by the Reviewer). The\nfact that (TaSe ) I is a “special” CDW material was noticed early, starting from Gruner’s\n4 2\nwork at UCLA. The CDW transport in this material was described by underdamped\noscillator. It appears that the CDW damping is weaker in this material. We have added\nsome considerations about the specifics of this material. However, no microscopic\ntransport theory exists to conclude. We made an important contribution toward it by\ndeveloping the first comprehensive phenomenological noise theory in CDW conductors\nwith low damping that the Reviewer can probably appreciate. The following sections\nhave been added.\n“We now address another fundamental question: Is the ultra-low noise of sliding condensate\nunique to the IC-CDW phase of (TaSe ) I, or is it a property of many quasi-1D CDW conductors?\n4 2\nThe early studies of the “broadband” noise in bulk CDW crystals focused on the threshold noise\n15–17 and never considered the possibility of noise reduction below the normal electron limit. A few\nrecent studies for TaS and NbSe found that beyond V, the noise returns to the normal electron\n3 3 t\nlimit 37,48, in contrast to our observation of continuous noise reduction in the sliding regime of\n13 | P age\n(TaSe ) I devices. Other potential candidates for ultra-low noise transport should include fully-\n4 2\ngapped CDW materials where the collective, strongly-correlated current of the sliding electron-\nlattice condensate is dominant. We experimented with nanowires of NbS -II, which exists in the\n3\nsame protocols as with (TaSe ) I nanowire devices. Figure 6 (a-f) shows transport and noise\n4 2\ncharacteristics for a representative NbS -II nanowire device. The most important observation is\n3\nthat the normalized spectral density in the NbS -II device reduces below the normal electron limit,\n3\nsimilar to that in the (TaSe ) I devices. The noise spectral density scaling with current is also\n4 2\nconsistent, i.e., noise reduces with increasing current. It reveals a similar, 𝑆 ⁄𝐼21/𝐼, dependence\n𝐼\nfor the current above 10 A. We note that the noise reduction in NbS -II is achieved at\n3\ntemperatures above RT. The latter is important for practical applications, i.e. in interconnects.\nThere is one difference in the noise behavior in the two materials: it appears the noise spectral\ndensity in NbS -II nanowires shows saturation behavior, even though the noise level is below the\n3\nnormal electron limit. This suggests that we may be reaching the intrinsic electron-lattice\ncondensate noise in NbS -II nanowires but not in (TaSe ) I nanowire devices (see additional data\n3 4 2\nin Supplementary Materials).\nThe origin of the ultra-low-noise collective current in (TaSe ) I nanowire, and the lack of noise\n4 2\nsaturation, may be related to the fact that the CDW transport in this specific material is described\nby the under-damped oscillator model 49,50. Compared to other materials like NbSe and o-TaS ,\n3 3\n(TaSe ) I has the largest energy gap and heaviest CDW effective mass, 𝑚∗/𝑚 ≈ 104, yielding\n4 2\nthe lowest CDW damping. The non-linear CDW conductivity in this material continues to increase\nrather than saturate due to damping, in the examined bias voltages. The current in NbS -II shows\n3\nsigns of saturation at higher bias voltages (see Figure 6 (b)), indicating the onset of stronger CDW\ncurrent damping, which likely translates to higher intrinsic CDW noise and eventual flattening of\nthe total noise level (see Figure 6 (e) and (f)). Based on the recent realization that (TaSe ) I is a\n4 2\nWeyl semimetal 30,31, one may also argue that topological scattering suppression has relevance to\nthe noiseless current of the sliding condensate in this material.”\n14 | P age\n-Eq. 2 is the combination of Eq.(5.6) and 5.14) from Ref.1\nThe first term results from the “single particle” model – classical particle moving in a\nsinusoidal washboard potential -in the case where the voltage is regulated yields j a\n(E2-E2t)1/2 , (however in the part but” Electronic transport and noise measurement in\nthe (TaSe4)2I nanowires” it seems that measurements were performed with a regulated\ncurrent. In that case the threshold is manifested by an infinite negative differential\nresistance that is not the case in Fig.1d).\nThe second term provides from a different model due to dynamical critical phenomenon\nwith the CDW velocity or the CDW current is a power law (E-Et) with an exponent 3/2 to\nbe compared with ½ in the single particle model (however see and 2 in eq.2). No\nreal justification is made for the addition of two different threshold current with non-linear\nvoltages with different exponents, even a third one - see figure caption of Extended data\nFig.6).\nRESPONSE: \nThere are several points here. The first is about writing the non-linear current 𝐼 = 𝐼 +\n𝑐 𝑐1\n𝐼 = Γ [ (𝑉2 −𝑉2)𝑎 +(𝑉−𝑉)2𝑎 ]𝜃(𝑉−𝑉) as a linear combination of the two non-\n𝑐2 𝑐 𝑡 𝑡 𝑡\nlinear current terms: 𝐼 = Γ (𝑉2 −𝑉2)𝑎𝜃(𝑉−𝑉) and 𝐼 = Γ (𝑉−𝑉)2𝑎𝜃(𝑉 −𝑉). These\n𝑐1 𝑐 𝑡 𝑡 𝑐2 𝑐 𝑡 𝑡\ntwo equations are based on the two equations noted above but combined in a way to\nminimize the number of free parameters. Thus, there is only one unknown exponent 𝑎,\none proportionality constant Γ , and one threshold voltage 𝑉. The threshold voltage is\n𝑐 𝑡\npretty obvious from the data, so there are really just two fitting parameters 𝑎 and Γ . The\n𝑐\nvalue for 𝑎 for 2 different samples of different geometries is 0.9 with all parameters\nlisted in the table on p. 16 of the Supplemental Material. While either 𝐼 or 𝐼 give a\n𝑐1 𝑐2\npretty good fit to the data, both are needed to match the derivative over the full bias\nrange. Near 𝑉, 𝐼 gives rise to the small discontinuous turn-on feature right above 𝑉 in\nt\n𝑡 𝑐1 𝑡\nFigs. 1e and 4e. This term also gives rise to the singular peak in the noise at 𝑉.\nThe\n𝑡\nsecond term gives the 1/𝐼 scaling of 𝑆/𝐼2 at higher bias. Without this term, the noise\nwould fall off even faster than 1/𝐼. The contributions of the individual terms in the\n15 | P age\nresulting noise expression (now Eq. (4) of the main text) are shown in Fig. S6 of the\nSupplement. Thus, the two non-linear current expressions with 2 free parameters are\nneeded to fit both the current and the noise, and their underlying form is guided by the\nprior current expressions.\nThe second point concerns the biasing. The biasing is described on p. 4 of Methods.\nThe devices are voltage biased – corresponding to the 𝐸 = 𝑐𝑜𝑛𝑠𝑡 condition for which the\n𝑡\n(𝐸2 −𝐸2)1/2 form is derived. The second form (𝐸 −𝐸 )𝜉 captures the larger biases well.\n𝑡 𝑡\nBoth are needed to match the current and the noise over the full bias range.\nThe final point is related to comment ‘even a third one - see figure caption of Extended\ndata Fig.6’. I think there is possible confusion caused by bad notation here. There are\nonly two components to the nonlinear current, 𝐼 and 𝐼 . Taking the derivative, sqaring,\n𝑐1 𝑐2\nand taking the ensemble averages, as now shown in Eqs. (3) and (4) of the main text\n(and also below), gives rise to four terms in the noise and three terms proportional to\n𝛿𝑉. These three terms were denoted as 𝑉 , 𝑉 , and 𝑉 in the original Extended Data\nt\n𝑡 𝑡1 𝑡2 𝑡3\nFig. 6. We now denote these three contributions resulting from fluctuations in the\nthreshold voltage as 𝑆 /𝐼2, 𝑆 /𝐼2, and 𝑆 /𝐼2. We emphasize that there is only one\nt1 𝑡2 𝑡3\nthreshold voltage and only two current components.\nIn the revised text, we added the following.\n“The general phenomenological theory of noise in CDW condensate conductors\nTo further understand the unusual noise reduction in fully-gapped CDW nanowires, we developed\na comprehensive phenomenological noise theory in non-linear approximation. It has been\naccomplished by extending the “two-fluid” approach to include explicitly the non-linear voltage\ndependence of the collective current in (TaSe ) I. The total current, 𝐼 = 𝐼 +𝐼 , consists of a\n4 2 𝑛 𝑐\nnormal current 𝐼 = 𝐺 𝑉, and the sliding CDW current modeled as 40:\n𝑛 𝑛\n𝐼 = 𝐼 +𝐼 = Γ [ (𝑉2 −𝑉2)𝑎 +(𝑉 −𝑉)2𝑎 ]𝜃(𝑉−𝑉). (2)\n𝑐 𝑐1 𝑐2 𝑐 𝑡 𝑡 𝑡\n16 | P age\nHere, 𝜃(𝑉 −𝑉) is the unit step function that ensures the CDW contribution begins only after\nt\ndepinning. We denote the two components of the collective current with different non-linear\nvoltage dependencies as 𝐼 = Γ (𝑉2 −𝑉2)𝑎𝜃(𝑉 −𝑉) and 𝐼 = Γ (𝑉−𝑉)2𝑎𝜃(𝑉 −𝑉), and Γ\n𝑐1 𝑐 𝑡 𝑡 𝑐2 𝑐 𝑡 𝑡 𝑐\nis a generalized conductance relating the current to the non-linear voltage. The transport behavior\nof our device exhibits a distinct kink at V. Similar changes in the bias dependence near threshold\nt\nwere previously attributed to the CDW creep before the onset of condensate sliding 41–43. Our\nphenomenological model incorporates two nonlinear terms with distinct power dependencies to\naccount for the initial depinning dynamics and the subsequent sliding behavior of the CDW beyond\nthe threshold. The accuracy of the description is confirmed by the excellent fit to the experimental\ndata of all examined devices (Figure 4 (e) and Supplementary Materials).\nAssuming uncorrelated, zero-mean fluctuations in the physical parameters, 𝐺 , Γ , and 𝑉, the total\nn 𝑐 𝑡\nfirst-order fluctuation in current can be expressed as:\n𝜕𝐼 𝜕𝐼 𝜕𝐼\n𝛿𝐼 = 𝛿𝐺 + 𝛿𝛤 + 𝛿𝑉\nn 𝑐 𝑡\n𝜕𝐺𝑛 𝜕𝛤𝑐 𝜕𝑉𝑡\n= 𝑉𝛿𝐺𝑛+{ Γ𝑐[(𝑉2 − 𝑉 𝑡 2) 𝑎 + (𝑉 − 𝑉𝑡 )2𝑎]𝛿Γ𝑐 − 2𝑎 𝛤𝑐𝑉𝑡[ 𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)]𝛿𝑉𝑡 } 𝜃(𝑉−𝑉).\n𝑡\n𝐺𝑛 Γ𝑐 𝑉𝑡\nDividing by the total current, squaring, and taking the ensemble average, we obtain the normalized\nnoise power expressed as:\n2\n〈𝛿𝐼2〉\n=\n⟨𝛿𝐺𝑛 2⟩\n(\n𝐼𝑛) 2\n+{\n⟨𝛿𝛤𝑐 2⟩\n(\n𝐼𝑐) 2\n+\n⟨𝛿𝑉\nt\n2⟩\n[\n2𝑎 𝛤𝑐{𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)}\n]\n}𝜃(𝑉−𝑉)\n𝐼2 𝐺2 𝐼 𝛤2 𝐼 𝑉2 𝐼 𝑡\nn 𝑐 𝑡\n= 〈𝛿𝐺𝑛 2〉𝐼𝑛 2 + { 〈𝛿Γ𝑐 2〉𝐼𝑐 2 +4𝑎2[ 𝑉 𝑡 4 𝐼𝑐 2 1 + 𝑉 𝑡 2 𝐼𝑐 2 2 + 2𝑉 𝑡 3 𝐼𝑐1 𝐼𝑐2] 〈𝛿𝑉 𝑡 2〉 }𝜃(𝑉−𝑉). (4)\n𝐺𝑛 2 𝐼2 Γ 𝑐 2 𝐼2 (𝑉2−𝑉 𝑡 2) 2 𝐼2 (𝑉−𝑉𝑡 )2 𝐼2 (𝑉2−𝑉 𝑡 2)(𝑉−𝑉𝑡 ) 𝐼2 𝑉 𝑡 2 𝑡\nThe first term is the noise of the normal current, the second term is the noise due to fluctuations of\nthe generalized CDW conductance, Γ , and the third term is the noise due to fluctuations in the\nc\nthreshold voltage, 𝑉 . At large voltages (𝑉 ≫ 𝑉), the terms proportional to 〈𝛿𝑉 2〉⁄𝑉2 are\nt 𝑡 𝑡 𝑡\nsuppressed, and Eq. (4) reduces to a form similar to Eq. (1). At 𝑉 = 𝑉+, the normalized noise\nt\ndue to fluctuations in 𝑉 is singular for 𝑎 < 1, but in the limit of large 𝑉 ≫ 𝑉 , it falls off rapidly.\nt 𝑡\n17 | P age\nLet us look closer at Figure 4 (f), which shows the noise calculated from Eq. (4) superimposed on\nthe experimental data. The fitting to the experimental I-Vs (see Figure 4 (e)), allows one to extract\nthe parameters, which are then used as inputs to the noise model. The first observation is that our\ntheory accurately describes the noise peak due to fluctuations in 𝑉 and how it rolls off as 1/𝐼.\nt\nThe bias dependence of the weighted normal electron noise and the CDW condensate noise are\ndescribed as [(〈𝛿𝐺 2〉⁄𝐺 2)(𝐼2⁄𝐼2)] and [(〈𝛿Γ 2〉⁄Γ 2)(𝐼2⁄𝐼2)], respectively. The noise of the\nn 𝑛 𝑛 𝑐 𝑐 𝑐\nnormal electrons dominates below the threshold voltage, V. Both noise components are constant\nt\nat lower and higher bias regimes, when 𝐼 ≈ 𝐼 and 𝐼 ≈ 𝐼 , respectively. In the high-bias regime\nn 𝑐\nwhere CDW noise emerges, it can saturate the total noise if its amplitudes grow large enough. The\nbias dependence of the threshold field fluctuations, 〈𝛿𝑉2〉⁄𝑉2, becomes pronounced near V. The\nt 𝑡 t\ntotal contribution from the threshold fluctuations to the noise contains the three distinct\ncomponents within the square brackets of Eq. (4), each with a characteristic bias dependence.\nThe first one, 𝑆 /𝐼2 = {4𝑎2𝑉4⁄(𝑉2 −𝑉2)2}(𝐼2 ⁄𝐼2)(〈𝛿𝑉2〉⁄𝑉2) dominates near threshold and\nt1 𝑡 𝑡 𝑐1 𝑡 𝑡\nproduces a noise peak that rapidly drops as 1⁄𝐼2. The second one, 𝑆 /𝐼2 =\nt2\n{4𝑎2𝑉2⁄(𝑉2 −𝑉2)2}(𝐼2 ⁄𝐼2)(〈𝛿𝑉2〉⁄𝑉2) dominates at 𝑉 ≫ 𝑉, resulting in noise scaling as 1/𝐼.\nt 𝑡 𝑐2 𝑡 𝑡 𝑡\nThe third one, 𝑆 /I2 = [8𝑎2𝑉3⁄{(𝑉2 −𝑉2)(𝑉−𝑉)}]{(𝐼 𝐼 )⁄𝐼2}(〈𝛿𝑉2〉⁄𝑉2) drops\nt3 𝑡 𝑡 𝑡 𝑐1 𝑐2 𝑡 𝑡\napproximately as 1⁄𝐼1.5 in the intermediate regime. Together, these components capture accurately\nthe evolution of the noise across the depinning transition and into the CDW sliding regime. The\ndetails of the model fitting and resulting noise behavior across different bias regimes are presented\nin the Supplemental Materials.”\nFrom their Eq.2 the authors derive the noise spectral density formed of 3 terms: noise of\nthe normal current, noise of the fluctuations of the threshold voltage (near Vt) and\nfluctuations of the CDW conductance. They found that the noise spectra can be\naccurently fitted with only the noise amplitude of the normal current and that of\nthreshold fluctuations but consequently without noise for the CDW conductance.\nHowever, the values of different parameters such these three values of Vt are not\nindicated.\n18 | P age\nRESPONSE: In the original submission, the values of the fitting were indicated in the\nExtended Data Figure 6 (see the labels in the figures with noise amplitudes and\ncaptions with the description). In the revised manuscript, we provided all the details of\nthe theory and fitting (see the main text and Supplementary Materials sections). We\nalso included another material system to add support to our claims. All derivation details\nof the theory were provided in the Supplement (see below).\n“The phenomenological theory of 𝟏/𝒇 noise in non-linear approximation\nWe now derive the general phenomenological theory of 1/𝑓 noise in CDW conductors using a\nnon-linear expression for current that captures transport characteristics of CDW conductors with\nlow damping. The initial expression for the current can be chosen to match the specific material\nof interest closely. Here, we start with the equation for (TaSe ) I , a Weyl CDW conductor with\n4 2\nlow damping. The total current can be best described by collective current with two non-linear\ncomponents:\n𝐼 = 𝐼 +𝐼 = 𝐼 +(𝐼 +𝐼 )= 𝐺 𝑉+Γ [ (𝑉2−𝑉2)𝑎+(𝑉−𝑉)2𝑎 ]𝜃(𝑉−𝑉) (S1)\nn 𝑐 𝑛 𝑐1 𝑐2 𝑛 𝑐 𝑡 𝑡 𝑡\nHere, the exponent α = 2𝑎, 𝐼 and 𝐼 represents two collective current components with different\n𝑐1 𝑐2\nnon-linear voltage dependencies.\nLet us assume uncorrelated, zero-mean fluctuations for, 𝛿𝐺 ,𝛿𝛤,𝛿𝑉. Then\nn 𝑐 𝑡\n𝜕𝐼 𝜕𝐼 𝜕𝐼\n𝛿𝐼 = 𝛿𝐺 + 𝛿𝛤 + 𝛿𝑉 (S2)\nn 𝑐 𝑡\n𝜕𝐺𝑛 𝜕𝛤𝑐 𝜕𝑉𝑡\n𝜕𝐼\n= 𝑉 (S3)\n𝜕𝐺𝑛\n𝜕𝐼 = (𝑉2 − 𝑉2)𝑎 + (𝑉 − 𝑉)2𝑎 (S4)\nt 𝑡\n𝜕𝛤𝑐\n𝜕𝐼 = −2𝑎 𝛤 [𝑉 (𝑉2 − 𝑉2)(𝑎−1) + (𝑉 − 𝑉)(2𝑎−1)] (S5)\n𝑐 𝑡 𝑡 𝑡\n𝜕𝑉𝑡\nWe can write the current fluctuation as:\n𝛿𝐼 =𝑉𝛿𝐺 +[(𝑉2 − 𝑉2)𝑎 + (𝑉 − 𝑉)2𝑎]𝛿Γ −2𝑎 𝛤 [𝑉 (𝑉2 − 𝑉2)(𝑎−1)+ (𝑉 − 𝑉)(2𝑎−1)]𝛿𝑉 (S6)\nn 𝑡 𝑡 𝑐 𝑐 𝑡 𝑡 𝑡 𝑡\n19 | P age\n𝛿𝐼 = 𝑉𝛿𝐺𝑛 + Γ𝑐[(𝑉2 − 𝑉 𝑡 2) 𝑎 + (𝑉 − 𝑉𝑡 )2𝑎]𝛿Γ𝑐 −2𝑎 𝛤 𝑉𝑡[ 𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)]𝛿𝑉𝑡 (S7)\n𝑐\n𝐺𝑛 Γ𝑐 𝑉𝑡\n𝛿𝐼 = 𝛿𝐺 𝑉𝐺𝑛 +𝛿Γ Γ𝑐[(𝑉2 − 𝑉 𝑡 2) 𝑎 + (𝑉 − 𝑉𝑡 )2𝑎] − 2𝑎 𝛤𝑐[𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) (𝑎−1) + 𝑉𝑡 (𝑉−𝑉𝑡 )(2𝑎−1)]𝛿𝑉𝑡 (S8)\nn 𝑐\n𝐺𝑛 Γ𝑐 𝑉𝑡\n𝛿𝐼 = \n𝛿𝐺𝑛𝐼\n+ \n𝛿Γ𝑐𝐼\n−[𝛿𝑉 ×2𝑎 𝛤\n{𝑉\nt\n2 (𝑉2 − 𝑉\nt\n2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)}\n]\n(S9)\nn 𝑐 𝑡 𝑐\n𝐺𝑛 Γ𝑐 𝑉𝑡\nDividing by the total current, we obtain:\n𝛿𝐼\n= (\n𝛿𝐺𝑛)( 𝐼𝑛)\n+ (\n𝛿𝛤𝑐)( 𝐼𝑐)\n− (\n𝛿𝑉𝑡)[\n2𝑎𝛤\n{𝑉\nt\n2 (𝑉2 − 𝑉\nt\n2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)}\n]\n(S10)\n𝑐\n𝐼 𝐺𝑛 𝐼 𝛤𝑐 𝐼 𝑉𝑡 𝐼\nAs the next step, we square, take the ensemble average, and assume no cross correlation to get to\nthe final expression for noise in this non-linear model:\n2\n〈𝛿𝐼2〉\n=\n⟨𝛿𝐺𝑛 2⟩\n(\n𝐼𝑛) 2\n+\n⟨𝛿𝛤𝑐 2⟩\n(\n𝐼𝑐) 2\n+\n⟨𝛿𝑉\nt\n2⟩\n[2𝑎 𝛤\n{𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)}\n]\n(S11)\n𝐼2 𝐺2 𝐼 𝛤2 𝐼 𝑉2 𝑐 𝐼\nn 𝑐 𝑡\n〈𝛿𝐼2〉 = ⟨𝛿𝐺𝑛 2⟩ ( 𝐼𝑛) 2 + ⟨𝛿𝛤𝑐 2⟩ ( 𝐼𝑐) 2 + ⟨𝛿𝑉 𝑡 2⟩ 4𝑎2 𝛤2 [𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)] 2 (S12)\n𝐼2 𝐺2 𝐼 𝛤2 𝐼 𝑉2 𝑐 𝐼2\nn 𝑐 𝑡\n〈𝛿𝐼2〉\n=\n⟨𝛿𝐺𝑛 2⟩\n(\n𝐼𝑛) 2\n+\n⟨𝛿𝛤𝑐 2⟩\n(\n𝐼𝑐) 2\n+\n⟨𝛿𝑉\nt\n2⟩\n×𝑓(𝑉,𝑉) (S13)\n𝐼2 𝐺2 𝐼 𝛤2 𝐼 𝑉2 𝑡\nn 𝑐 𝑡\nWhere,\n𝑓(𝑉,𝑉) = 4𝑎2 𝛤2\n[𝑉\nt\n2 (𝑉2 − 𝑉\nt\n2) (𝑎−1) + 𝑉𝑡 (𝑉 − 𝑉𝑡 )(2𝑎−1)] 2\n(S14)\nt 𝑐 𝐼2\n𝑓(𝑉,𝑉) = 4𝑎2 𝛤2\n[𝑉\nt\n4 (𝑉2 − 𝑉\nt\n2) 2(𝑎−1) +𝑉\nt\n2(𝑉 − 𝑉𝑡 )2(2𝑎−1)+2𝑉\nt\n3 (𝑉2 − 𝑉\nt\n2) (𝑎−1) (𝑉 − 𝑉𝑡 )(2𝑎−1)]\n(S15)\nt 𝑐 𝐼2\n𝑓(𝑉,𝑉) = \n4𝑎2[𝑉4𝛤𝑐 2(𝑉2 − 𝑉\nt\n2) 2(𝑎−1) +𝑉2𝛤𝑐 2(𝑉 − 𝑉𝑡 )2(2𝑎−1) +2𝑉3𝛤𝑐 2(𝑉2 − 𝑉\nt\n2) 2(𝑎−1) (𝑉 − 𝑉𝑡 )(2𝑎−1)\n]\n(S16)\nt 𝑡 𝐼2 𝑡 𝐼2 𝑡 𝐼2\n𝑓(𝑉,𝑉) = \n4𝑎2[𝑉4𝛤𝑐 2(𝑉2 − 𝑉\nt\n2) 2𝑎 +𝑉2𝛤𝑐 2(𝑉 − 𝑉𝑡 )4𝑎 +2𝑉3𝛤𝑐 2(𝑉2 − 𝑉\nt\n2) 2𝑎 (𝑉 − 𝑉𝑡 )4𝑎\n]\n(S17)\nt 𝑡 (𝑉2 − 𝑉 𝑡 2) 2 𝐼2 𝑡 (𝑉 − 𝑉𝑡 )2𝐼2 𝑡 (𝑉2 − 𝑉 𝑡 2)(𝑉 − 𝑉𝑡 )𝐼2\n𝑓(𝑉,𝑉) = 4𝑎2[𝑉4 {Γ𝑐 (𝑉2 − 𝑉 𝑡 2) 𝑎 } 2 +𝑉2{Γ𝑐 (𝑉 − 𝑉𝑡 )2𝑎} 2 +2𝑉3 Γ𝑐 {(𝑉2 − 𝑉 𝑡 2) 𝑎 ×Γ𝑐 (𝑉 − 𝑉𝑡 )2𝑎} 2 ] (S18)\nt 𝑡 (𝑉2 − 𝑉 𝑡 2) 2 𝐼2 𝑡 (𝑉 − 𝑉𝑡 )2𝐼2 𝑡 (𝑉2 − 𝑉 𝑡 2)(𝑉 − 𝑉𝑡 )𝐼2\n20 | P age\n𝑓(𝑉,𝑉) = 4𝑎2[𝑉4\n𝐼𝑐 2\n1 +𝑉2\n𝐼𝑐 2\n2 +2𝑉3\n𝐼𝑐1 𝐼𝑐2\n] (S19)\nt 𝑡 (𝑉2 − 𝑉 𝑡 2) 2 𝐼2 𝑡 (𝑉 − 𝑉𝑡 )2𝐼2 𝑡 (𝑉2 − 𝑉 𝑡 2)(𝑉 − 𝑉𝑡 )𝐼2\n𝑓(𝑉,𝑉) = 4𝑎2[ \n𝑉 𝑡 4 𝐼𝑐 2 1 + \n𝑉 𝑡 2 𝐼𝑐 2 2 + 2𝑉 𝑡 3 𝐼𝑐1 𝐼𝑐2] (S20)\nt (𝑉2 − 𝑉 𝑡 2) 2 𝐼2 (𝑉 − 𝑉𝑡 )2 𝐼2 (𝑉2 − 𝑉 𝑡 2)(𝑉 − 𝑉𝑡 ) 𝐼2\nBy adding Eq. (M20) in Eq. (M13), the total noise fluctuation becomes,\n〈𝛿𝐼2〉 = ⟨𝛿𝐺𝑛 2⟩ ( 𝐼𝑛) 2 + ⟨𝛿𝛤𝑐 2⟩ ( 𝐼𝑐) 2 + ⟨𝛿𝑉 𝑡 2⟩ 4𝑎2[ \n𝑉 𝑡 4 𝐼𝑐 2 1 + \n𝑉 𝑡 2 𝐼𝑐 2 2 2𝑉 𝑡 3 𝐼𝑐1 𝐼𝑐2]𝜃(𝑉−𝑉)\n𝐼2 𝐺 𝑛 2 𝐼 𝛤 𝑐 2 𝐼 𝑉 𝑡 2 (𝑉2 − 𝑉 𝑡 2) 2 𝐼2 (𝑉 − 𝑉𝑡 )2 𝐼2 (𝑉2 − 𝑉 𝑡 2)(𝑉 − 𝑉𝑡 ) 𝐼2 𝑡\n(S21)\nThis represents the noise model equation for (TaSe ) I device, where,\n4 2\n𝑆𝑡1 = 4𝑎2 \n𝑉 𝑡 4 𝐼𝑐 2 1 \n⟨𝛿𝑉 𝑡 2⟩ (S22)\n𝐼2 (𝑉2 − 𝑉 𝑡 2) 2 𝐼2 𝑉 𝑡 2\n𝑆𝑡2 = 4𝑎2 \n𝑉 𝑡 2 𝐼𝑐 2 2 \n⟨𝛿𝑉 𝑡 2⟩ (S23)\n𝐼2 (𝑉 − 𝑉𝑡 )2 𝐼2 𝑉\nt\n2\n𝑆𝑡3 = 4𝑎2 \n2𝑉 𝑡 3 𝐼𝑐1 𝐼𝑐2 \n⟨𝛿𝑉 𝑡 2⟩ (S24)\n𝐼2 (𝑉2 − 𝑉\nt\n2)(𝑉 − 𝑉𝑡 ) 𝐼2 𝑉\nt\nt 𝑡\nof large 𝑉 ≫ 𝑉 , it falls off rapidly.\nt\nAt 𝑉 ≫ 𝑉,\nt\n𝑆𝑡1, 𝑆𝑡2& 𝑆𝑡3\nfalls off as\n1\n,\n1\n&\n1\nrespectively, which corresponds to\n1\n,\n1\n&\n1\n.\n𝐼2 𝐼2 𝐼2 𝑉4 𝑉2 𝑉3 𝐼2 𝐼 𝐼1.5\nFitting parameters from the I-V model\nThe I-V model equation is fitted with experimental data. The obtained fitting parameters are as\nfollows:\nDevice 𝐺 (𝑆) Γ 𝑉 (𝑉) 𝑎\nn 𝑐 𝑡\n1 1.947×10−6 6.72×10−6 0.08241 0.9\n21 | P age\n2 3.912×10−6 3.556×10−6 0.04751 0.89393\n3 1.638×10−6 4.31×10−6 0.04392 0.79445\nFitting parameters for the noise model\nTo fit the noise model equation with the experimental data, we plugged in values of the I-V fitted\nparameters (𝐺 ,Γ ,𝑉 & 𝑎). The noise amplitudes are not calculated from the first principles but rather\nn 𝑐 𝑡\ndetermined by comparing the experimental data with the model. For that, we calculated the average\nconstant noise level, 𝑆 ⁄𝐼2, of the normal carrier noise from the experimental data since 𝑆 ⁄𝐼2 =\n𝐼 𝐼\n𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. The average noise is taken as the value of the linear noise amplitude, 〈𝛿𝐺 2〉⁄𝐺 2. The CDW\nn 𝑛\nnoise amplitude, 〈𝛿Γ 2〉⁄Γ 2 cannot be determined accurately since the experimental noise never saturates.\n𝑐 𝑐\nWe took the upper-bound value before the noise could be saturated at higher biases. For the threshold field\nnoise amplitude, 〈𝛿𝑉 2〉⁄𝑉 2, we plugged in different values and superimposed the noise model output\nt 𝑡\nwith the experimental data. The threshold amplitude was determined based on the best fit of the model\nwith the experimental data at higher biases, where the noise follows, 𝑆 ⁄𝐼2 ∝ 1/𝐼 .\n𝐼\nThe table contains the values used for the fluctuation terms to fit the noise model.\nDevice 〈𝛿𝐺 2〉 〈𝛿Γ2〉 〈𝛿𝑉2〉\nn 𝑐 𝑡\n𝐺 𝑛 2 Γ 𝑐 2 𝑉 𝑡 2\n1 4.9×10−9 1.0×10−12 3.5×10−7\n2 9.5×10−9 1.0×10−12 1.9×10−6\n3 1.3×10−9 1.0×10−12 1.45×10−6\nCalculation of the transport parameters and weighted noise contributions\nIn the incommensurate phase, total device current combines linear current and CDW current, such\nthat: 𝐼 = 𝐼 +𝐼 , where, 𝐼 is normal current, 𝐼 is the CDW current, 𝐺 is the total conductance,\nn 𝑐 𝑛 𝑐\n22 | P age\nwhich combines linear and CDW terms, such as, 𝐺 = 𝐺 +𝐺 . Here, 𝐺 is normal carrier\nn 𝑐 𝑛\nconduction, which is calculated as the slope of I-V data in the linear regime, such as, 𝐺 =\nd𝐼𝑛\nand\nn\nd𝑉\nd𝐼 𝑑𝐼\n𝐺 = −𝐺 , is the CDW carrier conductance, where 𝐺 = . The CDW current is calculated as,\nc 𝑛\nd𝑉 𝑑𝑉\n𝐼 = 𝐼 −𝐼 = 𝐼 −𝑉𝐺 . We now derive Eq. (1) of the main text. In the two-fluid model, total\nc 𝑛 𝑛\ndevice current is given by the equation: 𝐼 = 𝑉𝐺 = 𝐼 +𝐼 . If the fluctuations in the two\nn 𝑐\nconductivities are independent and the cross-correlation is neglected, we get the total current\nfluctuations as:\n𝛿𝐼 = 𝛿𝐼 +𝛿𝐼 = 𝑉(𝛿𝐺 +𝛿𝐺 ). (S1)\nn 𝑐 𝑛 𝑐\nTaking the square of the fluctuation terms and neglecting the cross-correlated term, we have:\n𝛿𝐼2 = 𝑉2𝛿𝐺 2 +𝑉2𝛿𝐺 2 . (S2)\nn 𝑐\nTaking the ensemble average, we obtain:\n〈𝛿𝐼2〉 = 𝑉2〈𝛿𝐺 2〉+𝑉2〈𝛿𝐺 2〉 . (S3)\nn 𝑐\nWriting in terms of spectral density for fluctuations, we get:\n𝑆 = 𝑉2𝑆 +𝑉2𝑆 . (S4)\n𝐼 𝐺𝑛 𝐺𝑐\nDividing by the square of 𝐼, we write:\n𝑆𝐼\n= 𝑆\n𝑉2\n+𝑆\n𝑉2\n, (S5)\n𝐼2 𝐺𝑛 𝐼2 𝐺𝑐 𝐼2\n𝑆𝐼\n= 𝑆\n1\n+𝑆\n1\n. (S6)\n𝐼2 𝐺𝑛𝐺2 𝐺𝑐𝐺2\nTherefore, the total current fluctuations, can be written as:\n𝑆𝐼\n=\n𝑆𝐺𝑛 𝐺𝑛 2\n+\n𝑆𝐺𝑐 𝐺𝑐 2\n. (S7)\n𝐼2 𝐺𝑛 2(𝐺𝑛+𝐺𝑐)2 𝐺𝑐 2(𝐺𝑛+𝐺𝑐 )2\nHere,\n𝑆𝐼\nis the total normalized current noise;\n𝐺𝑛 2\n,\n𝐺𝑐 2\nare the normalized conductions;\n𝐼2 (𝐺𝑛+𝐺𝑐)2 (𝐺𝑛+𝐺𝑐)2\n𝑆𝐺𝑛\n,\n𝑆𝐺𝑐\nare the normal and the CDW carrier conduction noise components;\n𝑆𝐺𝑛 𝐺𝑛 2\n,\n𝐺𝑛 2 𝐺𝑐 2 𝐺𝑛 2(𝐺𝑛+𝐺𝑐)2\n𝑆𝐺𝑐 𝐺𝑐 2\nare the weighted contributions of the normal and CDW carrier conduction noise\n𝐺𝑐 2(𝐺𝑛+𝐺𝑐 )2\ncomponents.”\nI have taken consideration of this work but I consider that the conclusion of a CDW\ncondensate moving without noise is not justified based on the presented data. There\nare two many uncertainties either in the experimental measurements and in the\n23 | P age\ntheoretical and fitting process which raise some, in my sense, raisonable doubts on the\nconclusions.\nRESPONSE: We have added extra data and explanations to remove any doubts\nabout our claims. We are ready to demonstrate the measurement procedures in the\nlaboratory if needed.\nThus I will not recommend publication of this work in Nature Communications\nRESPONSE: We sincerely hope that Reviewer 2 will change his/her opinion after\nconsidering all the evidence we provided and additional data, including for a different\nmaterial.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nRESPONSE: We sincerely thank the co-reviewer for their valuable time and\nconstructive feedback, which greatly helped improve our manuscript.\nReviewer #4 (Remarks to the Author):\n24 | P age\nThe manuscript NCOMMS-25-46228-T by Subhajit Ghosh et al. describes a study of\nlow-frequency electronic noise suppression in quasi-one-dimensional charge-density-\nwave (CDW) systems, specifically in (TaSe4)2I nanowires, under conditions of sliding\nFrohlich condensate current.\nThe reported suppression of low-frequency 1/f noise in the sliding CDW regime is highly\nunusual and underlines a significant difference from conventional transport behavior in\nresistive materials. In this respect, the linear decrease of normalized noise spectral\ndensity, as opposed to the typical constant value in normal conductors, suggests a\nfundamentally different noise mechanism linked to collective transport. This is an\nindicator of new physics in the correlated transport regime. Moreover, the connection to\nreal-world technological relevance, emphasizing technological relevance to quantum\ndevices, low-noise communication, and sensing, strengthens the case for broad impact.\nRESPONSE: We thank the Reviewer for stating that our work “suggests a\nfundamentally different noise mechanism linked to collective transport,” and that our\nstudy reports “new physics in the correlated transport regime,” and that our work has a\nbroad impact. We take these comments as indications of the suitability of our paper for\nNature Communications.\nIn the context of existing literature, the submitted work appears scientifically novel in its\nexperimental observation of sub-thermal electronic noise in the sliding CDW regime, a\nbehavior not established in previous CDW literature where broadband 1/f noise has\nbeen usually interpreted as a consequence of CDW dynamics near depinning.\nMoreover, recent studies, exploring CDW systems in reduced dimensions (nanowires,\nthin films), are concerned with transport phenomena but not noise suppression, as the\ncase of the article here considered.\nRESPONSE: We thank the Reviewer for recognizing the novelty of our work.\n25 | P age\nAlthough the work appears sufficiently novel and innovative for publication in Nature\nCommunications, two important points deserve more clarification and should be strongly\ntaken into account before a resubmission.\n(I) The first point concerns the fact that the notion of using strong correlations to\nsuppress noise, especially leveraging a collective mode like Frohlich conduction, is still\nunderexplored and seems to be largely speculative. The authors need to better clarify\nthis aspect, demonstrating that the theory developed is based on solid principles and\nthe results obtained are not just material-specific observations. In this respect, further\nexperimental investigations showing an analogous noise behavior as the one here\nreported, performed on different systems undergoing a similar CDW regime, could be\nvery useful to address such a criticism.\nRESPONSE: Following the Reviewer 3 request, we performed noise measurements on\nanother interesting system, NbS , which reveals CDW phases at RT. We planned it as a\n3\nseparate publication, but based on the Reviewer’s comment, we included it in this\npaper. This material system is also a fully gapped CDW where one can achieve\ndominance of the current by the sliding CDW condensate. We observed a similar\nsuppression below the normal electron limit. However, there is one interesting\ndifference: we can see some noise saturation below the normal electron limit. Below is a\nsection added to the revised manuscript.\n“The electronic noise of the CDW condensate in NbS nanowires\nto the IC-CDW phase of (TaSe ) I, or is it a property of many quasi-1D CDW conductors?\n4 2\nThe early studies of the “broadband” noise in bulk CDW crystals focused on the threshold noise 15–17\nand never considered the possibility of noise reduction below the normal electron limit. A few\nrecent studies for TaS and NbSe found that beyond V, the noise returns to the normal electron\n3 3 t\nlimit 37,48, in contrast to our observation of continuous noise reduction in the sliding regime of\n26 | P age\n(TaSe ) I devices. Other potential candidates for ultra-low noise transport should include fully-\n4 2\ngapped CDW materials where the collective, strongly-correlated current of the sliding electron-\nlattice condensate is dominant. We experimented with nanowires of NbS -II, which exists in the\n3\nsame protocols as with (TaSe ) I nanowire devices. Figure 6 (a-f) shows transport and noise\n4 2\ncharacteristics for a representative NbS -II nanowire device. The most important observation is\n3\nthat the normalized spectral density in the NbS -II device reduces below the normal electron limit,\n3\nsimilar to that in the (TaSe ) I devices. The noise spectral density scaling with current is also\n4 2\nconsistent, i.e., noise reduces with increasing current. It reveals a similar, 𝑆 ⁄𝐼21/𝐼, dependence\n𝐼\nfor the current above 10 A. We note that the noise reduction in NbS -II is achieved at\n3\ntemperatures above RT. The latter is important for practical applications, i.e. in interconnects.\nThere is one difference in the noise behavior in the two materials: it appears the noise spectral\ndensity in NbS -II nanowires shows saturation behavior, even though the noise level is below the\n3\nnormal electron limit. This suggests that we may be reaching the intrinsic electron-lattice\ncondensate noise in NbS -II nanowires but not in (TaSe ) I nanowire devices (see additional data\n3 4 2\nin Supplementary Materials).\nThe origin of the ultra-low-noise collective current in (TaSe ) I nanowire, and the lack of noise\n4 2\nsaturation, may be related to the fact that the CDW transport in this specific material is described\nby the under-damped oscillator model 49,50. Compared to other materials like NbSe and o-TaS ,\n3 3\n(TaSe ) I has the largest energy gap and heaviest CDW effective mass, 𝑚∗/𝑚 ≈ 104, yielding\n4 2\nthe lowest CDW damping. The non-linear CDW conductivity in this material continues to increase\nrather than saturate due to damping, in the examined bias voltages. The current in NbS -II shows\n3\nsigns of saturation at higher bias voltages (see Figure 6 (b)), indicating the onset of stronger CDW\ncurrent damping, which likely translates to higher intrinsic CDW noise and eventual flattening of\nthe total noise level (see Figure 6 (e) and (f)). Based on the recent realization that (TaSe ) I is a\n4 2\nWeyl semimetal 30,31, one may also argue that topological scattering suppression has relevance to\nthe noiseless current of the sliding condensate in this material.”\n27 | P age\n(II) The second point, which is someway connected to the first one, regards an\nimprovement of the experimental robustness and generality of the work, in order to\nelevate its impact from a specialized CDW study to a wider platform as, for example, for\nquantum/nanoelectronics.\nRESPONSE: The general noise theory that we developed for CDW materials\n(something that has not been done after 50 years of studies) will have broad\nimplications for quantum and nanoelectronics. In the revised manuscript and\nSupplemental Materials, we provided all details. In the original manuscript, we had a\nparagraph on benchmarking against conventional electronics, which we believe\naddresses another part of the Reviewer’s question.\nAs minor comment, due to the high visibility and high impact of Nature Communications\njournal, my suggestion to the authors is to mention other systems and materials where\nan evident reduction of the 1/f noise level has been observed. This is the case, for\nexample, of low-dimensional systems and topological insulators. A detailed description\nof key similarities and key differences in noise suppression mechanisms could also be\ngiven.\nRESPONSE: The type of noise suppression we observed and noise scaling with the\ncurrent is unique for fully gapped CDW materials with relatively low damping. We\nproved it for two different materials. We are not aware of any other results like these for\nany other material systems. This confirms the novelty and suitability of our paper for a\nNature family publication. In the revised manuscript, we added a paragraph that clarifies\nthe significance of the observed noise scaling.\n“The fact that we observed a different dependence on the normalized noise spectral density in a\nCDW conducting channel is of great importance. It is well-known that the noise level, i.e.,\nnormalized noise spectral density, in passive resistors does not depend on current: 𝑆 ⁄𝐼2 constant\n𝐼\n(or 𝑆  𝐼2). This fundamental property originates in the linear response theory38 . In terms of\n𝐼\nphysics, it means that the current does not drive the resistance fluctuations in the sample but rather\n28 | P age\nmakes them visible via Ohm’s law. The fact that 𝑆  𝐼2 allowed for the introduction of the\n𝐼\nphenomenological Hooge formula, 𝑆 /𝐼2~𝛼 /𝑓𝑁, where 𝛼 is the Hooge parameter, and 𝑁 is the\n𝐼 𝐻 𝐻\nnumber of charge carriers in the resistor. There are known cases of deviation from 𝑆 ⁄𝐼2 constant\n𝐼\ndependence in specific electronic devices, like diodes39 . In this work, we report the first\nobservation of as 𝑆 ⁄𝐼21/𝐼 scaling of noise in a generic resistor with two metal contacts. This\n𝐼\nmeans that increasing the current density will simultaneously decrease the noise level, which can\nbecome a game-changing advantage for interconnect applications.”\nFor all the reasons specified above, in my opinion, the paper is not suitable for\npublication in Nature Communications in its present form. A major revision, by\naddressing all the points indicated, is needed before the manuscript can be\nreconsidered.\nRESPONSE: We sincerely hope that Reviewer 3 can now recommend our manuscript\nfor publication, considering all the evidence we provided and additional data, including\nfor a different material (at the Reviewer 3 request).\n29 | P age\nRESPONSE TO THE REVIEWERS’ COMMENTS\nReviewers' comments:\nReviewer #1 (Remarks to the Author):\nThe comments from my earlier review have been satisfactorily addressed. The revised\nmanuscript and supplement information are vastly improved from the previous version.\nResponse: We thank Reviewer #1 for stating that we have satisfactorily addressed the\ncomments from the previous submission. We appreciate Reviewer #1's statement that “revised\nmanuscript and supplement information are vastly improved from the previous version”.\nMy only comment is for the authors to consider adding their comments about the novelty of\nthis work compared to previous publication (NbSe3). They have provided the arguments in the\nresponse which I find to be satisfactory, but it will be helpful to have them prominently stated\nin the revised manuscript.\nResponse: As suggested by Reviewer #1, in the revised manuscript, we added several sentences\nclarifying the novelty and explaining the principal difference of the present work with NbSe . As\n3\nwe mentioned in our previous response, NbSe is a CDW material, which is not fully gapped.\nCDWs. As a result, in NbSe , we have not observed the noise reduction below the normal\n3\nelectron noise limit. Our new results constitute a major step forward. The novelty includes: (1)\nclear demonstration of the noise below the normal electron noise limit in two fully-gapped\nCDW materials; (2) scaling of the CDW condensate noise with bias voltage, which is different\nfrom scaling of the noise of normal electrons; (3) comprehensive phenomenological theory of\n1/f noise in CDW materials (something that has not been done before, after 50 years of\ninvestigations). In the revised manuscript, we added the following sentences.\nIt is interesting to compare the noise dependence on bias in (TaSe ) I and NbS nanowires with\n4 2 3\nthat in NbSe nanowires37. While the noise in (TaSe ) I and NbS nanowires decreases below the\n3 4 2 3\nnormal electron limit when the CDW sliding current component starts to dominate, the noise in\nNbSe nanowires saturates at the normal electron noise level. This important difference is\n3\nexplained by the fact that NbSe is not a fully-gapped CDW material. For this reason, even after\nCorrespondingly, the noise level does not decrease below the normal electron limit. This\nobservation adds to the importance of the present finding of noise reduction and unusual\n1\nscaling with the current in nanowires made of fully-gapped CDW materials such as (TaSe ) I and\n4 2\nNbS .\nResponse: We thank Reviewer #1 for recommending our paper for publication in Nature\nCommunications.\nReviewer #2 (Remarks to the Author):\nI acknowledge the authors for the improvements they made in the revised version of their\npaper including in particular data on the CDW compound NbS3 with its Peierls transition\ntemperature above room temperature and their phenomenological theory of 1/f noise in\nSupplementary.\nResponse: We thank Reviewer #2 for acknowledging the improvements we have made in the\nrevised manuscript, specifically that we added data for a new material system (this required\nsignificant additional work) and developed a complete phenomenological theory of 1/f noise in\nCDW materials (something that researchers have not managed to do in more than 50 years of\nresearch). We respectfully note that the main elements of the theory were presented in the\nMethods section of the main text rather than the Supplement. The Supplement contains details\nof the calculation and data extraction for the model.\nHowever I did not get convincing answers to the questions raised in my first report, specifically\nconcerning heating effects.\nThe fact that no softening of the phonon modes in Raman experiments is, for the authors, the\ndirect proof of the absence of Joule heating. But the more simple and direct proof is the electric\nfield dependence of the normal resistance which in the pinned CDW state below the threshold\nshould keep its normal value. The activated temperature dependence of the resistance in the\nnormal state of these fully gapped CDW materials is a very sensitive probe of possible heating\neffects.\nNo explanation has been given on the electric field dependence of the differential resistance\nbelow threshold as shown in Fig1d in a semilog plot in the original manuscript. Opportunely the\nupdated rescaled Fig1d in the revised manuscript has in some way erased this dependency.\nplot would be important).\nThis question is of a huge importance since the way to extract the CDW noise results from the\nsubtraction from the total noise of independent noise from the normal current and noise from\nfluctuations in the threshold voltage. Any change in the normal current noise may affect the\nresult.\nIt appears difficult to ascertain as the authors state that Joule heating does not play any role.\nI keep the conclusions of my first report\nResponse: We take responsibility for not grasping what Reviewer 2 was finding problematic in\nthe data. Our focus was on the larger bias data ( ), where all the interesting phenomena\nwere occurring, rather than on the low-bias data before the threshold. We now understand the\n𝑉𝑉 > 𝑉𝑉𝑡𝑡\nconcern of Reviewer 2 with respect to the low-bias curves. To address Reviewer #2's concerns,\nwe revised our manuscript accordingly.\nBelow, we explain in detail that there was no self-heating in our experiments and sort out the\nconfusion regarding Figure 1 (d). We understand now where the misunderstanding might have\noriginated.\nIronically, Figure 1 (d) (in the original and resubmitted version) is not even used in the transport\nand noise data analysis. Figure 1 (d) was meant as preliminary material characterization\nshowing that we do have the correct (TaSe ) I material with the transition temperature and\n4 2\nthreshold voltage. The I-V and conductivity that were used in the noise data analysis were\nshown in Figures 1 (e) and (f), and similar corresponding figures for other devices. These I-Vs\nwere taken during the noise measurements. We have clarified it in the revised manuscript.\nMoreover, even if there were some local heating, it would not affect the conclusion of the\npaper on the noise reduction below the normal electron limit, which was experimentally\ndemonstrated for many devices made of (TaSe ) I and, at Reviewer #3-#4’s request, for another\n4 2\nmaterial system, NbS -II. The possibility of having the total noise below the normal electron\n3\nlimit was also rigorously proven theoretically (it could have been a separate paper). The noise\nof CDWs was not determined by simply subtracting the noise at low bias but rather extracted\nvia a rigorous procedure, which included the use of analytical fitting to the actual experimental\ndata (please see the theory section of the paper).\nWe start by pointing out that we used very low bias voltages and current levels, insufficient for\nany local heating. There is a simple formula from Fourier’s law that allows one to estimate the\nself-heating, ∆T = [Power] × [Thermal Resistance]. There is no possibility of having noticeable\n3\nheating effects in our devices at the bias voltages (0.1 V) and current levels (nano-Ampere or\nfraction of micro-Ampere) across the depinning that we used. In order to have noticeable self-\nheating, one needs to apply voltages above 10 V and have order of magnitude larger currents.\nWe address this in detail below, including the approach proposed by Reviewer #2.\nLet us explain the source of confusion, for which we take responsibility. This paper was\noriginally submitted to Nature, and we had to put all data for the main message into three\npanels, which did not allow us to explain things easily and created issues with the original\nsubmission to Nature Communications (the manuscript was transferred “as is”). The first 4\npanels of Figure 1 were meant for materials characterization. Those panels should have been a\nseparate figure, but were combined with the measurement for Device 1. In Figure 2, we\ncombined the final noise data for Devices 1 and 2 (last panel).\nHere is the measurement protocol, a standard one in the noise community, that Subhajit Ghosh\n(the first author) used in this study.\nHe measured the I-Vs twice. First, he did a quick measurement of I-Vs to characterize the\nmaterial and verify that the device was working fine. Such representative I-Vs were used in\nFigure 1 (d), original and resubmitted versions. Then he placed the device structure in the noise\nmeasurement setup and started biasing the device at different voltages (from low to high), and\nmeasured noise at every voltage. This gives us another version of I-V (similar to the first one,\nbut the actual voltage bias points may be different). This I-V, from the second measurement, is\nused in the analysis because it is the most relevant, taken simultaneously with the noise\nmeasurements. There can be slight variations in the flatness of the curves because CDW\ndepinning is never completely coherent. There might be small variations compared to the first I-\nV measurement because the time of measurement is slower (one has to measure noise for each\nvoltage bias point). However, one can see in Figure 1 (f) that at low bias, the curve is nearly flat.\nThe noise data and analysis in Figure 2 (a – e) correspond to this device without any signs of\nheating. Some kinks and deviation from flatness may be related to incomplete coherence of\ndepinned CDW (the process does not start in the entire sample at once), CDW creep before the\nfull sliding, and numerical errors of differentiation of the I-V in Origin (for Figure 1 (d)). We\nmeasure I-Vs and differentiate them numerically in Origin without function smoothing. What\nmatters for the noise extraction and analysis is seen below (from the original submission and\nresubmission).\nmultiple top electrodes. Again, that was numerical differentiation of the first I-V measurement\ndata conducted to check the material quality and device function. This data was for a device\nwith multiple electrodes to verify that the measurements in a 4-probe configuration and a 2-\nprobe configuration give similar results. The proof of the same I-Vs for 2-probe and 4-probe was\nplaced in the Supplemental Figure 3 (a). Plotted on a log-linear scale, it gave an exaggerated\nimpression of a bump at low bias. The format was used to place several curves for different\ntemperatures on the same plot. The curves that had some bumping in the middle were at the\nlower temperatures, T= 120 K and below, where the samples were substantially more resistive\n(hundreds of MΩ). The focus of the paper was on noise analysis at higher temperature, T=180 K\n– 200 K, which is closer to RT and has more practical relevance. At this temperature, the\nresistance of the samples was significantly lower, as required for reliable noise measurements,\nand the low-bias curves were flat. The noise analysis for NbS was for RT. The curves at T=280 K\n3\n– 320 K are flat.\nAfter Reviewer #2's first report, we had to fabricate a new device structure (the old one had\ndied) to add more dependencies to Figure 1 (d). It had a slightly longer channel, which explains\nthe difference in V. We thought that the concern was about self-heating at the largest applied\nt\nbias voltage, and did not pay much attention to near-zero voltage. We replaced Figure 1 (d) to\nshow more curves, but have not changed anything with Figures 1 (e ) or 1 (f), which were used\nfor the noise data processing. We did not notice that the caption was not revised properly to\nindicate this change. Figure 1 has been corrected now: Figure 1 (d) is now for Device 1 (used in\nthe actual noise measurements) and plotted in a format allowing for seeing the low bias region.\nfor the same Device 1.\nBelow, we plotted numerically differentiated I-Vs specifically for Device 1 on a different, linear–\nlogarithmic scale to show the dependencies more clearly.\nHere is the same in log–log scale.\nfew well-known physical reasons to expect slight bending and kinks in the differential resistance\nof CDW nanowires below the threshold. The CDW sliding does not start simultaneously\nthroughout the sample volume, and the onset of collective current is gradual. There is another\nreason – the CDW creep transport regime, which is the partial movement of CDW before the\nsliding. It can also result in bending of the differential I-Vs. Our plots look “textbook-like”,\nsimilar to the ones reported for bulk crystals, where self-heating was not possible due to the\nlarge sample size. Depending on the resistance values, the deviations from a perfectly flat\nconductivity can be seen as more or less pronounced. We got consistent results for other\ndevices, and included Devices 2 and 3 in the revised submission.\nTo sum up, to address Reviewer #2’s concerns, we added an extra explanation of how I-Vs are\nmeasured (once for material characterization and device test, and the second during the noise\nmeasurements) and indicated the possible reasons for the derivative I-V deviations from\nperfectly flat dependence.\nIn response to Reviewer #2’s comments, in the revised manuscript, we added the sentences:\nThe dependence was obtained by numerical differentiation of measured I-Vs.\n[…] The V value shows a non-monotonic temperature dependence consistent with trends\nt\nreported for other quasi-1D CDW systems.1,3,4. Slight deviations from the perfectly flat\nderivative characteristics at low bias and kinks are attributed to CDW creep before the\ndepinning and not completely coherent CDW depinning and sliding.\n[…] These characteristics are measured for the second time during the noise measurements.\nThey are further used in the noise data analysis.\nWe now address the additional data that we included in the previous resubmission. We used\nRaman spectroscopy for temperature sensing. This is a conventional method, which one of us,\nAlexander Balandin, used in his prior Nature Communications publication: Z. Yan, G. Liu, J. M.\nKhan, and A. A. Balandin, “Graphene quilts for thermal management of high-power GaN\ntransistors,” Nature Commun., 3, 827, 2012. Raman thermometry is an established and\ncommonly accepted technique. We note that if there were self-heating effects, they would\nhave appeared prominently in NbS devices. There were none.\nlong paragraph added with this extra data in the resubmitted version. Neither the analytical\nmodel nor the detailed COMSOL simulation predicts any self-heating. One can understand why\nit is negligible in our case: we only use a bias of ~1 V and a current of ~1 micro-A in the sliding\nregime. In the low-voltage linear regime, the current is in the nano-A range with applied\nvoltages of only a few mV. At such low levels, Joule heating is physically impossible, regardless\nof sample dimensions.\nHere is what we wrote in the resubmitted manuscript.\n“In the data analysis, it is also important to exclude local heating effects. No Joule self-heating\neffects are expected in our test structures at the small bias voltages, about ~1 V, and low\n7\ncurrent range, about ~ 1 µA. One can readily estimate the temperature rise, , from an\nanalytical formula derived from Fourier’s law: . Here,\nΔ𝑇𝑇\nP is the dissipated power in the nanowire, R is the thermal resistance of the SiO layer, T is\nT 2 ox\nΔ𝑇𝑇 = 𝑃𝑃×𝑅𝑅𝑇𝑇 = 𝐼𝐼 ×𝑉𝑉 ×(𝑇𝑇𝑜𝑜𝑜𝑜/𝐾𝐾×𝐿𝐿×𝑊𝑊)\nthe thickness of the SiO layer, K is the thermal conductivity of SiO , L is the length of the\n2 ox 2\nchannel, and W is the width of the channel. The formula assumes that the bulk Si substrate is\nthe thermal sink, and the SiO layer with its low thermal conductivity is the thermal barrier.\nnm, we obtain for the thermal resistance R ~0.2 MK/W and, correspondingly, for the\nT\n𝐼𝐼 𝑉𝑉 𝑇𝑇𝑜𝑜𝑜𝑜 𝐾𝐾𝑜𝑜𝑜𝑜 𝑊𝑊\ntemperature rise ~0.2 K. We can elaborate the estimate and consider the thermal boundary\nresistance, , between the nanowire and SiO layer. It is well known that it is primarily defined\n2\nΔ𝑇𝑇\nby the interface quality rather than specific channel material. Assuming the worst-case scenario\nof low inter 𝑅𝑅 fa 𝐶𝐶 ce conductance, =15 MW/m2K44,45, we obtained an extra thermal resistance of\n~0.2 MK/W, which doubles the total resistance and temperature rise. On the other side,\n𝐺𝐺𝑇𝑇\nconduction to metal contacts along the channel may reduce the temperature rise. These simple\nbut reliable estimates show that in our devices, in the considered bias and current ranges, the\nlocal heating is negligible. One has to pass currents in the mA range, a factor of ×1000 larger, to\ninduce significant Joule heating. The temperature rise was also assessed using COMSOL\nsoftware tools for a given device structure and bias voltages. The maximum temperature\nincrease at the hot spot (center of the channel) is approximately 1 K at a voltage drop of 0.9 V.\nThe simulation results confirmed negligible self-heating effects (see Supplemental Materials).”\nHere we address another Reviewer #2’s comment: “The activated temperature dependence of\nthe resistance in the normal state of these fully gapped CDW materials is a very sensitive probe\nof possible heating effects.”\nWe assume Reviewer #2 meant “resistance of the normal current” rather than “resistance in\nthe normal state”, since in the “normal state”, there is no gap.\nFollowing the comment, we conducted additional analysis based on the low-bias ( )\ncurrent in the fully gapped CDW phase. From the relationship , if\n𝑉𝑉 < 𝑉𝑉𝑡𝑡\nchanges with voltage such that then\n𝑅𝑅𝑛𝑛 ∝ exp(Δ/𝑘𝑘𝑇𝑇) 𝑅𝑅𝑛𝑛(𝑉𝑉)\n𝑅𝑅1(𝑉𝑉1) < 𝑅𝑅0(𝑉𝑉 = 0),\n−1\n1 𝑅𝑅1 1\n𝑇𝑇1 =  +ln   ,\nwhere . We used this formula,\n𝑇𝑇\na0ssuming\n𝑅𝑅\n0tha\nT\ntΔ the changes in the current below the\nthreshold are related to self-heating (even though it is unlikely). The resulting was found to\n𝑇𝑇Δ = Δ/𝑘𝑘𝐵𝐵\nbe negligible. In all checked cases, the temperature rise was below 1 K. The formula works\n∆𝑇𝑇\nbelow the CDW depinning.\nWe thank Reviewer #2 for suggesting this method for assessing the temperature. The findings\nusing the Reviewer #2 method agree with our approaches. We believe we have done\neverything to address the issue of heating. We appreciate Reviewer #2’s comments that\nallowed us to rigorously exclude any possibility of self-heating in our devices at low bias levels.\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nReviewer #4 (Remarks to the Author):\nThe revised manuscript NCOMMS-25-46228A by Subhajit Ghosh et al. has been improved, with\nrespect to the original submission, and my criticisms have been adequately addressed. In\nparticular, additional data of measurements performed on a different system (NbS3 nanowires)\nundergoing a similar CDW regime than (TaSe4)2I nanowires have been included in this revised\nversion, showing that the results obtained are not just material-specific observations but are\nmore general.\nResponse: We thank Reviewers #3 and #4 for stating that “criticisms have been adequately\naddressed.” We are glad that the significant extra work that we carried out for a second\nmaterial system (which was planned as a separate paper) was appreciated by the Reviewers.\nHowever, I do not agree with one statement of the authors (end of Page 8 of the revised\nversion) where they assert that: “in this work, it is reported for the first time the observation of\na decreasing noise level by increasing the current density”. While, I remember other works in\nliterature reporting systems where a decrease of the noise level is observed with an increasing\nbias current. This is the case of disordered LBMO thin films or the case of systems undergoing\nweak-localization effects (https://doi.org/10.3390/coatings11010096). My suggestion to the\nauthors is to take into account also these experimental results already published.\nIn the same review paper (https://doi.org/10.3390/coatings11010096), moreover, there is also\na section reporting a detailed study of fluctuation mechanisms induced by charge density waves\nconduction in PCMO epitaxial thin films. The authors should consider these investigations by\nevidencing similarities or differences between their conclusions and the other ones already\npublished.\nResponse: As requested, we removed the word “first time”. We modified the sentence in the\ntext and cited the recommended paper published in the Coatings journal: Barone, C. & Pagano,\n(2021).\nWe do believe that our findings are significant and constitute a major advancement in the state\nof the art. We thank you again, Reviewers #3 and #4, for their constructive comments and\nsuggestions.\nREVIEWERS' COMMENTS\nReviewer #2 (Remarks to the Author):\nThe main text of this last revised version has been slightly amended with additions in\nSupplementary. To get a final point to this tangled question of Fig.1d, it is now\nrecognized that the data in this figure do not concern Device 1 and are just an\nillustration from depinning obtained from a different sample (that has to be indicated in\nthe figure caption). Note however that the new fig1d exbibits a T dependance (non-\nuniform) of the threshold field different of that in the former fig1d in the original and first\nrevised version.\nHeating effects may affect measurements for any bias voltage: above threshold they are\nhidden and included in the non-linear variation of the resistance; below threshold they\nare detected from the non constant differential resistance. These effects are always\npresent even if their consequences can have a limited effect.\nEven if I feel uncomfortable with some assumptions/conclusions in this paper, I\nacknowledge the work done and I would not oppose the publication of this manuscript.\nNevertheless, if this work meets some interest in the community, it can be refuted in\nfuture publications\nRESPONSE: We thank the Reviewer for his/her consent to publish our manuscript in\nNature Communications and his/her acknowledgment of the work we accomplished.\nFigure 1 (d) in the latest resubmission corresponds to Device 1. It has been indicated in\nthe caption. The V value shows a non-monotonic temperature dependence consistent\nt\nwith trends reported for other quasi-1D CDW systems. This has been explained in the\ntext. We have proven that our noise measurement results and conclusions are not\naffected by any local heating using four different methods. The details are provided in\nthe previous three responses.\nI list below some points that can be considered by the authors; it would be worth to\nintegrate their answers in a final text:\n- All the analysis is based on the assumption that the fluctuations of the normal and\ncollective fluctuations are independent without cross-correlation. The green line in fig.4c\nindicates the constant value of noise of electrons as a function of bias voltage. It might\nbe useful to ascertain this hypothesis and beyond the two-fluid model to take into\naccount the CDW-normal carrier interactions\nRESPONSE: Yes, the Reviewer is correct. We made a conventional assumption that\nthe fluctuations of the condensate and normal electrons are uncorrelated. We have\n1\nstated this in the earlier version and emphasized this in the final version. Inclusion of the\ninteractions of the CDW condensate and normal electrons via cross-correlated terms is\na separate theoretical task, which we reserve for future study. However, we are\nconfident that the inclusion of the cross-correlations will not change the conclusions\nabout the noise reduction and its current dependence. On the other hand, the\ncomprehensive phenomenological model, which we developed to analyze the\nexperimental results for CDW material systems, already constitutes a significant\nadvancement in the state-of-the-art, which has to be reported to the community. We\nnote that previous models focused on the noise near the depinning point, which is\nextrinsic by its nature. No prior models separated the three different terms contributing\nto noise. No prior models addressed the intrinsic noise of the sliding CDW condensate\nand its current dependence.\n- Eq.2 gathers two different models: one described as a single particle model which\nimplies a kink or a negative differential resistance at Et and the other one treated as a\ndynamical critical phenomena. It would be necessary to be more explicit on the\ntheoretical basis which leads to Eq. 2, beyond reference 40. To what extent of E the (E-\nEt) is valid?. How this equation is justified ?. Even if the fits with the experimental data\nlook of good quality, how to understand that the exponent, a, determined by the fits is\naround 1 away from the ½ value for the single particle model and 3/2 for the critical\nphenomenon.\nRESPONSE: Eq. (2) is a phenomenological representation of the CDW contribution to\nthe current over the experimentally accessible sliding regime. The asymptotic\nexponents 1/2 and 3/2 quoted by Grüner apply to idealized bulk limits with a single\ndepinning field. In our nanowire geometry, neither exponent provides an adequate\ndescription of the full experimental I-V curve.\nFirst, the 3/2 exponent does not reproduce the experimentally observed change in\nslope at 𝑉, as seen in Fig. 4e,f. Capturing the kink at threshold requires a sub-linear\nt\nterm with an exponent below unity.\nSecond, an exponent of 1/2 alone underestimates the current in the higher-bias region\n𝑉 ≫ 𝑉, where the condensate contribution grows more rapidly. Modeling the curve over\nt\nthe full ∼ 1 V bias range, therefore, requires a term with an exponent exceeding unity.\nSince neither parameter is correct over the full bias range, we allow the two exponents\nin Eq. (2) to be fitted parameters, while keeping the underlying form of the theory\nunchanged—one term describing the initial onset of sliding, and a second term\ndescribing the steeper high-bias response. To minimize free parameters, the ratio of the\ntwo exponents is fixed such that both terms are written using a single generalized\nconductance 𝛤; this keeps the parametrization minimal and avoids introducing two\n𝑐\nindependent prefactors.\ncould also produce a reasonable empirical fit, but it would not change the conclusions of\nthe noise analysis. The central result from Eq. (4), that the contribution of threshold-field\nfluctuations is suppressed as 1/(𝑉−𝑉)2 at large bias, is determined by the derivative\nt\nstructure of the model and is unaffected by replacing the fitted exponent 2𝑎 = 1.8 with\nthe theoretical value of 1.5.\nThus, the fitted exponents are best viewed as effective parameters that allow Eq. (2) to\nreproduce the measured nonlinear transport over the full voltage range, while the\nphysical conclusions of the noise analysis do not depend on their precise numerical\nvalues.\nWe added the following text to the final manuscript.\n“Eq. (2) provides a phenomenological description of the CDW contribution to the current\nin our nanowire devices. The two power-law terms originate from the standard CDW\ncurrent expressions quoted by Grüner40, but the exponents 𝑎 and 2𝑎 are employed as\neffective exponents that capture the voltage dependence of the sliding condensate over\nthe full experimental voltage range. We further constrain the ratio of the exponents of the\ntwo nonlinear terms to share a single generalized conductance 𝛤 with fixed\n𝑐\ndimensionality, which keeps the parametrization minimal and avoids introducing\nadditional prefactors. This yields an accurate minimal model that reproduces the full\nnonlinear I-V characteristics and forms the basis for the quantitative noise analysis\nbelow.”\n- With the maximum voltage used, only the noise due to fluctuations in Vt is observed.\nThat lead the authors to consider this picture as the behavior of ideal CDW sliding as a\ndissipation-less, noise free process. What about the ac field generated in the sliding?.\nRESPONSE: We have not claimed that we achieved “dissipation-less” current transport.\nWe stated in the first sentence of the original abstract, “While achieving dissipation-less\nFrohlich current of the electron–lattice condensate appears to be impossible in real\nmaterials […]”. We claim that we have not reached the intrinsic noise of the sliding CDW\ncondensate in (TaSe ) I nanowires within the examined bias range. We stated in the\n4 2\noriginal abstract, “No residual minimum noise level is reached for the current of the\ncondensate in (TaSe ) I nanowires.” In this sense, the CDW condensate sliding is\n4 2\nnoiseless. The noise near the depinning point is extrinsic, related to defects. We\nclarified this point further in the final manuscript. We do not see the need for the AC field\ndiscussion in this paper. The AC signal proves the sliding. It requires measurements in\na different frequency range. We clearly proved sliding by other means.\n- Concerning (TaSe4)2I and NbS3 the aim was to compared fully-gapped CDW\nmaterials. NbS3 in the phase II has 3 CDW transitions at TP0 = 460K, TP1 =360K and\nconductivity below TP1 is very weakly T dependent - see Fig.8 in PRB95, 0356110.\nThere are 8 chains in the unit cell and even if the structure is complicated it appears that\nthe electron condensation appears successively on distinct chains (similarly to NbSe3\nand monoclinic and orthorhombic TaS3). NbS3 below TP1 is not fully gapped CDW\nmaterial . The fully dielectric state is below TP2. The best compound with an unique\nCDW and a fully-gapped for comparison with (TaSe4)2I would be blue bronze\n(K0.3MoO3).\nRESPONSE: We thank the Reviewer for this comment, which allowed us to improve the\nquality of our manuscript. Our original goal (aim) for this paper was to study and report\nnoise specifically in (TaSe ) I nanowires. We did not plan to put results for various\n4 2\nmaterial systems into one already complicated paper, which has both experiment and\ntheory. However, another Reviewer requested that we do a separate study of noise in\nanother CDW material to see if we can observe a similar effect. There was also a\nrequest to compare the results with NbSe . We complied with both requests.\nNbS -II. However, what is important is that the contribution of normal electrons to\n3\ntransport in NbS -II is less than in NbSe . The noise returns to the normal electron limit\n3 3\nin NbSe because the transport is dominated by normal carriers. The noise reduces\n3\nbelow the normal electron limit in NbS but starts to saturate. The noise reduces below\n3\nthe normal electron limit in (TaSe ) I nanowires without signs of saturation, in the\n4 2\nexamined bias range. We refined the description of NbS in the final manuscript and\n3\nprovided additional references. The reference L. Sneddon, “Sliding charge-density\nwaves - dc properties,” Phys. Rev. B, 29, 719 (1984) describes a sharp contrast of NbS\n3\nwith NbSe in terms of normal electron contribution at low temperature. We also added\n3\na reference recommended by the Reviewer: S.G. Zybtsev, V. Ya. Pokrovskii, et al.,\n“NbS : A unique quasi-one-dimensional conductor with three charge density wave\n3\ntransitions,” Phys. Rev. B, 95, 035110 (2017).\nWe thank the Reviewer for the suggestion to study blue bronze (K0.3MoO3). We will\nattempt to investigate the noise in blue bronze and TaS for the next paper.\n3\n4"}]

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